Keywords: Contrôle aux limites, réseaux de trac à grande échelle, Hamilton-Jacobi, équations aux dérivées partielles Boundary control, large-scale trac networks, Hamilton-Jacobi, partial dierential equations

Ce travail du doctorat

considerably enhanced my abilities as a researcher. Without their guidance and persistent help this PhD thesis would not have been possible. I would like to thank my jury members Christophe Prieur, Nikolaos Bekiaris-Liberis, Christian Claudel, Ludovic Leclerq and Benedetto Piccoli for their valuable comments and questions asked during my PhD presentation. This allowed me to look at my results from dierent perspectives.

I would also like to express my thanks to the guys in our group whom I got to know during my PhD: Martin Rodriguez Vega, Ujjwal Pratap, Umar Niazi, Stéphane Mollier, Nicolas Martin, Vadim Bertrand and Leo Senique. Drinking coee, playing board games and sharing all ups and downs of a PhD life in general could have been nicer. The time that we have spent together will always make part of my beautiful memories.

A huge thank you goes to family, my mother Iryna and my stepfather Jean-Marc, for their constant patience and care. They have always supportive during this period in all aspects.

A special thanks goes to my husband Denis Nikitin, with whom I also shared the oce during my PhD. We had long and deep discussions on all possible scientic or irrelevant topics, which was often so inpiring and enjoyable. Our common trips all over Southern Europe will always be one of the most beautiful highlights of my PhD time.

This thesis was mainly funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERCAdG no. 694209, Scale-FreeBack, website: http://scale-freeback.eu/).

iii Résumé L'urbanisation continue de la planète contribue à une augmentation de la demande de transport, ce qui entraîne la formation de graves congestions. Cela a un impact négatif sur la vie quotidienne. Par conséquent, des solutions ecaces pour la gestion intelligente des transports sont d'une grande importance.

Dans ce travail de thèse, on traite principalement du contrôle et de la modélisation du trac sur des réseaux urbains de dimension arbitraire avec une application à la ville de Grenoble.

Le trac est considéré d'un point de vue macroscopique. Son état est prédit par une équation diérentielle partielle (EDP) qui décrit le trac en termes du ux et de la densité des véhicules dans une zone donnée. Dans cette approche de modélisation, on suppose qu'il existe une relation concave entre le ux et la densité, connue sous le nom de diagramme fondamental (Fundamental Diagram -FD). Ainsi, en fonction de la densité des véhicules, on distingue deux régimes du trac: le régime du ux libre (la densité est inférieure à la valeur critique) et le régime de la congestion (la densité est supérieure à la valeur critique).

On propose des techniques de la conception du contrôle qui reposent sur les propriétés intrinsèques des modèles du trac macroscopiques. Tout d'abord, nous résolvons des problèmes du contrôle aux limites unidimensionnelles (1D) pour le trac évoluant sur des routes simples.

Ensuite, divers problèmes du contrôle sont posés et résolus pour le trac sur des réseaux urbains de la taille arbitraire. La dynamique du trac est alors décrite par un modèle bidimensionnel (2D) de loi de conservation. Ce modèle 2D est évolutif, c'est-à-dire qu'il décrit le trac urbain par une seule EDP. Cependant, il n'est applicable qu'aux zones urbaines ayant une direction de mouvement préférée. Enn, nous présentons un nouveau modèle du trac continu en 2D qui peut capturer la véritable multidirectionnalité du trac. Ce modèle est formellement dérivé du concept d'ore et de demande à une intersection. Il se compose de quatre EDP qui décrivent l'évolution de la densité des véhicules par rapport aux directions cardinales: Nord, Est, Ouest et Sud (North-East-West-South -NEWS). Les performances du modèle NEWS ont été testées en utilisant des données simulées et réelles (provenant des capteurs installés dans la ville de Grenoble).

Les principaux résultats de cette thèse sont résumés ci-dessous.

Contrôle du trac sur les routes

Dans ce chapitre, l'évolution de la dynamique du trac est étudié sur une route unique de la longueur nie en utilisant le modèle de Lighthill-Whitham-Richards (LWR). Ce modèle est une EDP hyperbolique non linéaire du premier ordre qui représente une loi de conservation, le nombre de véhicules étant la quantité conservée. Deux problèmes du contrôle aux limites sont posés pour suivre une densité désirée qui est une trajectoire dépendante de l'espace et du temps. Ces dépendances sont capables de capturer des nombreuses situations réalistes lorsque v vi les conditions du trac changent rapidement. Par exemple, il est courant qu'une route ait des valeurs du ux entrant diérentes au cours d'une même journée (il peut y avoir plus de véhicules à 9 heures du matin lorsque les gens vont au bureau et moins de voitures la nuit).

En outre, un état d'équilibre souhaité ne représente qu'un prol général variant dans l'espace et dans le temps. Cependant, un tel prol cible général entraîne une dynamique d'erreur non triviale.

Tout d'abord, nous considérons que le trac est uniquement en régime congestionné. Cela simplie le modèle LWR qui devient une EDP linéaire. Ce modèle est cependant inhomogène, puisque nous ajoutons également une fonction de perturbation générale dans le domaine dépendant de l'espace. Cette fonction incorpore une éventuelle inadéquation entre le modèle et la réalité. Par exemple, il peut y avoir un ux non mesuré de véhicules provenant de routes secondaires ou de véhicules en stationnement. Pour ce système linéaire inhomogène, nous formulons des problèmes d'atténuation des perturbations tout en atteignant le prol souhaité en termes de normes spatiales L 2 et L ∞ . La trajectoire souhaitée est également restreinte au régime congestionné pour des raisons de simplicité, c'est-à-dire qu'elle est régie par le système linéaire homogène LWR. On traite la fonction de perturbation inconnue en utilisant la méthode des caractéristiques qui permet d'exprimer la fonction de perturbation par des variables connues (mesurées) telles que la densité des véhicules et les actions du contrôle appliquées au cours des étapes temporelles précédentes. Le contrôle conçu se compose d'une partie à action directe et d'une partie à rétroaction.

En outre, on considére également un problème plus complexe dans le cas où l'état et la trajectoire souhaitée sont régis par des modèles non linéaires LWR comme dans sa formulation originale (et sans la perturbation). Le principal dé est alors lié aux chocs (discontinuités), qui apparaissent dans de tels systèmes même pour des données initiales régulaires en temps ni. Cela rend l'analyse explicite fastidieuse, puisqu'il n'existe pas de solutions classiques, et nous ne devons les considérer que dans un sens faible et suivre la dynamique des chocs.

Un autre dé est lié aux conditions aux limites faibles, ce qui implique qu'aucune action de contrôle ne peut être imposée aux limites (on doit prendre en compte l'état actuel du système).

Pour traiter ces deux problèmes, nous traduisons LWR en EDP de Hamilton-Jacobi (H-J) qui représente sa forme intégrale. La solution de l'EDP H-J est exempte de discontinuités et, dans le pire des cas, elle peut seulement devenir non-diérenciable. Son état correspond au nombre cumulé de véhicules qui peut être obtenu en intégrant la densité des véhicules. Le système H-J peut être vu comme un problème de contrôle optimal, et sa solution est obtenue de manière semi-explicite comme le minimum de tous les chemins valides. Dans le cas d'un FD triangulaire, la solution est obtenue comme le minimum sur seulement trois chemins valides, chacun associé à la condition initiale ou aux conditions limites, respectivement. Pour analyser le comportement du système en temps asymptotique, on estime le temps minimal auquel il est garanti que les conditions initiales n'aectent plus la solution H-J. La solution est alors formulée comme un minimum de seulement deux chemins valides associés aux conditions aux limites. La formulation intégrale du système de la circulation de Hamilton-Jacobi ainsi que la possibilité d'exprimer exactement sa solution, nous permettent d'analyser explicitement les périodes du temps, lorsque les limites sont restreintes pour accepter des actions du contrôle en fonction de l'état réel de la circulation. Ces fonctions dites de restriction du contrôle vii permettent de diviser la dynamique d'erreur en trois régimes diérents en fonction des limites qui peuvent actuellement accepter des actions du contrôle. Nous montrons que même lorsque les limites sont parfois incapables d'accepter les contrôles proposés, le système converge vers la trajectoire souhaitée de manière exponentielle. Les résultats sont validés numériquement pour diérents gains de contrôle.

Trac unidirectionnel sur les réseaux

Ce chapitre est consacré au contrôle du trac sur les réseaux urbains de toute taille. Sa dynamique est décrite par le modèle LWR en 2D (2D LWR) qui représente une loi de conservation en deux dimensions. Le trac est traité comme un uide qui se propage maintenant sur un plan 2D continu.

Le modèle 2D LWR est inspiré de la modélisation de la foule, la seule diérence étant la restriction pour les véhicules de se déplacer sur des routes physiques. Ainsi, le modèle nécessite de supposer que le réseau urbain est susamment dense pour être considéré comme un domaine continu. Pour modéliser le trac, il faut disposer d'informations sur la géométrie et l'infrastructure du réseau urbain, c'est-à-dire l'emplacement des routes et des intersections, le nombre de voies de chaque route et ses limites de vitesse. Ces informations sont utilisées pour dénir la densité et les capacités maximales partout dans le domaine du continuum. En particulier, on applique la pondération inverse à la distance pour attribuer des valeurs aux variables partout en fonction de la distance aux routes. Tous ces paramètres étant spéciques aux diérents réseaux urbains sont incorporés dans un FD qui devient explicitement dépendant de l'espace. Une analyse directe d'un tel modèle est une tâche compliquée en raison de la dérivée seconde de l'espace. Il n'est pas non plus évident de savoir quel point limite doit être actionné pour aecter un point ou une zone spécique du domaine.

On trouve une approche pour analyser ce modèle de telle sorte que l'on puisse suivre les trajectoires du ux dans la zone urbaine. Ceci est possible, car la structure du modèle 2D LWR limite son applicabilité uniquement pour les réseaux constitués de routes unidirectionnelles.

Le champ de direction ne dépend que de la géométrie du réseau et non de l'état. S'il n'y a pas de boucles dans un réseau, on peut dénir une transformation de coordonnées curviligne.

Cette transformation de coordonnées traduit le système de trac 2D en un ensemble paramétré de systèmes 1D avec un FD dépendant de l'espace, ce qui est beaucoup plus facile à analyser.

Mathématiquement, cela signie qu'au lieu de deux dérivées partielles par rapport à l'espace, le système modié n'en a qu'une. Bien que cette transformation des coordonnées ait pu être dénie en raison des restrictions spéciques de 2D LWR, ce modèle peut néanmoins être utilisé pour prédire l'évolution du trac dans plusieurs situations fréquentes, par exemple lorsque, à l'heure de pointe du matin, tous les véhicules se dirigent vers le centre-ville où se trouvent la plupart des bureaux.

En outre, nous présentons plusieurs résultats obtenus en analysant 2D LWR en coordonnées curvilignes. Nous élaborons une technique permettant d'obtenir une distribution de véhicules dans un état d'équilibre uniquement en connaissant les données du ux entrant et du ux viii sortant d'une zone urbaine. Cette capacité d'analyser l'équation en 2D pour obtenir des équilibres admissibles est un résultat essentiel qui permet de formuler des tâches de contrôle pour la stabilisation du trac urbain. En outre, nous concevons un contrôleur de frontière pour atténuer les congestions dans une zone urbaine. Ainsi, pour simplier, le trac est limité au régime de la congestion.

Ensuite, un problème de contrôle aux limites visant à approximer la trajectoire désirée du véhicule est posé pour un régime de trac mixte en temps asymptotique. Pour cela, le modèle 2D LWR en coordonnées curvilignes est ensuite considéré dans le formalisme de Hamilton-Jacobi qui facilite la gestion des discontinuités pour la conception du contrôle aux limites.

Le problème du contrôle aux limites est résolu de la même manière que pour le cas 1D. La diculté supplémentaire est introduite par la dépendance spatiale explicite dans le diagramme fondamental, de sorte que la formule classique de Lax-Hopf ne peut être appliquée. Au lieu de cela, on applique la théorie de la viabilité élaborée pour le cas des hamiltoniens dépendants de l'espace. Pour un exemple numérique, nous prend la structure du centre-ville de Grenoble comme réseau urbain.

Enn, le modèle 2D LWR est utilisé en coordonnées curvilignes pour concevoir un contrôleur de limite de vitesse variable (Variable Speed Limit -VSL). Le contrôleur VSL est utilisé pour aecter directement le ux de trac en imposant des restrictions temporaires sur la vitesse autorisée, ce qui est souvent utilisé pour des situations spéciques telles que les accidents, les mauvaises conditions météorologiques, etc. Il s'agit d'un contrôleur intradomaine qui est appliqué d'une manière continue dans l'espace sur l'ensemble du domaine. Il agit comme une linéarisation par rétroaction de sorte que l'équation d'état perd sa structure de loi de conservation, ce qui facilite son analyse. Le contrôleur VSL peut être utilisé pour stabiliser le système 2D à n'importe quel équilibre souhaité variant dans l'espace. Si FD a une dépendance concave par rapport au contrôleur dans le régime de trac congestionné et une dépendance linéaire dans le régime du ux libre, le contrôleur est diérentiable presque partout dans l'espace. On conçoit également un état d'équilibre optimal qui correspond à la maximisation du ux obtenue pour le nombre maximal possible de voitures. Dans un exemple numérique, on utilise à nouveau la structure du centre-ville de Grenoble, puis on démontre comment le contrôleur VSL conçu fait converger la densité de véhicules vers l'équilibre souhaité.

Trac multidirectionnel sur les réseaux

Dans ce chapitre, nous proposons notre propre méthode pour traiter le trac multidirectionnel évoluant sur des réseaux urbains de taille arbitraire à un niveau macroscopique. Le trac multidirectionnel est beaucoup plus proche de la représentation du trac urbain dans des situations réalistes que le modèle 2D LWR. L'idée globale consiste à dériver un modèle EDP qui capture le comportement du trac évoluant dans un réseau urbain dans n'importe quelle direction avec des croisements de ux.

On commence par considérer un modèle du ux du trac à une intersection basé sur le ix modèle classique de transmission cellulaire. Chaque intersection est caractérisée par un certain nombre de routes entrantes et sortantes qui peuvent être orientées arbitrairement dans l'espace.

Ainsi, il y a un nombre diérent de paramètres à régler pour chaque intersection individuelle.

Comme un réseau peut être composé de milliers d'intersections, on trouve une approche uniée pour décrire le trac aux intersections indépendamment de leurs paramètres individuels. Ainsi, on suppose que la dynamique du trac multidirectionnel peut être représentée par seulement 4 couches de direction: Nord, Est, Ouest et Sud (North-East-West-South -NEWS). Dans le formalism du NEWS, on déploie des matrices de projection basées sur la géométrie pour appliquer le ux de trac le long de toute route dans les directions cardinales. Les poids de projection varient continuellement avec l'angle d'orientation de la route. Ensuite, le concept du ux partiels est introduit pour capturer divers modèles d'origine-destination aux intersections.

Ainsi, on obtiens un modèle du ux du trac qui prédit le taux de changement de l'accumulation de véhicules à l'intersection d'une manière uniée. Ensuite, la méthode de continuation est appliquée pour obtenir un modèle qui prédit l'évolution de la densité des véhicules à proximité d'une intersection. Cette méthode est utilisée pour transformer une EDO (modèle routier) en une EDP (modèle d'intersection), qui représente un modèle de continuum macroscopique pour une intersection. Comme chaque intersection a été décrite de la manière uniée, pondération inverse à la distance est appliquée pour dénir tous les paramètres d'intersection pour chaque point dans un plan continuum. La dérivation du modèle NEWS a été faite analytiquement en utilisant une seule hypothèse sur la structure du réseau. À savoir, les réseaux urbains doivent être bien conçus en termes de ux maximal, c'est-à-dire que si les véhicules se déplacent à un ux maximal avant une intersection, ils continuent à utiliser la capacité de la route au maximum après le virage.

En conséquence, on obtiens le modèle NEWS qui prédit l'évolution du trac dans quatre directions cardinales. La propagation du ux du trac dans chaque direction est pilotée par le concept d'ore et de demande qui s'appuie sur le diagramme fondamental. De plus, véhicules peuvent changer de couche de direction, c'est-à-dire qu'il existe un couplage entre diérentes couches, ce qui est un aspect important en raison de son ubiquité physique.

Les propriétés mathématiques du modèle NEWS dérivé sont également analysées. Le système d'EDP est hyperbolique pour tout ensemble de paramètres. Le fait de pouvoir classer un modèle comme une EDP hyperbolique simplie considérablement l'analyse pour les tâches futures, puisque de nombreux résultats analytiques ont déjà été élaborés pour ce type de systèmes. Il a également été démontré que le modèle représente une loi de conservation, la densité du trac étant la quantité conservée. De plus, il a été démontré que son état est borné, ce qui est une hypothèse réaliste importante pour la modélisation du trac.

Les résultats de la prédiction du modèle sont validés à l'aide de la microsimulateur Aimsun et de la plateforme expérimentale GTL Ville qui fournit des données en temps réel provenant d'un réseau des capteurs installés dans le centre-ville de Grenoble. Les résultats de validation révèlent que la distribution de la densité prédite par le modèle NEWS reste en bon accord avec la densité de référence, soit 90 % de similarité avec Aimsun et 80 % de similarité avec l'expérience réelle. La validation du modèle avec des données réelles est un projet open source, c'est-à-dire que les résultats sont reproductibles et peuvent être utilisés pour des études futures.

x Enn, le modèle NEWS est étudié du point de vue du contrôle, où on se limite au régime du trac congestionné pour des raisons de simplicité. On analyse la classe d'équilibres admissibles souhaités qui doivent satisfaire un certain système d'EDP. On pose et résolve le problème de la recherche d'un état d'équilibre qui permet de minimiser la congestion dans un réseau urbain sous la contrainte que sa gamme de valeurs doit rester dans le régime de congestion. De plus, on prouve la convergence exponentielle d'un état congestionné contrôlé depuis un limite vers cet équilibre désiré en utilisant des méthodes de Lyapunov. Le réseau du centre-ville de Grenoble est à nouveau utilisé pour démontrer la performance du contrôleur aux limites dérivé à l'aide d'un exemple numérique. On montre que le contrôleur fait converger la densité du trac vers l'équilibre optimal désiré en temps ni, ce qui est lié à la nature hyperbolique de l'EDP.

Perspectives et extensions

Sur la base des résultats de cette thèse de doctorat, je vois un grand nombre de directions intéressantes pour la recherche future. Les questions ouvertes suivantes semblent être les plus pertinentes:

Dans cette thèse, le trac a été décrit d'une manière assez simpliste, puisque LWR représente un modèle macroscopique le plus simple du trac. En général, il est bien connu que l'approche de la modélisation LWR présente plusieurs inconvénients, car elle ne prend pas en compte de nombreux phénomènes importants tels que l'accélération limitée ou la baisse de capacité due à la transition d'un régime ux libre à un régime congestionné. En outre, une façon possible d'aner la description du trac est de prendre en compte diérentes classes de conducteurs en fonction de leur vitesse (par exemple, rapide et lente). Ainsi, on pourrait étudier les problèmes du contrôle des limites à l'aide d'une approche de modélisation plus sophistiquée qui tient compte des limites du modèle LWR (modèles d'ordre supérieur et multi-classes).

Le modèle 2D LWR est limité à la description du trac sur les réseaux qui ont une direction de mouvement préférée, ce qui n'est pas réaliste pour le trac général. Par conséquent, comme une extension prometteuse de la recherche sur la modélisation macroscopique du trac urbain, on pourrait développer des algorithmes de partitionnement qui divisent un réseau urbain en zones ayant une direction de mouvement préférée.

Les contrôleurs aux limites sont conçus pour un trac évoluant sur un domaine rectangulaire continu qui se rapproche du réseau urbain sous-jacent. En conséquence, on a obtenu des lois de contrôle dénies sur une ligne continue, ce qui n'est pas directement interprétable. Il serait donc intéressant d'étudier ce problème et de mettre au point une méthode permettant de transformer les contrôleurs aux limites dénis sur des lignes continues en contrôleurs réglés sur des points ou des intervalles spéciques sur des routes réelles.

where F stays for free-ow and C for congested regime. . . . . . . . . . . . . . . 52 2.11 A possible error behaviour e(t) (thick black line). From left to right: divergence for g out (t) = 0 sometimes (in orange); exponential convergence in Regime 1 (in green); fast convergence for g in (t) = 0 sometimes (in blue); then divergence for e(t ) < 0 ∀t ∈ [t i , ti ]; exponential convergence in Regime 1.

Trac on roads

Trac models have been developed and studied in order to describe trac dynamics and predict the appearance of congestions since the beginning of the twentieth century. The origin of the trac ow theory takes us back to the thirties, when Greenshields [START_REF] Greenshields | The photographic method of studying trac behavior[END_REF] collected data from a highway road on the car headway distance (average distance of two consequative vehicles) and their average velocity. One year later, in [START_REF] Greenshields | A study of trac capacity[END_REF] he proposed a fundamental relation connecting the average car velocity with the vehicle density. This empirically established law became very famous later in the trac engineering community, and nowadays it is known as the Greenshields fundamental diagram. It can also be represented in terms of vehicle density ρ (average number of vehicles per unit length) and ow φ (average number of vehicles per time unit), which yields a concave relation Φ(ρ):

φ = Φ(ρ) = v max 1 - ρ ρ max ρ,
where ρ max is the vehicle density at the trac jam, and v max = Φ (0) is the maximal average density of vehicles on a freeway. Afterwards, many other possible shapes of fundamental diagrams have been proposed, see [START_REF] Michael | A Generic Characterization of Equilibrium Speed-Flow Curves[END_REF] for a detailed overview on ow-density curves. Nowadays, the most simplistic ow-density relation that is widely used for analysis of trac is the triangular (bilinear) fundamental diagram proposed by Daganzo in 1994 [START_REF] Daganzo | The cell transmission model: A dynamic representation of highway trac consistent with the hydrodynamic theory[END_REF].

Origins of trac modeling

The discovery of fundamental diagram plays an essential role in the history of trac ow modeling. It was the rst evidence that trac can be described and analyzed in terms of dynamic systems rather than considering it as a collection of independent vehicles. Thus, the ability to formulate a fundamental diagram gave rise to appearance of dierent trac ow models, see [START_REF] Van Wageningen-Kessels | Genealogy of trac ow models[END_REF] for a detailed review. In general, these can be categorized as microscopic and macroscopic trac models depending on the level of description detail. Microscopic models trace the behaviour of each individual vehicle. The main assumption of these models is that drivers adapt their behaviour to that of the leading vehicle, i.e., car-following models, see for example [START_REF] Pipes | An Operational Analysis of Trac Dynamics[END_REF][START_REF] Kometani | Dynamic behaviour of trac with a non-linear spacingspeed relationship[END_REF]. On the other hand, trac can be alternatively described from the macroscopic point of view. In this case, trac state is given in terms of aggregated variables 1 such as average density and average ow, while individual vehicles are not modeled. In this thesis, we describe trac in terms of macroscopic variables. This enables to analyze trac on some aggregated level that is a useful approach when it comes to large-scale transportation networks.

Macroscopic trac models are often compared to uid, since they describe trac ow as if it were a continuum. In the fties, the kinematic wave theory for trac has been formulated by Lighthill and Whitham [START_REF] Lighthill | On kinematic waves, II: A theory of trac ow on long crowded roads[END_REF] and, independently, Richards [START_REF] Richards | Shock waves on the highway[END_REF]. This so-called LWR model is a uido-dynamic model that prescribes the conservation of the number of vehicles. It describes the spatio-temporal evolution of vehicle density on an innite highway road as the following rst-order scalar hyperbolic partial dierential equation (PDE):

∂ t ρ(x, t) + ∂ x Φ(ρ) = 0, ∀(x, t) ∈ R × R + .
Its key assumption is the existence of a concave ow-density relation (fundamental diagram), which allows to consider this conservation law equation as a model for trac.

The LWR model was the rst macroscopic model in the history of trac modeling, and it has some physical limitations. For instance, according to the LWR model vehicles reach the new equilibrium velocity immediately after a change in the trac state, which implies innite acceleration. This problem was addressed in [START_REF] Lebacque | A Two Phase Extension of the LWR Model Based on the Boundedness of Trac Acceleration[END_REF][START_REF] Leclercq | A New Numerical Scheme for Bounding Acceleration in the LWR Model[END_REF], where the LWR model was extended to take the bounded acceleration into account. Another drawback of the LWR model is that the transition from the free-ow to the congested trac regime occurs at the same density and without capacity drop. This was addressed by [START_REF] Daganzo | A simple physical principle for the simulation of freeways with special lanes and priority vehicles[END_REF][START_REF] Jin | A kinematic wave theory of lane-changing trac ow[END_REF] by introducing lane changing. However, even despite the appearance of more sophisticated rst-order [START_REF] Daganzo | A behavioral theory of multi-lane trac ow. Part I: Long homogeneous freeway sections[END_REF][START_REF] Wong | A multi-class trac ow model an extension of LWR model with heterogeneous drivers[END_REF][START_REF] Benzoni-Gavage | An n-populations model for trac ow[END_REF] or even higher-order models [START_REF] Aw | Resurrection of second order models of trac ow? In[END_REF][START_REF] Greenberg | Congestion on multilane highways[END_REF] capable of covering more realistic trac behaviour, the LWR model remains the most used one to study due to its simplicity and ability to reproduce the most essential trac phenomena such as wave formation and propagation. LWR model was also shown to be consistent with car-following behaviors at the aggregated level [START_REF] Newell | Nonlinear Eects in the Dynamics of Car Following[END_REF].

Although being the most simple continuous trac model, the explicit analysis of the LWR equation is a tedious task. In general, such partial dierential equations are solved using the method of characteristics [START_REF] Evans | Partial dierential equations[END_REF]. However, the nonlinearity of the fundamental diagram introduces nonlinearities in the characteristic elds. Therefore, even with a smooth initial datum characteristic lines may intersect, which leads to discontinuities at intersection points.

This triggers a shock or a rarefaction wave depending on the state at the moment of intersection (Riemann problem). Then, the conservation law solution is not dened in the classical sense, and therefore needs to be considered in its weak formulation. This formulation yields multiple solutions, among which the entropy solution [START_REF] Ansorge | What does the entropy condition mean in trac ow theory?[END_REF] is recognized to be the physically reasonable one. Mathematical properties of hyperbolic conservation laws have been extensively studied, and an interested reader is referred to [START_REF] Bressan | Hyperbolic systems of conservation laws[END_REF][START_REF] Holden | Front tracking for hyperbolic conservation laws[END_REF][START_REF] Lax | Nonlinear hyperbolic equations[END_REF].

There is however a way to study the kinematic waves of trac without any need to deal with shocks in the solution. In [START_REF] Newell | A simplied theory of kinematic waves in highway trac, part I: General theory[END_REF][START_REF] Newell | A simplied theory of kinematic waves in highway trac, part II: Queueing at freeway bottlenecks[END_REF][START_REF] Newell | A simplied theory of kinematic waves in highway trac, part III: Multidestination ows[END_REF] Newell proposed an alternative way to consider trac on a macroscopic scale by numbering vehicles at the highway entry and following the evolution of vehicle numbers at every location and time. Thus, the trac state can be described in terms of cumulative number of vehicles M that evolves as

∂ t M (x, t) + Φ(∂ x M ) = 0, ∀(x, t) ∈ R × R + .
This equation has the structure of a Hamilton-Jacobi PDE, which represents an integral form of the LWR PDE. Its solution is a Lipschitz continuous function that is free of shocks (in the worst case it is only non-dierentiable), and it can obtained by solving a simple minimization problem.

A variational formulation of kinematic waves was studied in [START_REF] Daganzo | A variational formulation of kinematic waves: basic theory and complex boundary conditions[END_REF][START_REF] Daganzo | On the variational theory of trac ow: Well-posedness, duality and applications[END_REF], who showed that every well-posed trac problem with a concave ow-density relation can be solved as a set of shortest paths. In general, the explicit solution of Hamilton-Jacobi PDE can be obtained using the viability framework, which was rst shown for the case of convex conservation laws in [START_REF] Joseph | Explicit formula for the solution of convex conservation laws with boundary condition[END_REF][START_REF] Joseph | Solution of convex conservation laws in a strip[END_REF]. The viability framework is based on using Lax-Hopf formula that exploits the structure of a dynamic programming problem, and the solution is obtained as the minimum of all valid paths, see also [START_REF] Claudel | LaxHopf Based Incorporation of Internal Boundary Conditions Into HamiltonJacobi Equation. Part I: Theory[END_REF]. Several computational algorithms have been developped to obtain solutions of H-J PDE for some special cases in the context of trac modeling. Thus, [START_REF] Claudel | LaxHopf Based Incorporation of Internal Boundary Conditions Into Hamilton-Jacobi Equation. Part II: Computational Methods[END_REF] presented a computational method for any piecewise ane initial condition. The Lax-Hopf algorithm to compute the solution on a single link for any concave fundamental diagram has been suggested by [START_REF] Mazaré | Analytical and grid-free solutions to the LighthillWhithamRichards trac ow model[END_REF]. Its improved version with a lower computational time has been proposed in [START_REF] Simoni | A Fast LaxHopf Algorithm to Solve the LighthillWhithamRichards Trac Flow Model on Networks[END_REF].

In some cases, the exact solution to LWR PDE can be obtained using the wave-front tracking method [START_REF] Henn | A Wave-Based Resolution Scheme for the Hydrodynamic LWR Trac Flow Model[END_REF][START_REF] Lu | Explicit construction of entropy solutions for the LighthillWhithamRichards trac ow model with a piecewise quadratic owdensity relationship[END_REF][START_REF] Wong | An analytical shock-tting algorithm for LWR kinematic wave model embedded with linear speeddensity relationship[END_REF]. This method can also be used to prove the existence of solutions to conservation laws, see for example [START_REF] Colombo | Wave Front Tracking in Systems of Conservation Laws[END_REF]. The solution of a LWR PDE can also be numerically approximated using computational methods such as the Godunov scheme [START_REF] Godunov | A dierence method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF], or the Lax-

Friedrichs method [START_REF] Leveque | Numerical Methods for Conservation Laws[END_REF]. These are both nite dierence methods. The Godunov scheme deals with Riemann problems at each cell, and the Lax-Friedrichs method requires adding articial viscosity.

In the nineties, a time-discrete approximation of the LWR equation was introduced in [START_REF] Daganzo | The cell transmission model: A dynamic representation of highway trac consistent with the hydrodynamic theory[END_REF][START_REF] Daganzo | The cell transmission model, part ii: network trac[END_REF], which is now known as the cell transmission model (CTM). This model can be viewed as a

Godunov-type discretization of LWR, and it is based on approximating links (roads) by cells.

The amount of ux that is transmitted between cells is based on their current occupancy (the demand-supply concept). Nowadays, CTM is the most popular model in the trac community due to its simplicity and the ability of a straightforward extension to networks.

Road control

Continuing urbanization caused by ever-growing population of the planet implies a growing demand for transportation. This entails formation of severe congestions that cost people hundreds of hours per year and that also have a signicant negative impact on the environment.

For instance, the Urban Mobility Report [START_REF] Schrank | Urban Mobility Report[END_REF] summarized the major daily life problems caused by trac congestions in USA in 2017. According to this report, urban American drivers experience the following losses per year on average: 8.8 billion hours of time delay, 3.3 billion gallons of wasted fuel and an equivalent monetary cost of 179 billion dollars. This requires the development of ecient solutions for intelligent transportation management.

There are several common techniques to control trac. One of the most widely used techniques is a suitable application of a variable speed limit (VSL) along a highway road. It is applied such that the maximal allowed speed is decreased, which mitigates risk of accidents, as well as it results into a lower fuel consumption and reduced emissions. Previous works [START_REF] Abdel-Aty | Evaluation of variable speed limits for real-time freeway safety improvement[END_REF][START_REF] Li | Development of a Control Strategy of Variable Speed Limits to Reduce Rear-End Collision Risks Near Freeway Recurrent Bottlenecks[END_REF][START_REF] Zhang | Combined Variable Speed Limit and Lane Change Control for Truck-Dominant Highway Segment[END_REF] conrmed that the VSL control enhances trac safety and has a positive environmental impact. Reduced travel time is another positive eect of VSL controllers reported by [START_REF] Muralidharan | Computationally ecient model predictive control of freeway networks[END_REF][START_REF] Ramón | Hybrid model predictive control for freeway trac using discrete speed limit signals[END_REF]. The eect of VSL on the shape of the fundamental diagram was studied in [START_REF] Castelan Carlson | Local Feedback-Based Mainstream Trac Flow Control on Motorways Using Variable Speed Limits[END_REF][START_REF] Papageorgiou | Eects of Variable Speed Limits on Motorway Trac Flow[END_REF].

However, the improvement of travel time achieved with VSL control revealed inconsistencies in microscopic simulations and eld tests [START_REF] Kejun | Model predictive control for variable speed limit in freeway work zone[END_REF][START_REF] Torné Santos | Evaluation of speed limit management on c-32 highway access to barcelona[END_REF]. These inconsistencies are related to the fact that it is hard to precisely predict trac conditions at some localized congested bottleneck via the macroscopic modeling. Moreover, not every human driver adapts his/her velocity to numbers displayed on electronic trac signs.

Alternatively, trac can be regulated from the boundary, that is either from entry or exit of the corresponding highway road, e.g., by actuating on-and o-ramps. By managing onramp trac inows, the application of meters also reduces the travel time, harmful emissions and improves highway safety [START_REF] Levinson | Ramp meters on trial: Evidence from the Twin Cities metering holiday[END_REF]. Moreover, [START_REF] Zhang | Ramp metering and freeway bottleneck capacity[END_REF] reported that ramp metering helps reducing the average freeway delay.

However, in most of the cases, control for a trac road is designed using the discretized version of the corresponding trac model. According to [START_REF] Leveque | Numerical Methods for Conservation Laws[END_REF], such discretizations are known to alter essential phenomena predicted by the original macroscopic trac models and may lead to inconsistent discrete versions. Recall that from the mathematical viewpoint, macroscopic trac models such as LWR mostly represent conservation laws with dynamics governed by hyperbolic partial dierential equations. The theory on the exact controllability and exact observability was completely developed in [START_REF] Russell | Controllability and Stabilizability Theory for Linear Partial Dierential Equations: Recent Progress and Open Questions[END_REF][START_REF] Lions | Controlabilite Exacte, Perturbations et Stabilisation de Systemes Distribues[END_REF] for linear and in [START_REF] Li | Controllability and Observability for Quasilinear Hyperbolic Systems[END_REF] for 1D quasilinear hyperbolic PDEs. The results on exact controllability for nonlinear scalar conservation laws with a strictly convex ux function were discussed in [START_REF] Perrollaz | Exact Controllability of Scalar Conservation Laws with an Additional Control in the Context of Entropy Solutions[END_REF]. Classical techniques widely used for control of hyperbolic conservation laws are backstepping [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF], Lyapunov-based [START_REF] Blandin | Boundary stabilization of the inviscid Burgers equation using a Lyapunov method[END_REF][START_REF] Coron | A Strict Lyapunov Function for Boundary Control of Hyperbolic Systems of Conservation Laws[END_REF] and optimal control methods using adjoint-based calculus [START_REF] Ulbrich | A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms[END_REF][START_REF] Gugat | Optimal control for trac ow networks[END_REF][START_REF] Jacquet | Optimal control of scalar one-dimensional conservation laws[END_REF][START_REF] Alexandre | Network Congestion Alleviation Using Adjoint Hybrid Control: Application to Highways[END_REF]. Optimal control tasks for trac are considered to solve the most common problems of trac regulation, i.e., minimization of total travel time and fuel consumption, or throughput maximization.

However, the classical control methods mentioned above are not always well suited to handle shocks, since they require the knowledge on the internal shock dynamics. Tracking dynamics of shocks was done, e.g., in [START_REF] Blandin | Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws[END_REF], where the weak formulation and the Rankine-Hugoniot relation were used to stabilize solution of the Burgers equation to a constant equilibrium. In a recent work [START_REF] Bastin | Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation[END_REF] the problem of boundary control of solutions with jump discontinuities has been considered. In both [START_REF] Blandin | Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws[END_REF][START_REF] Bastin | Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation[END_REF], the desired state was stationary and the Lyapunov methods were applied. In [START_REF] Donadello | Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions[END_REF] the exact controllability of solutions to conservation laws to space-and time-dependent trajectories has been studied. Nevertheless, the problem of stabilizing a state with shocks to space-and time-dependent trajectories that may also contain shocks has never been considered before. We address this problem in Chapter 2.

Trac on urban networks

In its original formulation, the LWR-model is applicable only to single roads of innite length.

Extension to urban networks required developing a methodology for intersection modeling within the LWR framework. This was rst done at the end of the last century by [START_REF] Holden | A Mathematical Model of Trac Flow on a Network of Unidirectional Roads[END_REF], who considered a network of uni-directional roads. Later on, this formulation was rened to capture multi-directional trac, e.g., see [START_REF] Coclite | Trac Flow on a Road Network[END_REF]. The CTM has also been extended to networks in [START_REF] Daganzo | The cell transmission model, part ii: network trac[END_REF],

who considered networks as directed graphs consisting of links (roads) and nodes (junctions).

The general theory of trac ow on networks is presented in [START_REF] Garavello | Trac Flow on Networks[END_REF]. The Cauchy problem for complex networks (with more than two incoming and outgoing roads at junctions) was considered in [START_REF] Garavello | Conservation laws on complex networks[END_REF].

The most common way to control trac on a urban level is to optimize the time intervals of green signal at signalized intersections, see [START_REF] Chitour | Trac circles and timing of trac lights for cars ow[END_REF][START_REF] Ge | An alternative denition of dynamic user optimum on signalised road networks[END_REF]. There exist also other control techniques applied in transportation networks, such as routing of trac [START_REF] Gugat | Optimal control for trac ow networks[END_REF], ramp metering [START_REF] Papageorgiou | ALINEA: a local feedback control law for on-ramp metering[END_REF],

variable speed limits [START_REF] Papageorgiou | Eects of Variable Speed Limits on Motorway Trac Flow[END_REF][START_REF] Suyash | Variable Speed Limit and Ramp Metering Control of Highway Networks Using Lax-Hopf Method: A Mixed Integer Linear Programming Approach[END_REF], see also [START_REF] Papageorgiou | Review of road trac control strategies[END_REF] for a general review of trac control strategies.

The main challenge in this link-level (discrete) representation of trac networks is the large computational time. For instance, if we consider large urban networks consisting of thousands of links, the need to use much of trac data considerably exaggerates validation of control performance [START_REF] Ziliaskopoulos | Large-scale dynamic trac assignment: implementation issues and computational analysis[END_REF].

Another way to model trac on urban areas is again to consider continuous macroscopic models. They describe trac as a two-dimensional uid moving on a continuum plane that corresponds to a dense urban network. This approach has various advantages, e.g., the problem size does not depend on the number of roads, as well as less data are required for the model setup. Early works on continuous urban trac modeling [START_REF] Smeed | The road capacity of city centers[END_REF][START_REF] Thomson | Speeds and ows of trac in Central London: 2. Speed-ow relations[END_REF][START_REF] Herman | A two-uid approach to town trac[END_REF] presented static models with the focus on determining equilibria states in urban networks. However, due to the lack of any knowledge of a ow-density relation on a city level, these models failed in capturing trac dynamics during rush hours, see [START_REF] Ho | Two-dimensional Continuum Modeling Approach to Transportation Problems[END_REF] for a general review of such models.

The rst demonstration of existing macroscopic relation between density and ow should be recognized to [START_REF] Williams | Urban trac network ow models[END_REF], who used data from microsimulations. Later this relation was also observed during an experiment conducted in the congested region of Yokohama, Japan [START_REF] Geroliminis | Existence of urban-scale macroscopic fundamental diagrams: Some experimental ndings[END_REF],

and was then generalized in [START_REF] Daganzo | An analytical approximation for the macroscopic fundamental diagram of urban trac[END_REF]. This functional relation has the same physical meaning as the fundamental diagram for highway roads, but it was shown to exist also on urban areas.

The discovery of macroscopic fundamental diagram (MFD) plays an essential role in the development of trac models for urban areas. The empirical evidence of MFD led to appearance of reservoir models, which are also called accumulation models. These models predict the rate of change of the vehicle accumulation in some urban area (reservoir) that is determined by the dierence between its inow and outow, see [START_REF] Aghamohammadi | Dynamic trac assignment using the macroscopic fundamental diagram: A Review of vehicular and pedestrian ow models[END_REF] for a review on several MFD-based models. The network's MFD can be dened by collecting real trac data [START_REF] Geroliminis | Existence of urban-scale macroscopic fundamental diagrams: Some experimental ndings[END_REF] or by running numerical simulations [START_REF] Haddad | On the stability of trac perimeter control in two-region urban cities[END_REF]. In [START_REF] Laval | Stochastic approximations for the macroscopic fundamental diagram of urban networks[END_REF] it has also been shown that the MFD can be well approximated by a function of only two parameters: the density of trac lights and the mean red to green ratio across the network.

The MFD-based approach is intuitive and easy in use, which makes it popular for trac control design such as perimeter control [START_REF] Geroliminis | Optimal Perimeter Control for Two Urban Regions With Macroscopic Fundamental Diagrams: A Model Predictive Approach[END_REF]3], robust control [START_REF] Haddad | Robust Constrained Control of Uncertain Macroscopic Fundamental Diagram Networks[END_REF], etc. It is important to note that only homogeneously congested areas may have a well-dened MFD, see [START_REF] Geroliminis | Properties of a well-dened macroscopic fundamental diagram for urban trac[END_REF] for properties of well-dened MFDs. In general, there must exist only one ow value for a given number of vehicles. This feature is preserved only in regions that consist of links characterized by similar congestion levels, while this causes problems in case of regions with heterogeneous links. In this case, one can apply partitioning algorithms that divide a problematic area into multiple smaller areas each having a well-dened MFD for given trac conditions [START_REF] Hajiahmadi | Optimal dynamic route guidance: A model predictive approach using the macroscopic fundamental diagram[END_REF][START_REF] Leclercq | Macroscopic trac dynamics with heterogeneous route patterns[END_REF]. For the case of rapidly changing trac conditions (e.g., accident on a road), a dynamic clustering algorithm has been proposed by [START_REF] Saeedmanesh | Dynamic clustering and propagation of congestion in heterogeneously congested urban trac networks[END_REF]. The main drawback of MFD-based modeling is that it assigns only one value to characterize trac on some urban area being the current number of cars in this area. Thus, the level of precision to describe trac behaviour on a global level depends on the number of dened clusters, i.e., in some sense it acts like CTM in 2D. This leads to the loss of information during the process of congestion formation and dissolution in a transportation network.

Another way to describe the evolution of trac in urban areas is to use dynamic twodimensional continuum models. These share a lot of features with pedestrian models [START_REF] Hughes | A continuum theory for the ow of pedestrians[END_REF].

The main dierence is that crowds evolve in an open space, while vehicles are restricted to move on roads. In [START_REF] Della Rossa | A distributed model of trac ows on extended regions[END_REF] authors considered a model including a diusion term and a drift term that depends on the density. The direction of the drift vector is determined by the shape of the network. Other works [START_REF] Jiang | A dynamic trac assignment model for a continuum transportation system[END_REF][START_REF] Du | Revisiting Jiang's dynamic continuum model for urban cities[END_REF][START_REF] Jiang | Macroscopic modeling approach to estimate trac-related emissions in urban areas[END_REF] dene the ux function by solving Eikonal equations such that the ow follows the path of the lowest cost. For a review of 2D continuum models the reader is referred to [START_REF] Aghamohammadi | Dynamic trac assignment using the macroscopic fundamental diagram: A Review of vehicular and pedestrian ow models[END_REF]. A recent work [START_REF] Mollier | Two-dimensional macroscopic model for large scale trac networks[END_REF] introduced a direct extension of LWR model in two dimensions:

∂ρ(x, y, t) ∂t

+ ∇ • Φ(x, y, ρ) = 0, ∀(x, y, t) ∈ R 2 × R + .
Thereby, the ux function became a vector. Its direction is retrieved from the geometry of the underlying urban network, while the ux magnitude depends on network infrastructure parameters that are incorporated into the space-dependency of the fundamental diagram. In Chapter 3 we investigate this model to design boundary and in-domain control for mixedregime urban trac that admits shocks.

The aforementioned references however consider trac ow that has only one direction of motion. Several years ago a dynamic continuum model for multi-directional pedestrian ows was presented in [START_REF] Flurin | A dynamic network loading model for anisotropic and congested pedestrian ows[END_REF]. This model represents an extesion of CTM to pedestrian dynamics, which however does not take urban network geometry into account. The rst attempt to include multiple directions in 2D continuum models for vehicular trac has been made also only a few years ago by [START_REF] Lin | A predictive continuum dynamic user-optimal model for a polycentric urban city[END_REF] who deployed dynamic user-optimal principle for the path choice.

The drawback of this model is that the trac density may become unbounded (it is not based on a fundamental diagram). There exist also other works [START_REF] Mollier | A step towards a multidirectional 2D model for large scale trac networks[END_REF]4] proposing 2D multi-layer models with bounded densities. However, these models do not include mixing between dierent direction layers, i.e., vehicles can not change their direction of motion. Then, these models are also not necessarily hyperbolic, i.e., their equation type varies with parameters, which exaggerates its analysis and numerical simulation. Hyperbolicity for all parameters implies that it can be analysed like many other conservation law based models for trac. We x both of these aspects in Chapter 4 by elaborating a novel multi-directional model. Subsequently, this model is used for boundary control design of multi-directional urban trac.

Problem statements and contributions

This thesis is devoted to control of urban trac evolving on large-scale transportation networks. Trac is described in terms of ow and density of vehicles, i.e., from the macroscopic point of view. The dynamic equation that predicts the spatio-temporal evolution of trac corresponds to a PDE that has a structure of a conservation law. Thus, trac can be seen as a 2D uid that propagates along a continuum 2D plane with a total surface determined by the size of the underlying urban network. To manage urban trac, we use a purely model-based control design. This means that control is designed by analysing the intrinsic properties of the model. Therefore, the obtained controller is scalable and adaptive to changing trac conditions, as well as it is applicable to any urban network of arbitrary size.

In Chapter 2, we consider trac evolving along a single road of nite length with dynamics governed by the LWR PDE. Our main goal thereby is to derive a boundary control law such that the trac state tracks some desired space-and time-varying trajectory.

First, we consider a linear system with disturbance for congested trac, which can be solved using the characteristics method. The desired trajectory is achieved by actuating the downstream boundary of the road (published in [START_REF] Tumash | Robust tracking control design for uid trac dynamics[END_REF]). Then, we extend this problem to a mixed-regime trac governed by a full LWR PDE, for which no classical solutions exist. We solve the problem by analysing the system in its Hamilton-Jacobi formulation. Thus, the main contribution of Chapter 2 is the boundary control design for a mixed-regime trac with solution shocks that tracks the desired trajectory that also admits solution shocks for asymptotic time. This result can be seen as a general solution to any control problem that can be posed for LWR trac on nite roads, and it was published in [START_REF] Tumash | Boundary Control Design for Trac with Nonlinear Dynamics[END_REF].

Chapter 3 is devoted to control design for trac on urban networks. Thereby, trac dynamics are described by the LWR model that contains an additional space dimension. It considers trac as a uid moving on a 2D plane that represents a continuous approximation of the urban area under consideration. The network infrastructure parameters are embedded as an explicit space-dependency of the fundamental diagram. The direction of movement is determined by the network geometry. The main limitation of this model is that it is designed to describe trac on networks with uni-directional roads. Our main contribution in this chapter is to propose a holistic approach to solve any possible control task for uni-directional 2D trac, which was sent for a publication [START_REF] Tumash | Boundary and VSL Control for Large-Scale Urban Trac Networks[END_REF]. This is done by proposing a curvilinear coordinate transformation that allows to rewrite the 2D model such that it can be treated as a parametrized 1D problem, which can be explicitly analyzed. By analyzing this trac model in new coordinates, we solve a variety of control problems. First, we present a method to analytically estimate a steady-state knowing only network structure and inow data (published in [START_REF] Tumash | Equlibrium manifolds in 2D uid trac models[END_REF]). Then, this result is used for a boundary control design such that congested trac achieves the best equilibrium corresponding to the throughput maximization (published in [START_REF] Tumash | Topologybased control design for congested areas in urban networks[END_REF]). Moreover, we also use this 2D model in curvilinear coordinates to solve a trajectory tracking task for trac in a mixed regime in a similar way as in Chapter 2 but handling additional technical issues. Finally, we also analyze the system to design a variable speed limit control. These results on mixed-regime trac control were sent to a journal [START_REF] Tumash | Boundary and VSL Control for Large-Scale Urban Trac Networks[END_REF].

In Chapter 4, we deal with the main limitation of the preceding chapter that considered only uni-directional urban networks. Grenoble downtown. The model design and validation results have been sent to a publication [START_REF] Tumash | Multi-Directional Continuous Trac Model For Large-Scale Urban Networks[END_REF]. Finally, the new model was analyzed for a boundary control design that can mitigate congestions in multi-directional trac networks (the result was accepted for a publication [START_REF] Tumash | Boundary Control for Multi-Directional Trac on Urban Networks[END_REF]).

This thesis is organized as follows. In Chapter 2, we introduce the LWR model and discuss its properties and solutions, as well as the Hamilton-Jacobi theory and the Godunov scheme are presented. Then, we present the control results obtained for trac on a single road. In Chapter 3, we give details on the 2D LWR model and how it can be used to describe the dynamics of urban trac, which is then also compared to an MFD-based model. Afterwards, we introduce the curvilinear coordinate transformation for the 2D LWR and solve a variety of control tasks for uni-directional urban trac. In Chapter 4, a new model for trac with multiple directions is derived. We also discuss its properties and validate it with synthetic and real data. Each chapter is divided into sections and is concluded with the summary of the main results. In turn, each section starts with its main contributions and concludes with the discussion of the results. All contributions and perspectives are summarized and discussed in Conclusions and Perspectives. Finally, lists of symbols used throughout the thesis are given in Appendix A, and technical proofs of some theorems and lemmas used in the main body of this work are given in Appendix B.

Publications

Journal publications:

1. L. We introduce the following norms with respect to the space variable x. For a function ρ(x, t)

∈ [0, L] × R + the L 1 , L 2 and L ∞ norms are dened ∀t ∈ R + as ρ(•, t) 1 := L 0 |ρ(x, t)|dx, (1.1) ρ(•, t) 2 := L 0 ρ 2 (x, t)dx, (1.2) ρ(•, t) ∞ := sup x∈[0,L] |ρ(x, t)|, (1.3)
where sup (inf ) indicates the essential supremum (inmum).

The deviation of the state from the desired vehicle density trajectory is dened ∀

(x, t) ∈ [0, L] × R + as ρ(x, t) = ρ(x, t) -ρ d (x, t).
(

The cumulative deviation from the desired vehicle density trajectory along a road of length L (integral error term, or error in the number of vehicles) is dened ∀t ∈ R + as e(t) = L 0 (ρ(s, t) -ρ d (s, t)) ds.

(1.5)

Now let us also explain the notations used in Chapters 3 and 4 devoted to 2D trac problems:

We introduce the L (1.8)

The deviation from a 2D space-varying desired equilibrium (constant in time) is dened ∀(x, y, t) ∈ [x min , x max ] × [y min , y max ] × R + as ρ(x, y, t) = ρ(x, y, t) -ρ d (x, y).

(1.9)

The deviation from a 2D time-and space-varying desired trajectory is dened ∀(x, y, t) ∈ [x min , x max ] × [y min , y max ] × R + as ρ(x, y, t) = ρ(x, y, t) -ρ d (x, y, t).

(1.10)

Chapter 2

Trac Control on Roads

This chapter is devoted to trac control problems on single roads of nite length. We consider trac within the macroscopic modeling approach that incorporates the kinematic wave theory for trac that applies principles from uid dynamics to predict trac. Using intrisic properties of the model, we design a boundary control law to track a desired vehicle trajectory. Section 2.1 contains preliminaries that include explanation of LWR model and its mathematical properties (weak solutions, boundary conditions, etc.), the basic numerical scheme to approximate LWR, as well as an equivalent approach to describe trac in terms of Hamilton-Jacobi equation. In Section 2.2, we rst consider trac being only in the congested regime with some in-domain disturbance, i.e., the state is driven by an inhomogeneous linear PDE system. The desired trajectory is also restricted to congested trac regime for simplicity. In Section 2.3, we consider trac state and desired trajectory both not being restricted to any particular trac regime, i.e., boundary control design is performed for a nonlinear system, which is mathematically quite challenging to handle due to shocks that arise in a full LWR system.

Preliminaries

The kinematic wave theory for trac was formulated in the fties by Lighthill, Whitham and Richards, and it is now known as the LWR model [START_REF] Lighthill | On kinematic waves, II: A theory of trac ow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF]. Its main assumption is the existence of a concave relation between the vehicle density and its ow. This model remains the most popular macroscopic model for trac due to its simplicity, while it is still able to capture the most essential trac phenomena. In this section, we present the LWR framework in more details, as well as discuss its solution, boundary conditions, the numerical scheme and its integral formulation.

Lighthill-Whitham-Richards model

The LWR model is a scalar conservation law, where the conserved quantity is the number of vehicles in some area. This conservation law model corresponds to a rst-order hyperbolic PDE, which also implies that the information propagates at a nite speed. This model predicts 13 the spatio-temporal evolution of trac ∀(x, t) ∈ R × R + as follows

   ∂ρ(x, t) ∂t + ∂Φ(ρ(x, t)) ∂x = 0, ρ(x, 0) = ρ 0 (x), (2.1)
where ρ : R × R + → R + is the vehicle density with ρ 0 (x) being the initial data, and Φ(ρ) : [0, ρ max ] → R + is the ux function that relates vehicle ow φ(x, t) with vehicle density ρ(x, t).

This relation is an empirically established law [START_REF] Greenshields | A study of trac capacity[END_REF] known as fundamental diagram (FD).

Mathematically speaking, the ux function Φ(ρ(x, t)) is a Lipschitz continuous and concave function that admits a unique maximum φ max (capacity) attained at ρ c (critical density), while its minimum value is achieved in two cases: either if there are no vehicles, i.e., Φ(0) = 0, or if trac is fully congested, i.e., Φ(ρ max ) = 0, where ρ max is the trac jam density.

Fundamental diagram

The most simple ow-density relation corresponds to the triangular (bilinear) fundamental diagram proposed in [START_REF] Daganzo | The cell transmission model: A dynamic representation of highway trac consistent with the hydrodynamic theory[END_REF], see Figure 2.1a):

Φ(ρ) = vρ, ρ ∈ [0, ρ c ], -ω(ρ -ρ max ), ρ ∈ (ρ c , ρ max ], (2.2)
where v and ω are kinematic wave speeds in the free-ow regime (wave moving forwards) and in the congested regime (wave moving backwards), respectively. Notice that kinematic wave speeds are not related to velocities of individual vehicles that are determined on average as the ow divided by the density. For instance, in the trac jam (when ρ = ρ max ) velocities of vehicles are zero, while the kinematic wave propagates backwards with -ω.

Thus, the triangular FD has only two slopes, since kinematic waves can take only two values (v and ω). The critical density ρ c and the capacity φ max are dened as

ρ c = ω v + ω ρ max , φ max = vρ c . (2.3)
In this thesis, we will also use the Greenshields FD that was the rst ow-density relation proposed in 1935 in [START_REF] Greenshields | A study of trac capacity[END_REF], see Figure 2.1b). This relation is given by the following smooth concave function

Φ(ρ) = v max 1 - ρ ρ max ρ, (2.4)
with the critical density ρ c and capacity φ max given as

ρ c = ρ max 2 , φ max = v max ρ max 4 , (2.5)
and v max = Φ (0) is the maximal kinematic wave (free-ow) speed determined by the speed of a vehicle moving along a freeway with ρ = 0.

Throughout this thesis, we will use ρ c = 1/3ρ max for all cases when a triangular FD is assumed, and ρ c = 1/2ρ max for the case of Greenshields FD. Although in this work only triangular and Greenshields FDs are used to model the owdensity relation, there exist also many other types of FDs, see [START_REF] Michael | A Generic Characterization of Equilibrium Speed-Flow Curves[END_REF] for a review. Del Castillo [START_REF] Del Castillo | Three new models for the owdensity relationship: derivation and testing for freeway and urban data[END_REF] formulated properties that must hold for realistic fundamental diagrams:

ρ Φ(ρ) a) ρ c φ max 0 ρ max v -ω ρ Φ(ρ) a) v max ρ c φ max 0 ρ max
1. The velocity range should be v ∈ [0, v max ].

2. The vehicle density should be ρ ∈ [0, ρ max ].

3. Cars stop moving when the trac jam density is reached: v(ρ max ) = 0.

4. There is no trac ow if there are no cars (zero density), or if cars are stuck in a trac jam: Φ(0) = Φ(ρ max ) = 0.

5. Maximum velocity and congestion wave speed are the slopes of the fundamental relation at ρ = 0 and ρ = ρ max , respectively.

6. Flux is a concave function of density.

Unique solution

Let us discuss the Cauchy problem (2.1). It has a structure of a rst-order hyperbolic PDE that can be solved using the method of characteristics that yields lines along which the state remains constant (see [START_REF] Evans | Partial dierential equations[END_REF] and Appendix B.1 for details). Thus, the characteristics of (2.1) are straight lines:

x -

x 0 = Φ (ρ 0 (x 0 ))t, x 0 ∈ R + , (2.6)
where x 0 is a point in space, from which the characteristic line originates. The density is constant along each of these lines, that is for all (x, t) satisfying (2.6) we obtain ρ(x, t) = ρ 0 (x 0 ).

It can however happen that these lines intersect proposing two dierent values of ρ at the intersection point x s , i.e., a discontinuity arises in the solution ρ. Hence, in general no classical solution exists for (2.1), and one needs to consider its solution in the weak sense.

A function ρ : R × R + → R + is a weak solution of the Cauchy problem (2.1) if for any test function ψ (which is a C 1 function with compact support in R 2 ) the following equation holds:

+∞ 0 +∞ -∞ (ρψ t + Φ(ρ)ψ x ) dx dt + +∞ -∞ ρ 0 (x)ψ(x, 0) dx = 0. (2.7)
and the map t → ρ(•, t) is continuous from R + into L 1 loc (set of locally integrable functions).

Given that ρ is smooth around the point of discontinuity, we can integrate (2.7) to obtain the speed of discontinuity ẋs known as the Rankine-Hugoniot relation [126]:

ẋs = Φ(ρ r ) -Φ(ρ l ) ρ r -ρ l , (2.8)
where ρ r and ρ l are values of the right and of the left limit of ρ at the point of discontinuity.

It is important to note that the Cauchy problem (2.1) can have an innite number of weak solutions. Hence, the weak solution (2.7) must be completed by a uniqueness condition:

Φ (ρ r ) ≤ ẋs ≤ Φ (ρ l ), (2.9)
where Φ (ρ r ) and Φ (ρ l ) are the characteristic speeds to the right and to the left of the discontinuity, correspondingly. In the theory of hyperbolic conservation laws, equation (2.9) is known as the Lax admissibility condition [START_REF] Lax | Nonlinear hyperbolic equations[END_REF]. It selects the unique solution out of a set of weak solutions. The Lax condition has a simple geometrical interpretation. Namely, the unique solution is the particular weak solution for which the characteristics run into the shock: all characteristics must end at the discontinuity as illustrated in Figure 2.2.

Ansorge [START_REF] Ansorge | What does the entropy condition mean in trac ow theory?[END_REF] was the rst to consider this uniqueness condition (2.9) in the context of trac.

He interpreted it as driver's ride impulse, i.e., one starts driving when a trac light switches from red to green. Thus, a weak solution of (2.1) satisfying (2.9) is the physically relevant one. It is also known as the entropy solution.

In particular, considering piecewise-constant initial state ρ 0 (x) = ρ l for x < x s and ρ 0 (x) = ρ r for x > x s (known also as a Riemann problem), entropy solutions can be of the two following types:

1. If ρ l < ρ r , shock arises. The entropy solution remains piecewise-constant, and shock wave propagates through space with velocity determined by (2.8): ρ(x, t) = ρ l for x < x s (t) and ρ(x, t) = ρ r for x > x s (t).

2. If ρ l > ρ r , the entropy solution becomes continuous and propagates in a form of a rarefaction wave. For any straight line starting from (x s , 0) the solution is constant and is determined only by (x -x s )/t:

ρ(x, t) =      ρ l , if (x -x s )/t < Φ (ρ l ), Φ -1 ((x -x s )/t), if (x -x s )/t ∈ [Φ (ρ l ), Φ (ρ r )], ρ r , if (x -x s )/t > Φ (ρ r ). x t 0 ρ l ρ r

Boundary conditions

In its original formulation, the LWR model (2.1) describes trac for innitely long highways without any on-ramps or o-ramps, which is a serious physical limitation. Thus, one needs to include the boundary conditions for (2.1) in order to be able to include ramps and in general to consider roads of nite length (bounded domains).

Let us consider a road of length L. The conservation principle states that the evolution of each aggregated conserved quantity in some domain [0, L] depends only on the ows at its boundaries and exogenous ows. Thus, for a complete model describing trac evolution along some road, we need to specify boundary conditions ρ in and ρ out that are all assumed to be functions of bounded variation (as well as the initial condition ρ 0 ). Then, the initial-boundary value problem (IBVP) reads ∀(x, t)

∈ [0, L] × R +          ∂ρ(x, t) ∂t + ∂Φ(ρ(x, t)) ∂x = 0, ρ(x, 0) = ρ 0 (x), ρ(0, t) = ρ in (t), ρ(L, t) = ρ out (t).
(2.10)

The main feature of the boundary conditions in conservation laws is that they can not be applied strongly for all time, see [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF]. Thus, the boundary conditions should be viewed only as proposed signals.

In general, boundary conditions may only be prescribed for the boundary where the characteristics are incoming. Hence, if trac is in the free-ow regime at road's entry for some t, then the kinematic wave propagates forward and we can write ρ(0, t) = ρ in (t), or if trac is congested at the road exit, then ρ(L, t) = ρ out (t). However, this behaviour is not guaranteed for general nonlinear conservation law systems such as (2.10). In order to guarantee that the weak solution ρ(x, t) is the entropy one ∀(x, t) ∈ [0, L] × R + , one needs to consider weak boundary conditions. See Section 2.1.3 for the physical sense of entropy solution.

In [START_REF] Issam | Weak formulation of boundary conditions for scalar conservation laws: An application to highway trac modelling[END_REF] the weak boundary conditions were considered for the case of concave ux function.

Thus, for a weak formulation of the boundary conditions of system (2.10), one of the following conditions must hold for the upstream boundary:

ρ(0, t) = ρ in (t), or Φ (ρ(0, t)) ≤ 0 and Φ (ρ in (t)) ≤ 0, or Φ (ρ(0, t)) ≤ 0 and Φ (ρ in (t)) ≥ 0 and Φ(ρ(0, t)) ≤ Φ(ρ in (t)), (2.11)
and similarly for the downstream boundary:

ρ(L, t) = ρ out (t), or Φ (ρ(L, t)) ≥ 0 and Φ (ρ out (t)) ≥ 0, or Φ (ρ(L, t)) ≥ 0 and Φ (ρ out (t)) ≤ 0 and Φ(ρ(L, t)) ≥ Φ(ρ out (t)).
(2.12)

The demand-supply concept

In some works [START_REF] Lebacque | The godunov scheme and what it means for rst order trac ow models[END_REF][START_REF] Lebacque | First-order macroscopic trac ow models: Intersection modeling, network modeling[END_REF], the weak boundary conditions are modeled using the demand-supply concept. According to this concept, in case of concave ow-density function Φ(ρ) (e.g., triangular or Greenshields FD), the proposed trac ow at the upstream boundary is given by the demand function

D(ρ in ) = Φ(ρ in ), if 0 ≤ ρ in ≤ ρ c , φ max , if ρ c < ρ in ≤ ρ max , (2.13)
and the proposed ow at the downstream boundary is given by the supply function

S(ρ out ) = φ max , if 0 ≤ ρ out ≤ ρ c , Φ(ρ out ), if ρ c < ρ out ≤ ρ max . (2.14)
The boundary Riemann problem for the upstream boundary ow is then given ∀t ∈ R + by φ in (t) = min {D(ρ in (t)), S(ρ(0, t))} , (2.15) whereas the downstream boundary ow is dened as

φ out (t) = min {D(ρ(L, t)), S(ρ out (t))} . (2.16)
Notice that (2.15) and (2.16) are consistent with (2.11) and (2.12), i.e., these are weak boundary conditions in terms of ows. Thus, the amount of ow that can enter the domain φ in is constrained by the supply at road's entry, while the trac ow leaving the domain φ out is constrained by the demand at road's exit. This means that incoming cars can be blocked by congested trac at the entry of the road, as well as the outow control may not be imposed if there are only a few cars at the exit. In case of triangular fundamental diagram, the demand an supply functions are given by

D(ρ(t)) = min {vρ(t), φ max } , S(ρ(t)) = min {ω(ρ max -ρ(t)), φ max } .
(2.17)

Figure 2.3 illustrates the demand and the supply functions for the case of triangular FD.

The demand-supply concept is equivalent to the weak boundary conditions formulation introduced above in terms of densities (2.11)-(2.12), though being much simpler. This concept has important practical implications when dealing with numerical schemes to simulate the LWR model, as we are going to show later in this thesis.

The Godunov scheme

Now let us describe the most basic numerical method for approximating conservation laws such as the LWR model. The Godunov scheme proposed in [START_REF] Godunov | A dierence method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF] is a rst-order numerical method based on solutions to Riemann problems. The global idea of the Godunov scheme is to approximate the initial datum by a piecewise linear function, then to compute solutions to Riemann problems and then to piece these solutions together.

We start by dening a numerical grid in [0, L] × R + by setting n to be number of cells, ∆x = L/n to be the space cell size, ∆t to be the time cell size, (i∆x, k∆t) for i ∈ {1, . . . , n} and k ∈ Z + to be the grid points.

The mesh sizes ∆x and ∆t are chosen such that they satisfy the Courant-Friedrichs-Lewy (CFL) condition [START_REF] Courant | On the Partial Dierence Equations of Mathematical Physics[END_REF]:

∆t max ρ |Φ (ρ)| ≤ ∆x 2 ,
where max ρ |Φ (ρ)| corresponds to the maximal kinematic wave speed in the free-ow regime, e.g., v (v max ) in case of triangular (Greenshields) FD. This condition needs to be satised, since it provides the non-interaction of waves generated by dierent Riemann problems.

The discrete density is then ρ i (k), and according to the Godunov scheme, we update it as follows ∀(i, k) ∈ {1, . . . , n} × Z + :

                   ρ 1 (k + 1) = ρ 1 (k) + ∆t ∆x (ϕ in (k) -ϕ 2 (k)) , ρ i (k + 1) = ρ i (k) + ∆t ∆x (ϕ i (k) -ϕ i+1 (k)) , ρ n (k + 1) = ρ n (k) + ∆t ∆x (ϕ n (k) -ϕ out (k)) , (2.18)
where ϕ i (k) is the Godunov numerical ux between cells dened as

ϕ i (k) = min {D(ρ i-1 (k)), S(ρ i (k))} , (2.19)
with D(ρ i-1 (k)) and S(ρ i (k)) being the discretized demand and supply functions that can be taken as in (2.17). Thus, the amount of ow transmitted from the left cell i -1 to the right cell i corresponds to the minimum between the demand of i -1 and supply of i, see Figure 2.4 for the illustration of the concept.

i -1 Notice that the discrete version of LWR model given by (2.18) together with demand and supply functions corresponding to triangular FD (2.17) is known as the cell transmission model (CTM). The CTM is by far the most widely used trac model due to its simplicity and straightforward extension to networks.

D(ρ i-1 ) i S(ρ i )
The boundary ows ϕ in (k) and ϕ out (k) from (2.18) are respectively set by specifying the density at cells with indices i = 0 and i = n + 1. These are called ghost cells, since they do not belong to the domain but are used to denote state at the boundaries:

ϕ in (k) = min {D(ρ 0 (k)), S(ρ 1 (k))} , ϕ out (k) = min {D(ρ n (k)), S(ρ n+1 (k))} . (2.20)
In the uncontrolled case, we set ρ 0 (k) = ρ 1 (k) and ρ n+1 (k) = ρ n (k), which gives ϕ in (k) = ϕ 1 (k) and ϕ out (k) = ϕ n (k), thus the system evolves freely. In the controlled case, we set ρ 0 (k) = u in (k) for the free-ow regime and ρ n+1 (k) = u out (k) for the congested regime, where u in and u out represent boundary control laws.

Hamilton-Jacobi formulation

As we have seen previously in Section 2.1.3, the kinematic wave theory for trac incorporated by the LWR model can have complications that arise when characteristic lines with dierent ρ intersect at some (x, t). This led to the necessity to introduce shocks in order to guarantee the conservation of the number of vehicles across the pass, and handling shocks can sometimes become a tedious task. To simplify the issue of handling nonlinearities within the LWR formulation, an alternative formulation of highway trac ow on a macroscopic level was proposed by Newell [START_REF] Newell | A simplied theory of kinematic waves in highway trac, part I: General theory[END_REF][START_REF] Newell | A simplied theory of kinematic waves in highway trac, part II: Queueing at freeway bottlenecks[END_REF][START_REF] Newell | A simplied theory of kinematic waves in highway trac, part III: Multidestination ows[END_REF]. He proposed to describe the trac state in terms of Moskowitz function M (x, t) (or shortly, MF). The name of this function comes after Karl Moskowitz, an engineer who rst used it to investigate trac in [START_REF] Moskowitz | Discussion of freeway level of service as inuenced by volume and capacity characteristics by dr drew and cj keese[END_REF], although it was rst mentioned only some decades later in [START_REF] Newell | A simplied theory of kinematic waves in highway trac, part I: General theory[END_REF].

Physically, MF corresponds to the cumulative number of vehicles. Its value is obtained by numbering vehicles at highway's entry and following the isolines of the functions representing vehicle numbers at all times and locations. It is assumed that vehicles can not pass each other, thus the ordering of the vehicles is preserved everywhere. Recall that trac ow φ(x, t) is the rate at which vehicles pass some point x ∈ R + , and the trac density ρ(x, t) is dened as the number of vehicles per unit length of road. Then, the cumulative number of vehicles can be easily obtained by integrating ow in space or by integrating density in time. This relation is formalized as follows:

ρ(x, t) = - ∂M (x, t) ∂x , φ(x, t) = ∂M (x, t) ∂t .
(2.21)

Recall that the key assumption of the kinematic wave theory for trac is the existence of a concave relation between ow and density. Let us now rewrite this fundamental law Φ (ρ(x, t)) = φ(x, t) using (2.21) as

∂M (x, t) ∂t -Φ - ∂M (x, t) ∂x = 0, (2.22)
which is a Hamilton-Jacobi PDE with a Lipschitz continuous function M (x, t) : [0, L] × R + → R being its state. The corresponding boundary conditions for (2.22) will be added in Section 2.1.8, where we will again consider trac evolution along a road of length L. In terms of viability theory, M (x, t) can also be called the congestion function (see [START_REF] Aubin | Viability Theory: New Directions[END_REF]), since

(2.22) can be considered as an optimal control problem minimizing a congestion functional M (x, t). In particular, vehicles tend to minimize the trac congestion by adapting their individual (microscopic) velocities to the kinematic wave velocity (a macroscopic quantity).

∂ρ(x, t) ∂t + ∂Φ (ρ(x, t)) ∂x = 0 ⇔ ∂ 2 M (x, t) ∂x∂t - ∂ 2 M (x, t) ∂t∂x = 0.
The rigorous relation was shown in [START_REF] Joseph | Explicit formula for the solution of convex conservation laws with boundary condition[END_REF].

Note that (2.22) depends only on the derivatives of M (x, t). Therefore, for any solution M (x, t) adding any constant M 0 gives also a solution M (x, t) + M 0 . This is obvious, since we can start numeration of cars from any particular number. The existence of M (x, t) itself guarantees the conservation of number of vehicles. Being an integral form of the LWR PDE, the solution of HamiltonJacobi PDE is a continuous function that has no shocks. A shock in the vehicle density function corresponds to a discontinuity in the rst derivative of M (x, t). Then, the conservation equations are still valid if M (x, t) is continuous across the shock path, which also must be stable. This requirement corresponds to the entropy condition in the LWR framework, where the characteristics must run into the shock (Lax admissibility condition). Thus, if a kinematic wave problem such as (2.10) is a well-posed problem, then it has a unique solution with stable shocks.

Let us now express the Moskowitz function M (x, t) through inows φ in (t) and outows φ out (t) of the system. This can be simply done by using the denitions from (2.21). Namely, we can dene a conservative eld (-ρ(x, t), φ(x, t)), which is a gradient of M (x, t) ∀(x, t) ∈ [0, L] × R + (consider again a road of length L). By the gradient theorem, it follows that the value of the line integral of this eld does not depend on a particular chosen path, and equals to only the dierence between the values of the Moskowitz function between ending and starting points of the path in space-time. Since M (x, t) is an integral function that is dened up to a constant, we are free to assign a reference value to this function at some particular point in space-time. Let us choose a starting point (L, 0) corresponding to the end of the road at inital time. Then, we also set M (L, 0) = 0, since congestion functions are decreasing functions of position and increasing functions of time, see Chapter 14 of [START_REF] Aubin | Viability Theory: New Directions[END_REF]. Thus, taking the ending point of the path as (x, t), one possible integration path is:

M (x, t) = t 0 φ out (τ ) dτ + L x ρ(s, t) ds, (2.23)
or if the starting point is (0, 0), then the integration path is

M (x, t) = L 0 ρ 0 (s) ds + t 0 φ in (τ )dτ - x 0 ρ(s, t) ds.
(2.24)

Variational theory

Equation (2.22) is a scalar Hamilton-Jacobi PDE that can be solved semi-analytically using initial condition function M Ini (x), upstream M Up (t) and downstream M Down (t) boundary condition functions. Note that the boundary conditions should be consistent with the weak boundary conditions formulation (2.15)-(2.16). Thus, let us dene the following IBVP for the H-J PDE with weak boundary conditions ∀(x, t) ∈ [0, L] × R + :

               ∂M (x, t) ∂t -Φ - ∂M (x, t) ∂x = 0, M (x, 0) = M Ini (x), M (0, t) = M Up (t), M (L, t) = M Down (t).
(2.25)

For convenience, let us introduce the value condition function c(x, t) : Dom(c) → R + , where Dom(c) = ({0, L} × R + ) ∪ ((0, L) × {0}). It aggregates the initial and boundary conditions of (2.25) (as in [START_REF] Claudel | LaxHopf Based Incorporation of Internal Boundary Conditions Into HamiltonJacobi Equation. Part I: Theory[END_REF]):

c(x, t) =      M Ini (x), t = 0, M Up (t), x = 0, M Down (t), x = L. (2.26)
Let us determine this value condition function (2.26), which implies the calculation of M Up (t), M Down (t) and M Ini (x). The upstream boundary condition should be expressed through inow φ in . Thus, we obtain M Up (t) by considering (2.24) for x = 0, which results into M Up (t) = c(0, t) = (2.27)

Then, the downstream boundary condition M Down (t) can be expressed by considering (2.23) for x = L:

M Down (t) = c(L, t) = t 0 φ out (τ ) dτ, ∀t ∈ R + .
(2.28)

Finally, we can obtain the initial condition by considering either (2.24) or (2.23) for t = 0, which gives us

M Ini (x) = c(x, 0) = L x ρ 0 (s) ds, ∀x ∈ [0, L].
(2.29)

The solution of a well-posed IBVP (2.25) is a set of least-cost paths in space-time, as it was

shown in [START_REF] Daganzo | A variational formulation of kinematic waves: basic theory and complex boundary conditions[END_REF]. In order to obtain its analytical solution, one should treat (2.25) as a capacityconstrained optimization problem, which should be interpreted as follows. Trac ow at any point is upper bounded by the road capacity φ max at this point, which, in general, depends on the number of lanes and speed limits. A similar capacity constraint holds, if the road is viewed from a rigid reference frame (observer) moving along this road at a speed v ∈ [-ω, v] next to a trac stream that is characterized by density ρ and ow φ. Then, the maximum rate at which the observer attached to the frame can be passed by the trac stream is φ -ρv (the relative capacity).

The fundamental diagram from the observer's viewpoint becomes Φ(ρ(x, t)) -ρv , and its

relative capacity is ∀v ∈ [-ω, v], L(v ) = sup ρ∈[0,ρmax] Φ(ρ) -ρv , (2.30)
where v and -ω are related to kinematic wave speeds for zero density and for the trac jam density, respectively, i.e., v = Φ (0), -ω = Φ (ρ max ).

Note that L(v ) corresponds to the Legendre-Fenchel transform of the ux function Φ(ρ). Thus, L(v ) ≥ 0 is a convex and strictly decreasing function ∀v ∈ [-ω, v], see Figure 2.5. It achieves minimum if the observer tends to adapt his/her velocity to the maximal kinematic wave speed v, whereas its maximal value is achieved for v = -ω. Thus, L(v ) corresponds to the cost per unit time [START_REF] Daganzo | On the variational theory of trac ow: Well-posedness, duality and applications[END_REF], and the observer moves such that this cost is minimized. The observer traveling at time t along a valid space-time path with starting time t s ∈ R + can not perceive a change in its associated cumulative vehicle number greater than

cost v L(v ) φ max -ω v
∆M (x, t) = t ts L(v )dt = (t -t s ) L(v ).
In general, its associated cumulative vehicle number M (x, t) can not be larger than the value at its origin boundary c(x -(t -t s )v , t s ) (starting cost) plus the maximal possible change in its vehicle number ∆M = (t -t s ) L(v ) caused by other vehicles that have passed the observer:

M (x, t) ≤ inf c(x -(t -t s )v , t s ) + (t -t s )L(v ) s.t. (v , t -t s ) ∈ [-ω, v] × R + and (x -(t -t s )v , t s ) ∈ Dom(c).
(2.31)

According to the variational theory presented in [START_REF] Bellman | Dynamic Programming and Partial Dierential Equations[END_REF], this capacity constraint (2.31) is binding, i.e., the actual value of M (x, t) is the largest possible allowed by this constraint. We introduce a time interval T = t -t s to make the notations shorter. Thus, the unique solution to (2.25) is found as

M (x, t) = inf c(x -T v , t -T ) + T L(v ) s.t. (v , T ) ∈ [-ω, v] × R + and (x -T v , t s ) ∈ Dom(c).
(2.32)

This expression is known as the Lax-Hopf formula, which provides a semi-analytical unique solution to the Hamilton-Jacobi system (2.25). Thus, the unique solution M (x, t) is the inmum of the innite number of functions of the value condition (see also [START_REF] Aubin | Dirichlet problems for some Hamilton-Jacobi equations with inequality constraints[END_REF]).

With a slight abuse of notation, we introduce two-argument functions M Up (x, t), M Down (x, t) and M Ini (x, t) as solutions to the Lax-Hopf formula (2.32) for corresponding domains of the value condition function c, which are M Up (t) (2.27), M Down (t) (2.28) and M Ini (x) (2.29). Thus, M Up (x, t) comes from the upstream boundary with a given initial cost M Up (t), then M Down (x, t) comes from the downstream boundary with a known M Down (t), and M Ini (x, t) comes from the initial condition M Ini (x). For example, M Up (x, t) is obtained by

M Up (x, t) = inf c(x -T v , t -T ) + T L(v ) s.t. (v , T ) ∈ [-ω, v] × R + and (x -T v , t -T ) ∈ {0} × R + , (2.33)
and the same formula yields M Down (x, t) for (x -T v , t -T ) ∈ {L} × R + and M Ini (x, t) for

(x -T v , t -T ) ∈ [0, L] × {0}.
This enables us to restate the solution to the Hamilton-Jacobi problem (2.25) as a minimum of three possible solution candidates M Up (x, t), M Down (x, t) and M Ini (x, t) ∀(x, t) ∈ [0, L] × R + . By properties of inmum, the original Lax-Hopf formula (2.32) can be rewritten as M (x, t) = min {M Ini (x, t), M Up (x, t), M Down (x, t)} .

(2. [START_REF] Courant | On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Dierences[END_REF] In some special cases, e.g., for a triangular FD dened in (2.2), we can calculate the solution to the H-J PDE in explicit form as it is shown in Appendix B.2.

Robust tracking boundary control design

In this section, we address two seldom studied issues: tracking time-and space-varying desired proles (rather than stabilizing to an equilibrium), and dealing with uncertainties due to a possible model mismatch. In particular, we consider an optimal boundary control problem to track a desired vehicle density on a single road with a state being subject to unknown space-dependent disturbances. For instance, imagine trac evolving along a road, for which we want to achieve some desired density prole by controlling the vehicle ow at the boundary of this road. However, even if we know the ow-density relation for this road, the tracking control problem might be challenging due to unknown number of vehicles originating from minor roads. Thus, we solve the problem of controlling the vehicle density whose value we can not predict exactly. For this, we include the disturbance term to model the unknown change in the number of vehicles coming from minor roads.

Introductions

Most works devoted to trac control have addressed the homogeneous (ideal) case, where the discrepancies between model and system are ignored (like in classical LWR). Some of the studies related to the disturbance rejection problem were devoted to disturbance attenuation on a boundary by action from another boundary [START_REF] Morten | Disturbance rejection in 2 × 2 linear hyperbolic systems[END_REF][START_REF] Tang | Sliding mode control to the stabilization of a linear 2×2 hyperbolic system with boundary input disturbance[END_REF][START_REF] Annsen | Disturbance rejection in the interior domain of linear 2 × 2 hyperbolic systems[END_REF]. For example, in [START_REF] Tang | Sliding mode control to the stabilization of a linear 2×2 hyperbolic system with boundary input disturbance[END_REF] the sliding mode control is used to stabilize a hyperbolic system with boundary input disturbance. [START_REF] Morten | Disturbance rejection in 2 × 2 linear hyperbolic systems[END_REF] proposed a controller able to reject disturbance at the boundary where this disturbance acts.

Later on, [START_REF] Annsen | Disturbance rejection in the interior domain of linear 2 × 2 hyperbolic systems[END_REF] proposed a controller for disturbance rejection at an arbitrary point within the domain. A model reference adaptive control problem has been solved for hyperbolic PDEs in [START_REF] Annsen | Model Reference Adaptive Control of 2 × 2 Coupled Linear Hyperbolic PDEs[END_REF]. Therein, the authors considered harmonic disturbances with known frequencies and designed a lter-based control law. In a related work on robust control design for systems of conservation laws [START_REF] Prieur | Robust boundary control of systems of conservation laws[END_REF], the problem of stabilization to a steady-state prole was considered.

Boundary control design was addressed previously in [START_REF] Blandin | Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws[END_REF] for the problem of stabilizing the vehicle density to a constant equilibrium.

The main contribution of this section is the optimal boundary controller, which leads to attenuation of a general in-domain space-dependent disturbance. This is the rst result devoted to a robust controller tracking a space-and time-dependent desired trac density.

Space-and time-dependency of trac density is an important aspect to handle, since in realistic trac situations it is more likely to obtain non-stationary proles due to rapidly changing trac conditions.

The control design is based on two components. These are the feedforward control component used to track the trajectory, whereas the feedback control component is used to minimize spatial L 2 and L ∞ error norms for asymptotic time. We show that the optimal feedback term takes dierent forms according to the norm to be minimized. The feedback law is given in its implicit but computationally feasible form, thus we can apply it without any explicit knowledge about the disturbance. In addition, we also compute the L 2 and L ∞ gains resulting from the application of the respective control laws. The special thing about these gains is that they depend only on system physical parameters such as length of the road stretch and parameters of the fundamental diagram.

Preliminaries

The goal of this section is to design a robust boundary control law for a vehicle density governed by equation (2.1) with disturbance such that the state tracks a desired time-and spacedependent prole. Let us now describe the problem in more concrete terms by performing the following steps: present the state equation with disturbance, perform motion planning, analyze error dynamics and solve it explicitly by using the method of characteristics. We will also present the general structure of the boundary controller to be designed.

System with disturbance

We assume that the vehicle density is quite high, i.e., we consider a road being in the congested trac regime, which is a common problem arising during rush hours. Here we consider trac on a bounded domain (road of length L), which implies that boundary conditions must also be included as in IBVP (2.10). In case of trac that is restricted to the congested regime, the system (2.10) becomes linear (like a transport equation), since it is considered only for ρ(x, t) ∈ (ρ c , ρ max ] ∀(x, t) ∈ [0, L] × R + that lets us write ∂ x Φ(ρ(x, t)) = -ω∂ x ρ(x, t) (we assume a triangular FD). For this case, let us also introduce an unknown disturbance term δ(x) : [0, L] → R that is assumed to be bounded. Then, the inhomogeneous initial-boundary value problem reads:

Σ =          ∂ρ(x, t) ∂t -ω ∂ρ(x, t) ∂x = δ(x), ρ(x, 0) = ρ 0 (x), ρ(L, t) = u(t), (2.35)
where u(t) is a controller to be designed. The control action is applied at the downstream boundary, since in the congested regime the kinematic wave is moving backwards. In the free-ow regime the kinematic wave propagates forwards, and then we would control the state by actuating the upstream boundary x = 0. Note that the linear system Σ given by (2.35) allows us to use the fact that characteristics always propagate only in one direction. Thus, the downstream boundary conditions can always be enforced, which enables us to consider them in a strong sense.

Physically, the disturbance term δ(x) in (2.35) corresponds to the additional unknown vehicle density per time unit, an example is illustrated in Figure 2.6. Also notice that the actual control of trac at road boundaries can be done only in terms of vehicle ow, which can be changed by, e.g., appropriately timing the trac light signals. However, in this section, the boundary conditions are prescribed in terms vehicle density, which is the state of system Σ, since it makes the analysis of the eect of boundary values on the system solution pretty straightforward. Then, if we want to transfer the control result that is going to be designed for system Σ into real life, we should use the ow-density function that allows us to obtain the corresponding ow value to be achieved at the boundary, which is then translated into, e.g., time intervals during which the trac lights are green/red.

Motion planning

Let us now dene the desired trajectory for the vehicle density ρ d (x, t) that we want to achieve via the boundary control. An admissible desired trajectory ρ d (x, t) must be a solution of the following system Σ d :

Σ d =          ∂ρ d (x, t) ∂t -ω ∂ρ d (x, t) ∂x = 0, ρ d (x, 0) = ρ d 0 (x), ρ d (L, t) = ρ dout (t), (2.36)
where ρ dout is the desired state at the downstream boundary and ρ d 0 (x) is the initial state in the desired system. Notice that system Σ d is an IBVP that looks exactly like (2.35) but with δ(x) = 0, i.e., Σ d is a homogeneous system.

To guarantee that the system Σ d is well-posed, its state ρ d (x, t) must always remain in the congested trac regime along the road, i.e., ρ d (x, t)

: [0, L] × R + → (ρ c , ρ max ].

Error dynamics

Let us now determine the dynamic system for the error density ρ(x, t) dened as in (1.4). Systems Σ and Σ d given by (2.35) and (2.36) are linear. Thus, we can simply subtract Σ d from Σ and obtain the following IBVP for the error ρ(x, t):

Σ err =          ∂ ρ(x, t) ∂t -ω ∂ ρ(x, t) ∂x = δ(x), ρ(x, 0) = ρ0 (x), ρ(L, t) = u(t) -ρ dout (t).
(2.37)

Σ d FF Σ FB ρ d u f f + u ρ u f b + Figure 2.7: Control scheme.

Control design

We aim to design a boundary control law that can be schematically represented as in Figure 2.7. Thus, the input is a sum of feedforward u f f (t) and feedback u f b (ρ, t) terms:

u(ρ, t) = u f f (t) + u f b (ρ, t). (2.38)
For simplicity of notations, the arguments of the controls are omitted, and in the following they will be included only if not clear from the context.

Remark 2.1

Note that the feedforward term is needed to track the desired trajectory ρ d (x, t), while the feedback term is needed for disturbance attenuation.

Solution of Σ err

Let us now consider (2.37). To analyze its solution, we apply the method of characteristics presented in [START_REF] Evans | Partial dierential equations[END_REF]. The details are given in Appendix B.1, where we nd that the error term ρ(x, t) evolves along the characteristic lines as

ρ(x, t) =        ρ (x + ωt, 0) + ∆(x) -∆(x + ωt), ∀t ∈ 0, L -x ω , ρ L, t -L-x ω + ∆(x), ∀t ∈ L -x ω , +∞ , (2.39)
where ∆(x) is the integral of the disturbance (disturbance accumulated along the road stretch) dened as

∆(x) = 1 ω L x δ(s) ds.
(2.40)

Remark 2.2

Note that t ctr = L ω is the minimum time for control action to propagate to the end of the road from x = L to x = 0, that is why in the following we consider solutions only for t ≥ L ω , i.e., the second expression in (2.39).

Let us rewrite (2.39) using the expression for the downstream boundary in the error system (2.37), which reads ρ L, t -L-x ω = u t -L-x ω -ρ dout t -L-x ω . Thus, we get the following solution for the error system:

ρ(x, t) = u t - L -x ω -ρ dout t - L -x ω + ∆(x).
(2.41)

Note that the time dependency in the error solution (2.41) is caused by the time dependency of the desired trajectory ρ dout (t). If there would be no disturbance (∆(x) = 0), the desired trajectory could be achieved by using only the feedforward term. Therefore, from now on we set u f f (t) = ρ dout (t), and write the solution of (2.41) only as a function of a feedback term (since u -

u f f = u f b ): ρ(x, t) = u f b t - L -x ω + ∆(x).
(2.42)

Problem statement

We also introduce notations for the density error and feedback term in asymptotic time:

ρ∞ (x) = lim t→∞ ρ(x, t), u f b ∞ = lim t→∞ ρout (t),
where the latter denition comes from comparing (2.42) with (2.39). Then, the density error solution (2.42) is given by the following relation for t → ∞:

ρ∞ (x) = u f b ∞ + ∆(x). (2.43)
The role of u f b ∞ is thus to ensure that ρ(x, t) → ρ∞ (x), and u f b ∞ is such that the eect of the cumulated disturbance ∆(x) is minimized in the sense of L 2 -space norm (Problem 2.1) and L ∞ -space norm (Problem 2.2). This is formalized as follows:

Problem 2.1

Find the optimal control law u * composed of (2.38) such that:

(i) ρ(x, t) → ρ∞ (x), (ii) u * = argmin u ρ∞ (x) 2 2 .
Problem 2.2

Find the optimal control law u * composed of (2.38) such that:

(i) ρ(x, t) → ρ∞ (x), (ii) u * = argmin u ρ∞ (x) ∞ ,
For the denition of L 2 and L ∞ norms see (1.2) and (1.3). Note that in both Problems 2.1 and 2.2, the argmin is taken over a set of all possible control functions u such that the density from (2.35) remains in the congested regime.

Note that due to the presence of in-domain disturbance δ(x), we can not drive the error ρ(x, t) to zero as t → ∞ by acting only from the boundary.

Convergence to an equilibrium

Let us now consider the error system Σ err given by (2.37). We will rst prove that a feedback controller u f b ∞ that is constant for asymptotic time acting such that the error term converges to a steady-state (as in statements (i) in Problems 2.1 and 2.2). Then, we will also derive the optimal control law u * that satises statements (ii) of Problems 2.1 and 2.2.

Lemma 2.1. Let u(t) = u f f (t) + u f b ∞ with u f f (t) = ρ dout (t) and u f b ∞ being some constant.
Then, the following statement holds:

lim t→∞ ρ(x, t) -ρ∞ (x) 2 = 0.
Proof. Similar to [START_REF] Yu | Trac congestion control for AwRascleZhang model[END_REF], we dene the following Lyapunov function candidate

V (t) = 1 2 L 0 e ωx (ρ(x, t) -ρ∞ (x)) 2 dx, (2.44)
where e ωx plays the role of a weighting function. The time derivative of (2.44) is

V (t) = L 0 e ωx (ρ(x, t) -ρ∞ (x)) ∂ ρ(x, t) ∂t dx = L 0 e ωx (ρ(x, t) -ρ∞ (x)) δ(x) + ω ∂ ρ(x, t) ∂x dx,
where the last expression comes from the error dynamics given by (2.37).

From (2.40), (2.43), and the fact that u f b ∞ does not depend on x, the derivative of ρ∞ (x)

with respect to x is ∂ ρ∞ (x) ∂x = - 1 ω δ(x),
and thus we get

V (t) = ω L 0 e ωx (ρ(x, t) -ρ∞ (x)) ∂ (ρ(x, t) -ρ∞ (x)) ∂x dx.
Integration by parts yields

V (t) = ω 2 e ωx (ρ(L, t) -ρ∞ (L)) 2 - ω 2 (ρ(0, t) -ρ∞ (0)) 2 -ω 2 V (t) ω 2 e ωL u f b ∞ -ρ∞ (L) 2 -ω 2 V (t), (2.45)
where the last inequality comes from the fact that ρ(L, t) = u(t) -ρ dout = u f f (t) + u f b ∞ -ρ dout , and using that u f f (t) = ρ dout (t) we obtain ρ(L, t) = u f b ∞ , i.e., the error at the boundary becomes constant as well. It follows then from (2.43) and (2.40) for x = L that ∆(L) = 0, which results for (2.45) into

V (t) ≤ ω 2 e ωL ∆(L) 2 -ω 2 V (t) = -ω 2 V (t).
Lemma statement follows directly from ρ(x, t) -ρ∞ (x) 2 2 2V (t) e -ωL , thus V (t) → 0 implies ρ(x, t) -ρ∞ (x) 2 → 0.

Disturbance attenuation in sense of L 2 norm

The following theorem completes the previous result and gives the optimal form of u f b * that acts to minimize the L 2 norm of the error term as t → ∞ as specied in Problem 2.1(ii).

Although the feedback term depends on unmeasured disturbance, we will be still able to present it in a computationally feasible form.

Theorem 2.1

For the density error ρ(x, t) given by IBVP (2.37), the optimal boundary controller minimizing the limit of its L 2 norm as t → ∞ is given by

u * = u f f + u f b * , where u f f = ρ dout (t) and (2.46) u f b * (t) =    0, if 0 ≤ t < L/ω, -1 L L 0 ρ (x, t) -u * t -L-x ω dx, if t ≥ L/ω.
Proof. First, assume that u f b * (t) is a constant for t ≥ L/ω. Thus note that minimization over

u(t) = u f f (t)+u f b in Problem 2.1(ii) is equivalent to the minimization over u f b = u f b ∞ = const: u f b * = argmin u f b ρ∞ (x) 2 2 = argmin u f b u f b + ∆(x) 2 2 = argmin u f b L 0 u f b + ∆(x) 2 dx,
where we have used the relation (2.43) and the denition of L 2 norm from equation (1.2).

Expanding the quadratic form in the integral, we obtain

L 0 u f b + ∆(x) 2 dx = (u f b ) 2 L + 2u f b L 0 ∆(x)dx + L 0 ∆ 2 (x)dx.
(2.47)

In order to compute u f b * minimizing the quadratic form (2.47), we need to take the derivative of (2.47) with respect to u f b and set this expression to zero. This allows us to obtain the optimal feedback term:

u f b * = - 1 L L 0 ∆(x)dx.
(2.48)

This expression corresponds to the subtraction of the mean value of cumulative disturbance ∆(x). However, we should recall that ∆(x) is an unmeasured function. Using the solution of the error term ρ(x, t) (2.39) obtained by the method of characteristics, we can express the integral disturbance as

∆(x) = ρ(x, t) -ρ d (x, t) -u * t - L -x ω + ρ dout t - L -x ω ,
where the last two terms come from the denition of the boundary conditions in the error system (2.37). Recall that the desired density ρ d (x, t) satises system Σ d given by (2.36), which is a homogeneous transport equation. Then, we can again apply the method of characteristics and get ρ d (x, t) = ρ dout (t -L-x ω ), which results into the following expression for the integral disturbance:

∆(x) = ρ(x, t) -u * t - L -x ω .
( Thus, the optimal controller given by (2.46) should be seen as a compensator for the average eect of disturbance in number of vehicles within the whole road.

Corollary 2.1. The optimal controller (2.46) provides the following bound for the L 2 norm of the density error as t → ∞:

ρ∞ (x) 2 2 ≤ k δ(x) 2 2 , with k = L 2 2ω 2 .
Proof. Let us rst explicitly calculate the L 2 norm of the density error ρ∞ (x) for t → ∞ under the optimal feedback control, for which we make use of (2.48) and get

ρ∞ (x) 2 2 = u f b * + ∆(x) 2 2 = L 0 ∆ 2 (x) dx - 1 L   L 0 ∆(x) dx   2 .
Using the denition of the integral disturbance (2.40), we obtain

u f b * + ∆(x) 2 2 ≤ L 0 ∆ 2 (x) dx = 1 ω 2 L 0   L x δ(s) ds   2 dx.
Using the Cauchy-Schwartz inequality we can provide an upper bound for the latter expression:

u f b * + ∆(x) 2 2 ≤ 1 ω 2 L 0   (L -x) L x δ 2 (s) ds   dx ≤ 1 ω 2 δ(x) 2 2 L 0 (L -x) dx,
and nally we get

u f b * + ∆(x) 2 2 ≤ L 2 2ω 2 δ(x) 2 2 ,
which shows that there is an upper bound of the error norm as t → ∞ and concludes the proof.

Disturbance attenuation in sense of

L ∞ norm Theorem 2.2
For the density error ρ(x, t) given by IBVP (2.37), the optimal boundary controller minimizing the limit of its L ∞ norm as t → ∞ is given by

u * = u f f + u f b * , where u f f = ρ dout (t) and
(2.50)

u f b * (t) =        0, if 0 ≤ t < L ω , - sup x∈[0,L] ρ(x, t) -u * t - L -x ω + inf x∈[0,L] ρ(x, t) -u * t - L -x ω 2 , if t ≥ L ω .
Proof. Following the proof of Theorem 2.1, the minimization over u * in Problem 2.2(ii) is again equivalent to the minimization over

u f b = u f b ∞ = const: u f b * = argmin u f b ρ∞ (x) ∞ = argmin u f b u f b + ∆(x) ∞ = argmin u f b sup x∈[0,L] |u f b + ∆(x)|.
Expanding the supremum term, we get

sup x∈[0,L] |u f b + ∆(x)| = max sup x∈[0,L] u f b + ∆(x) , -inf x∈[0,L] u f b + ∆(x) = max u f b + sup x∈[0,L] ∆(x), -u f b -inf x∈[0,L]
∆(x) .

(2.51)

The rst argument in (2.51) is a monotonically increasing function with respect to u f b , while the second argument is a monotonically decreasing one. Thus, the minimum can be achieved only at the intersection point of both functions, i.e.,

u f b * = - 1 2 sup x∈[0,L] ∆(x) + inf x∈[0,L] ∆(x) . ( 2
sup x∈[0,L] ∆(x) = sup x∈[0,L]   1 ω L x δ(s) ds   ≤ sup x∈[0,L] L -x ω sup s∈[0,x] δ(s) ≤ sup x∈[0,L] L -x ω sup s∈[0,L] δ(s) ≤        0, if sup x∈[0,L] δ(x) ≤ 0, L ω sup x∈[0,L] δ(x), if sup x∈[0,L] δ(x) > 0.
(2.53)

For the inmum we proceed in the same way and obtain:

inf x∈[0,L] ∆(x) = inf x∈[0,L]   1 ω L x δ(s) ds   ≥ inf x∈[0,L] L -x ω inf s∈[0,x] δ(s) ≥ inf x∈[0,L] L -x ω inf s∈[0,L] δ(s) ≥      0, if inf x∈[0,L] δ(x) ≥ 0, L ω inf x∈[0,L] δ(x), if inf x∈[0,L] δ(x) < 0.
) ∞ = 1 2 sup x∈[0,L] ∆(x) -inf x∈[0,L] ∆(x) ≤ L 2ω sup x∈[0,L] δ(x) = L 2ω δ(x) ∞ . 2. Both sup x∈[0,L] δ(x) and inf x∈[0,L] δ(x) are negative. Then δ(x) ∞ = -inf x∈[0,L] δ(x) and ρ∞ (x) ∞ = 1 2 sup x∈[0,L] ∆(x) -inf x∈[0,L] ∆(x) ≤ - L 2ω inf x∈[0,L] δ(x) = L 2ω δ(x) ∞ .
3. The signs of sup

x∈[0,L] δ(x) and inf x∈[0,L] δ(x) are dierent. Then sup x∈[0,L] δ(x) -inf x∈[0,L] δ(x) 2 sup x∈[0,L] |δ(x)| = 2 δ(x) ∞ , which leads to ρ∞ (x) ∞ = 1 2 sup x∈[0,L] ∆(x) -inf x∈[0,L] ∆(x) ≤ L 2ω sup x∈[0,L] δ(x) -inf x∈[0,L] δ(x) ≤ L ω δ(x) ∞ .
Thus, we can provide the following bound for the L ∞ norm as t → ∞:

ρ∞ (x) ∞ L ω δ(x) ∞ .

Numerical simulation

To verify our theoretical results, we provide a numerical example, which intends to illustrate the performance of the feedback term u f b * used to minimize the L 2 and L ∞ norms of ρ(x, t) as t → ∞. For the simulation, we use the Godunov scheme (2.18) described in Section 2.1.6, and the numerical grid is divided into n = 500 cells.

Simulation setup

Notice that we simulate the system only in the congested regime, thus in (2.19) the minimum is always resolved to the benet of the supply function, i.e., ∀(i, k) ∈ {1, . . . , n} × Z + :

ϕ i (k) = S(ρ i (k)), ϕ in (k) = S(ρ 1 (k)).
We also set ρ n+1 (k) = ρ n (k) if no boundary conditions are specied (freely evolving system), and ρ n+1 (k) = u(k) in case of boundary control so that ϕ out (k) = S(u(k)).

For the simulation we set the following parameters, which are taken from real trac data [START_REF] Canudas De Wit | Grenoble Trac Lab: An Experimental Platform for Advanced Trac Monitoring and Forecasting [Applications of Control[END_REF]:

v =
δ i = -0.0002, if i ∈ {1, . . . , n 2 } 0.0006, otherwise.
There are three possible control strategies, which can be applied at the downstream boundary of the system:

1. No control action is performed. the smallest values are denoted by blue color, while the most congested zones are marked in red. The disturbance term acts so that the freely evolving system becomes entirely congested in the right part of the road, as we can see from Figure 2.8b).

2. Only feedforward control u(k) = u f f (k) is applied. 3. Full control u * (k) = u f f (k) + u f b * (k) is
In Figure 2.8c) we show what happens to the state ρ(x, t), when only the feedforward control is applied (u = u f f ) by setting ρ d (L, t) at the downstream boundary. This technique provides results that are already considerably better than just letting the state evolve freely. Finally, the state under the optimal control laws u f b * from (2.46) In Figure 2.9 we can see that the optimal control law applied to minimize ρ∞ (x) 2 in

Discussions

In this section, we have designed a feedback control law that minimizes the deviation of the state from the desired time-and space-dependent trajectory as time goes to innity in sense of L 2 and L ∞ spatial norms. The vehicle density, for which we were designing the boundary control, is restricted to the congested trac regime, which allowed us to deal with a linear problem. The control is actuated at road's downstream boundary, and physically it corresponds to controlling the amount of trac ow to leave the road. The congested trac state can become unpredictably worth (such as becoming a full trac jam with ρ(x, t) = ρ max ∀x ∈ [0, L]) due to the presence of in-domain disturbance that has been included into the LWR equation. This included disturbance is used to capture the contribution of vehicles originating from minor roads having a non-zero inow into the road to control. The desired trajectory solves an ideal (homogeneous) linear LWR in the same regime. The problem was posed and solved as the disturbance attenuation problem. The results have been veried by a numerical example, which clearly illustrates that the feedback plays an important role in the designed controller, which performs signicantly better for the error norm minimization than the one including only the feedforward part.

The obtained controllers for minimization of both L 2 and L ∞ norms are optimal. The corresponding norm to be minimized should be chosen in accordance with the available knowledge about the source of disturbance. For example, the controller to minimize the L 2 norm should be chosen if the disturbance comes from a large number of minor roads, then it makes sense to minimize the mean-square deviation from the desired state. The controller for L ∞ norm minimization should be chosen, if the maximal deviation from the desired state should be attenuated. This can be a good practical choice, if the disturbance source corresponds to vehicles coming from, e.g., another important road in case of merge intersections.

In the the following section, we extend our analysis to a more complex problem, i.e., the trac state is not restricted to any particular regime, which yields a fully nonlinear problem with all the technical challenges related to this nonlinearity, i.e., crossing characteristics and shocks in the solution.

Boundary control design for trac with nonlinear dynamics

In this section, we again consider a tracking problem to be solved by properly actuating road boundaries. This time, both the trac state and the desired trajectory are vehicle densities that can take any value from its range. We consider a mixed trac problem, e.g., trac is in the free-ow regime at one part of the road, and congested at the other part. Hence, we are going to analyze a fully nonlinear LWR model as in (2.10). The main technical challenge thereby occurs when characteristics intersect (as in the case of kinematic waves moving with dierent speeds), which causes the emergence of shocks, see Section 2.1.3. In order to handle shocks, we make use of the Hamilton-Jacobi formulation, which is an integral formulation of LWR that does not contain shocks, see Section 2.1.7 for more details.

Introduction

There exist many works that used the structure of the Hamilton-Jacobi PDE to solve control tasks for trac. For instance, optimal control methods for a trac network based on viability framework are proposed in [START_REF] Li | Optimal Control of Scalar Conservation Laws Using Linear/Quadratic Programming: Application to Transportation Networks[END_REF][START_REF] Li | Optimal trac control in highway transportation networks using linear programming[END_REF]. The framework has also been used to develop a convex optimization approach to reduce the fuel consumption in [START_REF] Zu | Convex optimization for energy-ecient trac control[END_REF]. Also [START_REF] Bekiaris | Nonlinear bilateral output-feedback control for a class of viscous HamiltonJacobi PDEs[END_REF] considered a H-J PDE with viscous term that allowed to perform a feedback linearization, which enabled tracking a desired time-dependent state on some xed space point. One of the most recent works [START_REF] Suyash | Variable Speed Limit and Ramp Metering Control of Highway Networks Using Lax-Hopf Method: A Mixed Integer Linear Programming Approach[END_REF] used the analytical solution to the LWR PDE to formulate an optimization control problem for trac on networks with variable speed limit and ramp metering control.

The main contributions of this section are the following:

Tracking space-and time-dependent trajectory: we extend the results presented in [START_REF] Blandin | Regularity and Lyapunov Stabilization of Weak Entropy Solutions to Scalar Conservation Laws[END_REF][START_REF] Bastin | Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation[END_REF] and present a controller able to drive a state with shocks to any time-and spacedependent vehicle density that also may contain shocks. This is the rst boundary controller in the trac community able to solve such general tasks. Moreover, if we compare it to [START_REF] Bekiaris | Nonlinear bilateral output-feedback control for a class of viscous HamiltonJacobi PDEs[END_REF], our analysis is done for the original LWR system without linearization and viscosity term.

The explicit solution of H-J PDE is used to provide conditions on when and which control can be applied: we consider a general case with weak boundary conditions, when it is not guaranteed that control can be imposed pointwise (see Section 2.1.4). To handle this limitation of control, we have formulated weak boundary conditions in terms of control restriction functions, and then we use them to show that even in case of non acceptance of boundary control laws the goal can still be achieved.

Preliminaries

We seek to design a feedback boundary control law that is able to track any desired spaceand time-dependent vehicle density. Unlike in the previous Section 2.2 where only congested trac has been considered, here we deal with a full nonlinear trac system as in (2.10).

Thus, prior to stating the boundary control problem, we will rst do the following: present the nonlinear state equation, perform motion planning to dene admissible trajectories to track, then we also mention assumptions that need to hold in this section, and then we give an explicit Hamilton-Jacobi solution describing the temporal evolution of the trac system.

Nonlinear trac system

Let us formulate a nonlinear IBVP for trac evolving along a single road with x ∈ [0, L] as in (2.10), but for convenience we now specify the boundary values in terms of ows φ in , φ out . Thus, we introduce the following IBVP ∀(x, t)

∈ [0, L] × R + :      ∂ t ρ(x, t) + ∂ x Φ (ρ(x, t)) = 0, Φ (ρ(0, t)) = φ in (t), Φ (ρ(L, t)) = φ out (t), ρ(x, 0) = ρ 0 (x).
(2.56)

Thereby, the ux function Φ(ρ) is again assumed to have a triangular shape as in (2.2) with ρ c = ρ max /3, and inows φ in (t) and outows φ out (t) are dened ∀t ∈ R + as

φ in (t) = min {u in (t), S(ρ(0, t))} , φ out (t) = min {D(ρ(L, t)), u out (t)} , (2.57)
where u in and u out denote the proposed ow values for the upstream and for the downstream boundary, correspondingly. Here, we treat u in and u out as control variables. By comparing (2.57) to the boundary Riemann problems for x = 0 and x = L given by (2.15) and (2.16), we establish that ∀t ∈ R + u in (t) = D(ρ in (t)), and u out (t) = S(ρ out (t)),

which means that u in physically corresponds to control of the demand to enter the road, i.e, we decide how much vehicles to let enter the domain, while u out should be viewed as control of the supply of the exit of the road, i.e., we decide how much vehicles to let leave the domain.

Notice that both u in and u out can not always be applied, i.e., these boundary conditions (2.57) are equivalent to the weak boundary conditions formulation in terms of densities given in [START_REF] Issam | Weak formulation of boundary conditions for scalar conservation laws: An application to highway trac modelling[END_REF],

see also Section 2.1.4. Thus, the problem given by (2.56) and (2.57) is well-posed.

Motion planning

Now we dene a desired space-and time-varying density ρ d (x, t) ∀(x, t) ∈ [0, L] × R + that should be tracked with the help of boundary control. In order to be admissible, ρ d (x, t) ∈ R + must be a weak entropy solution of the following system:

         ∂ρ d (x, t) ∂t + ∂Φ (ρ d (x, t)) ∂x = 0, Φ (ρ d (0, t)) = φ in d (t), Φ (ρ d (L, t)) = φ out d (t), ρ d (x, 0) = ρ 0 d (x), (2.58)
where inows and outows in the desired system must also satisfy φ in d (t) ≤ S(ρ d (0, t)) and φ out d (t) ≤ D(ρ d (L, t)) (weak boundary conditions). Notice that unlike Σ d given by (2.36) in Section 2.2, the system (2.58) is a nonlinear hyperbolic PDE system, in which discontinuities may evolve even for smooth initial data.

Thus, this section is devoted to nding ∀t ∈ R + boundary control laws u in (t) and u out (t) such that the density achieves a desired trajectory ρ d (x, t) as t → ∞. Thereby, they can take any values from their range, i.e., ρ ∈ [0, ρ max ] and ρ d ∈ [0, ρ max ].

Assumptions

Finally, throughout this section we make the following assumptions: Assumption 2.1

The initial conditions have left the system, thus, the solution of system (2.56) is determined by the values at the boundaries only.

Assumption 2.2

There exists ε > 0 such that φ in and φ out from system (2.56) satisfy the following inequalities in time average:

t+T t φ in (τ )dτ ≤T φ max -ε and t+T t φ out (τ )dτ ≤ T φ max -ε, where t > 0 and T = min L v , L ω .
Thus, ows can not hold their maximal values during the time interval given by T .

Note that Assumption 2.2 is needed to prove the exponential convergence to a desired vehicle density prole, see details in the proof of Theorem 2.3.

Remark 2.3

Note that if Assumption 2.2 is satised, then Assumption 2.1 holds trivially ∀t ∈ [t min , +∞), where t min is dened as

t min = min L v , L ω 1 + L ε (ρ max + ρ c ) , (2.59)
as it is shown in Appendix B.3.

Hamilton-Jacobi system

Note that control enters the system through the minimum function (2.57), and in several situations it can not be applied pointwise. Hence, control variables should be understood only as proposed functions. If trac is restricted only to the congested regime, then φ out (t) = u out (t) is always satised, as it was the case for the linear system (2.35) considered in the previous section. However, to solve control problems for a trac state in the mixed regime, we must handle (2.57), since the actual ow passing through the boundary is determined by the trac state. For instance, if for some time t there are only a few cars at the end of the road, we would obtain from (2.57)

φ out (t) = D (ρ(L, t)) ⇒ u out (t) is not imposed.
In order to enable analytical treatment of weak boundary conditions, we use the Hamilton-Jacobi formulation, which is an integral form of the LWR PDE that was explained in Section 2.1.7. In particular, we will be able to analyze, when and for how long the proposed control values are accepted by the system. This is possible due to a cumulative description of trac within the H-J approach, since the state corresponds to the cumulative number of vehicles also known as Moskowitz function M (x, t) (MF).

Let us recall the IBVP in the H-J formulation as in (2.25), i.e., ∀(x, t)

∈ [0, L] × R + :                ∂M (x, t) ∂t -Φ - ∂M (x, t) ∂x = 0, M (x, 0) = M Ini (x), M (0, t) = M Up (t), M (L, t) = M Down (t).
As already discussed in Section 2.1.8, the solution of this system can be obtained explicitly using the Lax-Hopf formula (2.32) for the case of a triangular FD (2.2), which is also the case here. The derivation of the solution is presented in Appendix B.2. Here we consider the solution of the system for large enough time, which equivalently means that the eect of initial conditions should have left the system (Assumption 2.1):

∀(x, t) ∈ [0, L] × [t min , +∞) : M (x, t) = min t-x v 0 φ in (τ ) dτ + L 0 ρ 0 (s) ds, t-L-x ω 0 φ out (τ ) dτ + ρ max (L -x) , (2.60)
where t min was estimated in Appendix B.3, which requires that Assumption 2.2 holds as well.

Problem statement in H-J formulation

The desired MF is obtained similar to (2.60):

∀(x, t) ∈ [0, L] × [t min , +∞) : M d (x, t) = min t-x v 0 φ in d (τ ) dτ + L 0 ρ d 0 (s) ds, t-L-x ω 0 φ out d (τ ) dτ + ρ max (L -x) .
(2.61)

Note that both M (x, t) and M d (x, t) are dened up to a constant since the starting point for the numeration of cars can be arbitrary. Therefore, we state our problem in Hamilton-Jacobi formulation as a pointwise convergence of Moskowitz functions M (x, t) to M d (x, t) as t → ∞. This is formalized as follows.

Problem 2.3

Given a desired trajectory ρ d (x, t) governed by system (2.58), nd ∀t ∈ R + boundary control laws u in (t) and u out (t) for system (2.56) such that

∃M 0 ∈ R : ∀x ∈ [0, L] lim t→∞ (M (x, t) -M d (x, t)) = M 0 .
The constant M 0 should be understood as some historical dierence in the cumulative number of vehicles in both systems, and it does not have any eect on the trac evolution.

After we have stated Problem 2.3 in H-J formulation, let us establish the link to the LWR formulation.

Lemma 2.2. Problem (2.3) is equivalent to the integral convergence of densities over arbitrarily small intervals, i.e., ∀a, b:

0 ≤ a < b ≤ L we obtain lim t→∞ b a (ρ(s, t) -ρ d (s, t)) ds = 0, (2.62)
where a and b can be arbitrarily close points in space.

Proof. By the denition of the Moskowitz function (2.23), we can write

M (a, t) -M (b, t) = L a ρ (s, t) ds - L b ρ (s, t) ds = b a ρ(s, t)ds, (2.63)
and

M d (a, t) -M d (b, t) = b a ρ d (s, t) ds.
(2.64)

For x = a and x = b in Problem (2.3) we get M (a, t) → M d (a, t) + M 0 and M (b, t) → M d (b, t) + M 0 . This is equivalent to M (a, t) -M (b, t) → M d (a, t) -M d (b,

Remark 2.4

Notice that pointwise convergence of two functions does not imply the convergence for their derivatives in any of L p norms. However, we pose Problem 2.3 to reach the equality of the densities over arbitrarily small intervals, which means that the state approximates the desired trajectory as time goes to innity.

Thus, the density approximates the desired trajectory as in (2.62) if we nd a control law for system (2.56) that solves Problem 2.3 stated in the Hamilton-Jacobi formulation.

Control law design

Theorem 2.3

(1) u in (t) = φ in d (t) -ke(t), (2)
u out (t) = φ out d (t) + ke(t), t ∈ R + where e(t) = L 0 (ρ(s, t) -ρ d (s, t)) ds and k > 0.
(2.65)

Remark 2.5

Physically, the control law given by (2.65) makes us control inows and outows such that all the excess cars, given by the integral dierence in densities, leave the domain, and then inows and outows match the desired ones. When the goal is achieved, the following conditions hold for ∀t ≥ t min :

(i) φ in (t) = φ in d (t), (ii) φ out (t) = φ out d (t), (iii) t-x v 0 (φ in (τ ) -φ in d (τ )) dτ + L 0 (ρ 0 (s) -ρ d 0 (s)) ds = t-L-x ω 0 (φ out (τ ) -φ out d (τ )) dτ.
(2.66)

The derivation of these conditions is given in Appendix B. Error e(t) is dened as the dierence in the overall number of cars in the real and the desired systems (1.5). Using (2.23) for x = 0 we can rewrite the denition of error as

e(t) = M (0, t) -M d (0, t) + t 0 (φ out d (τ ) -φ out (τ )) dτ,
which by using (2.24) to evaluate M (0, t) -M d (0, t) can be further modied as

e(t) = t 0 (φ in (τ ) -φ out (τ ) + φ out d (τ ) -φ in d (τ )) dτ + L 0 (ρ 0 (s) -ρ 0 d (s)) ds. (2.67)
Error dynamics are found as the time derivative of (2.67)

ė(t) = φ in (t) -φ out (t) -φ in d (t) + φ out d (t). (2.68)
Recall that the main challenge in controlling system (2.56) is related to the fact that the boundary ows φ in and φ out are not always equal to u in and u out , respectively. Thus, for some periods of time, we lose the ability to impose any control u in or u out on the boundaries.

Let us investigate this problem in more details.

Analysis of ow restrictions

By denition of the Moskowitz function (2.21), inows and outows are time derivatives of M (0, t) and M (L, t), respectively. Let us rst focus on the inow dened as φ in (t) = ∂M (0, t)/∂t, which allows us to express M (0, t) by taking the time integral of φ in (t) and get

M (0, t) = t 0 φ in (τ ) dτ + M (0, 0), (2.69)
where M (0, 0) = L 0 ρ 0 (s) ds is obtained from the denition (2.23) for the space-time point (x, t) = (0, 0).

From now on, let us consider only t ≥ t min with t min being dened in (2.59). Then, we can also consider the MF solution (2.60) for x = 0 and obtain

M (0, t) = min t 0 φ in (τ ) dτ + L 0 ρ 0 (s) ds, t-L ω 0 φ out (τ ) dτ + Lρ max .
(2.70) Combining (2.69) with (2.70), we obtain the following minimum problem: (2.71) From (2.71) one can see that the following inequality holds

t 0 φ in (τ )dτ + L 0 ρ 0 (s)ds = min t 0 φ in (τ )dτ + L 0 ρ 0 (s)ds, t-L ω 0 φ out (τ )dτ + Lφ max .
t 0 φ in (τ ) dτ + L 0 ρ 0 (s) ds ≤ t-L ω 0 φ out (τ ) dτ + Lφ max . (2.72)
In case of equality in (2.72), we must provide that the right-hand term grows more quickly than the left-hand term. Thus, by taking the time derivative of (2.71) we obtain φ in (t) ≤ φ out (t-L ω ).

Notice that if the left-hand term is strictly smaller than the right-hand term in (2.72), then the inow is less constrained and we should be able to set φ in (t) = u in (t).

All this can be combined in the following property. We dene a control restriction function for the upstream boundary

g in (t) = t-L ω 0 φ out (τ ) dτ + Lρ max - t 0 φ in (τ ) dτ - L 0 ρ 0 (s) ds,
which represents the dierence between two arguments of the minimum from (2.71). By (2.72)

we obtain that g in (t) ≥ 0 always. Moreover, using the denition of φ in (t) given by (2.57), the condition on inow restriction can be formulated as:

g in (t) > 0 : φ in (t) = u in (t), g in (t) = 0 : φ in (t) = min u in (t), φ out t - L ω .
(2.73)

Note that the notation of control restriction should be understood as saturation control with φ out t -L ω being the saturation time-varying threshold.

Similarly, we proceed by considering the MF solution (2.34) for x = L to analyze φ out (t)

for the downstream boundary, and get its control restriction function that reads

g out (t) = t-L v 0 φ in (τ ) dτ + L 0 ρ 0 (s) ds - t 0 φ out (τ ) dτ,
and the following condition on the outow restriction

g out (t) > 0 : φ out (t) = u out (t), g out (t) = 0 : φ out (t) = min u out (t), φ in t - L v .
(2.74)

Thus, any boundary control can be imposed if g in > 0 and g out > 0.

Dening R(t) = L 0 ρ(s, t)ds and R 0 = R(0), and using the equality of (2.23) and (2.24)

(independence of chosen integration path), we obtain for ∀t ∈ R + and x = 0

R(t ) = R 0 + t 0 φ in (τ ) dτ - t 0 φ out (τ ) dτ.
(2.75)

Thus, the control restriction functions can be rewritten as

g in (t) = Lρ max -R(t ) - t t-L ω φ out (τ ) dτ - t t φ in (τ ) dτ, g out (t) = R(t ) - t t-L v φ in (τ ) dτ - t t φ out (τ ) dτ.
(2.76)

Note also that inows and outows are upper bounded by the road capacity, i.e., φ in ≤ φ max and φ out ≤ φ max , where φ max = vρ c (see Figure 2.1). To nd a time interval, during which no control law can be imposed, we set g in (t) = 0 and then express R(t ) from (2.76):

R(t ) = Lρ max - t t-L ω φ out (τ ) dτ - t t φ in (τ ) dτ ≥ Lρ max - t t-L ω φ max dτ - t t φ max dτ = Lρ max - L ω φ max = Lρ c .
The same steps are performed for g out (t) = 0, and we get

g in (t) = 0 ⇒ R(t ) ≥ Lρ c ∀t ∈ t - L ω , t , g out (t) = 0 ⇒ R(t ) ≤ Lρ c ∀t ∈ t - L v , t .
(2.77)

This means that not any control law can be applied at the upstream boundary at time t, if during the preceding time interval t -L ω , t the mean density was bigger than the critical density (and inversely for the downstream boundary).

Using Assumption 2.2 and the fact that the critical density

ρ c = ρ max /3, we set g in (t) = 0 in (2.76), which implies R(t) ≥ Lρ c + ε and R(t -L/ω) ≥ Lρ c + ε, as well as g out (t) = 0 implies R(t) ≤ Lρ c -ε and R(t -L/v) ≤ Lρ c -ε.
Let us consider (2.77) to investigate whether it is possible that control can not be imposed at both boundaries simultaneously. We pick some time point t such that g in (t) = 0 and some time point t ∈ (2.77). This is a contradiction, since satisfying both R(t) ≥ Lρ c + ε and R(t) ≤ Lρ c at the same t is impossible. Thus, the time point t when g out (t ) = 0 can occur at least after the interval L/v has passed since the last g in (t) = 0.

[t, t + L/v] with g out (t ) = 0. As written above, g in (t) = 0 implies R(t) ≥ Lρ c + ε. However, t ∈ [t -L/v, t ], thus R(t) ≤ Lρ c for g out (t ) = 0 by
Moreover, if t ≥ t + L/v and g out (t ) = 0, then R(t -L/v) ≤ Lρ c -ε. The maximal inow is always bounded from above by φ max , therefore the dierence in the integral densities R(t -L/v) -R(t) can be passed at least in time t -L/v -t ≥ 2ε/φ max . Performing the same analysis for the inverse case, we conclude that

g in (t) = 0 ⇒ g out (t ) > 0 ∀t ∈ t, t + L v + 2ε φ max , g out (t) = 0 ⇒ g in (t ) > 0 ∀t ∈ t, t + L ω + 2ε φ max .
(2.78) Thus, it is impossible for two boundaries to be unable to accept the control simultaneously, and the periods of uncontrollability are separated in time by at least

L v + 2ε φmax or L ω + 2ε φmax .

Dynamics of e(t)

Thus, in Section 2.3.4.1 we have established that at each moment either one of control restriction functions or none of them is zero (2.78). Hence, we separate the dynamics of the integral error term e(t) (1.5) into three possible cases.

1. Assume both g in (t) > 0 and g out (t) > 0. Then all the boundary control terms u in and u out can be applied, which by (2.65) implies that φ in (t) = φ in d (t) -ke(t) and φ out (t) = φ out d (t) + ke(t). According to (2.68), the error dynamics are given by ė(t) = -2ke(t), (2.79) and, thus, e(t) converges exponentially to zero.

2. Assume g in (t) = 0. Then, the control can not be applied at the upstream boundary, i.e., φ in (t) ≤ φ in d (t) -ke(t) and φ out (t) = φ out d (t) + ke(t), which means ė(t) ≤ -2ke(t).

(2.80) Thus, a positive error e(t) > 0 implies even faster convergence to zero. If e(t) < 0, such dynamics can diverge from zero. However, it is possible to show that after a period of not being able to impose any control at the upstream boundary, the error will not be further away from zero than at the beginning of the period. Consider the control restriction function g in d (t)

for the upstream boundary of the desired system:

g in d (t) = Lρ max -R d (t ) - t t-L ω φ out d (τ ) dτ - t t φ in d (τ ) dτ ≥ 0, for t ∈ t - L ω , t . Using e(t ) = R(t ) -R d (t ), we obtain g in d (t) -g in (t) = t t-L ω (φ out (τ ) -φ out d (τ )) dτ + t t (φ in (τ ) -φ in d (τ )) dτ + e(t ) ≥ 0. (2.81) Using the properties φ in (t) ≤ φ in d (t) -ke(t) and φ out (t) = φ out d (t) + ke(t), we obtain e(t ) + k t t-L ω e(τ ) dτ -k t t e(τ ) dτ ≥ 0.
We substitute t = t to get the rst inequality, and then t = t -L ω to get the second one:

1) e(t)

+ k t t-L ω e(τ ) dτ ≥ 0, 2) e t - L ω -k t t-L ω e(τ )dτ ≥ 0,
and the sum of these inequalities yields e(t) + e t -L ω ≥ 0.

(2.82)

3. Assume g out (t) = 0. Then, the control can not be applied at the downstream boundary, i.e., φ out (t) ≤ φ out d (t) + ke(t) and φ in (t) = φ in d (t) -ke(t), which yields the following error dynamics ė(t) ≥ -2ke(t).

(2.83)

Using the same analysis as above for e(t) > 0, we obtain e(t) + e t -L v ≤ 0.

(2.84)

Proof that e(t) converges to zero

In Table 2.1 we have summarized three regimes of error dynamics. The regimes can alternate

1 g in (t) > 0, g out (t) > 0 ė(t) = -2ke(t) Regime 2 g in (t) = 0, g out (t ) > 0, ė(t) ≤ -2ke(t) ∀t ∈ t, t + L v + 2ε φmax Regime 3 g out (t) = 0, g in (t ) > 0, ė(t) ≥ -2ke(t) ∀t ∈ t, t + L ω + 2ε
φmax as depicted in Figure 2.10. In this part of the proof, we will show that the error can enter the second dynamic regime (Regime 2) only being positive, while it enter Regime 3 only being negative. Thus, if the error is positive in Regime 1 (green circle), it either remains there forever and the exponential convergence to zero is guaranteed by (2.79), or it enters Regime 2

(violet circle). Then, being positive, by (2.80) the error converges to zero even faster than in Regime 1. However, it can also become negative, and in this case the error might diverge from zero. Nevertheless, the divergence from zero can last only for a bounded time interval, and by

(2.82) the absolute value of the error term can not exceed the value it had some time ago. As this happens, the error enters again Regime 1 as a negative term. It either stays there forever, or switches to Regime 3, where it goes to zero even more quickly by (2.83). The rest can be described in a symmetric manner. Recall also that Regimes 2 and 3 are always separated in time by at least

L v + 2ε φmax or L ω + 2ε
φmax . Further, we provide a strict proof of the exponential convergence of the error term to zero.

Imagine a time axis split into three types of intervals corresponding to three dierent error dynamic regimes as shown in Figure 2.11. Recalling that Regimes 2 and 3 cannot occur in a row, we can observe, e.g., a sequence like this: 12121313121.... Thus, it is possible that after Regime 2 and then Regime 1, the second regime comes again, since nothing prohibits g in (t) to become zero again almost immediately. We denote this sequence of regimes as g in (t) = 0 sometimes , which is dened as the largest row of Regimes 1 and 2 that starts and ends with Regime 2 and does not contain any time interval with Regime 3. The same can be done with the regime sequence containing Regimes 1 and 3 called g out (t) = 0 sometimes . These sequences g in (t) = 0 sometimes and g out (t) = 0 sometimes alternate strictly, always having Regime 1 between them. Finally, for a time interval corresponding to the regime (or regime sequence) with index i we can dene entrance time t i and exit time ti . By (2.78) we see that t i -ti-1

≥ L ω + 2ε φmax if Regime i is g in (t) = 0 sometimes , and t i -ti-1 ≥ L v + 2ε φmax if Regime i is g out (t) = 0 sometimes .
Let us x i corresponding to g in (t) = 0 sometimes (the other case is symmetric). First of all, by (2.82) we obtain

g in (t i ) = 0 and g in ( ti ) = 0. Therefore e(t i ) + e t i - L ω ≥ 0 (2.85) e(t) = 0 e(t) > 0 e(t) < 0 313... 1 212... ti-1 ti -L ω t i ti t i 1 t e(t)
Figure 2.11: A possible error behaviour e(t) (thick black line). From left to right: divergence for g out (t) = 0 sometimes (in orange); exponential convergence in Regime 1 (in green); fast convergence for g in (t) = 0 sometimes (in blue); then divergence for e(t ) < 0 ∀t ∈ [t i , ti ]; exponential convergence in Regime 1. Blue empty circles are related to (2.86). and e( ti ) + e ti -L ω ≥ 0.

(

It is clear that t i -L ω ≥ ti-1 , which means that the dynamics of e in the interval [t i -L ω , t i ] are exponential. Thus, both e(t i ) and e t i -L ω have the same sign, and by (2.85) they are both positive. A similar analysis can be done for the regime sequence g out (t) = 0 sometimes , which means that from g out (t) = 0 to g in (t) = 0 the error term is positive and from g in (t) = 0 to g out (t) = 0 the error term is negative (and thus e( ti ) < 0). Consequently, inside each regime sequence i there should be a time point t i , when e(t i ) = 0. Now, by (2.86) and using that e( ti ) is negative, we see that e ti -L ω ≥ 0, which means that ti -L ω ≤ t i (see Figure 2.11).

During the time interval [ ti-1 , t i ] it is clear that the convergence is exponential (Regime 1). During the time interval [t i , t i ] the dynamics are ė ≤ -2ke, and e(t) ≥ 0. The time point ti -

L ω ∈ [ ti-1 , t i ], therefore e ti - L ω ≤ e( ti-1 ) exp -2k( ti -L ω -ti-1 ) ,
which is also valid for its absolute values

e ti - L ω ≤ |e( ti-1 )| exp -2k( ti -L ω -ti-1 )
.

Now from (2.86) and the fact that e( ti ) < 0 we see that |e( ti

)| ≤ e ti -L ω , thus |e( ti )| ≤ |e( ti-1 )| exp -2k( ti -L ω -ti-1 )
.

We can write titi-1

≥ t i -ti-1 ≥ L ω + 2ε φmax , which yields ti -L ω -ti-1 ti -ti-1 = 1 - L ω ti -ti-1 ≥ 1 - L ω L ω + 2ε φmax = 2ε φmax L ω + 2ε φmax = 2εω Lφ max + 2εω
, which allows us to bound time interval in the exponential function from below by

ti - L ω -ti-1 ≥ 2εω Lφ max + 2εω ( ti -ti-1 ) .
This nally leads to

|e( ti )| ≤ |e( ti-1 )| exp -2k 2εω Lφmax+2εω ( ti -ti-1 ) , (2.87)
which proves the exponential convergence to zero of e(t).

Proof that integral inows converge

In order to further proceed with the proof of Theorem 2.3, we need to show that the integral of inow dierence in the real (2.56) and in the desired system (2.58) has a limit, as it is required for the convergence of Moskowitz functions as stated in Problem 2.3 (see Section 2.3.4.5, where the existence of this limit is used for the introduction of constant M 0 ):

∃ lim t→∞ t 0 (φ in (τ ) -φ in d (τ )) dτ. (2.88)
By the Cauchy criterion for the convergence of functions, it suces to show that lim

t 1 ,t 2 →∞ t 2 t 1 (φ in (τ ) -φ in d (τ )) dτ = 0, ∀t 1 , t 2 : t 2 > t 1 .
(2.89)

First, we nd an upper bound for this limit (2.89). By combining (2.73) with (2.65) we obtain

φ in (t) ≤ φ in d (t) -ke(t) ∀t ∈ R + , thus, we can write ∀t 1 , t 2 → ∞ t 2 t 1 (φ in (τ ) -φ in d (τ )) dτ ≤ -k t 2 t 1 e(τ ) dτ → 0,
where we have used the exponential convergence result for the error term (2.87). This in turn implies that the upper bound is 0:

lim t 1 ,t 2 →∞ t 2 t 1 (φ in (τ ) -φ in d (τ )) dτ ≤ 0.
(2.90)

Now let us estimate the lower bound for the limit (2.89). Thereby, we distinguish two possible cases:

1. Assume g in (t) > 0. In this case, we can write

t t-L ω (φ in (τ ) -φ in d (τ )) dτ ≥ -k t t-L ω |e(τ )| dτ. 2. Assume g in (t) = 0. Using (2.81) for t = t -L ω , we get t t-L ω (φ in (τ ) -φ in d (τ )) dτ ≥ -e(t - L ω ) ≥ -e(t - L ω
) .

The combination of these two cases yields

t t-L ω (φ in (τ ) -φ in d (τ )) dτ ≥ -k t t-L ω |e(τ )| dτ -e(t - L ω
) .

(2.91) Now let us divide the time interval [t 1 , t 2 ] into equal subintervals of length L/ω. Thus, (2.91) can be rewritten for a larger time interval as

t 2 t 1 (φ in (τ ) -φ in d (τ )) dτ ≥ -k t 2 t 1 |e(τ )| dτ - t 2 -t 1 L/ω n=0 e t 1 + n L ω .
where the sum goes over intervals of size L/ω, i.e., t 1 , t 1 + L/ω, . . . , t 2 . If we take the time limit of the latter expression for t 1 , t 2 → ∞, both right-hand terms converge to zero, as for the sum term we can apply the integral test for convergence. Finally, we obtain the lower bound:

lim t 1 ,t 2 →∞ t 2 t 1 (φ in (τ ) -φ in d (τ )) dτ ≥ 0.
(2.92)

The combination of (2.92) and (2.90) provides that the limit is zero, which proves the existence of the limit of the integral dierence between inows in both systems as in (2.88).

Proof that Moskowitz functions converge

Finally we arrived at the last part of the proof of Theorem (2.

3). Let us dene two auxiliary

Moskowitz functions as

M 1 (x, t) = R 0 -R 0 d + t-x v 0 (φ in (τ ) -φ in d (τ )) dτ, (2.93) M 2 (x, t) = t-L-x ω 0 (φ out (τ ) -φ out d (τ )) dτ.
(2.94)

Notice that M 1 (x, t) and M 2 (x, t) correspond to the left-and to the right-hand side of (2.66)(iii), which must hold for large t. First of all, using e 0 = R 0 -R 0 d and the dynamics of e(t) given by (2.68), we obtain that ∀x ∈ [0, L]

lim t→∞ (M 1 (x, t) -M 2 (x, t)) = ∞ 0 (φ in (τ ) -φ in d (τ ) -φ out (τ ) + φ out d (τ )) dτ + e 0 = lim t→∞ e(t) = 0.
(2.95)

Moreover, as it was proven in the previous Section (2.3.4.4), M 1 (x, t) has a limit due to (2.88) and e 0 being constant in time. Therefore, we can dene

M 0 := lim t→∞ M 1 (x, t),
and by (2.95) we get

lim t→∞ M 2 (x, t) = M 0 .
We can also dene the MF error terms as

M1 (x, t) = M 1 (x, t) -M 0 and M2 (x, t) = M 2 (x, t) -M 0 .
Finally, recall that the desired MF given by (2.61) can be expressed as

M d (x, t) = min M Up d (x, t), M Down d (x, t) ,
since this is the general denition of a solution which is not aected by the initial conditions (see Assumption 2.1). Thus, by using the MF solution (2.60) together with M 1 (x, t) (2.93) and M 2 (x, t) (2.94), we obtain

M (x, t) = min M Up d (x, t) + M 1 (x, t), M Down d (x, t) + M 2 (x, t) , or M (x, t) = min M Up d (x, t) + M1 (x, t), M Down d (x, t) + M2 (x, t) + M 0 .
Minimum is a continuous function on both arguments, thus we obtain for t → ∞ that

M (x, t) → M d (x, t) + M 0 , ∀x ∈ [0, L], (2.96)
as stated in Problem 2.3, which nally concludes the proof.

Remark 2.6

Note that Assumption 2.2 is non-limiting. Indeed, it requires that the ow integral over time T is less than its maximum value by at least ε, which is always possible, except when vehicles enter and leave the system at maximum rate during T. Obviously, in this case, it is also possible to reach the goal.

The need to use Assumption 2.2 comes from the fact that at maximum ow by (2.77) both g in (t) = 0 and g out (t) = 0 for the same t, which means that with the slightest uctuation a boundary becomes uncontrollable. Thus, the state with maximum ows at both boundaries during time interval T is unstable, and therefore for this case it is impossible to prove the exponential convergence of the error term e(t).

Numerical example

We illustrate the eciency of the feedback boundary control law (2.65) by driving a state being initially almost completely in a trac jam to a desired vehicle density trajectory being in a mixed regime. This is done by providing results from a numerical simulation, for which we again use the standard Godunov scheme that is described in Section 2.1.6.

Simulation setup

As in Section 2.2.7, here we also consider a space interval [0, L] that is divided into n = 500 cells. The feedback term given by the integral error (1.5) is computed using the Riemann summation over cells i ∈ {2, . . . , n -1}. Thus, here we seek to demonstrate the eciency of using a state feedback for a road whose rst quarter is initially empty, and the trac jam is formed at the rest of the road:

ρ 0 (0 ≤ x < 1/4L) = 0 and ρ 0 (1/4L ≤ x ≤ L) = ρ max .
Thus, we consider here a system being almost completely in the trac jam as initial condition.

For the simulation, we use the same parameter set as in (2.55).

As a target state, we consider a vehicle density trajectory in a mixed trac regime (a space-and time-dependent function), whose evolution is given by ghost cells (which are set by copying the value from the neighbor cell) with ρ d in (t) = 0.04 + 0.04 sin (t/8) and ρ dout (t) = 0.1 + 0.06 sin (t/4) .

Here, for convenience, the boundary values are prescribed in terms of densities, since it allows a straightforward implementation of the Godunov scheme (2.18). Then, these density values are transformed into inows and outows by using the supply-demand formulation for the case of a triangular FD (2.17).

We will demonstrate how the feedback term given by -ke(t) for the upstream and +ke(t)

for the downstream boundary improves the result and provides the asymptotic convergence targeting the desired prole. Two control strategies are compared:

Simulation results

The simulation results are presented in Figure 2.12. Thereby, Figure 2 Comparing these plots, we can see that control including the feedback term performs considerably better. Without feedback the congested regime almost completely occupies the domain as time runs, while the feedback term makes the system approach the desired state after the time inferior to the minimal controllability time, which is t ctr = L v + L ω = 200.5 s. The convergence results in sense of L 1 norm are compared for dierent control gains (k = 0, k = 0.005 and k = 0.1) in Figure 2.12b). Thereby, we observe a faster convergence rate for the largest controller gain.

Note that an open-loop control (such as applying absorbing boundary conditions until the road becomes empty, and then applying desired inows and outows) will not achieve the goal at all due to the dierence in initial densities (2.67).

Discussion

In this section, we have designed boundary control laws that enable tracking a target spaceand time-dependent vehicle density on a single road. Both real and desired states are governed by LWR PDEs with triangular fundamental diagram, and they are allowed to be in a mixed trac regime. This means that the controller is activated at both road boundaries. It allows us to drive any state, being in a partly congested and partly free-ow regime or even being completely congested, to some desired state that is also governed by a fully nonlinear LWR PDE.

The main challenge in control design was related to the fact that we can not apply the boundary conditions pointwise in a general nonlinear LWR system, since one always has to deal with demand-supply concept. We could handle this issue by using the explicit solution formula to H-J PDE that was obtained using the properties of triangular FDs due to the convenient shape of its Legendre transform. The cumulative representation of trac in terms of number of vehicles within the H-J approach allowed us to formulate and to analyze the control restriction functions. These functions describe time periods, when a domain boundary can accept a proposed controller. These control restriction functions could be dened by exploiting the system evolution at previous times due to the integral structure of the H-J solution. The designed controller has a feedback term, which physically corresponds to the dierence between the given number of vehicles on a road and in the desired system multiplied by some controller gain. The numerical example veried the results and illustrated that feedback is absolutely necessary to achieve the goal.

a) ρ d (x, t) b) ||ρ -ρ d || L 1 c) ρ(x, t), no feedback d) |ρ -ρ d |, no feedback e) ρ(x, t), k = 0.1 f ) |ρ -ρ d |, k = 0.1

Chapter conclusions

In this chapter, we have investigated trac dynamics evolution on a single road of nite length. This was done within the LWR modeling approach that was described in Section 2.1.

Further, we have formulated tracking problems for a desired density being a space-and timedependent trajectory. These dependencies were included for a better approximation of some realistic trac situations when trac conditions change rapidly, e.g., it is common to have non constant inows and outows at dierent day times. Moreover, an equilibrium desired state is just a special case of a general space-and time-varying prole, which can also be covered by the theoretical results derived in this chapter. Such a general target prole results into a non-trivial error dynamics that have been analyzed in both Sections 2.2 and 2.3.

Then, we have also designed feedback boundary control laws on a single road for two dierent systems. The rst trac system that we have considered in Section 2.2 corresponds to a linear LWR model describing trac in a congested regime that incorporates a possible mismatch between the model and the reality in terms of unknown in-domain disturbance function. We formulated disturbance attenuation problems while reaching the desired prole in terms of L 2 and L ∞ spatial norms (Problems 2.1 and 2.2). We were able to handle unknown disturbance using characteristics method that allowed us to express the disturbance function through known (measured) variables such as vehicle density and control actions applied at previous time steps. The achieved results stated in Theorems 2.1 and 2.2 were validated with the help of a numerical simulation example, which illustrated considerable improvements of a trac state under the boundary controller compared to a freely evolving trac system with no boundary control. In particular, we were able to observe how easily a system can get into a complete trac jam along the whole road, if we do not apply at least the feedforward control, which is used to track the desired density. The feedforward control is able to considerably reduce the congestion level, although the desired state must remain in the congested regime for the well-posedness of the problem. Then, it was also shown that if in addition to the feedforward controller we also include the feedback part used, then we track the desired state even better. This became obvious from the temporal behaviour of L 2 and L ∞ norms of the deviation from the desired trajectory illustrated in Figure 2.9.

Further, in Section 2.3, we have considered a more complex problem for the case, when both the state and the desired trajectory are governed by full nonlinear LWR models as in its original formulation (also without the disturbance). The main challenge thereby was related to shocks (discontinuities), which arise in such systems for smooth initial data in nite time.

This makes an explicit analysis a tedious ask, since then we have to consider the solution only in a weak sense (no classical solutions any more) and track shocks dynamics. Another challenge to deal with was related to the weak boundary conditions, which implies the nonability to impose any boundary control. To handle both of these issues, we translated the LWR trac system into its integral form corresponding to the Hamilton-Jacobi PDE that is free of discontinuities, and in the worst case it can only become non-dierentiable. Its state corresponds to a cumulative number of vehicles that can be obtained by integrating the vehicle density. The H-J system can be seen as an optimal control problem, and its solution is obtained semi-explicitly as the minimum of all valid paths. In case of triangular FD that we assumed in this chapter, the solution is obtained as the minimum over only three valid paths each associated with one of the boundary conditions or with the initial condition. For the analysis of the system in asymptotic time, we were able to estimate the minimal time (2.59) upon which it is guaranteed that the initial conditions do not aect the H-J solution any more.

The solution can then just be formulated as a minimum of two valid paths associated with the boundary conditions. Thus, the integral formulation of Hamilton-Jacobi trac system, as well as the ability to express its solution exactly, allowed us to explicitly analyze the periods of time, when boundaries are restricted to accept control action as a function of the actual trac state. These so-called control restriction functions enabled to divide the error dynamics into three dierent regimes depending on which boundaries can currently accept control actions.

The main result of Section 2.3 is given by Theorem 2.3. Thereby, we have shown that even when boundaries are sometimes unable to accept proposed controls, the system converges to the desired trajectory exponentially. The designed boundary controller also consists of a feedforward and a feedback part, where the latter is an essential component to achieve the goal. The results have been validated numerically for dierent control gains.

In the next chapter, we are going to extend this result to a large scale, i.e., we will pose and solve various control tasks for the vehicle density dened in some urban area using a scalable modeling approach by considering a conservation law for trac in 2D.

Chapter 3

Uni-Directional Trac on Networks

This chapter is devoted to trac control problems in large-scale urban networks with a preferred direction of trac ow. The analysis and control design will be done within the same modeling approach as in the previous Chapter 2 but in two dimensions (2D).

Preliminaries

In case of trac modeling on large-scale urban networks, one needs to look for macroscopic approaches due to increasing computational complexity. However, prior to [START_REF] Williams | Urban trac network ow models[END_REF] who used data from microsimulations, there has been no evidence of any existence of macroscopic relation between density and ow on a city level as it was established on single roads by Greenshields [START_REF] Greenshields | A study of trac capacity[END_REF]. Later, in 2008, Geroliminis and Daganzo observed a similar relation during data collection in a real-life experiment conducted during a rush-hour in the city of Yokohama, Japan [START_REF] Geroliminis | Existence of urban-scale macroscopic fundamental diagrams: Some experimental ndings[END_REF][START_REF] Daganzo | An analytical approximation for the macroscopic fundamental diagram of urban trac[END_REF]. The discovery of the so-called macroscopic fundamental diagram (or shortly, MFD) gave rise to reservoir models, which track the number of cars in a urban area. MFD-based models are intuitive, simple, and do not require a high computational eort to be applied. For an MFD to be well-dened, there must exist only one ow value for a given number of vehicles.

This feature is preserved only in regions that consist of links that have similar congested levels, while this causes problems in case of regions with heterogeneous links. To overcome this problem, [START_REF] Hajiahmadi | Optimal dynamic route guidance: A model predictive approach using the macroscopic fundamental diagram[END_REF][START_REF] Leclercq | Macroscopic trac dynamics with heterogeneous route patterns[END_REF] presented partitioning algorithms that intend to split an urban area into multiple homogeneous zones each having its own well-dened MFD.

LWR model in 2D

Let us present here a macroscopic model presented in [START_REF] Mollier | Two-dimensional macroscopic model for large scale trac networks[END_REF] that corresponds to a conservation law on a two-dimensional plane, where the conserved quantity is the total number of vehicles in this plane. This model will be used to investigate the macroscopic trac behaviour in a urban network that is represented by a 2D continuum plane (x, y) ∈ Ω ∈ R 2 that is a bounded rectangular domain, i.e., Ω : [x min , x max ] × [y min , y max ]. The size of the rectangular domain is determined by the size of the urban network, i.e., x min is associated to the intersection with the minimal x space coordinate among all intersections (the rest is dened similarly).

This 2D model can be seen as an extension of the classical 1D LWR model (2.1) to two dimensions that describes the trac density evolution over a continuum plane ∀(x, y, t)

∈ 63 Ω × R + as:    ∂ρ(x, y, t) ∂t + ∇ • Φ(x, y, ρ(x, y, t)) = 0, ρ(x, y, 0) = ρ 0 (x, y), (3.1)
where ρ(x, y, t) : Ω × R + → R + is now a two-dimensional density that aggregates the number of vehicles per square meter, ρ 0 (x, y) is its value at initial time. The ux function in (3.1) is now an explicitly space-dependent vector function with the magnitude Φ(x, y, ρ) : E → [0, φ max ] with the set of departure being E = {(x, y, ρ) : (x, y) ∈ Ω, ρ ∈ [0, ρ max (x, y)]}.

The ux magnitude Φ(x, y, ρ) is again a concave Lipschitz continuous function that reects the empirically established law relating the average 2D ow with the average 2D density (fundamental diagram), i.e., φ(x, y, t) = Φ(x, y, ρ). The ux vector function is then dened as a product of the magnitude Φ(x, y, ρ) and the direction vector d θ (unit vector):

Φ(x, y, ρ) = Φ(x, y, ρ) d θ (x, y), (3.2)
where

d θ = cos(θ(x, y)) sin(θ(x, y)) (3.3)
is a vector that depends on the network geometry given by angle θ(x, y) : Ω → [0, 2π) that must be smooth enough (more details on its smoothness are given in Section 3.2). Angle θ(x, y)

is related to the orientation of roads in a urban network, thus, it determines the direction of trac ow. Hence, from now on, we will call d θ the direction eld to stress its physical meaning. The details on how to obtain this vector d θ (x, y) ∀(x, y) ∈ Ω are given in Section 3.1.3.1. Finally, the nabla operator in (3.1) is dened as

∇ = ∂ ∂x , ∂ ∂y .
Thus, the divergence term ∇ • Φ(x, y, ρ(x, y, t)) in (3.1) is a scalar. The existence and uniqueness of solutions for a system like (3.1) were shown in [START_REF] Kruzhkov | First order quasilinear equations in several independent variables[END_REF] (see p.223 for the conditions of uniqueness, and existence is discussed on p.230). The boundary conditions of (3.1) will be discussed later in Section 3.1.4.

As the 2D model (3.1) represents an extension of the standard 1D LWR model (2.1), their units and structure are compared in Table 3.1. In general, 2D models like (3.1) are not expected to describe very precisely the density evolution in space coordinates. They are rather used to capture the main trac features on a global scale such as the location and propagation of congested areas in a transportation network.

[ρ] = veh/m (scalar) [ρ] = veh/m 2 (scalar) velocity [v] = m/s (scalar) [ v] = m/s (vector) ux [Φ] = veh/s (scalar) [ Φ] = veh/(s•m) (vector) equation ∂ t ρ + ∂ x Φ(ρ) = 0 ∂ t ρ + ∇ • Φ(x, y, ρ) = 0

Space-dependent fundamental diagram

In general, the ux magnitude in 2D (3.1) is very similar to the ux function in 1D (e.g., it can be the triangular FD (2.2)) both being Lipschitz continuous concave functions. However, unlike in the 1D case, the 2D ux magnitude Φ(x, y, ρ) incorporates network infrastructure parameters by having an explicit space-dependency. Imagine a urban network containing roads that may have dierent speed limits and number of lanes, i.e., roads usually have dierent transportation capacities. For instance, compare some major three-lane roads with 50 km/h as a speed limit with minor single-lane roads with 30 km/h that can clearly accommodate less cars than the major roads. This kind of infrastructure dierences are captured by space-varying parameters ρ max (x, y), ρ c (x, y) and v(x, y). Thus, we can see that in a 2D representation of trac, the assumption that the FD parameters are identical everywhere, does not hold any more, since the network geometry should also be taken into account.

All these parameters still have the same physical meaning as in the 1D case introduced in Sections 2.1.1 and 2.1.2, but their units are consistent with those in Table 3 2 ), correspondingly. Thus, space-dependent fundamental diagrams are functions with space-dependent parameters, e.g., the triangular FD is dened ∀(x, y, ρ) ∈ E as:

Φ(x, y, ρ) = v(x, y)ρ, ρ ∈ [0, ρ c (x, y)], -ω(x, y)(ρ -ρ max (x, y)), ρ ∈ (ρ c (x, y), ρ max (x, y)], Φ(x, y, ρ) = v max (x, y) 1 - ρ ρ max (x, y)
ρ.

(3.5)

Note that all FDs can still be depicted as in Figures 2.1a) and 2.1b) having in mind that they can have dierent peaks and slopes for dierent space points (x, y) ∈ Ω.

Continuous approximation of parameters

In general, two-dimensional continuum models with a structure similar to (3.1) are commonly used in pedestrian (crowd) modeling [START_REF] Helbing | A uid dynamic model for the movement of pedestrians[END_REF][START_REF] Hughes | A continuum theory for the ow of pedestrians[END_REF]. It is however worth noting that crowds evolve in open spaces, and, unlike vehicles, pedestrians are not constrained to move on trac roads.

Here, we are going to use the 2D model (3.1) to predict the propagation of trac in a urban network, which represents a set of roads (links) and intersections (nodes). Thus, the equation (3.1) is a valid model for trac modeling, if we assume that the urban network is dense enough to be approximated as a continuum. As already mentioned above, we will use the network geometry to parametrize the model, e.g., we will estimate the values of velocity and direction eld as a function of the distance to physical roads. Let us explain parametrization on an example of a ux with the magnitude corresponding to the Greenshields FD:

Φ(x, y, ρ) = v max (x, y) d θ (x, y) 1 - ρ ρ max (x, y) ρ.

Inverse distance weighting

From (3.1) and (3.6) we can see that the trac ow direction is determined by the velocity eld, which is a product of direction eld d θ (3.3) and the maximal kinematic wave speed v max . Thus, we expect more trac to be concentrated in areas with densely located roads and along roads with high speed limits, e.g., highways. We achieve that by applying the inverse distance weighting method (IDW), which assignes larger weights to space points that are close to roads, see [START_REF] Mollier | Two-dimensional macroscopic model for large scale trac networks[END_REF] for a detailed explanation, while in the next paragraph we will give a brief idea.

Let us denote roads of the network by q ∈ {1, . . . , Q}. For the sake of computation, each road is parametrized by s ∈ {1, . . . , s max } such that variable s allows to progress along the road curvature from one intersection to the next one. Then, the velocity eld d θ (x, y) v max (x, y) can be computed ∀(x, y) ∈ Ω as:

d θ (x, y) v max (x, y) = Q q=1 smax s=1 w ( (x, y) -p(q, s) ) d θq v maxq Q q=1 smax s=1 w ( (x, y) -p(q, s) ) , (3.7)
where p(q, s) is the spatial coordinate of cell s of road q, and the weighting function w(l) : R + → R + is a decreasing function of the (Euclidean) distance, e.g., here we use the exponential function:

w(l) = e -µl and l = (x -p(q, s) x ) 2 + (y -p(q, s) y ) 2 , where µ is a weighting parameter that needs to be tuned according to the desired accuracy of reproducing the network structure in a 2D representation: for a small µ the velocity eld follows only the global trend of the network geometry, while for a large µ the velocity eld follows the roads in a detailed way. These two extreme cases are illustrated in Figure 3.1, where a small Manhattan grid area is taken as a network example.

In this work, we would like to capture the evolution of a 2D vehicle density quite accurately but without over-resolving the network geometry, for example, see Figure 3.2a) that illustrates the direction eld estimated for µ = 50 for a network representing the city center of Grenoble of the total area 1 × 1.4 km 2 . Thereby, we can also notice that the integral lines of the direction eld drawn in Figure 3.2b) do not cross. This results from the model structure, since the integral lines can be seen as unique solutions to the dierential equation governed by d θ . Moreover, we assume that there are no loops in the urban network, i.e., there exists a preferred direction of motion. Indeed, if there would be a loop, then there would be a point inside of every loop where θ is undened, since the direction lines cannot cross each other. Moreover, any loop would have no boundary, thus the cars following this path would never be created nor destroyed. The condition on not having loops plays an essential role in the coordinate transformation that will be explained in Section (3.2). In terms of integral lines, we require that any integral line of the directional eld d θ begins and ends at the boundary of the domain. In terms of network structure, we consider only networks (or urban areas) with uni-directional roads that are located such that no loops arise. This is the main restriction of the model (3.1), which limits its usability for general trac applications although being still useful in several situations. For example, imagine a rush hour (e.g., at 9 am), when many people are driving to the business center of the city located in some particular point, then the assumption of the preferred direction of motion is realistic. Thus, equation (3.7) constructs a velocity eld d θ (x, y) v max (x, y) at any point in the domain as a normalized weighted average of the road directions d θq v maxq such that the trac ow direction at some point is mostly impacted by the nearest roads to this point.

Kernel density estimation

To complete the denition of FD parameters, we also need to determine density-related parameters of the fundamental diagram, i.e., the critical density ρ c (x, y) and the maximal density ρ max (x, y) ∀(x, y) ∈ Ω. Let us rst concentrate on the maximal density ρ max (x, y), and then it will be straightforward to determine ρ c (x, y), if we know the particular FD shape, e.g., from real trac measurements.

In a 2D representation of trac, the maximal density depends not only on the number of lanes of particular roads but can also increase in areas with high concentration of roads. In order to estimate ρ max (x, y) ∀(x, y) ∈ Ω, we ll each road of the network by placing a vehicle at a minimum headway distance of 6 m, since this is an approximate distance between two consecutive vehicles in a trac jam. Thus, we place vehicles as close as possible to determine the maximal density by using the kernel density estimation (KDE).

The idea of this method is that each individual vehicle contributes to the total vehicle density as a Gaussian function with a kernel located around the vehicle position. The total estimated density then corresponds to the superposition of all their contributions. Let the position of a vehicle v ∈ {1, . . . , V (t)} be denoted by (x v (t), y v (t)) at some time t.

Then, the vehicle density can be estimated as follows:

ρ(x, y, t) = 1 2πd 2 0 V (t) v=1 e -1 2d 2 0 ((x-xv) 2 +(y-yv) 2 ) , (3.8)
where d 0 is a standard deviation of a Gaussian function. Note that Gaussians are used to preserve the conservation of vehicles, since density integrals are normalized to 1.

Parameter d 0 in (3.8) determines the range of impact of the Gaussian kernel that has to be chosen. For example, in Figure 3.3b) we can see how equidistant vehicles on a road contribute to the global density by its Gaussian functions. In the upper plot, each car has an impact on the density in the range of d 0 = 25 m around its position, which results into a constant density along the road. The lower plot illustrates the situation when the range of impact is set to d 0 = 100 m, which is too high, since then the reconstructed density has a bell shape due to boundary eects. There are several works regarding the optimal choice of this parameter, see [START_REF] Shimao | Data-tted generic second order macroscopic trac ow models[END_REF][START_REF] Fan | Comparative model accuracy of a data-tted generalized Aw-Rascle-Zhang model[END_REF]. The authors rely on the idea that the parameter d 0 should be chosen such that equidistant cars should provide constant density.

The same qualitative eects can be observed also in 2D, see [START_REF] Mollier | Two-dimensional macroscopic model for large scale trac networks[END_REF] for more details. Note that the maximal density ρ max can be estimated by KDE (3.8), with the only dierence being that all the vehicles are placed as densely as possible. An application example of KDE is illustrated in Figure 3.3a), where the density (colormap value) is obtained by using KDE from vehicles' positions denoted by blue dots. The positions of vehicles were generated using commercial software Aimsun that takes any network geometry as input and produces microsimulations of trac on this network with the possibility to specify boundary inows. Note that unlike in previous density representation in 1D, the colormap in Figure 3.3a) is used to denote the ratio of the density value to the maximal density over the whole network (absolute maximal density), i.e., ρ max = max (x,y)∈Ω ρ max (x, y). Moreover, due to the space-dependency of FD, the critical density ρ c (x, y) is dierent ∀(x, y) ∈ Ω.

Boundary conditions

To complete the 2D LWR model (3.1) that describes trac dynamics on a bounded domain, we need to introduce the boundary conditions, as we did in Section 2.1.4 for the 1D case.

Dene a set Γ ⊂ Ω as the boundary of a rectangular domain Ω. The boundary consists of two subsets Γ = Γ in ∪ Γ out . Thereby, Γ in is a set of boundary points (x, y) for which n(x, y) • d θ (x, y) > 0, where n(x, y) is a unit normal vector to the boundary oriented inside the domain. In a similar way, we also dene Γ out such that ∀(x, y) ∈ Γ out : n(x, y) • d θ (x, y) < 0. Now let us x boundary ows φ in (x, y, t) and φ out (x, y, t) for the 2D system given by (3.1) and formulate the following IBVP: where inows φ in (x, y, t) and outows φ out (x, y, t) are dened as φ in (x, y, t) = min {D(ρ in (x, y, t)), S (ρ(x, y, t))} , (x, y) ∈ Γ in φ out (x, y, t) = min {D (ρ(x, y, t)) , S (ρ out (x, y, t))} , (x, y) ∈ Γ out

               ∂ρ(x, y, t) ∂t + ∇ • Φ(x, y, ρ(x, y, t)) = 0, Φ(x, y, t) = φ in (x, y, t) d θ (x, y), ∀(x, y) ∈ Γ in φ out (x, y, t) d θ (x, y), ∀(x, y) ∈ Γ out ρ(x, y, 0) = ρ 0 (x, y),
(3.10)
where D(ρ) and S(ρ) are demand and supply functions dened as in Section 2.1.5 but depending on space and in two dimensions. The well-posedness of IBVP given by (3.9) and (3.10) will be discussed in Section 3.2.

The upstream Γ in and downstream Γ out boundaries are the ones that should be actuated when it comes to control applications. As an illustrative example, these boundaries are indicated by black arrows in Figure 3.4.

Comparison between 2D LWR and MFD-based models

This section is devoted to the comparison between the newly introduced continuum model in 2D (3.9) to reservoir models based on a macroscopic fundamental diagram, which are very popular in trac applications due to their simplicity. We seek to show that 2D conservation law models such as (3.9) have their own advantages. By running the same trac scenario on a Manhattan grid network with these two dierent approaches and comparing the steady state results to those predicted by microsimulator Aimsun, we will motivate the use of the 2D LWR model that can be a reasonable choice for many trac control applications in large urban networks.

Macroscopic fundamental diagram

As described in [3], MFD-based models (also known as reservoir models) describe the evolution of accumulation of vehicles in some urban zone. Let us consider a heterogeneous network partitioned into N reservoirs, e.g., N = 4 as illustrated in Figure 3.5. Let n i (t) be the accumulation of vehicles in reservoir i at time t. The main assumption of reservoir models is the existence of MFD φ i (n i (t)), which relates the number of cars in a reservoir i with the outow from this reservoir φ i,out . Let us also dene N in,i as a set of neighboring reservoirs, from which cars can directly reach reservoir i, and N out,i as a set of neighboring reservoirs that can be directly reached by cars from reservoir i, as illustrated in Figure 3.5. Then, the rate of change in the number of cars n i (t) in reservoir i is given by the dierence in its inow and outow, that is:

dn i (t) dt = φ in,i (t) -φ out,i (t), with φ in,i (t) = j∈N in,i r ji min {D j , S i } and φ out,i (t) = j∈N out,i r ij min {D i , S j } , (3.11)
where r ji and r ij are numbers of roads leading from reservoir j to reservoir i and from reservoir i to reservoir j, respectively. Demand D i and supply S i functions are dened as

D i (n i (t)) = φ i (n i (t)), if n i < n c,i φ max,i , if n i ≥ n c,i S i (n i (t)) = φ max,i , if n i ≤ n c,i φ i (n i (t)), if n i > n c,i , (3.12
) thereby, n c denotes the critical car number that has the same implication as the critical density ρ c in (3.4), i.e., we observe the free-ow regime if n i ≤ n c , otherwise it indicates that the congestion has occurred. We compute MFD for each reservoir by using the GPS data (velocities) from the microsimulator Aimsun at each t:

φ i (n i (t)) =   1 n i n i (t) m=1 v im        n i Q i q=1 L iq      = 1 Q i q=1 L iq n i (t) m=1 v im ,
where L iq corresponds to the length of road q ∈ {1, . . . , Q i } in reservoir i, and v im is the velocity of vehicle m in reservoir i. Note this expression is the product of average velocity and density in reservoir i. Having data as (n i , φ i ), we t a cubic polynomial as it was done in [3], and extract the maximal ow φ max,i . The nal step to complete the denition of MFD is to get the maximal car number obtained by counting cars placed in a reservoir at the minimal headway distance (6 meters).

[3] presented a method to perform a network partition depending on the trac state such that each part has its own well-dened MFD (low scattering of the MFD curve). However, to Manhattan grid network into N = 16 equal parts each having its own MFD, and then compare the steady states achieved by using two dierent models with the steady state obtained by using the microsimulator Aimsun for the same inow and outow data.

A steady state in a reservoir model is reached when the accumulation of vehicles stops changing its value over time, i.e., dn i (t)/dt = 0 ∀i ∈ {1, . . . , N }. Further, by (3.11) we obtain for each reservoir i that the number of cars is preserved whenever inow equals to outow:

φ * in,i (t) = φ * out,i (t), (3.13)
where the asterisk is used to denote a steady state.

We will compare steady states predicted by both models (3.1) and (3.11) with the one obtained with microsimulator Aimsun, which simulates the dynamics of vehicle positions in a given urban area. The vehicle positions are then used to reconstruct the density using the kernel density estimation method that was presented in Section 3.1.3.2.

To enable a quantitative comparison of steady states, we will compute the L 2 norm of the deviation of the density predicted by one of the models ρ * pred (x, y) from the ground true distribution ρ * sim (x, y) obtained by Aimsun in the steady state, i.e., ρ * (x, y) 2 , where ρ * (x, y) = ρ * pred (x, y) -ρ * sim (x, y).

The L 2 norm in 2D is computed as in (1.7). Note that in case of MFD-based model, ρ * pred (x, y) is a piecewise constant function obtained from the accumulation of vehicles in a zone multiplied by its area.

Scenario description

We consider a 10 × 10 Manhattan grid network with a total surface of 1 km 2 , which is drawn by grey lines on all three plots of Figure 3.6. Positions of nodes (intersections) are slightly disordered with white noise of standard deviation that equals 10 m. We assume that all roads are single-lane and are globally oriented towards the North-East direction (grey arrows in Figure 3.6 are used to point the direction of trac on each road). The network contains a topological bottleneck in the middle, e.g., a river with some bridges. The speed limits on most of the roads are set to 30 km/h, and there are also two roads with 50 km/h.

In order to obtain a non-trivial congestion pattern in the steady state, we create a congestion formation scenario in the Western part of the network, while the rest of the network should remain in the free-ow regime. We achieve that by setting appropriate inows (demand functions). The domain contains 15 incoming roads in total: 8 roads are coming from the North and 7 are coming from the West. We can identify the incoming roads from Figure We set the inow demand at the upstream boundary for the numerical simulation of the 2D model (3.9) by deploying the 1D kernel density estimation method. Namely, KDE is used to reconstruct the density created by vehicles entering the domain through the continuous boundary line. The numerical scheme for 2D LWR system given by (3.9) was discussed in [START_REF] Mollier | Two-dimensional macroscopic model for large scale trac networks[END_REF]. The brief idea is to perform dimensional splitting, and then for each dimension the numerical ux is computed using the Godunov scheme. However, the simulation result in Figure 3.6 was obtained with our own numerical method for this model that will be presented later in the next section. To produce the result depicted in Figure 3.6b), we perform a numerical simulation of vehicle density governed by a 2D model, until the steady-state is reached. It is also worth noting that the supply of the downstream boundary is set to φ max so that cars can freely leave the domain. Notice that the result in Figure 3.6b) was obtained with a low weighting parameter µ = 20 for the continuous approximation of velocity eld (see Section 3.1.3.1).

Thus, we run a dynamic scenario on Aimsun for 2 hours of simulation time setting the time-constant inow values indicated above. Thereby, we see that the shape of a congested zone does not change much after a certain simulation time indicating that the steady-state was reached. In order to set up a simulation, Aimsun requires also to dene turning ratios at each intersection. A turning ratio is assigned to a pair of roads i and j connected by a junction, and it denotes a percentage of vehicles that turn from road i to road j. On a global scale, turning ratios determine the overall trac ow direction. Since the applicability of the 2D LWR model (3.9) is limited to networks that have a preferred direction of motion, we set the turning ratios accordingly. Thus, at each 2 × 2 intersection, 75% of vehicles will turn and the rest 25% continue moving straightforward, while at each 1 × 2 intersection the turning ratios are set to 50%.

ρ * mfd (x, y) = n * i s i , where i : (x, y) ∈ R i , (3.14)
where s i is the area (in m 2 ) of reservoir with index i, and R i is the domain taken by this reservoir.

By comparing Figures 3.6a There exist also other arguments to prefer the 2D continuum model to MFD-based models in several situations. Thus, MFD-based models are discrete in space and, by their nature, they do not really enable us to develop model-based control approaches. This is related to the fact that the result of the network partitioning algorithm as in [3] depends on the current trac state for MFD-based models, since the main point thereby is to dene zones consisting of roads with similar congestion levels. Thus, if trac conditions change, e.g., a higher inow comes through roads that did not provide a high inow before, this might lead to invalid MFDs, which causes the necessity to perform the network partitioning again. Moreover, even for stationary inows, the performance of the model degrades as reservoirs' areas enlarge.

Thus, we have shown that the 2D LWR model is a benecial representation of trac, especially in a urban network with multiple congestion zones that may relocate in time. The 2D LWR model is a scalable model and it does not cost a high computational eort to be applied. Moreover, unlike MFD-based models, it is able to track the shape of congestion evolution quite well without the necessity to perform network partitioning. In the next Section 3.2 we will present a method to translate the 2D LWR model in a form that can be easily analyzed for a large variety of model-based control design tasks that will be considered in the current chapter.

Curvilinear coordinate transformation

The structure of the 2D LWR model (3.9) implies that the direction eld of trac ow d θ

given by (3.3) depends only on the network geometry and not on the state. This enables us to describe the trac ow trajectories that do not change with time. These trajectories are obtained by building tangents to the direction eld d θ . This gives us the integral lines illustrated in Figure 3.2b).

In the following, we will perform a curvilinear coordinate transformation that translates these integral curves into a set of straight parallel lines as illustrated in Figure 3.7.

Afterwards, a trac state evolving along a straight line can be treated as a 1D system, which would signicantly simplify any analysis of the 2D system (3.9).

General idea

Let us assume that angle θ ∈ C 1 (Ω). We introduce new coordinates (ξ, η) in a dierential form:

dξ dη = C(x, y)R θ (x, y) dx dy , (3.15)
where R θ (x, y) is a rotation matrix given by R θ (x, y) = cos (θ(x, y)) sin (θ(x, y)) -sin (θ(x, y)) cos (θ(x, y)) ,

and C(x, y) is a diagonal scaling matrix given by

C(x, y) = α(x, y) 0 0 β(x, y) , (3.17)
where α(x, y) and β(x, y) are positive and bounded scaling parameters needed for the existence of the coordinate transformation (will be dened later in this section).

Thus, matrix R θ (x, y) provides the rotation of the integral lines in (x, y)-plane, and the scaling matrix C(x, y) acts such that these lines have the same metric, see Figure 3.7. In Figure 3.7a) we have used the topological structure of Grenoble downtown (the same as in Figure 3.2) where the direction at each road is set such that loops and ow crossings are impossible, i.e., all roads need to be uni-directional and there exists a preferred direction of motion. Thus, on a global scale, the motion on this network is oriented towards North-East of the city.

Intuition: straight lines

In the case of straight lines depicted in Figure 3.7b) we have θ = 0 ∀(x, y) ∈ Ω, which implies that the rotation (3.16) and scaling matrices (3.17) become identity matrices, i.e., C = R θ = I. Then, by (3.15) the new coordinates (ξ, η) would completely coincide with (x, y) up to a constant shift. In this case, the direction eld dened in (3.3) becomes d θ (ξ, η) = (1, 0), and by (3.2) we obtain:

Φ = Φ(ξ, η, ρ) 1 0 , (3.18)
which can be inserted into the divergence term in (3.9) resulting into:

∂ ∂ξ , ∂ ∂η 1 0 Φ(ξ, η, ρ) = ∂Φ(ξ, η, ρ) ∂ξ . (3.19)
Notice that in case of straight lines, the divergence (3.19) contains only one term instead of two as it was in the original system with curved trajectories (3.9). Thus, the trac ow evolves only along ξ coordinates, which are tangent to the ow motion. At the same time there is no motion in the orthogonal direction of η, which can be treated as a parameter (a label numbering the ow path). Afterwards, we can treat each such line of constant η as a 1D

equation, for which we will be able to solve dierent control tasks.

Curvilinear coordinate transformation

After providing an intuitive explanation on how this coordinate transformation should work, let us rst dene scaling parameters α and β from (3.17). Then, we will be able to perform the coordinate transformation of the original 2D system (3.9) in order to turn it into a continuous Thus, α(x, y) and β(x, y) being functions of angle θ(x, y) only, can be computed from the network geometry. In Figure 3.8 we illustrate the role of these parameters in the coordinate transformation by considering a single line of constant η. As we can see, α and β are used to scale the distances between the lines of constant ξ and between the lines of constant η, respectively. In (ξ, η)-space the ow evolves only along lines of constant η as in (3.19).

Model in (ξ, η)-space

According to Chapter 2 of [9], we can apply the divergence formula to calculate ∇ • Φ in (ξ, η)-space:

∇ • Φ(ξ, η, ρ) = 1 h ξ h η   ∂ Φ ξ h η ∂ξ + ∂ Φ η h ξ ∂η   , (3.24)
where h ξ and h η are known as Lamé coecients, which correspond to the lengths of the basis vectors in (ξ, η)-space: (3.26)

h ξ = ∂x ∂ξ , ∂y
The combination of (3.26) and (3.25) yields the basis vectors in (ξ, η)-space:

h ξ = 1 α cos θ sin θ , h η = 1 β
-sin θ cos θ .

(3.27)

We then nd the lengths of these vectors (3.27), which gives us Lamé coecients:

h ξ = | h ξ | = 1 α , h η = | h η | = 1 β . ( 3
where e x and e y are the normalized basis vectors of (x, y)-space, and e ξ and e η are the normalized basis vectors of (ξ, η)-space.

Let us now rewrite vector Φ given by (3.2) in (ξ, η)-space. Notice that in (x, y)-space this vector reads:

Φ(x, y, ρ) = Φ(x, y, ρ) cos(θ(x, y)) e x + Φ(x, y, ρ) sin(θ(x, y)) e y .

(

Then, by using (3.29) we obtain:

Φ(ξ, η, ρ) = Φ(ξ, η, ρ) e ξ . (3.31)
Having Lamé coecients (3.28) and the ux vector in (ξ, η)-space (3.31), we nalize the calculation of the divergence term (3.24) in (ξ, η)-space as:

∇ • Φ(ξ, η, ρ) = α(ξ, η)β(ξ, η)
∂(Φ(ξ, η, ρ)/β) ∂ξ . For simplicity, we also introduce some new functions by scaling density, ows, demand and supply functions as:

ρ = ρ αβ , Φ = Φ β , φin = φ in β , φout = φ out β , S = S β , D = D β . (3.33)
Finally, the last thing that needs to be claried prior to rewriting the 2D LWR system (3.9) in (ξ, η)-space, is the denition of a spatial domain, on which the system in new coordinates will evolve. Thus, the new spatial domain Ω is a compact domain dened as:

Ω = {(ξ, η) : ∃ (x, y) ∈ Ω, ξ = ξ(x, y), η = η(x, y)} .
Then, the domain boundary in (x, y)-space can be uniquely projected into the boundary in (ξ, η)-space, i.e., Γ Ω → ΓΩ. In particular, ΓΩ consists of points (ξ min (η), ξ max (η)) such that ξ min (η) = min Now, using the divergence term in (ξ, η)-space (3.32), we can rewrite the 2D LWR system (3.9) that now reads ∀(ξ, η, t)

∈ Ω × R + :                ∂ ρ(ξ, η, t) ∂t + ∂ Φ(ξ, η, ρ) ∂ξ = 0, φin (η, t) = min D (ρ in (η, t)) , S (ρ (ξ min (η), η, t)) , φout (η, t) = min D (ρ (ξ max (η), η, t)) , S (ρ out (η, t)) , ρ(ξ, η, 0) = ρ0 (ξ, η), (3.34)
where Φ(ξ, η, ρ) is now a scalar function that preserves all the FD properties such as being Lipschitz continuous and concave, e.g., consider the Greenshields FD in (ξ, η)-space:

Φ(ξ, η, ρ) = vmax (ξ, η) 1 - ρ ρmax (ξ, η) ρ, where vmax = αv max , ρmax = ρ max αβ . (3.35)
The general rule in scaling functions is the following: all functions that have ow units (φ in , φ out , S(ρ), D(ρ), Φ(ρ), φ max ) have to be divided by β, all density-related functions (ρ, ρ max and ρ c ) must be divided by αβ, and velocity-related functions (v max in case of Greenshields FD and v, ω in case of triangular FD) must be multiplied by α. Note that also here the demand D(ρ) and supply S(ρ) functions are dened as in Section 2.1.5 but depending on (ξ, η)-space and in two dimensions.

We can see that the trac ow evolves now only along lines of constant η in (ξ, η)-space.

Thus, the system in new coordinates (3.34) should be seen as a continuous set of 1D LWR equations each following a path parametrized by η. This means that we can also analyze its solution in the same way as we do it in case of 1D LWR. Namely, in system (3.34) shocks arise when characteristics cross at some point of space, and thus we need to consider its solution in a weak sense, and then the unique (entropy) solution is the one that satises the Lax condition (see Section 2.1.3). Moreover, to guarantee that the weak solution ρ(ξ, η, t) is the entropy one ∀(ξ, η, t) ∈ Ω × R + , one needs to consider the boundary conditions in the weak sense, see Section 2.1.4. Notice, that the boundary conditions φin (η, t) and φout (η, t) in (3.34) are formulated using the demand-supply concept (see Section 2.1.5 for the explanation). Thus, the initial boundary value problem (3.34) is well-posed, see more details in [START_REF] Bürger | Dierence schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic trac ow model[END_REF][START_REF] Wu | Steady-state trac ow on a ring road with up-and down-slopes[END_REF] for entropy conditions for inhomogeneous LWR model.

Let us now summarize all the steps that need to be performed in order to be able to describe the evolution of trac in (ξ, η)-space by system (3.34) in some urban area.

1. As an input, we get some urban network as a collection of roads and junctions with known coordinates in (x, y)-space, speed limits and the number of lanes.

2. We dene a rectangular plane Ω such that the corresponding urban network is contained in it. Then, we apply approximations to nd all variables and FD parameters ∀(x, y) ∈ Ω: the inverse distance weighting to nd the vector eld d θ (x, y)v max (x, y) (see Section 3.1.3.1), then the kernel density estimation to obtain ρ max (x, y) (see Section 3.1.3.2).

3. Apply 1D kernel density estimation to the boundary conditions given as inows and outows of particular roads of the city.

4. Given road orientation angle θ(x, y) ∀(x, y) ∈ Ω obtained as arctangent of the velocity eld, calculate functions α(x, y) and β(x, y) by solving PDEs (3.20) and (3.21).

5. Using α(x, y) and β(x, y), calculate new coordinates (ξ, η) ∈ Ω by numerical integration of (3.15).

6. Rescale all the FD parameters, density-and ow-related variables as in (3.33). For example, demand and supply functions at the boundaries should be rescaled as:

Din (η) = D in (η) β(ξ min (η), η) , Sout (η) = S out (η) β(ξ max (η), η)
.

For the rest of this chapter we will be always referring to the system written in (ξ, η)-space (3.34). Thus, with a slight abuse of notations, we will omit bars for all the variables from (3.34), however leaving the notations for domains Ω and its boundary Γ ∈ Ω.

Numerical scheme

Since the system (3.34) is essentially just a set of 1D LWR equations, its numerical solution is found using the same Godunov scheme as described in Section 2.1.6. The dierences emerge from the dependency on the additional dimension η and also from the space-dependency of FD parameters ρ max (ξ, η) and v max (ξ, η). For convergence results for the Godunov scheme applied to kinematic wave systems with space-dependent fundamental diagrams see [START_REF] Bürger | Dierence schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic trac ow model[END_REF].

We start by dening a numerical grid in Ω × R + by setting m to be number of cells to discretize η dimension, ∆η = (η max -η min )/m to be the space cell size in η dimension, η j = η min + j∆η to be the grid point in η dimension for j ∈ {1, ..., m}, ∆ξ to be the space cell size in ξ dimension, n j = (ξ max (η j ) -ξ min (η j ))/∆ξ to be the number of cells in ξ dimension for particular η = η j for j ∈ {1, ..., m}, ξ ij = ξ min (η j ) + i∆ξ to be the grid point in ξ dimension for i ∈ {1, ..., n j } and j ∈ {1, ..., m}, ∆t to be the time cell size, t k = k∆t for k ∈ Z + to be the grid point in time.

Similarily to Section 2.1.6, the mesh sizes ∆ξ and ∆t are chosen sucht that they satisfy the Courant-Friedrichs-Lewy condition [START_REF] Courant | On the Partial Dierence Equations of Mathematical Physics[END_REF]:

∆t max (ξ,η,ρ) |Φ (ξ, η, ρ)| ≤ ∆ξ 2 .
The discrete density is then ρ i,j (k), and according to the Godunov scheme, we update it ∀j ∈ {1, . . . , m}, ∀i ∈ {1, . . . , n j } and ∀k ∈ Z + as follows:

                   ρ 1,j (k + 1) = ρ 1,j (k) + ∆t ∆ξ (ϕ in,j (k) -ϕ 2,j (k)) , ρ i,j (k + 1) = ρ i,j (k) + ∆t ∆ξ (ϕ i,j (k) -ϕ i+1,j (k)) , ρ n j ,j (k + 1) = ρ n j ,j (k) + ∆t ∆ξ ϕ n j ,j (k) -ϕ out,j (k) , (3.36)
where ϕ i,j (k) is the Godunov numerical ux dened as

ϕ i,j (k) = min {D(ρ i-1,j (k)), S(ρ i,j (k))} , (3.37)
with D(ρ i-1,j (k)) and S(ρ i,j (k)) being the discretized demand and supply functions same as in Section 2.1.6 except that the numerical ux in (3.36) has space-dependent parameters ρ max (ξ, η) and v max (ξ, η), which should be used for the computation of D(ρ i-1,j (k)) and S(ρ i,j (k)) in (3.37).

The boundary ows ϕ in,j (k) and ϕ out,j (k) from (3.36) are determined by specifying the density at the ghost cells with indices i = 0 and i = n j + 1 for j ∈ {1, . . . , m}:

ϕ in,j (k) = min {D(ρ 0,j (k)), S(ρ 1,j (k))} , ϕ out,j (k) = min D(ρ n j ,j (k)), S(ρ n j +1,j (k)) . (3.38)

Hamilton-Jacobi formulation

Let us now consider the parametrized set of 1D LWR equations with space-dependent FD (3.34) (2D LWR in curvilinear coordinates) in Hamilton-Jacobi formulation. The Hamilton-Jacobi formalism here is quite similar to the one presented in Section 2.1.7. The only dierence here is that we must carefully handle the space-dependency of FD and the additional space parameter η ∈ [η min , η max ] that is just used as a label of the ow path rather than the second space dimension. Thus, let us again consider the cumulative vehicle number function M (ξ, η, t) that is an integral function of ow in time or density in a 2D space (see (2.21) for 1D). This integral function can be expressed through domain outows as in (2.23), if the starting point of the integration is set to the downstream boundary of ow path η at initial time, i.e., M (ξ max (η), η, 0) = 0. Then, the Moskowitz function in the 2D plane is dened

∀(ξ, η, t) ∈ Ω × R + as M (ξ, η, t) = t 0 φ out (η, τ )dτ + ξmax(η) ξ ρ( ξ, η, t)d ξ.
(3.39) Also we can express the Moskowitz function through domain inows and initial density distribution as in (2.24), if the starting point is set to the upstream boundary of η-line at initial time, i.e., M (ξ min (η), η, 0) = 0. In this case we obtain:

M (ξ, η, t) = ξmax(η) ξ min (η) ρ 0 ( ξ, η)d ξ + t 0 φ in (η, τ )dτ - ξ ξ min (η) ρ( ξ, η, t)d ξ. (3.40)
The relation of H-J formulation to the LWR formulation is the same as discussed in Section 2.1.7. Thus, the H-J PDE with space-dependent Lipschitz continuous Hamiltonian can be obtained from the space-dependent ow-density relation Φ(ξ, η, ρ) = φ(ξ, η, t), where the ow and density functions are then replaced by formulas similar to (2.21).

Let us introduce the following initial boundary value problem ∀(ξ, η, t) ∈ Ω × R + in Hamilton-Jacobi formulation:

               ∂M (ξ, η, t) ∂t -Φ ξ, η, - ∂M (ξ, η, t) ∂ξ = 0, M (ξ, η, 0) = M Ini (ξ, η), M (ξ min (η), η, t) = M Up (η, t), M (ξ max (η), η, t) = M Down (η, t).
(3.41)

The main advantage of the H-J PDE is that we can indeed formulate its solution in terms of a minimization problem. For several shapes of Hamiltonian (for example, triangular FD), the solution to the minimization problem can be found explicitly.

Solution of the H-J IBVP (3.41) can be obtained analytically in accordance with the variational principle using only its boundary and initial conditions, which can be encoded in the general value condition function c. For 1D it was already done in (2.26), however, in 2D it has a dierent set of departure, i.e., c(ξ, η, t) : Dom(c) → R + , where

Dom(c) = (ξ, η, t) : η ∈ [η min , η max ], ξ ∈ {ξ min (η), ξ max (η)} , t ∈ R + ∪ {(ξ, η, 0) : η ∈ [η min , η max ], ξ ∈ [ξ min (η), ξ max (η)]} .
This function generalizes the initial M Ini (ξ, η) and boundary conditions M Up (η, t) and M Down (η, t) of (3.41) that are then used for the computation of the inmum problem:

c(ξ, η, t) =     
M Ini (ξ, η), t = 0, M Up (η, t), ξ = ξ min (η), M Down (η, t), ξ = ξ max (η).

(3.42)

Now we specify the value condition function (3.42) by calculating M Up (η, t), M Down (η, t) and M Ini (ξ, η). We proceed in a similar way as in Section 2.1.8. Thus, the upstream boundary condition M Up (η, t) can be obtained by considering (3.40) for ξ = ξ min (η), which results into

M Up (η, t) = c(ξ min (η), t) = t 0 φ in (η, τ )dτ + ξmax(η) ξ min (η) ρ 0 ( ξ, η)d ξ, ∀(η, t) ∈ [η min , η max ] × R + . (3.43)
Then, the downstream boundary condition M Down (η, t) can be expressed from (3.39) for ξ = ξ max (η):

M Down (η, t) = c(ξ max (η), t) = t 0 φ out (η, τ )dτ, ∀(η, t) ∈ [η min , η max ] × R + . (3.44)
Finally, the initial condition M Ini (ξ, η) can be expressed from either (3.40) or (3.39) for t = 0, ∀v ∈ [-ω(ξ, η), v(ξ, η)] :

L(ξ, η, v ) = sup ρ∈[0,ρmax(ξ,η)] (Φ(ξ, η, ρ) -v ρ), (3.46)
where v(ξ, η) and -ω(ξ, η) are related to the maximal and minimal kinematic wave speeds in free-ow and congested trac regime (not necessarily as in triangular FD). This function achieves minimum, if an observer moving in a trac stream adapts his/her individual speed to the maximal kinematic wave speed (see the discussion in Section 2.1.8).

Finally, the closed-form solution to (3.41) corresponding to the inmum among all viable evolutions that start at initial time t -T and arrive at (ξ, η) at terminal time t reads as:

M (ξ, η, t) = inf (T,v )∈S   c ξ(0), η, t -T + T 0 L ξ(τ ), η, v (τ ) dτ   , (3.47)
where the inmum is taken over domain S dened as:

S = (T, v ) T ∈ R + , v (•) ∈ L 1 (0, T ), ξ(τ ) = v (τ ), ξ(T ) = ξ, v (τ ) ∈ -ω ξ(τ ), η , v ξ(τ ), η , ξ(0), η, t -T ∈ Dom(c) . (3.48)
Here ξ(τ ) denotes the trajectory of an observer moving along a trac stream with possibly non-constant speed v (τ ) unlike in Lax-Hopf formula (2.32), since now we consider spacedependent FDs that include inhomogeneity of the network infrastructure. Trajectory ξ(τ ) originates at τ = 0 on a boundary of the domain of c and arrives at the point ξ at terminal time τ = T .

As already mentioned, in case of a triangular FD (3.4) the solution to H-J PDE (3.41) can be found explicitly. We show the derivation of the explicit solution in Appendix B.5, where the solution was considered for large enough time such that the eect of initial conditions is negligible. This result will be then used in Section 3.5, where we consider a boundary control problem for trac in a mixed regime evolving on a large urban network and prove its exponential convergence to the desired trajectory.

Equilibrium manifolds

Analysis of steady states emerging in large urban networks is of a key importance in understanding trac dynamics. In particular, steady states need to be studied to enable comparison of dierent models or to solve optimal control tasks of driving a state to some desired equilibrium. This section is devoted to the model-based estimation of steady states for trac density evolving on arbitrary large-scale urban networks. The trac state is governed by a two-dimensional conservation law (3.9).

In the previous Section 3.2, we presented the curvilinear coordinate transformation for the 2D conservation law model that could be translated into a parametrized inhomogeneous 1D LWR system (3.34), i.e., each such 1D system incorporates space-dependency in the ux function that is related to bottlenecks and varying speed limits along the trac ow path.

This section demonstrates the rst analytic result that can be easily obtained for this kind of systems. In particular, we present the rst model-based steady state estimation result for large trac networks, which became possible due to this curvilinear coordinate transformation.

Thereby, we will use only the information about the network geometry and infrastructure parameters, as well as demand and supply at network boundaries. For a 1D inhomogeneous case (one road with bottlenecks) this was done in [START_REF] Wu | Steady-state trac ow on a ring road with up-and down-slopes[END_REF], who used the wave entropy conditions derived in [START_REF] Zhang | Hyperbolic conservation laws with space-dependent ux: I. Characteristics theory and Riemann problem[END_REF] to ensure the physically relevant solution. We will rely on this previous result [START_REF] Wu | Steady-state trac ow on a ring road with up-and down-slopes[END_REF] to extract the correct density from the steady state ow, which provides the entropy solution of system (3.34).

Problem statement

Let us consider the parametrized inhomogeneous PDE system (3.34). Assume that demand at the upstream boundary and the supply at the downstream boundary are constant in time,

i.e., D(ρ in (η)) and S(ρ out (η)) are given ∀η ∈ Ω. We seek to develop a technique that yields the steady state of (3.34) analytically. This is formalized as follows:

Problem 3.1

Given system (3.34) with constant demand and supply functions D(ρ in (η)) and S(ρ out (η)) ∀η ∈ Ω, nd a time-invariant density distribution ρ * (ξ, η) such that that is:

∂Φ * (ξ, η, ρ * ) ∂ξ = 0, ∀(ξ, η) ∈ Ω.
φ * (η) := Φ * (ξ, η, ρ * ).
Recall that, in general, trac operates at maximum eciency when Φ(ξ, η, ρ) = φ max (ξ, η), i.e., when the trac conditions allow to exploit roads at their capacity. However, φ max (ξ, η) can not be the steady state ow, since it should not depend on ξ. The equilibrium trac ow can not exceed the capacity of the worst bottleneck along its path (line of constant η). If trac conditions do not allow that (e.g., congested trac along the whole η line), we need to mind the boundary conditions as well. Assume that demand at the upstream boundary D(ρ in (η)) and supply at the downstream boundary S(ρ out (η)) are given ∀η ∈ [η min , η max ]. In accordance with the analysis performed in [START_REF] Wu | Steady-state trac ow on a ring road with up-and down-slopes[END_REF], we obtain that the steady state ow along its path is the minimum of three functions

φ * (η) = min{D(ρ in (η)), φ min max (η), S(ρ out (η))}, (3.50)
where φ min max (η) is the transportation capacity at the strongest bottleneck along the η-line dened as:

φ min max (η) = min ξ∈[ξ min (η),ξmax(η)] φ max (ξ, η), (3.51)
and the point where the minimum is achieved is the location of the strongest bottleneck denoted by ξ * (η):

ξ * (η) = argmin ξ∈[ξ min (η),ξmax(η)]
φ max (ξ, η).

(3.52)

If there are several points ξ * or it is an interval, then we take the left-most value (the rst point on the ow path), i.e., ξ * = ξ * Thus, the steady state trac ow along a line of constant η is the minimum between the demand at its entry, the supply at its exit and the minimum bandwidth (3.51), which is determined by the strongest bottleneck of the η-line. On a physical road, such a bottleneck can be caused by a reduced number of lanes or by a lower speed limit. Recall that lines of constant η are used to describe the trac ow path, and they are dierent from the physical roads as these lines are dened on a continuum 2D plane using approximations of all parameters (see Sections 3.1.3.1 and 3.1.3.2). Thus, the strongest bottleneck is determined by the level of compression of roads, i.e., the smallest capacity is achieved in areas with highly compressed roads. This dependency is incorporated in the scaling factor β(ξ, η), since recall that every term in (3.50) was divided by it (as in (3.33), and then the bars were omitted to simplify the notations).

As a next step, we need to nd the corresponding steady state density. Due to the concavity of the fundamental diagram (see Figure 2.1), for each ow value (except the maximal ow φ max ), there exist two densities corresponding to this ow: the lower value denotes the freeow trac regime, and the higher value denotes the congested regime.

Based on the result obtained by solving the minimum (3.50), we can distinguish three possible cases:

1. φ(η) * = φ min max (η). Then, the steady state ρ * should be chosen to guarantee the congested regime ∀ξ ∈ [ξ min (η), ξ * (η)), while it must provide the free-ow regime ∀ξ ∈ (ξ * (η), ξ max (η)]. This is the only solution satisfying the wave entropy condition for inhomogeneous roads (space-dependent FDs) as presented in [START_REF] Wu | Steady-state trac ow on a ring road with up-and down-slopes[END_REF][START_REF] Zhang | Hyperbolic conservation laws with space-dependent ux: I. Characteristics theory and Riemann problem[END_REF]. This means that the strongest bottleneck creates congestion, and after passing it, vehicles can move freely. As mentioned above, such bottlenecks can be caused by highly compressed roads (characterized by a high scaling parameter β(ξ, η)), low maximal density ρ max (ξ, η) (e.g., on a river's bridge), or low speed limits v max (ξ, η).

2. φ(η) * = D(ρ in (η)). This implies that the demand to enter this road is too small, and all cars can pass through the system freely. Therefore, the whole domain is in the free-ow trac regime and ξ * (η) = ξ min (η).

3. φ(η) * = S(ρ out (η)). This implies that the supply at the exit of this road is too low, and the cars get blocked there. The strongest bottleneck is at the exit of η-line, i.e., ξ * (η) = ξ max (η). Therefore, the whole domain is in the congested trac regime.

Notice that the steady state ρ * is obtained by taking the inverse of the fundamental diagram, and the correct trac regime providing the entropy solution of (3.34) should be set depending on the steady state ow (3.50), as discussed above for three possible cases. As a nal step, we need to rescale the density back as:

ρ(ξ, η) = ρ(ξ, η)α(ξ, η)β(ξ, η),
which allows us to compare steady states obtained by a numerical simulation of (3.9) and by performing a model-based analysis of (3.34) (see next Section 3.3.3).

Steady state example

Let us now demonstrate a steady state example that can be obtained by following the steps described in Section 3.3.2. For this, let us take a synthetic 10 × 10 Manhattan network as described earlier in Section 3.1.6. The demand at the upstream boundary D(ρ in (η)) is also set as in Section 3.1.6, and the supply at the downstream boundary S(ρ out (η)) = φ max (ξ max (η), η), i.e., all vehicles can leave the domain freely. Further, we discretize η dimension into m = 180 cells. Following the steps described in Section 3.3.2, we obtain a steady state for a parametrized inhomogeneous 1D LWR system (3.34) shown in Figure 3.9b). The continuous approximation was again performed for a low weighting parameter µ = 20. We thus seek to capture only the global trend of the velocity eld in this example.

We compare now this steady state to the one obtained by running a numerical simulation of trac density governed by a 2D LWR model (3.9), which is illustrated in Figure 3.9a).

Thereby, we use the Godunov scheme in 2D presented in Section 3.2.5 for an unscaled system (3.34), which is the same system as (3.9) with the only dierence that it is written in dierent coordinates. Additionally, we also compare the obtained steady state distribution to the one that results from running the microsimulator Aimsun, see Figure 3.9c). Recall that Aimsun produces vehicle trajectories, and then we use the kernel density estimation (see Section

Discussions

In this section, we demonstrated the rst result that can be obtained by analysing the 2D LWR model in (ξ, η)-coordinates. The rewritten model (3.34) represents a parametrized 1D LWR model with a space-dependent FD, where the second dimension is used to label the trac ow path. We described how to obtain its steady state that corresponds to a space-varying density distribution by following two steps: rst, solving the minimum (3.50) between the demand ow at the upstream boundary, the supply ow at the downstream boundary and the capacity at the strongest bottleneck, and second, by extracting the density satisfying the wave entropy We then provided an example for a steady state that can be obtained by this model-based analysis, which was then compared to the previous results obtained by running a congestion formation scenario for 2 hours of simulation time. For this, we referred to Section 3.1.7, where the steady-state distribution obtained by simulating a trac density governed by the 2D LWR (3.34) in (x, y)-coordinates was compared to the reference steady state distribution predicted by microsimulator Aimsun. Thus, the analytically obtained steady state from the model in (ξ, η)-coordinates (3.34) appeared to provide quite accurate results by capturing the shape of trac congestion even better in comparison to the result obtained numerically. There are two main advantages of model-based steady state prediction: rst, it saves a lot of computational time, since there is no more need to run simulations until the steady state is achieved, and second, being an explicit result it can be used to solve control related tasks for trac in large urban networks. This will be shown in the following section, where the explicitly estimated steady state will be used as a desired equilibrium to reach via a boundary control.

Boundary control for congested areas

In this section, we seek do design a boundary control for a congested area within a large urban network using the same modeling approach as in Section 3.3. Thus, we again describe the trac state by its density whose temporal evolution is given by the 2D LWR model rewritten in curvilinear coordinates (3.34). We will consider a urban network that includes congested areas that will be controlled from their downstream boundary as shown in Figure 3.10. The control should drive the trac system to the equilibrium that provides the maximal throughput of the system. This stabilized system is then characterised by a reduced average latency and a higher average velocity. Our main contribution here is to suggest a model-based control design technique that requires only the knowledge about the network geometry and its infrastructure, i.e., speed limits and transportation capacities. This is the rst work of this kind for two-dimensional trac systems providing an explicit solution to the problem.

First, we will discuss the desired equilibrium to be achieved in a congested urban area.

Then, the boundary control result will be presented. Finally, the theoretical results will be veried with the help of a numerical example, where we demonstrate the performance of the designed controller.

Optimal equilibrium

In this section, our main goal is to design a boundary controller that can drive a 2D trac system governed by (3.34) to a steady state providing the maximal throughput of the system.

Thereby, we rely on the steady state analysis from the previous Section 3.3.

General steady states

Recall that a steady state ρ * (ξ, η) implies space-independent φ * (η), which can be achieved only for time constant D(ρ in (η)) and S(ρ out (η)). We obtained that the steady state ow along the line of constant η (3.50) is the minimum of demand at the entry, supply at the exit and the capacity of the strongest bottleneck located at ξ * (η) (3.51). By bottlenecks we mean permanent capacity constraints in the network itself, e.g., a road segment with low speed limit or with a few lanes (see Figure 3.11).

Thus, from the steady state ow given by (3.50), we need to extract the steady state density ρ * that provides the physically relevant solution (entropy solution), which was already discussed in Section 3.3.2. Here we need to consider the minimum (3.50) of only two functions, as the demand at the upstream boundary D(ρ in (η)) is always larger than capacity at the bottleneck. This happens, since ρ in (η) is assumed to be very high for all η-lines, which implies by (2.13) that D(ρ in (η)) = φ max (ξ min (η), η), ∀η ∈ [η min , η max ].

(3.53)

If supply at the downstream boundary is also larger S(ρ out (η)) > φ min max , then ρ * should be chosen to provide the congested regime ∀ξ ∈ [ξ min , ξ * ), and the free-ow regime occurs ∀ξ ∈ (ξ * , ξ max ]. If there are several such ξ * (or it is an interval), then we take the left-most value, i.e., ξ * = ξ * 1 in Figure 3.11. If S(ρ out (η)) is smaller than the capacity at the bottleneck, then the whole domain is in the congested trac regime.

Optimal steady state

Here we consider congested urban areas, and thus the inow demand is assumed to be very high as in (3.53). This also means that the minimum function in the demand-supply problem (2.16) is resolved to the supply at the domain exit, which is treated as a control variable.

Thus, we control the area outow from its downstream boundary, i.e., u(η) = S (ρ out (η)) ∀η ∈ [η min , η max ] (as it was done in Section 2.2 but now it is on a 2D domain).

From (3.50) it is clear that the maximal throughput of the system in the equilibrium is achieved for φ * (η) = φ min max (η) for all η-lines. In order to provide a steady state that ensures the maximal throughput, we can actuate the downstream boundary accordingly, i.e., u(η) = φ min max (η) . However, this control would lead to the violation of the congested regime, since the wave entropy condition prescribes the free-ow trac regime ∀ξ ∈ (ξ * (η), ξ max (η)], where ξ * (η) is given by (3.52) (see (3.50) and the discussion above). This is a situation that we would like to avoid, since this section deals exclusively with congested areas for mathematical simplicity.

Thus, we would like to dene a desired steady state ow to be as close as possible to the maximal possible steady state ow (determined by the capacity at the strongest bottleneck) that still respects the constraint on the congested trac regime in the whole area. For this purpose, we introduce a small constant > 0 (small ow), and then the desired steady state ow can be dened as

φ d (η) = φ min max (η) -, ∀η ∈ [η min , η max ]. (3.54)
By setting the control variable u(η) = φ d (η), we translate the bottleneck to the end of the η-line, i.e., ξ * (η) = ξ max (η). In this case, we guarantee that the congested trac regime is preserved within the whole interval [ξ min (η), ξ max (η)], see Figure 3.11. This allows us to control the system from the exit, and this control is applied in the strong sense, since the whole system is assumed to operate in one trac regime (as in Section 2.2).

From the practical viewpoint, subtraction of does not change much the desired state, since can be set to an arbitrarily small value. Thus, in the following we will call the desired state an -optimal equilibrium w.r.t. throughput maximization. Note that controlling the domain exit can be physically realized by installing, e.g., trac lights.

ξ max ξ * 1 ξ * 2 ξ min ρ Φ(ρ) φ min max - ρ Φ(ρ) ρ Φ(ρ)
S(ρ out ) The desired -optimal equilibrium ρ d (ξ, η) w.r.t. the throughput maximization is dened ∀(ξ, η) ∈ Ω as

ρ d (ξ, η) = ρ max (ξ, η) 2 + ρ 2 max (ξ, η) 4 - ρ max (ξ, η) v max (ξ, η) φ d (η), (3.55)
where φ d (η) is dened in (3.54) and > 0, see Figure 3.11.

Note that (3.55) was obtained by taking the inverse of (3.35) for Φ(ξ, η, ρ d ) = φ d (η), which leads us to the quadratic formula with two possible roots. To provide the congested trac regime, we need to pick the plus sign. where ρ(ξ, η, t) is the L 2 norm of the deviation from the desired equilibrium (3.55). The L 2 norm and the error term are dened as in (1.7) and (1.9), respectively, but in (ξ, η)-space.

Boundary control design

Theorem 3.1

The boundary control problem of driving a congested urban area to the desired -optimal equilibrium (3.55), as formulated in Problem 3.2, is solved with (3.60)

u(η) = φ d (η), where φ d (η) = φ min max (η) -, ∀η ∈ [η min , η max ]. ( 3
We consider the most right-hand-side term and linearize the ux function around the desired state as follows:

Φ (ξ, ρ d + ρ) ≈ Φ (ξ, ρ d ) + ∂Φ (ξ, ρ d ) ∂ρ ρ, (3.61)
which being inserted in (3.60) yields

∂ ρ(ξ, t) ∂t = - ∂Φ(ξ, ρ d ) ∂ξ - ∂ (Φ (ξ, ρ d ) ρ) ∂ξ , (3.62)
where the prime denotes Φ = ∂Φ/∂ρ.

Recall that, in general, the conservation law prescribes that ∂ρ(ξ, t) ∂t = -∂Φ(ξ, ρ) ∂ξ .

Hence, if we consider a time-constant density ρ d (ξ), then by the conservation law we obtain ∂Φ(ξ, ρ d )/∂t = 0. This allows us to simplify (3.62) to

∂ ρ(ξ, t) ∂t = - ∂ (Φ (ξ, ρ d ) ρ(ξ, t)) ∂ξ . (3.63)
To simplify the notations, we omit the arguments of ρ(ξ, t) and insert (3.63) into (3.59), which yields 2 ∂ξ dξ.

V (t) = - ξmax ξ min e ξ ρ ∂ (Φ (ξ, ρ d ) ρ) ∂ξ dξ = - ξmax ξ min e ξ 2Φ (ξ, ρ d ) 2Φ (ξ, ρ d ) ρ ∂ (Φ (ξ, ρ d ) ρ) ∂ξ dξ = - ξmax ξ min e ξ 2Φ (ξ, ρ d ) ∂ (Φ (ξ, ρ d ) ρ)
(3.64)

We now consider Φ (ξ, ρ d ), which is obtained by taking a derivative of (3.35) w.r.t. density:

Φ (ξ, ρ d ) = v max (ξ) 1 - 2ρ d (ξ) ρ max (ξ) . (3.65)
In order to estimate an upper bound of (3.64), we will evaluate the derivative Φ at ξ * , which is the location of the bottleneck. Note that being the derivative of a concave function, Φ

achieves its maximum at the bottleneck in the congested regime (in the free-ow regime it is vice versa).

First, let us obtain the desired density at ξ * using (3.35). Recall that by (2.5), in general, the capacity is given by φ max = v max ρ max /4, which lets us write:

Φ (ξ * , ρ d (ξ * )) = φ max (ξ * ) -⇒ Φ (ξ * , ρ d (ξ * )) = v max (ξ * )ρ max (ξ * ) 4 -.
By using (3.35), this can be further rewritten as:

v max (ξ * ) ρ d (ξ * ) - v max (ξ * ) ρ 2 d (ξ * ) ρ max (ξ * ) = v max (ξ * ) ρ max (ξ * ) 4 - ⇒ ρ 2 d (ξ * ) -ρ d (ξ * ) ρ max (ξ * ) + ρ 2 max (ξ * ) 4 - ρ max (ξ * ) v max (ξ * ) = 0 ⇒ ρ d (ξ * ) = ρ max (ξ * ) 2 + ρ max (ξ * ) v max (ξ * ) . (3.66)
Recall that in the solution of the quadratic equation, we need to choose the plus sign to respect the congested trac regime. Thus, we insert (3.66) into (3.65) and introduce a variable ν used to denote Φ at the bottleneck:

Φ (ξ * ) = - v max (ξ * ) ρ max (ξ * ) = -ν. (3.67)
Notice that Φ in (3.67) has the same physical meaning as velocity, which can be seen from its physical units by having in mind that is measured in [veh/s], see (3.54).

Let us now again use the arguments of ρ and η. We can bound (3.64) from above using (3.67):

V (η, t) ≤ 1 2ν ξmax(η) ξ min (η)
e ξ ∂ (Φ (ξ, η, ρ d ) ρ(ξ, η, t)) 2 ∂ξ dξ.

(3.68)

Integration by parts of (3.68) yields

V (η, t) = e ξmax(η) 2ν(η) Φ 2 (ξ max (η), η, ρ d ) ρ2 (ξ max (η), η, t) - e ξ min (η) 2ν(η) Φ 2 (ξ min (η), η, ρ d ) ρ2 (ξ min (η), η, t) - 1 2ν(η) ξmax(η) ξ min (η) e ξ Φ (ξ, η, ρ d ) ρ (ξ, η, t)
2 dξ.

(3.69)

The last term in (3.69) can be again bounded by ν(η) as follows:

-

1 2ν(η) ξmax(η) ξ min (η) e ξ Φ (ξ, η, ρ d ) ρ (ξ, η, t) 2 dx ≤ - ν(η) 2 ξmax(η) ξ min (η)
e ξ ρ2 (ξ, η, t) dξ = -ν(η)V (η, t).

V (η, t) = - e ξ min (η) 2ν(η) Φ 2 (ξ min (η), η, ρ d ) ρ2 (ξ min (η), η, t) -ν(η)V (η, t).
Thus, we have proved the L 2 convergence of ρ (ξ, η, t) to the desired -optimal equilibrium ρ d (ξ, η) as t → ∞ ∀η ∈ [η min , η max ]. It also follows that the pointwise convergence in η is achieved, which implies the L ∞ convergence in η. In bounded spaces (which is the case for η-space) this also implies the L 2 convergence in η. This proves the asymptotic L 2 convergence in the whole (ξ, η)-space.

Numerical example

Now let us demonstrate how this boundary control law (3.57) provides the convergence to the desired equilibrium with the help of a numerical example. For this purpose, we will again take a synthetic Manhattan grid network as in Section 3.1.6. The only dierence is a larger noise Thus, we will apply the boundary control to a fully congested urban area with the initial density distribution given by ρ 0 (ξ, η) = ρ max (ξ, η), ∀(ξ, η) ∈ Ω.

There are a lot of vehicles at the upstream boundary of this area, i.e., ρ in (η, t) = ρ max (ξ min (η), η) ∀(η, t) ∈ [η min , η max ] × R + . These vehicles permanently provide a maximal possible inow into the system, that is D(ρ in (η)) = φ max (ξ min (η), η). This trac jam distribution is illustrated in Figure 3.12b). Thereby, the dierences in the heatmap are caused by the variation of ρ max (ξ, η) along the domain. Thus, more yellow zones are the those characterized by a low maximal density, which is usually achieved in areas with low concentration of roads.

The desired -optimal steady state given by (3.55) is illustrated in Figure 3.12a) for = 10 -5 veh/s. Notice that the desired density distribution is space-dependent, which is caused by the variety of the infrastructure in the considered domain, e.g., inhomogeneous distribution of roads, dierent speed limits, etc. Recall that this desired distribution provides the maximal possible throughput of the system at equilibrium up to a small constant that is introduced to guarantee that the vehicle density is always larger than the critical value (congested trac regime).

For the numerical simulation, we rst discretize the domain by η into m = 180 cells, and then the Godunov scheme (3.36) is implemented for every constant η. The boundary conditions are assigned to the ghost cells, which are the cells that do not belong to the domain (see Section 3.2.5 for more details). Thus, for the congested system the boundary ows (3.38) in the numerical scheme are set to φ in,j = S(ρ 1,j ), φ out,j = u(j), ∀j ∈ {1, . . . , m}, where u j is the boundary controller (3.57) that was shown to provide the convergence to the desired steady state. The performance of this controller is shown in Figure 3.12c), from which we observe that the spatial L 2 norm of the error from the desired equilibrium converges to zero in nite time. The nite time convergence can be explained by the fact that in a linear trac system (obtained if we consider only one regime, as we did here and in Section 2.2), the boundary condition is propagated in the whole domain with the characteristic line that has a nite propagation speed.

Discussions

In this section, we considered large-scale urban networks from the control point of view. In particular, we again used the 2D LWR model rewritten in curvilinear coordinates (3.34), and demonstrated how it can be used for control design. The control goal was to drive a fully congested area to the equilibrium state characterized by the maximal throughput of the system, which also implies shorter traveling times. The maximal throughput at each line of constant η (ow path in a continuum plane) is constrained from above by the capacity of its strongest bottleneck. For instance, imagine a road (or η-line in our terms) that consists of segments characterized by dierent speed limits, e.g., 30 km/h and 50 km/h. Then, the steady state ow is constant along the road, and its value is determined by the capacity of the road segment with the lowest speed limit.

To simplify the problem mathematically, we restricted this part to trac being only in the congested regime. This allows us to consider a linear problem (as it was done in Section 2.2), which is a set of transport PDEs parametrized by η with space-dependent FD. This simplication allows us to consider boundary conditions in a strong sense, and moreover, we do not have to handle discontinuities in the solution. We provide the congested regime by adding a small constant , and subtract it from the desired equilibrium ow, which corresponds to the maximal throughput minus . Hence, we call the desired state the -optimal state w.r.t. throughput maximization. Notice that this constant was introduced for mathematical simplicity, and its value can be arbitrarily small. Thus, the desired state can still be seen as the equilibrium of (almost) maximal throughput.

The control design should be realized by actuating only the downstream boundary of the congested domain. It again relies on the model (as in the previous section) and requires only the information about the network geometry and its infrastructure parameters. The controller (3.57) includes only the feedforward component, since the curvilinear coordinate transformation and the restriction to only one trac regime allowed us to considerably simplify the 2D network control problem. Lyapunov methods were used to prove the exponential convergence to the desired equilibrium. Finally, we demonstrated the performance of the boundary controller on an example of a heavily congested network with a large inow demand at its entry. The L 2 norm of the error term showed a nite time convergence.

In the next section, we are going to extend the boundary control problem to the mixed regime trac, which implies considering a nonlinear PDE for a urban network (3.34).

Boundary control for mixed regime trac

In this section, we consider control problem for large-scale urban networks with mixed regime trac. Thereby, we again rely on the 2D LWR model resulting from the curvilinear coordinate transformation (3.34), i.e., an assumption on uni-directional trac must still hold. Unlike in the previous Section 3.4, here the trac state satises a fully nonlinear PDE system without being restricted to any particular regime. Thus, we now consider a much more general problem that poses a lot of technical issues to handle due to discontinuities in the solutions and weak boundary conditions.

We design a boundary control law for some uni-directional urban transportation area explicitly by relying only on intrinsic model properties and network geometry. The main contribution of this section is to present the rst explicitely derived boundary controller for a 2D conservation law model that is able to track a space-and time-dependent trajectory that admits discontinuities in its solutions. To make this possible, we use the Hamilton-Jacobi framework as it was done in Section 2.3, but extending it to 2D and handling space-dependency of the fundamental diagram, see Section 3.2.6 for a general theory on Hamilton-Jacobi PDE with space-dependent Hamiltonians. This means that instead of the classical Lax-Hopf formula (2.32), we have to apply the viability theory to the solution of a Hamilton-Jacobi-Moskowitz problem with a space-dependent Hamiltonian explained in [START_REF] Désilles | Viability approach to Hamilton-Jacobi-Moskowitz problem involving variable regulation parameters[END_REF][START_REF] Aubin | Viability Theory: New Directions[END_REF].

Problem statement Problem 3.3

Our objective is to design boundary control laws u in (η, t) and u out (η, t) ∀(η, t) ∈ [η min , η max ]× R + such that the vehicle density ρ(ξ, η, t) given by the system (3.34) tracks a desired trajectory as t → ∞.

In Section 2.3 a similar problem was posed for a single homogeneous road (see Problem 2.3 and Lemma 2.2) but there was no space-dependency in the FD. Here we extend this result for a large urban area whose infrastructure is captured by the space-dependency in the FD, which makes its solution more technically involved. Throughout this section we make the following assumptions:

Assumption 3.1

Inows φ in (η, t) and outows φ out (η, t) of the 2D trac system (3.34) must satisfy the follow-

ing inequalities ∀(η, t) ∈ [η min , η max ] × R + φ in (η, t) ≤ φ min max (η), φ out (η, t) ≤ φ min max (η), (3.72)
where φ min max (η) is the transportation capacity at the strongest bottleneck along the η-line dened in (3.51).

Moreover, there exists ε > 0 such that φ in (η, t) and φ out (η, t) additionally satisfy:

t+tctr(η) t φ in (η, τ )dτ ≤ t ctr (η)φ min max (η) -ε and t+tctr(η) t φ out (η, τ )dτ ≤ t ctr (η)φ min max (η) -ε, (3.73)
where t ctr (η) is the minimal controllability time for η-line, i.e., the time needed for a solution evolving from one end of η-line to reach the opposite end: Here we consider a vehicle density given by system (3.34). To analyze this system for a boundary control task (as it was done in the proof of Theorem 2.3), one can obtain its solution in explicit form for the equivalent H-J system (3.41

t ctr (η) = min      ξmax(η) ξ min (η) 1 v( ξ, η) d ξ, ξmax(η) ξ min (η) 1 ω( ξ, η) d ξ     . ( 3
  1 +      1 ξmax(η) ξ min (η) ρ max ( ξ, η) + ρ c ( ξ, η) d ξ        ,
M (ξ, η, t) = min t-Tv(ξ,η) 0 φ in (η, τ ) dτ + ξmax(η) ξ min (η) ρ 0 ( ξ, η) d ξ, t-Tω(ξ,η) 0 φ out (η, τ ) dτ + ξmax(η) ξ ρ max ( ξ, η) d ξ , (3.75)
where

T v (ξ, η) = ξ ξ min (η) 1 v( ξ, η) d ξ, T ω (ξ, η) = ξmax(η) ξ 1 ω( ξ, η) d ξ.
(3.76)

Recall that t ∈ [t min , +∞) implies that the eect of initial conditions has left the system (see Remark 3.1).

Remark 3.2

We widely use the solution (3.75) obtained in H-J formalism to analyze the properties of system (3.34) in order to design the boundary control. The major reason lies in weak boundary conditions given by (3.37), which imply that not any control can be imposed at the boundaries at any time. Thus, (3.75) is used to estimate time periods during which controls might not be accepted by the system in terms of control restriction functions, as it was done in Section 2.3.

Boundary control design

u in (η, t) = D (ρ in (η, t)) , u out (η, t) = S (ρ out (η, t)) .
Proof of Theorem 3.2. The proof shows that the MF solutions converge pointwise up to a constant shift as t → ∞, see Lemma 2.2 and Remark 2.4 for implications of this convergence.

Thus, the procedure here is the same as in the proof of Theorem 2.3 for a problem in 1D apart from a few dierences listed in Appendix B.6.

Remark 3.3

Note that the integral convergence of densities stated in Theorem 3.2 implies that the state approximates the desired trajectory as time goes to innity, since a and b can be arbitrarily close in space, i.e., ρ ≈ ρ d as t → ∞.

Numerical example

Here we demonstrate the eciency of our boundary controller (3.77) applied to trac evolving on a urban network with geometry as in Grenoble downtown. The total surface of the chosen Grenoble area is approximately 1.4 × 1 km 2 . We track a desired density prole that is spacedependent and periodic in time. The geometry of the studied area in Grenoble is shown in grey in Figure 3.13. The directions of trac motion on roads were however modied for this example (numerically) such that all roads are uni-directional. Thus, there exists some global direction of trac ow towards North-East of the city and no loops are allowed, which is exactly how it is illustrated in Figure 3.7. The speed limits on roads are taken from real Grenoble network data: some roads can be driven with 30 km/h, and others can be driven with 50 km/h.

We dene a numerical grid in Ω × R + and deploy the Godunov scheme in 2D, as described in Section 3.2.5. First, discretize the η dimension into m = 180 cells. Then, we use the 2D Godunov scheme (3.36) for every j ∈ {1, . . . , m} with a discretization step ∆ξ = 5 m (space cell size in ξ dimension). We also set the time cell size ∆t = 0.1 s, which provides that the CFL condition is satised. In order to compute the integral related to the feedback term in (3.77) we perform the Riemann summation for every j ∈ {1, . . . , m} over all ξ cells, i.e., i ∈ {1, . . . , n j }, where n j is the number of ξ cells contained in each cell j.

Recall that triangular FD is characterized by ρ c = ρ max /3. The initial vehicle density distribution is given ∀(ξ, η) ∈ Ω by ρ 0 (ξ, η) = ρ max (ξ, η).

We set the inow demand D (ρ in d (η, t)) and the outow supply S (ρ out d (η, t)) in the desired system to be time-periodic functions: u in (η, t) = φ in d (η, t) and u out (η, t) = φ out d (η, t).

D (ρ in d (η, t)) =
In Figure 3.13 the evolution of trac density within the time interval of 2τ = 4800 seconds is shown, i.e., 2 time periods of ρ d (ξ, η, t). The middle column illustrates the evolution of density controlled with the gain k = 5 • 10 -5 , i.e., strategy 1). The left column corresponds to the density evolution using only the boundary conditions of the desired system, i.e., strategy 2).

The right column is related to the time-periodic desired density trajectory with the boundary conditions as described above. We can observe the convergence to the desired proles for the case with feedback that becomes visible already at t = 2τ , while this does not happen for the case without feedback. Notice that all the density distributions are drawn in (x, y)-coordinates in Figure 3.13, i.e., we had to rescale the functions and to perform the back transformation from (ξ, η)-space to (x, y)-space.

In Figure 3.14, the L 1 norm of the error in the number of cars is depicted as a function of time for dierent control gains. The density error ρ(x, y, t) is dened as in (1.10), and its L 1 norm can be computed as in (1.6). We can clearly see that a higher control gain k = 10 -3 provides a higher convergence speed in comparison to a controller with a smaller gain k = 5 • 10 -5 . On the contrary, k = 0 will not achieve the goal even if we would start from an empty city without any cars. This could work only if there is absolutely no dierence in the initial conditions with the desired prole, which is hardly ever possible.

a) t = 0 b) t = 0 c) t = 0 d) t = 0.5τ e) t = 0.5τ f ) t = 0.5τ g) t = 1τ h) t = 1τ i) t = 1τ j) t = 2τ k) t = 2τ l) t = 2τ

Discussions

In this section, a boundary control technique was presented for a mixed regime trac density evolving on a large urban network with a preferred direction of motion. The control goal was formulated in a similar way as in Section 2.3 but in two dimensions, which caused additional technical diculties. The viability solution of the Hamilton-Jacobi PDE with space-dependent Hamiltonian (3.75) was used to prove Theorem 3.2 stating that the desired trajectory is approximated even if controls can not be directly imposed at the boundaries, i.e., we are able to handle weak boundary conditions in 2D using control restriction functions as in Section 2.3.

Approximating desired density trajectory implies that the number of vehicles tracks pointwise the desired number of vehicles. Thus, from the practical view point, this control goal has even more sense than pointwise tracking of the desired density.

The controller (3.77) is applied at all boundaries of the urban area, and it acts as to track a space-and time-dependent trajectory that can be in any trac regime. Its performance has been veried with the help of a numerical example using the geometry of an area in Grenoble downtown. Thereby, the initial density distribution corresponded to a trac jam.

We compared two control strategies: without the feedback part and with it. As expected from the theoretical results, feedback plays an essential role in tracking the desired density prole in the mixed trac regime. Moreover, the control gain aects the convergence speed.

Trac control using variable speed limit

Let us now demonstrate how to solve control tasks using a variable speed limit (VSL) in a 2D-plane by stating a new problem in (ξ, η)-space.

We consider the following IBVP on the same bounded domain (ξ, η, t)

∈ Ω × R + :                ∂ρ(ξ, η, t) ∂t + ∂Φ(ξ, η, ρ(ξ, η, t), u) ∂ξ = 0, φ in (η, t, u) = min {D (ξ min , η, ρ in (η, t), u) , S (ξ min , η, ρ (ξ min , η, t) , u)} , φ out (η, t, u) = min {D (ξ max , η, ρ (ξ max , η, t) , u) , S (ξ max , η, ρ out (η, t), u)} , ρ(ξ, η, 0) = ρ 0 (ξ, η), (3.78)
where the ux function Φ now depends also on a control parameter u ∈ [0, 1] that represents the variable speed limit ratio: no VSL is applied if u = 1, and no movement is allowed if u = 0.

Applying variable speed limits should be understood as a exible (temporary) restriction on speed at which vehicles can drive on a given stretch of road. The speed limit varies according to the current environmental and road conditions and is displayed on electronic trac signs.

Setting u = 1 implies that vehicles can drive at speeds bounded by the legal maximum (e.g., 130 km/h on French highways, or by the comfort zone of drivers on German highways).

Note that ux Φ is still a concave function with respect to ρ, and Φ is continuous in u.

Moreover, Φ(ξ, η, ρ, 0) = Φ(ξ, η, 0, u) = 0. One should see u as the in-domain controller that aects the trac ow. It is applied in the whole domain including its boundaries. Therefore, the demand and supply functions in (3.78) have u as an additional argument.

Contributions

The material presented in this section was inspired by a previous work [START_REF] Karafyllis | Feedback control of scalar conservation laws with application to density control in freeways by means of variable speed limits[END_REF]. However, there are four major points that were not considered in [START_REF] Karafyllis | Feedback control of scalar conservation laws with application to density control in freeways by means of variable speed limits[END_REF], and thus will be addressed here:

1. 2D systems : this is the rst time that VSL control is applied on a large transportation network directly using the intrinsic properties of the model only. Hence, the VSL controller is designed by analysing the structure of a 2D conservation law (3.78) without any discretization that needs to be done to obtain a numerical solution.

2. Space-dependent diagrams : we extend the result of [START_REF] Karafyllis | Feedback control of scalar conservation laws with application to density control in freeways by means of variable speed limits[END_REF] by considering space-dependent diagrams, which imply space-dependent desired equilibrium proles.

3. Realistic FDs : in [START_REF] Karafyllis | Feedback control of scalar conservation laws with application to density control in freeways by means of variable speed limits[END_REF] it was assumed that ∂Φ(ξ, η, ρ, u)/∂u > 0 holds, see Figure 3.15a).

This assumption was made for simplicity to avoid multi-valued functions, i.e., there is only one value of u for each ow φ. In this section, we omit this condition by allowing more general forms of FD. In general, applying speed limits (u < 1) can cause a shift of the critical density towards larger values in realistic fundamental diagrams. This is

ρ Φ(ρ) a) 0 ρ c ρ max 0 ρ Φ(ρ) b) ρ c 1 ρ c 2 ρ c 3 ρ max f (ρ) φ 1 φ 2 Figure 3
.15: Fundamental diagrams and their dependence on speed limits: a) monotonic dependence ∂Φ(ξ, η, ρ, u)/∂u > 0 used in [START_REF] Karafyllis | Feedback control of scalar conservation laws with application to density control in freeways by means of variable speed limits[END_REF]; b) dependence we assume here, i.e., possible increase of ρ c when stronger speed limits are applied (from real data, see [24]). Blue line: u = 1. Red line: u = 0.7. Green line: u = 0.5. Bold dashed line: maximal ow function dened in (3.80). schematically depicted in Figure 3.15b), see red FD for u = 0.7 and green FD for u = 0.5 and compare ρ c3 and ρ c2 with ρ c1 achieved with u = 1. This means that applying speed limits can increase the range of vehicle density, for which the free-ow regime is preserved. There it is also shown how VSL can enhance trac ow for some given densities in the congested regime, e.g., compare ows φ 2 with φ 1 that can be achieved with dierent speed limits for the same vehicle density ρ c2 . These VSL eects on the shape of FD were revealed from data obtained due to a real-life experiment conducted on a European VSL-equipped motorway, see [24]. In general, we have no restrictions on how FD must depend on VSL apart from Φ(ξ, η, ρ, 0) = 0, i.e., the ux function is zero if there is no movement allowed. 4. Investigate the smoothness of VSL controller : considering such a general class of fundamental diagrams may lead to irregular control policies. We investigate whether any conditions must be imposed on the functional dependence of FD on VSL in order to provide smoothness.

Problem statement

Let us rst introduce the following notations: ,

and now we can formulate the stabilization problem as follows.

Problem 3.4

Given ∀(ξ, η) ∈ Ω the fundamental diagram Φ(ξ, η, ρ, u) and the initial density ρ 0 (ξ, η) with dynamics governed by (3.78), nd a VSL controller u(ξ, η, t) such that

lim t→∞ ρ(ξ, η, t) = 0, ∀(ξ, η) ∈ Ω (3.79)
where ρ(ξ, η, t) is the deviation from a desired equilibrium ρ d (ξ, η) ∈ (0, ρ max (ξ, η)).

VSL control design

Let us dene a maximal ow function f (ξ, η, ρ), which is the maximum possible ow that can be achieved at a given space point for a given vehicle density over all the VSL values (see the thick dashed line in Figure 3.15):

f (ξ, η, ρ) = max u∈[0,1]
Φ(ξ, η, ρ, u).

(3.80)

We also introduce a multi-valued function G(ξ, η, ρ, φ), which is the inverse image of the fundamental diagram with respect to the speed limit: where the control gain γ is a positive constant dened as

G(ξ, η, ρ, φ) = {u ∈ [0, 1] : Φ(ξ, η, ρ, u) = φ}.
0 < γ < min η    ξmax(η) ξ min (η) ρ max ( ξ, η)d ξ   -1
.

Then there exists c = c(γ, ρ 0 ) > 0 such that for every ρ 0 ∈ C 1 ( Ω) the system (3.78) with initial condition ρ(ξ, η, Note that the VSL in-domain controller u(ξ, η, ρ) depends on the state, i.e., it is a feedback control law. Let us give several comments on the controller structure:

0) = ρ 0 (ξ, η) has a unique solution ρ ∈ C 1 ( Ω × R + ), which satises max (ξ,η)∈ Ω |ρ(ξ, η, t)| ≤ e -ct max (ξ,η)∈ Ω |ρ(ξ, η, 0)|, ∀t ∈ R + , ( 3
1. Such a controller choice (3.82) assures that the system ow is immediately set to the desired ow φ d , i.e.:

Φ (ξ, η, ρ, u) = φ d (ξ, η, ρ), ∀(ξ, η, t) ∈ Ω × R + .
2. The desired ow φ d (ξ, η, ρ) is designed such that it does not exceed the maximal ow function f (ξ, η, ρ) in any point, i.e., ∀(ξ, η, t) ∈ Ω × R + . The space-dependency of the desired ow is incorporated into function B(ξ, η, ρ).

3. The function B(ξ, η, ρ) is constructed in such a way that it acts as a feedback linearization for system (3.78). Thus, the system loses the conservation law structure, and we do not have to handle discontinuities in the solution. This will be shown later in the proof of Theorem 3.3.

4.

The lower and upper bound on control gain γ are set such to guarantee that function B(ξ, η, ρ) is positive, i.e., B : Ω × R + → R + . The upper bound on γ is required for situations when the density error ρ has a negative value, which can appear since we design a general controller that drives any state to any desired equilibrium.

Proof of Theorem 3.3. First of all, we need to prove that the controller given by (3.82) is welldened. Namely, we will show that the set G(ξ, η, ρ, φ d (ξ, η, ρ)) is not empty, i.e., the desired ow takes values in a bounded range that can be achieved by the VSL control. Indeed, for all (ξ, η) ∈ Ω we get from (3.82) that

φ d (ξ, η, ρ) B(ξ, η, ρ) = min ξ f (ξ , η, ρ) B(ξ , η, ρ) ≤ f (ξ, η, ρ) B(ξ, η, ρ) , (3.85)
and, thus, by the positivity of function B(ξ, η, ρ) (see item 4 in Remark 3.4), we get φ d (ξ, η, ρ) ∈ [0, f (ξ, η, ρ)] ∀(ξ, η, t) ∈ Ω × R + . This interval exactly corresponds to the range of the ux function Φ(ξ, η, ρ, u(ξ, η, ρ)) w.r.t. u, therefore the set function G(ξ, η, ρ, φ d (ξ, η, ρ))

is not empty. Now we substitute the constructed ux function

Φ(ξ, η, ρ, u(ξ, η, ρ)) = B(ξ, η, ρ) min ξ f (ξ , η, ρ) B(ξ , η, ρ) (3.86)
into IBVP (3.78) and obtain:

∂ ρ(ξ, η, t) ∂t + min ξ f (ξ , η, ρ) B(ξ , η, ρ) ∂B(ξ, η, ρ) ∂ξ = 0.
Then, if we insert the denition of function B(ξ, η, ρ) from (3.82), this equation can be further simplied as

∂ ρ(ξ, η, t) ∂t = -γ ρ(ξ, η, t) min ξ f (ξ , η, ρ(ξ , η, t)) B(ξ , η, ρ) . (3.87)
This equation does not contain any partial space derivatives, and thus the controller really acts such that the conservation law structure is lost. Moreover, this dynamic equation has a stable equilibrium at zero. By [START_REF] Karafyllis | Feedback control of scalar conservation laws with application to density control in freeways by means of variable speed limits[END_REF], we obtain an exponential convergence to the desired equilibrium with rate c > 0, which depends on controller gain γ and the maximal ow function f dened in (3.80).

Finally, we see that the convergence of densities ∀(ξ, η) ∈ Ω ρ(ξ, η, t) → ρ d (ξ, η) as t → +∞ implies that function B(ξ, η, ρ) → 1, and thus (3.86) results into

Φ(ξ, η, ρ(ξ, η, t), u(ξ, η, ρ)) → min ξ f (ξ , η, ρ d (ξ , η)),
which coincides with (3.84), and thus concludes the proof. Remark 3.5 Property (3.84) means that the highest possible equilibrium constant ow is achieved for a given ρ d (ξ, η). Namely, by denition of (3.80), the following double inequality holds ∀η ∈

[η min , η max ] min ξ Φ(ξ, η, ρ d , 1) ≤ min ξ f (ξ, η, ρ d ) ≤ φ min max (η), (3.88)
where φ min max (η) is the capacity at the strongest bottleneck along the η-line (3.51). The left inequality in (3.88) implies that the same or higher trac ow can be achieved with lower speed limits than for u = 1. Thus, any VSL controller in the system can provide at most the ow min ξ f (ξ, η, ρ d ), which is indeed achieved by controller (3.82) due to the property (3.84).

Smoothness of VSL controller

The VSL controller (3.82) is dened via inclusion, and in general it can result in a discontinuous function in space. For example, imagine that two dierent speed limits are able to provide the desired trac ow. In this case, our fear would be that the speed limits jump from one value to another along the road innitely many times. However, if we assume additional properties on how the ux function should depend on the speed limit, we will obtain that u(ξ, η, ρ) is dierentiable almost everywhere. Theorem 3.4 Assume that ∀(ξ, η) ∈ Ω, ∀ρ ∈ [0, ρ max (ξ, η)] and ∀u ∈ [0, 1] the ux function Φ(ξ, η, ρ, u) is dierentiable. Moreover, assume that it is either twice dierentiable and strictly concave in u (congested regime) or monotonic in u and reaches its maximum at u = 1 (free-ow regime). Then using controller provided in Theorem 3.3 and assuming ρ ∈ C 1 ( Ω) ∀t > 0, we can choose the speed limit function u(ξ, η, ρ) such that it is dierentiable almost everywhere w.r.t. ξ.

Remark 3.6

This additional assumption on the functional dependence of Φ(ξ, η, ρ, u) on u can be interpreted as follows. When trac is in the congested regime and speed limit decreases, the trac ow can rst increase for a xed value density as illustrated in Figure 3.16b), and then it drops to zero as the speed limit approaches zero. On the contrary, when trac is in the free-ow regime, the ow of vehicles is maximal if there are no speed limits (u = 1), and when speed limits are applied the ow decreases monotonically as u decreases, see Figure 3.16a).

Remark 3.7

Notice that by Theorem 3.3, vehicle density is a dierentiable function ρ ∈ C 1 ( Ω) ∀t ∈ R + if the initial condition function of system (3.78) is dierentiable, i.e., ρ 0 ∈ C 1 ( Ω).

Proof of Theorem 3.4. For the proof, we x time t and line η. Let us consider an interval of all possible ξ values and split it in two subsets H 1 and H 2 as:

[ξ min (η), ξ max (η)] =H 1 ∪ H 2 ,
where

H 1 = ξ ∈ [ξ min (η), ξ max (η)] ∂Φ(ξ, ρ(ξ), u(ξ, ρ)) ∂u = 0 , H 2 = ξ ∈ [ξ min (η), ξ max (η)] ∂Φ(ξ, ρ(ξ), u(ξ, ρ)) ∂u = 0 .
We introduce also further subsets of H 1 and H 2 that correspond to their interiors:

E 1 = int(H 1 ), E 2 = int(H 2 ).
Moreover, we introduce a complementary subset E 0 as

E 0 = (H 1 \ E 1 ) ∪ (H 2 \ E 2 ), such that E 0 ∪ E 1 ∪ E 2 = H 1 ∪ H 2 = [ξ min (η), ξ max (η)].
It is clear that sets E 1 and E 2 have the same Lebesgue measure as sets H 1 and H 2 , respectively. Thus, the set E 0 is of measure zero. This means that showing that the controller function u(ξ) = u(ξ, ρ(ξ)) is dierentiable on sets E 1 and E 2 would imply that it is dierentiable almost everywhere. Let us rst consider set E 1 with the following function dened from (3.86):

F 1 (ξ, u) = Φ(ξ, ρ(ξ), u) -B(ξ, ρ(ξ))k, where k = min ξ f (ξ , ρ(ξ )) B(ξ , ρ(ξ ))
.

This function is dierentiable by the assumptions made in Theorem 3.4 and is equal to zero by (3.86). Moreover, the derivative of Φ(ξ, ρ(ξ), u) with respect to u is non-zero on set E 1 by its denition. This immediately implies that the derivative of F 1 (ξ, u) with respect to u is also non-zero. Therefore, we can use the Implicit Function Theorem, which assures that there exists a dierentiable function u(ξ) on this set satisfying (3.86).

In the second set E 2 we dene another function as

F 2 (ξ, u) = ∂Φ(ξ, ρ(ξ), u) ∂u .
Notice that F 2 (ξ, u) is zero by the denition of set E 2 , and it has a negative derivative with respect to u, since we assumed concavity of the ux function for the congested trac regime (in a pure free-ow regime set E 2 would be empty). This means that we can use the Implicit Function Theorem again, thus a dierentiable function u(ξ) exists on set E 2 as well.

Finally, combining these results, we obtain that the controller function u(ξ) is dierentiable on E 1 ∪ E 2 , i.e., almost everywhere. Proposition 3.1. In case of concave dependence of FD on speed limits, u(ξ, η, ρ) can sometimes be chosen from two values G(ξ, η, ρ, φ d ) for ρ being in congested regime, see Figure 3.16 b). Then, the most appropriate choice from the practical point of view is the minimal value, since it provides the free-ow trac regime: u(ξ, η, ρ) := min{G(ξ, η, ρ, φ d )}.

As an example, consider the intersection point (black dot) in Figure 3.15b) corresponding to the ow-density pair that can be achieved using either u = 1 or u = 0.7. In this case, we should choose u = 0.7, since this provides the free-ow regime and, thus, a more smooth trac motion.

To conclude, we have shown that the VSL controller is dierentiable almost everywhere if the fundamental diagram depends on u is a special way, i.e., monotonically increasing function of u in free-ow regime and a concave function of u in congested regime. In order to be able to apply the designed VSL controller (3.82) in practice (or in our case, it will be a numerical example), we should rst discuss ux functions depending on u by suggesting an explicit relation satisfying assumptions made in Theorem 3.4.

Parametrization of fundamental diagram

Let us assume that the basic shape of FD is triangular as in (3.4), which should be modied due to the dependence on speed limits. We denote v 1 (ξ, η) and ω 1 (ξ, η) as kinematic wave Chapter 3. Uni-Directional Trac on Networks speeds for u = 1 in the free-ow and in the congested regime, respectively. We can assume a linear dependence of kinematic wave speeds on speed limits, e.g., v(ξ, η, u) = u v 1 (ξ, η), ω(ξ, η, u) = ω 1 (ξ, η) + (1 -u)ω add (ξ, η), (3.89) where ω add (ξ, η) is an additional value expressing the eect of speed limit on the kinematic wave speed in the congested regime. This value is bounded and will be dened later. Thus, if speed limits are high (u 1), drivers are moving slowly, and therefore start braking late (larger safety distance for lower speeds). Let us estimate the range of reasonable values for

ω add (ξ, η) such that ∀(ξ, η) ∈ Ω ∂φ max (ξ, η, u) ∂u ≥ 0. (3.90)
Condition (3.90) means that it is not possible to enhance the transportation capacity by applying speed limits, see (3.88). This comes from the fact that φ max is determined by the number of lanes and free-ow kinematic wave speed, which depends on the legal maximum speed that takes specic values depending on a country and road type. Thus, the transportation capacity is a property of urban network geometry and it should not be changed with a variable speed limit. In the following, we skip the dependence on (ξ, η) for simplicity of notations. We insert ω(u) and v(u) from (3.89) into the denition of φ max for triangular FD (2.3) and get

φ max (u) = v 1 ρ max u (ω 1 + (1 -u)ω add ) ω 1 + v 1 u + (1 -u)ω add . (3.91)
We take the partial derivative of (3.91) w.r.t. u and obtain

∂φ max (u) ∂u = v 1 ρ max (ω 1 + (1 -u) ω add ) 2 -u 2 v 1 ω add (ω 1 + v 1 u + (1 -u)ω add ) 2 .
(3.92)

In accordance with condition (3.90), we need to nd such range of ω add that (3.92) is positive.

We distinguish two dierent cases, for which the nominator of (3.92) takes non-negative values ∀u ∈ [0, 1]:

1. Case ω add ≤ 0. Then, obviously ∂φ max (u)/∂u > 0 holds always.

2. Case ω add > 0. Then, we must provide that

ω 1 + (1 -u) ω add ≥ u √ ω add v 1 ⇒ ω 1 + ω add ≥ u (ω add + √ ω add v 1 ) .
In the worst case, this inequality must be satised for u = 1, which results into

ω add ≤ ω 2 1 v 1 .
This expression yields the upper bound for ω add . By the denition (3.89) and the fact that ω(u) should be non-negative, the lowest bound is -ω 1 . Thus, the reasonable range reads

ω add ∈ -ω 1 , ω 2 1 v 1 .
For the numerical example, we will pick the largest possible value ω add = ω 2 1 /v 1 , since by (3.92) this value provides ∂φ max (u)/∂u = 0 at u = 1. From the physical viewpoint, this choice implies the largest possible inuence of VSL on FD in the congested regime (the largest possible surface enclosed by the blue line in the congested regime and the thick dashed line in Figure 3.17).

Optimal equilibrium

The controller given by (3.82) can be applied to achieve any type of desired equilibrium ρ d (ξ, η) ∈ (0, ρ max (ξ, η)) ∀(ξ, η) ∈ Ω. However, for the following numerical example, we seek to achieve an optimal equilibrium ρ opt d that corresponds to the throughput maximization and, at the same time, to the density maximization, i.e., the highest possible number of cars should be able to pass the system at maximal ow. Thereby, the number of cars in a urban area is directly related to the vehicle density in it that can be increased due to the change in the shape of fundamental diagram caused by u(ξ, η, ρ), as it is shown in Figure 3.17.

The method to compute exactly equilibrium proles providing the maximal ow in the system was presented in Section 3.4.1. However, there it was done for u = 1, i.e., no speed limits were applied. With the help of speed limits, we are now able to extend the result of Section 3.4.1 by maximizing also the number of vehicles that can pass the system at maximal ow. In particular, we seek to nd ∀(ξ, η) ∈ Ω speed limits u opt (ξ, η) such that φ max (ξ, η, u opt ) = φ min max (η, 1),

where φ min max (η, 1) is the maximal possible steady state ow determined by the capacity at the strongest bottleneck along the η-line (3.51). Thus, the VSL controller must provide that this steady state ow is achieved, and at the same time

ρ opt d (ξ, η) = ρ c (ξ, η, u opt ).
Thus, the desired equilibrium density corresponds to the critical density achieved for u opt . In terms of Figure 3.17, this means that if φ min max (u = 1) = φ max (u opt ) for some (ξ, η) ∈ Ω, then u opt is such that ρ opt d = ρ c (u opt ). Speaking in terms of Theorem 3.3, the desired ow φ d = φ max (u opt ). Hence, the controller should act such to provide the same maximal possible ow, while the density is increased, since ρ opt d > ρ 1 . Notice that due to the change of FD shape, at the desired equilibrium trac operates only at critical density, i.e., there are no congestions in the whole area.

Let us again skip (ξ, η) in the notations for simplicity. In order to nd u opt ∀(ξ, η) ∈ Ω, we use (3.91) and (2.3), and obtain

φ max (u opt ) = v 1 v 1 + ω 1 ω 1 ρ c u opt ω 1 + (1 -u opt )ω add ω 1 + v 1 u opt + (1 -u opt )ω add , (3.93)
where ρ c corresponds to the critical density as in (2.3) Further, we use ρ c v 1 = φ max 1 with φ max 1 being the highest possible ow for some (ξ, η) ∈ Ω

for v = v 1 and ω = ω 1 . ρ Φ(ρ) 0 ρ max ρ opt d ρ 1 φ min max (u opt )
reached with u = 1, and ω add = ω 2 1 /v 1 to rewrite (3.93) as

φ max (u opt ) = φ max 1 u opt v 1 + (1 -u opt )ω 1 ω 1 + (v 1 -ω 1 ) u opt . (3.94)
Let us now introduce a coecient κ ∈ (0, 1] to denote the ratio of the ow at the strongest bottleneck along the η-line to the maximal possible ow at space point (ξ, η) for u = 1:

κ(ξ, η) = φ min max (η, 1) φ max (ξ, η, 1)
.

From (3.94) we get the following equation ∀(ξ, η) ∈ Ω to be solved for u opt :

κ = u opt v 1 + (1 -u opt )ω 1 ω 1 + (v 1 -ω 1 ) u opt ,
which can be further expanded as

u opt 2 + u opt κ v 1 ω 1 -1 - v 1 ω 1 -1 + κ = 0.
This is a quadratic equation with respect to u opt , which yields two possible solutions. We pick the one with the minus sign, since this guarantees that u opt remains below 1:

u opt = µ + 1 -κ(ν -1) -(ν + 1 -κ(ν -1)) 2 -4κ 2 , (3.95)
with ν = v 1 /ω 1 . Finally, the optimal equilibrium is the critical density dened in (2.3) obtained for u opt from (3.95):

ρ opt d = ω(u opt ) v(u opt ) + ω(u opt ) ρ max , (3.96)
where v(u opt ) and ω(u opt ) can be taken from (3.89) for u = u opt and ω add = ω 2 1 /v 1 .

Numerical example

As a network we again take the downtown of Grenoble. All the infrastructure parameters and the two-dimensional discretization scheme are exactly the same as described in the numerical example for 2D boundary control presented in Section 3.5.3.

It is again assumed that the critical density in triangular FD is ρ c = ρ max /3. The initial datum is given ∀(ξ, η) ∈ Ω by ρ 0 (ξ, η) = 3ρ max (ξ, η)/4, thus, it is in the congested trac regime. The inow demand and the outow supply are set to the maximal possible steady-state ows for u = 1, that is

D (ξ min , η, ρ in (η), u) = φ min max (η, 1), S (ξ max , η, ρ out (η), u) = φ min max (η, 1),
which are the only possible values, if we want to maximize the throughput of the system.

The desired optimal steady state (3.96) is constructed following the steps described above, and it is depicted in Figure 3.18b). This state is characterized by the maximal possible ow through the system achieved for the maximal possible number of vehicles. The numerical scheme needed to discretize the PDE system (3.78) is again the Godunov scheme in 2D that was described in Section 3.2.5. The only dierence is that the for every grid point in space and time ∀(i, j, k) ∈ {1, . . . , m} × {1, . . . , n j } × Z + , the ux function must include dependence on VSL controller as in (3.89) for u = u opt from (3.95).

Note that in (3.82) there exists an upper bound for the controller gain γ that guarantees that B(ξ, η, ρ) > 0 ∀(ξ, η, t) ∈ Ω × R + . However, one can accelerate the convergence rate by choosing the maximal possible γ(η, t) for each line of constant η and for each time. Thus, we will compare the control results obtained with two dierent control gains:

1. A constant control gain γ = 0.14 that is the largest possible value for a given urban network (Grenoble downtown) that matches the bounds stated in Theorem 3.3.

2.

A time-and space-varying control gain γ(η, t):

γ(η, t) = 1 - max -min ξ ξ ξ min (η) ρ( ξ, η, t)d ξ, δ , (3.97)
where δ > 0 is chosen to get γ > 0 even if the minimum is positive (since the arbitrarily large γ can be used), and > 0 provides the lower bound for B(ξ, η, ρ).

Notice that Theorem 3.3 was proved for the case of constant γ (as in item 1). However, the convergence can be accelerated also with γ that depends on η and t as in (3.97). The only issue is that function B must be always positive, and also that γ can not depend on dimension ξ, since in this case the feedback linearization would not work such that the dynamic equation turns into (3.87) due to an additional derivative term w.r.t. ξ. = 0.01 and δ = 0.1. Thereby, at every time step, the demand and supply functions at domain boundaries are set to the maximal possible throughput corresponding to the desired ow in the system. We observe that the state converges to the desired equilibrium, which becomes visible already after t = 2 hours of simulation time. Remark 3.8 Notice that at the desired equilibrium the critical density at each point of space will be higher than at initial time, since the VSL control changes the FD shape and aects the desired density is as in (3.96). Therefore, the results presented in Figure 3.18 may look like driving the trac state towards more congested regime, although it is still the free-ow (recall that in the desired equilibrium the trac operates at critical density, which becomes higher with VSL control).

The trac ow corresponds to the maximal possible steady state ow that is only determined by the network geometry (capacities at strongest bottlenecks).

Further, we compute the L 1 norm of the error in the number of cars as in (1.6) with ρ opt d (ξ, η) from (3.96) being the desired state. Its temporal evolutions for two dierent control gains are shown in Figure 3.18a). As in the previous example, we again observe that a larger control gain (3.97) provides a higher convergence speed in comparison to the constant γ = 0.14.

Recall that as soon as we start applying control, the trac system is completely set to the free-ow regime, since we always choose the minimal VSL value (see Proposition 3.1).

Discussions

In this section, we designed an in-domain controller for the trac state evolving on urban networks with dynamics governed by (3.78). This controller is a parameter incorporated into the ux function, and it should be interpreted as the ratio of a variable speed limit to the regular maximal allowed speed, i.e., u < 1 means that the speed limit is applied. Real data conrmed however that the trac ow can be enhanced for a given density in the congested regime, i.e., applying VSL might be an ecient solution to manage congestions. It also revealed that the VSL changes the shape of the fundamental diagram such that the critical density is increased, i.e., applying speed limits may result into setting the trac state to the free-ow regime. This, in general, results into a more smooth trac motion without sudden breaking, which has also a positive ecological impact.

The VSL controller is presented in Theorem 3.3. It is applied continuously in space and time, and the controller aects the state such that the desired ow is immediately achieved, and the vehicle density converges to the desired equilibrium as in (3.87). Thus, the controller changes the structure of the PDE system (3.78) such that it is not a conservation law, i.e., the controller performs a feedback linearization. This considerably simplies the analysis, since for continuous initial datum the solution to the state equation is a continuous function (and no weak formulation is required). Thus, it was shown that the exponential convergence to the desired state is guaranteed under the proposed VSL controller (3.82).

Further, we analyzed the structure of the proposed controller in Theorem 3.4. Thereby, it was assumed that FD depends on speed limits such that it monotonically increases with u for the free-ow trac regime and it reveals a concave dependence for the congested trac regime.

Under these assumptions, the controller was shown to be dierentiable almost everywhere with respect to ξ-dimension, which determines the ow motion.

Then, we suggested a specic way to parametrize the triangular FD with u such that all these assumptions hold. This analysis allowed us to obtain the explicit form of FD, which was then analyzed to obtain the controller (3.95) providing the optimal steady state (3.96). This desired equilibrium corresponds to the throughput maximization for the maximal possible number of vehicles, i.e., as many vehicles as possible pass a urban area at maximal ow (for the same total traveling time). This is guaranteed by the modications in the shape of FD introduced by the VSL controller that shifts the critical density such that the desired state corresponds to the critical density. Thus, the trac is in the free-ow regime everywhere in the domain, while the maximum throughput is experienced by the maximal possible number of vehicles. The performance of the designed VSL controller was demonstrated on a numerical example, where a congested trac is driven to the optimal equilibrium. The convergence to the desired state was observed after 2 hours. Notice that the convergence speed is determined by the controller gain that can be chosen larger, although its upper bound must not be violated. The value of the upper bound depends on the network infrastructure. The convergence happens also faster if the state is close to the desired equilibrium.

Chapter conclusions

This chapter was devoted to control of trac on urban networks of any size whose dynamics are described by a conservation law in two dimensions such as 2D LWR (3.1). Trac is viewed from the macroscopic point of view within this modeling approach. As in the 1D case, trac is treated as a uid that now propagates on a continuum 2D plane.

In Section 3.1 the 2D LWR model was presented. The model is inspired from crowd modeling with the only dierence being the restriction for vehicles to move on real physical roads. Thus, the model requires to assume that the urban network is dense enough to be viewed as a continuum plane. This plane is bounded by the size of the considered urban area. To model trac, one needs to have information about geometry and infrastructure of the urban network under study, i.e., the location of roads and intersections, number of lanes at each road and its speed limits. This information is used to dene the maximal density and capacities on the network. Then, all these parameters are approximated on a continuum plane by applying the inverse distance weighting method that assigns values to variables everywhere as a function of the distance to real roads. All these parameters being specic for dierent urban networks are incorporated into the fundamental diagram that becomes an explicitly space-dependent function.

The 2D LWR model was compared to an MFD-based model in Section 3.1.5 using a steady-state vehicle density predicted by commercial microsimulator Aimsun as a reference distribution. It appeared that the 2D LWR model is able to predict steady-states even more accurately than the MFD-based model. Moreover, it tracks more precisely shapes of congested areas, which may play an important role for localized congestion mitigation control tasks.

Thus, the 2D LWR model was justied as a reasonable choice for model-based control design.

However, a direct analysis of such a model is a complicated task due to the second space derivative. It is also unclear which boundary point should be actuated to aect a specic in-domain point or area.

We had to nd an approach to analyze this model such that one gets information about the trajectories followed by vehicles in the urban area. Such analysis became possible, since the structure of the 2D LWR model limits its applicability only for networks that consist of uni-directional roads. This ability to analyze equation in 2D to obtain admissible equilibria is an essential result that enables formulating control tasks for stabilization of trac evolving on large-scale urban networks. Further, this result was directly used in Section 3.4, where the model was analyzed for a boundary control design to mitigate congestions in some urban area. Thereby, trac was restricted to the congested regime for mathematical simplicity, since otherwise one would have to deal with solution discontinuities.

The Hamilton-Jacobi formalism enabled to handle discontinuities for the boundary control design in Section 3.5, where the 2D LWR model in curvilinear coordinates was considered.

There, the problem of approximating the desired vehicle trajectory has been posed for a mixed trac regime in asymptotic time. The problem was solved in a similar way as it was done for the 1D case in Section 2.3. The additional diculty was introduced due to the explicit space-dependency in the fundamental diagram such that the classical Lax-Hopf formula could not be applied. Instead, we had to apply the viability theory elaborated for the case of space-dependent Hamiltonians. For a numerical example, we took the structure of Grenoble downtown as a urban network. Simulation results revealed that the feedback part in the boundary controller is an essential component that makes the convergence to the desired vehicle trajectory possible.

Finally, we used the 2D LWR model in curvilinear coordinates to design a variable speed limit controller in Section 3.6. The VSL controller is used to directly aect the trac ow by imposing temporary restrictions on allowed speed, which is often used for specic situations such as accidents, bad weather conditions, etc. This is an in-domain controller that is applied continuously in space in the whole domain. It acts as a feedback linearization such that the state equation loses its conservation law structure, which exempts us from considering the solution in the weak formulation due to shocks if the initial datum is continuous. The VSL controller can be used to stabilize the 2D system to any desired space-varying equilibrium.

If FD has a concave dependence on controller in the congested trac regime and a linear one in the free-ow regime (which is a physically intuitive assumption), then the controller is dierentiable almost everywhere in space. The smoothness of the VSL controller has been studied, since the desired vehicle ow can sometimes be achieved for several speed limit ratio values. We have also investigated how to design an optimal steady state that corresponds to the throughput maximization achieved for the maximal possible number of cars. In a numerical example, we again used the structure of Grenoble downtown, and then demonstrated how the VSL controller makes the vehicle density converge to the desired equilibrium.

In the next Chapter 4, we will extend all these results to capture urban trac that admits multiple directions and ow crossings, since assuming a network without loops was the main limitation of this chapter.

Chapter 4

Multi-Directional Trac on Networks

This chapter is devoted to modeling and control of multi-directional trac evolving on large-scale networks. Here we directly address the main limitation of the previous Chapter 3, which was the assumption on the existence of some preferred direction of motion on a network level. We propose a novel model that is able to describe trac evolving in multiple directions on a urban network on a macroscopic level in Section 4.1. Thereby, we provide the derivation of this model step-by-step from the CTM at one intersection. Then, in Section 4.2, the model is validated using synthetic data from microsimulator, as well as using real data that we get from real sensors installed in the center of Grenoble. Finally, in Section 4.3, we design a boundary controller for trac governed by our new model that acts to mitigate congestion.

Multi-directional continuous trac model

In this section, we propose a new multi-directional two-dimensional continuous trac model.

It is called the NEWS model, since it consists of four PDEs that describe the evolution of vehicle density in four cardinal directions: North, East, West and South. This model can be applied to predict trac evolution on a general urban network of arbitrary size by knowing only the information about its boundary ows, as well as network topology, turning ratios at each intersection and infrastructure parameters. The literature review on existing works in this direction is given in Chapter 1.

The contribution of this section is the formal derivation of a macroscopic model describing trac propagation in a large trac urban network of arbitrary size by using the classical CTM at each intersection. The resulting NEWS model is a hyperbolic system with bounded densities in each layer. It will also be shown that the model also corresponds to a conservation law with the conserved quantity being the vehicle density in the domain. The main novelty of our model is that it includes mixing between dierent density layers, i.e., it allows cars to change their original direction of movement. For example, imagine a car going to the North that changed its direction and turned to the East. Thus, there is a non-zero ow from one layer to another, which is captured by our model. We present a method allowing to transform trac evolving on arbitrarily sparse networks into a continuum model, i.e., PDE. This is a benecial form when it comes to modeling on a large scale, since it allows to describe trac in terms of aggregated variables rather than tracking the motion of each individual vehicle.

First of all, we review the CTM for one intersection in Section 4.1.1, thereby introducing 121

φ in 2 φ in 1 ρ out 2 ρ out 1 φ out 2 φ out 1 (x 1 , y 1 ) (x 2 , y 2 ) (x 3 , y 3 )

Trac model for one intersection

In this section, we seek to derive a multi-directional macroscopic trac model that is able to predict the temporal evolution of trac density. To achieve this, we need rst of all to derive a trac ow model for one intersection. During this derivation we will be able to dene several important variables that will be later used to derive a continuous model for the whole network. In particular, we use the cell transmission model (CTM) [START_REF] Daganzo | The cell transmission model: A dynamic representation of highway trac consistent with the hydrodynamic theory[END_REF] at one intersection to introduce the concept of partial ows from one road to another. Partial ows will then be used to express the trac ow directions as a function of the network topology (more details are given below).

Let us consider an intersection located at (x 1 , y 1 ) with two incoming and two outgoing roads (as illustrated in Figure 4.1), and show a step-by-step derivation of the trac model at this intersection. Then, the trac model will be generalized for an intersection with an arbitrary number of roads.

Flows at intersections: example

We use the demand-supply concept described in Section 2.1.5 to derive a trac model for the intersection at (x 1 , y 1 ) as illustrated in Figure 4.1. In particular, we need to determine inows φ in (t) and outows φ out (t) for this intersection that stay in balance

φ in 1 + φ in 2 = φ out 1 + φ out 2 .
Notice that unlike in previous chapters of this thesis, we now indicate the inow and outow w.r.t. some particular intersection (and not a domain boundary as it was meant, e.g., in (2.57)). Thus, we use a subscript to number roads, and a superscript is used to indicate whether this particular road is incoming or outgoing, e.g., φ in max,1 is the capacity of incoming road number 1.

Assume that the ow-density relation at any road is given by a triangular FD as in (2.2).

Then, the demand and supply functions are given by (2.17).

Remark 4.1

Notice that, in general, the derivation of the model relies only on the demand-supply concept, which is applicable also for a more general FD shape (not only triangular) as long as it is a concave function of density. We assumed the triangular shape only to gain more clarity during the upcoming step-by-step model derivation.

Each incoming road has its own ow demand to enter the intersection (illustrated in Figure 4.1) that reads with (2.17):

D 1 = min{v in 1 ρ in 1 , φ in max,1 }, D 2 = min{v in 2 ρ in 2 , φ in max,2 }. (4.1)
A part of the ow entering the intersection goes to the rst outgoing road and the other part goes to the second outgoing road. These ows are split according to the turning ratios

(TR) α ij ∈ [0, 1]
, where i is the index of the incoming road and j is the index of the outgoing road. For instance, if α 11 = 0.6 and α 12 = 0.4, then 60% of the cars from the rst incoming road turn to the rst outgoing road, and 40% turn to the second outgoing road. Note also that the sum of turning ratios for each incoming road must be 1, i.e.,

α 11 + α 12 = 1, α 21 + α 22 = 1.
The concept of TR was discussed, for example, in [START_REF] Daganzo | The cell transmission model, part ii: network trac[END_REF] for the case of diverging intersections.

Let us now introduce the concept of partial demands. A partial demand refers to the demand ow of an incoming road to enter a particular outgoing road. These are equal to the overall demands (4.1) (demand to enter an intersection) multiplied by the corresponding TR:

D 11 = min{α 11 v in 1 ρ in 1 , α 11 φ in max,1 }, D 12 = min{α 12 v in 1 ρ in 1 , α 12 φ in max,1 }, D 21 = min{α 21 v in 2 ρ in 2 , α 21 φ in max,2 }, D 22 = min{α 22 v in 2 ρ in 2 , α 22 φ in max,2 },
where the rst number in the subscript of D is related to the incoming road, and the second number is related to the outgoing road.

In accordance with [START_REF] Daganzo | The cell transmission model, part ii: network trac[END_REF], each outgoing road provides supply for the ow coming from an intersection, which in case of triangular FD (2.17) reads:

S 1 = min{ω out 1 (ρ out max,1 -ρ out 1 ), φ out max,1 }, S 2 = min{ω out 2 (ρ out max,2 -ρ out 2 ), φ out max,2 }. (4.2)
Let us also assume that each outgoing road has a particular supply for each incoming road, e.g., S 1 is split into S 11 and S 21 (recall that the rst number is referred to an incoming road). In order to dene these partial supplies, we introduce supply ratios (SR)

β ij ∈ [0, 1]
used to denote the proportion of supply of outgoing road j that it provides for the maximal ow coming from a particular incoming road i relative to the supply it provides for all the incoming roads. The supply ratio β ij is thus dened as

β ij = α ij φ in max,i n in k=1 α kj φ in max,k , (4.3)
where n in is the overall number of incoming roads for some intersection, here n in = 2. Notice that for each outgoing road the sum of its SR must be 1, i.e.,

β 11 + β 21 = 1, β 12 + β 22 = 1.
With the denition of supply ratios (4.3), we are now ready to formulate partial supplies as the overall (intersection-related) supply given by (4.2) multiplied by the corresponding SR:

S ij = β ij S j = min{β ij ω out j (ρ out max,j -ρ out j ), β ij φ out max,j }.
Under the assumption of SR, we can also dene partial ows as the minimum between partial demand and partial supply, e.g., φ 11 = min{D 11 , S 11 } yields:

φ 11 = min{α 11 v in 1 ρ in 1 , β 11 ω out 1 (ρ out max,1 -ρ out 1 ), α 11 φ in max,1 , β 11 φ out max,1 }.
Finally, the intersection-related ows from incoming and to outgoing roads are expressed as sums of the corresponding partial ows, i.e.,

φ in 1 = φ 11 + φ 12 , φ in 2 = φ 21 + φ 22 , φ out 1 = φ 11 + φ 21 , φ out 2 = φ 12 + φ 22 .
Notice that the sum of ows before and after the intersection is always conserved, i.e.,

φ in 1 + φ in 2 = φ out 1 + φ out
2 . Thus, we have established a trac ow model at one particular intersection from Figure 4.1 by explicitly deriving expressions for its inows and outows.

Flows at intersections: generalization

We can now generalize the calculations from above to any intersection with n in incoming roads with densities ρ in i and ows φ in i for i ∈ {1, . . . , n in }, and n out outgoing roads with densities ρ out j and ows φ out j for j ∈ {1, . . . , n out }.

Every incoming road i has its own ow demand D i to enter its source intersection:

D i = min{v in i ρ in i , φ in max,i }.
Then, we dene partial demands from road i to road j as

D ij = α ij D i = min{α ij v in i ρ in i , α ij φ in max,i }.
Supply S j of the outgoing road j is simply given by S j = min{ω out j (ρ out max,j -ρ out j ), φ out max,j }.

Partial ow φ ij from incoming road i towards outgoing road j is given by

φ ij = min{D ij , S ij } = = min{α ij v in i ρ in i , β ij ω out j (ρ out max,j -ρ out j ), α ij φ in max,i , β ij φ out max,j }. (4.4)
Finally, the ow from incoming road φ in i is the sum over all the ows exiting this road, and the ow into outgoing road φ out j is the sum over all the ows coming into this road:

φ in i = nout j=1 φ ij , φ out j = n in i=1 φ ij . (4.5)
For a better overview, we have summarized all the notations introduced in this section in Appendix A.3.1.

NEWS framework

We seek to develop a model capable of predicting the evolution of multi-directional trac in a large-scale network that may consist of thousands of intersections. The main challenge thereby is that roads at every intersection may be oriented arbitrarily. Hence, we would like to obtain a model in terms of ows that are parallel to the cardinal directions: North (N),

East (E), West (W) and South (S). This will enable us to formulate the model in macroscopic terms, if every intersection will be described in a unied way. Let us call it the NEWS-model.

Its state variables should be denoted by bars, and they represent 4-dimensional vectors, e.g., φin = ( φin

N , φin E , φin W , φin S ) T .
Notice that the resulting model is intended to describe the evolution of densities in four direction layers, although an urban area can in general be represented as a 2D plane (x and y). The reason to consider trac evolution in opposite directions (e.g., North and South) independently is related to the idea to preserve ow values positive, since we want to keep information about the number of vehicles moving in each direction.

Projection from roads to NEWS

In order to formulate the trac model in terms of NEWS, we will use only the geometric properties of the network, such as angles of the road orientations with respect to the East direction counter-clockwise denoted by θ that ranges from 0 to 2π, see Figure 4.2. Thereby, roads 1 in and 2 out are oriented towards North-East, and roads 2 in and 1 out are oriented towards South-East.

Let us consider the projection of ows into the North. We calculate the ow to the North as a weighted sum of all ows on the roads that have angles less than π/2 with the North direction, i.e., these are roads 1 in and 2 out in Figure 4.2. This also means that, in general, an angle of road's direction with non-zero projection to the North is bounded to the range θ ∈ (0, π), while for non-zero projections to the East, West and South the angle must be θ ∈ (0, π 2 ) ∪ ( 3π 2 , 2π), θ ∈ (π/2, 3π/2) and θ ∈ (π, 2π), respectively. Then, outows in NEWS formulation can be found from the road formulation by applying the following projection:

φout N = p N
Notice that these properties are dened for the North direction, while the same holds also for other directions. The simplest choice for the projection coecients p θ , satisfying all these properties, is

p N θ =      sin(θ) | cos(θ)| + | sin(θ)| , θ ∈ (0, π), 0, elsewhere, p E θ
=      cos(θ) | cos(θ)| + | sin(θ)| , θ ∈ (0, π 2 ) ∪ ( 3π 2 , 2π), 0, elsewhere.
p

W θ =      -cos(θ) | cos(θ)| + | sin(θ)| , θ ∈ ( π 2 , 3π 2 ), 0, elsewhere, p S θ =
     -sin(θ) | cos(θ)| + | sin(θ)| , θ ∈ (π, 2π), 0, elsewhere, (4.6)
where θ is a positive angle between the direction of the road and the East direction.

Notice that, in general, each road can have non-zero weights with at most two directions.

For example, in

Flows in NEWS formulation

Flows at each intersection in NEWS formulation should be given by vectors φin = ( φin N , φin E , φin W , φin S ) T and φout = ( φout N , φout E , φout W , φout S ) T . This allows us to establish the following relation with ows from the original road formulation given by (4.5):

φin =        φin N φin E φin W φin S        =         p N θ in 1 p N θ in 2 p E θ in 1 p E θ in 2 p W θ in 1 p W θ in 2 p S θ in 1 p S θ in 2         φ in 1 φ in 2 and φout =        φout N φout E φout W φout S        =         p N θ out 1 p N θ out 2 p E θ out 1 p E θ out 2 p W θ out 1 p W θ out 2 p S θ out 1 p S θ out 2         φ out 1 φ out 2 .
For a general case of n in incoming and n out outgoing roads, we introduce projection matrices P in ∈ R 4×n in and P out ∈ R 4×nout consisting of coecients p θ in i and p θ out j , respectively. In general, φin N is the ow on incoming roads going to the North direction before the intersection, and φout N is the ow on outgoing roads going to the North after the intersection, see Figure 4.3 for the illustration of this concept. They can also be represented by the sums over partial ows in the NEWS formulation : φin N = φNN + φNE + φNW + φNS , where, for example, φNE is the ow consisting of cars going to the North before the intersection and to the East after they have passed this intersection, as it is illustrated in Figure 4.3. Thus, φin N (4.8) is composed of all such ows that were going to the North before the intersection and then continued their way either to the North or changed to the East, West or South after passing the intersection.

In the NEWS formulation, partial ows are dened from the road formulation as follows:

φEN = n in i=1 nout j=1 p E θ in i p N θ out j φ ij , (4.10)
where p θ are the projection coecients from (4.6). Notice that the correctness of this denition of partial ows can be veried by inserting (4.10) into (4.9):

φout N = nout j=1 p N θ out j n in i=1 p N θ in i + p E θ in i + p W θ in i + p S θ in i φ ij = nout j=1 p N θ out j n in i=1 φ ij = nout j=1 p N θ out j φ out j ,
whereby we have used the fact that the sum of projection coecients over all cardinal directions is 1 (see property 3 in the denition of p θ and (4.5)).

To gain more insight into the concept of partial ows, let us consider an example of an intersection that has one incoming and one outgoing road, as shown in

φin =       φin N φin E φin W φin S       =       0 φEN + φEE 0 φSN + φSE       .
Thereby, we see that φin N = φin W = 0, since the incoming road has a zero weight with respect to both North and West direction, while it has non-zero weights with South and East directions.

The outgoing road has non-zero weights only with North and East direction, which results into φin S = φSN + φSE and φin E = φEN + φEE . Hence, the ow on the outgoing road yields:

φout =       φout N φout E φout W φout S       =       φSN + φEN φSE + φEE 0 0      
. Also note that in Figure 4.4 there is no ow in the West direction, therefore all the ows containing at least one W are zero, e.g., φNW = φSW = 0, etc.

1 in E S 1 out E N

Turning and supply ratios in NEWS formulation

Similar to the trac model in road formulation given by (4.5) and (4.4), we would like to dene partial ows in the NEWS formulation using the demand-supply concept as in (2.15) and (2.16). For this, we will need to dene turning ᾱ and supply ratios β in the NEWS formulation. Moreover, we will also have to dene the parameters of triangular FD v, ω, ρmax in the NEWS formulation to be able to derive the complete model.

Demand D ∈ R 4×1 and supply S ∈ R 4×1 functions from (2.17) can be formulated in terms of NEWS using coecient matrices P in , P out as in (4.7):

D = P in min{v in ρ in , φ in max }, S = P out min{ω out (ρ out max -ρ out ), φ out max }. (4.11)
Now, without loss of generality, let us consider the partial ow from East to North φEN , which we would like to be able to express using demand and supply as in (4.4):

φEN = min{ᾱ EN DE , βEN SN }, (4.12)
where ᾱEN is the TR from East to North, and βEN is the SR of the North provided for vehicles arriving from the East, i.e., the same as β ij from (4.3) but in the NEWS formulation.

The coecients ᾱEN and βEN need to be determined, which can be done using (4.10), in which we substitute (4.4) that yields φEN =

n in i=1 nout j=1 p E θ in i p N θ out j min α ij v in i ρ in i , β ij ω out j (ρ out max,j -ρ out j ), α ij φ in max,i , β ij φ out max,j .
This expression is a sum over minimum functions, which is tedious to handle. We make the following approximation : change the order of taking the minimum and the summations. This leads to the minimum over just four arguments as in the demand-supply concept (4.4):

φEN ≈ min    nout j=1 p N θ out j n in i=1 p E θ in i α ij v in i ρ in i , nout j=1 p N θ out j n in i=1 p E θ in i β ij ω out j (ρ out max,j -ρ out j ), . . .    .
Notice that the dierence between putting minimum inside and outside the summation is decreasing as the level of the homogeneity in the congestion of roads increases. This approximation is exact if all roads in the network are in the same trac regime, i.e., either all roads are in free-ow or congested.

We set the latter expression equal to (4.12) for φ = φ max , and get the coecients ᾱEN and βEN that read

ᾱEN = nout j=1 p N θ out j n in i=1 α ij p E θ in i φ in max,i n in i=1 p E θ in i φ in max,i , (4.13

FD parameters and densities in NEWS formulation

Consider demand and supply functions in the NEWS formulation. From one side, we can calculate them using the projection matrices P in and P out as in (4.11). From the other side, we would like to be able to calculate demand and supply using a fundamental diagram as in (2.17), which should enable us to describe trac ow in a unied way for any intersection.

Recall that FD parameters depend on a specic road, while another road might already have a dierent speed limit or capacity.

Thus, we are going to dene a unied FD in NEWS formulation such that the FD is dened for each direction separately. This equivalently means that the parameters of FD will all become 4-dimensional vectors or 4 × 4 diagonal matrices. Let us consider the FD for the North direction, while similar steps should be done for other directions. That is, for DN and SN we would like to nd kinematic wave speeds vin N and ωout N and density transformations ρin N and ρout

N such that the following relations would hold:

DN = n in i=1 p N i min{v i ρ i , φ max,i } ≈ min{v in N ρin N , φin max,N }, SN = nout j=1 p N j min{ω j (ρ max,j -ρ j ), φ max,j } ≈ min{ω out N (ρ out max,N -ρout N ), φout max,N }. (4.15)
Note that in the case when there are much more than 4 roads, we can use only approximations of the fundamental diagram.

By approximating sum of minimum functions as a minimum of sums and writing the conditions on maximal ows together, we get a system of two equations

n in i=1 p N i v i ρ c,i = vin N ρin c,N , nout j=1 p N j ω j (ρ max,j -ρ c,j ) = ωout N (ρ out max,N -ρout c,N ). (4.16)
System (4.16) is undetermined, since it consists of two equations that have ve unknowns

(v in N , ωout N , ρin c,N , ρout c,N , ρout max,N ).
In general, we get the coordinates of each road, its number of lanes and speed limits as network data. Speed limits are directly related to the kinematic wave speeds v j , while the maximal density ρ max,j on each road j (either incoming or outgoing) is determined by its number of lanes and the minimal car-to-car distance (we again assume it to be 6 m). Knowing ρ max,j for every road, we can easily obtain the critical density ρ c,j from the shape of the fundamental diagram (recall that we have assumed ρ c = ρ max /3). Negative kinematic wave speeds ω j can be obtained from the speed limits v j and critical density ρ c,j using (2.3) as

ω j = ρ c,j v j ρ max,j -ρ c,j
.

Both incoming and outgoing roads contribute to the vehicle density in some neighborhood of the intersection. Moreover, since we want to have a general model, which is symmetric with respect to incoming and outgoing roads, and in order to dene each parameter only once, we assume symmetry ρin N = ρout N = ρN , vin N = vout N = vN and ωin N = ωout N = ωN . Assume further that densities are transformed into NEWS formulation in the same way as it is done for the ows (4.7), i.e.:

ρN = n in i=1 p N i ρ i + nout j=1 p N j ρ j , (4.17)
which then also holds for maximal ρmax,N and critical ρc,N densities. After we have dened all the densities, using symmetry assumption we can express the velocities from (4.16) as vN =

n in i=1 p N i v i ρ c,i + nout j=1 p N j v j ρ c,j ρc,N , ωN = n in i=1 p N i ω i (ρ max,i -ρ c,i ) + nout j=1 p N j ω j (ρ max,j -ρ c,j ) ρmax,N -ρc,N .
Recall that all these calculations are not limited to the particular triangular shape of FD, and thus can be performed in the same way for any type of FD as long as it is a concave function of density as it is also assumed in the LWR model (see Remark 4.1). The only thing that would have changed for dierent FD shapes are formulas for its parameters (??), since each FD can have a dierent set of parameters.

For a better overview, we have summarized all the notations introduced in this and next sections in Appendix A.3.2.

Derivation of the NEWS model

Our main goal here is to derive the macroscopic NEWS model for multi-directional trac that can describe the evolution of trac in terms of density (as in case of 1D and 2D LWR).

For the moment, we can already describe trac ow for any intersection in the unied way, which became possible due to the concept of partial ows in the NEWS formulation given by (4.12) with (4.13) and (4.14). The dynamic NEWS model in terms of density will be derived by considering an intersection and its outgoing roads that should be viewed as incoming roads for the neighboring intersections. In the end, we will be able to formulate a valid model for the whole urban area due to a unied description of trac behavior at any intersection. This unied description will be obtained using the continuation method that was introduced in [START_REF] Nikitin | A Continuation Method for Large-Scale Modeling and Control: from ODEs to PDE, a Round Trip[END_REF].

Continuation

Previously, we considered inows φ in and outows φ out with respect to some intersection.

However, for the derivation of the macroscopic continuum model, we consider inows and outows with respect to roads that will be denoted by ψ in and ψ out as in Figure 4.5.

Recall that θ is an angle between the road orientation and the East direction. Denote the ow in the direction θ as ψ θ . Essentially, there are two ows with direction θ: inow ψ in θ which is a sum of all ows entering a road with direction θ, and outow ψ out θ which is a sum of all ows outgoing from this road. Notice that, in the following, we will deal only with outgoing roads. Thus, we skip the superscript in the notation of angle, i.e., θ out j = θ j . Now consider some road j of length l j that is an outgoing road for the intersection located at (x 1 , y 1 ), see Figure 4.5. The density evolution on road j that is connecting the intersection at (x 1 , y 1 ) and the intersection at (x 2 , y 2 ) is given by

(x 1 , y 1 ) (x 2 , y 2 ) l j ψ in θ j ψ out θ j θ j x2 -x1 y2 -y1
∂ρ j ∂t = 1 l j ψ in θ j (x 1 , y 1 ) -ψ out θ j (x 2 , y 2 ) ,
where θ j = atan[(y 2 -y 1 )/(x 2 -x 1 )] as in Figure 4.5. Notice that there are no bars here in the notations, since we again refer to the road formulation.

The equation written above depends on two dierent space points (x 1 , y 1 ) and (x 2 , y 2 ).

However, we would like to obtain an equation that is given for a unique point of space. In order to achieve that, we can perform continuation at the beginning of the road (x 1 , y 1 ).

In its simplest form, the continuation method corresponds to the rst-order term of Taylor expansion in spatial coordinates, which reads

ψ out θ j (x 2 , y 2 ) ≈ ψ out θ j (x 1 , y 1 ) + (x 2 -x 1 ) ∂ψ out θ j ∂x + (y 2 -y 1 )
∂ψ out θ j ∂y ,

and assuming this approximation to be an equality, we get the following model

∂ρ j ∂t = 1 l j ψ in θ j (x 1 , y 1 ) -ψ out θ j (x 1 , y 1 ) -(x 2 -x 1 ) ∂ψ out θ j ∂x -(y 2 -y 1 )
∂ψ out θ j ∂y , or simply

∂ρ j ∂t = 1 l j ψ in θ j (x 1 , y 1 ) -ψ out θ j (x 1 , y 1 ) -cos θ j ∂ψ out θ j ∂x -sin θ j ∂ψ out θ j ∂y .
At the same time, by performing continuation at the end of the road (x 2 , y 2 ) we arrive at

∂ρ j ∂t = 1 l j ψ in θ j (x 2 , y 2 ) -ψ out θ j (x 2 , y 2 ) -cos θ j ∂ψ in θ j ∂x -sin θ j ∂ψ in θ j
∂y .

Since the density along the road is assumed to be constant, both continuous models can be used to represent the original one. The rst model is dened in terms of the beginning of the road (x 1 , y 1 ) and contains spatial derivatives of ψ out θ j

, whereas the second model is dened in terms of the end of the road (x 2 , y 2 ) and contains spatial derivatives of ψ in θ j

. However, performing continuation not at the end points but somewhere in between can result into a more general form that unies these two models.

Let us perform continuation of the model for some arbitrary point along the road (x, y) whose coordinates lie between two endpoints (x 1 , y 1 ) and (x 2 , y 2 ):

x = x 1 γ + x 2 (1 -γ), y = y 1 γ + y 2 (1 -γ),
where γ ∈ [0, 1]. Thus, by performing continuation at (x, y), we arrive at

∂ρ j ∂t = 1 l j ψ in θ j (x, y) -ψ out θ j (x, y) -cos θ j ∂((1 -γ)ψ in θ j + γψ out θ j ) ∂x -sin θ j ∂((1 -γ)ψ in θ j + γψ out θ j ) ∂y . (4.18)
Now let the vector-ow on road j be Ψ θ j = ψ θ j cos θ j sin θ j , where ψ θ j = (1 -γ)ψ in θ j + γψ out θ j .

Then, the model (4.18) can be rewritten as

∂ρ j ∂t = 1 l j ψ in θ j (x, y) -ψ out θ j (x, y) -∇ • Ψ θ j (x, y), (4.19)
where ∇ is a nabla operator dened as ∇ = ( ∂ ∂x , ∂ ∂y ).

This model (4.19) predicts the dynamics of the vehicle density at some outgoing road j with direction θ j . Equation (4.19) has the same form for any intersection located at (x k , y k ), where k ∈ {1, . . . , K} is an index used to label intersections in the network. Notice that parameter γ was introduced only for the derivation purposes, it will not explicitly appear in the nal model, see details below.

The NEWS model

We would like to translate the model given in road formulation (4.19) into NEWS formulation, which allows to describe trac ow direction at any intersection in a unied way independently of the number of its outgoing roads. Recall that densities in every direction layer are transformed as in (4.17). Let us again consider the North direction for simplicity, while the same steps should be performed for all other directions. Thus, multiplying the equation (4.19) by p N θ j and taking the summation, we get the model that predicts the evolution of vehicle density in the North direction on outgoing roads of an intersection located at (x k , y k ) that reads

∂ ρN ∂t = nout j=1 p N θ j 1 l j ψ in θ j -ψ out θ j -∇ •   nout j=1 p N θ j Ψ θ j   . (4.20)
We cannot further simplify the equation (4.20) towards the NEWS formulation, since the summations contain additional index-dependent coecients such as 1/l j , sin θ j and cos θ j (embedded in Ψ θ j ). Let us then approximate the system (4.20) by averaging road lengths l j such that the mean length of outgoing roads conserves the maximum number of cars:

L = nout j=1 ρ max,j l j nout j=1 ρ max,j
.

Further, we also approximate sine and cosine in (4.20) as

cos θ N = nout j=1 p N θ j cos θ j φ max,j nout j=1 p N θ j φ max,j , sin θ N = nout j=1 p N θ j sin θ j φ max,j nout j=1 p N θ j φ max,j
.

Substituting these approximations into (4.20), we get

∂ ρN ∂t = 1 L nout j=1 p N θ j ψ in θ j -ψ out θ j -∇ •   nout j=1 cos θ N sin θ N p N θ j (1 -γ)ψ in θ j + γψ out θ j   , or simply ∂ ρN ∂t = 1 L ψin N -ψ out N -∇ • cos θ N sin θ N (1 -γ) ψin N + γ ψout N , (4.21)
where we can further dene ψN = (1 -γ) ψin

N + γ ψout N .
Notice that this model (4.21) is the macroscopic NEWS model, since it does not depend on road index j any more. In some sense, the model (4.21) is dened for any particular space point in the vicinity of an intersection. Therefore, it makes no more sense to have separate notations for ows related to intersections φ and roads ψ. Thus, for convenience and consistency with other parts of this manuscript, we will again use the notation φ for ows.

The model (4.21) can be further simplied in order to get rid of spatial derivatives over multi-directional ows, since otherwise the PDE can lose hyperbolicity and, moreover, we want to eliminate the parameter γ.

Model simplication

The term under the space derivative in (4.21

) is φN = (1 -γ) φin N + γ φout N .
Recall that by (4.8) and (4.9) we can express inows and outows at any point as sums over partial ows: Physically, this assumption means that if vehicles move at maximal possible ow before an intersection, they continue to use roads' transportation capacities at maximum after the intersection.

φin N = φNN + φEN + φW N + φSN , φout N = φNN + φNE + φNW + φNS .
The proof that (4.23) holds under the assumption of a well-designed network (4.24), being rather technical, is shifted to Appendix B.7, where we show that there exists parameter γ such that (4.23) holds. Thus, the transported term under the derivative in (4.21) can be approximated by a standard ow in the demand-supply formulation that depends only on the density of the same direction. Hence, the full system of equations can be written as

                           ∂ ρN ∂t = 1 L φin N -φout N - ∂(cos θ N φN ) ∂x - ∂(sin θ N φN ) ∂y , ∂ ρE ∂t = 1 L φin E -φout E - ∂(cos θ E φE ) ∂x - ∂(sin θ E φE ) ∂y , ∂ ρW ∂t = 1 L φin W -φout W - ∂(cos θ W φW ) ∂x - ∂(sin θ W φW ) ∂y , ∂ ρS ∂t = 1 L φin S -φout S - ∂(cos θ S φS ) ∂x - ∂(sin θ S φS ) ∂y , (4.25)
where the term φinφout is given by

    φin N -φout N φin E -φout E φin W -φout W φin S -φout S     =     φEN + φW N + φSN -φNE -φNW -φNS φNE + φW E + φSE -φEN -φEW -φES φNW + φEW + φSW -φW N -φW E -φW S φNS + φES + φW S -φSN -φSE -φSW     ,
where partial ows between two direction layers are obtained as demand-supply problem, e.g.:

φEN = min{ᾱ EN DE , βEN SN }.
This system of equations describes the density evolution in the vicinity of one intersection.

Thus, the density ρ(x, y, t) and the ow φ(x, y, t) are space-and time-dependent functions, whereas all the parameters are constant ( ᾱ, β, L, v, ω, ρmax , cos θ, sin θ).

Notice that the term φinφout is responsible for mixing between dierent density layers, e.g., φin N = φSN + φW N + φEN accounts for vehicles that were moving to the South, West and First, we calculate ᾱ, β, L, v, ω, ρmax , cos θ, sin θ for all K intersections in the network.

Then, we apply the inverse distance weighting method described in Section 3.1.3.1 to approximate all these parameters over a continuum plane, e.g., the value of an average road length can be dened ∀(x, y) ∈ Ω

L(x, y) = K k=1 L(x k , y k )w k e -µ √ (x-x k ) 2 +(y-y k ) 2 K k=1 w k e -η √ (x-x k ) 2 +(y-y k ) 2 , ( 4.26)
where µ is a weighting parameter used to denote the sensitivity of the estimated variables to the distance from the intersections (and not roads as it was meant in Section 3.1.3.1). Thus, the inuence of the intersection parameter, e.g. L(x k , y k ), decreases exponentially with the distance to this intersection.

Further, w k in (4.26) is the weight of intersection k, which can be used to assign larger weights to intersections with important roads. If one wants to emphasize the main transportation arteries of the city (its most used roads), w k can be set to larger values for particular intersections (i.e., w k > 1). The main arteries are revealed from the historical (TomTom) data (see [START_REF] Andrea | A functional road classication with data mining techniques[END_REF] for Functional Road Classication). Note that the most used roads are not necessarily the roads with largest transportation capacities due to a non-optimal design of a urban network. For example, imagine a road with many lanes that does not connect any important points in a city, and thus it is not used at maximum. At the same time, another road with the same transportation capacity connecting important locations has been used more extensively according to historical data, and thus one can assign larger weights to intersections that are connected to this road directly. Otherwise, setting all w k = 1 assures that pure network infrastructure data are used. Notice that the intersection weights w k change only the interpolation procedure and not the network and model parameters itself, and it can be calculated as:

w k = 1 n in,k + n out,k n in,k i=1 w q,i + n out,k j=1 w q,j , ∀k ∈ {1, . . . , K}, (4.27)
where w q are weights of roads based on their importance classes. There are 7 road classes in total, see Figure 4.11. We assign w q = 2 for all roads of classes 1 and 2 (all major roads of high importance), and for any other case (classes from 3 to 7) we set w q = 1.

Thus, we dene all the geometrical and FD parameters ∀(x, y) ∈ Ω using inverse distance weighting (IDW) as in (4.26). As a result, we obtain a continuous PDE system that looks like (4.25) with time-and space-dependent density ρ(x, y, t) and ow φ(x, y, ρ), while all parameters are obtained using (4.26), which makes them space-dependent functions, i.e., ᾱ(x, y), β(x, y), v(x, y), etc. We consider some road j, and the exterior vehicles penetrate its entry at ow ψ source θ j (we use again ψ for ow, since it is here formulated in terms of roads). We take this additional ow of vehicles into account by adding it into equation (4.19) for road j, which yields

∂ρ j ∂t = 1 l j ψ in θ j -ψ out θ j -∇ • Ψ θ j + 1 l j ψ source θ j . (4.28)
In general, when we want to specify inow for some road, we can only propose it in terms of demand function. This is equivalent to the weak boundary conditions, see Section 2.1.5.

Then, the amount of ow entering this road depends on its supply, which is in turn determined by the trac state of this road:

ψ source θ j = min{D source θ j
, S θ j (ρ j )}.

We can rewrite (4.28) in NEWS formulation by performing the transformations described in Section 4.1.3.2, which leads us to the extended NEWS model (with sinks also included): we specify it in terms of roads of the network. In contrast to all other variables obtained by (4.26), the overall number of incoming cars should be conserved. Thus, we choose Gaussian kernel for the approximation of demand and supply functions:

                           ∂ ρN ∂t = 1 L φin N -φout N + φsource N -φsink N - ∂(cos θ N φN ) ∂x - ∂(sin θ N φN ) ∂y , ∂ ρE ∂t = 1 L φin E -φout E + φsource E -φsink E - ∂(cos θ E φE ) ∂x - ∂(sin θ E φE ) ∂y , ∂ ρW ∂t = 1 L φin W -φout W + φsource W -φsink W - ∂(cos θ W φW ) ∂x - ∂(sin θ W φW ) ∂y , ∂ ρS ∂t = 1 L φin S -φout S + φsource S -φsink S - ∂(cos θ S φS ) ∂x - ∂(sin θ S φS ) ∂y , (4.29)
Dsource N (x, y) = K k=1 Dsource N (x k , y k )G σ (x -x k , y -y k ),
where G d 0 (x, y) is a two-dimensional symmetric Gaussian kernel with standard deviation d 0 : Notice that from now on, we will consider all the variables and parameters only in terms of NEWS formulation. Therefore, for the remaining part of this thesis, we omit bars everywhere in the notations of NEWS variables keeping in mind that these are 4-dimensional vectors, e.g., density and FD, or 4 × 4 diagonal matrices, e.g., TR and SR matrices. We keep the overlines only for cos θ and sin θ, since these do not preserve the sense of cosine and sine functions in NEWS formulation (the sum of their squares is not necessarily equal to 1).

G d 0 (x, y) = 1 2πd 0 2 e -1 2d 0 2 (x 2 +y 2 ) ,

Mathematical properties of NEWS model

Let us now investigate the properties of the NEWS model. For its explicit analysis, we take system (4.25) that does not include any source terms. In this section, we will check whether our system represents a conservation law, then we will discuss the boundedness of its state ρ, and, nally, we will show that the model represents a hyperbolic PDE system.

Conservation law

The overall density in the network is the sum over the densities in all four directions, that is

ρ = ρ N + ρ E + ρ W + ρ S .
By taking its time derivative we get

∂ρ ∂t = ∂ρ N ∂t + ∂ρ E ∂t + ∂ρ W ∂t + ∂ρ S ∂t ,
and for each of these terms we can substitute the corresponding PDE from our model (4.25).

It appears that all the mixing terms (φ in -φ out ) cancel each other, and we simply get

∂ρ ∂t = -∇ • Φ, (4.30)
where

Φ = cos θ N sin θ N φ N + cos θ E sin θ E φ E + cos θ W sin θ W φ W + cos θ S sin θ S φ S ,
which has a form of a conservation law, where the conserved quantity is the overall density in the network.

Boundedness of the state

The boundedness of the density ρ ∈ [0, ρ max ] is not violated in the model given by (4.25), since the terms under the derivatives are resolved using the standard Godunov scheme, i.e., trac ow in each direction is determined by the minimum between demand and supply, as in LWR formalism. For example, consider the North direction, then the term under the spatial derivative in (4.25) is just φ N = min{D N , S N }.

Mixing terms with a positive sign (these are φ SN , φ W N and φ EN in the equation for ρ N ) depend on the supply of N , e.g.,

φ EN = min{α EN D E , β EN S N }. If ρ N = ρ max,N , then S N = 0 ⇒ φ EN = 0 ⇒ ∂ρ N ∂t ≤ 0,
which means that positive terms can not contribute to the increase of density, whenever it has reached ρ max,N .

Let us now consider negative mixing terms. These depend on the demand of the North direction, e.g.,

φ N E = min{α N E D N , β N E S E }, which in case of ρ N = 0 ⇒ D N = 0 yields φ N E = 0 ⇒ ∂ρ N ∂t ≥ 0.
This implies that negative terms do not contribute to the decrease of density when it is already zero.

Hyperbolicity

Let us now investigate whether the model (4.25) is a hyperbolic PDE (as it is the case for 1D LWR (2.1) but not the general case for multi-directional 2D LWR [START_REF] Mollier | A step towards a multidirectional 2D model for large scale trac networks[END_REF]). Hyperbolicity is a fundamental property determining the behavior of solutions, which also plays an important role in the choice of the corresponding numerical scheme. For example, if we show that the model is a hyperbolic PDE, then we can apply the Godunov scheme for numerical simulation, as it is usually done for hyperbolic models such as 1D LWR.

In contrast to other types of partial dierential equations, in a hyperbolic PDE any disturbance made in the initial data travels along the characteristics of the equation with a nite propagation speed. Although the denition of hyperbolicity is fundamentally a qualitative one, there are precise criteria with which one can dene whether a partial dierential equation is hyperbolic. Thus, equation (4.25) can be written in a following general form:

∂ t ρ + ∂ x F x (ρ, x, y) + ∂ y F y (ρ, x, y) = g(ρ, x, y), (4.31)
where F x and F y are the ow matrices dened from (4.25) as

F x =     cos θ N φ N 0 0 0 0 cos θ E φ E 0 0 0 0 cos θ W φ W 0 0 0 0 cos θ S φ S     ,
and

F y =     sin θ N φ N 0 0 0 0 sin θ E φ E 0 0 0 0 sin θ W φ W 0 0 0 0 sin θ S φ S     .
The right-hand side term g(ρ, x, y) from (4.31) corresponds to the vector containing all the mixing terms from (4.25):

g(ρ, x, y) = 1 L     φ EN + φ W N + φ SN -φ N E -φ N W -φ N S φ N E + φ W E + φ SE -φ EN -φ EW -φ ES φ N W + φ EW + φ SW -φ W N -φ W E -φ W S φ N S + φ ES + φ W S -φ SN -φ SE -φ SW     .
The spatial derivatives of ow matrices from (4.31) can be written as a chain rule

∂ x F x (ρ, x, y) = ∂ ρ F x (ρ, x, y) • ∂ x ρ + ∂ x F x (ρ,
x, y), and

∂ y F y (ρ, x, y) = ∂ ρ F y (ρ, x, y) • ∂ y ρ + ∂ y F y (ρ, x, y),
which is further inserted into equation (4.31) that yields

∂ t ρ + ∂ ρ F x (ρ, x, y) • ∂ x ρ + ∂ ρ F y (ρ, x, y) • ∂ y ρ = b(ρ, x, y), (4.32)
where b(ρ, x, y) = g(ρ, x, y) -∂ x F x (ρ, x, y) -∂ y F y (ρ, x, y). According to Section 3.1 of [START_REF] Serre | Systems of Conservation Laws I: Hyperbolicities, Entropies, Shock Waves[END_REF], the right-hand side part of (4.32) b(ρ, x, y) does not play any signicant role for the analysis of the equation. Thus, we simply omit it by setting b(ρ) = 0.

Let us further rewrite (4.32) as

∂ t ρ + A x ∂ x ρ + A y ∂ y ρ = 0, (4.33)
where A x = ∂F x /∂ρ and A y = ∂F y /∂ρ represent matrices of ow derivatives:

A x =      cos θ N ∂φ N ∂ρ 0 0 0 0 cos θ E ∂φ E ∂ρ 0 0 0 0 cos θ W ∂φ W ∂ρ 0 0 0 0 cos θ S ∂φ S ∂ρ     
, and

A y =      sin θ N ∂φ N ∂ρ 0 0 0 0 sin θ E ∂φ E ∂ρ 0 0 0 0 sin θ W ∂φ W ∂ρ 0 0 0 0 sin θ S ∂φ S ∂ρ     
.

The system (4.33) is symmetrisable hyperbolic, since matrices A x and A y are both symmetric.

This implies that the system (4.33) is hyperbolic [START_REF] Serre | Systems of Conservation Laws I: Hyperbolicities, Entropies, Shock Waves[END_REF], which equivalently means that our model given by (4.25) is a hyperbolic PDE system.

Discussions

In this section, we have rigorously derived a new macroscopic trac model to predict trac evolution from the cell transmission model on one intersection. The main challenge thereby was to nd a unied approach to describe the direction of trac ow at intersections that have dierent number of incoming and outgoing roads and might be oriented arbitrarily. We faced this challenge by introducing the NEWS formulation of trac in Section 4.1. While deriving the NEWS model (4.25), we had to introduce several new concepts. The main of them is the concept of partial ows that allowed us to capture the truly multidirectional behaviour of trac. Thus, the model includes trac ow that changes its original direction of movement. For example, there is a non-zero ow from the North direction layer to the East, since there might be vehicles that move to the North, and then they reach an intersection and turn to the East. This modeling phenomenon is equivalent to including turning ratio information at intersections in NEWS formulation. Thus, for every intersection we consider 16 origin-destination pairs, as shown in Figure 4.3. One of the major assumptions that were made is that the urban network is well-designed in terms of road capacities. Thus, road capacities must be such that cars do not get blocked when they turn, i.e., high demand to enter roads with large capacities (e.g., highways or major trac roads) and low demand to enter minor roads.

The NEWS model corresponds to a conservation law, where the conserved quantity is the number of vehicle density in a network. The state of this model was shown to be bounded, and moreover, the NEWS model represents a hyperbolic PDE. In the next Section 4.2, we will validate this model by using synthetic data from microsimulator, as well as by using real data from sensors installed in Grenoble downtown.

Validation of NEWS model

This section is devoted to validation of the NEWS model that was derived and analyzed in the previous Section 4.1. First, we will discuss the numerical method used to simulate trac with NEWS model. Then, the similarity measure will be introduced to enable a quantitative For the numerical simulation of (4.25), we use the Godunov scheme in two dimensions.

We start by dening a numerical grid in Ω × R + by setting n x to be number of cells to discretize x dimension, n y to be number of cells to discretize y dimension, ∆x = (x max -x min )/n x to be the space cell size in x dimension, ∆y = (y max -y min )/n y to be the space cell size in y dimension, ∆t to be the time cell size, (i∆x, j∆y, k∆t) for i ∈ {1, . . . , n x }, j ∈ {1, . . . , n y } and k ∈ Z + to be the grid point.

For the simulation of trac on this area of Grenoble, we set n x = 60 and n y = 60, i.e., the 2D plane is divided into n x × n y = 3600 cells. Similarily to Section 2.1.6, the mesh sizes ∆x and ∆y and time step ∆t are chosen such that they satisfy the Courant-Friedrichs-Lewy condition.

The discrete density vector is then ρ k (i, j) = ρ k N (i, j), ρ k E (i, j), ρ k W (i, j), ρ k S (i, j)

F k x,s (i, j) = cos θ s (i, j) + cos θ s (i -1, j) 2∆x min D k s (i -1, j), S k s (i, j) - cos θ s (i, j) + cos θ s (i + 1, j) 2∆x min D k s (i, j), S k s (i + 1, j) ,
while the derivative term w.r.t. y dimension is similarly computed as F k y,s (i, j) = sin θ s (i, j) + sin θ s (i, j -1) 2∆y min D k s (i, j -1), S k s (i, j) -sin θ s (i, j) + sin θ s (i, j + 1) 2∆y min D k s (i, j), S k s (i, j + 1) .

Notice that F k x,s (i, j), F k y,s (i, j) are obtained using the upwind scheme [START_REF] Courant | On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Dierences[END_REF] for cos θ s (i, j) > 0, sin θ s (i, j) > 0. The upwind scheme is used to guarantee the correct direction of information propagation in a ow eld, which needs to be reversed if cos θ s (i, j) < 0 for F k x,s (i, j) and sin θ s (i, j) < 0 for F k y,s (i, j).

Finally, H k s (i, j) includes source and sink terms, thus it is computed as

H k s (i, j) = 1 L(i, j) min D source,k s (i, j), S k s (i, j) -min D k s (i, j), S sink,k s (i, j) .

Structural similarity measure

In order to enable a quantitative comparison between two density distributions, we use the structural similarity measure (SSIM) [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF]. This should be understood as an index used to measure the similarity between two images. Thereby, three dierent image properties are compared: luminance, contrast and structure.

In general, the SSIM between two density distributions ρ 1 (i, j) (NEWS prediction) and ρ 2 (i, j) (reference distribution) ∀(i, j) ∈ {1, . . . , n x } × {1, . . . , n y } can be calculated as:

SSIM (ρ 1 , ρ 2 ) = (2ζ 1 ζ 2 + c) (2σ 12 + c) ζ 2 1 + ζ 2 2 + c σ 2 1 + σ 2 2 + c , ( 4
σ(ρ 1 , ρ 2 ) = 1 n x 1 n y nx i=1 ny j=1 S s=N ρ s,1 (i, j) -ζ 1 S s=N ρ s,2 (i, j) -ζ 2 .
Finally, c > 0 in (4.34) is a constant that needs to be small, e.g., we take c = 1 • 10 -13 for the computation. This constant prevents instability, when the denominator is close to zero. The

range of SSIM is [-1, 1]
, where 1 is achieved if two images are identical, whereas -1 means that one image is the inverse of the second image.

The main advantage of using SSIM is that it is a perception-based metric used to detect structural changes in the image, while, for example, the mean square error evaluates only the absolute error rather than the dierences in congestion patterns. Thus, even if two density distributions are characterized to have the same number of cars, the SSIM is still able to detect whether congested zones have dierent shapes.

Model validation with Aimsun

We run a scenario of congestion formation in the selected area of Grenoble downtown (see In general, we have access to the following network data: (x, y) coordinates of all intersections and its corresponding roads, as well as speed limits and number of lanes for each road.

Using these data, we compute the parameters of the fundamental diagram v, ω, ρ max and the intersection parameters α, β, L, cos θ, sin θ in the NEWS framework for all the intersections as follows. For each road j in the selected Grenoble area, we read the free-ow velocity v j from its speed limit data. Then, the maximal density ρ max,j is computed by placing a car every 6 m at every road, and then KDE is used as described in Section 3.1.3.2. Here we assume that each vehicle has inuence within d 0 = 70 m range around its position, see (3.8). Further, we assume that the critical density ρ c,j = ρ max,j /3 everywhere (triangular FD), which allows us also to calculate the negative kinematic wave speed ω j and the road capacities φ max,j .

Then, these parameters are translated into NEWS formulation using the network geometry, see Section 4.1.2.4 for more details.

In order to determine the trac ow direction, we use turning ratios α ij for each road i towards road j that are estimated as a function of road capacities:

α ij = φ max,j nout q=1 φ max,q
.

Then, supply ratios β ij are calculated using (4.3). Both ratios α and β are translated into NEWS formulation as in (4.13) and (4.14). Further, the coordinates of road's both ends are used to determine its length l j and orientation angle θ j , from which we then obtain L, cos θ, sin θ in NEWS formulation as described in Section 4.1.

Then, all these intersection and FD parameters are approximated for every grid point First of all, we load the Grenoble network into Aimsun (see Figure 4.6b)), and let vehicles enter through its boundaries by specifying inows. We choose inows such that the main stream of vehicles comes from the South of the area. The microsimulations evolve for 2.5 min, and then the state is saved and later used as an initial condition for both Aimsun and numerical simulation of the NEWS model. Afterwards, we continue the microsimulation on Aimsun until we do not perceive any structural changes in the state, which indicates that a steady state has been achieved. The results are saved as vehicle positions at all time instants.

Therefore, we use KDE to transform the standard Aimsun data into a density distribution.

KDE in 1D is also used to smooth inows such that they enter the domain in a continuous line rather than at discrete points of space. We set constant inows in order to let the system converge to a steady state. We then perform a numerical simulation of the NEWS model as described in Section 4.2.1 using the initial conditions from Aimsun.

The results are depicted in Figure 4.7, where the comparison of both density distributions is shown for t ∈ [0, 50] min. We see that in both cases the distributions look quite similar but not identical, which might be caused by several things. In Aimsun, vehicles are restricted to move only on real physical roads, while more freedom of movement is perceived in a PDEdriven system. Moreover, in Aimsun, turning ratios indicate the probability with which a car turns to one or another road, whenever it reaches an intersection at some time instant. Thus, TR in Aimsun should be understood as mathematical expectation rather than deterministic values. Hence, it often appears that scenarios in Aimsun, although having the same initial condition and inows, converge to dierent density distributions. Vehicles might get stuck in dierent parts of the city, while this is unlikely to happen during the numerical simulation of NEWS density, where cars move on a continuum space. However, on a global scale trac regimes seem to be reproduced correctly in most parts of the city.

Let us now compute the structural similarity measure (4.34) to compare two density distributions from Figure 4.7. For that the domain is divided into 9 windows of equal size, as drawn in Figure 4.8a). We do this in order to be able to compare density distributions zone-by-zone.

The zones are numbered from top left to bottom right, as shown in Figure 4.8a). The SSIM SSIM l with l = {1, . . . , 9}.

of the whole domain is then calculated as the mean value over all zones:

SSIM (ρ 1 , ρ 2 ) = N l=1 SSIM l ζ l (ρ 2 ) N l=1 ζ l (ρ 2 ) , (4.36)
where N = 9 is the total number of zones in the domain, SSIM l is referred to zone l each given by (4.34), and ζ l (ρ 2 ) is the average of the density distribution used to assign larger weights to zones that are strongly occupied in the reference distribution (here, ρ 2 is the total density in Aimsun). Thus, the fewer cars a zone has, the smaller is its weight. The weights are assigned in order to avoid giving too much importance to zones that are currently almost empty. Notice that ζ l (ρ 2 (t)) is a time-dependent parameter.

In its original formulation, SSIM values range from -1 to 1. In order to facilitate the interpretation of this index in the context of density comparison, we make its range to be SSIM ∈ [0, 1] by doing (SSIM + 1)/2. Thus, SSIM = 1 implies that two distributions are identical, and SSIM = 0 means that one distribution is completely the opposite of the second one (inverted image).

The SSIM of corresponding zones in both distributions is depicted as a function of time in Figure 4.8b). It seems that the most problematic zones are the most empty ones that are concentrated in the upper part of the domain (zones 2 and 3), while the best captured zones are the most congested ones (zones 4 and 9). This can be explained by the fact that the main stream of vehicles enters the domain from the South (since this is where the largest inows are set), where they build the most congested areas. Thus, cars might not have reached the upper part in Aimsun, since they got stuck in the Southern part of the area.

Finally, in order to unable a quantitative comparison of the density in the whole Grenoble area, the SSIM is averaged over all zones by using (4.36), and we obtain the result depicted in

Model validation with real data

For the model validation with real data, we make use of Grenoble Trac Lab for Urban Networks known as GTL Ville, see http://gtlville.inrialpes.fr/. This is an experimental platform for real-time collection of trac data coming from a network of stationary ow sensors installed in Grenoble downtown, see Figure 4.10. This platform also provides real-time trac indicators oriented towards the users of the city, trac operators and researchers [START_REF] Canudas De Wit | Grenoble Trac Lab: An Experimental Platform for Advanced Trac Monitoring and Forecasting [Applications of Control[END_REF]. The collected data and computed indicators are available for download at the GTL website.

Moreover, for model validation with real data we use the available information on road importance that is obtained from historical TomTom data, see intersections of Grenoble during a measurement campaign lasting one week. These gures are taken from [START_REF] Rodriguez-Vega | Optimal sensor placement and density estimation in large-scale trac networks[END_REF].

network parameters.

The maximal densities at every road ρ max,j , capacities φ max,j , road lengths l j and orientations θ j are the same as described above, since these parameters are dened by the network topology, which remains the same for the real-life experiment. However, the free-ow speed data are now taken from oating car data reported from several vehicles that are equipped with devices such as a GPS navigator. The free-ow speed is estimated as the maximal speed of a vehicle in the absence of other cars, and it starts decreasing as the density of surrounding cars increases. It is worth noting that, in general, the free-ow speed is lower than the corresponding speed limit value, since in reality cars lose their average velocity, e.g., by stopping at trac lights. Thus, our data indicate that the free-ow speed is approximately equal to 60 -70% of road's speed limit. Now let us explain how do we get turning ratios α ij . These data are obtained from automatic vehicle identiers using Bluetooth devices that were installed at adjacent incoming and outgoing roads of 12 intersections in total, see their location in Figure 4.10b). These identiers are able to detect vehicles equipped with another Bluetooth device, which enables to assign origin and destination roads of individual vehicles. For the estimation of the remaining turning ratios (since there are more than 12 intersections in total), the information on road importance is used (FRC), and then the optimization problem minimizing the deviation of predicted and actual ows is solved.

Finally, we also get the estimated density values for every road ρ j for every minute of the 8th of January 2021 from 6 am to 9 pm, as well as inows and outows at domain boundaries.

Notice that in this scenario inows are time-dependent functions. Estimation of free-ow speed, turning ratios, vehicle density and boundary ows is described in more details in [START_REF] Rodriguez-Vega | Urban network trac state estimation using a data-based approach[END_REF]. In order to get density values all over the continuum plane, i.e., at every point in Grenoble downtown (not only at physical roads), we divide each road into 10 parts, and at the boundary between each part we set a group of vehicles. Thus, there is a known number of vehicles at 10 points of every road. We then assume that all these cars contribute to the global density around d 0 = 70 m from its positions using KDE method, see Section 3.1.3.2. We also use KDE for the inow values, as it was done in the previous example.

The results are depicted in Figure 4.12, where the comparison of two density distributions is shown. Again, we see that in both cases the distributions look quite similar. The rst possible reason for these distributions to be non-identical is the probabilistic nature of turning ratios in reality opposed to deterministic nature in numerical simulation. Another reason is that the NEWS model does not include trac lights, as well as it is not able to capture accidents or the eect of pedestrians crossing a road. Moreover, the NEWS model does not take into account parking lots. Thus, in reality parking vehicles are seen as stationary objects that do not contribute to the trac ow any more, while in NEWS-driven system these vehicles stay in the domain and create congestions, since the NEWS model is a conservation law.

Another source of mismatch could be induced by data on inows and outows. The problem is that the data represent estimated measurements of the ows in the city that we can not enforce in our system, since there is always a demand-supply problem that needs to where ext is used in the subscript to highlight that these functions depend on what happens outside the domain. Thus, the available data are not related to demand and supply at domain boundaries but to actual inow φsource and outow φsink of the system (hats are used to denote the measurement data).

To understand which problems can be provoked by these issues, let us consider some measured outow φsink , which in turn is also just a result of solving the minimum between demand and supply, i.e., φsink = min{D(ρ), S ext }, (4.37) where demand D(ρ) depends on the measured density, which might be something dierent than the one we get from the numerical simulation of NEWS-driven density.

For the numerical simulation, the best thing we can do with the measured outow data However, it follows from (4.37) that φsink ≤ S ext , where the equality holds in case of congested trac. If the trac is not congested, then setting the external supply to be equal to measured outow might lead to blocking vehicles at domain exit instead of letting them come out.

Two distributions are again compared by using the weighted SSIM averaged over 9 zones as in the previous case using (4.36) and (4.34). The result is depicted in Figure 4.13a), while Figure 4.13b) is referred to SSIM for each zone computed using (4.34). Notice that the zone numbering here is preserved the same as in Figure 4.8a). The worst captured zones are 1 and 2 located on the upper part of the city, and the best results are achieved for zones 5, 4 and 8.

A possible reason might be the fact that the cars get stuck at the bottom of the city in the real experiment, while they move more freely in a PDE governed system (as in Section 4.2.3). In general, notice that the best results are achieved for the time when the congestion level is the highest, as we can see from Figure 4.13a). This is related to the weighting parameters used for the calculation of SSIM (4.36). Weights tend to introduce more noisiness into computation, when there are only a few cars in the city. Finally, recall that the real-life data are also an approximation, since these densities are obtained by the estimation procedure that is not error-free due to the lack of sensors at every road. On average, the total SSIM is around 0.8 (80% accuracy), which indicates that two density distributions are still quite close.

Reproducibility of the results

It is worth noting that the source code used for model validation is an open source project that you can nd here: https://github.com/Lyurlik/multidirectional-traffic-model.

The README.md le contains all the essential information about the code structure and the data les such that anyone can get use of it for dierent purposes (e.g., prediction of trac density for a dierent urban network). Thus, the results are made to be reproducible. This code is used to produce two dierent vehicle density distributions: by running a numerical simulation of NEWS model (4.25), and the other one is the reconstructed density from data obtained from real sensors.

In order to run the code, one needs to have the following les: Network topology 1. ../ModelValidation/IntersectionTable.csv contains information about intersections:

x and y coordinates of every intersection (columns 1 and 2), its ID (column 3) and whether it is a node on border (column 4), which means that this intersection is located at domain boundary through which vehicles may enter (inows), or exit (outows); 2. ../ModelValidation/RoadTable.csv contains information about roads: ID1 and ID2

(columns 3 and 4) are the id's of corresponding intersections that the road is connecting, ID_road (column 5) is the road's ID, max_vel (column 6) is its free-ow limit estimated from real measurements, then we have number of lanes (column 7) and road's length (column 8); ID1 of incoming road (column 1), ID2 of outgoing road (column 2) and the turning ratio between these roads (column 5).

Data from real sensors

Code structure

The main le of the project is mainwindow.cpp: in its constructor we specify the le names to be loaded, simulation starting time (line 27) and simulation step size (line 29).

The paths to les containing network and density data are also specied there. We can also change the weighting parameter µ used to approximate parameters for every cell (line 4), and parameter d 0 (line 5) is the standard deviation parameter used for Gaussian kernel estimation.

Other important classes are:

UrbanNetwork, which contains all the network geometry information (this is the place, where all the network les are read). This network is used for both density predictions.

In its function loadRoads, one needs to specify the minimum distance between the heads of two consequential vehicles, which is then used for the computation of ρ m ax (we set it to 6 m).

NEWSModel, which contains translation procedure of all network and intersection parameters into NEWS formulation (function processIntersections ). After all parameters are dened in NEWS terms, it calls constructInterpolation function that approximates these parameters on every grid point in space using their known values at every intersection. Then update is performed, where the Godunov numerical scheme is applied for the state update using NEWS model. There is also a function getSSIMDi_mean_weighted used to compute the weighted SSIM between two densities (4.36).

GrenobleData, where all the data estimated from the real-life experiments are loaded.

In function reconstructDensity, the density initially given for each road is dened for every cell. Thereby, every road is divided in 10 parts, and density values are presented as points on the border between these parts. Then, Gaussian kernel estimation is used to determine density for every cell in the domain.

TracSystem, which implements concurrent threads for a parallel NEWS simulation relative to the main visualization thread.

Discussions

In this section, we demonstrated how the NEWS model predicts the trac state compared to the ground true results. For this purpose, we deployed the Godunov scheme in 2D to run a numerical simulation of vehicle density governed by NEWS model. The predicted density was then compared to the reference density distribution obtained from two dierent sources:

microsimulator Aimsun (Figure 4.7) that produced vehicle trajectories using synthetic data, and estimated density from data coming from sensors installed in Grenoble downtown (Figure 4.12). These data from real-life measurements are available for download at our experimental platform GTL Ville. To enable a quantitative comparison of vehicle densities, we implemented the measure of structural similarity, which is an index used to reveal to which extent are two images similar.

In both cases, the predicted density looked quite similar to the reference density distribution, i.e., the index revealed at least 80% similarity in both cases. In the rst case (NEWS vs Aimsun), the predicted densities are similar to 90%. There are several sources of mismatch. The rst one to name is that in Aimsun as well as in a real urban network, vehicles are restricted to move along physical roads, while in a PDE-driven system this is not that strict, since the underlying surface is a 2D continuum plane. Another reason is that, in reality, turning ratios are only expectation values rather than deterministic values as in NEWS model. Thus, the proportion of cars turning to one or another road may be dierent from the given xed proportion due to the nite duration of experiment. The NEWS framework assumes also that there are 4 possible directions for trac at every intersection. If there are at least two roads with dierent orientation angles θ going approximately in the same direction (e.g., North-East), then the NEWS framework introduces mismatches due to approximations it makes with projection matrices. Another assumption of NEWS relies on optimally designed networks in terms of capacities. This assumption means that the usage of roads is related to its capacities, i.e., more cars tend to turn to highways than to roads of minor importance.

In the second case (NEWS vs real data), we had to deal with some additional problems.

The rst problem to name is that not all the roads in Grenoble downtown are equipped with sensors due to high economical cost. Thus, the data available for every road come from the estimation procedure that is not free of errors. The second problem to be mentioned is related to boundary ows. Namely, the measured ow data can not be enforced in our NEWS model.

They can only be suggested as boundary conditions, which are fullled only if the trac state at the boundary points admits it (demand-supply problem). Finally, we can also name the parking lots as sources of errors. In the real-life experiment, cars stop at parking lots and are not detected by sensors, thus, the reconstructed density does not take them into account.

On the contrary, the NEWS model is based on a conservation law. Thus, if vehicles stop at parking lots before exiting the domain, they are counted as obstacles that create congestions.

However, the validation results revealed a good agreement with the prediction. The NEWS model appeared to be a good approximation of multi-directional trac in urban networks, which was conrmed by real-life measurement data. The code used for model validation is

available as an open source project, and the NEWS model can be deployed to predict trac on any urban network of interest in further research projects. In the next section, we will suggest a boundary control technique used to mitigate congestions in a urban network with multi-directional trac governed by NEWS model.

Boundary control for multi-directional trac

In this section, we design a boundary control law for a multi-directional urban trac governed by NEWS model. This controller acts such that initially congested trac achieves the best possible desired equilibrium. This steady state provides congestion mitigation, which equivalently implies throughput maximization of the transportation network. Thus, the control goal here is similar to what has already been done for uni-directional urban trac in Section 3.4.

Our main contribution here is an extensive analysis of possible desired space-varying proles that the system can achieve. In this section, we will see that nding an admissible steady state is far from being trivial for multi-directional trac systems. Moreover, we will also use Lyapunov methods to show the exponential convergence of the trac state to the desired equilibrium.

NEWS model for congested trac regime

Let us consider the NEWS model (4.25) for a special case of congested trac regime. Restricting to one trac regime allows to considerably simplify the system for analysis, as it was done in Sections 3.4 and 2.2. A congested trac in some urban area will be controlled from its downstream boundary such that the level of congestion is minimized under the constraint that ρ(x, y, t) ≥ ρ c (x, y) ∀(x, y, t) ∈ Ω × R + for all 4 directions. Recall that Ω ∈ R

φ EN = min{α EN v E ρ E , α EN φ max,E , β EN ω N (ρ max,N -ρ N ), β EN φ max,N }. (4.39)
In the congested trac regime, the minimum in demand-supply problem is always resolved to the benet of supply. This in turn implies for (4.39) that

φ EN = β EN ω N (ρ max,N -ρ N ).
Using this expression and xing ρ 0 (x, y) ∈ [ρ c (x, y), ρ max (x, y)] ∀(x, y) ∈ Ω as initial density distribution condition, we can now introduce the following IBVP for the NEWS model (4.25) North in blue (u N ), East in dark red (u E ), West in green (u W ) and South in orange (u S ).

that describes the dynamics of congested trac density on some urban area:

         ∂ρ ∂t = 1 L (I -B)W (ρ max -ρ) - ∂[C W (ρ max -ρ)] ∂x - ∂[S W (ρ max -ρ)] ∂y , ρ(x, y, t) = u(x, y, t), ∀(x, y) ∈ Γ out , ρ(x, y, 0) = ρ 0 (x, y), (4.40)
where Γ out ⊂ Ω represents a set of boundary points (x, y) associated with the domain exit (downstream boundary):

Γ out = (y max , x max , x min , y min ) T .

The congested trac state governed by the NEWS system (4.40) is controlled at the downstream boundary Γ out by specifying the control vector u = (u N , u E , u W , u S ) T . See Figure 4.14, where the arrows are used to denote the boundaries to be activated for control of trac in each direction.

Finally, C, S, W and B in (4.40) are all 4 × 4 matrices such that C and S are diagonal matrices, W is a positive-denite diagonal matrix, and B is a non-negative matrix:

C =     cos θ N 0 0 0 0 cos θ E 0 0 0 0 cos θ W 0 0 0 0 cos θ S     , S =     sin θ N 0 0 0 0 sin θ E 0 0 0 0 sin θ W 0 0 0 0 sin θ S     , W =     ω N 0 0 0 0 ω E 0 0 0 0 ω W 0 0 0 0 ω S     , B =     β N N β N E β N W β N S β EN β EE β EW β ES β W N β W E β W W β W S β SN β SE β SW β SS     .

Desired equilibrium

Let the error from the desired equilibrium be denoted by ρ(x, y, t), which is dened as in (1.9). To keep everything simple, the desired space-varying equilibria take values only in the congested regime range, i.e., ρ d (x, y) ≥ ρ c (x, y) ∀(x, y) ∈ Ω.

The time derivatives of state and error coincide, which in combination with (4.40) yields

∂ ρ ∂t = 1 L (I -B)W (ρ max -ρ d -ρ) - ∂[C W (ρ max -ρ d -ρ)] ∂x - ∂[S W (ρ max -ρ d -ρ)] ∂y . (4.41)
Let us also write the dynamics of the desired density that is a time-constant function:

∂ρ d ∂t = 0 = 1 L (I -B)W (ρ max -ρ d ) - ∂[C W (ρ max -ρ d )] ∂x - ∂[S W (ρ max -ρ d )]
∂y .

(4.42)

If we subtract (4.42) from (4.41), we also obtain the pure error term dynamics:

∂ ρ ∂t = 1 L (B -I)W ρ + ∂[C W ρ] ∂x + ∂[S W ρ] ∂y . (4.43)
We seek to nd a desired density distribution that corresponds to congestion minimization, and then to design a boundary control that stabilizes the trac state to that desired equilibrium. Thereby, the desired density prole must remain in the congested regime, i.e., ρ d (x, y) ≥ ρ c (x, y) ∀(x, y) ∈ Ω, and its values at the boundaries should be proportional to the maximal densities at the corresponding boundary coordinates, i.e., ∃γ ∈ [1/3, 1] such that

ρ d (x, y) = γρ max (x, y), ∀(x, y) ∈ Γ out . (4.44)
The range of constant γ is related to the requirement for ρ d to stay in the congested regime, since with γ = 1/3 the desired state equals the critical density ρ c (recall that ρ c = 1/3ρ max in case of triangular FD). Thus, we need to determine constant γ that provides congestion minimization given (4.44

L (I -B)W ρ = ∂[C W ρ] ∂x + ∂[S W ρ] ∂y . (4.46)
Then, we compute the desired state ρ d (x, y) ∀(x, y) ∈ Ω by performing the following steps:

1. Initial guess: set the desired density at the downstream boundary Γ out equal to the corresponding critical values, i.e., pick the lowest possible γ = 1/3, which leads to ρ(x, y) = 2 3 ρ max (x, y), ∀(x, y) ∈ Γ out .

2. Consider the same area in Grenoble city center as in all previous sections, also see 3. Discretize the PDE system given by (4.46). For convenience, we consider the PDE for the density in the North direction, and then the same steps should be done for the remaining directions. In accordance with the upwind scheme [START_REF] Courant | On the Solution of Nonlinear Hyperbolic Differential Equations by Finite Dierences[END_REF] used to provide the correct direction of information propagation in a ow eld, the discretization scheme of cos θ N ω N ρN and sin θ N ω N ρN depends on the signs of cos θ N and sin θ N , that is ∀(i, j) ∈ {1, . . . , n x } × {1, . . . , n y }:

cos θ N (i, j) > 0 : cos θ N (i + 1, j)ω N (i + 1, j)ρ N (i + 1, j) -cos θ N (i, j)ω N (i, j)ρ N (i, j) ∆x , cos θ N (i, j) < 0 : cos θ N (i, j)ω N (i, j)ρ N (i, j) -cos θ N (i -1, j)ω N (i -1, j)ρ N (i -1, j) ∆x .
Therefore, we can dene 4 × 4 diagonal matrices Q x , Q y , R x and R y that capture the upwind scheme as cos θ N (i, j) > 0 :

Q x N N (i, j) = cos θ N (i + 1, j)ω N (i + 1, j) ∆x , R x N N (i, j) = 0, else : Q x N N (i, j) = 0, R x N N (i, j) = - cos θ N (i -1, j)ω N (i -1, j) ∆x ,
and the same can be written for sin θ N and y-direction, for which we x i and vary j.

4. Dene also a 4 × 4 matrix P as:

P (i, j) = 1 L(i, j) (B(i, j) -I)W (i, j) - |C(i, j)|W (i, j) ∆x - |S(i, j)|W (i, j) ∆y .
Using the denition of matrices P , Q x , Q y , R x and R y , we can now write the PDE system for ρ given by (4.46) in a discretized form that reads ∀(i, j) ∈ {1, . . . , n x } × {1, . . . , n y }:

P (i, j)ρ(i, j) + Q x (i, j)ρ(i + 1, j) + Q y (i, j)ρ(i, j + 1) + R x (i, j)ρ(i -1, j) + R y (i, j)ρ(i, j -1) = 0. (4.47)
Notice that ρ(0, j), ρ(n x + 1, j), ρ(i, 0), ρ(i, n y + 1) take the values from the boundary conditions (ghost cells).

5. System (4.47) is solved using the alternating direction implicit method, which is equivalently known as dimensional splitting, see [START_REF] Peaceman | The numerical solution of parabolic and elliptic dierential equations[END_REF]. At each iteration, rst x and then y steps are performed. At each x step, the terms ρ(i, j -1) and ρ(i, j + 1) take xed values from the previous iteration, while ρ(i -1, j) and ρ(i + 1, j) are xed for each y step. At x step, our system (4.47) is solved for every j by the block tridiagonal matrix algorithm, while at y step this algorithm is applied for every column i.

6. Thus, we have described the numerical method to obtain a solution ρ for the PDE (4.47), which is not necessarily optimal. Since this PDE (4.47) is a linear system, αρ for α ∈ [0, 1] is also its solution. Let us estimate the parameter α * that provides the optimal equilibrium as in Problem 4.1.

Consider again the desired state ρ d that is obtained from (4.45) as

ρ d = ρ max -αρ. (4.48)
By choosing α = 0 we obtain ρ d = ρ max , while by choosing α = 1 we achieve ρ d = ρ c at the boundaries (see step 1 and use ρ c = 1/3ρ max ). This implies that by taking an intermediate value of α, we guarantee the congested trac regime at the boundaries. Let us calculate α * that provides for the desired state ρ d to be as close as possible to ρ c while staying in the congested regime (see Remark 4.2), for which in general we can write:

ρ d ρ c ≥ 1 ⇒ ρ max -αρ 1/3ρ max ≥ 1 ⇒ 3 -3α ρ ρ max ≥ 1 ⇒ α ≤ 2 3 ρ max ρ , ∀(x, y) ∈ Ω.
From the discussion above, it follows that the optimal state is achieved if ∃(x * , y * ), for which α * = min (x,y)∈Ω s∈{N,E,W,S} 2 3 ρ max,s (x, y) ρs (x, y) .

(4.49)

Thus, the optimal desired prole in the whole domain Ω can be obtained from (4.48) for optimal α = α

* ρ d (x, y) = ρ max (x, y) -α * ρ(x, y), (x, y) ∈ Ω, (4.50)
with α * being given by (4.49). To get an expression for the optimal desired prole ρ d (x, y) at the boundary ∀(x, y) ∈ Γ out , we take ρ from step 1 and insert it into (4.50), which yields

ρ d (x, y) = γ * ρ max (x, y), with γ * = 1 - 2 3 α * , ∀(x, y) ∈ Γ out . (4.51)
As a result, we have derived the expression for optimal desired equilibrium (4.50) that corresponds to the state of minimal congestion. As we can see, it depends on the solution ρ of system (4.47) that can be found numerically using alternation direction implicit method.

We could also get an explicit formula for optimal equilibrium at the domain boundary Γ out given by (4.51). This is a useful expression, since it directly determines the boundary control variables u(x, y) from (4.40), see details below. In order to prove the convergence to the desired prole, we have to assume that the main directions of transportation coincide with the cardinal directions, which for example holds for a Manhattan grid type of trac networks.

Boundary control design

Assumption 4.1

The matrices C and S from (4.40) are constant in space, e.g., they can be dened as:

cos θ N = 0, , cos θ E = 1 cos θ W = -1, cos θ S = 0, sin θ N = 1, sin θ E = 0, sin θ W = 0, sin θ S = -1. (4.52)
In general, the further analysis requires these variables to be just constant in space, but we choose (4.52) for simplicity. We also make an assumption on supply ratios:

u(x, y) =     ρ d,N (x, y max ) ρ d,E (x max , y) ρ d,W (x min , y) ρ d,S (x, y min )     , ∀(x, y) ∈ Γ out , (4.53)
then ∃K, k > 0 such that

ρ(t) -ρ d 2 L 2 e -kt K ρ(0) -ρ d 2 L 2 ,
i.e., the state ρ(x, y, t) exponentially converges to the desired equilibrium ρ d (x, y) ∀(x, y) ∈ Ω as t → ∞.

Remark 4.4

Although for simplicity of the proof we assumed a regular Manhattan grid structure (Assumption 4.1), the feedforward boundary controller (4.53) can be applied to a more general network, as will be shown on a numerical example, for which we take the network of Grenoble downtown. Let us also introduce a diagonal 4×4 matrix composed by exponential functions as follows:

E =     e y 0 0 0 0 e x 0 0 0 0 e -x 0 0 0 0 e -y     .
This matrix is used as weights in each direction that helps achieving exponential convergence.

We dene the following Lyapunov function candidate:

V = xmax x min ymax y min ρT W DE ρ dydx = xmax x min ymax y min (ρ 2 N ω N D N e y + ρ2 E ω E D E e x + ρ2 W ω W D W e -x + ρ2 S ω S D S e -y ) dydx, (4.55)
where D N , D S , D W and D E are the diagonal elements of matrix D.

V = xmax x min ymax y min 1 L (W ρ) T DE(B -I) + (B T -I)DE W ρ dydx + 2 xmax x min ymax y min (W ρ) T DE ∂[C W ρ] ∂x + ∂[S W ρ] ∂y dydx. (4.57)
Let us now denote the rst term of (4.57) as V1 and the second term as V2

ω q (x, y) xmax x min ymax y min e x D E ω E ρ2 E + e -x D W ω W ρ2 W + e y D N ω N ρ2 N + e -y D S ω S ρ2 S dydx, (4.60)
where we have used the fact that the kinematic wave speed is positive by denition, i.e., ω > 0.

The integral on the right-hand side of (4.60) coincides with the Lyapunov function (4.55).

This means that by inserting (4.59) into (4.58) and also by using the bound from (4.60), we can write

V = V1 + V2 ≤ V2 ≤ -kV,
where k ∈ R + is a positive constant k = min (x,y)∈Ω q∈{N,E,W,S} ω q (x, y).

One can also prove that the state ρ converges to zero in L 2 norm exponentially. Indeed, note that the Lyapunov function V from (4.55) denes an equivalent norm on the density space:

m ρ 2 L 2 V M ρ 2 L 2 with m = min (x,y)∈Ω q∈{N,E,W,S} ω q (x, y)D q E q (x, y), M = max (x,y)∈Ω q∈{N,E,W,S} ω q (x, y)D q E q (x, y).
By the exponential convergence of the Lyapunov function we have

V (t) e -kt V (0), therefore ∀(x, y) ∈ Ω ρ(t) 2 L 2 e -kt M m ρ(0) 2 L 2 .
Remark 4.5

Assumption 4.2 on space-independent B can be relaxed, if it is possible to nd such a matrix D that satises inequality (4.54), and whose elements D E (y) and D W (y) may depend on y, while D N (x) and D S (x) may depend on x.

Numerical example

Finally, we demonstrate how a boundary control given by (4.53) works in practice using a selected area Grenoble downtown with a total surface of around 1.4 × 1 km 2 , which is the same area as in all previous sections of this chapter, e.g., see density. Therefore, we also applied the continuation method to be able to dene the model predicting the evolution of vehicle density in the vicinity of an intersection. This method was used to turn an ODE into a PDE, which allowed us to obtain a macroscopic continuum model for one intersection. Since every intersection was described in a unied way, we nally applied the inverse distance weighting to dene all the intersection parameters for every point in a continuum plane. The derivation of the NEWS model was done analytically using only one assumption on network structure. Namely, urban networks under consideration must be well-designed in terms of maximal ows, i.e., if vehicles move at maximal ow before an intersection, they continue using the road capacity at maximum after the turn.

As a result, in Section 4.1, we derived the NEWS model (4.29) that predicts the evolution of trac in four cardinal directions. The propagation of trac ow in each direction is driven by the demand-supply concept that uses a fundamental diagram. Moreover, vehicles moving in some layer can switch to another layer, i.e., there is a mixing between dierent layers, which is an important aspect due to its physical ubiquity.

The mathematical properties of the NEWS model have also been analyzed. The PDE system was shown to be hyperbolic for any parameter set, as it is often the case for conservation law based trac models. Being able to classify a model as a hyperbolic PDE signicantly simplies the analysis for future tasks such as explicit control design or steady state estimation, since a lot of analytical results have already been elaborated for this type of systems. It was also shown that the model represents a conservation law with trac density being the conserved quantity. Moreover, it was shown that its state is bounded, which is a realistic assumption for trac modeling, since vehicles can not be located innitely dense.

The model prediction results have been validated in Section (4.2) using microsimulation Aimsun, and experimental platform GTL Ville that provides real-time data from a network of real sensors installed in Grenoble downtown. The validation results revealed that the density distribution predicted by NEWS model stays in a good agreement with the reference density, i.e., 90% of similarity with Aimsun and 80% similarity with the real-life experiment. Although the validation results proved a high prediction quality with the NEWS model, it is however not completely error-free, since it is based on several assumptions that do not necessarily hold for a general trac situation in reality. The model validation with real data was made to be an open source project such that the results are reproducible and can be used for future studies.

In the last Section 4.3 of this chapter, the NEWS model has been investigated from the control perspective, whereby we restricted to the congested trac regime for simplicity. We have analyzed the class of admissible desired equilibria that must satisfy a certain system of PDEs. We have posed and solved the problem of nding an equilibrium state that corresponds to the state of congestion minimization in a urban network under the constraint that its range must remain in the congested regime. Further, we proved the exponential convergence of a congested state controlled from the boundary to this desired equilibrium using Lyapunov methods. Thereby, for the proof, we had to assume several restrictive assumptions such as Manhattan grid like topology of underlying urban networks, and similar turning ratio patterns at intersections. These assumptions were introduced only to simplify the proof and are not necessarily real restrictions needed to provide the functionality of the suggested controller, which was shown in a numerical example. Finally, we again used the same area in Grenoble downtown as in Section 4.2 to demonstrate the performance of the derived feedforward boundary controller with the help of a numerical example. It was shown that the controller acts such that the trac density converges to the desired optimal equilibrium in nite time, which is related to the hyperbolic nature of NEWS PDE.

An appealing direction for future studies might be nding equilibria proles that admit mixed trac regimes. Another possible extension may include elaborating a boundary controller under the constraint that the activation boundary is a set of points on real roads rather than a continuous line, as it is assumed to be in a PDE-driven model. More research perspectives are given in Chapter Conclusions and Perspectives.

Conclusions and Perspectives

Summary of contributions

This thesis was devoted to trac control on urban networks. Thereby, we have used the macroscopic modeling approach that enables characterizing trac as a uido-dynamic system, and its state is described in terms of vehicle density. This is a benecial form to analyze trac on large-scale networks, since it allows us to consider trac as a single dynamic object rather than a collection of vehicles. By considering trac on a macroscopic scale, the model-based control design becomes scalable and easy to validate even for arbitrarily large urban networks.

We were mainly interested in predicting congestion formations in large transportation networks and in dissolving them through the boundary control, i.e., by setting appropriate on-and oramps. Reaching such a goal implies taking a big step towards the development of intelligent transportation networks. In this work, we proceeded towards the global goal of trac control on any urban network step-by-step. First, trac on single roads was considered and analyzed for control. Then, we looked at trac on urban networks with a preferred direction of motion.

Finally, we developed our own approach to modeling trac with any direction of movement.

Let us summarize the main contributions that have been achieved at each of these stages.

Trac control on roads

We considered trac evolving on single roads of nite lengths using the LWR approach in Chapter 2. The goal of this chapter was to design a boundary control law that acts such that any desired space-and time-varying vehicle density prole is tracked for asymptotic time. To be admissible, the desired vehicle density must be governed by the LWR PDE as well. A space-and time-dependent desired state can be seen as a generalization of any desired state.

Stabilization of trac to some desired equilibrium is just a special case of trajectory tracking, and can be achieved by applying the same control law. Also notice that non-stationary proles are more frequent in real trac situations.

There are however two major diculties that arise for LWR-driven trac control design.

The rst diculty is that there are no classical solutions to the LWR PDE due to the nonlinearity of the fundamental diagram even for a smooth initial datum. Characteristic lines propagate with dierent speeds, and whenever they cross, the discontinuities in the solution arise. This requires considering solutions in the weak sense, and the unique solution is the one that satises the Lax entropy condition. Thus, treating discontinuities is a tedious procedure that gets even more complicated, if we design control to track some trajectory that also does not exist in the classical sense.

The second diculty is that considering trac governed by LWR PDE on nite roads requires to include boundary conditions into the consideration. One needs to consider these conditions also in the weak sense, since in general they can not be imposed for all time, which equivalently means that the demand-supply problem needs to be solved. This triggers a general problem, when the boundary conditions need to be designed for trac control. We can not even analyze how the system evolves under the eect of control in a straightforward way, since the controller may not be accepted by the system.

Thus, we rst considered a more simple case by restricting only to congested trac in Section 2.2. Considering trac only in the congested regime considerably facilitates the structure of the trac system that becomes linear, as well as boundary conditions can be imposed in a strong sense. We also added a general unknown in-domain disturbance to the system.

This was done to capture the unmeasured inuence of vehicles that come from minor roads and aggravate congestion. Thus, we designed the optimal boundary controller that acts to minimize the deviation from the desired trajectory while attenuating disturbance. The deviation is minimized in sense of L 2 norm if the boundary controller from Theorem 2.1 is applied. We also achieved the minimization of the deviation in sense of L ∞ norm by applying the boundary control law from Theorem 2.2. When it comes to practical applications, the minimization norm should be chosen based on the available knowledge of the disturbance source, i.e., L 2 norm should be minimized if the unmeasured ow of vehicles originates from many minor roads, while we choose the L ∞ minimization if the disturbance is related to a stream of vehicles originating from a single major road. In both cases, the designed controllers have feedback parts used to attenuate the disturbance. Although the controllers are optimal, in general, the in-domain disturbance can not be completely rejected by acting only from the boundary due to a nite propagation speed of information, which is a general property of hyperbolic systems. The material of Section 2.2 was published in [START_REF] Tumash | Robust tracking control design for uid trac dynamics[END_REF].

In Section 2.3, we extended this result by considering the full LWR system (without disturbance), for which no classical solutions exist. The trac state as well as the desired trajectory are governed by the LWR PDE, which implies that they can be in dierent trac regimes. Solving a boundary control task for such a system is a much more general result, since real-life trac usually builds non-trivial density patterns, e.g., it can be partially in the free-ow trac regime and partially congested. We could solve this problem by considering the LWR system in Hamilton-Jacobi formulation that represents its integral (cumulative) form.

This formulation enabled us to obtain a continuous solution to the LWR system explicitly for large enough time. It was possible due to a convenient shape of the triangular fundamental diagram. The solution to the LWR system in H-J formulation was then used to analyze the boundary conditions in terms of control restrictive functions. The cumulative H-J formulation thus enabled us to estimate the time periods during which the road boundaries can not accept the proposed controller values. The main result of this section is the boundary control law with feedback term in Theorem 2.3, and we can also say that it is the main result of the whole Chapter 2. We have shown that even with time periods during which no control can be imposed, the system exponentially converges to the desired trajectory in Hamilton-Jacobi formulation. This result implies that under this control law, the number of vehicles on the controlled trac road converges to the desired number of vehicles pointwise. This also means that the vehicle density converges to the desired trajectory in the integral sense over arbitrarily small intervals. Thus, we suggested a general approach to solve any control task for LWR-driven mixed-regime trac on a single road by acting only from the road boundary, and it was published in [START_REF] Tumash | Boundary Control Design for Trac with Nonlinear Dynamics[END_REF].

Uni-directional trac on networks

After the boundary control problem for a general trac state was solved for a single road, we were seeking to nd a holistic approach to solve any control tasks for urban trac in Chapter 3.

This was done within the same modeling approach but in two dimensions, i.e., we used the 2D LWR model to predict the evolution of trac on a urban network by approximating it as a 2D continuum plane. Namely, trac is again seen as a uid that propagates along a 2D plane that is a rectangular domain whose size is determined by the size of the underlying network that represents a set of roads and intersections. The network infrastructure is incorporated as an explicit space-dependency of the fundamental diagram that captures various speed limits and transportation capacities along the roads of urban network. The FD parameters are approximated everywhere in the continuum, and its values are mostly inuenced by the parameters of the closest roads. The direction of trac ow propagation is determined by the direction eld that depends on the network geometry. The structure of the model implies that the integral lines of the direction eld do not cross. Moreover, to be well-dened on a continuum plane, we can apply this model only to urban networks that contain no loops. These two requirements (no crossing lines and no loops) makes this model applicable only to uni-directional networks, i.e., there must be a preferred direction of motion. This restriction makes it dicult to use this model for a general multi-directional trac. However, the 2D LWR model can be useful in a variety of situations. For example, many people driving simultaneously to the business district (e.g., at 9 am on a weekday) create a uni-directional trac pattern.

The second space dimension makes it dicult to analyze the 2D LWR model in its original form for the control design. For example, it was not clear which boundary point should be actuated such that some area inside the domain is aected. In Section 3.2, we elaborated a technique that turns this 2D model into a parametrized system of 1D LWR equations with an explicitly space-dependent FD. This technique is the main contribution of Chapter 3. It is based on the curvilinear coordinate transformation that scales and rotates the space such that it is then treated as a continuum, in which trac propagates along straight lines (as in Figure 3.7). Thus, the 2D LWR model was rewritten in new coordinates (3.34), and it is seen as a continuum plane composed of inhomogeneous roads. The big advantage of this system is that we could apply similar control techniques as in the previous chapter but handling the explicit space-dependency of FD and an additional space parameter.

In Section 3.3, we discussed the equilibria that can be achieved in urban networks given inows and network structure (published in [START_REF] Tumash | Equlibrium manifolds in 2D uid trac models[END_REF]). Then, this result was used in Section 3.4 to obtain the optimal equilibrium state corresponding to congestion mitigation. Thereby, we considered a simplistic case of trac being only in the congested regime. The boundary controller is given in Theorem 3.1, and it was shown that it is able to drive congested urban trac to an equilibrium of maximal throughput (published in [START_REF] Tumash | Topologybased control design for congested areas in urban networks[END_REF]). Next, in Section 3.5, we solved the problem of boundary control design for mixed-regime urban trac such that it tracks any desired space-and time-dependent trajectory. This is the same problem as considered for 1D case, and thus it could also be solved using the H-J formulation. This was however more technically involved and not that straightforward due to the space-dependency of the Hamiltonian. The designed boundary controller is presented in Theorem 3.2. This result is a general solution to the boundary control problem for any urban network that has a preferred direction of motion that can achieve any time-and space-dependent prole governed by the 2D LWR model.

Finally, in Section 3.6, control was designed in a dierent way than in all the preceding parts. Namely, we demonstrated the ability to control urban trac using the variable speed limit applied continuously in space and time (in-domain controller). The VSL controller given in Theorem 3.3 is able to drive urban trac to any desired equilibrium. The dierence with respect to the previous boundary control result is that this desired equilibrium is not even restricted to satisfy the conservation law equation. This is possible, since the controller is designed such that the closed-loop system loses the conservation law structure, i.e., it feedback linearizes the system. Thus, the desired equilibrium is bounded only by the maximal density that is determined by the network structure. It was shown that the controller is smooth in space for some special dependencies of FD on VSL in Theorem 3.4. Then we also designed an equilibrium that provides that the system is used at its maximal theoretical throughput by the maximal possible number of drivers. The material presented in Sections 3.5 and 3.6 was sent for a publication [START_REF] Tumash | Boundary and VSL Control for Large-Scale Urban Trac Networks[END_REF].

Multi-directional trac on networks

In Chapter 4, we addressed the main limitation of the previous results that have been developed only for uni-directional trac. A new model for trac with multiple directions was introduced in Section 4.1. This model is explicitly derived from the demand-supply concept for one intersection. Since a urban network usually contains much more than one intersection, we had to develop a new framework that can describe trac in a unied way for all intersections.

The main diculty was introduced by the fact that every intersection may have an arbitrary number of incoming and outgoing roads.

We suggested to introduce a projection matrix that assigns weights to every road with respect to 4 cardinal directions, which are North, East, West and South (NEWS). Thus, every intersection is approximated as if it would have 4 incoming and 4 outgoing roads in each direction, i.e., 16 pairs of origin-destination ows can be dened. This enabled us to formulate a trac ow model at every intersection in a unique way such that it predicts the rate of change in the number of vehicles at intersection given inows and outows in NEWS formulation. Further, the continuation method was applied to translate this ODE model into a PDE that describes the evolution of vehicle density in the vicinity of an intersection. This was done, since our goal was to derive a macroscopic continuous model, as in the previous chapters but allowing any possible direction. Thus, we obtained a unique model that describes the evolution of density in 4 direction layers at every intersection in the same way. Using approximation methods, we dened also the parameters of FD everywhere in the continuum.

Therefore, we obtained the NEWS model (4.25) that is the main contribution of Chapter 4.

It consists of mixing and transportation terms. The mixing term is responsible for modeling of inter-layer trac ow, which is an essential phenomenon that allows to capture turning ratios correctly. The transportation term describes the spatial propagation of trac ow in each direction layer that depends only on demand and supply functions of the corresponding direction.

In Section 4.1.4, the properties of this new model have been studied. It was shown that this model represents a conservation law. Its state is always positive and bounded by the trac jam density that is determined by the network topology. Then, we also showed that this model corresponds to a hyperbolic partial dierential equation for any possible parameter set. Being able to classify a trac model as a hyperbolic conservation law allows to consider the properties of its solution in the same way as it is done for other hyperbolic conservation laws such as 1D LWR, as well as the same nite dierence approximation method can be applied for numerical simulations.

The ability to consistently predict trac evolution on large-scale networks using the NEWS model was validated in Section 4.2. First, we used synthetic trac data obtained by running a congestion formation scenario on a commercial microsimulator Aimsun. It predicts the trajectories of individual vehicles given some network with dened road and intersection parameters as well as the inow rate. The same scenario and parameters were taken for the numerical simulation of trac with NEWS-driven dynamics. The results were compared using the structural similarity index that is a perception-based measure of similarity between two images (density distributions). The index revealed 90% of similarity meaning that the NEWS model is able to predict the evolution of trac with the accuracy of 90%. The main factor explaining that two distributions are not identical is a pure continuous nature of the NEWS model, which does not strictly prohibit cars to move outside of real roads.

Then, the NEWS prediction results were also validated with real data obtained from the experimental platform GTL Ville (Grenoble Trac Lab) that collects trac data from a set of real sensors installed in Grenoble downtown. These data are related to inows and outows at stationary points, turning ratios were obtained with Bluetooth devices, and TomTom data provide velocities. It is important to note that due to economic cost sensors are not installed at every road. The rest of the trac state in Grenoble downtown was reconstructed using heuristic algorithms. Thus, the NEWS trac density was compared to the trac density reconstructed from real measurements, and the similarity index revealed 80% of similarity.

The results are presented in Figure 4.12, which is the major contribution of Section 4.2 (and one of the main contributions of this whole chapter). An additional source of distribution mismatch comes through the disability to enforce the same inow and outow data as in the real-life experiment due to the demand-supply problem. The derivation of NEWS model and its validation with synthetic and real data was sent for a publication [START_REF] Tumash | Multi-Directional Continuous Trac Model For Large-Scale Urban Networks[END_REF].

Finally, we analyzed the NEWS model for the case of multi-directional trac in the congested regime, and designed a boundary control law to manage this trac in Section 4.2 (sent for a publication [START_REF] Tumash | Boundary Control for Multi-Directional Trac on Urban Networks[END_REF]). The control goal thereby was again to drive the system to the best equilibrium proving the maximal throughput under the constraint that mathematically trac could still be described by the NEWS system in congested trac regime. The most non-trivial thing was to analyze admissible equilibria, for which some PDEs had to be solved. The designed boundary controller is presented in Theorem 4.1. For its proof, we had to assume a Manhattan grid structure of network and a similar supply ratio pattern at every intersection, which are however not necessarily real restrictions that need to hold in order to achieve the convergence to the desired state with the boundary controller.

Perspectives and extensions

Based on the results of this PhD thesis, I see a plenty of appealing directions for the future research. The following open questions seem to be the most relevant ones:

In this thesis, trac was described in a quite simplistic way, since LWR model represents the most simple macroscopic model of trac. In general, it is well known that LWR modeling approach has several drawbacks, since it does not take many important phenomena such as bounded acceleration or capacity drop due to the transition from free-ow to congested trac regime. Moreover, a possible way to rene the description of trac is to take into account dierent driver classes based on their velocity (fast and slow). Thus, one could investigate the boundary control problem to track a desired space-and time-varying prole using a more sophisticated modeling approach that addresses limitations of LWR model (higher-order and multi-class models).

The 2D LWR model is restricted to describe trac on networks that have a preferred direction of motion, which is not realistic for general trac. A similar problem was encountered for MFD-based models. Recall that MFD becomes ill-dened in zones with heterogeneously congested roads, and partitioning algorithms had to be developed to divide a network into homogeneously congested zones. Thus, as a promising extension of research on macroscopic urban trac modeling, one could develop partitioning algorithms that would divide a urban network into zones that have a preferred direction of motion.

In both Chapters 3 and 4, the boundary controllers were designed for trac evolving on a continuum rectangular domain that approximates the underlying urban network.

As a result, we obtained control laws dened on a continuum line, which is not directly interpretable physically. It would be thus interesting to investigate this problem and to develop a method to map the boundary controllers dened on continuous lines into actuators that are set on specic points or intervals on real roads.

In Chapter 4 the NEWS model was derived. This model is a system of only four PDEs that is able to predict the evolution of multi-directional trac on urban networks quite accurately (which was conrmed with experimental data). It is important for future studies to rigorously characterize the mathematical properties of its solutions. Moreover, the last Section 4.3 presented the rst control result for trac governed by the NEWS model, however for a simplied case of congested trac. It would be thus appealing to extend this result to capture mixed-regime trac.

A. We dene y = x -tv and take the inmum over y:

M Ini (x, t) = Note that the space coordinates should not lie outside the road stretch, i.e., we must provide that x ∈ [0, L], which is achieved in four possible cases: In general, the inmum value is related to the number of crossings of critical density, and it cannot be exactly calculated for a general case unless additional assumptions on initial conditions are imposed.

B.2.4 Unique solution

In order to obtain the unique solution to H-J system (2.25), we need to nd the minimum of

B.3 H-J solution for large time

The solution to the Hamilton-Jacobi system is given by (B.5), which is divided into four dierent cases depending on the value of time t. Let us determine the solution for large time, where ε > 0 and T = min L v , L ω . This enables us to provide a further lower bound of (B.7):

M Ini (x, t) -M Up (x, t) ≥ t -L v T ε -L (ρ max + ρ c ) .

(B.8)

Finally, we use (B.8) to determine the lowest t, for which the term M Ini (x, t) -M Up (x, t) becomes non-negative ∀x ∈ [0, L]:

t ≥ L v + L ε (ρ max + ρ c ) min L v , L ω .
(B.9)

Following the same steps, we obtain that M Ini (x, t) -M Down (x, t) ≥ 0 for all such t that are not smaller than

t ≥ L ω + L ε (ρ max + ρ c ) min L v , L ω .
(B.10)

The earliest time after which we can neglect the eect of the initial condition on the solution is thus the minimum of (B.9) and (B.10): which is the solution of the Hamilton-Jacobi system for all t ≥ t min . This expression can be used to study the asymptotic behavior of systems governed by H-J PDEs with a triangular FD being their Hamiltonian.

t min = min L v , L ω 1 + L ε (ρ max + ρ c ) .

B.4 Necessary conditions for tracking desired state

In accordance with Problem 2.3, we should nd u in (t) and u out (t) ∀t ∈ R + such that the equality of M (x, t) and M d (x, t) up to some constant M 0 is guaranteed ∀x ∈ [0, L] as t → ∞.

For the equality of two minimum functions (2.60) and (2.61), it is sucient to provide the

B.5.1 Upstream boundary condition

The solution candidate M Up (ξ, t) is related to the cumulative vehicle number originating from the upstream boundary ξ min at initial time. where the inmum is taken over domain S Up that is dened exactly as in (3.48) but with ξ(0), t -T ∈ Dom(c Up ), where c Up = M Up (t) as in (3.42).

With the expression for the upstream boundary condition (3.43), the inmum problem can be rewritten as M Up (ξ, t) = inf where the inmum is now taken over domain S Down dened as in (3.48) with ξ(0), t -T ∈ Dom(c Down ), where c Down = M Down (t) as in (2.26).

Again using Assumption 3.1, we obtain that the inmum is achieved for the minimal traveling time interval T , which corresponds to:

T ω (ξ) = Finally, we dene the time when initial conditions will leave the system as a whole as t min , which is not dependent on η. Therefore, it should be computed as the maximum possible value of all t min (η) for particular η:

t min = max η∈[η min ,ηmax]
t min (η). Combining these results all together, we show the desired property: Hence, if we assume that γ can be manipulated independently for every pairwise ow, we can summarize the discussion above in the formula: (1 -γ) φEN + γ φNE = min{ᾱ N E DN , βEN SN }. This leads to the following transformation of (4. [START_REF] Donadello | Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions[END_REF]

Q(0) ≤ βEN SN , Q (1)
B =         β N N -1 β N E β N W β N S β EN β EE -1 β EW β ES β W N β W E β W W -1 β W S β SN β SE β SW β SS -1         . (B.

  Blue empty circles are related to (2.86). . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.12 a) Desired prole in space-time, b) L 1 error as a function of time for dierent control gains. Spatio-temporal evolution of the density (left) and of the absolute dierence between the real and the target state (right) for: c),d) k = 0; e),f ) k = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.1 Direction eld estimation (blue arrows) for: a) small µ = 10, b) large µ = 100. The gure is taken from Chapter 2 of [101]. Grey arrows indicate the direction of real roads in a Manhattan-grid network. . . . . . . . . . . . . . . . . . . . . . intermediate value of the weighting parameter µ = 50: a) estimated direction eld (blue arrows), b) integral lines of trac ow direction (tangent of d θ ). Grey lines represent real roads in Grenoble city center, arrows indicate the direction of trac. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3 Density reconstruction from car positions by KDE: a) 2D density is reconstructed from vehicle positions (blue dots) that move along a network with the geometry of Grenoble city center (grey arrows), b) 1D density estimation from equidistant vehicles with Gaussians having dierent standard deviations: d 0 = 25 m (upper plot) and d 0 = 100 m (lower plot). . . . . . . . . . . . . . . . 69 3.4 The vehicle density in a 2D domain with indicated upstream and downstream boundaries. The underlying network geometry corresponds to Grenoble city center (grey arrows). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.5 Schematic illustration of a network divided into N = 4 zones. The variables are dened with respect to reservoir i, which has its own MFD φ i (n i ) (tted data, in red) with the maximum ow φ max,i attained with the critical number of cars n c,i . The change in vehicle's accumulation n i (t) is determined by ows from N in,i = {j} and by ows into N out,i = {k}. . . . . . . . . . . . . . . . . . . 72 3.6 Steady-states obtained by: a) MFD-based model, b) numerical simulation of 2D LWR, c) microsimulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.7 Coordinate transformation mapping: (a) curved trajectories in Grenoble downtown in (x, y)-plane into (b) straight lines in (ξ, η)-plane. . . . . . . . . . . . . . 76 3.8 Coordinate transformation for one single line of constant η that is parametrized by ξ(η) ∈ [ξ min (η), ξ max (η)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.9 Steady state obtained by: a) numerical simulation of density governed by the 2D LWR system (3.9), b) model-based analysis of (3.34), c) density reconstruction from vehicle positions predicted by Aimsun. . . . . . . . . . . . . . . . . . . . . 89 3.10 A sketch of a urban network that contains a congested area (grey Manhattan greed) to be controlled from its downstream boundary (in red). . . . . . . . . . 90 3.11 A single η-line with inhomogeneous capacities. Its worst bottleneck occupies a road segment ξ * = [ξ * 1 , ξ * 2 ]. The ow-density relation Φ(ρ) is Greenshields FD: green and red areas indicate free-ow and congested trac regimes, respectively. 92 3.12 Control of urban trac from the downstream boundary: a) desired steady state distribution, b) initial state of trac jam, c) L 2 norm of the density error as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2 . 1 :

 21 Figure 2.1: Fundamental diagrams: a) triangular, b) Greenshields. Free-ow and congested trac regimes correspond to green and red areas, respectively.

Figure 2 . 2 :

 22 Figure 2.2: Geometrical interpretation of Lax admissibility condition. Thick black line: shock curve.

Figure 2 . 3 :

 23 Figure 2.3: Demand D(ρ) (in blue) and supply S(ρ) functions (in orange) for triangular FD.

Figure 2 . 4 :

 24 Figure 2.4: Schematic illustration of the demand-supply concept.

t 0 φ

 0 in (τ ) dτ + L 0 ρ 0 (s) ds, ∀t ∈ R + .

Figure 2 . 5 :

 25 Figure 2.5: Legendre-Fenchel transform of triangular FD.

Figure 2 . 6 :

 26 Figure 2.6: Example of a highway road with exit/entry minor roads (left) and the corresponding disturbance function (right).

(2. 54 )

 54 From the bounds on supremum (2.53) and inmum (2.54), we distinguish three possible cases: 1. Both sup x∈[0,L] δ(x) and inf x∈[0,L] δ(x) are positive. Then, δ(x) ∞ = sup x∈[0,L] δ(x) and ρ∞ (x

Figure 2 .Figure 2 . 9 :

 229 Figure 2.9a) andρ∞ (x) ∞ in Figure2.9b) performs better in both cases with respect to the case with no feedback. The feedback controller is a constant that is switched on after the minimal controllability time t ctr ≈ 140 s has passed. Then, in both cases the norms achieve their minimal values already after 2t ctr , since another time period equal to the minimal controllability time must pass for the control action to propagate to the end of the road.

3 Figure 2 . 10 :

 3210 Figure 2.10: Diagram of regimes illustrating how they can alternate. Arrows denote possible regime switches. FC, CC and FF are used to denote regimes at both boundaries, where F stays for free-ow and C for congested regime.

Figure 2 .

 2 Figure 2.12: a) Desired prole in space-time, b) L 1 error as a function of time for dierent control gains. Spatio-temporal evolution of the density (left) and of the absolute dierence between the real and the target state (right) for: c),d) k = 0; e),f ) k = 0.1.

Figure 3 . 1 :

 31 Figure 3.1: Direction eld estimation (blue arrows) for: a) small µ = 10, b) large µ = 100. The gure is taken from Chapter 2 of [101]. Grey arrows indicate the direction of real roads in a Manhattan-grid network.

Figure 3 . 2 :

 32 Figure 3.2: Result for an intermediate value of the weighting parameter µ = 50: a) estimated direction eld (blue arrows), b) integral lines of trac ow direction (tangent of d θ ). Grey lines represent real roads in Grenoble city center, arrows indicate the direction of trac.

Figure 3 . 3 :

 33 Figure 3.3: Density reconstruction from car positions by KDE: a) 2D density is reconstructed from vehicle positions (blue dots) that move along a network with the geometry of Grenoble city center (grey arrows), b) 1D density estimation from equidistant vehicles with Gaussians having dierent standard deviations: d 0 = 25 m (upper plot) and d 0 = 100 m (lower plot).

(3. 9 )Figure 3 . 4 :

 934 Figure 3.4: The vehicle density in a 2D domain with indicated upstream and downstream boundaries. The underlying network geometry corresponds to Grenoble city center (grey arrows).

Figure 3 . 5 :

 35 Figure 3.5: Schematic illustration of a network divided into N = 4 zones. The variables are dened with respect to reservoir i, which has its own MFD φ i (n i ) (tted data, in red) with the maximum ow φ max,i attained with the critical number of cars n c,i . The change in vehicle's accumulation n i (t) is determined by ows from N in,i = {j} and by ows into N out,i = {k}.

  3.6a), where the green and red arrows are assigned to points, through which vehicles enter the domain. Thereby, we provide a large demand in the South-Western area by setting D in = 1200 veh/h on 8 incoming roads (red arrows), while a lower demand D in = 300 veh/h is created for the remaining boundary roads (green arrows). Notice that veh/h is the basic Aimsun unit for trac ow, while this should be veh/s in the 2D LWR model, which we can easily get by dividing by 3600. Although we have marked these arrows only for the results related to the MFD-based simulation, the same inow values are set at the same points in Aimsun (Figure3.6c)).

  During the Aimsun simulation, we save the position of all cars at each time step, i.e., generate car trajectories. Finally, from the vehicle positions (blue dots in Figure3.6c)) we reconstruct a two dimensional density using KDE (see Section 3.1.3.2). The density in Figure3.6c) was estimated with a Gaussian standard deviation d 0 = 50 m, i.e., we assume that every car contributes to the total density in 50 m range around its position. Finally, the state governed by the MFD-based model (3.11) was updated using the forward Euler method with the time step ∆t = 0.01 until the convergence to the steady state (3.13) was reached.3.1.7 Comparison of steady statesIn Figure3.6 we present the steady state results predicted by the MFD-based model (panel a)), the numerical simulation of 2D LWR (panel b)) and those obtained by running a simulation on Aimsun (panel c)). For the case with MFD, we performed a partition into 16 zones (black dashed lines). Then, we used Aimsun velocity data to dene MFD for each zone as described in Section 3.1.5.1, and using (3.11) we nd the number of cars for each zone n * i , as depicted in Figure3.6a). Then, the vehicle density in each reservoir i ∈ {1, . . . , N } is obtained by:

  ) and 3.6c), we can see that the MFD-based model captures quite well the phenomenon of trac congestion in zones where it arises, although it provides only 16 values in our case. To enable a quantitative comparison, we use(3.14) to compute the L 2 norm of the deviation from Aimsun and obtain ρ * (x, y) 2 = 0.58, where ρ * (x, y) is the dierence in the steady state densities predicted by MFD-based model and Aimsun.By comparing Figures 3.6b) and 3.6c), we can observe that both steady states look very similar, the congestion shape reproduced by the 2D LWR model (3.9) looks even better than in the case of MFD-based model(3.11). For a quantitative comparison, we again compute the L 2 norm and obtain ρ * (x, y) 2 = 0.38, which is a way smaller value than in the case with MFD. Thus, the steady state obtained by numerical simulation of 2D LWR captures the spatial distribution of congestion signicantly better than the result predicted by numerical integration of MFD-based model.

Figure 3 . 6 :

 36 Figure 3.6: Steady-states obtained by: a) MFD-based model, b) numerical simulation of 2D LWR, c) microsimulator.

Figure 3 . 7 :

 37 Figure 3.7: Coordinate transformation mapping: (a) curved trajectories in Grenoble downtown in (x, y)-plane into (b) straight lines in (ξ, η)-plane.

1DFigure 3 . 8 :

 38 Figure 3.8: Coordinate transformation for one single line of constant η that is parametrized by ξ(η) ∈ [ξ min (η), ξ max (η)].

1 α cos θ -1 β sin θ 1 α

 11 of (3.25), we invert the Jacobian (3.15) and get: dx dy = sin θ 1 β cos θ dξ dη .

. 28 )

 28 Using (3.28), we are able to normalize the basis vectors by dividing (3.27) by their length (3.28): e ξ = e x cos θ + e y sin θ, e η = -e x sin θ + e y cos θ,

(3. 32 )

 32 Thus, we have shown that our curvilinear coordinate transformation (3.15) does really reformulate the 2D divergence term into 1D. This means that the temporal change of vehicle density in a 2D plane is caused by the change of trac ow along only one coordinate in (ξ, η)-space, as we were showing by(3.19) for the case of straight lines.

  (x,y)∈Ω, η(x,y)=η ξ(x, y), ξ max (η) = max (x,y)∈Ω, η(x,y)=η ξ(x, y), and we can also dene the maximal and minimal values of η as η min = min{η | ∃ξ : (ξ, η) ∈ Ω}, η max = max{η | ∃ξ : (ξ, η) ∈ Ω}.

M

  Ini (ξ, η) = c(ξ, η, 0) = ξmax(η) ξ ρ 0 ( ξ, η)d ξ.

( 3 . 45 )

 345 Further, we introduce a Legendre-Fenchel transform of the space-dependent ux function Φ(ξ, η, ρ) as:

(3. 49 ) 3 . 3 . 2

 49332 Steady state densityStationary solutions to system (3.34) might be space-varying functions ρ * (ξ, η) due to the space-dependency of the fundamental diagram. By the mass conservation law, the steady state trac ow in(3.34) should be constant along its evolution path (lines of constant η),

3. 1 . 3 . 2 )

 132 to reconstruct the 2D density from vehicle positions. For the density reconstruction, we again use the standard deviation of the Gaussian d 0 = 50 m as in Section 3.1.6. Notice also that Figures 3.9a) and 3.9c) are exactly the same as Figures 3.6b) and 3.6c), since we use here the same congestion formation scenario as in Section 3.1.6. Thus, we can observe that the analytical steady state solution presented in Figure 3.9b) captures quite well the spatial distribution of congested and free-ow areas compared to the ground true steady state density obtained from Aimsun (Figure 3.9c)). In particular, in plots b) and c) the lines separating congested and free-ow areas in the South-Western part are very similar, while in case of steady state density obtained by the numerical simulation (Figure 3.9a)) this line lies notably lower. The L 2 norm of the deviation from the Aimsun density yields ρ * (x, y) 2 = 0.4 for b) , which is almost the same as for a). Thus, the modelbased steady state calculation yields quite accurate results, which are obtained analytically without any need to run long simulations as required in the case of 2D LWR model in Figure 3.9a) (2 hours of simulation time).

Figure 3 . 9 :

 39 Figure 3.9: Steady state obtained by: a) numerical simulation of density governed by the 2D LWR system (3.9), b) model-based analysis of (3.34), c) density reconstruction from vehicle positions predicted by Aimsun.

Figure 3 . 10 :

 310 Figure 3.10: A sketch of a urban network that contains a congested area (grey Manhattan greed) to be controlled from its downstream boundary (in red).

Figure 3 .

 3 Figure 3.11: A single η-line with inhomogeneous capacities. Its worst bottleneck occupies a road segment ξ * = [ξ * 1 , ξ * 2 ]. The ow-density relation Φ(ρ) is Greenshields FD: green and red areas indicate free-ow and congested trac regimes, respectively.

Problem 3. 2

 2 Given a urban network and its infrastructure parameters v max (ξ, η), ρ max (ξ, η) and φ max (ξ, η)∀(ξ, η) ∈ Ω with initially congested trac ρ 0 (ξ, η) ∈ (ρ c (ξ, η), ρ max (ξ, η)]whose dynamics are governed by (3.34) with Greenshields FD (3.35), and given large constant inow demand at domain entry D(ρ in (η)) = φ max (η) ∀η ∈ [η min , η max ], design a boundary control law u(η) = S out (η) such that ∀(ξ, η) ∈ Ω: lim t→∞ ρ(ξ, η, t) 2 = 0, (3.56)

(3. 70 )

 70 Inserting (3.70) into (3.69), we see that the only positive term is the rst one, which can be eliminated by providing ρ (ξ max (η), η, t) = 0, i.e., ρ (ξ max (η), η, t) = ρ d (ξ max (η), η). This can be achieved by accordingly adjusting the boundary controlu(η) = φ d (η), where φ d (η) = φ min max (η) -, ∀η ∈ [η min , η max ],(3.71)as stated in Theorem 3.1. Note that the control term is dierent for each η. Thus, with (3.71) and (3.70), we can rewrite (3.69) as

Figure 3 . 12 :

 312 Figure 3.12: Control of urban trac from the downstream boundary: a) desired steady state distribution, b) initial state of trac jam, c) L 2 norm of the density error as a function of time.

  ) and 3.12b)). For a continuous approximation of the velocity eld v max d θ , we use the weighting parameter µ = 20 (the same as in Sections 3.3.3 and 3.1.6).

  which corresponds to equations (B.40) and (B.41) that were derived in Appendix B.5.4.

Figure 3 . 13 :

 313 Figure 3.13: Trac control in Grenoble downtown. Right column: desired density ρ d (x, y, t); middle column: evolution of ρ(x, y, t) with k = 5 • 10 -5 ; left column: evolution of ρ(x, y, t)with k = 0. All the plots represent snapshots made at: a), b), c) t = 0; d), e), f ) t = 0.5τ ; g), h), i) t = 1τ ; j), k), l) t = 2τ .

Figure 3 . 14 :

 314 Figure 3.14: The L 1 norm of the density error as a function of time for dierent control gains.

  min η min η∈[η min ,ηmax] , min ξ min ξ∈[ξ min (η),ξmax(η)]

(3. 81 )

 81 In general, it is possible that several values of speed limits u provide the same ow value, see the black dot in Figure 3.15. Therefore, G(ξ, η, ρ, φ) for a xed set of parameters represents a set, not a single value. Theorem 3.3 Let the controller u = u(ξ, η, ρ) be given ∀(ξ, η) ∈ Ω and for ρ = ρ(ξ, η, t) by the following inclusion u(ξ, η, ρ) ∈ G(ξ, η, ρ, φ d (ξ, η, ρ)), with φ d (ξ, η, ρ) = B(ξ, η, ρ) min ξ f (ξ , η, ρ(ξ , η, t)) B(ξ , η, ρ) and B(ξ, η, ρ) = 1 + γ ξ ξ min (η) ρ( ξ, η, t)d ξ, (3.82)

. 83 )

 83 and moreover, ∀(ξ, η) ∈ Ω lim t→∞ Φ(ξ, η, ρ(ξ, η, t), u(ξ, η, ρ)) = min ξ f (ξ , η, ρ d (ξ , η)).

1 Figure 3 . 16 :

 1316 Figure 3.16: FD as a function of u: a) monotonic dependence for a xed ρ in the free-ow regime, b) concave dependence for a xed ρ in the congested regime.

Figure 3 . 17 :

 317 Figure 3.17: Blue line: FD for u = 1. Red line: FD for u = u opt .

Figure 3 .

 3 Figure 3.18c) -f ) illustrates the temporal evolution of trac density under the VSL control (3.95) with a time-varying gain given by (3.97) with

a) L 1

 1 Figure 3.18: a) L 1 norm of density error as a function of time for dierent control gains, b) the desired optimal equilibrium as in (3.96). Trac ow control by VSL in Grenoble downtown. Density ρ(x, y, t) at: c) t = 0, d) t = 10 min, e) t = 30 min, f ) t = 2 hours.

Figure 4 . 1 :

 41 Figure 4.1: Example of a small trac network consisting of 3 intersections. We consider the intersection lled in blue.

4. 1 .

 1 Multi-directional continuous trac model 123 4.1.1.

Figure 4 . 2 :

 42 Figure 4.2: Idea of NEWS framework: map road original directions into North, East, West and South directions, and then trac ow can be described in terms of 4 direction layers.

Figure 4 .out 1 > 0 and p E θ out 1 >

 411 2 the ow along the rst outgoing road 1 out has non-zero weights with South and East direction, i.e., p S θ 0.

Figure 4 . 3 :

 43 Figure 4.3: Schematic explanation of ow directions in NEWS formulation.

  + φEN + φW N + φSN ,

Figure 4 . 4 .

 44 Hence, we dene the incoming ow in NEWS formulation from Figure 4.4:

Figure 4 . 4 :

 44 Figure 4.4: Sketch of an intersection with one incoming road 1 in and one outgoing road 1 out .

Figure 4 . 5 :

 45 Figure 4.5: Illustration of notations used for derivation of the NEWS model.

  Therefore, we can insert this denition into φN and getφN = φNN + (1 -γ) φEN + γ φNE + + (1 -γ) φW N + γ φNW + (1 -γ) φSN + γ φNS .

( 4 . 22 )

 422 This means that (4.21) requires taking spatial derivatives over multi-directional ows. However, the model (4.21) would be considerably simplied if each term under the spatial derivative could be written only as a function of demand and supply of the corresponding direction, i.e., φN = min{ DN , SN }.

( 4 . 23 )

 423 Now we make an assumption that the network is well-designed in terms of maximal ows, that is ᾱNE φmax,N = βNE φmax,E .

  East, and then turned to the North. System (4.25) together with a 4-dimensional fundamental diagram (that can be any concave Lipschitz continuous vector function) represents the NEWS model, which is one of the main results of this whole chapter. It models the evolution of vehicle density on outgoing roads of an intersection in all cardinal directions: North, East, West and South.The last step that needs to be taken is to obtain a continuous PDE system describing trac ow propagation in the whole network. This can be done by approximating the entire parameters of system (4.25) over the whole continuum domain. Let us again dene a bounded rectangular domain Ω ∈ R 2 : [x min , x max ] × [y min , y max ], as it was done for 2D LWR model in Section 3.1.1. The size of Ω is determined by the size of the considered urban network.

4. 1 . 3 . 4

 134 Extended model with source and sink terms In a urban network of nite size there exist roads, through which cars can enter or exit the domain. Such roads are called sources and sinks, respectively. The boundary conditions for the PDE system (4.25) are thus directly determined by these sources and sinks, i.e., upstream and downstream boundary conditions are specied for sources and sinks, respectively. It appears that they can be trivially captured by the NEWS model, which will also allow us to design boundary control for multi-directional urban trac in Section 4.3. Let us now show how sources are implemented into the model (4.25), while the implementation of sinks can be done in the same way.

  sink θ j . Further, one needs to approximate Dsource N and Ssink N in the whole domain, since originally

  which is very similar to the kernel density estimation(3.8). The dierence is that here the function depends on the positions of intersections. Parameter d 0 can be tuned to change the range of inuence of demand and supply functions around the intersection. Note that such a choice of G d 0 (x, y) provides that its integral over the whole domain equals 1, therefore the overall incoming demand in (4.29) is the same as in the original network model (4.28) (road formulation).

Figure 4 . 6 :

 46 Figure 4.6: Selected area in Grenoble downtown.

.34) where ζ 1

 1 and ζ 2 are the mean values of distributions ρ 1 and ρ 2 over the domain that are computed as:

(4. 35 ) 2 ,

 352 This term is used to compare luminance of two images. Then, σ 1 and σ 2 in (4.34) are the standard deviations of density distributions used to compare the signal contrasts: i, j) -ζ(ρ) and σ 12 is the correlation coecient of two density distributions used to measure the similarity of their structures:

Figure 4 . 6 )

 46 Figure 4.6). For this, we use microsimulator Aimsun and perform also a numerical simulation of trac density governed by NEWS model (4.29). For the numerical simulation we deploy the Godunov scheme in two dimensions described in Section 4.2.1. Then, the obtained steady states are compared, as it was done for the comparison of 2D LWR model and MFD-based model with Aimsun in Section 3.1.7. Recall that Aimsun takes network, turning ratios and inows as input, and produces microsimulations of vehicle trajectories. We then reconstruct the density distribution from vehicle positions in Aimsun and compare it to the state predicted by NEWS model. The density reconstruction is done using KDE method, see Section 3.1.3.2.

(

  i, j) ∈ {1, . . . , n x } × {1, . . . , n y } using IDW method, see Section 4.1.3.3. Thereby, we do not deploy intersection weights, i.e., all w k = 1 in (4.26), since here the pure network infrastructure data are used. Weights are assigned to intersections with important roads for the validation with real data in Section 4.2.4. For the results presented here, we choose the weighting parameter µ = 20, which is a relatively low value meaning that only the global trend in the network geometry is reproduced, see Figure3.1 for more intuition.

Figure 4 . 7 :

 47 Figure 4.7: Congestion formation in Grenoble downtown for t ∈ [0, 50] min: numerical simulation of density governed by NEWS model (left plots) and Aimsun (right plots). Weighting parameter: µ = 20. Black dashed lines separate Grenoble in zones used for the calculation of SSIM.

Figure 4 .

 4 Figure 4.8: a) Zone numbering in Grenoble network, b) structural similarity zone-by-zone:

Figure 4 . 9 :

 49 Figure 4.9: Mean value over zones of SSIM computed by (4.36) between densities obtained with Aimsun and numerical simulation of NEWS as a function of time.

Figure 4 . 9 .

 49 Figure 4.9. Thereby, we can see that the overall SSIM is approximately equal to 0.9 (≈ 90% accuracy), which indicates that the congested steady state is close to be reproduced correctly by our model (4.25).

Figure 4 .

 4 Figure 4.10: Sensor location in Grenoble downtown: a) xed ow sensors: R denote radars and L denote induction loops, b) automatic vehicle identiers using Bluetooth installed at 12

Figure 4 .

 4 Figure 4.11: Functional Road Classication of Grenoble downtown. The image is taken from [120].

Figure 4 . 12 :

 412 Figure 4.12: Evolution of trac density in Grenoble downtown on 8th of January, 2021 from t = 6 am to t = 9 pm: numerical simulation of NEWS model (left plots) and real data (right plots). Weighting parameter µ = 20.

Figure 4 .

 4 Figure 4.13: a) Mean SSIM (4.36) between the density ρ 1 predicted by numerical simulation of NEWS model and the density ρ 2 estimated from real data as a function of time, b) similarity zone-by-zone: SSIM l with l = {1, . . . , 9}. Weighting parameter: µ = 20.

Figure 4 . 14 :

 414 Figure 4.14: Vehicle density in a 2D continuum plane that incorporates Grenoble downtown. Downstream boundaries for control of multi-directional trac are indicated by colorful arrows:

Figure 4 . 6 .

 46 Figure 4.6. Dene a numerical grid for this area in Ω as in Section 4.2.1 but without discretization of time, since we deal with a time-constant PDE.

Proof of Theorem 4 . 1 .

 41 Let us rst analyze matrix B -I. Its non-diagonal elements are positive, and its diagonal elements are negative. Moreover, B -I has one eigenvalue equal to zero and all others are negative, as it is shown in Appendix B.8. Therefore, B -I is a negative singular M -matrix with one zero eigenvalue. Thus, there exists a positive-denite diagonal 4 × 4 matrix D such that D(B -I) + (B T -I)D 0.

Figure 4 . 6 .

 46 For the numerical simulation of trac density evolution governed by NEWS system (4.40) in the congested regime, we again deploy the Godunov scheme in 2D as described in 4.2.1. The downstream boundary conditions in (4.40) are set to the desired optimal density as in (4.53), while the upstream boundary conditions are initialised with the maximal possible ow as in Section 3.4.3, where the control of congested uni-directional trac was considered. We will thus demonstrate how the boundary controller (4.53) performs for congestion mitigation purposes given the initial state ρ 0 (x, y) = ρ max (x, y), ∀(x, y) ∈ Ω. The results of control performance on a congested trac in Grenoble downtown are shown in Figure 4.15. The continuous approximation of FD and network parameters using IDW method (4.26) was done with a low weighting parameter µ = 5, such that only the global trend of motion is reproduced.

Figure 4 .

 4 Figure 4.15a) illustrates the initial vehicle density that indicates the state of a trac jam. The optimal desired equilibrium prole ρ d obtained by following all the steps in Section

Figure 4 . 15 :

 415 Figure 4.15: Boundary control of congested trac in Grenoble downtown: a) initial congested state ρ 0 , b) desired equilibrium ρ d ; controlled state after: c) t = 5 min, d) t = 20 min, e) t = 50 min; d) SSIM between the state and the desired density as a function of time.

l i length of road i m and the second

  term as -tρ c v = x x-tv -ρ c ds, which leads us to M Ini (x, t) = L x ρ 0 (s) ds + tφ max + inf v ∈[-ω,v]

L x ρ 0

 0 (s) ds + tφ max + inf y∈[x-tv,x+tω]

:MM:

  M Ini (x, t) = H(x, t) + inf y∈[x-tv,x+tω] Ini (x, t) = H(x, t) + inf y∈[x-tv,L] Ini (x, t) = H(x, t) + inf y∈[0,x+tω] M Ini (x, t) = H(x, t) + inf y∈[0,L] (s) -ρ c ) ds   ,with H(x, t) = L x ρ 0 (s) ds + tφ max . (B.4)

(B. 2 )MMφMφ

 2 , (B.3) and (B.4) as in(2.34). Thus, depending on the values of t, the explicit solution to H-J system (2.25) can be divided into four dierent cases ∀x ∈[0, L] (see a) -d) below). (x, t) = H(x, t) + inf y∈[x-tv,x+tω] (x, t) = min H(x, t) + inf y∈[x-tv,L] out (τ ) dτ + ρ max (L -x) , (x, t) = min H(x, t) + inf y∈[0,L] out (τ ) dτ + ρ max (L -x) , with H(x, t) = L x ρ 0 (s) ds + tφ max . (B.5)As already mentioned above, in all the cases a) -d) the information on crossings of the critical value by the initial density is required in order to solve the inmum problem.

(B. 11 )B. 3 . 2 Hφ

 1132 -J solution for t ≥ t min Thus, we have estimated the minimal time t min (B.11) needed for the initial conditions to leave the system. Thus, in the H-J solution given by (B.5), the term M Ini can be excluded from the minimum operator ∀t ∈ [t min , +∞): M (x, t) = min out (τ ) dτ + ρ max (L -x) ,

From the denition of 0 v 0 φ

 00 the value condition function(3.42), we get c( ξ(0), t-T ) = M Up (t-T ) in(3.47). The upstream boundary condition is assigned to the upstream boundary ξ min , which implies the following start and end points of the observer trajectory that starts traveling from the upstream boundary with non-constant speed v (τ ):ξ(0) = ξ min , ξ(t) = ξ min + t (τ ) dτ, where v (τ ) ∈ -ω ξ(τ ) , v ξ(τ ) . (B.[START_REF] Bekiaris | Nonlinear bilateral output-feedback control for a class of viscous HamiltonJacobi PDEs[END_REF] Using (B.13) and (3.47), we formulate the following problem associated to the solution that originates from this boundary:M Up (ξ, t) = inf (T,v )∈S Up   M Up (t -T ) + T max ξ(τ ) dτ -T 0 ρ c ( ξ(τ ))v (τ ) dτ   ,

φ 0 v

 0 max ξ(τ ) dτ -T 0 ρ c ξ(τ ) v (τ ) dτ . (B.17) Now let us consider in more details the last term T 0 ρ c ξ(τ ) v (τ )dτ . By denition d ξ = v (τ )dτ , which allows us to perform the following change of variables: T 0 ρ c ξ(τ ) v (τ ) dτ = ξ ξ min ρ c ( ξ) d ξ =: R c (ξ), (B.18)where R c (ξ) is a new variable that denotes the cumulative critical density. Further, we can decompose the integrals in (B.17) ast-T 0 φ in (τ )dτ + T 0 φ max ( ξ(τ ))dτ = t 0 φ in (τ )dτ + T 0 φ max ( ξ(τ )) -φ in (t -T + τ ) dτ. (B.19)B.5.2 Downstream boundary conditionAs the second step, we need to obtain the solution M Down (ξ, t) that is related to the down- stream boundary ξ max . Notice that viable evolutions related to this boundary are characterized by the following start and end points of traveling:ξ(0) = ξ max , ξ(t) = ξ max + t (τ ) dτ, where v (τ ) ∈ -ω ξ(τ ) , v ξ(τ ) . (B.25)As in the previous case, we use the expression for the downstream boundary condition M Down (t) from (3.44) and the result from (B.19), and write the following inmum problem M Down (ξ, t) = inf (T,v )∈S Down T 0 φ max ( ξ(τ )) -φ out (t -T + τ ) dτ -T 0 ρ c ( ξ(τ ))v (τ ) dτ + t 0 φ out (τ ) dτ, (B.26)

ξmax ξ 1 ω

 1 ( ξ) d ξ and v = -ω. (B.27) We use (B.27) and φ max = ρ c v to solve the inmum problem (B.26), which yields:M Down (ξ, t) = t-Tω(ξ) 0 φ out (τ ) dτ + Tω(ξ) 0 ρ c ( ξ(τ ))v( ξ(τ )) dτ + Tω(ξ) 0 ρ c ( ξ(τ ))ω( ξ(τ )) dτ. (B.28)From denition of the critical density for triangular FD (2.3) we getρ c = ρ max ω v + ω ⇒ ρ max ω = ρ c (v + ω),which is then inserted into (B.28):M Down (ξ, t) = t-Tω(ξ) 0 φ out (τ ) dτ + Tω(ξ) 0 ρ max ( ξ(τ ))ω( ξ(τ )) dτ.

Finally, we perform the change of variables Tω(ξ) 0 ρρ

 0 max ( ξ(τ ))ω( ξ(τ )) dτ = ξmax ξ ρ max ( ξ) d ξ, with > 0, and t ctr (η) is the minimal controllability time dened in (3.74), which can be used to rewrite (B.36) asM Ini (ξ, η, t) -M Up (ξ, η, t) ≥ t -T v (ξ max (η)) t ctr (η) -ξmax(η) ξ min (η) ρ max ( ξ, η) + ρ c ( ξ, η) d ξ. (B.37)Now we can determine the minimal time, after which the right-hand side of (B.37) is nonmax ( ξ, η) + ρ c ( ξ,

(B. 38 )ρ

 38 Afterwards, the same steps are performed to obtain the minimal time, after whichM Ini (ξ, η, t) -M Down (ξ, η, t) ≥ 0 holds: max ( ξ, η) + ρ c ( ξ, η) d ξ    t ctr (η).

(B. 39 )ρ

 39 Then, t min (η) is the minimum between (B.38) and (B.39) ∀η ∈ [η min , η max ]:t min (η) = t ctr (η) max ( ξ, η) + ρ c ( ξ,

(B. 41 )B. 5 . 5

 4155 Unique solutionThe nal solution M (ξ, η, t) of the H-J system (3.41) can nally be found as a minimum of solutions associated with the upstream (B.24) and downstream (B.29) boundary conditions ∀t ∈ [t min , +∞), thus, the eect of initial conditions is negligible:M (ξ, η, t) = min t-Tv(ξ,η) 0 φ in (η, τ ) dτ + ξmax(η) ξ min (η) ρ 0 ( ξ, η) d ξ, t-Tω(ξ,η) 0 φ out (η, τ )dτ +ξmax(η) ξ ρ max ( ξ, η) d ξ .

From the second inequality for γ = 1

 1 we obtain Q(1) = φNE = min{ᾱ N E DN , βNE SE } = ᾱNE DN , and from the third inequality we get Q(1) > βEN SN .

  > βEN SN , ⇒ ∃γ ∈ [0, 1) : Q(γ) = βEN SN = min{ᾱ N E DN , βEN SN }.

5 .

 5 ᾱEN DE ≤ βEN SN , ᾱNE DN ≤ βNE SE and ᾱNE DN ≤ βEN SN . The analysis here is the same as in case (3): we take γ = 1, which results into Q(1) = min{ᾱ N E DN , βNE SN }.

6 .

 6 ᾱEN DE ≥ βEN SN , ᾱNE DN ≥ βNE SE and ᾱNE DN > βEN SN . Here we should proceed as we did in case (4): taking γ = 0 results into Q(0) = βEN SN . Further, by the second condition we obtain Q(1) ≤ ᾱNE DN . Therefore, there exists γ ∈ [0, 1] such that Q(γ) = min{ᾱ N E DN , βEN SN }.

  22): φN = φNN + min{ᾱ N S DN , βSN SN }+ + min{ᾱ N W DN , βW N SN } + min{ᾱ N E DN , βEN SN }.

Finally, using the

  approximation by replacing the sum of minima with the minimum of sums, we can write φN = min{ᾱ N N DN + ᾱNS DN + ᾱNW DN + ᾱNE DN , βNN SN + βSN SN + βW N SN + βEN SN } = min{ DN , SN }, which is exactly the property we wanted to prove (4.23).B.8 Eigenvalues of matrix B -ILet us now analyze eigenvalues of matrix B -I, where B is the SR matrix from (4.40). To simplify the notations, we introduce B = B -I that reads

  The main contribution of Chapter 4 is to propose a novel macroscopic model for multi-directional trac. The model is rigorously derived from the CTM at one intersection by solely relying on the demand-supply concept. As a result, we obtain a system of four PDEs each describing the propagation of vehicle

density in North, East, South and West direction, respectively. This model is applicable to any urban networks with arbitrarily oriented roads. It includes interactions between dierent direction layers, i.e., direction is determined by turning ratios at intersections.

Our model is validated using real data provided by Grenoble Trac Lab, which is an experimental platform that collects data from a network of real sensors installed in

Table 2 .

 2

1: Summary of error regimes Regime

  Thereby, the left column shows the evolution of trac under the control action, whereas on the right column one can see the corresponding evolution of the absolute dierence between the real and the desired states, i.e., L 1 spatial norm dened in (1.1). Figures 2.12c) and 2.12d) illustrate the result if no feedback is applied at the boundaries, while plots e) and f ) depict the situation if feedback with gain k = 0.1 is applied. The corresponding error behaviour for dierent gains is shown in Figure2.12b).

.12a) illustrates the evolution of a desired density trajectory being in a mixed trac regime. The results of achieving this state with and without feedback are shown below, i.e., see Figures 2.12c) -f ).

Table 3 .

 3

	1: Comparison of 2D and 1D LWR models
	1D LWR	2D LWR
	density

  .57) Proof. Let us dene the following Lyapunov function candidate ∀η ∈ [η min , η max ]

				ξmax(η)
		V (η, t) =	1 2	e ξ ρ2 (ξ, η, t)dξ,	(3.58)
				ξ min (η)
			ξmax
				e ξ ρ(ξ, t)	∂ ρ(ξ, t) ∂t	dξ.	(3.59)
			ξ min
	To simplify (3.59), we use the time-independence of ρ d as:
	∂ ρ(ξ, t) ∂t	=	∂ρ(ξ, t) ∂t	≡ -	∂Φ (ξ, ρ) ∂ξ	= -	∂Φ (ξ, ρ d + ρ) ∂ξ	.

where e ξ is a weighting function used to provide the exponential convergence of the Lyapunov function (similar as in (2.44)). For simplicity of notations, we neglect variable η as an argument. The time derivative of (3.58) is

V (t) =

  Note that if Assumption 3.1 is satised, then Assumption 3.2 holds trivially by taking t ≥ t min , where t min is the largest time, after which it is guaranteed that the initial conditions will have left the domain Ω. The value of t min is given by

	Assumption 3.2
	The solution of IBVP (3.34) is determined by the boundary conditions only, i.e., the initial
	conditions have left the system.
	Remark 3.1
	t min =	max η∈[η min ,ηmax]	t min (η), where
	t min (η) = t ctr (η)

.

[START_REF] Jiang | A dynamic trac assignment model for a continuum transportation system[END_REF]

It means that inows and outows for each η-line are not allowed to exceed the capacity of the strongest bottleneck of the corresponding line instantly (3.72), and (3.73) means that they must be strictly lower during the time interval given by t ctr (η)

(3.74)

. This assumption is necessary for the proof of Theorem 3.2.

  ). Let us assume that the space-dependent ow-density relation in(3.34) has a triangular shape(3.4), and then apply the variational principle(3.47) to calculate the solution to(3.41). The derivation of its solution is quite technical and, therefore, we shift it to Appendix B.5. Thus, if Assumptions 3.1 and 3.2 hold, the solution M (ξ, η, t) reads ∀(ξ, η, t) ∈ Ω × [t min , +∞) :

  ∀a, b: ξ min (η) ≤ a < b ≤ ξ max (η) we obtain ∀η ∈ [η min , η max ]Notice that the boundary controllers u in (η, t) and u out (η, t) are applied by changing the demand at domain entry and the supply at domain exit, respectively. The control functions

	b
	lim t→∞	ρ( ξ, η, t) -ρ d ( ξ, η, t) d ξ = 0.
	a
	enter the system (3.34) through the minimum function (and therefore are not necessarily
	fullled pointwise):
	Theorem 3.2
	ξmax(η)	(3.77)
	ξ min (η)

Consider a vehicle density function ρ(ξ, η, t) governed by system (3.34) ∀(ξ, η, t) ∈ Ω × R + , for which Assumptions 3.1 and 3.2 hold, and the corresponding Hamilton-Jacobi solution given by

(3.75)

. Assume also the desired density prole ρ d (ξ, η, t) and boundary ows φ in d (η, t) and φ out d (η, t) that are also given by

(3.34)

. Then, if ∀(η, t) ∈ [η min , η max ] × R + the boundary controllers in

(3.34)

are set to

[START_REF] Morten | Disturbance rejection in 2 × 2 linear hyperbolic systems[END_REF]

u in (η, t) = φ in d (η, t) -ke(η, t), (2) u out (η, t) = φ out d (η, t) + ke(η, t),

where e(η, t) = ρ( ξ, η, t) -ρ d ( ξ, η, t) d ξ and k > 0, then

  Hence, these boundary ow functions are guaranteed to be smaller than the minimal capacity on each line of constant η. Note that these functions were chosen such to generate a mixed regime desired trajectory ρ d (ξ, η, t) with a period of τ = 2400 seconds. Such a desired trajec-

	φ min max (η) 0.6 + 0.4 sin 2π	t 1200	+ 2	η -η min η max -η min	,
	S (ρ out d (η, t)) = φ min max (η) 0.6 + 0.4 sin 2π	t 2400	+ 2	η -η min η max -η min	.

tory is generated on purpose, since the biggest advantage of boundary controllers

(3.77)

is the ability to handle mixed trac regimes, which is mathematically a tricky case.

We demonstrate here, how the boundary control law enhances the trac state if there is a feedback, i.e., k > 0 in (3.77). The controller is applied at both upstream and downstream boundaries of the domain, and it physically corresponds to demand at the entry and supply of the exit, as illustrated in Figure

3

.4. Thus, we will compare two possible strategies:

1. Both feedforward and feedback terms are used, i.e., ∀(η, t) ∈ [η min , η max ] × R + : u in (η, t) = φ in d (η, t) -ke(η, t) and u out (η, t) = φ out d (η, t) + ke(η, t).

2. Only feedforward term is used (no feedback), i.e., ∀(η, t) ∈ [η min , η max ] × R + :

  [START_REF] Abdel-Aty | Evaluation of variable speed limits for real-time freeway safety improvement[END_REF], where ow and densities are formulated such if there would exist only four principal directions of trac: North, East, West and West (which gave the abbreviation NEWS). To enable such a formulation, we introduced projection matrices that use the intersection geometry to determine, in which direction it has more impact. For instance, we classify a road to be oriented towards North-East, if its orientation has non-zero projection weights for North and East directions. From a unied model for one intersection (that can be seen just as a point in space), we moved to the continuation of this model. This is a special technique that helped us to translate an ODE-like model into a PDE-like model. As a result, we could formulate the model (4.25) for vehicle density evolving in 4 directions on outgoing roads in the vicinity of intersections. Then, we applied inverse distance weighting to dene the parameters everywhere in a continuum plane that may incorporate any urban network of interest.

  {N, E, W, S} is updated ∀(i, j, k) ∈ {1, . . . , n x }×{1, . . . , n y }×Z +

					T	. The
	density in each direction s = as
	ρ k+1 s	(i, j) =ρ k s (i, j) + ∆t E k s (i, j) + F k x,s (i, j) + F k y,s (i, j) + H k s (i, j) ,
	where E k s (i, j) corresponds to the mixing term between direction layers
	E k s (i, j) =	1 L(i, j)	4 r =s r=1	min α rs (i, j)D k r (i, j), β rs (i, j)S k s (i, j)
			-min α sr (i, j)D k s (i, j), β sr (i, j)S k r (i, j) ,
	and F k x,s (i, j), F k y,s (i, j) correspond to derivative terms. The derivative term w.r.t. x dimension
	can be computed as

  "../ModelValidation/AllOutows.csv" contains outow values (in veh/hour) for every road for every time step (one minute). If road is incoming into intersection that is not on border, then the outow value is zero.

	5. "../ModelValidation/Timestamp.csv" contains time in seconds at which the data are
	given (unix timestamp), the time step equals to one minute;
	6. "../ModelValidation/Density.csv" contains estimated density from real sensors: rst
	number is road_id followed by its density (that is assumed to be constant within one
	road) at all time instants, then the next road_id with its density data for each time
	instant and so on;

7. "../ModelValidation/AllInows.csv" contains inow values (in veh/hour) for every road for every time step (one minute). If road is outgoing from intersection that is not on border, then the inow value is zero; 8.

  ).Minimizing congestion means ndingρ d (x, y) ≥ ρ c (x, y) ∀(x, y) ∈ Ω such that ρ d (•) -ρ c (•) ∞ is minimized. The L ∞ space norm is dened as in (1.8).In order to nd a desired prole satisfying Problem 4.1, we need to solve the PDE(4.42) that describes its structural dependence on (x, y). First of all, we need to introduce a change of variables ρ(x, y) ∀(x, y) ∈ Ω as

	Remark 4.2
	Remark 4.3
	Physically, a proportional relation of density values at the boundaries to the corresponding
	maximal densities (4.44) implies that boundaries are lled in a homogeneous way. This might
	be useful in a situation when vehicles concentrated in a city center tend to leave it simultane-
	ously, e.g., when people drive back home from their oces.
	(4.45)
	which being inserted into (4.42) yields
	1
	Problem 4.1

Find the desired space-varying density ρ d (x, y) ∀(x, y) ∈ Ω that corresponds to the state of minimal congestion under the constraints that ρ d (x, y) ≥ ρ c (x, y) ∀(x, y) ∈ Ω, and boundary values being proportional to maximal densities (4.44). ρ(x, y) = ρ max (x, y) -ρ d (x, y),

  Assumption 4.2 SR matrix B is constant in space, which in turn implies that every intersection has the same turning ratio pattern.Let us now show that setting the boundary controller equal to the desired state at the boundary under Assumptions 4.1 and 4.2 provides the exponential convergence of trac state to the desired equilibrium in the whole domain. This is formalized as follows.

	Theorem 4.1
	Under Assumptions 4.1 and 4.2, let the boundary controller be dened as

  The function (4.55) is obviously positive-denite, since matrix W DE > 0.

						Let us now
	take its time derivative, which yields
	xmax	ymax
	V =		2	∂ ρT ∂t	W DE ρ dydx,	(4.56)
	x min	y min
	where the error dynamics ∂ ρ/∂t should be taken from (4.43), which allows us to further expand
	(4.56) as:

  . The term V1 is negative due to (4.54) and the fact that matrix E is non-negative, i.e., We further consider V2 by inserting the values of matrices C and S (4.52) from Assumption ( D W ω W ρW ) 2 -e x ( D E ω E ρE ) 2 D E ω E ρE ) 2 -e -x ( D W ω W ρW ) 2 ( D S ω S ρS ) 2 -e y ( D N ω N ρN ) 2 D N ω N ρN ) 2 -e -y ( D S ω S ρS ) 2

	V1 = (W ρ) 4.1 xmax x min ymax y min 1 L V2 = 2 xmax ymax ω E ρE D E e x ∂(ω E ρE ) ∂x V2 = ymax y min e -x x=x min + ymax y min e x ( x=xmax dy + xmax x min e -y y=y min dx dy + xmax x min e y ( y=ymax dx -xmax x min ymax y min e (4.59) and one ensures that the rst four integrals in (4.58) go to zero. The last term in (4.58) can be bounded as follows xmax x min ymax y min e ≤ -min (x,y)∈Ω -ω This expression is then integrated by parts, which yields: q∈{N,S,W,E}
	x min	y min

T DE(B -I) + (B T -I)DE W ρ dydx < 0. W ρW D W e -x ∂(ω W ρW ) ∂x + ω N ρN D N e y ∂(ω N ρN ) ∂y -ω S ρS D S e -y ∂(ω S ρS ) ∂y dydx. x D E (ω E ρE ) 2 + e -x D W (ω W ρW ) 2 + e y D N (ω N ρN ) 2 + e -y D S (ω S ρS ) 2 dydx.

(4.58) By setting the boundary controller u(x, y) as in (4.53), we achieve that ∀t ∈ R + ρN (x, y max , t) = 0, ρS (x, y min , t) = 0, ∀x ∈ [x min , x max ], ρW (x min , y, t) = 0, ρE (x max , y, t) = 0, ∀y ∈ [y min , y max ], x D E (ω E ρE ) 2 + e -x D W (ω W ρW ) 2 + e y D N (ω N ρN ) 2 + e -y D S (ω S ρS ) 2 dydx

  3 Multi-directional trac on a 2D plane A.3.1 Road formulation

	Variable	Meaning	Units
	ρ(x, y, t)	vehicle density	veh/m
	Φ(x, y, ρ)	ow function	veh/s
	v(x, y)	kinematic wave speed in free-ow regime	m/s
	ω(x, y)	kinematic wave speed in congested regime	m/s
	ρ c (x, y)	critical vehicle density	veh/m
	φ max (x, y)	ow capacity	veh/s
	D(ρ)	demand function	veh/s
	S(ρ)	supply function	veh/s
	φ in i	inow to intersection from road i	veh/s
	φ out j	outow from intersection to road j	veh/s
	ψ in j	inow into road j	veh/s
	ψ out j	outow from road j	veh/s
	n in	number of incoming roads for intersection	-
	n out	number of outgoing roads from intersection	-
	φ ij	ow from road i to road j	veh/s
	α ij	turning ratio from road i to road j
			veh/s
	S ij	supply of road j for ow coming from road i	veh/s
	θ i	angle that road i builds with the East direction	rad

β ij

supply coecient of road j for the ow from road i -D ij ow demand of road i to enter road j

(3.4) and, similarly, the Greenshields space-dependent FD is dened ∀(x, y, ρ) ∈ E as:

(3.6)

in Figure 3.11.

The same can be written for sin θ N and for y-direction, for which we x i and vary j.

Acknowledgements

I would like to express the deepest appreciation to my supervisors, Carlos Canudas-de-Wit and Maria Laura Delle Monache. From them I have learned a lot of valuable lessons that

est eectué dans le cadre du projet European Research

List of Tables 2.1 Summary of error regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.3.2 is illustrated in Figure 4.15b). Recall that the desired state corresponds to congestion minimization under the constraints formulated in Problem 4.1. The desired state is found by rst solving the PDE for ρ given by (4.47) and then using (4.50), where we use α * = 0.51 that was obtained using (4.49). Further, we show the impact of boundary controller (4.53) on the congested trac state after t = 5 min, t = 20 min and t = 50 min in Figures 4.15c), 4.15d) and 4.15e), respectively. We can see that the controlled state at t = 50 min is identical to the desired equilibrium from Figure 4.15b).

We could quantitatively measure the similarity between two density distributions by deploying the mean SSIM (denoted by SSIM ), see Section 4.2.2 and (4.36). Thereby, the Grenoble area was again divided into 9 zones to compute SSIM using (4.36). Notice that SSIM as a function of time is shown in Figure 4.15f ), where range of SSIM is preserved as in its original formulation, i.e., SSIM ∈ [-1, 1].

Chapter conclusions

In this chapter, we suggested our own way to deal with multi-directional trac evolving on urban networks of arbitrary size. Multi-directional trac is much more close to represent urban trac in realistic situations compared to 2D LWR approach considered in the previous Chapter 3. The global idea was to derive a PDE model that captures the trac behaviour evolving in a urban network in any direction with ow crossings.

We started elaborating a modeling approach for multi-directional trac by considering a trac ow model at one intersection based on the classical CTM. Each intersection is characterized by a certain number of incoming and outgoing roads that may be arbitrarily oriented in space. Thus, the trac ow model has a dierent number of parameters to tune for each individual intersection. As a network may consist of thousands of intersections, we had to nd a unied approach to describe trac at intersections regardless of their individual parameters.

Thereby, we assumed that the dynamics of multi-directional trac can be represented by only 4 direction layers: North, East, West and South. This led to the formulation of NEWS framework that deploys geometry-based projection matrices to map the trac ow along any road into the nearest cardinal directions. The projection weights vary continuously with road's orientation angle. For instance, if a road goes exactly to the North, the projection weight for the North direction is equal to 1 (maximal possible angle), while it is equal to 0.5 if the road is oriented perfectly towards North-East or North-West. We have also introduced the concept of partial ows to capture various origin-destination patterns at intersections. For instance,

φ N E is the ow of vehicles that were moving along a road oriented to the North, and then at intersection turned to the road oriented to the East direction.

Thus, we were able to obtain a trac ow model that predicts the rate of change of vehicle accumulation at intersection in a unied way. However, since our goal was the derivation of Here we provide the details on how to obtain a solution ρ(x, t) to a transport equation with a disturbance term (2.37). Since in this case we are dealing with a linear system (2.37), we can apply the method of characteristics to nd its solution ρ(x, t)

to the method of characteristics, nding a solution is equivalent to nding an integral surface S such that the coecient vector eld V = (1, -ω, δ(x)) is tangent at each point for any curve Γ ∈ S [START_REF] Evans | Partial dierential equations[END_REF].

Let us introduce a variable s used for the parametrization of Γ. Thus, we need to nd a curve Γ = (x(s), y(s), z(s)) ∈ R 3 such that the following system of ODEs is satised:

By eliminating s from the rst two ODEs, we obtain the projection of the characteristic curve in (x, t)-plane : t -L-x ω = const. Note that we consider only t ≥ L-x ω , since the control action has a nite propagation time t ctr = L ω .

Now let us estimate ρ(x, t) from the third ODE. For this, we need to parametrize the line passing through two points (x, t) and (L, t -L-x ω ): in the rst variable it is x(y) = y as y varies from x to L, and in the second variable it is t(y) = (t -(y-x) ω ). If z has to be on the integral curve, then z(y) = ρ(y, t -(y-x) ω ). Therefore, the third ODE becomes -ω dz dy = δ(y).

Integration from L to x of both sides yields

Notice that we need to perform similar steps in order to obtain a solution ρ(x, t) for t < L-x ω . In this case, the projection of the characteristic curve in (x, t)-plane is x+ωt = const.

B.2 Solution of a Hamilton-Jacobi system

Here we explicitly derive the analytic solution of a H-J problem as in (2.25) for the special case of Hamiltonian being a triangular FD. In this case, the convex transform (2.30) of a triangular

and it is illustrated in Figure 2.5.

In order to obtain a unique solution M (x, t) ∀(x, t) ∈ [0, L] × R + for the case of triangular FD, we should explicitly calculate solution candidates M Up (x, t), M Down (x, t) and M Ini (x, t), and then extract the minimum of these functions as in (2.34).

B.2.1 Upstream boundary condition

The function M Up (x, t) denotes the solution of the Lax-Hopf formula (2.32) that originates from the upstream boundary x = 0 at time t -T given the initial cost M Up (T -t) (a more detailed explanation is given in 2.1.8).

By looking at the value condition function c dened in (2.26), we establish that the initial cost in (2.33) is given by c(x -T v , t -T ) = M Up (t -T ). The equality to zero of the rst argument of function c is achieved for T = x v , where v ∈ [-ω, v]. Since T can only be positive by denition, the minimal value of the time interval should be T min = x v . Using (B.1), this results into the following inmum problem:

Using the expression for the upstream boundary condition (2.27), the inmum problem reads

which by using T φ max = T t-T φ max dτ can be rewritten as

The inmum is achieved for T = T min = x v , i.e., the correct solution is related to the path along which the vehicle has the maximal velocity. Substituting T = T min , we get ∀x ∈ [0, L] t < x v :

where for t < x/v the value of M Up (x, t) is undened, thus, we set it to innity.

B.2.2 Downstream boundary condition

Further, we need to calculate M Down (x, t), which is related to the downstream boundary x = L.

The space argument in the value condition function now becomes x -

Thus, the smallest value of the time interval should be T min = x-L -ω . The calculation is done performing the same steps as for M Up (x, t), and we obtain ∀x ∈ [0, L] t < L -x ω :

B.2.3 Initial condition

Finally, we calculate the function M Ini (x, t) determined by the vehicle with known label at t = 0 (2.29). The equality to zero of the time argument in the value condition function is provided by T = t. This yields

Using the denition of M Ini (x) from (2.29), we obtain

We decompose the integral as

which is then used for the analysis of the trac system behaviour for t → ∞. Notice that the solution candidate M Ini (x, t) associated with the initial condition given by (B.4) is the most confusing term, since it contains an unresolved inmum problem. However, the only time-depending term in (B.4) is tφ max , thus dM Ini /dt = φ max . Taking the time derivative for other terms yields dM Up /dt = φ in (t -x/v) and dM Down /dt = φ out (t -(L -x)/ω). By the capacity constraint φ(ρ(•, •)) ≤ φ max , we establish that the term M Ini grows faster than the others (or in some special cases equally fast).

B.3.1 Time when initial conditions leave the system

Let us estimate the minimal time t min , after which the initial condition does not aect the solution of a H-J system (2.25), i.e., ∀(x, t) ∈ [0, L] × [t min , +∞): M Ini (x, t) ≥ M Up (x, t) or M Ini (x, t) ≥ M Down (x, t). Let us rst establish the earliest time, for which M Ini (x, t) ≥ M Up (x, t), then we will do the same for M Down (x, t), and then the nal value will be the minimum of two cases.

Thus, using M Up (x, t) from (B.2) and M Ini (x, t) from (B.4) with tφ max = t-

Further, using Assumption 2.2 we can make another lower bound:

(B.12) Firstly, by taking the time derivative of (B.12), we see that in the steady-state φ in (t) ≡ φ in d (t) and φ out (t) ≡ φ out d (t).

Secondly, by expressing M 0 from both parts of (B.12), we obtain the necessary condition (2.66) to track ρ d .

B.5 Solution of a H-J PDE with space-dependent Hamiltonian

Here we consider the initial boundary value problem in Hamilton-Jacobi formulation given by (3.41) for a trac system evolving on a large urban network. This problem contains an explicit space-dependency in the fundamental diagram that captures the network infrastructure. We nd its solution explicitly for the case of space-dependent triangular FD using the variational principle (3.47).

In the following, we will skip writing η in the arguments to make the notations less heavy. Let us here assume that we solve the H-J PDE explicitly for each line of constant η. The Legendre transform (2.30) of the triangular FD is

We need to calculate the viability episolutions M Up (ξ, t), M Down (ξ, t) and M Ini (ξ, t) asso- ciated with given value conditions M Up (t), M Down (t) and M Ini (ξ), respectively, using (3.43), (3.44), (3.45) and (3.47). Notice that these viability episolutions are equivalent to solution candidates that were discussed in Section 2.1.8. Finally, the unique solution of (3.41) corresponds to the minimum of three functions:

which is similar to (2.34) but in (ξ, η)-space.

Notice that, in the following, we will consider only solutions for large enough time

Thus, using (B.18) and (B. [START_REF] Blandin | Boundary stabilization of the inviscid Burgers equation using a Lyapunov method[END_REF]) we can rewrite (B.17) as This implies that the solution is assigned to a traveler that moves with the maximal speed at each space point, i.e., (B.16) becomes

where v is the maximal kinematic wave speed. Thus, in the inmum, T is the solution to (B.21) for t = T : We rewrite the rst term on the right-hand side of (B.23) as

Using (B.21), we can perform the change of variables in the latter integral as

With this result, two R c (ξ) terms with opposite signs in (B.23) cancel each other, and we obtain the solution associated with the upstream boundary

and thus obtain the solution associated with the downstream boundary:

(B.29)

B.5.3 Initial condition

As the third step, we need to calculate M Ini (ξ, t) that is related to the vehicle with known label at initial time that follows the path of viable evolution (see (2.29)).

We can already establish that T = t, since the viability evolution starts its path at initial time. Thus, using the variational principle (3.47) with initial condition given by (3.45), we can state the inmum problem as

where domain S Ini is dened as in (3.48) for T = t:

In the rst term of the right-hand side of (B.30), the integral runs from ξ0 used to dene the coordinate from which the viable evolution starts its path at initial time:

Again we use the change of variables such that v (τ )dτ = d ξ and rewrite (B.30) as

We can not further simplify (B.33), unless some specic information about the initial conditions is known. However, we can estimate the lower bound of (B.33) term by term:

where φ min max is the capacity at the strongest bottleneck along the η-line dened in (3.51).

As already mentioned above, the unique solution (B.14) is the minimum of three functions.

In the following section, we will show that starting from some time t min , the initial conditions will have left the system and thus can be excluded from the minimum operator.

B.5.4 Time when initial conditions leave the system

Here we aim to estimate the minimal time t min (η) such that ∀(ξ, η, t) ∈ Ω × [t min (η), +∞):

M Ini (ξ, η, t) ≥ M Up (ξ, η, t) or M Ini (ξ, η, t) ≥ M Down (ξ, η, t). This was already done for the 1D case in Appendix B.3.1. However, now this minimal time is dierent for each η-line. Therefore, we will again write the dependence on η in the notations to gain more clarity.

First of all, we will estimate the time after which M Ini (ξ, η, t) ≥ M Up (ξ, η, t), then we do the same for M Ini (ξ, η, t) ≥ M Down (ξ, η, t). Finally, t min (η) is found as the smallest value of these two results.

We combine the result for M Up (ξ, η, t) (B.24) with the lower bound for M Ini (ξ, η, t) (B. 34), and write which yields 3. Equation (2.77) in the proof of Theorem 2.3 should be rewritten as:

We obtain (B.42) by using the following upper bound:

B.7 Proof that φN = min{ DN , SN }

Here we prove that the ow in some direction (here North) can be written as a function of demand and supply of the same direction: φN = min{ DN , SN }, which allows to simplify the model (4.21). Thus, here we seek to prove that equation (4.23) holds. The main assumption that needs to be made thereby is that the urban network is well-designed in terms of maximal ows, see (4.24).

Let us consider the term (1 -γ) φEN + γ φNE from (4.22). Using the denition of partial ows (4.12), we can write

Recall that by denition of the demand-supply formulation, if DE < φmax,E , then SE = φmax,E and vice versa, see Figure 2.3. The same holds for DN and SN . To simplify the notations, let us denote Q(γ) = (1 -γ) φEN + γ φNE . We will prove that there always exists γ such that Q(γ) = min{ᾱ N E DN , βEN SN }. In total, there are six dierent cases to consider for partial ows φEN and φNE :

1. ᾱEN DE < βEN SN and ᾱNE DN > βNE SE . From the rst inequality we obtain ᾱEN DE < βEN SN ≤ βEN φmax,N = ᾱEN φmax,E , where the last equality comes for the assumption that the network is well-designed (4.24). Thus, we get that DE < φmax,E .

From the other side, if we consider the second inequality, we get βNE SE < ᾱNE DN ≤ ᾱNE φmax,E ⇒ SE < φmax,E .

According to the demand-supply formulation, it is however not possible that DE < φmax,E and SE < φmax,E hold at the same time. Thus, this case can be excluded from consideration.

2. ᾱEN DE > βEN SN and ᾱNE DN < βNE SE . This case is also impossible, since from the rst inequality we get SN < φmax,N and from the second inequality we get DN < φmax,N , which violates the demand-supply formulation. Let us consider λ( B) = 0 with x being the corresponding eigenvector:

x T B = 0 = x T λ( B).

Using the denition of matrix B, we further get

x T (B -I) = 0 ⇒ x T B = x T .

Thus, it follows that x is also the eigenvector of matrix B associated with the eigenvalue λ(B) = 1.

Note that matrix B is a positive matrix, i.e., β ij > 0 for 1 ≤ i, j ≤ 4 (assume we have no zero turning ratios). Then, by Perron-Frobenius theorem, λ(B) = 1 is a Perron root (since all columns of B sum to 1), and thus it is a simple root. It follows that all the eigenvalues of matrix B = B -I are strictly negative and only one eigenvalue is zero.