H Px, Bvqq

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' % ´Bt v ´µi B 2 v `H px, Bvq " V rms, t P p0, T q, x P Γ i z tOu , i " 1, N , B t m ´µi B 2 m `B pmB p

Keywords: Problèmes de contrôle optimal, équation de Hamilton-Jacobi, équations aux dérivées partielles elliptiques et paraboliques, solutions de viscosité, solutions faibles, réseaux, jeux à champ

T pxq, m px, 0q " m 0 pxq, ż G m 0 pxqdx " 1, x P G. (0.0.3) v La description et l'interprétation de ce système est semblable au cas du chapitre précédent pour un problème en horizon fini, ce qui se traduit par des équations d'évolution à la place d'un problème stationnaire. Les principales différences dans les hypothèses concernent les équations d'HJB. Nous devons ici considérer des hamiltoniens H qui sont globalement lipschitzien par rapport au gradient (et donc sous-linéaires) et un couplage qui est régularisant (et donc nonlocal).

Le premier résultat est de pose le problème et d'expliquer l'obtention du système d'EDP aisin que les conditions aux jonctions. Ensuite, nous montrons quelques résultats utiles pour une équation de la chaleur modifiée avec des conditions de Kirchhoff générales et pour des équations de FP avec des conditions de transmission spéciales. Ces résutat utilisent des méthodes de Galerkin type. Notons que les solutions faibles sont définies en utilisant une paire appropriée d'espaces de fonctions Sobolev V et W définis sur le réseau. La principale difficulté dans ce travail est comment d'obtenir la régularité pour la fonction valeur v sur le réseau. L'idée est de dériver l'équation HJB pour v et de prouver une estimation de régularité parabolique pour la solution du problème dérivé. Enfin, nous donnons les preuves des principaux résultats de l'existence et l'unicité du système MFG des EDP en utilisont là aussi des théorèmes de point fixe.

This dissertation would not have been completed without the help, encouragements and supports from many people who all deserve my sincerest gratitude and appreciation.

In fact, I truly think that no words can express my gratitude to my three advisors, whose personalities, expectations and working styles are very different; however, what they have in common is their dedication in teaching and growing me without which I would not be here writing this acknowledgement.

First of all, my deepest gratitude goes to my doctoral advisor Professor Olivier Ley, who has always been there with me since my master internship, sharing with me his valuable advices, supporting me during the darkest moments of my Ph.D. study and inspiring me by simply being himself. I will never forget the time that he reminded me a thousand times to bring my passport on a travel (I did forget it anyways) or the days when we had to go back and forth a hundred times on a paper and he still patiently bore with me and improved it little by little. I still keep those manuscripts with his hand-writing comments as a reminder of a caring, devoted and respectable advisor.

I wish to express my gratitude most strongly to Professor Yves Achdou for having offered me such an interesting Ph.D. project and more importantly, a chance to discover mathematics in a very different way. To be honest, during my first year, I always felt nervous before every single meeting with him; however throughout the years, I found it more and more interestingly challenging thanks to which I have learned and matured a lot. He provided me the optimal space and also "pressure" to continuously pushing and developing myself and at the same time, a lifeboat when things did not happen as we expected. The most touching moment was when I informed him of my post-doc interview, he replied me with very detailed email instructing me what to do and what not to do before and during the interview. I am, always have been, and will always be, proud to be his student.

A special mention goes to Professor Nicoletta Tchou. If I was allowed to modify the function appearing after her name on the cover of this dissertation, it would be "co-directrice de thèse". Beside being a great advisor, for me she was also the best moderator for our projects and meetings who connected not only ideas, but also the people, us. Additionally, she was usually the one who paid the closest attention to my work and pointed out the most obvious (and silly) mistakes of mine, from which I have learned not only how to do mathematics, but also how to make people understand how we do it in the most comprehensive and appealing manner.

I greatly appreciate Professor Francisco Silva and Professor Claudio Marchi, the two reviewers of this dissertation, who have been so kind to spend time to read, evaluate my works and give me valuable comments. I am also thankful to Professor Pierre Cardaliaguet and Professor Hasnaa Zidani who accepted to be in my thesis committee. Having my thesis committee including these admirable experts in viscosity solutions and mean field games is my great honor.

I am grateful to Institute of Mathematical Research of Rennes (IRMAR, University of Rennes 1) and Center Henri Lebesgue who have financially supported my doctoral study during the past three years. My special thanks to the secretaries Xhenxila Lachambre, Carole Wosiak, Marie-Aude Verger, Florian Rogowski and Chantal Halet who have endlessly supported me even before the first day I was in France till the last. I also acknowledge the partial support from the ANR i project HJnet "Hamilton-Jacobi equations on heterogeneous structures and networks" and ANR project "Mean field games". Working in the National Institute of Applied Sciences of Rennes (INSA Rennes) was a wonderful time in my Ph.D. period. I am thankful for the equipment and facilities I fully took advantage of.

I would also like to thank Prof. Boualem Djehiche who offered me the Post-doc position even before when I submitted my thesis, which made me feel much more "secure" and motivated to finish the last phase of my PhD life.

I am truly thankful to my "master" and also my friend Nguyen Quang Huy for the mathematical discussions that we had and career orientations he has shared. A sincere thank goes to Salomé Oudet for her supports during my master internship. I'm grateful to one of my close friends, Nguyen Thi Tuyen for her help and encouragements from my first days in France until now.

A special thanks must go to my PUF close friends: Tran Minh Quan for being friends with another-me-that-no-one-can-see; Truong Kim Tien for being my partner in crime and for the unforgettable EuroTrips; Dang Phuoc Nhat for everything we have been through; and Pham Thi Da Cam for her caring.

Besides, I have also been lucky to meet and make friends with many nice people in Rennes. Thank you, my French friends who share the experience of Ph.D. life and who are many times my saviors: Tangi Migot, Emilie Joannopoulos, Audrey Poterie, and Samuel Corre. Thank you, my mathematics friends, Nguyen Van Trinh, Chau Thi Tuyet Trang, Nguyen Duc Tho, Nguyen Dinh Duong, Nguyen Thi Minh Phuong, Nguyen Thi Hoai Thuong for our friendship, our mathematical discussions and the parties that we can freely talk about anything on earth. To be honest, thank to all of you and what we have together, I sometimes forget that I am living in France. And thank you, my Vietnamese friends in France: Nguyen Hoang Sang, Nguyen Thi Lan Anh, Nguyen Hoai Nam, Nguyen Tran Dan Ngoc, Nguyen Ha Thanh, and many others... for the moral and sometimes physical support.

Last but not least, I would like to express all my gratitude to my family for constant encouragement and unconditional love. Especially, I dedicate this thesis to my wife, who gave up all she had in Vietnam to come with me to France. I cannot forget all the moments when she tried to proofread all my manuscripts, making them much more easily understandable and giving me useful advices even if she is not at all a mathematician. All my stress is gone when I am home, having dinner with her and watching our favorite TV shows/movies. Thank her again for being and bearing with me in this adventure, and the others to come.

Équations de Hamilton-Jacobi et jeux à champ moyen sur les réseaux Resumé

Cette thèse porte sur l'etude de problèmes des équations de Hamilton-Jacobi-Bellman (HJB) associées à des problèmes de contrôle optimal et jeux à champ moyen sur les réseaux, c'est-à-dire, des ensembles constitués d'arêtes connectées à des sommets. Différentes dynamiques et coûts sont autorisés dans chaque arête du réseau. Sur la Figure 0.1, on présente deux exemples de réseau. Dans le chapitre 2, on considère un problème de contrôle optimal sur les réseaux dans l'esprit des travaux d'Achdou, Camilli, Cutrì & Tchou [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF] et Imbert, Moneau & Zidani [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF]. Plus précisément, nous considérons un problème de contrôle optimal dans lequel nous ajoutons des coûts d'entrée (ou de sortie) aux sommets du réseau et étudions les équations HJB associeés. L'effet des coûts d'entrée/sortie est de rendre discontinue la fonction valeur du problème. Pour simplifier le problème, nous étudions seulement le cas de la jonction, c'est-à-dire, un réseau de la forme G " Y N i"1 Γ i avec N arêtes Γ i et un seul sommet O. Nos hypothèses à propos de la dynamique et des coûts sont similaires à ceux faits dans le travail de Achdou, Oudet & Tchou [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF], avec des coûts supplémentaires c i pour entrer dans l'arête Γ i à partir de O ou d i pour quitter Γ i en O. La fonction valeur est continue sur Gz tOu, mais est en général discontinue en O. Par conséquent, au lieu de considérer la fonction valeur v, nous la remplaçons par la collection pv i q 1ďiďN , où v i est la restriction de v à l'arête Γ i z tOu prolongée par continuité en O. Dans le cas des coûts d'entrée par exemple, notre premier résultat principal est de trouver la relation iii entre v pOq, v i pOq et v j pOq`c j pour 1 ď i, j ď N . Nous en déduisons que les fonctions pv i q 1ďiďN sont solutions de • Le terme min j‰i tu j pOq `cj u prend en compte les situations dans lesquelles la trajectoire entre dans Γ i 0 où u i 0 pOq `ci 0 " min j‰i tu j pOq `cj u.

• Le terme H ì ˆO, du i dx i pOq ˙prend en compte les situations dans lesquelles la trajectoire ne sort pas de Γ i . ' ' ' % ´µi B 2 v `H px, Bvq `ρ " V rms , x P Γ i z tOu , i " 1, N , µ i B 2 m `B pmB p H px, Bvqq " 0, x P Γ i z tOu , i " 1, N ,

N ÿ i"1 γ i µ i B i v pOq " 0, N ÿ i"1 " µ i B i m pOq `Bp H i ´O, B i vpOq ¯mi pOq ı " 0, B i v pν i q " 0, µ i B i m pν i q `Bp H i ´νi , B i vpν i q ¯mi pν i q " 0, i " 1, N , v i pOq " v j pOq , m i pOq γ i " m j pOq γ j , i, j " 1, N , ż G v pxq dx " 0, ż G m pxq dx " 1, m ě 0. (0.0.2)
Ici tγ i u est un ensemble de constantes positives et tµ i u est une constante de viscosité correspondant à Γ i . Commentons le système MFG (0.0.2).

• Les constantes positives pγ i q sont reliées aux probabilités d'entrée dans les arrêtes du processus stochastique sous-jacent sur le réseau qui décrit la dynamique d'un joueur "moyen".

• Les constantes positives pµ i q sont les coefficients de diffusion dun processus stochastique dans les arêtes.

• Les EDP de la première ligne sont les équations de Hamilton-Jacobi-Bellman (HJB) ergodiques dans les arêtes, associées au problème de contrôle optimal du joueur typique du problème MFG. Les hypothèses principales sont que nous considèrons des Hamiltoniens sous-quadratiques par rapport au gradient et un couplage V très général. Ce dernier peut être local, seulement borné inférieurement et strictement croissant (pour obtenir l'unicité).

• Les EDP de la deuxième ligne sont des équations de Fokker-Planck (FP) dans les arêtes qui décrivent la distribution m de l'ensemble de joueurs du problème MFG.

• La troisième ligne est une condition de Kirchhoff pour la fonction valeur du problème de contrôle au sommet.

• La quatrième équation est une condition de transmission pour m au sommet.

• La cinquième ligne traduit les conditions de Kirchhoff pour v aux bords du réseau (qui se réduisent à des conditions de Neumann) et les conditions de transmission pour m aux bords du réseau (qui se réduisent à des conditions de Robin). Ici, dans le cas de la jonction, le bord du réseau est constitué du but des arêtes qui ne contiennent pas O.

• La sixième ligne du système exprime les conditions de continuité pour v et les conditions de saut pour m (ces dernières représentant une des originalité et difficulté du problème).

• La dernière ligne contient des conditions de normalisation pour v et m (qui est une densité de probabilité).

Le premier résultat est de poser le problème et d'expliquer l'obtention du système d'EDP et des conditions de jonctions. Ensuite, nous montrons quelques résultats préliminaires utiles, d'abord sur certains problèmes aux limites linéaires elliptiques équations, puis pour deux d'équations linéaires de Kolmogorov et de Fokker-Planck en dualité. L'existence de solutions faibles est obtenue en appliquant le théorème de Banach-Necas-Babuška à une paire spéciale d'espaces Sobolev appelés V et W ci-dessous et l'alternative de Fredholm. L'unicité vient d'un principe maximum. À l'aide de cas résultats nous prouvons que le système (0.0.2) est bien posé par des arguments de point fixe.

Dans le dernier chapitre, nous considérons le système MFG à horizon fini sur les réseaux suivant:

1 Introduction

General introduction

The aim of this dissertation is to study Hamilton-Jacobi equations associated with optimal control problems and mean field games problems in the case when the state space is a network.

Hamilton-Jacobi equations and optimal control problems

In this section, we recall general optimal control problems, and their connection to Hamilton-Jacobi equations.

First of all, we consider a classical problem in calculus of variation, in which a single agent tries to optimize his path in space time with respect to a fixed cost function. Specifically, suppose that the agent is at x P R n at time t " 0 and moves with a given velocity (or dynamic) f , possibly subject to some random noise. The trajectory of the agent is solution to a controlled ordinary differential equation (when µ " 0) or stochastic differential equation (when µ ą 0), namely dy psq " f py psq , α psqq ds `a2µ dW s , y p0q " x, (

where W t is a F t -adapted Wiener process in a reference probability system pΩ, pF t q tě0 , Pq. The F t -progressively measurable function α : r0, `8q Ñ A is called the control and the set of all admissible control functions is denoted by T . Under mild hypotheses, for a given control α, (1.1.1) has a unique solution y " y x,α , which is an absolutely continuous function (when µ " 0) or a continuous simple path (when µ ą 0). Depending on the situation, we will consider several costs associated to a given trajectory. The first one is related to the so-called infinite horizon control problem, for which J px, αq :" E x "ż 8 0 py x,α ptq, αptqq e ´λt dt  , (

with is a given running cost and λ is a positive constant. The case of finite horizon control problem is when one runs the trajectory for a finite range of time T ´t, starting at x at time t J px, t, αq :" E x,t

""ż T t py x,α psq , αpsqq ds `vT py x,α pT qq

* , (1.1.3)
where v T is called the terminal cost. In the deterministic case µ " 0, there is no expectation.

The optimal control problem is to find an optimal control α P T such that the total cost is minimized. In 1950s, to study these problems, Bellman [START_REF] Bellman | Dynamic programming[END_REF] developed a Dynamical Programming approach. The first step is to introduce the value function of the problem vpxq " inf αp¨qPT Jpx, αq, or vpx, tq " inf αp¨qPT Jpx, t, αq, (1.1.4) which is the optimal cost of the optimization problem.

Chapter 1. Introduction

Then the key idea is that v satisfies a functional equation, called the Dynamical Programming Principle (DPP), from which we obtain that v is a solution of a Hamilton-Jacobi-Bellman (HJB) equation, namely λv pxq ´µ∆u `H px, ∇v pxqq " 0, (1.1.5) in the infinite horizon case, and # ´Bt v px, tq ´µ∆u `H px, Bvq " 0, vpx, T q " v T pxq, (1.1.6) in the finite horizon case. The Hamiltonian H is defined by H px, pq " sup aPA t´p ¨f px, aq ´ px, aqu .

We can see that the HJB equations (1.1.5) and (1.1.6) contain all the relevant information to compute the value function and to design the optimal control strategy.

Unfortunately, the value function is not differentiable in general and therefore to make rigorous the previous approach, solutions to the HJB equations need to be considered in a weak sense. In this thesis, we will consider viscosity solutions (Chapter 2) and classical weak solutions (Chapters 3 and Chapter 4).

Weak solutions are well-known and we refer to classical books of Lions [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF], Brezis [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF] and Evans [START_REF] Evans | Partial differential equations[END_REF] for details. As far as viscosity solutions are concerned, the history is more recent. From the 1970's, there are considerable breakthroughs dealing with non-smooth value functions. We refer the reader to the books of Aubin & Cellina [START_REF] Aubin | of Grundlehren der Mathematischen Wissenschaften[END_REF], Aubin & Frankowska [START_REF] Aubin | Set-valued analysis, volume 2 of Systems & Control: Foundations & Applications[END_REF], Clarke [START_REF] Clarke | Methods of dynamic and nonsmooth optimization[END_REF][START_REF] Clarke | Optimization and nonsmooth analysis[END_REF][START_REF] Clarke | Nonsmooth analysis and control theory[END_REF] and references therein. At the beginning of the 1980's, Crandall & Lions [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF] and Crandall, Evans & Lions [START_REF] Crandall | Some properties of viscosity solutions of Hamilton-Jacobi equations[END_REF] introduced viscosity solutions, which appear to be well-adapted to solve PDEs like (1.1.5) and (1.1.6). Main references on the subject are Lions [START_REF] Lions | Solutions de viscosité des équations de Hamilton-Jacobi du premier ordre et applications[END_REF], Crandall, Ishii & Lions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF], Barles [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF], Bardi & Capuzzo-Dolcetta [START_REF] Bardi | Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations[END_REF], Fleming & Soner [START_REF] Fleming | Controlled Markov processes and viscosity solutions[END_REF], Bardi, Crandall, Evans & Soner [START_REF] Bardi | Viscosity solutions and applications[END_REF], Achdou, Barles, Ishii & Litvinov [START_REF] Achdou | Hamilton-Jacobi equations: approximations, numerical analysis and applications[END_REF].

Mean field games

In this section, we shall give an overview introduction to the mean field games.

Recently, an important research activity on mean field games (MFG for short) has been initiated since the pioneering works [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen. II. Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Mean field games[END_REF] of Lasry & Lions. Related ideas have been developed independently in the engineering literature by Huang, Caines & Malhamé, see for example [START_REF] Huang | Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle[END_REF][START_REF] Huang | Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized -Nash equilibria[END_REF][START_REF] Huang | An invariance principle in large population stochastic dynamic games[END_REF]. It aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number N of agents tends to infinity. Previously, the concept was developed in economic literature under the terminology of heterogeneous agent models, see [START_REF] Aiyagari | Uninsured idiosyncratic risk and aggregate saving[END_REF][START_REF] Bewley | Stationary monetary equilibrium with a continuum of independently fluctuating consumers[END_REF][START_REF] Huggett | The risk-free rate in heterogeneous-agent incomplete-insurance economies[END_REF][START_REF] Krusell | Income and wealth heterogeneity in the macroeconomy[END_REF]. In the asymptotic behavior of stochastic differential games, it is assumed that the agents are all identical and that an individual agent can hardly influence the outcome of the game. Moreover, each individual strategy is influenced by some averages of functions of the states of the other agents. In the limit when N Ñ `8, a given agent feels the presence of the others through the statistical distribution of the states. Since perturbations of the strategy of a single agent do not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. The delicate question of the passage to the limit is one of the main topics of the book of Carmona & Delarue [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications I-II[END_REF]. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward in time Kolmogorov or Fokker-Planck (FP) equation and a backward HJB equation. The unknown of this system is a pair of two functions:

1.1. General introduction

• the value function of the stochastic optimal control problem solved by a representative agent. The associated SDE is (1.1.1) with f " α, namely dy " αpsqds `a2µ dW s ,

• the density of the distribution of states.

In the infinite horizon limit, one obtains a system of two stationary PDEs. In classical control problems as described in Section 1.1.1, a single agent has his/her own fixed cost to minimize. In MFG, the model is generalized by allowing the cost of the representative agent to also depends on an interaction term between the agents: the cost functional depends on the probability density function m of all agents. More precisely, an typical agent controls the SDE (1.1.1) and we aim to minimize the following cost functional

J px, t, αq " E x,t
"ż T t p py x,α psq , α psqq `V rm p¨, sqs py x,α psqqq ds `vT py x,α pT q , T q

 .
The interaction term V may have different meanings. If V is increasing, then the model means that the agent prefers to be away from the other agents, which leads to a repulsive effect. Conversely, a decreasing V leads to an attractive effect.

Let v be the value function of the problem. From Section 1.1.1, if v is smooth enough, then v is a solution of a viscous HJB equation

´Bt v ´µ∆v `H px, ∇vq " V rms , v px, T q " v T pxq . (1.1.7)
The equation (1.1.7) is a backward equation since the agents' decisions are based on their goals in the future. The optimal control is heuristically given in feedback form α ‹ px, tq " ´Bp H px, ∇v px, tqq. Now if all agents argue in this way, their repartition will move with a velocity which is due, on the one hand, to the diffusion, and, one the other hand, on the drift term ´Bp H px, ∇v px, tqq. This leads to the FP equation ´Bt m `µ∆m `div pB p H px, ∇vq mq " 0, m px, 0q " m 0 .

(1.1.8)

The forward equation (1.1.8) represents where the agents actually end up, based on their initial distribution. To summarize, the non-stationary mean field games equation becomes

$ ' ' & ' ' % ´Bt v ´µ∆v `H px, ∇vq " V rms in R n ˆp0, T q , ´Bt m `µ∆m `div pB p H px, ∇vq mq " 0 in R n ˆp0, T q , m p0q " m 0 , v px, T q " v T pxq .
(1.1.9)

Let us discuss the coupling term V in MFG system (1.1.9). When the coupling V is such that there is F : R `Ñ 0 with V rmspx, tq " F pmpx, tqq, we say that V is a local coupling. But the coupling may also be nonlocal, for instance when V rmspx, tq " rmp¨, tq ‹ gp¨, tqspxq where ‹ stands for the space convolution.

Since the seminal works of Lasry & Lions and Caines, Huang & Malhamé, the subject has been growing quickly. A very nice introduction to the theory of MFGs is supplied in the notes of Cardaliaguet [START_REF] Cardaliaguet | Notes on mean field games[END_REF] and Achdou [START_REF] Achdou | Mean field games: additional notes[END_REF]. We also refer to the survey paper of Gomes & Saúde [START_REF] Gomes | Mean field games models-a brief survey[END_REF] and the books of Bensoussan, Frehse & Yam [START_REF] Bensoussan | Mean field games and mean field type control theory[END_REF], Gomes, Pimentel & Voskanyan [START_REF] Gomes | Regularity theory for mean-field game systems[END_REF] and Carmona & Delarue [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications I-II. Probability Theory and Stochastic Modelling[END_REF] for a general presentation.

Chapter 1. Introduction

Optimal control problems on networks and Hamilton-Jacobi equations on networks

In this section, we consider optimal control problems and HJ equations on networks and state the results obtained in Chapter 2. There are two main challenges that we have not encountered yet. Firstly, a network is a somewhat complicated space and we shall define the set of admissible controls which make the trajectories remained on networks. Secondly, the Hamiltonian on networks is generally defined branch by branch and so, it may be discontinuous at the vertices, which inhibits us from applying directly the classical techniques to deal with the HJ equations.

The first issue already appears in state constraints control problem and the second one is a natural issue in the case of Hamilton-Jacobi equations with discontinuities in space. We describe now these two cases before tackling the issue of Hamilton-Jacobi equations on networks, which is our main interest in Chapter 2.

State constrained optimal control problems

In optimal control problems with state constraints, we study the trajectories of the controlled dynamical system which are confined in a given set. More specifically, we add the following constraint to the problem (1.1.4): y x,α ptq P K, for all t P r0, T s in the deterministic case µ " 0, and y x,α ptq P K, for all t P r0, T s, P-a.e. in Ω in the stochastic case µ ą 0, where K Ă R d is a given closed subset. It means the set of admissible controls α : r0, T s Ñ A has to be restricted to the subset T x which keeps the solution of ( assuming classical continuity assumptions on J. Now, since T depends on the starting point of the trajectory, the above computation is not true anymore and continuity of v is more involved. The value function is no longer continuous unless a special controllability assumption is added to modify the dynamics on the boundary of state constraints. The characterization of the value function as the unique solution of a HJB equation is also another big issue. In [START_REF] Soner | Optimal control with state-space constraint[END_REF][START_REF] Soner | Optimal control with state-space constraint[END_REF], Soner characterized the value function of optimal control problems with state constraints as the unique constrained viscosity solutions of the related HJ equation, i.e., viscosity solution "inside K" and viscosity supersolution on the boundary of K. It was then developed by Capuzzo-Dolcetta & Lions [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF] and Ishii & Koike [START_REF] Ishii | A new formulation of state constraint problems for first-order PDEs[END_REF]. As noted above, we need a special controllability assumption to ensure the continuity of the value function, like the "inward pointing qualification condition (IQ)" in [START_REF] Soner | Optimal control with state-space constraint[END_REF]. It means that at each point of K, there exists a control such that the dynamic points inward K.

In such a case, the value function is the unique continuous viscosity solution to an appropriate HJB equation, see [START_REF] Soner | Optimal control with state-space constraint[END_REF][START_REF] Soner | Optimal control with state-space constraint[END_REF][START_REF] Ishii | A new formulation of state constraint problems for first-order PDEs[END_REF][START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF] and the work of Motta [START_REF] Motta | On nonlinear optimal control problems with state constraints[END_REF]. However, in some cases, we do 1.2. Optimal control problems on networks and Hamilton-Jacobi equations on networks not have the condition IQ and hence the continuity of the value function is no longer ensured. The "outward pointing qualification condition (OQ)" was therefore introduced by Blanc [START_REF] Blanc | Deterministic exit time control problems with discontinuous exit costs[END_REF], Frankowska & Plaskacz [START_REF] Frankowska | Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints[END_REF], which assume that every point on the boundary of K can be reached by a trajectory coming from the interior of K. Under this assumption, one can characterize the value function as the discontinuous bilateral viscosity solution of a HJB equation. There are other works dealing with such kind of problems under weaker conditions, see Bokanowski, Forcadel & Zidani [START_REF] Bokanowski | Deterministic state-constrained optimal control problems without controllability assumptions[END_REF], Frankowska & Mazzola [START_REF] Frankowska | Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints[END_REF][START_REF] Frankowska | On relations of the adjoint state to the value function for optimal control problems with state constraints[END_REF].

HJB approach for problems with discontinuity in state

This section is concerned with optimal control problems and HJB equations on multi-domains, which means that the control problems and the corresponding Hamiltonians in each domains are different. In the interior of each domain, we have a classical HJB equation like (1.1.5) and (1.1.6). The point is to determine the relevant conditions at the interfaces, as in the state constraints problems at the boundary.

To explain more precisely the problem, we consider a simple example λv pxq `H px, ∇v pxqq " 0,

x P R, (1.2.2)
where λ is a positive constant and the Hamiltonian H : R ˆR Ñ R is defined by

H px, pq " $ & % H 1 ppq , if x ă 0, H 2 ppq , if x ě 0, (1.2.3)
where H 1 , H 2 are convex and coercive. The Hamiltonian H is definitely discontinuous at x " 0 if H 1 ‰ H 2 . In [START_REF] Ishii | Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets[END_REF], Ishii introduces discontinuous viscosity solutions to solves this problem. A locally bounded function u is a discontinuous viscosity solution of (1.2.2) provided u ˚is a subsolution and u ˚is a supersolution in the following sense 1 . For all x P R and ϕ P C 1 pRq such that x is a maximum (resp. minimum) point of u ˚´ϕ (resp. u ˚´ϕ), then λu ˚pxq `H1 `ϕ1 pxq ˘ď 0 (resp. λu ˚pxq `H1 `ϕ1 pxq ˘ě 0), x ă 0, λu ˚pxq `H2 `ϕ1 pxq ˘ď 0 (resp. λu ˚pxq `H2 `ϕ1 pxq ˘ě 0), x ą 0, and, if x " 0 then

λu ˚pxq `min H 1 `ϕ1 p0q ˘, H 2 `ϕ1 p0q ˘( ď 0, x " 0, (resp. λu ˚pxq `max H 1 `ϕ1 p0q ˘, H 2 `ϕ1 p0q ˘( ě 0, x " 0),
Note that the first conditions are classical viscosity inequalities in the open sets tx ă 0u and tx ą 0u, whereas the last condition is a mixed condition at the interface. Here it means that at least one of the left or right inequality has to hold at x " 0.

To see the difficulty to obtain uniqueness, let us try to prove a comparison principle, i.e., to prove that any USC subsolution u is below any LSC supersolution v (uniqueness follows easily from such a result). Assuming by contradiction that pu ´vqpx 0 q ą 0 for some x 0 , we use the "doubling variable technique" (e.g., [START_REF] Achdou | Hamilton-Jacobi equations: approximations, numerical analysis and applications[END_REF]), which consists in considering 0 ă sup x,yPR φ ε px, yq, where φ ε px, yq " upxq ´vpyq ´|x ´y| 2 ε 2 .

Chapter 1. Introduction

We assume here for simplicity that the supremum is achieved at px ε , y ε q. By the choice of the penalization term in φ ε , it follows that x ε and y ε tend to the same x as ε Ñ 0. In the easy case, when x " 0, both x ε and y ε are in the same branch for small ε, for instance x ε , y ε ą 0. Applying the definition of viscosity solution, we get 0 ă λupx ε q ´λvpy ε q " λupx ε q ´λvpy ε q `H2 ˆ2px ε ´yε q

ε 2 ˙´H 2 ˆ2px ε ´yε q ε 2 ˙ď 0,
which is a contradiction. But, if x " 0 (meaning that 0 is a maximum point of u ´v), then it may happen that x ε and y ε are in different branches for all ε, let us say x ε ă 0 and y ε ą 0. In this case, the definition of viscosity solution gives only

λupx ε q ´λvpy ε q `H1 ˆ2px ε ´yε q ε 2 ˙´H 2 ˆ2px ε ´yε q ε 2 ˙ď 0,
and we cannot conclude since

H 1 ´2pxε´yεq ε 2 ¯ " H 2 ´2pxε´yεq ε 2 ¯.
In this thesis, the problem is not specific to this example, but appears when the Hamiltonian is discontinuous with respect to the state variable. Hence, although a comparison principle for viscosity sub and supersolution is proved in [START_REF] Ishii | Hamilton-Jacobi equations with discontinuous Hamiltonians on arbitrary open sets[END_REF], we will see later that a viscosity solution in the sense of Ishii is not enough to characterize a value function of an optimal control problem on networks. To overcome this difficulty, it is necessary to impose certain transmission conditions on the interfaces, which will lead to new notions of viscosity solutions.

Soravia [START_REF] Soravia | Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian[END_REF][START_REF] Soravia | Degenerate eikonal equations with discontinuous refraction index[END_REF] and Soravia & Garavello [START_REF] Garavello | Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost[END_REF][START_REF] Garavello | Representation formulas for solutions of the HJI equations with discontinuous coefficients and existence of value in differential games[END_REF] were among the first to tackle this issue in the framework of viscosity solutions both from the HJB and optimal control point of view. In [START_REF] Garavello | Optimality principles and uniqueness for Bellman equations of unbounded control problems with discontinuous running cost[END_REF], the authors study an optimal control with a discontinuous cost with respect to the state variable, which leads to HJB equations with discontinuity in state. However, their value function is not the unique solution of this HJB equation. They can only characterize the minimal and the maximal solution of the HJB equation, using some sub-and super-optimality principles from the optimal control problem. Note that some sub-and super-optimality principles together with additional transmission conditions were later used in a fruitful way in [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF][START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] and in Chapter 2 (see below for details).

Some transmission conditions appear in the work of Bressan & Hong [START_REF] Bressan | Optimal control problems on stratified domains[END_REF] about HJB equations in stratified domains. They introduce HJB tangential equations on the interfaces, which allow them to prove a comparison principle using control arguments.

After that, Barles, Briani & Chasseigne [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF] solve the problem (1.2.2)-(1.2.3) in general twodomains regional control problems. They consider two transmission conditions on the interface, so-called singular and regular dynamics, according to the behavior of dynamics on the interface. More precisely, let f i and A i be the dynamic and the control set corresponding to Ω i (Ω 1 " p´8, 0s " R `e1 and Ω 1 " r0, `8q " R `e2 ). Then the singular case means f i p0, a i q ă 0, while the regular case means f i p0, a i q ě 0 for all a i P A i . Then, there are two different value functions U ´and U `. The first one is obtained when allowing all kind of controlled strategies (with singular and regular dynamic) while the last one is obtained by forbidding singular dynamics. The authors use the Ishii's notion of solutions and prove that U ´and U `are , respectively, the minimal and maximal solution of (1.2.2)-(1.2.3). Hence, they consider additional properties to obtain the characterization result. A comparison principle and a stability results for both value functions are also established. The idea for uniqueness comes from the fact that, in some senses the singular strategies are not encoded in the equations (1.2.2)-(1.2.3), while it is the case for the regular ones.

The article of Rao & Zidani [START_REF] Rao | Hamilton-Jacobi-Bellman Equations on Multi-Domains[END_REF] and Rao, Siconolfi & Zidani [START_REF] Rao | Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations[END_REF] are also in line with [START_REF] Bressan | Optimal control problems on stratified domains[END_REF]. The problems studied are also close to those considered in [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF]. Indeed, the authors are interested 1.2. Optimal control problems on networks and Hamilton-Jacobi equations on networks

ν 2 ν 3 ν 4 ν 1 ν 6 ν 5 Γ 1 Γ 2 Γ 5 Γ 3 Γ 4 O Figure 1.1: On left, a general network Γ; on right, a junction G.
in problems where they consider different HJB equations in each region Ω i of space instead of studying the equations at the junction. The authors propose a junction condition involving a Hamiltonian H E which is called essential Hamiltonian. The equation with the essential Hamiltonian are stronger for the characterization of sub and supersolutions, but the value function satisfies this equation on the interfaces. Finally, in their work, the comparison principle between USC subsolutions and LSC supersolutions (they are continuous on the interface) is obtained.

Let us also mention some works dealing with HJ equations without using control arguments. In [START_REF] Camilli | Hamilton-Jacobi equations with measurable dependence on the state variable[END_REF][START_REF] Camilli | Time-dependent measurable Hamilton-Jacobi equations[END_REF], Camilli & Siconolfi propose a theory of HJ equations with Hamiltonians Hpx, pq only measurable with respect to the state variable, convex (or quasi-convex in the autonomous case) and coercive respect to p. Another important work is [START_REF] Coclite | Viscosity solutions of Hamilton-Jacobi equations with discontinuous coefficients[END_REF] in which Coclite & Risebro study the discontinuous HJ equations that may appear in 3-dimension reconstruction problems from shadows or the "shape from shading". Finally, in [START_REF] Giga | Hamilton-Jacobi equations with discontinuous source terms[END_REF], Giga & Hamamuki study HJ equations with intermittent source term, which appears for example in training models of crystals.

Hamilton-Jacobi equations and optimal control on networks

In this section, we consider the optimal control problems on networks and the associated HJ equations. A network (or a graph) is a set of items, referred to as vertices (or nodes/crosspoints) which are denoted in this work by ν i (in general case), or O (in junction case-a network has only one vertex). The connections between them referred to as edges, which are denoted by Γ α , see example in Figure 1.1.

As explained in Section 1.2.2, to characterize the value function of the optimal control problem on network as the unique viscosity solution of a suitable HJB equation, one needs to find the right transition conditions at the vertices.

One of the first articles about optimal control problems on networks appeared in 2013. Achdou, Camilli, Cutri & Tchou [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF] derived the HJB equation associated to an infinite horizon optimal control on a network and proposed a suitable notion of viscosity solution. At the same time, Imbert, Monneau & Zidani [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] proposed an equivalent notion of viscosity solution for studying a Hamilton-Jacobi approach to junction problems and traffic flows. Both [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF] and [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] contain first results on comparison principles which were fundamental for several developments that follow. It is also worth mentioning the work by Schieborn & Camilli [START_REF] Schieborn | Viscosity solutions of Eikonal equations on topological networks[END_REF], in which the authors focus on eikonal equations on networks and on a less general notion of viscosity solution. We also refer to the work [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF] where Camilli, Marchi & Schieborn study elliptic equations on the edges with Kirchhoff-type conditions at the vertices; after that, they prove the definition of solutions, which Chapter 1. Introduction is defined in [START_REF] Schieborn | Viscosity solutions of Eikonal equations on topological networks[END_REF], is consistent with the vanishing viscosity method. In the particular case of eikonal equations, Camilli & Marchi [START_REF] Camilli | A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks[END_REF] establish the equivalence between the definitions given in [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF][START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF][START_REF] Schieborn | Viscosity solutions of Eikonal equations on topological networks[END_REF].

Since 2012, several proofs of comparison principles for HJB equations on networks, giving uniqueness of the solution, have been proposed. 1. Following [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF], Achdou, Oudet & Tchou [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] prove the comparison principle for a stationary HJB equation arising from an optimal control with infinite horizon, by mixing arguments from the theory of optimal control and PDE techniques. Their proof was inspired by the works of Barles, Briani & Chasseigne [START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF] on regional optimal control problems with discontinuous dynamics and costs.

2. A different and more general proof, using only arguments from the theory of PDEs was obtained by Imbert & Monneau in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]. The proof works for quasi-convex Hamiltonians, and for stationary and time-dependent HJB equations. It relies on the construction of suitable vertex test functions which are designed to take into account the transition condition at the vertices.

3.

A simple and elegant proof, working for non convex Hamiltonians, has been very recently given by Lions & Souganidis [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF][START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF].

The works [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] and [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF][START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF] have been particularly influential in our work in Chapter 2 and we will give more details in Section 1.2.4.

Results in Chapter 2: Hamilton-Jacobi equation on networks with switching costs

In Chapter 2, we consider an optimal control problem on a network the setting of which is close to [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF]. In addition to this work, we suppose that there are entry (or exit) costs at the boundary of each edge of the network. Our goal is to characterize the value solution as the unique viscosity solution of an appropriate HJB equation.

For simplicity, we only consider a network with only one vertex from which start N semiinfinite straight edges. We call such a network G a junction. The edges are denoted by pΓ i q i"1,N . See Figure 1.1 for an example of junction on R 2 .

We consider infinite horizon optimal control problems which have different dynamics and running costs in each edge. For i " 1, N , the control sets, dynamics and running cost corresponding to Γ i are, respectively, denoted by A i , f i and i . This means that, if a trajectory moves inside Γ i , we use a control taking values in A i , the velocity is f i and we pay the running cost i . Interestingly, if we come to O, it is possible to use a control taking values in all A i .

Precise assumptions are a little technical so we refer the reader to the set of assumptions [H] in Chapter 2 for details. We prefer to give a flavour of the model.

We consider the infinite horizon optimal control problems (1.1.1) on the junction G with µ " 0. Moving on G, beside paying the cost as in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF], one has to pay the additional entry cost c i e ´λt ik . Here c i is a positive entry costs corresponding to Γ i and t ik P K i is the time the trajectory enters Γ i z tOu. We define a cost functional with entry costs:

J px, αq " ż `8 0 py x,α pζq , α pζqq e ´λt dt `N ÿ i"1 ÿ kPK i ĂN
c i e ´λt ik (cost functional with entry cost).

Optimal control problems on networks and Hamilton-Jacobi equations on networks

Here T x is the set of admissible controlled trajectories starting from x P Γ (it depends on x for the same reason as in Section 1.2.1). The value function of the infinite horizon optimal control problem with entry costs is defined by: v pxq " inf αPTx J px, αq (value function with entry cost).

We only focus on the case with entry costs here, but it is also possible to consider the case with exit costs, see Chapter 2. By the definition of the value function, we are only interested in control laws α such that J px, αq ă `8. Hence, we only consider the case such that the state cannot switch edges infinitely many times in finite time, otherwise the cost functional is obviously infinite.

As noted above, our definition of the value function is similar to one made in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] except the additional costs c i for entering the edge Γ i at O. This makes the value function possibly discontinuous contrary to [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] where it is continuous. For a better illustration, let us give the following simple example.

Example 1.2.1. Consider the same junction as in problem (1.2.2), G " Γ 1 Y Γ 2 where Γ 1 " R `e1 and Γ 2 " R `e2 where e 1 " ´1 and e 2 " 1. The control sets are A i " r´1, 1sˆtiu with i P t1, 2u. Set f i px, pa i , iqq " a i and 1 " 1, 2 px, pa 2 , 2qq " 1 ´a2 . An easy computation gives the explicit formula for the value function u without entry cost,

u pxq " $ & % 0, in Γ 2 , 1 ´e´λ|x| λ , in Γ 1 .
Note that u is continuous. Now, if we add some positive entry costs c 1 and c 2 in the edges, the value function can again be computed:

1. If c 2 ě 1 λ , then v pxq " $ & % 0 if x P Γ 2 z tOu , 1 λ if x P Γ 1 . 2. If c 2 ă 1 λ , then v pxq " $ & % 0 if x P Γ 2 z tOu , 1 ´e´λ|x| λ `c2 e ´λ|x| if x P Γ 1 .
In this case, v is discontinuous, see Figure 1.2. From the above formulas, since 1 ě 2 , if the trajectory starts at x P Γ 2 , the bad strategy is moving to O, paying entry cost and entering Γ 1 . Hence, c 1 does not appear in the formula for the value function v.

Before stating the main results, let us have a quick look at our assumption rHs. In this assumption, the control sets A i are disjoint; the dynamics are bounded, Lipschitz continuous; the running cost are bounded uniformly continuous. Additionally, we suppose that a controllability assumption holds near O. It means that it is always possible to find a trajectory connecting two points sufficiently close to O. It follows that the restriction v| Γ i ztOu of v is Lipschitz continuous and therefore it may be extended to O in each edge. This extension is denoted by v i . Note that v i p0q may not be equal to v j p0q if i " j since, in general, v is not continuous at O.

We are ready to introduce the first theorem.

Chapter 1. Introduction

Γ 1 Γ 2 upxq Γ 2 Γ 1 vpxq c 2 1 λ 1 λ O O Figure 1
where H T O " max i"1,N max a i PA O i t´ j pO, a j qu " ´min i"1,N min a i PA O i t j pO, a j qu. (1.2.5)
This theorem makes the link between the value of the original value function v and the values of the extensions v i at the junction. The equality in (1.2.4) means that if the trajectory begins at O, the optimal strategy is either to stay at O for all time or to enter immediately the edge Γ i that has the lowest possible cost. Theorem 1.2.3. (Theorem 2.6.7) Let v be a value function with entry cost. Then pv 1 , . . . , v N q is a viscosity solution of the following Hamilton-Jacobi system

λu i pxq `Hi ˆx, du i de i pxq ˙" 0 if x P Γ i z tOu , i " 1, N , λu i pOq `max " ´λ min j‰i tu j pOq `cj u , H ì ˆO, du i dx i pOq ˙, H T O * " 0 if x " O. (1.2.6)
Inside every edge Γ i ztOu, we recognize a classical Hamilton-Jacobi with a Hamiltonian H i corresponding to the edge Γ i , namely

H i px, pq " max aPA i t´pf i px, aq ´ i px, aqu .
Here, the Hamiltonian at O with respect to i is defined by

H ì pO, pq " max aPA ì t´pf i px, aq ´ i px, aqu ,
where A ì " ta P A i : f i pO, aq ě 0u. At the vertex O, there is a special Hamiltonian or transmission condition, which takes into account all the possible strategies near the vertex:

• The term min j‰i tu j pOq `cj u accounts for situations in which the trajectory enters Γ i 0 where u i 0 pOq `ci 0 " min j‰i tu j pOq `cj u. ). Under the hypothesis rHs, let u " pu 1 , . . . , u N q be a bounded viscosity subsolution of (1.2.6) and w " pw 1 , . . . , w N q be a bounded viscosity supersolution of (1.2.6); then u ď w in G, namely u i ď w i in Γ i for all i " 1, N .

We provide two different proofs of this theorem, see a sketch of proof below. The first proof is inspired from [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF]. The authors focus on optimal control problems with independent dynamics and running costs in the edges, and after that they show that some arguments of [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF] can be adapted to yield a simple proof of a comparison principle. The second proof is based on the work of Lions & Souganidis [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF] and uses only PDEs tools. More specifically, the authors build a simple but very useful test-function on networks, and adapt it to doubling variable technique.

Let us mention that the existence of a unique viscosity solution follows directly from the comparison principle and Theorem 2.6.7. It is also possible to build a viscosity solution directly from the comparison principle and Perron's method, as in [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF].

To prove the comparison principle, we start as in the example in Section 1.2.2. We assume by contradiction that u i px 0 q ´wi px 0 q ą 0 for some i and x 0 . If the u i ´wi attains the supremum inside Γ i , the proof is done by classical arguments. Hence, we only focus on the case u i pOq ´wi pOq " max (1.2.9)

xPΓ i tu i pxq ´wi pxqu ą 0, ( 1
and the comparison principle for (1.2.9) is proved in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF]. Thus, to obtain the contradiction with (1.2.8), it suffices to prove that w i is a viscosity supersolution of (1.2.9) with j replaced by i. Now, since λu i pOq `HT O ď 0, we have λw i pOq `HT O ă 0. We now consider the two following cases.

Case 1: If w i pOq ă min j‰i tw j pOq `cj u, by (1.2.6), then w is a viscosity supersolution of (1.2.9) and it leads us to a contradiction.

Case 2: If w i pOq ě min j‰i tw j pOq `cj u, then there exists j 0 ‰ i such that w j 0 pOq `cj 0 " min j"1,N tw j pOq `cj u " min j‰i tw j pOq `cj u ď w i pOq .

Therefore, w j 0 pOq ă min j‰j 0 tw j pOq `cj u. We also have λu i pOq ´λ min j‰i tu j pOq `cj u ď 0. Thus w j 0 pOq ă u j 0 pOq. Repeating the proof of Case 1 with j 0 , we reach a contradiction.

A sketch of proof of Theorem 1.2.4 inspired by Lions & Souganidis [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF]. As in the example in Section 1.2.2, the key idea of this proof is building an admissible test-function ϕ ε such that upxq ´wpyq ´ϕε px, yq attains a maximum point px ε , y ε q P Γ i ˆΓi , then using the viscosity inequalities to obtain the contradiction. We consider the function Let ε tend to 0 and γ tend to 0, we obtain that u i pOq´w i pOq ď 0, the desired contradiction.

Ψ i,ε : Γ i ˆΓi ÝÑ R px,

Mean field games on networks

The section is devoted to the introduction of our works on infinite horizon (Chapter 3) and finite horizon (Chapter 4) Mean Field Games on general bounded networks. For simplicity, in this section, we only consider the model case of the junction G with N bounded edges. The other endpoint of Γ i is denoted by ν i and the edges Γ i are oriented from ν i to O.

Similarly to the problems in Section 1.2.3, what makes the MFG on networks more challenging comparing to classical MFG (Section 1.1.2) is stochastic optimal control problems with state constraints and discontinuity in states. To deal with this, one needs to answer the important question: What is a suitable transition condition at the vertices?

Camilli & Marchi [START_REF] Camilli | Stationary mean field games systems defined on networks[END_REF] is one of the first articles on infinite horizon MFGs on networks. They consider a particular type of Kirchhoff condition at the vertices for the value function. This condition comes from an assumption which can be informally stated as follows: if, at time τ , 1.3. Mean field games on networks the controlled stochastic process X t associated to a given agent hits O, then the probability that X τ `belongs to Γ i is proportional to the diffusion coefficient in Γ i . Under this assumption, it can be seen that the density of the distribution of states m is continuous at O. In our work, the assumption mentioned above is no longer valid. Therefore, it will be seen later that the value function v satisfies more general Kirchhoff conditions, and accordingly, the density of the distribution of states m is no longer continuous at O; the continuity condition is then replaced by suitable compatibility conditions on the jumps across the vertex. Accordingly, the weak solutions spaces V and W of the uncoupled HJB and FP equations (see (1.3.19)- (1.3.20)) are not the same. To overcome this difficulty, it is essential to consider an isomorphism from V to W , which is then used to build a suitable test-function for each uncoupled equation. See more details in Section 3.1.4.

Finally, under suitable assumptions, we will prove the existence, uniqueness and regularity for the both MFG systems. See Section 3.4 and Section 4.4.1.

Derivation of Mean field games system on networks

In this section, we consider the derivations of both cases: infinite horizon and finite horizon MFG systems on G. The main ideas for both systems are quite similar. However, the technique for finite horizon MFG system is more difficult than the other one. Hence, in this section, we mainly focus on the derivation of finite horizon MFG system.

Consider a real valued function a P P CpGq, where P CpGq contains all piecewise functions on G which are continuous except at O where it can be extended by continuity. Let us consider the linear partial differential operator:

Lu pxq " L i u pxq :" µ i B 2 u pxq `ai pxq Bu pxq , if x P Γ i , (1.3.1)
with domain D pLq :"

# u P C 2 pGq : N ÿ i"1 γ i µ i B i u pOq " 0, for all i " 1, N + . (1.3.2)
Freidlin and Sheu proved in [START_REF] Freidlin | Diffusion processes on graphs: stochastic differential equations, large deviation principle[END_REF] that the operator L is the infinitesimal generator of a Feller-Markov process on G with continuous sample paths. The operators L i and the transmission conditions at the vertices

N ÿ i"1 γ i µ i B i upOq " 0 (1.3.3)
define such a process in a unique way, see also [START_REF] Freidlin | Diffusion processes on graphs and the averaging principle[END_REF]Theorem 3.1]. The process can be written pX t , i t q where X t P Γ it . Moreover, there exist

E pM t |X s q " M s , for all 0 ď s ă t ď T. (1.3.6)
The goal is to derive the boundary value problem satisfied by the law of the stochastic process X t . Since the derivation here is formal, we assume that the law of the stochastic process X t is a measure which is absolutely continuous with respect to the Lebesgue measure on G and regular enough so that the following computations make sense. Let m px, tq be its density. We have By integration by parts, we get 0 "

E rvpX t , tqs " ż G v px,
N ÿ i"1 ż Γ i `Bt mpx, tq ´µi B 2 mpx, tq `Bpamqpx, tq ˘upx, tqdx `N ÿ i"1 ra| Γ i pν i , tqm| Γ i pν i , tq ´µi B i mpν i , tqs u| Γ i pν i , tq ´N ÿ i"1 ra| Γ i pO, tqm| Γ i pO, tq `µi B i mpO, tqs u| Γ i pO, tq ´N ÿ i"1 µ i m| Γ i pO, tqB i upO, tq. (1.3.8)

Mean field games on networks

We choose first, for i " 1, N , a smooth function u which is compactly supported in pΓ i z tO, ν i uqr 0, T s. Hence u| Γ i pν i , tq " u| Γ i pO, tq " 0 and B i upν i , tq " B i upO, tq " 0 for i " 1, N . Notice that up¨, tq P DpLq. It follows that m satisfies `Bt m ´µi B 2 m `B pmaq ˘px, tq " 0, for x P Γ i z tOu, t P p0, T q, i " 1, N .

(1.3.9)

For a smooth function χ : r0, T s Ñ R compactly supported in p0, T q, we may choose for every i P t1, . . . , N u, a smooth function u such that upν j , tq " χptqδ i,j and upO, tq " 0 for all t P r0, T s, j " 1, N and B j upO, tq " 0 for all t P r0, T s, j " 1, N , we infer a boundary condition for m a| Γ i pν i , tqm| Γ i pν i , tq ´µi B i mpν i , tq " 0, i " 1, N , t P p0, T q.

Next, we choose a smooth function u such that B j upO, tq " 0 for all t P r0, T s, j " 1, N , we infer a condition for m at O:

N ÿ i"1 a| Γ i pO, tqm| Γ i pO, tq `µi B i mpO, tq " 0, t P p0, T q.
Finally, for a smooth function χ : r0, T s Ñ R compactly supported in p0, T q, we choose u such that

• up¨, tq P DpLq

• B i upO, tq " χptq{p i , B j upO, tq " ´χptq{p j , B k upO, tq " 0 if k ‰ i, j.
Using such a test-function in (1.3.8) B t m ´µi B 2 m `B pmaq " 0, px, tq P pΓ i z tOuq ˆp0, T q, i " 1, N ,

N ÿ i"1 µ i B i m pO, tq `a| Γ i pO, tqm| Γ i pO, tq " 0, t P p0, T q, µ i B i m pν i , tq ´a| Γ i pν i , tqm| Γ i pν i , tq " 0, t P p0, T q, i " 1, N , m| Γ i pO, tq γ i " m| Γ j pO, tq γ j , t P p0, T q, i, j " 1, N , mpx, 0q " m 0 pxq, x P G. (1.3.10)
Consider a continuum of indistinguishable agents moving on the network G. Under suitable assumptions, the theory of MFGs asserts that the distribution of states is absolutely continuous with respect to Lebesgue measure on G. Hereafter, m stands for the density of the distribution of states: m ě 0 and ş G mpx, tqdx " 1 for t P r0, T s. The state of a representative agent at time t is a time-continuous controlled stochastic process X t in G, as defined previously, where the control is the drift a t , supposed to be of the form a t " apX t , tq.

For a representative agent, the optimal control problem is of the form

v px, tq " inf a E xt "ż T t
pL pX s , a s q `V rmp¨, tqs pX s qq ds `vT pX T q  , (1. 3.11) where E xt stands for the expectation conditioned by the event X t " x. The functions and operators involved in (1.3.11) will be described below. Let us assume that there is an optimal feedback law, i.e., a function a ‹ defined on G ˆr0, T s which is sufficiently regular in the edges of G such that the optimal control at time t is given by a ‹ t " a ‹ pX t , tq. An informal way to describe the behavior of the process at the vertices is as follows: if X t hits O, then it enters Γ i , i " 1, N with probability p i ą 0.

Let us discuss the ingredients in (1.3.11). The running cost depends separately on the control and on the distribution of states. The contribution of the distribution of states involves the coupling cost operator V , which may be either nonlocal and regularizing, i.e., V : P pGq Ñ C 2 pGq for example, or local, i.e. V rmspxq " F pmpxqq where F : R `Ñ R is a continuous function.

The contribution of the control involves the Lagrangian L, i.e., a real valued function defined on pY iPA Γ i zVq ˆR. If x P Γ i zV and a P R, Lpx, aq " L i px, aq, where L i is a continuous real valued function defined on Γ i ˆR. We assume that lim |a|Ñ8 inf xPΓ i L i px,aq |a| " `8. The last one is the terminal cost v T . Further assumptions on L, V and v T will be made below.

Under suitable assumptions, Ito calculus as in [START_REF] Freidlin | Diffusion processes on graphs and the averaging principle[END_REF][START_REF] Freidlin | Diffusion processes on graphs: stochastic differential equations, large deviation principle[END_REF] and the dynamic programming principle lead to the following HJB equation on G, more precisely the following boundary value problem:

$ ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' % ´Bt v ´µi B 2 v `H px, Bvq " V rmp¨, tqspxq, in pΓ i z tOuq ˆp0, T q , i " 1, N , v| Γ i pO, tq " v| Γ j pO, tq t P p0, T q i, j " 1, N , N ÿ i"1 γ i µ i B i v pO, tq " 0, t P p0, T q, B i v pν i , tq " 0, t P p0, T q, i " 1, N , v px, T q " v T pxq , x P G. (1.3.12)
We refer to [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF] Lasry | Mean field games[END_REF] for the interpretation of the value function v. Let us comment the different equations in (1.3.12):

1. The Hamiltonian H is a real valued function defined on `YN i"1 Γ i z tOu ˘ˆR. For x P Γ i z tOu and p P R, H px, pq " sup a t´ap ´Li px, aqu ,

The Hamiltonians H| Γ i ˆR are supposed to be C 1 and coercive with respect to p uniformly in x.

2. The second condition means in particular that v is continuous at the vertices.

3. The third equation in (1.3.12) is a Kirchhoff transmission condition (or Neumann boundary condition if ν i P BG); it is the consequence of the assumption on the behavior of X s at vertices.

If (1.3.11) has a smooth solution, it provides a feedback law for the optimal control problem, i.e., a ‹ px, tq " ´Bp H px, Bv px, tqq .

According to the previous part, the density mpx, tq of the law of the optimal stochastic process X t satisfies (1.3.10) (where a is replaced by a ‹ ). Finally, replacing a ‹ px, tq by the value

Mean field games on networks

´Bp H px, Bv px, tqq, we obtain the system

$ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' % ´Bt v ´µi B 2 v `H px, Bvq " V rms, t P p0, T q, x P Γ i z tOu , i " 1, N , B t m ´µi B 2 m `B pmB p H px, Bvqq " 0, t P p0, T q, x P Γ i z tOu , i " 1, N , N ÿ i"1 γ i µ i B i v pO, tq " 0, t P p0, T q, N ÿ i"1 " µ i B i m pO, tq `Bp H i ´O, B i vpO, tq ¯mi pO, tq ı " 0, t P p0, T q, B i v pν i , tq " 0, µ i B i m pν i , tq `Bp H i ´νi , B i vpν i , tq ¯mi pν i , tq " 0, t P p0, T q, i " 1, N , v i pO, tq " v j pO, tq , m i pO, tq γ i " m j pO, tq γ j , t P p0, T q, i, j " 1, N , v px, T q " v T pxq, m px, 0q " m 0 pxq, ż G m 0 pxqdx " 1, x P G.
(1.3.13) At a vertex O, the transmission conditions for both v and m consist of N linear relations, which is the appropriate number of relations to have a well posed problem. If ν i P BG, there is of course only one condition.

To end this part, we introduce the infinite horizon MFG system. Since the idea and the technique for derivation of this system are similar to (even simpler than) the finite case, we shall not explain them in details.

The first goal is also to derive the boundary value problem satisfying the law of the stochastic process X t . Consider the invariant measure associated with the process X t and assume that it is absolutely continuous with respect to the Lebesgue measure on G. Let m be its density:

ErupX t qs :" ż G
upxqmpxqdx, for all u P P CpGq.

(1.3.14)

Taking the time-derivative of each member of (1.3.14), choosing appropriate test-functions u P DpLq Ă P CpGq step by step as in the finite case, we get the following boundary value problem

for m $ ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' % ´µi B 2 m ´B pbmq " 0, in Γ i z tOu , m i pOq γ i " m j pOq γ j , i, j " 1, N , N ÿ i"1 rb pOq m Γ i pOq `µi B i m pOqs " 0, µ i B i m pν i q `Bp H i ´νi , B i vpν i q ¯mi pν i q " 0, i " 1, N , (1.3.15)
In the ergodic case, we aim to minimize the average cost

ρ :" inf as lim inf T Ñ`8 1 T E x "ż T 0 L pX s , a s q `V rm p¨, sq pX s qs ds  , (1.3.16)
where E x stands for the expectation conditioned by the event X 0 " x. The functions and operators involved in (1.3.16) were described in the derivation for finite horizon MFG system above. Under suitable assumptions, the Ito calculus and the dynamic programming principle

$ ' ' ' ' & ' ' ' ' % ´µi B 2 v `H px, Bvq `ρ " f, if x P Γ i z tOu , v i pOq " v j pOq, i, j " 1, N , N ÿ i"1 γ i µ i B i v pOq " 0, B i v pν i q " 0, i " 1, N .
(1.3.17)

If (1.3.16) has a smooth solution, then it provides a feedback law for the optimal control problem, i.e., a ‹ pxq " ´Bp Hpx, Bvpxqq. At the MFG equilibrium, m is the density of the invariant measure associated with the optimal control feedback law, so it satisfies (1.3.15), where a is replaced by a ‹ . To summarize, we get the following system

$ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' % ´µi B 2 v `H px, Bvq `ρ " V rms , x P Γ i z tOu , i " 1, N , µ i B 2 m `B pmB p H px, Bvqq " 0, x P Γ i z tOu , i " 1, N , N ÿ i"1 γ i µ i B i v pOq " 0, N ÿ i"1 " µ i B i m pOq `Bp H i ´O, B i vpOq ¯mi pOq ı " 0, B i v pν i q " 0, µ i B i m pν i q `Bp H i ´νi , B i vpν i q ¯mi pν i q " 0, i " 1, N v i pOq " v j pOq , m i pOq γ i " m j pOq γ j , i, j " 1, N , ż G v pxq dx " 0, ż G m pxq dx " 1, m ě 0.
(1.3.18)

Results in Chapter 3: A Class of Infinite Horizon Mean Field Games on Networks

After obtaining the transmission conditions at the O for both the value function and the density, we shall define a weak solution for MFG system in suitable Sobolev spaces on networks as follows:

V : " H 1 pGq " v P C pGq : v i P H 1 pΓ i q for i " 1, N ( , (1.3.19)
W : " " w P P CpGq :

w i P H 1 pΓ i q and w i pOq γ i " w j pOq γ j for all i, j " 1, N * . (1.3.20)
These functions spaces are indeed suitable since if we multiply the first equation in (1.3.18) with w P W , integrate over G and use integration by part for each Γ i , one gets

N ÿ i"1 ż Γ i pµ i BvBw `Hi px, Bvq w `ρwq dx ´N ÿ i"1 µ i w i pν i q B i v pν i q `N ÿ i"1 µ i w i pOq B i v pOq " 0.
(1.3.21) The second term is 0 because of Neumann boundary condition. By the property of the functions space W , the last term becomes

N ÿ i"1 µ i γ i w i pOq γ i B i v pOq " w 1 pOq γ 1 N ÿ i"1 µ i γ i B i v pOq , 1.
3. Mean field games on networks and it also vanishes by the Kirchhoff condition. Thus, from (1.3.21), one gets

N ÿ i"1 ż Γ i pµ i BvBw `Hi px, Bvq w `ρwq dx " 0. (1.3.22)
Similarly, multiply the second equation in (1.3.18) with u P V , integrate over G, use integration by part for each Γ i , apply the boundary conditions, transition conditions for m and the continuity of u, one gets

N ÿ i"1 ż Γ i pµ i BmBw `Bp H i px, Bvq wBuq dx " 0. (1.3.23)
These computations motivate the following definition of weak solutions for the MFG system on networks. We introduce our assumption in Chapter 3

(Hamiltonian) We assume that

H i P C 1 pΓ i ˆRq ; (1.3.24) H i px, ¨q is convex in p for each x P Γ i ; (1.3.25) H i px, pq ě C 0 |p| q ´C1 for px, pq P Γ i ˆR; (1.3.26) |B p H i px, pq| ď C 2 ´|p| q´1 `1¯f or px, pq P Γ i ˆR. (1.3.27)
for some constants C 0 , C 1 and C 2 .

(Coupling term) We assume that V r ms pxq " F pm pxqq with F P C pr0, `8q ; Rq , (1.3.28) for all m which are absolutely continuous with respect to the Lebesgue measure and such that d m pxq " m pxq dx. We shall also suppose that F is bounded from below, i.e., there exists a positive constant M such that F prq ě ´M, for all r P r0, `8q .

(1.3.29)

Theorem 1.3.2. Under assumption (1.3.24)-(1.3.29)
, there exists a weak solution of (1.3.18) pv, m, ρq P V ˆW ˆR. Moreover, v P C 2 pGq and m i P C 1 pΓ i q for all i. Finally, the uniqueness of (1.3.18) holds under assumption strictly increasing of F .

Let us give a sketch of proof for this theorem. First of all, we study the well-posedness for weak solution of the uncoupled HJB and FP equations in suitable Sobolev spaces V and W . More precisely, we study some linear boundary value problems with elliptic equations (Step 1), then on a pair of linear Kolmogorov and Fokker-Planck equations in duality (Step 1 and

Step 2). By and large, the existence of weak solutions is obtained by applying Banach-Necas-Babuška theorem to the Sobolev spaces V and W and Fredholm's alternative. Uniqueness comes from a maximum principle. In [START_REF] Camilli | Stationary mean field games systems defined on networks[END_REF], the authors apply Lax-Milgram lemma instead of Banach-Necas-Babuška theorem to get existence and uniqueness. However, it is impossible to apply Lax-Milgram lemma in our work since our solution space and the test-function space are not Chapter 1. Introduction the same, which results from the "jump condition". Using Step 1, we obtain the well-posedness for classical solution of the HJB (Step 3) and apply it to prove the well-posedness for classical solution of the ergodic problem (Step 4). Then, using the well-posedness for weak solution of the uncoupled ergodic equation and FP equation, we establish the existence result for the MFG system by a fixed point argument and a truncation technique (Step 5). Uniqueness is proved when the coupling is increasing, i.e., the function F in (1.3.18) is increasing for local couplings (in the case of nonlocal coupling, the increasing condition is replaced by (1.3.42)). Finally, classical arguments will then lead to the regularity and uniqueness of the solutions.

Step 1 : [Section 3.2.1, Section 3.2.2 and a part of Section 3.2.3] Consider a Kolmogorov's equation, for b P P C pΓq, g P W 1 and λ ě 0:

$ ' ' ' ' & ' ' ' ' % ´µi B 2 v `bBv `λv " g, in Γ i z tOu , v i pOq " v j pOq, i, j " 1, N , N ÿ i"1 γ i µ i B i v pOq " 0.
(1.3.30)

A weak solution of (1.3.30) is a function v P V such that A ‹ pv, wq :"

N ÿ i"1 ż Γ i pµ i BvBw `bBvwq dx " ż G
gwdx, for all w P W.

If λ ą 0, (1.3.30) has a unique weak solution. If λ " 0 and g " 0, the set of solutions of (1.3.30) is the set of constant functions on Γ.

Step 2 : [Section 3.2.3] Let λ 0 ą 0 and h P V 1 , we consider the following problem

$ ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' % λ 0 m ´µi B 2 m ´B pbmq " h, in Γ i z tOu , m i pOq γ i " m j pOq γ j , i, j " 1, N , N ÿ i"1 rb pOq m Γ i pOq `µi B i m pOqs " 0, µ i B i m pν i q `Bp H i ´νi , B i vpν i q ¯mi pν i q " 0, i " 1, N , (1.3.31) with m ě 0, ż G mdx " 1. (1.3.32)
A weak solution of (1.3.31) is a function m P W such that

A λ 0 pm, vq :" N ÿ i"1 ż Γ i rλ 0 mv `pµ i Bm `bmq Bvs dx " ż Γ hvdx, for all v P V.
For λ 0 large enough, we prove that there exists a constant C such that inf

wPW sup vPV A λ 0 pw, vq }v} V }w} W ě C, inf vPV sup wPW A λ 0 pw, vq }w} W }v} V ě C,
by Babuska's lemma (or inf sup lemma), we obtain the well-posedness for (1.3.31).

Mean field games on networks

Next, we prove the well-posedness for Fokker-Planck equation on G, namely (1.3.31) with λ 0 " 0.

Step 2 allows us to define a linear operator:

T : L 2 pGq ÝÑ W ãÑ L 2 pGq, T pmq " m,
where m is the solution of (1.3.31)-(1.3.32) with h " λ 0 m. Using the uniformly estimate for (1.3.31) and applying the fixed point theory, there exists a solution m for FP equation.

Next, from Step 1, the set of solutions of (1.3.30), with λ 0 " 0 and g " 0, is the set of constant functions on G. Hence, applying the Fredholm alternative, the set of solutions of FP equation is 1-dimensional. By the normalization condition (1.3.32), we obtain the uniqueness for FP equation. Moreover, by the comparison principle, m is strictly positive.

Step : [Section 3.3.1] We study the Hamilton-Jacobi equation for all λ ą 0

$ ' ' ' ' & ' ' ' ' % ´µi B 2 u λ `H px, Bu λ q `λu λ " f, in Γ i z tOu , u iλ pOq " u jλ pOq, i, j " 1, N , N ÿ i"1 γ i µ i B i v pOq " 0, B i v pν i q " 0, i " 1, N . (1.3.33)
We first deal with the bounded Hamiltonian. From Step 1, this allows us to define a linear operator

T : V ÝÑ V, T puq " v,
with b " 0 and g " Hpx, Buq. Using the fixed point theory, we obtain the existence of (1.3.33). The uniqueness is a consequence of the comparison principle. Now, to deal with quadratic Hamiltonian, we use the truncation technique combining with the previous process (with bounded Hamiltonian). This step is adapted the classical proof of Boccardo, Murat & Puel [START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique[END_REF].

Step : [Section 3.3.2] We are ready to solve the ergodic problem

$ ' ' ' ' & ' ' ' ' % ´µi B 2 v `H px, Bvq `ρ " f, if x P Γ i z tOu , v i pOq " v j pOq, i, j " 1, N , N ÿ i"1 γ i µ i B i v pOq " 0, B i v pν i q " 0, i " 1, N , (1.3.34) and ż G vdx " 0. (1.3.35)
From Step 3 and the quadratic Hamiltonian (1.3.27), we can obtain some uniform estimate for (1.3.33). Hence, the existence of (1.3.34) is deduced by letting λ to 0. The uniqueness results from the maximum principle of the classical equations on 1 dimension and the normalization condition (1.3.35).

Step : [Section 3.4] In the last step, using the well-posedness for weak solution of the uncoupled FP equation (Step 2) and ergodic equation (Step 4), we establish the existence for the MFG system with the bounded coupling term F . Then using the truncation technique based on the energy estimate, the existence with the general coupling term F is obtained. Finally, the uniqueness is deduced by using the increasing coupling term.

Results in Chapter 4: A Class of Finite Horizon Mean Field Games on Networks

Camilli & Marchi [START_REF] Camilli | Stationary mean field games systems defined on networks[END_REF] introduce the MFG systems on networks in the finite time horizon case (1.3.13) from two different points of view: either as the characterization of a Pareto equilibrium for dynamic games with a large number of indistinguishable players or as the optimality conditions for optimal control problems whose dynamic is governed by a PDE. They also show that two models lead to the same transition conditions. However, they only study for the infinite horizon MFG system (1. Let us introduce our assumptions in Chapter 4.

(Hamiltonian) We assume that for a constant C 0 independent of i.

H i P C 1 pΓ i ˆRq , ( 1
(Coupling operator) We assume that V is a continuous map from L 2 pGq to L 2 pGq, such that for all m P L 2 pGq, }V rms} L 2 pGq ď Cp}m} L 2 pGq `1q. (1.3.41)
Note that such an assumption is satisfied by local operators of the form V rmspxq " F pmpxqq where F is a Lipschitz-continuous function.

(Initial and terminal data) v T P H 1 pGq and m 0 P L 2 pGq.

(Stronger assumption for Hamiltonian and coupling term) For i " 1, N , B p H i px, pq is Lipschitz continuous on Γ i ˆR and the coupling V maps the topological dual of W to H 1 b pΓq; more precisely, V defines a Lipschitz map from W 1 to H 1 b pΓq. Note that such an assumption is not satisfied by local operators.

The above set of assumptions, except the last one, is referred to as [A], will be the running assumptions hereafter. We will also say that the coupling V is strictly increasing if, for any

m 1 , m 2 P M X L 2 pGq, ż G pm 1 ´m2 qpV rm 1 s ´V rm 2 sqdx ě 0 (1.3.42)
and equality implies m 1 " m 2 .

Definition 1.3.3. (solutions of the MFG system) A weak solution of the MFG system (1.3.13) is a pair pv, mq such that V rmp¨, tqspxqw pxq dx, for all w P W , a.e. t P p0, T q, vpx, T q " v T pxq for a.e. x P G, and m satisfies

v P L 2 `0, T ; H 2 pGq ˘X Cpr0, T s; V q, B t v P L 2 `0, T ; L 2 pGq ˘, m P L 2 p0, T ; W q X Cpp0, T s; L 2 pGq X Mq, B t m P L 2 `0, T ; V 1 ˘,
$ ' ' ' & ' ' ' % N ÿ i"1 ż Γ i
rB t m px, tq v pxq dx `µi Bm px, tq Bv pxq `Bp H px, Bv px, tqq m px, tq Bv pxqs dx " 0, for all v P V , a.e. t P p0, T q, mpx, 0q " m 0 pxq for a.e. x P G, where V and W are introduced in Section 1.3.2.

We ready to introduce the main theorem of Chapter 4.

Theorem 1.3.4 (Theorem 4.1.11). Under assumptions [A],

(i) (Existence) There exists a weak solution pv, mq of (1.3.13).

(ii) (Uniqueness) If V is strictly increasing, then the solution is unique.

(iii) (Regularity) If V satisfies furthermore the stronger assumptions for coupling term then v P C 2,1 pG ˆr0, T sq.

Moreover, if the Hamiltonian H i satisfies the stronger assumptions for Hamiltonians, and if m

0 P W , then m P Cpr0, T s; W q X W 1,2 p0, T ; L 2 pGqq X L 2 p0, T ; H 2 b pGqq.
Let us give the idea of the proof for Theorem 1.3.4. We first study a modified heat equation and FP equation on G, whose existence results are obtained by using the Galerkin's method to construct solutions of certain finite-dimensional approximations to these equations. The uniqueness is a direct consequence of the energy estimate. Next, we shall establish the existence result for weak solutions of MFG system (1.3.13) by a fixed point argument. Uniqueness will also be proved for strictly increasing couplings.

Step 1 : [Section 4.2] Consider a modified heat equation on G with general Kirchhoff condition

$ ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' % ´Bt v ´µi B 2 v " h, in pΓ i z tOuq ˆp0, T q, i " 1, N , v i pO, tq " v j pO, tq, t P p0, T q, i, j " 1, N , N ÿ i"1 γ i µ i B i v pO, tq " 0, t P p0, T q, B i v pν i , tq " 0, t P p0, T q, i " 1, N , vpx, T q " v T pxq, x P G, (1.3.43) for h P L 2 p0, T ; W 1 q and v T P L 2 pGq. A weak solution of (1.3.43) is a function v P L 2 p0, T ; V q X Cpr0, ts; L 2 pGqq such that B t v P L 2 p0, T ; W 1 q and $ ' ' & ' ' % ´xB t v ptq , wy W 1 ,W `N ÿ i"1 ż Γ i µ i BvBwdx " xh, wy W 1 ,
W for all w P W and a.e. t P p0, T q, v px, T q " v T pxq.

(1.3.44)

Chapter 1. Introduction

We use the Galerkin's method, namely we construct solutions of some finite-dimensional approximations to (1.3.44), to prove the existence of (1.3.44). Moreover, if v T P V , we get more regularity for v: v P L 2 p0, T ; H 2 pGqq X Cpr0, T s; V q and B t v P L 2 p0, T ; L 2 pGqq. The uniqueness and stability is a direct consequence of the energy estimate.

Step 2 : [Section 4.3] In this step, we study a boundary value problem including a Fokker-Planck equation

$ ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' % B t m ´µi B 2 m ´B pbmq " 0, in pΓ i z tOuq ˆp0, T q , i " 1, N , m i pO, tq γ i " m j pO, tq γ j , t P p0, T q , i, j " 1, N , N ÿ i"1 µ i B i m pO, tq ´bi pO, tq m i pO, tq " 0, t P p0, T q , ÿ iPA i µ i B i m pν i , tq `bi pν i , tq m i pν i , tq " 0, t P p0, T q , i " 1, N , m px, 0q " m 0 pxq , x P G, (1.3.45)
where b P P C pG ˆr0, T sq and m 0 P L 2 pGq. A weak solution of (1.3.45) is a function

m P L 2 p0, T ; W q X Cpr0, T s; L 2 pGqq such that B t m P L 2 p0, T ; V 1 q and $ ' ' & ' ' % xB t m, vy V 1 ,V `N ÿ i"1 ż G µ i BmBvdx `żG
bmBvdx " 0 for all v P V and a.e. t P p0, T q, m p¨, 0q " m 0 .

(1.3.46) Similarly to Step 1, the existence is deduced from the Galerkin's method and the uniqueness and stability is a consequence of the energy estimate. If b i is Lipschitz continuous on Γ i , we get more regularity for m: m P L 2 p0, T ; H 2 b pGqq X Cpr0, T s; W q and B t m P L 2 p0, T ; L 2 pGqq.

T : L 2 p0, T ; V q ÝÑ L 2 p0, T ; V q, T puq " v,
with h " f ´Hpx, Buq. The existence is obtained by using the fixed point theory,. The uniqueness is a consequence of the comparison principle. Next, to deal with sublinear Hamiltonian, we use the truncation technique combining with the previous process (with bounded Hamiltonian).

Step 4 : [Section 4.4.2] The main difficulty in this work is how to obtain the regularity for the boundary value problem including a HJ equation on G. Let us explain formally the idea. We derive the HJB equation for v and prove some regularity estimate for the solution of the derived problem. More precisely, u :" Bv satisfies the following PDE

´Bt u ´µi B 2 u `B pHpx, uqq " Bf,
with terminal condition upx, T q " Bv T pxq. From the Kirchhoff conditions in the uncoupled HJ equations, we obtain a transition condition of Dirichlet type for u,

N ÿ i"1
µ i γ i u| Γ i pO, tq " 0, t P p0, T q.

Mean field games on networks

Now, by extending continuously the HJB uncoupled until the vertex O in the branches Γ i and Γ j , i, j " 1, N and using the continuity condition of v, one gets

´µi B 2 v| Γ i `H| Γ i pν i , Bv| Γ i pν i , tqq ´f | Γ i pν i , tq " ´µj B 2 v| Γ j `H| Γ j pν i , Bv| Γ j pν i , tqq ´f | Γ j pν i , tq.
This gives a second transition condition for u

µ i Bu| Γ i pν i , tq´H| Γ i pν i , u| Γ i pν i , tqq`f | Γ i pν i , tq " µ j Bu| Γ j pν i , tq´H| Γ j pν i , u| Γ j pν i , tqq`f | Γ j pν i , tq.
Hence, we shall study the following nonlinear boundary value problem for u " Bv,

$ ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' % ´Bt u ´µi B 2 u `Bp H px, uq Bu " G px, tq , pΓ i z tOuq ˆp0, T q , i " 1, N , ÿ iPA i γ i µ i u| Γ i pO, tq " 0, t P p0, T q , µ i Bu| Γ i pO, tq ´H| Γ i pO, u| Γ i pO, tqq `f | Γ i pO, tq " µ j Bu| Γ j pO, tq ´H| Γ j pO, u| Γ j pO, tqq `f | Γ j pO, tq , t P p0, T q , i, j " 1, N , u px, T q " u T pxq , x P G, (1.3
.47) where G P L 2 `0, T ; L 2 pGq ˘and u T P H 1 b pGq. After obtaining the existence, uniqueness and regularity for (1.3.47) (by using also the Galerkin's method and energy estimate), the regularity of the HJB equations will follow by proving u " Bv in case G " Bf ´Bx Hpx, Bvq and u T " Bv T .

Step 5 : [Section 4.5] In the last step, using Step 2 and Step 3 and the fixed point theory, we can obtain the existence for weak solution pv, mq of MFG system (1.3.13). The uniqueness is a consequence of the increasing coupling term. Finally, we get the regularity of (1.3.13) from Step 4 and the stronger assumptions for Hamiltonian and coupling term.

We end by comparing the results in the stationary and the non-stationary case. Assumptions in the non-stationary case are more restrictive. We only focus on the more basic assumptions, globally Lipschitz Hamiltonian, instead of subquadratic ones in the stationary case, and rather strong assumptions on the coupling cost. This will allow us to concentrate on the difficulties induced by the Kirchhoff conditions. Therefore, this work should be seen as a first and necessary step to deal with more difficult situations, for example with quadratic or subquadratic Hamiltonians. We believe that treating such cases will possible by combining the results contained in the present work with methods that can be found in [START_REF] Ladyzenskaja | Linear and quasilinear equations of parabolic type[END_REF][START_REF] Lieberman | Second order parabolic differential equations[END_REF].

Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

Abstract:

We consider an optimal control on networks in the spirit of the works of Achdou, Camilli, Cutrì & Tchou [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF] and Imbert, Monneau & Zidani [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF]. The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou, Oudet & Tchou [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] and one based on partial differential equations techniques inspired by a recent work of Lions & Souganidis [START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF].

Introduction

A network (or a graph) is a set of items, referred to as vertices or nodes, which are connected by edges (see Figure 2.1 for example). Recently, several research projects have been devoted to dynamical systems and differential equations on networks, in general or more particularly in connection with problems of data transmission or traffic management (see for example Garavello & Piccoli [START_REF] Garavello | Traffic flow on networks[END_REF] and Engel, Fijavvz, Nagelm & Sikolya [START_REF] Engel | Vertex control of flows in networks[END_REF]).

An optimal control problem is an optimization problem where an agent tries to minimize a cost which depends on the solution of a controlled ordinary differential equation (ODE). The ODE is controlled in the sense that it depends on a function called the control. The goal is to find the best control in order to minimize the given cost. In many situations, the optimal value of the problem as a function of the initial state (and possibly of the initial time when the horizon of the problem is finite) is a viscosity solution of a Hamilton-Jacobi-Bellman partial differential equation (HJB equation). Under appropriate conditions, the HJB equation has a unique viscosity solution characterizing by this way the value function. Moreover, the optimal control may be recovered from the solution of the HJB equation, at least if the latter is smooth enough.

The first articles about optimal control problems in which the set of admissible states is a network (therefore the state variable is a continuous one) appeared in 2012: in [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF], Achdou, Camilli, Cutrì & Tchou derived the HJB equation associated to an infinite horizon optimal control on a network and proposed a suitable notion of viscosity solution. Obviously, the main difficulties arise at the vertices where the network does not have a regular differential structure. As a result, the new admissible test-functions whose restriction to each edge is C 1 are applied. Independently and at the same time, Imbert, Monneau & Zidani [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] proposed an equivalent notion of viscosity solution for studying a Hamilton-Jacobi approach to junction problems and traffic flows. Both [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF] and [START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF] contain first results on comparison principles which were improved Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs later. It is also worth mentioning the work by Schieborn & Camilli [START_REF] Schieborn | Viscosity solutions of Eikonal equations on topological networks[END_REF], in which the authors focus on eikonal equations on networks and on a less general notion of viscosity solution. In the particular case of eikonal equations, Camilli & Marchi established in [START_REF] Camilli | A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks[END_REF] the equivalence between the definitions given in [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF][START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF][START_REF] Schieborn | Viscosity solutions of Eikonal equations on topological networks[END_REF].

Since 2012, several proofs of comparison principles for HJB equations on networks, giving uniqueness of the solution, have been proposed. [START_REF] Achdou | Mean field games: additional notes[END_REF]. In [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF], Achdou, Oudet & Tchou give a proof of a comparison principle for a stationary HJB equation arising from an optimal control with infinite horizon, (therefore the Hamiltonian is convex) by mixing arguments from the theory of optimal control and PDE techniques. Such a proof was much inspired by works of Barles, Briani & Chasseigne [START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF], on regional optimal control problems in R d , (with discontinuous dynamics and costs).

2. A different and more general proof, using only arguments from the theory of PDEs was obtained by Imbert & Monneau in [START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF]. The proof works for quasi-convex Hamiltonians, and for stationary and time-dependent HJB equations. It relies on the construction of suitable vertex test functions.

3. A very simple and elegant proof, working for non convex Hamiltonians, has been very recently given by Lions & Souganidis [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF][START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF].

The goal of this paper is to consider an optimal control problem on a network in which there are entry (or exit) costs at each edge of the network and to study the related HJB equations. The effect of the entry/exit costs is to make the value function of the problem discontinuous. Discontinuous solutions of Hamilton-Jacobi equation have been studied by various authors, see for example Barles [START_REF] Barles | Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit[END_REF], Frankowska & Mazzola [START_REF] Frankowska | Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints[END_REF], and in particular Graber, Hermosilla & Zidani [START_REF] Graber | Discontinuous solutions of Hamilton-Jacobi equations on networks[END_REF] for different HJB equations on networks with discontinuous solutions.

To simplify the problem, we first study the case of junction, i.e., a network of the form G " Y N i"1 Γ i with N edges Γ i (Γ i is the closed half line R `ei ) and only one vertex O, where tOu " X N i"1 Γ i . This can be easily generalized into networks with an arbitrary number of vertices. In the case of the junction described above, our assumptions about the dynamics and the running costs are similar to those made in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF], except that additional costs c i for entering the edge Γ i at O or d i for exiting Γ i at O are added in the cost functional. Accordingly, the value function is continuous on G, but is in general discontinuous at the vertex O. Hence, instead of considering the value function v, we split it into the collection pv i q 1ďiďN , where v i is continuous function defined on the edge Γ i . More precisely,

v i pxq " $ & % v pxq if x P Γ i z tOu , lim δÑ0 `v pδe i q if x " O.
Our approach is therefore reminiscent of optimal switching problems (impulsional control): in the present case the switches can only occur at the vertex O. Note that our assumptions will ensure that v| Γ i ztOu is Lipschitz continuous near O and that lim δÑ0 `v pδe i q does exist. In the case of entry costs for example, our first main result will be to find the relation between v pOq, v i pOq and v j pOq `cj for i, j " 1, N .

This will show that the functions pv i q 1ďiďN are (suitably defined) viscosity solutions of the following system

λu i pxq `Hi ˆx, du i dx i pxq ˙" 0 if x P Γ i z tOu, λu i pOq `max " ´λ min j‰i tu j pOq `cj u , H ì ˆO, du i dx i pOq ˙, H T 2.2.
Optimal control problem on junction with entry/exit costs

Γ 1 Γ 2 Γ 5 Γ 3 Γ 4 O Figure 2.1: The junction G with 5 edges on R 2
Here H i is the Hamiltonian corresponding to edge Γ i . At vertex O, the definition of the Hamiltonian has to be particular, in order to consider all the possibilities when x is close to O. More specifically, if x is close to O and belongs to Γ i then:

• The term min j‰i tu j pOq `cj u accounts for situations in which the trajectory enters Γ i 0 where u i 0 pOq `ci 0 " min j‰i tu j pOq `cj u.

• The term H ì ˆO, du i dx i pOq ˙accounts for situations in which the trajectory does not leave Γ i .

• The term H T O accounts for situations in which the trajectory stays at O.

The most important part of the paper will be devoted to two different proofs of a comparison principle leading to the well-posedness of (2.1.1): the first one uses arguments from optimal control theory coming from [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF] and [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF]; the second one is inspired by Lions & Souganidis [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF] and uses arguments from the theory of PDEs.

The paper is organized as follows: Section 2.2 deals with the optimal control problems with entry and exit costs: we give a simple example in which the value function is discontinuous at the vertex O, and also prove results on the structure of the value function near O. In Section 2.3, the new system of (2.1.1) is defined and a suitable notion of viscosity solutions is proposed. In Section 2.4, we prove our value functions are viscosity solutions of the above mentioned system. In Section 2.5, some properties of viscosity sub and supersolution are given and used to obtain the comparison principle. Finally, optimal control problems with entry costs which may be zero and related HJB equations are considered in Section 2.6.

Optimal control problem on junction with entry/exit costs 2.2.1 The geometry

We consider the model case of the junction in R d with N semi-infinite straight edges, N ą 1. The edges are denoted by pΓ i q i"1,N where Γ i is the closed half-line R `ei . The vectors e i are two by two distinct unit vectors in R d . The half-lines Γ i are glued at the vertex O to form the junction G Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

G " N ď i"1 Γ i .
The geodetic distance d px, yq between two points x, y of G is

d px, yq " $ & % |x ´y| if x,

The optimal control problem

We consider infinite horizon optimal control problems which have different dynamic and running costs for each and every edge. For i " 1, N ,

• the set of control on Γ i is denoted by A i

• the system is driven by a dynamics f i

• there is a running cost i .

Our main assumptions, referred to as rHs hereafter, are as follows:

rH0s (Control sets) Let A be a metric space (one can take A " R d ). For i " 1, N , A i is a nonempty compact subset of A and the sets A i are disjoint.

rH1s (Dynamics) For i " 1, N , the function f i : Γ i ˆAi Ñ R
F pxq " $ & % F i pxq if x P Γ i z pOq , Ť N i"1 F i pOq if x " O.
For x P G, the set of admissible trajectories starting from x is

Y x " # y x P Lip `R`; G ˘ˇˇˇˇ9 y x ptq P F py x ptqq for a.e. t ą 0 y x p0q " x.
+ .

According to [8, Theorem 1.2], a solution y x can be associated with several control laws. We introduce the set of admissible controlled trajectories starting from x

T x " " py x , αq P L 8 loc `R`; M ˘: y x P Lip `R`; G ˘and y x ptq " x `ż t 0 f py x psq , α psqq ds * .
Notice that, if py x , αq P T x then y x P Y x . Hereafter, we will denote y x by y x,α if py x , αq P T x . For any y x,α , we can define the closed set T O " tt P R `: y x,α ptq " Ou and the open set T i in R `" r0, `8q by T i " tt P R `: y x,α ptq P Γ i z tOuu. The set T i is a countable union of disjoint open intervals

T i " ď kPK i ĂN T ik " $ & % r0, η i0 q Y Ť kPK i ĂN ‹ pt ik , η ik q if x P Γ i z tOu , Ť kPK i ĂN ‹ pt ik , η ik q if x R Γ i z tOu ,
where K i " t1, . . . , nu if the trajectory y x,α enters Γ i n times and K i " N if the trajectory y x,α enters Γ i infinite times.

Remark 2.2.3. From the above definition, one can see that t ik is an entry time in Γ i z tOu and η ik is an exit time from Γ i z tOu . Hence y x,α pt ik q " y x,α pη ik q " O.

Let C " tc 1 , c 2 , . . . , c N u be a set of entry costs and D " td 1 , d 2 , . . . , d N u be a set of exit costs. We underline that, except in Section 2.6, entry and exist costs are positive.

In the sequel, we define two different cost functionals (the first one corresponds to the case when there is a cost for entering the edges and the second one corresponds to the case when there is a cost for exiting the edges):

Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs Definition 2.2.4 (The cost functionals and value functions with entry/exit costs). The costs associated to trajectory py x,α , αq P T x are defined by J px; py

e ´λt ik c i " ´M λ `ci `8 ÿ k"1 e ´λt ik " `8,
in contradiction with J px, αq ă `8. This means that the state cannot switch edges infinitely many times in finite time, otherwise the cost functional is obviously infinite.

The following example shows that the value function with entry costs is possibly discontinuous (The same holds for the value function with exit costs).

Example 2.2.6. Consider the network G " Γ 1 YΓ 2 where Γ 1 " R `e1 " p´8, 0s and Γ 2 " R `e2 " r0, `8q. The control sets are A i " r´1, 1s ˆtiu with i P t1, 2u. Set pf px, aq , px, aqq " $ & % pf i px, pa i , iqq e i , i px, pa i , iqqq if x P Γ i z tOu and a " pa i , iq P A i , pf i pO, pa i , iqq e i , i pO, pa i , iqqq if x " O and a " pa i , iq P A i , 2.2. Optimal control problem on junction with entry/exit costs where f i px, pa i , iqq " a i and 1 " 1, 2 px, pa 2 , 2qq " 1 ´a2 . For x P Γ 2 z tOu, then v pxq " v 2 pxq " 0 with optimal strategy consists in choosing α ptq " p1, 2q. For x P Γ 1 , we can check that v pxq " min

# 1 λ , 1 ´e´λ|x| λ `c2 e ´λ|x| + .
More precisely, for all x P Γ 1 , we have

v pxq " $ ' ' ' & ' ' ' % 1 λ if c 2 ě 1 λ
, with the optimal control α ptq " p´1, 1q,

1 ´e´λ|x| λ `c2 e ´λ|x| if c 2 ă 1 λ
, with the optimal control α ptq "

$ & % p1, 1q if t ď |x| , p1, 2q if t ě |x| .
Summarizing, we have the two following cases

1. If c 2 ě 1{λ, then v pxq " $ & % 0 if x P Γ 2 z tOu , 1 λ if x P Γ 1 .
The graph of the value function with entry costs c 2 ě 1{λ " 1 is plotted in the left of Figure 2.2.

2. If c 2 ă 1{λ, then v pxq " $ & % 0 if x P Γ 2 z tOu , 1 ´e´λ|x| λ `c2 e ´λ|x| if x P Γ 1 .
The graph of the value function with entry costs c 2 " 1{2 ă 1 " 1{λ is plotted in the right of Figure 2.2. Lemma 2.2.7. Under assumptions rH1s and rH4s, there exist two positive numbers r 0 and C such that for all x 1 , x 2 P B pO, r 0 q X G, there exists

Γ 1 Γ 2 vpxq Γ 2 Γ 1 vpxq c 2 1 λ 1 λ O O
`yx 1 ,αx 1 ,x 2 , α x 1 ,x 2 ˘P T x 1 and τ x 1 ,x 2 ď Cd px 1 , x 2 q such that y x 1 pτ x 1 ,x 2 q " x 2 .
Proof of Lemma 2.2.7. This proof is classical. It is sufficient to consider the case when x 1 and x 2 belong to same edge Γ i , since in the other cases, we will use O as a connecting point between x 1

Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs and x 2 . According to Assumption rH4s, there exists a P A i such that f i pO, aq " δ. Additionally, by the Lipschitz continuity of f i ,

|f i pO, aq ´fi px, aq| ď L |x| ,
hence, if we choose r 0 :" δ{2L ą 0, then f i px, aq ě δ{2 for all x P B pO, r 0 q X Γ i . Let x 1 , x 2 be in B pO, r 0 q X Γ i with |x 1 | ă |x 2 |: there exist a control law α and τ x 1 ,x 2 ą 0 such that α ptq " a if 0 ď t ď τ x 1 ,x 2 and y x 1 ,α pτ x 1 ,x 2 q " x 2 . Moreover, since the velocity f i py x 1 ,α ptq , α ptqq is always greater than δ{2 when t ď τ x 1 ,x 2 , then τ

x 1 ,x 2 ď 2{δd px 1 , x 2 q . If |x 1 | ą |x 2 |
, the proof is achieved by replacing a P A i by a P A i such that f i pO, aq " ´δ and applying the same argument as above.

Some properties of value function at the vertex

H T O " max i"1,N max a i PA O i t´ j pO, a j qu " ´min i"1,N min a i PA O i t j pO, a j qu, (2.2.1)
where A O i " ta i P A i : f i pO, a i q " 0u . The relation between the values vpOq, v i pOq and H T O will be given in the next theorem. Hereafter, the proofs of the results will be supplied only for the value function with entry costs v, the proofs concerning the value function with exit costs v are totally similar. Since α is chosen arbitrarily and i is bounded by M , we get

v i pxq ď M τ x,O `e´λτ x,O v pOq .
Let x tend to O then τ x,O tend to 0 from Lemma 2.2.7. Therefore, v i pOq ď v pOq. Since the above inequality holds for i " 1, N , we obtain that max i"1,N tv i pOqu ď v pOq .

Prove that v pOq ď min i"1,N tv i pOq `ci u. For i " 1, N ; we claim that v pOq ď v i pOq `ci . Consider x P Γ i z tOu with |x| small enough and any control law ᾱx such that py x, ᾱx , ᾱx q P T x . From Lemma 2.2.7, there exists a control law α O,x connecting O and x and we consider

α psq " $ & % α O,x psq if s ď τ O,x , ᾱx ps ´τO,x q if s ą τ O,x .
It means that the trajectory goes from O to x using the control law α O,x then proceeds with the control law ᾱx . Therefore

v pOq ď J pO, αq " c i `ż τ O,x 0 i py O,α psq , αpsqq e ´λs ds `e´λτ O,x J px, ᾱx q .
Since α x is chosen arbitrarily and i is bounded by M , we get

v pOq ď c i `M τ O,x `e´λτ O,x v i pxq
Let x tend to O then τ O,x tends to 0 from Lemma 2.2.7, then v pOq ď c i `vi pOq . Since the above inequality holds for i " 1, N , we obtain that

v pOq ď min i"1,N tv i pOq `ci u .
Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs Lemma 2.2.12. The value functions v and v satisfy

v pOq , v pOq ď ´HT O λ (2.2.2)
where

H T O is defined in (2.2.1).
Proof of Lemma 2.2.12. From (2.2.1), there exists j P t1, . . . , N u and a j P A O j such that

H T O " ´min i"1,N min a i PA O i t i pO, a i qu " ´ j pO, a j q
Let the control law α be defined by α psq " a j for all s, then v pOq ď J pO, αq " ż `8 0 j pO, a j q e ´λs ds "

j pO, a j q λ " ´HT O λ .
We are ready to prove Theorem 2.2.9.

Proof of Theorem 2.2.9. According to Lemma 2.2.11 and Lemma 2.2.12, v pOq ď min

" min i"1,N tv i pOq `ci u , ´HT O λ * . Assuming that v pOq ă min i"1,N tv i pOq `ci u , ( 2.2.3)
it is sufficient to prove that v pOq " ´HT O λ . By (2.2.3), there exists a sequence tε n u nPN such that ε n Ñ 0 and v pOq `εn ă min i"1,N tv i pOq `ci u for all n P N.

On the other hand, there exists an ε n -optimal control α n , v pOq `εn ą J pO, α n q. Let us define the first time that the trajectory y O,αn leaves O

t n :" inf i"1,N T n i ,
where T n i is the set of times t for which y O,αn ptq belongs to Γ i z tOu. Notice that t n is possibly `8, in which case y O,αn psq " O for all s P r0, `8q. Extracting a subsequence if necessary, we may assume that t n tends to t P r0, `8s when ε n tends to 0.

If there exists a subsequence of tt n u nPN (which is still noted tt n u nPN ) such that t n " `8 for all n P N, then for a.e. s P r0, `8q Choose a subsequence tε n k u kPN of tε n u nPN such that for some i 0 P t1, . . . , N u, c ipn k q " c i 0 for all k. By letting k tend to 8, recall that lim kÑ8 t n k " t, we have three possible cases

$ & % f py O,αn psq , α n psqq " f pO, α n psqq " 0, py O,
1. If t " `8, then v pOq ě ´HT O λ
. By Lemma 2.2.12, we obtain v pOq " ´HT O λ .

2. If t " 0, then v pOq ě c i 0 `vi 0 pOq. By (2.2.3), we obtain a contradiction.

3. If t P p0, `8q, then v pOq ě 1 ´e´λt λ

`´H T O ˘`rc i 0 `vi 0 pOqs e ´λt . By (2.2.3), c i 0 `vi 0 pOq ą v pOq, so v pOq ą 1 ´e´λt λ `´H T O ˘`v pOq e ´λt .
This yields v pOq ą ´HT O {λ, and finally obtain a contradiction by Lemma 2.2.12.

Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

The Hamilton-Jacobi systems. Viscosity solutions 2.3.1 Test-functions

Definition 2.3.1. A function ϕ : Γ 1 ˆ. . . ˆΓN Ñ R N is an admissible test-function if there exists pϕ i q i"1,N , ϕ i P C 1 pΓ i q, such that ϕ px 1 , . . . , x N q " pϕ 1 px 1 q , . . . , ϕ N px N qq. The set of admissible test-function is denoted by R pGq. We now introduce the Hamilton-Jacobi system for the case with entry costs

Definition of viscosity solution

λu i pxq `Hi ˆx, du i dx i pxq ˙" 0 if x P Γ i z tOu, λu i pOq `max " ´λ min j‰i tu j pOq `cj u , H ì ˆO, du i dx i pOq ˙, H T O * " 0 if x " O, (2.3.1)
for all i " 1, N and the Hamilton-Jacobi system with exit costs

λû i pxq `Hi ˆx, dû i dx i pxq ˙" 0 if x P Γ i z tOu, λû i pOq `max " ´λ min j‰i tû j pOq `di u , H ì ˆO, dû i dx i pOq ˙, H T O ´λd i * " 0 if x " O,
(2.3.2) for all i " 1, N and their viscosity solutions.

Definition 2.3.3 (Viscosity solution with entry costs).

' A function u :" pu 1 , . . . , u N q where u i P U SC pΓ i ; Rq for all i " 1, N , is called a viscosity subsolution of (2.3.1) if for any pϕ 1 , . . . , ϕ N q P R pGq, any i " 1, N and any x i P Γ i such that u i ´ϕi has a local maximum point on Γ i at x i , then

λu i px i q `Hi ˆx, dϕ i dx i px i q ˙ď 0 if x i P Γ i z tOu, λu i pOq `max " ´λ min j‰i tu j pOq `cj u , H ì ˆO, dϕ i dx i pOq ˙, H T O * ď 0 if x i " O.
' A function u :" pu 1 , . . . , u N q where u i P LSC pΓ i ; Rq for all i " 1, N , is called a viscosity supersolution of (2.3.1) if for any pϕ 1 , . . . , ϕ N q P R pGq, any i " 1, N and any x i P Γ i such that u i ´ϕi has a local minimum point on Γ i at x i , then

λu i px i q `Hi ˆxi , dϕ i dx i px i q ˙ě 0 if x i P Γ i z tOu, λu i pOq `max " ´λ min j‰i tu j pOq `cj u , H ì ˆO, dϕ i dx i pOq ˙, H T O * ě 0 if x i " O.
2.4. Connections between the value functions and the Hamilton-Jacobi systems.

' A functions u :" pu 1 , . . . , u N q where u i P C pΓ i ; Rq for all i " 1, N , is called a viscosity solution of (2.3.1) if it is both a viscosity subsolution and a viscosity supersolution of (2.3.1).

Definition 2.3.4 (Viscosity solution with exit costs).

' A function û :" pû 1 , . . . , ûN q where ûi P U SC pΓ i ; Rq for all i " 1, N , is called a viscosity subsolution of (2.3.2) if for any pψ 1 , . . . , ψ N q P R pGq, any i " 1, N and any y i P Γ i such that ûi ´ψi has a local maximum point on Γ i at y i , then λû i py i q `Hi ˆyi , dψ i dx i py i q ˙ď 0 if y i P Γ i z tOu,

λû i pOq `max " ´λ min j‰i tû j pOqu ´λd i , H ì ˆO, dψ i dx i pOq ˙, H T O ´λd i * ď 0 if y i " O.
' A function û :" pû 1 , . . . , ûN q where ûi P LSC pΓ i ; Rq for all i " 1, N , is called a viscosity supersolution of (2.3.2) if for any pψ 1 , . . . , ψ N q P R pGq, any i " 1, N and any y i P Γ i such that u i ´ψi has a local minimum point on Γ i at y i , then

λû i py i q `Hi ˆyi , dψ i dx i py i q ˙ě 0 if y i P Γ i z tOu, λû i pOq `max " ´λ min j‰i tû j pOqu ´λd i , H ì ˆO, dψ i dx i pOq ˙, H T O ´λd i * ě 0 if y i " O.
' A functions û :" pû 1 , . . . , ûN q where ûi P C pΓ i ; Rq for all i " 1, N , is called a viscosity solution of (2.3.2) if it is both a viscosity subsolution and a viscosity supersolution of (2.3.2).

Remark 2.3.5. This notion of viscosity solution is consitent with the one of [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF]. It can be seen in Section 2.6 when all the switching costs are zero, our definition and the one of [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] coincide.

Connections between the value functions and the

Hamilton-Jacobi systems.

Let v be the value function of the optimal control problem with entry costs and v be a value function of the optimal control problem with exit costs. Recall that v i , vi : Γ i Ñ R are defined in Lemma 2.2.8 by

$ & % v i pxq " v pxq if x P Γ i z tOu , v i pOq " lim Γ i ztOuQxÑO v pxq ,
and

$ & % vi pxq " v pxq if x P Γ i z tOu , vi pOq " lim Γ i ztOuQxÑO v pxq .
We wish to prove that v :" pv 1 , v 2 , . . . , v N q and v :" pv 1 , . . . , vN q are respectively viscosity solutions of (2.3.1) and (2.3.2). In fact, since Gz tOu is a finite union of open intervals in which the classical theory can be applied, we obtain that v i and vi are viscosity solutions of λu pxq `Hi px, Du pxqq " 0 in Γ i z tOu .

Therefore, we can restrict ourselves to prove the following theorem. in the viscosity sense. Let a i P A i be such that f i pO, a i q ą 0. Setting α ptq " a i then py x,α , αq P T x for all x P Γ i . Moreover, for all x P Γ i z tOu, y x,α ptq P Γ i z tOu (the trajectory cannot approach O since the speed pushes it away from O for y x,α P Γ i X B pO, rq). Note that it is not sufficient to choose a i P A i such that f pO, a i q " 0 since it can lead to f px, a i q ă 0 for all x P Γ i z tOu.

Next, for τ ą 0 fixed and any x P Γ i , if we choose

α x ptq " $ & % α ptq " a i 0 ď t ď τ, α pt ´τ q t ě τ, (2.4.1)
where α is arbitrary, then y x.αx ptq P Γ i z tOu for all t P r0, τ s. It yields

v i pxq ď J px, α x q "
ż τ 0 i py x,α psq , a i q e ´λs ds `e´λτ J py x,α pτ q , αq .

Since this holds for any α (α x is arbitrary for t ą τ ), we deduce that By letting τ tend to 0, we obtain that ´fi pO, a i q dϕ dx i pOq ď i pO, a i q ´λv i pOq .

v i pxq ď ż τ 0 i py x,
Hence,

λv
then there exist τ ą 0, r ą 0 and ε 0 ą 0 such that for any x P pΓ i z tOuq X B pO, rq, any ε ă ε 0 and any ε´optimal control law α ε,x for x, y x,αε,x psq P Γ i z tOu , for all s P r0, τ s .

Remark 2.4.4. Roughly speaking, this lemma takes into account the case λv i `Hì ˆx, dv i dx i pOq ˙ď 0, i.e., the situation when the trajectory does not leave Γ i , see introduction.

Proof of Lemma 2.4.3. Suppose by contradiction that there exist sequences tε n u , tτ n u Ă R ànd tx n u Ă Γ i z tOu such that ε n OE 0, x n Ñ O, τ n OE 0 and a control law α n such that α n is ε n -optimal control law and y xn,αn pτ n q " O. This implies that v i px n q `εn ą J px n , α n q " ż τn 0 py xn,αn psq , α n psqq e ´λs ds `e´λτn J pO, α n p¨`τ n qq . (2.4.4) Since is bounded by M by rH1s, then v i px n q `εn ě ´τn M `e´λτn v pOq . By letting n tend to 8, we obtain v i pOq ě v pOq . However, v pOq " min min j tv j pOq `cj u , ´HT O {λ ( by Theorem 2.2.9. Therefore, v pOq " v i pOq `ci ą v i pOq, which is a contradiction with (2.4.5).

Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs Lemma 2.4.5. The function v i is a viscosity supersolution of (2.3.1) at O. Proof of Lemma 2.4.5. We adapt the proof of Oudet [START_REF] Oudet | Hamilton-Jacobi equations for optimal control on multidimensional junctions[END_REF] and start by assuming that

v i pOq ă min " min j‰i tv j pOq `cj u , ´HT O λ * .
We need to prove that λv i pOq `Hì ˆO, dv i dx i pOq ˙ě 0 in the viscosity sense. Let ϕ P C 1 pΓ i q be such that 0 " v i pOq ´ϕ pOq ď v i pxq ´ϕ pxq for all x P Γ i , (2.4.6)

and tx ε u Ă Γ i z tOu be any sequence such that x ε tends to O when ε tends to 0. From the dynamic programming principle and Lemma 2.4.3, there exists τ such that for any ε ą 0, there exists py ε , α ε q :" py xε,αε , α ε q P T xε such that y ε pτ q P Γ i z tOu for any τ P r0, τ s and

v i px ε q `ε ě ż τ 0 i py ε psq , α ε psqq e ´λs ds `e´λτ v i py ε pτ qq .
Then, according to (2.4.6)

v i px ε q ´vi pOq `ε ě ż τ 0 i py ε psq , α ε psqq
v i px ε q ´vi pOq " o ε p1q , v i pOq ´1 ´e´λτ ¯" o pτ q `τ λv i pOq ,
where the notation o ε p1q is used for a quantity which is independent on τ and tends to 0 as ε tends to 0. For a positive integer k, the notation opτ k q is used for a quantity that is independent on ε and such that opτ k q{τ k Ñ 0 as τ Ñ 0. Finally, Opτ k q stands for a quantity independent on ε such that Opτ k q{τ k remains bounded as τ Ñ 0. From (2.4.7), we obtain that

τ λv i pOq ě ż τ 0 i py ε psq , α ε psqq ds `ϕ py ε pτ qq ´ϕ pOq `τ o ε p1q `o pτ q `oε p1q . ( 2

Comparison Principle and Uniqueness

Inspired by [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF][START_REF] Barles | A Bellman approach for regional optimal control problems in R N[END_REF], we begin by proving some properties of sub and super viscosity solutions of (2.3.1). The following three lemmas are reminiscent of Lemma 3.4, Theorem 3.1 and Lemma 3.5 in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF].

Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs Lemma 2.5.1. Let w " pw 1 , . . . , w N q be a viscosity supersolution of (2.3.1). Let x P Γ i z tOu and assume that

w i pOq ă min " min j‰i tw j pOq `cj u , ´HT O λ * . (2.5.1)
Then for all t ą 0,

w i pxq ě inf α i p¨q,θ i ˆż t^θ i 0 i `yi
x psq , α i psq ˘e´λs ds `wi `yi x pt ^θi q ˘e´λpt^θ i q ˙, where α i P L 8 p0, 8; A i q, y i x is the solution of y i x ptq " x `"ş t 0 f i

`yi

x psq , α i psq ˘ds ı e i and θ i satisfies y i x pθ i q " 0 and θ i lies in rτ i , τ i s, where τ i is the exit time of y i x from Γ i z tOu and τ i is the exit time of y i

x from Γ i .

Proof of Lemma 2.5. (2.5.2)

Hence, we can apply the result in [8, Lemma 3.4]. We refer to [START_REF] Barles | A Bellman approach for two-domains optimal control problems in R N[END_REF] for a detailed proof. The main point of that proof uses the results of Blanc [START_REF] Blanc | Deterministic exit time control problems with discontinuous exit costs[END_REF][START_REF] Blanc | Comparison principle for the Cauchy problem for Hamilton-Jacobi equations with discontinuous data[END_REF] on minimal supersolutions of exit time control problems.

Lemma 2.5.2 (Super-optimality).

Under assumption rHs, let w " pw 1 , . . . , w N q be a viscosity supersolution of (2.3.1) that satisfies (2.5.1); then there exists a sequence tη k u kPN of strictly positive real numbers such that lim kÑ8 η k " η ą 0 and a sequence x k P Γ i z tOu such that lim kÑ8 x k " O, lim kÑ8 w i px k q " w i pOq and for each k, there exists a control law α k i such that the corresponding trajectory y x k psq P Γ i for all s P r0, η k s and

w i px k q ě ż η k 0 i ´yx k psq , α k i psq ¯e´λs ds `wi py x k pη k qq e ´λη k .
Proof of Lemma 2.5.2. According to (2.5.1) ŵi pOq ă ´HT O {λ. Hence, this proof is complete by applying the proof of in [8, Theorem 3.1]. Lemma 2.5.3. Under assumption rHs, let u " pu 1 , . . . , u N q be a viscosity subsolution of (2.3.1). Then u i is Lipschitz continuous in B pO, rq X Γ i . Therefore, there exists a test function ϕ i P C 1 pΓ i q which touches u i from above at O. Proof of Lemma 2.5.3. Since u is a viscosity subsolution of (2.3.1), u i is a viscosity subsolution of (2.5.2). Recal that H i px, ¨q is coercive for any x P Γ i X B pO, rq, we can apply the proof in [8, Lemma 3.2], which is based on arguments due to Ishii and contained in [START_REF] Ishii | A short introduction to viscosity solutions and the large time behavior of solutions of Hamilton-Jacobi equations[END_REF].

Lemma 2.5.4 (Sub-optimality).

Under assumption rHs, let u " pu 1 , . . . , u N q be a viscosity subsolution of (2.3.1). Consider i " 1, N , x P Γ i z tOu and α i P L 8 p0, 8; A i q. Let T ą 0 be such that y x ptq " x `"ş t 0 f i py x psq , α i psqq ds ı e i belongs to Γ i for any t P r0, T s, then We give two proofs of Theorem 2.5.6. The first one is inspired by [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] and uses the previously stated lemmas. The second one uses the elegant arguments proposed in [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF].

u i pxq ď ż T 0 i py x psq
Proof of Theorem 2.5.6 inspired by [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] . We focus on u and w, the arguments used for the comparison of û and ŵ are totally similar. Suppose by contradiction that there exists x P Γ i such that u i pxq ´wi pxq ą 0. By classical comparison arguments for the boundary value problem, see [START_REF] Barles | An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications[END_REF], sup BΓ i tu i ´vi u `ě sup Γ i tu i ´vi u `, so we have

u i pOq ´wi pOq " max xPΓ i tu i pxq ´wi pxqu ą 0.

By definition of viscosity subsolution

λu i pOq `HT O ď 0. (2.5.3)
This implies λw i pOq `HT O ă 0. We now consider the two following cases.

Case 1: If w i pOq ă min j‰i tw j pOq `cj u, from Lemma 2.5.2 (using the same notations),

w i px k q ě ż η k 0 i ´yx k psq , α k i psq ¯e´λs ds `wi py x k pη k qq e ´λη k .
Moreover, according to Lemma 2.5.4, we also have

u i px k q ď ż η k 0 i ´yx k psq , α k i psq ¯e´λs ds `ui py x k pη k qq e ´λη k .
This yields u i px k q ´wi px k q ď ru i py x k pη k qq ´wi py x k pη k qqs e ´λη k ď ru i pOq ´wi pOqs e ´λη k .

By letting k tend to 8, one gets u i pOq ´wi pOq ď ru i pOq ´wi pOqs e ´λη .

This implies that u i pOq ´wi pOq ď 0 and leads to a contradiction.

Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs Case 2: If w i pOq ě min j‰i tw j pOq `cj u, then there exists j 0 ‰ i such that w j 0 pOq `cj 0 " min j"1,N tw j pOq `cj u " min j‰i tw j pOq `cj u ď w i pOq , because c i ą 0. Since c j 0 is positive The comparison principle can also be obtained alternatively, using the arguments which were very recently proposed by Lions & Souganidis in [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF]. This new proof is self-combined and the arguments do not rely at all on optimal control theory, but are deeply connected to the ideas used by Soner [START_REF] Soner | Optimal control with state-space constraint[END_REF][START_REF] Soner | Optimal control with state-space constraint[END_REF] and Capuzzo-Dolcetta & Lions [START_REF] Capuzzo-Dolcetta | Hamilton-Jacobi equations with state constraints[END_REF] for proving comparison principles for state-constrained Hamilton-Jacobi equations.

w j 0 pOq
Proof of Theorem 2.5.6 inspired by [START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF]. We start as in first proof. We argue by contradiction without loss of generality, assuming that there exists i such that u i pOq ´wi pOq " max Γ i tu i pxq ´wi pxqu ą 0.

Therefore w i pOq ă ´HT

O {λ. We now consider the two following cases. Case 1: If w i pOq ă min j‰i tw j pOq `cj u, then w i is a viscosity supersolution of (2.5.2). Recall that by Lemma 2.5.3, there exists a positive number L such that for i " 1, N , u i is Lipschitz continuous with Lipschitz constant L in Γ i X Bp0, rq. We consider the function

Ψ i,ε : Γ i ˆΓi ÝÑ R px, yq ÝÑ u i pxq ´wi pyq ´1 2ε r´|x| `|y| `δ pεqs 2 ´γ p|x| `|y|q ,
where δ pεq " pL `1q ε and γ P p0, 1{2q. It is clear that Ψ i,ε attains its maximum M ε,γ at px ε,γ , y ε,γ q P Γ i ˆΓi . By classical techniques, we check that x ε,γ , y ε,γ Ñ O and that px ε,γ ´yε,γ q 2 {ε Ñ 0 as ε Ñ 0. Indeed, one has Since u i pOq ´vi pOq ą 0, the term in (2.5.8) is positive when ε is small enough. We also deduce from the above inequality and from the boundedness of u i and w i that, maybe after the extraction of a subsequence, x ε,γ , y ε,γ Ñ x γ as ε Ñ 0, for some x γ P Γ i . From (2.5.7),

u i px ε,γ q ´wi py ε,γ q ´r´|x ε,γ | `|y ε,γ | `δ
u i px ε,γ q´w i py ε,γ q´p |x ε,γ | ´|y ε,γ |q 2 2ε ´p´|x ε,γ | `|y ε,γ |q δ pεq ε ě max Γ i tu i pxq ´wi pxq ´2γ |x|u .
Taking the lim sup on both sides of this inequality when ε Ñ 0,

u i px γ q ´wi px γ q ´2γ |x γ | ě max Γ i tu i pxq ´wi pxq ´2γ |x|u `lim sup εÑ0 p|x ε,γ | ´|y ε,γ |q 2 2ε ě u i pOq ´wi pOq `lim sup εÑ0 p|x ε,γ | ´|y ε,γ |q 2 2ε ě u i pOq ´wi pOq `lim inf εÑ0 p|x ε,γ | ´|y ε,γ |q 2 2ε ě u i pOq ´wi pOq .
Recalling that u i pOq ´wi pOq " max Γ i pu i ´wi q, we obtain from the inequalities above that x γ " O and that lim εÑ0 p|x ε,γ | ´|y ε,γ |q 2 2ε " 0.

(2.5.9)

We claim that if ε ą 0, then x ε,γ ‰ O. Indeed, assume by contradiction that x ε,γ " O:

1. if y ε,γ ą 0, then M ε,γ " u i pOq ´wi py ε,γ q ´1 2ε r|y ε,γ | `δ pεqs 2 ´γ |y ε,γ | ě u i py ε,γ q ´wi py ε,γ q ´δ2 pεq 2ε ´2γ |y ε,γ | .
Since u i is Lipschitz continuous in B pO, rq X Γ i , we see that for ε small enough

L |y ε.γ | ě u i pOq ´ui py ε,γ q ě |y ε,γ | 2 2ε `|y ε,γ | δ pεq ε ´γ |y ε,γ | ě |y ε,γ | δ pεq ε ´γ |y ε,γ | .
Therefore, if y ε,γ " O, then L ě L`1´γ which gives a contradiction since γ P p0, 1{2q.

Otherwise, if y ε,γ " O, then

M ε,γ " u i pOq ´wi pOq ´δ2 pεq 2ε ě u i pεe i q ´wi pOq ´1 2ε r´ε `δ pεqs 2 ´γε.

Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs Since u i is Lipschitz continuous in B pO, rq X Γ i , we see that for ε small enough,

Lε ě u i pOq ´ui pεe i q ě |y ε.γ | 2 2ε `|y ε.γ | δ pεq ε ´2γ |y ε.γ | ě |y ε.γ | δ pεq ε ´2γ |y ε.γ | .
This implies that L ě ´1{2 `L `1 ´γ, which gives a contradiction since γ P p0, 1{2q.

Therefore the claim is proved. It follows that we can apply the viscosity inequality for u i at x ε,γ . Moreover, notice that the viscosity supersolution inequality (2.5.2) holds also for y ε,γ " 0 since H i pO, pq ď H ì pO, pq for any p. Therefore u i px ε,γ q `Hi ˆxε,γ , ´xε,γ `yε,γ `δ pεq ε `γ˙ď 0, w i py ε,γ q `Hi ˆyε,γ , ´xε,γ `yε,γ `δ pεq ε ´γ˙ě 0.

Subtracting the two inequalities,

u i px ε,γ q´w i py ε,γ q ď H i ˆyε,γ , ´xε,γ `yε,γ `δ pεq ε `γ˙´H i ˆxε,γ ,
´xε,γ `yε,γ `δ pεq ε ´γ˙.

(2.5.10)

Using rH1s and rH2s, it is easy to see that there exists M i ą 0 such that for any x, y P Γ i and p, q P R |H i px, pq ´Hi py, qq| ď |H i px, pq ´Hi py, pq| `|H i py, pq ´Hi py, qq|

ď M i |x ´y| p1 `|p|q `M i |p ´q| .
It yields

u i px ε,γ q ´wi py ε,γ q ď M i " |x ε,γ ´yε,γ | ˆ1 `ˇˇˇ´x ε,γ `yε,γ `δ pεq ε ´γˇˇˇˇ˙`2 |γ|  ď M i « |x ε,γ ´yε,γ | ˆγ `1 `δ pεq ε ˙`|x ε,γ ´yε,γ | 2 ε `2 |γ| ff .
Applying (2.5.9), let ε tend to 0 and γ tend to 0, we obtain that u i pOq ´wi pOq ď 0, the desired contradiction.

Case 2: w i pOq ě min j‰i tw j pOq `cj u " w j 0 pOq `cj 0 . Using the same arguments as in Case 2 of the first proof, we get

w j 0 ă min " min j‰j 0 tw j pOq `cj u , ´HT O λ *
and w j 0 pOq ă u j 0 pOq. Repeating Case 1, replacing the index i by j 0 , implies that w j 0 pOq ě u j 0 pOq, the desired contradiction.

Corollary 2.5.7 (Uniqueness). If v is the value function (with entry costs) and pv 1 , . . . , v N q is defined by

v i pxq " $ & % v pxq if x P Γ i z tOu , lim δÑ0 `v pδe i q if x " O,
2.6. A more general optimal control problem then pv 1 , . . . , v N q is the unique bounded viscosity solution of (2.3.1).

Similarly, if v is the value function (with exit costs) and pv 1 , . . . , vN q is defined by vi pxq "

$ & % v pxq , if x P Γ i z tOu , lim δÑ0 `v pδe i q , if x " O,
then pv 1 , . . . , vN q is the unique bounded viscosity solution of (2.3.2).

Remark 2.5.8. From Corollary 2.5.7, we see that in order to characterize the original value function with entry costs, we need to solve first the Hamilton-Jacobi system (2.3.1) and find the unique viscosity solution pv 1 , . . . , v N q. The original value function v with entry costs satisfies

v pxq " $ & % v i pxq , if x P Γ i z tOu , min ! min i"1,N tv i pOq `ci u , ´HT O {λ ) , if x " O.
The characterization of v pOq follows from Theorem 2.2.9. The characterization of the original value function with exit costs v is similar.

A more general optimal control problem

In what follows, we generalize the control problem studied in the previous sections by allowing some of the entry (or exit) costs to be zero. The situation can be viewed as intermediary between the one studied in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] when all the entry (or exit) costs were zero, and that studied above when all the entry or exit costs were positive. Accordingly, every result presented below will mainly be obtained by combining the arguments proposed above with those used in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF]. Hence, we will present the results and omit the proofs.

To be more specific, we consider the optimal control problems with non-negative entry cost C " tc 1 , . . . c m , c m`1 , . . . c N u where c i " 0 if i ď m and c i ą 0 if i ą m, keeping all the assumptions and definitions of Section 2.2 unchanged. The value function associated to C will be denoted by V. Similarly to Lemma 2.2.8, V| Γ i ztOu is continuous and Lipschitz continuous near O: therefore, it is possible to extend V| Γ i ztOu at O. This extension will be noted V i . Moreover, one can check that V i pOq " V j pOq for all i, j ď m, which means that V| Y m i"1 Γ i is a continuous function which will be noted V c hereafter.

Combining the arguments in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF] and in Section 2.2 leads us to the following theorem.

Theorem 2.6.1. The value function V satisfies

max i"m`1,N tV i pOqu ď V pOq " V c pOq ď min " min i"m`1,N tV i pOq `ci u , ´HT O λ * .
Remark 2.6.2. In the case when c i " 0 for i " 1, N , V is continuous on G and it is exactly the value function of the problem studied in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF].

We now define a set of admissible test-function and the Hamilton-Jacobi equation that will characterize V.

Definition 2.6.3. A function ϕ : pY

m i"1 Γ i q ˆΓm`1 ˆ. . . ˆΓN Ñ R N ´m`1 of the form ϕ px c , x m`1 , . . . , x N q " pϕ c px c q , ϕ m`1 px m`1 q , . . . , ϕ N px N qq is an admissible test-function if
Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

• ϕ c is continuous and for i ď m, ϕ c | Γ i belongs to C 1 pΓ i q,
• for i ą m, ϕ i belongs to C 1 pΓ i q,

• the space of admissible test-function is noted R pGq.

Definition 2.6.4. A function U " pU c , U m`1 , . . . , U N q where U c P U SC `Ym j"1 Γ j ; R ˘, U i P U SC pΓ i ; Rq is called a viscosity subsolution of the Hamilton-Jacobi system if for any testfunction pϕ c , ϕ m`1 , . . . , ϕ N q P R pGq:

1. if U c ´ϕc has a local maximum at x c P Y m j"1 Γ j and if • x c P Γ j z tOu for some j ď m, then λU c px c q `Hj ˆx, dϕ c dx j px c q ˙ď 0,

• x c " O, then λU c pOq `max " ´λmin jąm tU j pOq `cj u , max jďm " H j ˆO, dϕ c dx j pOq ˙* , H T O * ď 0;
2. if U i ´ϕi has a local maximum point at x i P Γ i for i ą m, and if

• x i P Γ i z tOu, then λU i px i q `Hi ˆx, dϕ i dx i px i q ˙ď 0, • x i " O, then λU i pOq `max " ´λ min jąm,j‰i tU j pOq `cj u , ´λU c pOq , H ì ˆO, dϕ i dx i pOq ˙, H T O * ď 0.
A function U " pU c , U m`1 , . . . , U N q where U c P LSC `Ym j"1 Γ j ; R ˘, U i P LSC pΓ i ; Rq is called a viscosity supersolution of the Hamilton-Jacobi system if for any pϕ c , ϕ m`1 , . . . , ϕ N q P R pGq:

1. if U c ´ϕc has a local maximum at x c P Y m j"1 Γ j and if • x c P Γ j z tOu for some j ď m, then λU c px c q `Hj ˆx, dϕ c dx j px c q ˙ě 0, where H O pp 1 , . . . , p N q is defined in [8, page 6]. This means that, in the case when all the entry costs c j vanish, we recover the notion of viscosity solution proposed in [START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF].

• x c " O,
We now study the relation between the value function V and the Hamilton-Jacobi system.

Theorem 2.6.7. Let V be the value function corresponding to the entry costs C, then the vector-valued function pV c , V m`1 , . . . , V N q is a viscosity solution of the Hamilton-Jacobi system.

Let us state the comparison principle for the Hamilton-Jacobi system.

Theorem 2.6.8. Let U " pU c , U m`1 , . . . , U N q and W " pW c , W m`1 , . . . , W N q be a bounded viscosity subsolution and a viscosity supersolution, respectively, of the Hamilton-Jacobi system. The following holds: U ď W in G, i.e., U c ď W c on Y m j"1 Γ j , and

U i ď W i in Γ i for all i ą m.
Proof of Theorem 2.6.8. Suppose by contradiction that there exists i P t1, . . . , N u and

x P Γ i such that $ & % U c pxq ´Wc pxq ą 0, if i ď m, U i pxq ´Wi pxq ą 0, if i ą m, then $ & % U c pOq ´Wc pOq " max Y m j"1 Γ j tU c ´Wc u ą 0, if i ď m, U i pOq ´Wi pOq " max Γ i tU i ´Wi u ą 0, if i ą m,
since the case where the positive maximum is achieved outside the junction leads to a contradition by classical comparison results.

Case 1: U c pOq ´Wc pOq " max

Y m i"1 Γ i pU c ´Wc q ą 0
Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

Sub-case 1-a: W c pOq ă min jąm tW j pOq `cj u. Since W c pOq ă U c pOq ď ´HT O {λ, the function W c is a viscosity supersolution of $ ' ' & ' ' % λW c pxq `Hi ˆx, dW c dx i pxq ˙" 0 if i ď m, x P Γ i z tOu , λW c pOq `Hc ˆdW c dx 1 pOq , . . . , dW c dx m pOq ˙" 0 if x " O.
where H c pp 1 , . . . , p m q " max iďm H ì pO, p i q. Applying Lemma 2.7.3 in the Appendix, we obtain that U c pOq ď W c pOq in contradiction with the assumption.

Sub-case 1-b: W c pOq ě min jąm tW j pOq `cj u " W i 0 pOq `ci 0 . Since c i 0 ą 0, we first see that W i 0 pOq ă min min jąm tW j pOq `cj u , W c pOq , ´HT O {λ ( . Hence, W i 0 is a viscosity supersolution of (2.5.2) replacing i by i 0 . Moreover, since

U i 0 pOq `ci 0 ě min jąm pU j pOq `cj q ě U c pOq ą W c pOq ą W i 0 pOq `ci 0 ,
then U i 0 pOq ą W i 0 pOq. Applying the same argument as Case 1 in the second proof of Theorem 2.5.6 replacing i by i 0 , we obtain that U i 0 pOq ď W i 0 pOq, which is contradictory.

Case 2: U i pOq ´Wi pOq " max Γ i pU i ´Wi q ą 0 for some i ą m. Using the definition of viscosity subsolutions and Case 1, we see that

W i pOq ă U i pOq ď U c pOq ď W c pOq.
Sub-case 2-a: W i pOq ă min jąm tW j pOq `cj u. Since U i pOq ă ´HT O {λ, we first see that W i pOq ă min min jąm tW j pOq `cj u , W c pOq , ´HT O {λ ( . Hence, W i is a viscosity supersolution of (2.5.2). Applying the same argument as in Case 1 in the second proof of Theorem 2.5.6, we see that U i pOq ď W i pOq, which is contradictory.

Sub-case 2-b: W i pOq ě min jąm tW j pOq `cj u " W i 0 pOq `ci 0 . Since c i 0 ą 0, we can check that W i 0 pOq ă min min jąm tW j pOq `cj u , W c pOq , ´HT O {λ ( . Hence, W i 0 is a viscosity supersolution of (2.5.2) replacing i by i 0 . Moreover, since U i 0 pOq `ci 0 ě min jąm pU j pOq `cj q ě U c pOq ą W i pOq ą W i 0 pOq `ci 0 , then U i 0 pOq ą W i 0 pOq. Applying the same argument as Case 1 in the second proof of Theorem 2.5.6 replacing i by i 0 , we obtain that U i 0 pOq ď W i 0 pOq which is contradictory.

Appendix

Lemma 2.7.1. For any a P A ì , there exists a sequence ta n u such that a n P A i and

f i pO, a n q ě δ n ą 0, |f i pO, a n q ´fi pO, aq| ď 2M n , | i pO, a n q ´ i pO, aq| ď 2M n .

Appendix

Proof of Lemma 2.7.1. From assumption rH4s, there exists a δ P A i such that f i pO, a δ q " δ.

Since FL i pOq is convex (by assumption rH3s), for any n P N, a P A ì 1 n pf i pO, a δ q e i , i pO, a δ qq `ˆ1 ´1 n ˙pf i pO, aq , i pO, aq e i q P FL i pOq .

Then, there exists a sequence ta n u such that a n P A i and 1 n pf i pO, a δ q , i pO, a δ qq `ˆ1 ´1 n ˙pf i pO, aq , i pO, aqq " pf i pO, a n q , i pO, a n qq P FL i pOq .

(2.7.1) Notice that f i pO, aq ě 0 since a P A ì , this yields

f i pO, a n q ě f i pO, a δ q n " δ n ą 0.
From (2.7.1), we also have

|f i pO, a n q ´fi pO, aq| " 1 n |f i pO, a δ q ´fi pO, aq| ď 2M n ,
and

| i pO, a n q ´ i pO, aq| " 1 n | i pO, a δ q ´ i pO, aq| ď 2M n .
We can state the following corollary of Lemma 2.7.

˙.

The function Ψ j,ε attains its maximum at px j,ε,γ , y j,ε,γ q P Γ j ˆΓj . Applying the same argument as in the second proof of Theorem 2.5.6, we have x j,ε,γ , y j,ε,γ Ñ O and px j,ε,γ ´yj,ε,γ q 2 ε Ñ 0 as ε Ñ 0. Moreover, for any j " 1, m, x j,ε,γ ‰ O. We claim that y j,ε,γ must be O for ε small enough . Indeed, if there exists a sequence ε n such that y j,εn,γ P Γ j z tOu, then applying viscosity inequalities, we have U c px j,εn,γ q `Hj ˆxj,εn,γ , ´xj,εn,γ `yj,εn,γ `δ pε n q ε n `γ˙ď 0, W c py j,εn,γ q `Hj ˆyj,εn,γ , ´xj,εn,γ `yj,εn,γ `δ pε n q ε n ´γ˙ě 0.

Subtracting the two inequalities and using (2.5.10) with H j , we obtain U c px j,εn,γ q ´Wc py j,εn,γ q ď M j |x j,εn,γ ´yj,εn,γ | ˆ1 `ˇˇˇ´x j,εn,γ `yj,εn,γ `δ pε n q ε n ´γˇˇˇˇ˙`M j 2γ.

Recall that we already have px j,εn,γ ´yj,εn,γ q 2 ε n Ñ 0 as n Ñ 8. Let n tend to 8 and γ tend to 0 then we obtain U c pOq ´Wc pOq ď 0. It leads us to a contradiction. So this claim is proved. Define the function Ψ :

Ť m j"1 Γ j Ñ R by Ψ| Γ i pyq " 1 2ε ÿ j‰i ! r´|x i,ε,γ | `δ pεqs 2 ´γ |x i,ε,γ | ) `1 2ε r´|x i,ε,γ | `|y| `δ pεqs 2 `γ p´|x i,ε,γ | `|y|q .
We can see that Ψ is continuous on Ť m j"1 Γ j and belongs to C 1 pΓ j q for j " 1, m. Moreover, for j " 1, m and for ε small enough, y j,ε,γ =O then the function Ψ + W c has a minimum point at O. It yields

λW c pOq `Hc ˆ´x 1,ε,γ `δ pεq ε , . . . , ´xm,ε,γ `δ pεq ε ˙ě 0.
By definition of H c , there exists j 0 P t1, . . . , mu such that

λW c pOq `Hj 0 ˆO, ´xj 0 ,ε,γ `δ pεq ε ˙ě 0.
This implies λW c pOq `Hj 0 ˆO, ´xj 0 ,ε,γ `δ pεq ε ˙ě 0

A Class of Mean Field Games on Networks. Part One: the Ergodic Case

Abstract: We consider stochastic mean field games for which the state space is a network. In the ergodic case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are the invariant measure m, a value function v, and the ergodic constant ρ. The function v is continuous and satisfies general Kirchhoff conditions at the vertices. The invariant measure m satisfies dual transmission conditions: in particular, m is discontinuous across the vertices in general, and the values of m on each side of the vertices satisfy special compatibility conditions. Existence and uniqueness are proven, under suitable assumptions.

Introduction and main results

Recently, an important research activity on mean field games (MFGs for short) has been initiated since the pioneering works [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen. II. Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Mean field games[END_REF] of Lasry & Lions (related ideas have been developed independently in the engineering literature by Huang, Caines & Malhamé, see for example [START_REF] Huang | Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle[END_REF][START_REF] Huang | Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized -Nash equilibria[END_REF][START_REF] Huang | An invariance principle in large population stochastic dynamic games[END_REF]): it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number N of agents tends to infinity. In these models, it is assumed that the agents are all identical and that an individual agent can hardly influence the outcome of the game. Moreover, each individual strategy is influenced by some averages of functions of the states of the other agents. In the limit when N Ñ `8, a given agent feels the presence of the others through the statistical distribution of the states. Since perturbations of the strategy of a single agent do not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. The delicate question of the passage to the limit is one of the main topics of the book of Carmona & Delarue, [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications I-II[END_REF]. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward in time Kolmogorov or Fokker-Planck (FP) equation and a backward Hamilton-Jacobi-Bellman (HJB) equation. The unknown of this system is a pair of functions: the value function of the stochastic optimal control problem solved by a representative agent and the density of the distribution of states. In the infinite horizon limit, one obtains a system of two stationary PDEs.

A very nice introduction to the theory of MFGs is supplied in the notes of Cardaliaguet [START_REF] Cardaliaguet | Notes on mean field games[END_REF]. Theoretical results on the existence of classical solutions to the previously mentioned system of PDEs can be found in [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen. II. Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Mean field games[END_REF][START_REF] Gomes | On the existence of classical solutions for stationary extended mean field games[END_REF][START_REF] Gomes | Time-dependent mean-field games in the subquadratic case[END_REF][START_REF] Gomes | Local regularity for mean-field games in the whole space[END_REF]. Weak solutions have been studied in [START_REF] Lasry | Mean field games[END_REF][START_REF] Porretta | Weak solutions to Fokker-Planck equations and mean field games[END_REF][START_REF] Porretta | On the weak theory for mean field games systems[END_REF][START_REF] Achdou | Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games[END_REF]. The numerical approximation of these systems of PDEs has been discussed in [START_REF] Achdou | Mean field games: numerical methods[END_REF][START_REF] Achdou | Finite difference methods for mean field games[END_REF][START_REF] Achdou | Convergence of a finite difference scheme to weak solutions of the system of partial differential equations arising in mean field games[END_REF].

A network (or a graph) is a set of items, referred to as vertices (or nodes/crosspoints), with connections between them referred to as edges. In the recent years, there has been an increasing Chapter 3. A Class of Mean Field Games on Networks. Part One: the Ergodic Case interest in the investigation of dynamical systems and differential equations on networks, in particular in connection with problems of data transmission and traffic management (see for example [START_REF] Garavello | Traffic flow on networks[END_REF][START_REF] Engel | Vertex control of flows in networks[END_REF][START_REF] Forcadel | Homogenization of second order discrete model with local perturbation and application to traffic flow[END_REF]). The literature on optimal control in which the state variable takes its values on a network is recent: deterministic control problems and related Hamilton-Jacobi equations were studied in [START_REF] Achdou | Hamilton-Jacobi equations constrained on networks[END_REF][START_REF] Imbert | A Hamilton-Jacobi approach to junction problems and application to traffic flows[END_REF][START_REF] Achdou | Hamilton-Jacobi equations for optimal control on junctions and networks[END_REF][START_REF] Imbert | Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks[END_REF][START_REF] Lions | Viscosity solutions for junctions: well posedness and stability[END_REF][START_REF] Lions | Well-posedness for multi-dimensional junction problems with Kirchoff-type conditions[END_REF]. Stochastic processes on networks and related Kirchhoff conditions at the vertices were studied in [START_REF] Freidlin | Diffusion processes on graphs and the averaging principle[END_REF][START_REF] Freidlin | Diffusion processes on graphs: stochastic differential equations, large deviation principle[END_REF].

The present work is devoted to infinite horizon stochastic mean field games taking place on networks. The most important difficulty will be to deal with the transition conditions at the vertices. The latter are obtained from the theory of stochastic control in [START_REF] Freidlin | Diffusion processes on graphs and the averaging principle[END_REF][START_REF] Freidlin | Diffusion processes on graphs: stochastic differential equations, large deviation principle[END_REF], see Section 3.1.3 below. In [START_REF] Camilli | Stationary mean field games systems defined on networks[END_REF], the first article on MFGs on networks, Camilli & Marchi consider a particular type of Kirchhoff condition at the vertices for the value function: this condition comes from an assumption which can be informally stated as follows: consider a vertex ν of the network and assume that it is the intersection of p edges Γ 1 , . . . , Γ p , ; if, at time τ , the controlled stochastic process X t associated to a given agent hits ν, then the probability that X τ `belongs to Γ i is proportional to the diffusion coefficient in Γ i . Under this assumption, it can be seen that the density of the distribution of states is continuous at the vertices of the network. In the present work, the above mentioned assumption is not made any longer. Therefore, it will be seen below that the value function satisfies more general Kirchhoff conditions, and accordingly, that the density of the distribution of states is no longer continuous at the vertices; the continuity condition is then replaced by suitable compatibility conditions on the jumps across the vertex. Moreover, as it will be explained in Remark 3.1.14 below, more general assumptions on the coupling costs will be made. Mean field games on networks with finite horizon will be considered in Chapter 4.

After obtaining the transmission conditions at the vertices for both the value function and the density, we shall prove existence and uniqueness of weak solutions of the uncoupled HJB and FP equations (in suitable Sobolev spaces). We have chosen to work with weak solutions because it is a convenient way to deal with existence and uniqueness in the stationary regime, but also because it is difficult to avoid it in the nonstationary case, see Chapter 4 for finite horizon MFGs. Classical arguments will then lead to the regularity of the solutions. Next, we shall establish the existence result for the MFG system by a fixed point argument and a truncation technique. Uniqueness will also be proved under suitable assumptions.

The present work is organized as follows: the remainder of Section 3.1 is devoted to setting the problem and obtaining the system of PDEs and the transmission conditions at the vertices. Section 3.2 contains useful results, first about some linear boundary value problems with elliptic equations, then on a pair of linear Kolmogorov and Fokker-Planck equations in duality. By and large, the existence of weak solutions is obtained by applying Banach-Necas-Babuška theorem to a special pair of Sobolev spaces referred to as V and W below and Fredholm's alternative, and uniqueness comes from a maximum principle. Section 3.3 is devoted to the HJB equation associated with an ergodic problem. Finally, the proofs of the main results of existence and uniqueness for the MFG system of PDEs are completed in Section 3.1.

Networks and function spaces

The geometry A bounded network Γ (or a bounded connected graph) is a connected subset of R n made of a finite number of bounded non-intersecting straight segments, referred to as edges, which connect nodes referred to as vertices. The finite collection of vertices and the finite set of closed edges

Introduction and main results

are respectively denoted by V :" tν i , i P Iu and E :" tΓ α , α P Au, where I and A are finite sets of indices contained in N. We assume that for α, β P A, if α " β, then Γ α X Γ β is either empty or made of a single vertex. The length of Γ α is denoted by α . Given ν i P V, the set of indices of edges that are adjacent to the vertex ν i is denoted by A i " tα P A : ν i P Γ α u. A vertex ν i is named a boundary vertex if 7 pA i q " 1, otherwise it is named a transition vertex. The set containing all the boundary vertices is named the boundary of the network and is denoted by BΓ hereafter.

The edges Γ α P E are oriented in an arbitrary manner. In most of what follows, we shall make the following arbitrary choice that an edge Γ α P E connecting two vertices ν i and ν j , with i ă j is oriented from ν i toward ν j : this induces a natural parametrization π α : r0, α s Ñ Γ α " rν i , ν j s:

π α pyq " α ´y α ν i `y α ν j for y P r0, α s. (3.1.1)
For a function v : Γ Ñ R and α P A, we define v α : p0, α q Ñ R by v α pxq :" v| Γα ˝πα pxq, for all x P p0, α q.

Remark 3.1.1. In what precedes, the edges have been arbitrarily oriented from the vertex with the smaller index toward the vertex with the larger one. Other choices are of course possible.

In particular, by possibly dividing a single edge into two, adding thereby new artificial vertices, it is always possible to assume that for all vertices ν i P V, either π α pν i q " 0, for all α P A i or π α pν i q " α , for all α P A i . (

This idea was used by Von Below in [START_REF] Below | Classical solvability of linear parabolic equations on networks[END_REF]: some edges of Γ are cut into two by adding artificial vertices so that the new oriented network Γ has the property (3.1.2), see Figure 3.1 for an example. In Sections 3.1.2 and 3.1.3 below, especially when dealing with stochastic calculus, it will be convenient to assume that property (3.1.2) holds. In the remaining part of the paper, it will be convenient to work with the original network, i.e. without the additional artificial vertices and with the orientation of the edges that has been chosen initially.

Γ 1 ν 1 ν 2 Γ 2 ν 3 Γ 3 ν 4 Γ 4 Γ1 ν1 ν2 Γ2 ν5 Γ5 ν3 Γ3 ν4 Γ4

Function spaces

The set of continuous functions on Γ is denoted by CpΓq and we set P C pΓq :" # v : Γ Ñ R : for all α P A, ˇˇˇˇvα P Cp0, α q v α can be extended by continuity to r0, α s.

+ .

By the definition of piecewise continuous functions v P P CpΓq, for all α P A, it is possible to extend v| Γα by continuity at the endpoints of Γ α : if Γ α " rν i , ν j s, we set

v| Γα pxq " $ ' ' ' ' & ' ' ' ' % v α `π´1 α pxq ˘, if x P Γ α zV, v α p0q :" lim yÑ0 `vα pyq , if x " ν i , v α p α q :" lim yÑ ά v α pyq , if x " ν j . (3.1.3)
For m P N, the space of m-times continuously differentiable functions on Γ is defined by C m pΓq :" tv P C pΓq : v α P C m pr0, α sq for all α P Au , and is endowed with the norm }v} C m pΓq :"

ÿ αPA ÿ kďm › › ›B k v α › › › L 8 p0, αq .
For σ P p0, 1q, the space C m,σ pΓq, contains the functions v P C m pΓq such that B m v α P C 0,σ pr0, α sq for all α P A; it is endowed with the norm For a positive integer m and a function v P C m pΓq, we set for k ď m

B k v pxq " B k v α `π´1 α pxq ˘if x P Γ α zV. (3.1.4)
Notice that v P C k pΓq is continuous on Γ but that the derivatives B l v, 0 ă l ď k are not defined at the vertices. For a vertex ν, we define B α v pνq as the outward directional derivative of v| Γα at ν as follows:

B α v pνq :" $ ' & ' % lim hÑ0 `vα p0q ´vα phq h , if ν " π α p0q , lim hÑ0 `vα p α q ´vα p α ´hq h , if ν " π α p α q . (3.1.5)
For all i P I and α P A i , setting

n iα " # 1 if ν i " π α p α q, ´1 if ν i " π α p0q, (3.1.6)
we have B α vpν i q " n iα Bv| Γα pν i q " n iα Bv α pπ ´1 α pν i qq.

Introduction and main results

If for all α P A, v α is Lebesgue-integrable on p0, α q, then the integral of v on Γ is defined by

ż Γ v pxq dx " ÿ αPA ż α 0 v α pyq dy.
For p P r1, 8s, L p pΓq : " tv : v α P L p p0, α q for all α P Au " tv : v| Γα P L p pΓ α q for all α P Au , is endowed with the norm

}v} L p pΓq :" ˜ÿ αPA }v α } p L p p0, αq ¸1 p , if 1 ď p ă `8, and max αPA }v α } L 8 p0, αq , if p " 8.
We shall also need to deal with functions on Γ whose restrictions to the edges are weaklydifferentiable: we shall use the same notations for the weak derivatives. Let us introduce Sobolev spaces on Γ.

Definition 3.1.3. For any integer s ě 1 and any real number p ě 1, the Sobolev space W s,p pΓq is defined as follows:

W s,p pΓq :" tv P C pΓq : v α P W s,p p0, α q for all α P Au , and endowed with the norm

}v} W s,p pΓq " ˜s ÿ k"1 ÿ αPA › › ›B k v α › › › p L p p0, αq `}v} p L p pΓq ¸1 p .
We also set H s pΓq " W s,2 pΓq.

A class of stochastic processes on Γ

After rescaling the edges, it may be assumed that α " 1 for all α P A. Let µ α , α P A and p iα , i P I, α P A i be positive constants such that ř αPA i p iα " 1. Consider also a real valued function a P P CpΓq. As in Remark 3.1.1, we make the assumption (3.1.2) by possibly adding artificial nodes: if ν i is such an artificial node, then 7pA i q " 2, and we assume that p iα " 1{2 for α P A i . The diffusion parameter µ has the same value on the two sides of an artificial vertex. Similarly, the function a does not have jumps across an artificial vertex.

Let us consider the linear differential operator:

Lu pxq " L α u pxq :" µ α B 2 u pxq `a| Γα pxq Bu pxq , if x P Γ α , ( 3.1.8)
with domain D pLq :"

# u P C 2 pΓq : ÿ αPA i p iα B α u pν i q " 0, for all i P I + . ( 3
ÿ αPA i p iα B α upν i q " 0 (3.1.10)
define such a process in a unique way, see also [START_REF] Freidlin | Diffusion processes on graphs and the averaging principle[END_REF]Theorem 3.1]. The process can be written pX t , α t q where X t P Γ αt . If X t " ν i , i P I, α t is arbitrarily chosen as the smallest index in A i . Setting x t " π αt pX t q defines the process x t with values in r0, 1s.

2. There exist a) a one dimensional Wiener process W t , b) continuous non-decreasing processes i,t , i P I, which are measurable with respect to the σ-field generated by pX t , α t q, c) continuous non-increasing processes h i,t , i P I, which are measurable with respect to the σ-field generated by pX t , α t q, such that

dx t " ? µ αt dW t `aαt px t qdt `d i,t `dh i,t , (3.1.11)
i,t increases only when X t " ν i and x t " 0, h i,t decreases only when X t " ν i and x t " 1.

3. The following Ito formula holds: for any real valued function u P C 2 pΓq:

upX t q "upX 0 q `ÿ αPA ż t 0 1 tXsPΓαzVu ´µα B 2 upX s q `apX s qBupX s qds `a2µ α BupX s qdW s ÿ iPI ÿ αPA i p iα B α upν i qp i,t `hi,t q.
(3.1.12) Remark 3.1.5. The assumption that all the edges have unit length is not restrictive, because we can always rescale the constants µ α and the piecewise continuous function a. The Ito formula in (3.1.12) holds when this assumption is not satisfied.

Consider the invariant measure associated with the process X t . We may assume that it is absolutely continuous with respect to the Lebesgue measure on Γ. Let m be its density:

E ru pX t qs :" ż Γ u pxq m pxq dx,
for all u P P CpΓq.

(3.1.13)

We focus on functions u P D pLq. Taking the time-derivative of each member of (3.1.13), Ito's formula (3.1.12) and (3.1.10) lead to:

E " 1 tXtRVu `aBupX t q `µB 2 upX t q ˘‰ " 0.

This implies that ż Γ `apxqBupxq `µB 2 upxq ˘mpxqdx " 0.

B λ pv, ŵq `λ0 pv, ŵq ě µ ψ 2 }v} 2 V ě µ ψ 2C ψ }v} V } ŵ} W ,
using the fact that, from Remark 3.1.10, there exists a positive constant C ψ such that }vψ} W ď C ψ }v} V for all v P V . This yields

inf vPV sup wPW B λ pv, wq `λ0 pv, wq }v} V }w} W ě µ ψ 2C ψ . ( 3.2.7)
Using a similar argument for any w P W and v " wφ, where φ is given in Definition 3.1.9, we obtain that for λ 0 large enough, there exist a positive constant C φ such that inf

wPW sup vPV B λ pv, wq `λ0 pv, wq }w} W }v} V ě µ φ 2C φ . ( 3.2.8)
From (3.2.7) and (3.2.8), by the Banach-Necas-Babuška lemma (see [START_REF] Ern | Theory and practice of finite elements[END_REF] or [START_REF] Brezzi | Mixed and hybrid finite element methods[END_REF]), for λ 0 large enough, there exists a positive constant C such that for any f P W 1 , there exists a unique solution v P V of (3.2.5) and }v} V ď C }f } W 1 . Hence, our claim is proved. Now, we fix λ 0 large enough and we define the continuous linear operator R λ 0 : W 1 Ñ V where R λ 0 pf q " v is the unique solution of (3.2.5). Since the injection I from V to W 1 is compact, then I ˝Rλ 0 is a compact operator from W 1 into W 1 . By the Fredholm alternative (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]), one of the following assertions holds: There exists v P W 1 z t0u such that `Id ´λ0 `I ˝Rλ 0 ˘˘v " 0.

(

For any g P W 1 , there exists a unique v P W 1 such that `Id ´λ0 `I ˝Rλ 0 ˘˘v " g. (3.2.10)

We claim that (3.2.10) holds. Indeed, assume by contradiction that (3.2.9) holds. Then there exists v ‰ 0 such that

$ & % v P V, I ˝Rλ 0 v " v λ 0 . Therefore $ ' & ' % v P V, B λ ˆv λ 0 , w ˙`λ 0 ˆv λ 0 , w ˙" pv, wq , for all w P W .
This yields that B λ pv, wq " 0 for all w P W and by Lemma 3.2.5, we get that v " 0, which leads us to a contradiction. Hence, our claim is proved. Next, from (3.2.10), we can check that for any g P V Ă W 1 , there exists a unique v P V such that `Id ´λ0 `I ˝Rλ 0 ˘˘v " g. It allows us to define

`Id ´λ0 `I ˝Rλ 0 ˘˘´1 | V : V ÝÑ V Ă W 1 .
Let us consider the operator T :" `Id ´λ0 `I ˝Rλ 0 ˘˘´1 | V ˝Rλ 0 : W 1 ÝÑ V . We claim that for any f P W 1 , T pf q is a solution of (3.2.1). Indeed, set g :" R λ 0 pf q and v :" T pf q " `Id ´λ0 `I ˝Rλ 0 ˘˘´1 | V pgq. Then B λ pg, wq `λ0 pg, wq " pf, wq , for all w P W , • (Existence) There exists a solution p m P W of (3.2.12)-(3.2.13) satisfying

} p m} W ď C, 0 ď p m ď C, (3.2.16)
where the constant C depends only on }b} 8 and tµ α u αPA . Moreover, p m α P C 1 p0, α q for all α P A. Hence, p m P W.

• (Uniqueness) p m is the unique solution of (3.2.12)-(3.2.13).

• (Strictly positive solution) p m is strictly positive.

Proof of existence in Theorem 3.2.13. We divide the proof of existence into three steps:

Step 1: Let λ 0 be a large positive constant that will be chosen later. We claim that for m P L 2 pΓq and h :" λ 0 m P L 2 pΓq Ă V 1 , (3.2.14) has a unique solution m P W . This allows us to define a linear operator as follows:

T : L 2 pΓq ÝÑ W, m Þ ÝÑ m,
where m is the solution of (3.2.14) with h " λ 0 m. We are going to prove that T is well-defined and continuous, i.e, for all m P L 2 pΓq, (3.2.14) has a unique solution that depends continuously on m. For w P W , set p v :" wφ P V where φ is given by Definition 3.1.9. We have

A λ 0 pw, p vq " ÿ αPA ż Γα " λ 0 φw 2 `pµ α Bw `bwq B pwφq ‰ dx " ÿ αPA ż Γα " pλ 0 φ `bBφq w 2 `pµ α Bφ `bφq wBw `µα φ pBwq 2 ı dx.
It follows that when λ 0 is large enough (larger than a constant that only depends on b, φ and µ α ), A λ 0 pw, p vq ě p C λ 0 }w} 2 W for some positive constant p C λ 0 . Moreover, by Remark 3.1.10, there exists a positive constant p C φ such that for all w P W , we have }wφ} V ď C φ }w} W . This yields

inf wPW sup vPV A λ 0 pw, vq }v} V }w} W ě p C λ 0 C φ .
Using similar arguments, for λ 0 large enough, there exist two positive constants C λ 0 and C ψ such that

inf vPV sup wPW A λ 0 pw, vq }w} W }v} V ě C λ 0 C ψ .
From Banach-Necas-Babuška lemma (see [START_REF] Ern | Theory and practice of finite elements[END_REF] or [START_REF] Brezzi | Mixed and hybrid finite element methods[END_REF]), there exists a constant C such that for all m P L 2 pΓq, there exists a unique solution m of (3.2.14) with h " λ 0 m and }m} W ď C }m} L 2 pΓq . Hence, the map T is well-defined and continuous from L 2 pΓq to W .

Step 2: Let K be the set defined by K :"

" m P L 2 pΓq : m ě 0 and ż Γ mdx " 1 * .
We claim that T pKq Ă K which means ş Γ m " 1 and m ě 0. Indeed, using v " 1 as a test function in ( We can see that the right hand side is non-negative. Moreover, for λ 0 large enough (larger than the same constant as above, which only depends on b, φ and µ α ), the left hand side is non-positive. This implies that m ´" 0, and hence m ě 0. Therefore, the claim is proved.

Step 3: We claim that T has a fixed point. Let us now focus on the case when m P K. Using mφ as a test function in (3.2.14) yields

ÿ αPA ż Γα " pλ 0 φ `bBφq m 2 `µα pBmq 2 φ `pµ α Bφ `bφq m pBmq ı dx " ż Γ λ 0 mmφdx. (3.2.17)
Since H 1 p0, α q is continuously embedded in L 8 p0, α q, there exists a positive constant C (independent of m P K) such that

ż Γ mmφdx ď ż Γ mdx }m} L 8 pΓq φ " }m} L 8 pΓq φ ď C }m} W .
Hence, from (3.2.17), for λ 0 large enough, there exists a positive constant

C 1 such that C 1 }m} 2 W ď λ 0 C }m} W . Thus }m} W ď λ 0 C C 1 . ( 3.2.18)
Therefore, T pKq is bounded in W . Since the bounded subsets of W are relatively compact in L 2 pΓq, T pKq is compact in L 2 pΓq. Moreover, we can see that K is closed and convex in L 2 pΓq. By Schauder fixed point theorem, see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Corollary 11.2], T has a fixed point p m P K which is also a solution of (3.2.12) and } p m} W ď λ 0 C{C 1 . Finally, from the PDE in (3.2.12), for all α P A, we have p p m 1 α `bα p m α q 1 " 0 on p0, α q. This implies that there exists a constant C α such that p m 1 α `bα p m α " C α , for all x P p0, α q.

It follows that p m 1 α P Cpr0, α sq, for all α P A.

Hence p m α P C 1 pr0, α sq for all α P A. Thus, p m P W.

Remark 3.2.14. Let m P W be a solution of (3.2.12). If b, Bb P P C pΓq, standard arguments yield that m α P C 2 p0, α q for all α P A. Moreover, by Theorem 3.2.13, there exists a constant C which depends only on }b} 8 , t}Bb α } 8 u αPA and µ α such that }m α } C 2 p0, αq ď C for all α P A.

Proof of the positivity in Theorem 3.2.13. From (3.2.13), p m is non-negative on Γ. Assume by contradiction that there exists x 0 P Γ α for some α P A such that p m| Γα px 0 q " 0. Therefore, the minimum of p m over Γ is achieved at x 0 P Γ α . If x 0 P Γ α zV, then B p mpx 0 q " 0. In (3.2.19), we thus have C α " 0, and hence p m α satisfies p m 1 α `bα p m α " 0, on r0, α s , with p m α `π´1 α px 0 q ˘" 0. It follows that p m α " 0 and p m| Γα pν i q " p m| Γα pν j q " 0 if Γ α " rν i , ν j s.

Preliminary: A class of linear boundary value problems

Therefore, it is enough to consider x 0 P V. Now, from Remark 3.1.1, we may assume without loss of generality that x 0 " ν i and π β pν i q " 0 for all β P A i . We have the following two cases.

Case 1: if x 0 " ν i is a transition vertex, then, since p m belongs to W , we get p m| Γ β pν i q " γ iβ γ iα p m| Γα pν i q " 0, for all β P A i . (3.2.21) This yields that ν i is also a minimum point of p m| Γ β for all β P A i . Thus B β p m pν i q ď 0 for all β P A i . From the transmission condition in (3.2.12) which has a classical meaning thanks to (3.2.20), B β p m pν i q " 0, since all the coefficients µ β are positive. From (3.2.19), for all β P A i , we have C β " p m 1 β p0q `bβ p0q p m β p0q " 0.

Therefore, p m 1 β pyq `bβ pyq p m β pyq " 0, for all y P r0, β s with p m β p0q " 0. This implies that p m β " 0 for all β P A i . We can propagate the arguments from the vertices connected to ν i . Since Γ is connected, we obtain that p m " 0 on Γ. Case 2: if x 0 " ν i is a boundary vertex, then the Robin condition in (3.2.12) implies that B α p m pν i q " 0 since µ α is positive. From (3.2.19), we have C α " 0. Therefore, p m 1 α pyq bα pyq p m α pyq " 0, for all y P r0, α s with p m α p0q " 0. This implies that p m pν j q " 0 where ν j is the other endpoint of Γ α . We are back to Case 1, so p m " 0 on Γ. Finally, we have found that p m " 0 on Γ, in contradiction with ş Γ p mdx " 1.

Now we prove uniqueness for (3.2.12)-(3.2.13).

Proof of uniqueness in Theorem 3.2.13. The proof of uniqueness is similar to the argument in [32, Proposition 13]. As in the proof of Lemma 3.2.6, we can prove that for λ 0 large enough, there exists a constant C such that for any f P V 1 , there exists a unique w P W which satisfies

A λ 0 pw, vq " xf, vy V 1 ,V for all v P V. (3.2.22)
and }w} W ď C }f } V 1 . This allows us to define the continuous linear operator

S λ 0 : L 2 pΓq ÝÑ W, f Þ ÝÑ w,
where w is a solution of (3.2.22). Then we define R λ 0 " J ˝Sλ 0 where J is the injection from W in L 2 pΓq, which is compact. Obviously, R λ 0 is a compact operator from L 2 pΓq into L 2 pΓq. Moreover, m P W is a solution of (3.2.12) if and only if m P ker pId ´λ0 R λ 0 q. By Fredholm alternative, see [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF], dim ker pId ´λ0 R λ 0 q " dim ker `Id ´λ0 R ‹ λ 0 ˘. In order to characterize R ‹ λ 0 , we now consider the following boundary value problem for

g P L 2 pΓq Ă W 1 : $ ' ' ' ' & ' ' ' ' % λ 0 v ´µα B 2 v `bBv " g, in Γ α zV, α P A, v| Γα pν i q " v| Γ β pν i q α, β P A i , i P I, ÿ αPA i γ iα µ α B α v pν i q " 0, i P I.
2 ψ pBv n q 2 `c2 2µ α ψ pBv n q 2 v 2 n  dx,
where we have used Young inequalities. Since λ ą 0 and ψ ą 0, we deduce that

ÿ αPA ż Γα e Kv 2 n " ´µα 2 ψ ¯pBv n q 2 `2ψ ˆµα K ´c2 4µ α ˙v2 n pBv n q 2 `pµ α Bψq v n Bv n  dx ď c 2 4λ ż Γ e Kv 2 n ψdx. (3.3.11) Next, choosing K ą p1 `c2 {4µq{µ yields that ÿ αPA ż Γα e Kv 2 n " µ α 2 ψ pBv n q 2 `2ψv 2 n pBv n q 2 `pµ α Bψq v n Bv n ı dx ď C
for a positive constant C independent of n, because v n is bounded by c{λ. Since ψ is bounded from below by a positive number and Bψ is piecewise constant on Γ, we infer that

ÿ αPA ż Γα e Kv 2 n v 2 n pBv n q 2 ď r C,
where r C is a positive constant independent on n. Using this information and (3.3.11) again, we obtain that ş Γ pBv n q 2 is bounded uniformly in n. There exists a constant C such that }v n } V ď C for all n.

Hamilton-Jacobi equation and the ergodic problem

Next, set u λ :" v λ ´min Γ v λ . We see that u λ is the unique classical solution of

$ ' ' ' ' & ' ' ' ' % ´µα B 2 u λ `H px, Bu λ q `λu λ `λ min Γ v λ " f, in Γ α zV, α P A,
u| Γα pν i q " u| Γ β pν i q, α, β P A i , i P I, ÿ αPA i γ iα µ α B α u λ pν i q " 0, i P I.

(3.3.18)

Before passing to the limit as λ tends 0, we need to estimate u λ in C 2 pΓq uniformly with respect to λ. We do this in two steps:

Step 1: Estimate of }Bu λ } L q pΓq . Using ψ as a test-function in (3.3.18), see Definition 3.1.9, and recalling that λu λ `λ min Γ v λ " λv λ , we see that

ÿ αPA ż Γα µ α Bu λ Bψdx `żΓ pH px, Bu λ q `λv λ q ψdx " ż Γ f ψdx.
From (3.1.27) and (3.3.17),

ÿ αPA ż Γα µ α Bu λ Bψdx `ÿ αPA ż Γα C 0 |Bu λ | q ψdx ď ż Γ pf `C `C1 q ψdx.
On the other hand, since q ą 1, ψ ě ψ ą 0 and Bψ is bounded, there exists a large enough positive constant C 1 such that

ÿ αPA ż Γα µ α Bu λ Bψdx `1 2 ÿ αPA ż Γα C 0 |Bu λ | q ψdx `C1 ą 0, for all λ ą 0.
Subtracting the two inequalities, we get

C 0 2 ψ ż Γ |Bu λ | q dx ď ż Γ pf `C `C1 q ψdx `C1 .
Hence, for all λ ą 0, we have }Bu λ } L q pΓq ď r C, (3.3.19) where r C :" "`2 ş Γ p|f | `C `C1 q ψdx `2C 1 ˘{pC 0 ψq ‰ 1{q .

Step 2: Estimate of }u λ } C 2 pΓq . Since u λ " v λ ´min Γ v λ , there exists α P A and x λ P Γ α such that u λ px λ q " 0. For all λ ą 0 and x P Γ α , we have

|u λ pxq| " |u λ pxq ´uλ px λ q| ď ż Γ |Bu λ | dx ď }Bu λ } L q pΓq |Γ| q{pq´1q .
From (3.3.19) and the latter inequality, we deduce

}u λ | Γα } L 8 pΓαq ď r C |Γ| q{pq´1q .
Let ν i be a transition vertex which belongs to BΓ α . For all β P A i , y P Γ β ,

|u λ pyq| ď |u λ pyq ´uλ pν i q| `|u λ pν i q| ď 2 r C |Γ| q{pq´1q .
Since the network is connected and the number of edges is finite, repeating the argument as many times as necessary, we obtain that there exists M P N such that T is continuous. Let m n , m P K be such that }m n ´m} Mσ Ñ 0 as n Ñ `8; set m n " T pm n q , m " T pmq. We need to prove that m n Ñ m in M σ . Let pv n , ρ n q , pv, ρq be the solutions of (3.3.12)-(3.3.13) corresponding respectively to f " F pm n q and f " F pmq. Using estimate (3.3.14), we see that up to the extraction of a subsequence, we may assume that pv n , ρ n q Ñ pv, ρq in C 1 pΓq ˆR. Since F pm n q | Γα Ñ F pmq | Γα in C pΓ α q, H α py, pBv n q α q Ñ H α py, Bv α q in C pr0, α sq, and since it is possible to pass to the limit in the transmission and boundary conditions thanks to the C 1 -convergence, we obtain that pv, ρq is a weak (and strong by Remark 3.3.3) solution of (3.3.12)- (3.3.13). By uniqueness, pv, ρq " pv, ρq and the whole sequence pv n , ρ n q converges.

}u λ } L 8 pΓq ď M r C |Γ| q{pq´1q .
Next, m n " T pm n q , m " T pmq are respectively the solutions of (3.2.12)-(3.2.13) corresponding to b " B p H px, Bv n q and b " B p H px, Bvq. From the estimate (3.2.16), since B p H px, Bv n q is uniformly bounded in L 8 pΓq, we see that m n is uniformly bounded in W . Therefore, up to the extraction of subsequence, we have

$ & % m n á p m in W , m n Ñ p m in M σ ,
because W is compactly embedded in M σ for σ P p0, 1{2q. It is easy to pass to the limit and find that p m is a solution of (3.2.12)-(3.2.13) with b " B p H px, Bvq. From Theorem 3.2.13, we obtain that m " p m, and hence the whole sequence m n converges to m. The image of T is precompact. Since F P C 0 pR `; Rq is a uniformly bounded function, we see that F pmq is bounded in L 8 pΓq uniformly with respect to m P K. From Theorem 3.3.7, there exists a constant C such that for all m P K, the unique solution v of (3.3.12)-(3.3.13) with f " F pmq satisfies }v} C 2 pΓq ď C. From Theorem 3.2.13, we obtain that m " T pmq is bounded in W by a constant independent of m. Since W is compactly embedded in M σ , for σ P p0, 1{2q we deduce that T has a precompact image.

End of the proof. We can apply Schauder fixed point theorem (see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Corollary 11.2]) to conclude that the map T admits a fixed point m. By Theorem 3.2.13, we get m P W. Hence, there exists a solution pv, m, ρq P C 2 pΓq ˆW ˆR to the mean field games system (3.1.24). If F is locally Lipschitz continuous, then v P C 2,1 pΓq from the final part of Theorem 3.3.7.

Proof of uniqueness in Theorem 3.4.1. We assume that F is strictly increasing and that there exist two solutions pv 1 , m 1 , ρ 1 q and pv 2 , m 2 , ρ 2 q of (3.1.24). We set v " v 1 ´v2 , m " m 1 ´m2

Proof of the main result

Then F n is continuous, bounded below by ´M as in (3.1.31) and bounded above by some constant C n . By Theorem 3.4.1, for all n P N, there exists a unique solution pv n , m n , ρ n q P C 2 pΓq ˆW ˆR of the mean field game system (3.1.24) where F is replaced by F n . We wish to pass to the limit as n Ñ `8. We proceed in several steps:

Step 1: ρ n is bounded from below. Multiplying the HJB equation in (3.1.24) by m n and the Fokker-Planck equation in (3.1.24) by v n , using integration by parts and the transmission conditions, we obtain that In what follows, the constant C may vary from line to line but remains independent of n. From (3.1.26), we see that B p H px, Bv n q Bv n ´H px, Bv n q ě ´Hpx, 0q ě ´C. Therefore

ÿ αPA ż Γα µ α Bv n Bm n dx `żΓ H px, Bv n q m n dx `ρn " ż Γ F n pm n q m n dx, ( 3
ρ n ě ż Γ F n pm n q m n dx ´C ż Γ m n dx " ż Γ F n pm n q m n dx ´C. (3.4.7)
Hence, since F n `M ě 0 and ş Γ m n dx " 1, we get that ρ n is bounded from below by ´M ´C independently of n.

Step 2: ρ n and ş Γ F n pm n q dx are uniformly bounded. By Theorem 3.2.13, there exists a positive solution w P W of (3.2.12)-(3. Multiplying the HJB equation of (3.1.24) by w, using integration by parts and the Kirchhoff condition, we get

ÿ αPA ż Γα µ α Bv n Bwdx looooooooooomooooooooooon "0 `żΓ H px, Bv n q wdx `ρn ż Γ wdx loomoon "1 " ż Γ F n pm n q wdx.
This implies, using (3.1.27), (3.2.16) and F n `M ě 0,

ρ n " ż Γ F n pm n q wdx ´żΓ H px, Bv n q wdx ď }w} L 8 pΓq ż Γ pF n pm n q `M q dx ´M ´żΓ pC 0 |Bv n | q ´C1 q wdx ď C ż Γ F n pm n q dx `C ´żΓ C 0 |Bv n | q wdx. ( 3
´M ´C ď ż Γ F n pm n q m n dx ´C ď ρ n ď C ż Γ F n pm n q dx `C. (3.4.9)
Let K ą 0 be a constant to be chosen later. We have

ż Γ F n pm n q dx ď ż mněK pF n pm n q `M qdx `żmnďK F n pm n q dx ď 1 K ż mněK rF n pm n q `M s m n dx `sup 0ďrďK F n prq ż mnďK dx ď 1 K ż mněK rF n pm n q `M s m n dx `sup 0ďrďK F prq ż mnďK dx (3.4.10) ď 1 K ż Γ F n pm n q m n dx `M K `CK , ( 3
ż E F n pm n qdx ď 1 K ż EXtmněKu rF n pm n q `M sm n dx `sup 0ďrďK F prq ż EXtmnďKu dx ď C `M K `CK η,
where the last inequality comes from ş E F n pm n qm n dx ď C and C K " | sup 0ďrďK F prq|. Therefore, for all ε ą 0, we may choose K such that pC `M q{K ď ε{2 and then η such that C K η ď ε{2 and get ż E F n pm n q dx ď ε, for all E which satisfies |E| ď η, which proves the uniform integrability of tF n pm n qu n . Next, since ρ n and ş Γ F n pm n q dx are uniformly bounded, we infer from (3.4.8) that Bv n is uniformly bounded in L q pΓq. Since by the condition ş Γ v n dx " 0, there exists x n such that v n px n q " 0, we infer from the latter bound that v n is uniformly bounded in L 8 pΓq. Using the HJB equation in (3.1.24) and Remark 3.1.7, we get

µ α |B 2 v n | ď |Hpx, Bv n q| `|F n pm n q| `|ρ n | ď C q p|Bv n | q `1q `|F n pm n q| `|ρ n |.
We obtain that B 2 v n is uniformly bounded in L 1 pΓq, which implies that v n is uniformly bounded in C 1 pΓq. Therefore the sequence of functions C q p|Bv n | q `1q `|F n pm n q| `|ρ n | is uniformly integrable, and so is B 2 v n . This implies that Bv n is equicontinuous. Hence, tv n u is relatively 3.4. Proof of the main result compact in C 1 pΓq by Arzelà-Ascoli's theorem. Finally, from the Fokker-Planck equation and Theorem 3.2.13, since B p H px, Bv n q is uniformly bounded in L 8 pΓq, we obtain that m n is uniformly bounded in W .

Step 4: Passage to the limit From Step 1 and 2, since tρ n u is uniformly bounded, there exists ρ P R such that ρ n Ñ ρ up to the extraction of subsequence. From Step 3, there exists m P W such that m n á m in W and m n Ñ m almost everywhere, up to the extraction of subsequence. Also from Step 3, since F n pm n q is uniformly integrable, from Vitali theorem, we have Finally, we prove the regularity for the solution of (3.1.24). Since m P W , we get F pmq P C 0,σ pΓq for some constant σ P p0, 1{2q. By Theorem 3.3.7, v P C 2 (v P C 2,σ pΓq if F is locally Lipschitz continuous). Then, by Theorem 3.2.13, we get m P W. We also obtain that v satisfy the Kirchhoff condition and transition condition in (3.1.24). The proof is done.

A Class of Mean Field Games on Networks. Part two: Finite Horizon Games

Abstract:

We consider stochastic mean field games for which the state space is a network. In the non-stationary case, they are described by a system coupling a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation, whose unknowns are an measure m and a value function v. The function v is continuous and satisfies general Kirchhoff conditions at the vertices. The measure m satisfies dual transmission conditions: in particular, m is discontinuous across the vertices in general, and the values of m on each side of the vertices satisfy special compatibility conditions. Existence and uniqueness are proven, under suitable assumptions.

Introduction and main results

The present work is devoted to finite horizon stochastic mean field games taking place on networks. The most important difficulty will be to deal with the transition conditions at the vertices. The latter are obtained from the theory of stochastic control in [START_REF] Freidlin | Diffusion processes on graphs and the averaging principle[END_REF][START_REF] Freidlin | Diffusion processes on graphs: stochastic differential equations, large deviation principle[END_REF], see Section 4.1.3 below. In [START_REF] Camilli | Stationary mean field games systems defined on networks[END_REF], the first article on MFGs on networks, Camilli & Marchi consider a particular type of Kirchhoff condition at the vertices for the value function: this condition comes from an assumption which can be informally stated as follows: consider a vertex ν of the network and assume that it is the intersection of p edges Γ 1 , . . . , Γ p , ; if, at time τ , the controlled stochastic process X t associated to a given agent hits ν, then the probability that X τ `belongs to Γ i is proportional to the diffusion coefficient in Γ i . Under this assumption, it can be seen that the density of the distribution of states is continuous at the vertices of the network. In the present work, the above mentioned assumption is not made any longer. Therefore, it will be seen below that the value function satisfies more general Kirchhoff conditions, and accordingly, that the density of the distribution of states is no longer continuous at the vertices; the continuity condition is then replaced by suitable compatibility conditions on the jumps across the vertices. A complete study of the system of differential equations arising in infinite horizon mean field games on networks with at most quadratic Hamiltonians and very general coupling costs has been supplied in a previous work, see [START_REF] Achdou | A class of infinite horizon mean field games on networks[END_REF].

In the present work, we focus on the more basic case, namely finite horizon MFG with globally Lipschitz Hamiltonian with rather strong assumptions on the coupling cost. This will allow us to concentrate on the difficulties induced by the Kirchhoff conditions. Therefore, this work should be seen as a first and necessary step in order to deal with more difficult situations, for example with quadratic or subquadratic Hamiltonians. We believe that treating such cases will possible by combining the results contained in the present work with methods that can be found in [START_REF] Ladyzenskaja | Linear and quasilinear equations of parabolic type[END_REF][START_REF] Lieberman | Second order parabolic differential equations[END_REF]. Part two: Finite Horizon Games After obtaining the transmission conditions at the vertices for both the value function and the density, we shall prove existence and uniqueness of weak solutions of the uncoupled HJB and FP equations (in suitable space-time Sobolev spaces), and regularity results.

The present work is organized as follows: the remainder of Section 4.1 is devoted to setting the problem and obtaining the system of partial differential equations and the transmission conditions at the vertices. Section 4.2 contains useful results on a modified heat equation in the network with general Kirchhoff conditions. Section 4.3 is devoted to the Fokker-Planck equation. Weak solutions are defined by using a special pair of Sobolev spaces of functions defined on the network referred to as V and W below. Section 4.4 is devoted to the HJB equation supplemented with the Kirchhoff conditions: it addresses the main difficulty of the work, consisting of obtaining regularity results for the weak solution (note that, to the best of our knowledge, such results for networks and general Kirchhoff conditions do not exist in the literature). Finally, the proofs of the main results of existence and uniqueness for the MFG system of partial differential equations are completed in Section 4.5.

Networks and function spaces

The geometry A bounded network Γ (or a bounded connected graph) is a connected subset of R n made of a finite number of bounded non-intersecting straight segments, referred to as edges, which connect nodes referred to as vertices. The finite collection of vertices and the finite set of closed edges are respectively denoted by V :" tν i , i P Iu and E :" tΓ α , α P Au, where I and A are finite sets of indices contained in N. We assume that for α, β P A, if α " β, then Γ α X Γ β is either empty or made of a single vertex. The length of Γ α is denoted by α . Given ν i P V, the set of indices of edges that are adjacent to the vertex ν i is denoted by A i " tα P A : ν i P Γ α u. A vertex ν i is named a boundary vertex if 7 pA i q " 1, otherwise it is named a transition vertex. The set containing all the boundary vertices is named the boundary of the network and is denoted by BΓ hereafter.

The edges Γ α P E are oriented in an arbitrary manner. In most of what follows, we shall make the following arbitrary choice that an edge Γ α P E connecting two vertices ν i and ν j , with i ă j is oriented from ν i toward ν j : this induces a natural parametrization π α : r0, α s Ñ Γ α " rν i , ν j s:

π α pyq " α ´y α ν i `y α ν j for y P r0, α s. (4.1.1)
For a function v : Γ Ñ R and α P A, we define v α : p0, α q Ñ R by v α pxq :" v| Γα ˝πα pxq, for all x P p0, α q.

Remark 4.1.1. In what precedes, the edges have been arbitrarily oriented from the vertex with the smaller index toward the vertex with the larger one. Other choices are of course possible.

In particular, by possibly dividing a single edge into two, adding thereby new artificial vertices, it is always possible to assume that for all vertices ν i P V, either π α p0q " ν i , for all α P A i or π α p α q " ν i , for all α P A i .

Introduction and main results

Γ 1 ν 1 ν 2 Γ 2 ν 3 Γ 3 ν 4 Γ 4 Γ1 ν1 ν2 Γ2 ν5 Γ5 ν3 Γ3 ν4 Γ4

Function spaces related to the space variable

The set of continuous functions on Γ is denoted by CpΓq and we set P C pΓq :" # v : Γ Ñ R : for all α P A, ˇˇˇˇvα P Cp0, α q v α can be extended by continuity to r0, α s.

+ .

By the definition of piecewise continuous functions v P P CpΓq, for all α P A, it is possible to extend v| Γα by continuity at the endpoints of Γ α : if Γ α " rν i , ν j s, we set

v| Γα pxq " $ ' ' ' ' & ' ' ' ' % v α `π´1 α pxq ˘, if x P Γ α zV, v α p0q :" lim yÑ0 `vα pyq , if x " ν i , v α p α q :" lim yÑ ά v α pyq , if x " ν j . (4.1.3)
For m P N, the space of m-times continuously differentiable functions on Γ is defined by C m pΓq :" tv P C pΓq : v α P C m pr0, α sq for all α P Au , and is endowed with the norm }v} C m pΓq :" max αPA max 0ďkďm }B k v α } L 8 p0, αq . For σ P p0, 1q, the space C m,σ pΓq, contains the functions v P C m pΓq such that B m v α P C 0,σ pr0, α sq for all α P A; it is endowed with the norm

}v} C m,σ pΓq :" }v} C m pΓq `sup αPA sup y‰z y,zPr0, αs |B m v α pyq ´Bm v α pzq| |y ´z| σ .
For a positive integer m and a function v P C m pΓq, we set for k ď m,

B k v pxq " B k v α `π´1 α pxq ˘if x P Γ α zV. (4.1.4)
Notice that v P C k pΓq is continuous on Γ but that the derivatives B l v, 0 ă l ď k are not defined at the vertices. For a vertex ν, we define B α v pνq as the outward directional derivative of v| Γα at ν as follows: For all i P I and α P A i , setting

B α v pνq :" $ ' & ' % lim hÑ0 `vα p0q ´vα phq h , if ν " π α p0q , lim hÑ0 `vα p α q ´vα p α ´hq h , if ν " π α p α q . ( 4
n iα " # 1 if ν i " π α p α q, ´1 if ν i " π α p0q, (4.1.6)
we have B α vpν i q " n iα Bv| Γα pν i q " n iα Bv α pπ ´1 α pν i qq. We say that v is Lebesgue-integrable on Γ α if v α is Lebesgue-integrable on p0, α q. In this case, for all x 1 , x 2 P Γ α ,

ż rx 1 ,x 2 s v pxq dx :" ż π ´1 α px 2 q π ´1 α px 1 q v α pyq dy. ( 4
ÿ αPA › › ›B k v α › › › p L p p0, αq `}v} p L p pΓq ¸1 p .
For s P Nzt0u, we also set H s b pΓq " W s,2 b pΓq and H s pΓq " CpΓq X H s b pΓq.

Finally, when dealing with probability distributions in mean field games, we will often use the set M of probability densities, i.e., m P L 1 pΓq, m ě 0 and ş Γ mpxqdx " 1.

Some space-time function spaces

The space of continuous real valued functions on Γ ˆr0, T s is denoted by CpΓ ˆr0, T sq.

Let P CpΓ ˆr0, T sq be the space of the functions v : Γ ˆr0, T s Ñ R such that 1. for all t P r0, T s, vp¨, tq belongs to P CpΓq 2. for all α P A, v| Γαˆr0,T s is continuous on Γ α ˆr0, T s;

Introduction and main results

For a function v P P CpΓ ˆr0, T sq, α P A, we set v α px, tq " v| Γαˆr0,ts pπ α pxq, tq for all px, tq P r0, α s ˆr0, T s.

For two nonnegative integers m and n, let C m,n pΓ ˆr0, T sq be the space of continuous real valued functions v on Γ ˆr0, T s such that for all α P A, v| Γαˆr0,T s P C m,n pΓ α ˆr0, T sq. For σ P p0, 1q, τ P p0, 1q, we define in the same manner C m`σ,n`τ pΓ ˆr0, T sq Useful results on continuous and compact embeddings of space-time function spaces are given in Appendix 4.6.

A class of stochastic processes on Γ

After rescaling the edges, it may be assumed that α " 1 for all α P A. Let µ α , α P A and p iα , i P I, α P A i be positive constants such that ř αPA i p iα " 1. Consider also a real valued function a P P CpΓ ˆr0, T sq, such that, for all α P A, t P r0, T s, a| Γα p¨, tq belongs to C 1 pΓ α q.

As in Remark 4.1.1, we make the assumption (4.1.2) by possibly adding artificial nodes: if ν i is such an artificial node, then 7pA i q " 2, and we assume that p iα " 1{2 for α P A i . The diffusion parameter µ has the same value on the two sides of an artificial vertex. Similarly, the function a does not have jumps across an artificial vertex.

Consider a Brownian motion pW t q defined on the real line. Following Freidlin & Sheu ([55]), we know that there exists a unique Markov process on Γ with continuous sample paths that can be written pX t , α t q where X t P Γ αt (if X t " ν i , i P I, α t is arbitrarily chosen as the smallest index in A i ) such that, defining the process x t " π αt pX t q with values in r0, 1s,

• dx t " a 2µ αt dW t `aαt px t , tqdt `d i,t `dh i,t , (4.1.9)
• i,t is continuous non-decreasing process (measurable with respect to the σ-field generated by pX t , α t q) which increases only when X t " ν i and x t " 0,

• h i,t is continuous non-increasing process (measurable with respect to the σ-field generated by pX t , α t q) which decreases only when X t " ν i and x t " 1, and for all function v P C For what follows, it will be convenient to set D :"

# u P C 2 pΓq : ÿ αPA i p iα B α u pν i q " 0, for all i P I + . ( 4

Introduction and main results

where H α :" H| ΓαˆR . At a vertex ν i , i P I, the transmission conditions for both v and m consist of d ν i " 7pA i q linear relations, which is the appropriate number of relations to have a well posed problem. If ν i P BΓ, there is of course only one condition.

Assumptions and main results

Before giving the precise definition of solutions of the MFG system (4.1.24) and stating our result, we need to introduce some suitable functions spaces.

Function spaces related to the Kirchhoff conditions

The following function spaces will be the key ingredients in order to build weak solutions of (4.

$ ' ' & ' ' % ϕ α is affine on p0, α q , ϕ| Γα pν i q " γ iα , if α P A i ,
ϕ is constant on the edges Γ α which touch the boundary of Γ. Note that ϕ is positive and bounded. We set ϕ " max Γ ϕ, ϕ " min Γ ϕ. Remark 4.1.8. One can see that v P V Þ ÝÑ vϕ is an isomorphism from V onto W and w P W Þ ÝÑ wϕ ´1 is the inverse isomorphism. Definition 4.1.9. Let the function space W Ă W be defined as follows:

W :" " m : Γ Ñ R : m α P C 1 pr0, α sq and m| Γα pν i q γ iα " m| Γ β pν i q γ iβ for all i P I, α, β P A i * .

(4.1.27)

Running assumptions

(Diffusion constants) pµ α q αPA is a family of positive numbers.

(Jump coefficients) pγ iα q αPA i is a family of positive numbers such that

ÿ αPA i γ iα µ α " 1.
(Hamiltonian) The Hamiltonian H is defined by the collection H α :" H| ΓαˆR , α P A: we assume that We propose to send n to `8 and show that a subsequence of tv n u converges to a solution of (4.2.1). Hence, we need some uniform estimates for tv n u.

H α P C 1 pΓ α ˆRq , ( 4

Lemma 4.2.3.

There exists a constant C depending only on Γ, pµ α q αPA , T and ϕ such that

}v n } L 8 p0,T ;L 2 pΓqq `}v n } L 2 p0,T ;V q `}B t v n } L 2 p0,T ;W 1 q ď C ´}h} L 2 p0,T ;W 1 q `}v T } L 2 pΓq ¯. Proof of
ş Γ v 2 n px, T q ϕdx is bounded by ϕ ş Γ v 2 T dx from (4.2.6), it follows that ż Γ v 2 n px, sqϕdx `ż T s ż Γ v 2 n ϕdxdt `ż T s ż Γ µpBv n q 2 ϕdxdt ď 2e λT ˆC2 µ }h} 2 L 2 p0,T ;W 1 q `ϕ ż Γ v 2 T dx ˙. ( 4

.2.9)

Estimate of v n in L 8 `0, T ; L 2 pΓq ˘and L 2 p0, T ; V q. From (4.2.9), it is straightforward to see that

}v n } L 8 p0,T ;L 2 pΓqq `}v n } L 2 p0,T ;V q ď C ´}h} L 2 p0,T ;W 1 q `}v T } L 2 pΓq ¯, ( 4
} W ď C }w} W , it follows that }B t v n ptq} W 1 ď C p}h ptq} W 1 `µ }v n ptq} V q ,
for almost every t, and therefore, from (4.2.10), we obtain

}B t v n ptq} 2 L 2 p0,T ;W 1 q ď C ´}h} 2 L 2 p0,T ;W 1 q `}v T } 2 L 2 pΓq ¯,
for a constant C independent of n.

Theorem 4.2.4.

There exists a unique solution v of (4.2.1), which satisfies

}v} L 8 p0,T ;L 2 pΓqq `}v} L 2 p0,T ;V q `}B t v} L 2 p0,T ;W 1 q ď C ´}h} L 2 p0,T ;W 1 q `}v T } L 2 pΓq ¯, (4.2.12)
where C is a constant that depends only on Γ, pµ α q αPA , T and ϕ.

Proof of Theorem 4.2.4. From Lemma 4.2.3, the sequence pv n q nPN is bounded in L 2 p0, T ; V q and the sequence pB t v n q nPN is bounded in L 2 p0, T ; W 1 q. Hence, up to the extraction of a subsequence, there exists a function v such that v P L 2 p0, T ; V q , B t v P L 2 p0, T ; W 1 q and

# v n á v weakly in L 2 p0, T ; V q , B t v n á B t v weakly in L 2 p0, T ; W 1 q . (4.2.13)
Fix an integer N and choose a function v P C 1 pr0, T s ; V q having the form

v ptq " N ÿ k"1 d k ptq v k , ( 4
pΓq ď C `}B t v} L 2 p0,T ;W 1 q `}v} L 2 p0,T ;V q ˘.
We are now going to prove v pT q " v T . For all v P C 1 pr0, T s ; V q of the form (4. We know that v n pT q Ñ v T in L 2 pΓq. Then, using (4.2.13), we obtain

ż Γ v pT q v T ϕdx " ż Γ v pT q v pT q ϕdx.
Since the functions of the form ř N k"1 d k pT q v k are dense in L 2 pΓq, we conclude that v pT q " v T . In order to prove the energy estimate (4.2.12), we use ve λt ϕ as a test function in (4.2.2) and apply similar arguments as in the proof of Lemma 4.2.3 for λ large enough, we get (4.2.12).

Finally, if h " 0 and v T " 0, by the energy estimate for v in (4.2.12), we deduce that v " 0. Uniqueness is proved. Theorem 4.2.5. If v T P V and h P L 2 pΓ ˆp0, T qq, then the unique solution v of (4.2.1) satisfies v P L 2 `0, T ; H 2 pΓq ˘X Cpr0, T s; V q and B t v P L 2 `0, T ; L 2 pΓq ˘. Moreover, }v} L 8 p0,T ;V q `}v} L 2 p0,T ;H 2 pΓqq `}B t v} L 2 p0,T ;L 2 pΓqq ď C ´}h} L 2 p0,T ;L 2 pΓqq `}v T } V ¯, (4.2.17) for a positive constant C that depends only on Γ, pµ α q αPA , T and ϕ. Using similar arguments as in Section 4.2, in particular a Galerkin method, we obtain the following result, the proof of which is omitted. Theorem 4.3.2. If b P L 8 pΓ ˆp0, T qq and m 0 P L 2 pΓq, there exists a unique function m P L 2 p0, T ; W q X Cpr0, T s; L 2 pΓqq such that B t m P L 2 p0, T ; V 1 q and (4.3.2). Moreover, there exists a constant C which depends on pµ α q αPA , }b} 8 , T and ϕ, such that }m} L 2 p0,T ;W q `}m} L 8 p0,T ;L 2 pΓqq `}B t m} L 2 p0,T ;V 1 q ď C }m 0 } L 2 pΓq .

(4.3.3) Remark 4.3.3. If m 0 P M, which will be the case when solving the MFG system (4.1.24), then mp¨, tq P M for all t P r0, T s. Indeed, we use v " 1 P V as a test-function for (4.3.1). Since Bv " 0, integrating (4.3.2) from 0 to t, we get ş t 0 ş Γ B t mpx, sqdxds " 0. This implies that ż Γ mpx, tqdx " ż Γ m 0 pxqdx " 1, for all t P p0, T s.

Setting m

´" ´1tmă0u m, we can also use v " ϕ ´1m ´e´λt as a test-function for λ P R `. Indeed, the latter function belongs to L 2 p0, T ; V q. Taking λ large enough and using similar arguments as for the energy estimate (4.3.3) yield that m ´" 0, i.e., m ě 0.

We end this section by stating a stability result, which will be useful in the proof of the main Theorem. The sequence pm ε q converges to m in L 2 p0, T ; W qXL 8 `0, T ; L 2 pΓq ˘, and the sequence pB t m ε q converges to pB t mq in L 2 p0, T ; V 1 q.

Proof of Lemma 4.3.4 From the previous proof for bounded Hamiltonians, for all n, there exists a solution v n P L 2 `0, T ; H 2 pΓq ˘X Cpr0, T s; V q X W 1,2 `0, T ; L 2 pΓq ˘of (4.4.1), where H is replaced by H n . We propose to send n to `8 and to show a subsequence of tv n u converges to a solution of (4.4.1). Hence, we need some uniform estimates for tv n u. As in the proof of Proposition 4.4.3, using ´vn e λt ϕ as a test-function, integrating from 0 to T and noticing that H is sublinear, see In the following lines, the constant C above will vary from line to line and will depend only on pµ α q αPA , C H , T and ϕ. Taking λ large enough leads to the following estimate: Therefore, tH n px, Bv n q ´f u is uniformly bounded in L 2 `0, T ; L 2 pΓq ˘. From Theorem 4.2.5, we obtain that pv n q nPN is uniformly bounded in L 2 `0, T ; H 2 pΓq ˘XCpr0, T s; V qXW 1,2 `0, T ; L 2 pΓq ˘. By the Aubin-Lions theorem (see Lemma 4.6.1), pv n q n is relatively compact in L 2 p0, T ; V q (and bounded in C pr0, T s; V q). Hence, up to the extraction of a subsequence, there exists v P L 2 p0, T ; V q X W 1,2 `0, T ; L 2 pΓq ˘such that v n Ñ v, in L 2 p0, T ; V q (strongly), B t v n á B t v, in L 2 pΓ ˆp0, T qq (weakly). (4.4.8)

}v n } L 2 p0,T ;V q ď C ´}f } L 2 p0
Hence, H n px, Bv n q Ñ H px, Bvq a.e. in Γ ˆp0, T q. Note also that we can apply Lebesgue dominated convergence theorem to H n px, Bv n q because H n px, Bv n q ď H px, Bv n q ď C 0 p1`|Bv n |q.

Therefore, H n px, Bv n q Ñ H px, Bvq in L 2 pΓ ˆp0, T qq. Thus, it is possible to pass to the limit in the weak formulation satisfied by v n and obtain that for all w P W , χ P C c p0, T q, Therefore, v satisfies (4.4.2). From Theorem 4.2.4, v n pT q " v T for all n. Since for all α P A, pv n q n tends to v in L 2 pΓ α p0, T qq strongly and in W 1,2 pΓ α ˆp0, T qq weakly, v n | Γαˆtt"T u converges to v| Γαˆtt"T u in L 2 pΓ α q strongly. Passing to the limit in the latter identity, we get (4.4.3). We have proven that v is a weak solution of (4.4.1).

We end the section with a stability result for the Hamilton-Jacobi equation. Lemma 4.4.4. Let pv T ε q ε , pf ε q ε be sequences of functions satisfying

v T ε ÝÑ v T in V, f ε ÝÑ f in L 2 pΓ ˆp0, T qq .
Let v ε be the weak solution of (4.4.1) with data v T ε , f ε , then pv ε q ε converges in L 2 `0, T ; H 2 pΓq ˘X Cpr0, T s; V q X W 1,2 `0, T ; L 2 pΓq ˘to the weak solution v of (4.4.1) with data v T , f . Proof of Lemma 4.4.4. Subtracting the two PDEs for v ε and v, multiplying by pv ε ´vq e λt ϕ ´1, taking the integral on Γ ˆp0, T q and using similar computations as in the proof of Proposition 4.4.3, we obtain }v ε ´v} L 2 p0,T ;V q ď C ´}f ε ´f } L 2 pΓˆp0,T qq `}v T ε ´vT } L 2 pΓq ¯, for λ large enough and C independent of ε. This proves the convergence of v to v in L 2 p0, T ; V q. Then, the convergence in L 2 `0, T ; H 2 pΓq ˘X Cpr0, T s; V q X W 1,2 `0, T ; L 2 pΓq ˘results from the assumption that H is Lipschitz with respect to its second argument, and from stability results for the linear boundary value problem (4.2.1) which are obtained with similar arguments as in the proof of Theorem 4.2.5.

The Hamilton-Jacobi equation

Proof of Lemma 4.4.13. The proof follows the same lines as the one of Lemma 4.2.2 but it is more technical since we obtain a system of nonlinear differential equations. For n ě 1, we consider the symmetric n by n matrix M n defined by

pM n q k " ż Γ f k f ψdx.
Since ψ is positive and bounded and since pψf k q 8

k"1 is a Hilbert basis of L 2 pΓq, we can check that M n is a positive definite matrix and there exist two constants c, C independent of n such that where

c |ξ| 2 ď n ÿ k, "1 pM n q k ξ k ξ ď C |ξ| 2 ,
• B k " ş Γ µBf Bpψf k qdx • H i pY q " ´şΓ
Hpx, Y T F qBpf i ψqdx with F " pf 1 , ¨¨¨, f n q T and Y T F " ř y n f " u n

• G i ptq " ´şΓ f px, tqBpf i ψqdx for all i P 1, ¨¨¨, n.

Since the matrix M is invertible and the function H is Lipschitz continuous by (4.1.31), the system (4.4.25) has a unique global solution. This ends the proof of the lemma.

We start by giving some estimates for the approximation u n . }u n } L 8 p0,T ;F q `}u n } L 2 p0,T ;H 2 b pΓqq `}B t u n } L 2 pΓˆp0,T qq ď C. Proof of Lemma 4.4.14. We divide the proof into two steps:

Step 1: Uniform estimates of u n in L 8 p0, T ; L 2 pΓqq, L 2 p0, T ; F q and W 1,2 p0, T ; E 1 q. Multiplying (4.4.23) by y n k ptq f k e λt ψ where λ is a positive constant to be chosen later, summing for k " 1, . . . , n and using (4.4.21), we get ´żΓ B t u n u n e λt ψdx `żΓ ´µBu n ´Hpx, u n q ¯B ´un e λt ψ ¯dx " ´żΓ f Bpu n ψe λt qdx.

In the following lines, C will be a constant that may vary from lines to lines. Since H satisfies (4. 1.30) Let J iα ppq be the primitive function of p Þ Ñ H α pν i , pq such that J iα p0q " 0: Let us end the proof by proving (4.4.32). The function φ " u n | Γα p¨, tq is in H 1 pΓ α q. By the continuous embedding H 1 pΓ α q ãÑ CpΓ α q, we can define φ in the pointwise sense (and even at two endpoints of any edges, see Part two: Finite Horizon Games so up to the extraction of a subsequence, we may assume that u n Ñ u in L 2 p0, T ; F q and almost everywhere. Moreover, from the compactness of the trace operator from W 1,2 pΓ α ˆp0, T qq to L 2 pBΓ α ˆp0, T qq, u n | BΓαˆp0,T q Ñ u| BΓαˆp0,T q in L 2 pBΓ α ˆp0, T qq and for almost every t P p0, T q.

H α pν i ,
Similarly, u n | Γαˆtt"T u Ñ u| Γαˆtt"T u in L 2 pΓ α q and almost everywhere in Γ α . Then, using the Lipschitz continuity of H with respect to its second argument, and similar arguments as in the proof of Theorem 4.2.4, we obtain the existence of a solution of (4.4.12) satisfying (4.4.18) by letting n Ñ `8. Since H 2 pΓ α q Ă C 1`σ pΓ α q for some σ P p0, 1{2q, up¨, tq P C 1`σ pΓ α q for all α P A and a.a. t. Finally, the proof of uniqueness is a consequence of the energy estimate (4.4.18) for u.

Next, we want to prove that, if u is the solution of (4.4.12) and v is the solution (4.4.1), then Bv " u. It means that we have to define a primitive function on the network Γ. Proof of Lemma 4.4.16. First, it is sufficient to prove ş L u pζ, tq dζ " 0 for all L P Ý Ñ xx. Secondly, if a given edge is browsed twice in opposite senses, the two related contributions to the integral sum to zero. It follows that, without loss of generality, we only need to consider loops in Ý Ñ xx such that all the complete edges that it contains are browsed once only. It is also easy to see that we can focus on the case when x P V. To summarize, we only need to prove that ż L u pζ, tq dζ " 0 when ν i 0 P VzBΓ and L " ν i 0 Ñ ν i 1 Ñ . . . Ñ ν im Ñ ν i 0 , where ν i k ‰ ν i for k " l.

The following conditions 1. e| Γα " 0 on each edge Γ α not contained in L 2. for all k " 0, . . . m ´1, e| Γα k " 1 i k ăi k`1 ´1i k ąi k`1 if Γ α k is the edge joining ν i k and ν i k`1 3. e| Γα m " 1 imăi 0 ´1imąi 0 if Γ αm is the edge joining ν im and ν i 0

5 Γ 1 Γ 2 Γ 5 Γ 3 Γ 4 OFigure 0 . 1 :

 51253401 Figure 0.1: À gauche un réseau géneral Γ; à droite, une jonction G.

Chapter 1 .

 1 Introduction lead to the following ergodic HJ equation on G

3 . 18 )

 318 without jump condition of density m. The first result in Chapter 4 also explains where the MFG system (1.3.13) comes from, see Section 1.3.1 or Sections 4.1.2-4.1.3.

Figure 2 . 2 :

 22 Figure 2.2: Left: The value function v with entry cost c 2 ě 1{λ " 4. Right: The value function v with entry cost c 2 " 2 ă 1{λ " 4.

  By letting δ tend to 0, v pOq `εn ě ż tn 0 py O,αn psq , α n psqq e ´λs ds `cipnq e ´λtn `e´λtn v ipnq pOq . Note that y O,αn psq " O for all s P r0, t n s, i.e., f pO, α n psqq " 0 a.e. s P r0, t n q. Hence v pOq `εn ě ż tn 0 pO, α n psqq e ´λs ds `cipnq e ´λtn `e´λtn v ipnq pOq ě ż tn 0 `´H T O ˘e´λs ds `cipnq e ´λtn `e´λtn v ipnq pOq " 1 ´e´λtn λ `´H T O ˘`c ipnq e ´λtn `e´λtn v ipnq pOq .

Figure 3 . 1 :

 31 Figure 3.1: Left: the network Γ in which the edges are oriented toward the vertex with larger index (4 vertices and 4 edges). Right: a new network Γ obtained by adding an artificial vertex (5 vertices and 5 edges): the oriented edges sharing a given vertex ν either have all their starting point equal ν, or have all their terminal point equal ν.

Chapter 3 .

 3 A Class of Mean Field Games on Networks. Part One: the Ergodic Case

  }v} C m,σ pΓq :" }v} C m pΓq `sup αPA sup y‰z y,zPr0, αs |B m v α pyq ´Bm v α pzq| |y ´z| σ .

( 3 . 1 . 7 )

 317 Remark 3.1.2. Note that in (3.1.5), changing the orientation of the edge does not change the value of B α vpνq.

.1. 9 ) 3 . 1 . 4 .

 9314 Remark Note that in the definition of D pLq, the condition at boundary vertices boils down to a Neumann condition. Chapter 3. A Class of Mean Field Games on Networks. Part One: the Ergodic Case 1. The operator L is the infinitesimal generator of a Feller-Markov process on Γ with continuous sample paths. The operators L α and the transmission conditions at the vertices

( 3 . 1 . 14 )When λ 0 ě µ α 2 `µα Bψ 2 2ψ 2

 3114222 Chapter 3. A Class of Mean Field Games on Networks. Part One: the Ergodic Case for all α P A, we obtain that

3. 2 .Theorem 3 . 2 . 13 .

 23213 Preliminary: A class of linear boundary value problems For any b P P C pΓq,

( 3 .

 3 2.23)A weak solution of (3.2.23) is a function v P V such that T λ 0 pv, wq :" ÿ αPA ż Γα pλ 0 vw `µα BvBw `bwBvqdx " ż Γ gwdx, for all w P W.

  2.13) with b " 0. It yields BwBudx " 0, for all u P V, ş Γ wdx " 1.

FΓFΓF

  n pm n q wdx " ż pmq wdx, for all w P W.From Step 3, up to the extraction of subsequence, there exists v P C 1 pΓq such that v n Ñ v in C 1 pΓq. Hence, pv, ρ, mq satisfies the weak form of the MFG system:ÿ αPA ż Γα µ α BvBwdx `żΓ pH px, Bvq `ρq wdx " ż pmq wdx, for all w P W, and ÿ αPA ż Γα µ α BmBṽdx `żΓ B p H px, Bvq mBṽdx " 0, for all ṽ P V.

( 4 . 1 . 2 )

 412 This idea was used by Von Below in[START_REF] Below | Classical solvability of linear parabolic equations on networks[END_REF]: some edges of Γ are cut into two by adding artificial vertices so that the new oriented network Γ has the property (4.1.2), see Figure4.1 for an example.

Figure 4 . 1 :

 41 Figure 4.1: Left: the network Γ in which the edges are oriented toward the vertex with larger index (4 vertices and 4 edges). Right: a new network Γ obtained by adding an artificial vertex (5 vertices and 5 edges): the oriented edges sharing a given vertex ν either have all their starting point equal ν, or have all their terminal point equal ν.

.1. 5 )

 5 Chapter 4. A Class of Mean Field Games on Networks. Part two: Finite Horizon Games

( 4 . 1 . 7 ) 4 . 1 . 2 .

 417412 Remark Changing the orientation of the edge does not change the value of B α vpνq in (4.1.5).

.1. 13 )

 13 Remark 4.1.4. Note that in (4.1.10), the condition at boundary vertices boils down to a Neumann condition.

4. 3 .Definition 4 . 3 . 1 .

 3431 The Fokker-Planck equation where b P P C pΓ ˆr0, T sq and m 0 P L 2 pΓq. A weak solution of (4.3.1) is a function m P L 2 p0, T ; W q X Cpr0, T s; L 2 pΓqq such that B t m P L 2 p0, T ; V 1 q and $ & % xB t m, vy V 1 ,V `A pm, vq " 0 for all v P V and a.e. t P p0, T q, m p¨, 0q " m 0 , (4.3.2) where A : W ˆV Ñ R is the bilinear form A pv, wq "

Lemma 4 . 3 . 4 .

 434 Let m 0ε , b ε be sequences of functions satisfyingm 0ε ÝÑ m 0 in L 2 pΓq , b ε ÝÑ b in L 2 pΓ ˆp0, T qq ,and for some positive number K independent of ε, }b} L 8 pΓˆp0,T qq ď K, }b ε } L 8 pΓˆp0,T qq ď K.Let m ε (respectively m) be the solution of (4.3.2) corresponding to the datum m 0ε (resp. m 0 ) and the coefficient b ε (resp. b).

2 nH

 2 e λt ϕ `µ |Bv n | 2 e λt ϕ `µBv n v n e λt Bϕ n px, Bv n q v n e λt ϕdxdt `n |q |v n | e λt ϕdxdt `

Γ 2 0 2 L 2

 222 ,T ;L 2 pΓqq `}v T } L 2 pΓq `1¯, (4.4.7) 4.4. The Hamilton-Jacobi equation and thus, from (4.1.30) again, we also obtain ż T 0 ż |H n px, Bv n q| 2 dxdt ď ´|Bv n | 2 `1¯d xdt ď C ´}f } 2 L 2 p0,T ;L 2 pΓqq `}v T } pΓq `1¯.

Lemma 4 . 4 . 14 .

 4414 Under the assumptions made in Theorem 4.4.11, there exists a constant C depending only on Γ, T , ψ, }u T } F , }Bf } L 2 pΓˆp0,T qq }f } CpΓˆr0,T sq and }B t f } L 2 p0,T ;H 1 b pΓqq such that

  (4.1.3)). For all α P A and x, y P Γ α , we have φpxq " φpyq `şry,xs Bφpξqdξ. It follows |Γ α |φpxq " first estimate setting x " ν i . The second estimate is obtained in the same way replacing φ by φ 2 and using the fact that W 1,1 pΓ α qss is continuously embedded in CpΓ α q.Proof of Theorem 4.4.11. From Lemma 4.4.14, up to the extraction of a subsequence, thereexists u P L 2 `0, T ; H 2 b pΓq ˘X W 1,2 pΓ ˆp0, T qq such that $ & % u n á u, in L 2 `0, T ; F X H 2 b pΓq ˘,B t u n á B t u, in L 2 pΓ ˆp0, T qq .

( 4 . 4 . 33 )

 4433 Moreover, by Aubin-Lions Theorem (see Lemma 4.6.1),L 2 `0, T ; F X H 2 b pΓq ˘X W 1,2 `0, T ; L 2 pΓq ˘compact ãÑ L 2 p0, T ; F q ,Chapter 4. A Class of Mean Field Games on Networks.

  viscosité du système suivant λu i pxq `Hi ˆx, est le hamiltonien correspondant l'arête Γ i . Le point important est l'expression du hamiltonien en O, qui prend en compte toutes les stratégiés possibles dans voisinage de O. Plus précisément, si la trajectoire est proche de O et appartient à Γ i alors:

	λu i pOq `max	" ´λ min j‰i	du i dx i tu j pOq `cj u , H pxq ˙" 0 ì ˆO, du i dx i	O pOq ˙, H T	*	" 0	si x P Γ i z tOu, si x " O,	(0.0.1)
	où H i

  .2: Left: The value function u with λ " 1{4. Right: The value function v with entry cost c 2 " 2 and λ " 1{4.

	Theorem 1.2.2 (Theorem 2.2.9 and Lemma 2.2.11). Under assumption rHs,
			"		*
	max i"1,N	tv i pOqu ď v pOq " min	min i"1,N	tv i pOq `ci u ,	´HT O λ	,	(1.2.4)

  a fixed time τ ą 0 such that for any x P pΓ i z tOuq and near O, we can find an "almost optimal" control which makes the cost functional close to vpOq. Moreover, with this control, the trajectory starts at x still remains on Γ i z tOu in the time interval r0, τ s. Using this property and applying [89, the proof of Lemma 2.2], we deduce the viscosity super-inequality for vpxq and H ì px, pq where x P Γ i z tOu is near O. Let x Ñ O and the proof for supersolution is done.

	1.2. Optimal control problems on networks and Hamilton-Jacobi equations on networks
	• The term H	ì ˆO,	du i dx i	pOq ˙accounts for situations in which the trajectory does not leave
	Γ i .
	• The term H T O accounts for situations in which the trajectory stays at O.
	Sketch of proof of Theorem 1.2.3. From (1.2.4) and (1.2.5), it suffices to prove that v i pOq satis-
	fies	λv i pOq `Hì ˆO,	dv i dx i	pOq ˙ď 0, in the viscosity sense.
	Let a P A ì such that f i pO, aq ą 0. For all x P Γ i and near O, using the strong controllability at
	O, we get
	λv i pxq ´fi px, aq	dv i dx i	px, aq ´ i px, aq ď 0, in the viscosity sense,
	Let x Ñ O and the proof for subsolution is done.
	The characterization for supersolutions is more difficult. We need to prove the following
	property (which is Lemma 2.4.3 in Chapter 2): if
							"	*
			v i pOq ă min	min j‰i	tv j pOq `cj u ,	´HT O λ	,	(1.2.7)
	then there exist The main result of Chapter 2 is
	Theorem 1.2.4 (Comparison Principle, Theorem 2.5.6

  ia B i v pO, tq " 0, B i v pν i , tq " 0, i " 1, N , t P r0, T s ,

	Chapter 1. Introduction
	and for all function v P C 2,1 pG ˆr0, T sq such that
	N
	ÿ
	i"1
	the process
	M t " vpX t , tq	´ż t	`Bt vpX s , sq `µis B 2 vpX s , sq `ais pX s , sqBvpX s , sq ˘ds	(1.3.5)
		0
	is a martingale, namely
	1. a one dimensional Wiener process W t ,
	2. continuous non-decreasing processes i,t , i " 1, N , which are measurable with respect to
	the σ-field generated by pX t , i t q,
	3. continuous non-increasing processes h i,t , i " 1, N , which are measurable with respect to
	the σ-field generated by pX t , i t q, such that
		dx t "	a 2µ it dW t `ait px t qdt `d i,t `dh i,t ,	(1.3.4)
				i,t increases only when X t " O,
				h i,t decreases only when X t " ν i ,

p

  tq m px, tq dx, for all v P P C pG ˆr0, T sq . `Bt upX s , sq `µis B 2 upX s , sq `ais pX t , tqBupX t , tq ˘ .

			(1.3.7)
	Consider u P C 2,1 pG ˆr0, T sq such that for all t P r0, T s, u p¨, tq P D pLq. Then from (1.3.5)-
	(1.3.6), we see that
	E rupX t , tqs " E rupX 0 , 0qs	`E "ż t
		0
	Taking the time-derivative of each member of (1.3.7), we obtain
	ż
	B t pumq px, tq dx " E `Bt upX t , tq `µis B 2 upX t , tq `ais pX t , tqBupX t , tq ˘.
	G
	Using again (1.3.7), we get
	ż	ż
	`µB 2 upx, tq `apx, tqBupx, tq ˘mpx, tqdx "	upx, tqB t m px, tq dx.
	G	G

  y belong to the same egde Γ i , |x| `|y| if x, y belong to different edges Γ i and Γ j .

  Remark 2.2.1. The assumption that the sets A i are disjoint is not restrictive. Indeed, if A i are not disjoint, then we define Ãi " A i ˆtiu and fi px, ãq " f i px, aq , ˜ i px, ãq "The function f is continuous on M since the sets A i are disjoint. (The speed set and the admissible control set). The set F pxq which contains all the "possible speeds" at x is defined by

	Definition 2.2.2

is continuous and bounded by M . Moreover, there exists L ą 0 such that |f i px, aq ´fi py, aq| ď L |x ´y| for all x, y P Γ i , a P A i .

Hereafter, we will use the notation F i pxq for the set tf i px, aq e i : a P A i u.

rH2s (Running costs) For i " 1, N , the function i : Γ i ˆAi Ñ R is a continuous function bounded by M ą 0. There exists a modulus of continuity ω such that | i px, aq ´ i py, aq| ď ω p|x ´y|q for all x, y P Γ i , a P A i .

rH3s (Convexity of dynamic and costs) For x P Γ i , the following set FL i pxq " tpf i px, aq e i , i px, aqq : a P A i u is non-empty, closed and convex.

rH4s (Strong controllability)

There exists a real number δ ą 0 such that r´δe i , δe i s Ă F i pOq " tf i pO, aq e i : a P A i u . i px, aq with ã " pa, iq with a P A i . The assumption rH3s is made to avoid the use of relaxed control. With assumption rH4s, one gets that the Hamiltonian which will appear later is coercive for x close to the O. Moreover, rH4s is an important assumption to prove Lemma 2.2.7 and Lemma 2.5.3.

2.2. Optimal control problem on junction with entry/exit costs

Let

M " px, aq : x P G, a P A i if x P Γ i z tOu , and a P Y

N i"1 A i if x " O ( .

Then M is closed. We also define the function on M by for all px, aq P M, f px, aq "

$ & % f i px, aq e i if x P Γ i z tOu and a P A i , f i pO, aq e i if x " O and a P A i .

  x,α , αqq "

		ż `8	py x,α ptq , α ptqq e ´λt dt`N ÿ	ÿ	c i e ´λt ik (cost functional with entry cost),
		0			i"1	kPK i
	and
	Ĵ px; py x,α , αqq "	ż `8	py x,α ptq , α ptqq e ´λt dt`N ÿ	ÿ	d i e ´λη ik (cost functional with exit cost),
		0			i"1	kPK i
	where the running cost : M Ñ R is
				$
				&
			px, aq "
				%
		J px, αq ě	´M λ	``8 ÿ k"1

i px, aq if x P Γ i z tOu and a P A i , i pO, aq if x " 0 and a P A i .

Hereafter, to simplify the notation, we will use J px, αq and Ĵ px, αq instead of J px; py x,α , αqq and Ĵ px; py x,α , αqq, respectively. The value functions of the infinite horizon optimal control problem are defined by:

v pxq " inf pyx,

α,αqPTx J px; py x,α , αqq (value function with entry cost), and v pxq " inf pyx,α,αqPTx Ĵ px; py x,α , αqq (value function with exit cost). Remark 2.2.5. By the definition of the value function, we are mainly interested in a control law α such that J px, αq ă `8. In such a case, if |K i | " `8, then we can order tt ik , η ik : k P Nu such that t i1 ă η i1 ă t i2 ă η i2 ă . . . ă t ik ă η ik ă . . . , and lim kÑ8 t ik " lim kÑ8 η ik " `8. Indeed, assuming if lim kÑ8 t ik " t ă `8, then

  Under assumption rHs, v| Γ i ztOu and v| Γ i ztOu are continuous for any i " 1, N . Moreover, there exists ε ą 0 such that v| Γ i ztOu and v| Γ i ztOu are Lipschitz continuous in pΓ i z tOuqX B pO, εq. Hence, it is possible to extend v| Γ i ztOu and v| Γ i ztOu at O into Lipschitz continuous functions in Γ i X B pO, εq. Hereafter, v i and vi denote these extensions.Proof of Lemma 2.2.8. The proof of continuity inside the edge is classical by using rH4s, see[START_REF] Achdou | Hamilton-Jacobi equations on networks[END_REF] for more details. The proof of Lipschitz continuity is a consequence of Lemma 2.2.7. Indeed, for x, y belong to Γ

	Lemma 2.2.8.

i X B p0, εq, by Lemma 2.2.7 and the definition of value function, we have

v pxq ´v pzq " v i pxq ´vi pzq ď ż τx,z 0 i `yx,αx,z ptq , α x,z ptq ˘e´λt dt `vi pzq ´e´λτx,z ´1¯. Since i is bounded by M from rH2s, v i is bounded in Γ i X B pO,

εq and e ´λτx,z ´1 is bounded by τ x,y , there exists a constant C such that v i pxq ´vi pzq ď Cτ x,z ď CC |x ´z| . The last inequality follows from the Lemma 2.2.7. The inequality v i pzq ´vi pxq ď CC |x ´z| is obtained in a similar way. The proof is done. Let us define the tangential Hamiltonian H T O at vertex O by

Theorem 2.2.9.

  Under assumption rHs, the value functions v and v satisfy .2. Optimal control problem on junction with entry/exit costs Remark 2.2.10. Theorem 2.2.9 gives us the characterization of the value function at vertex O. The proof of Theorem 2.2.9, makes use of Lemma 2.2.11 and Lemma 2.2.12 below. Lemma 2.2.11 (Value functions v and v at O). Under assumption rHs, then Proof of Lemma 2.2.11. We divide the proof into two parts. Prove that max i"1,N tv i pOqu ď v pOq. First, we fix i P t1, . . . , N u and any control law α such that py O, ᾱ, ᾱq P T O . Let x P Γ i z tOu such that |x| is small. From Lemma 2.2.7, there exists a control law α x,O connecting x and O and we consider

	max	tv i pOqu ď v pOq ď min	tv i pOq `ci u ,
	i"1,N				i"1,N
	and
	max	tv i pOq ´di u ď v pOq ď min	tv i pOqu .
	i"1,N				i"1,N
			$
	α psq "	&	α x,O psq	if s ď τ x,O ,
			%
				"
	v pOq " min	min	tv i pOq `ci u ,	´HT
				i"1,N

O λ * , and v pOq " min

" min i"1,N tv i pOqu , ´HT O λ * . 2ᾱ ps ´τx,O q if s ą τ x,O .

It means that the trajectory goes from x to O with the control law α x,O and then proceeds with the control law ᾱ. Therefore v pxq " v i pxq ď J px, αq " ż τ x,O 0 i py x,α psq , αpsqq e ´λs ds `e´λτ x,O J pO, ᾱq .

  αn psq , α n psqq " pO, α n psqq . By letting n tend to 8, we get v pOq ě ´HT O {λ. On the other hand, since v pOq ď ´HT O {λ by Lemma 2.2.12, this implies that v pOq " ´HT O {λ. Let us now assume that 0 ď t n ă `8 for all n large enough. Then, for a fixed n and for any positive δ ď δ n where δ n small enough, y O,αn psq still belongs to some Γ ipnq z tOu for all s P pt n , t n `δs. We have v pOq `εn ą J pO, α n q py O,αn psq , α n psqq e ´λs ds `cipnq e ´λtn `ż tn`δ tn ipnq py O,αn psq , α n psqq e ´λs ds `e´λptn`δq J py O,αn pt n `δq , α n p¨`t n `δqq py O,αn psq , α n psqq e ´λs ds `cipnq e ´λtn `ż tn`δ tn ipnq py O,αn psq , α n psqq e ´λs ds `e´λptn`δq v py O,αn pt n `δqq py O,αn psq , α n psqq e ´λs ds `cipnq e ´λtn `ż tn`δ tn ipnq py O,αn psq , α n psqq e ´λs ds `e´λptn`δq v ipnq py O,αn pt n `δqq .

	In this case, α n psq P Y N i"1 A O i for a.e. s P r0, `8q. Therefore, for a.e. s P r0, `8q 0 pO, α n psqq e ´λs ds ě ż `8 0 `´H T O ˘e´λs ds " " ż tn 0 ě ż tn 0 " ż tn py ż `8 0	´HT

O,αn psq , α n psqq " pO, α n psqq ě ´HT O , 2.2. Optimal control problem on junction with entry/exit costs and v pOq `εn ą J pO, α n q " O λ .

  Definition 2.3.2 (Hamiltonian). We define the Hamiltonian H i : Γ i ˆR Ñ R by H i px, pq " max aPA i t´pf i px, aq ´ i px, aqu and the Hamiltonian H ì pO, ¨q : R Ñ R by H ì pO, pq " max aPA ì t´pf i pO, aq ´ i pO, aqu , where A ì " ta i P A i : f i pO, a i q ě 0u. Recall that the tangential Hamiltonian at O, H T O , has been defined in (2.2.1).

  Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs in the viscosity sense. The function vi satisfies For i " 1, N , the function v i is a viscosity subsolution of (2.3.1) at O.

	λv i pOq `max	" ´λ min j‰i	tv j pOq `di u , H	ì ˆO,	dv i dx i	pOq ˙, H T O ´λd i	*	" 0
	in the viscosity sense.
	The proof of Theorem 2.4.1 follows from Lemmas 2.4.2 and 2.4.5 below. We focus on v i since
	the proof for vi is similar.
	Lemma 2.4.2. Proof of Lemma 2.4.2. From Theorem 2.2.9,
					"				*
	λv i pOq `max	´λ min j‰i	tv j pOq `cj u , H T O	ď 0.
	It is thus sufficient to prove that
			λv i pOq `Hì ˆO,	dv i dx i	pOq ˙ď 0
	Theorem 2.4.1. For i " 1, N , the function v i satisfies
			"
	λv i pOq `max	´λ min

j‰i tv j pOq `cj u , H ì ˆO, dv i dx i pOq ˙, H T O * " 0

  Lt , 2.4. Connections between the value functions and the Hamilton-Jacobi systems.for t P r0, τ s, yielding that y x,αx ptq tends to y O,α O ptq when x tends to O. Hence, from (2.4.2), by letting x Ñ O, we obtain O psq , a i q e ´λs ds `e´λτ v i py O,α O pτ qq .Let ϕ be a function in C 1 pΓ i q such that 0 " v i pOq ´ϕ pOq " max Γ i pv i ´ϕq. This yields

	ż τ
	v i pOq ď i py O,α ϕ pOq ´ϕ py O,α O pτ qq 0 τ ď 1 τ ż τ 0 i py O,α O psq , a i q e ´λs ds	``e ´λτ ´1˘v i py O,α O pτ qq τ	.
	|x|	`L ż t	|y x,α psq ´yO,α psq| ds,
		0
	where α 0 satisfies (2.4.1) with x " O. According to Grönwall's inequality,

αx psq , a i q e ´λs ds `e´λτ v i py x,αx pτ qq . (2.4.2) Since f i p¨, aq is Lipschitz continuous by rH1s, we also have for all t P r0, τ s, |y x,αx ptq ´yO,α O ptq| " ˇˇˇx `ż t 0 f i py x,α psq , a i q e i ds ´ż t 0 f i py O,α psq , a i q e i ds ˇˇď |y x,αx ptq ´yO,α O ptq| ď |x| e

  py εn psq , α εn psqq e i " f i pO, α εn psqq `L |y εn psq| " f i pO, α εn psqq e i `on p1q `om p1q , i py εn psq , α εn psqq e i " i pO, α εn psqq `ω p|y εn psq|q " i pO, α εn psqq e i `on p1q `om p1q . On the other hand, from Lemma 2.4.3, y εn psq P Γ i z tOu for all s P r0, τ m s. This yields

													2.5. Comparison Principle and Uniqueness
	Moreover, ϕ px ε q ´ϕ pOq " o ε p1q and that	dϕ dx i	py ε psqq "	dϕ dx i	pOq `oε p1q `O psq. Thus
	λv i pOq	´1 τ	ż τ 0	"	i py ε psq , α ε psqq	`dϕ dx i	pOq f i py ε psq , α ε psqq		ds ě o ε p1q	`o pτ q τ	`oε p1q τ	.
													(2.4.10)
	Let ε n Ñ 0 as n Ñ 8 and τ m Ñ 0 as m Ñ 8 such that
	pa mn , b mn q :"	ˆ1 τ m	ż τm 0	f i py εn psq , α εn psqq e i ds,	1 τ m	0 ż τm	i py εn psq , α εn psqq ds ˙ÝÑ pa, bq P Re i	ˆR
	as n, m Ñ 8. By rH1s and rH2s
	$
	& % f i It follows that
	pa mn , b mn q " f aPA ż τm ˆ1 τ m 0 "ż τn f y εn pτ m q " 0 ì " ´fi pO, aq dϕ dx i pOq ´ i pO, aq	*	ě λv i pOq `"´f i pO, aq	dϕ dx i	pOq ´ i pO, aq		ě 0.
	ż τ 0 py Hence, from (2.4.8) dϕ dx i py ε psqq 9 y ε psq ds " ż τ 0 dϕ dx i
	τ λv i pOq	´ż τ	"
					0
													(2.4.9)

.4.8) Since y ε pτ q P Γ i for all ε, one has ϕ py ε pτ qq ´ϕ px ε q " ε psqq f i py ε psq , α ε psqq ds. i py ε psq , α ε psqq `dϕ dx i py ε psqq f i py ε psq , α ε psqq  ds ě τ o ε p1q `o pτ q `oε p1q . i pO, α εn psqq e i ds, 1 τ m ż τm 0 i pO, α εn psqq ds ˙`o n p1q `om p1q P FL i pOq `on p1q `om p1q , since FL i pOq is closed and convex. Sending n, m Ñ 8, we obtain pa, bq P FL i pOq so there exists a P A i such that lim m,nÑ8 ˆ1 τ m ż τm 0 f i py εn psq , α εn psqq e i ds, 1 τ m ż τm 0 i py εn psq , α εn psqq ds ˙" pf i pO, aq e i , i pO, aqq . (2.4.11) i py εn psq , α εn psqq ds  e i `xεn . Since |y εn pτ m q| ą 0, then 1 τ m ż τm 0 f i py εn psq , α εn psqq ds ě ´|x εn | τ m .

Let ε n tend to 0, then let τ m tend to 0, one gets f i pO, aq ě 0, so a P A ì . Hence, from (2.4.10) and (2.4.11), replacing ε by ε n and τ by τ m , let ε n tend to 0, then let τ m tend to 0, we finally obtain λv i pOq `max

  1. According to (2.5.1), the function w i is a viscosity supersolution of the following problem in Γ i

	$ ' ' & λw i pxq `Hi ˆx,	dw i dx
	'
	'
	%

i pxq ˙" 0 if x P Γ i z tOu , λw i pOq `Hì ˆO, dw i dx i pOq ˙" 0 if x " O.

  Since u is a viscosity subsolution of (2.3.1), u i is a viscosity subsolution of (2.5.2) and satisfies u i pOq ď ´HT O {λ. Hence, we can apply the proof in [8,Lemma 3.5 ].

			2.5. Comparison Principle and Uniqueness
	Proof of Lemma 2.5.4. Remark 2.5.5. Under assumption rHs, Lemmas 2.5.1, 2.5.2, 2.5.3 and 2.5.4 hold for vicosity sub
	and supersolution û and ŵ respectively, of the exit cost control problem if (2.5.1) replaced by
		"			*
	ŵi pOq ă min	min j‰i	t ŵj pOqu `di ,	´HT O λ	`di	.
	Theorem 2.5.6 (Comparison Principle). Under assumption rHs, let u be a bounded viscosity
	subsolution of (2.3.1) and w be a bounded viscosity supersolution of (2.3.1); then u ď w compo-
	nentwise. This theorem also holds for viscosity sub and supersolution û and ŵ, respectively, of
	the exit cost control problem (2.3.2).

, α i psqq e ´λs ds `ui py x pT qq e ´λT .

  ă min

				j‰j 0	tw j pOq `cj u .		(2.5.4)
	Next, by Lemma 2.5.3, there exists a test function ϕ i in C 1 pJ i q that touches u i from above
	at O, it yields
	λu i pOq ´λ min j‰i	tu j pOq `cj u
	ď λu i pOq `max	" ´λ min j‰i	tu j pOq `cj u , H	ì ˆO,	dϕ i dx i	0 pOq ˙, H T	*
	ď 0.

Therefore w j 0 pOq `cj 0 ď w i pOq ă u i pOq ď min j‰i tu j pOq `cj u ď u j 0 pOq `cj 0 .

Thus

w j 0 pOq ă u j 0 pOq . (

2.5.5)

Replacing index i by j 0 in (2.5.3), we get

λw j 0 pOq `HT O ă 0. (2.5.6)

By (2.5.4) and (2.5.6), (2.5.1) holds true. Repeating the proof of Case 1 with j 0 , we reach a contradiction with (2.5.5). It ends the proof.

  " pU c , U 1 , . . . , U m q where U c P C pY jďm Γ j ; Rq and U i P C pΓ i ; Rq for all i ą m is called a viscosity solution of the Hamilton-Jacobi system if it is both a viscosity subsolution and a viscosity supersolution of the Hamilton-Jacobi system. Remark 2.6.6. In the case when c i " 0 for i " 1, N , i,e., m " N , the term ´λ min jąm U j pOq `cj vanishes. This implies that

										2.6. A more general optimal control problem
		• x i " O for i ą m then
		λU i pOq `max	" ´λ min jąm,j‰i	tU j pOq `cj u , ´λU c pOq , H	ì ˆO,	dϕ i dx i	pOq ˙, H T O	*	ě 0.
	A function U Remark 2.6.5. The term ´λH C pOq in the above definition accounts for the situation in which
	the trajectory enters Y m j"1 Γj. The term max jďm	" H	j ˆO,	dϕ c dx j	pOq ˙* accounts for the situation
	in which the trajectory enters Γ i 0 where H ì0 ˆO,	dϕ c dx j	pOq ˙" max jďm	" H	j ˆO,	dϕ c dx j	pOq ˙*.
	max	" ´λmin	jďm	" H	j ˆO,	dϕ c dx j	pOq ˙* , H T O	*	" max j"1,N	" H	j ˆO,	dx j dϕ c	pOq ˙*
														"H O	ˆdϕ c dx 1	pOq , . . . ,	dx N dϕ c	pOq ˙.
		then
		λU c pOq `max	" ´λmin jąm	tU j pOq `cj u , max jďm	" H	j ˆO,	dϕ c dx j	pOq ˙* , H T O	*	ě 0;
	2. if U dϕ i dx i	px i q ˙ě 0,

i ´ϕi has a local minimum point at x i P Γ i for i ą m, and if

• x i P Γ i z tOu, then λU i px i q `Hi ˆx, jąm tU j pOq `cj u , max

  max aPA i s.t. f i pO,aqě0 t´f i pO, aq p i ´ i pO, aqu " sup aPA i s.t. f i pO,aqą0t´f i pO, aq p i ´ i pO, aqu . Assume that there exists x P Γ i where 1 ď i ď m and U c pxq´W c pxq ą 0. By classical comparison principle for the boundary problem on Γ Chapter 2. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs Applying again classical comparison principle for the boundary problem for each edge Γ j U c pOq ´Wc pOq " max

	Lemma 2.7.3. If U c and W c are respectively viscosity sub and supersolution of λU c pxq `Hi ˆx, dU c dx i pxq ˙ď 0 if x P Γ i z tOu , λU c pOq `Hc ˆdU c dx 1 pOq , . . . , dU c dx m pOq ˙ď 0 if x " O, and λW c pxq `Hi ˆx, dW c dx i pxq ˙ě 0 if x P Γ i z tOu , λW c pOq `Hc ˆdW c dx 1 pOq , . . . , dW c dx m pOq ˙ě 0 if x " O, then U c pxq ď W c pxq for all x P Ť m i"1 Γ i . i"1 Γ i tU c pxq ´Wc pxqu ą 0. For j " 1, N , we consider the function Ψ j,ε,γ : Γ j ˆΓj ÝÑ R px, yq ÝÑ U c pxq ´Wc pyq ´1 2ε r´|x| `|y| `δ pεqs 2 ´γ p|x| `|y|q , Proof of Lemma 2.7.3. Ť m 1 where δ pεq " pL `1q ε, γ P ˆ0, 2

1: Corollary 2.7.2. For i " 1, N and p i P R, i , one gets U c pOq ´Wc pOq " max Γ i tU c pxq ´Wc pxqu ą 0.

  Chapter 3. A Class of Mean Field Games on Networks. Part One: the Ergodic Case Notice that m ´P W and m ´φ P V , where φ is given by Definition 3.1.9. Using m ´φ as a test function in (3.2.14) yields `bBφq pm ´q2 `µα pBm ´q2 φ `pµ α Bφ `bφq m ´Bm ´‰ dx " ż

	ÿ αPA ´żΓα	"	pλ 0 φ
		3.2.14), we have	ş Γ mdx "	ş	Γ mdx " 1. Next, let
					$
				m ´pxq "	&	0	if m pxq ě 0,
					%

´m pxq if m pxq ă 0. Γ λ 0 mm ´φdx.

  3.3. Hamilton-Jacobi equation and the ergodic problemBy Lemma 3.3.5, for all n P N, since H n px, pq is continuous and bounded by C 2 `1 `n2 ˘, there exists a classical solution v n P C 2 pΓq for the following boundary value problem `Hn px, Bvq `λv " 0, x P Γ α zV, α P A, v| Γα pν i q " v| Γ β pν i q, for all α, β P A i , i P I, ÿWe wish to pass to the limit as n tend to `8; we first need to estimate v n uniformly in n, successively in L 8 pΓq, H 1 pΓq and H 2 pΓq.Estimate in L 8 pΓq. Since |H n px, pq| ď c ´1 `|p| 2 ¯for all x, p, then ϕ " ´c{λ and ϕ " c{λ are respectively a sub and supersolution of(3.3.10). Therefore, from Lemma 3.3.6, we obtain|λv n | ď c.Estimate in V . For a positive constant K to be chosen later, we introduce w n :" e Kv 2 n v n ψ P W , where ψ is given in Definition 3.1.9. Using w n as a test function in (3.3.10) leads to Bv n Bw n `λv n w n q dx " ´żΓ H n px, Bv n q w n dx." pµ α ψq pBv n q 2 `pµ α 2Kψq v 2 n pBv n q 2 `pµ α Bψq v n Bv n `λψv 2

						$
						' ' ' ' & ' ' ' ' %	γ iα µ α B α v pν i q " 0, ´µα B 2 v αPA i	i P I.	(3.3.10)
							ż
						ÿ
	αPA pµ α Since |H n px, pq| ď c `1 `p2 ˘, we have Γα
		ÿ	ż	e Kv 2 n					n	ı	dx
		αPA	Γα
		ż
	ď			e Kv 2 n |H n px, Bv n q| |v n ψ| dx
		Γ
		ż
	ď			ce Kv 2 n ψ |v n | dx	`żΓ	cψe Kv 2 n |v n | ψ pBv n q 2 dx
		Γ
	ď	ż	Γ	e Kv 2 n ˆλψv 2 n	`ψ c 2 4λ	˙dx	`ÿ αPA	Γα ż	e Kv 2 n	"	µ α

  3.4. Proof of the main result which, endowed with the norm }m} Mσ " }m} L 8 pΓq `max and observe that K is a closed and convex subset of M σ . We define a map T : K Ñ K as follows: given m P K, set f " F pmq. By Theorem 3.3.7, (3.3.12)-(3.3.13) has a unique solution pv, ρq P C 2 pΓq ˆR. Next, for v given, we solve (3.2.12)-(3.2.13) with b p¨q " B p H p¨, Bv p¨qq P P CpΓq. By Theorem 3.2.13, there exists a unique solution m P K X W of (3.2.12)-(3.2.13). We set T pmq " m; we claim that T is continuous and has a precompact image. We proceed in several steps:

		αPA	sup y,zPr0, αs,y‰z	|m α pyq ´mα pzq| |y ´z| σ	,
	is a Banach space. Now consider the set
		"	ż	*
	K "	m P M σ : m ě 0 and	mdx " 1
			Γ

  pm n q m n dx `żΓ rB p H px, Bv n q Bv n ´H px, Bv n qs m n dx.

					.4.4)
	and	ż
	ÿ
		µ α Bv n Bm n dx	`żΓ	B p H px, Bv n q m n Bv n dx " 0.	(3.4.5)
	αPA	Γα
	Subtracting the two equations, we obtain
	ż
	ρ n "				(3.4.6)

Γ

F n

  .4.11) where C K :" | sup 0ďrďK F prq| is independent of n. Choosing K " 2C where C is the constant in (3.4.9), we get by combining (3.4.11) with (3.4.9) that Step 3: Prove that F n pm n q is uniformly integrable and v n and m n are uniformly bounded respectively in C 1 pΓq and W . Let E be a measurable with |E| " η. By (3.4.10) with Γ is replaced by E, we have

	ż
	F n pm n qm n ď C.
	Γ
	Using (3.4.11) again, we obtain
	ż
	Hence, from (3.4.9), we conclude that |ρ n |	`| ş

Γ F n pm n qdx ď C. Γ F n pm n qdx| ď C.

  , sq `µαs B 2 v pX s , sq `a| Γα s pX s , sq Bv pX s , sq ¯ds

			(4.1.10)
	the process
	M t " vpX t , tq	´ż t	´Bt v pX s (4.1.11)
		0
	is a martingale, i.e.,
			EpM

2,1

pΓ ˆr0, T sq such that ÿ αPA i p iα B α v pν i , tq " 0, for all i P I, t P r0, T s, t |X s q " M s , for all 0 ď s ă t ď T.

(4.1.12)

  1.24). Let the function ϕ P W be defined as follows:

	Definition 4.1.6. We define two Sobolev spaces: V :" H 1 pΓq, and
	W :"	" w : Γ Ñ R : w P H 1 b pΓq and	w| Γα pν i q γ iα	"	w| Γ β pν i q γ iβ	for all i P I, α, β P A i	*	, (4.1.25)
	which is a subspace of H 1 b pΓq.
	Definition 4.1.7.

  Lemma 4.2.3. Multiplying (4.2.7) by y n k ptq e λt for a positive constant λ to be chosen later, summing for k " 1, . . . , n and using the formula (4.2.5) for v n , we get ´żΓ B t v n v n e λt ϕdx `żΓ µBv n B ´vn e λt ϕ ¯dx " e λt xhptq, v n ϕy W 1 ,W , pBv n q 2 e λt ϕdx `żΓ µBv n v n e λt Bϕdx " e λt xhptq, v n ϕy.

	and
	´żΓ µ Integrating both sides from s to T , we obtain " B t ˆv2 n 2  e λt ˙´λ 2 v 2 n e λt ϕdx `żΓ
				ż Γ	ˆv2 n px, sq 2	e λs ´v2 n px, T q 2	e λT ˙ϕdx	`λ 2	ż T s	ż Γ	v 2 n e λt ϕdxdt
				`ż T	ż	µpBv n q 2 e λt ϕdxdt	`ż T	ż	µBv n v n e λt Bϕdxdt
							s		Γ					s	Γ
				ż T
			"				e λt xhptq, v n ptqϕydt
					s
						ż T
			ď C				e λt }hptq} W 1 }v n ptq} V dt
							s
			ď	1 2	ż T s	ż Γ	µ `pBv n q 2 `v2 n	˘eλt ϕdxdt	`C2 2µ	ż T s	e λt }hptq} 2 W 1 dt,
	where C is positive constant depending on ϕ, because of Remark 4.1.8. Therefore,
	e λs	ż Γ	v 2 n px, sq 2	ϕdx	`1 4	ż T s	ż Γ	µpBv n q 2 ϕe λt dxdt	`˜λ 2 ´µ 2	´µ }Bϕ} 2 L 8 pΓq ϕ 2	¸ż T s	ż Γ	v 2 n e λt ϕdxdt
	ď e λT	ż Γ	v 2 n px, T q 2	ϕdx	`C2 2µ	e λT	ż T s	}hptq} 2 W 1 dt.
	Choosing λ ě 1{2 `µ `2µ||Bϕ|| 2 L 8 pΓq {ϕ 2 and noticing that

  Chapter 4. A Class of Mean Field Games on Networks. Part two: Finite Horizon Games for some constant C depending only on pµ α q αPA , ϕ and T .EstimateB t v n in L 2 p0, T ; W 1 q. Consider the closed subspace G 1 of W defined by G 1 " w P W : ş Γ v k wdx " 0 for all k ď n (. It has a finite co-dimension equal to n. Consider also the n-dimensional subspace G 2 " span tv 1 ϕ, . . . , v n ϕu of W . The invertibility of the matrix M n introduced in the proof of Lemma 4.2.2 implies that G 1 X G 2 " t0u. This implies that W " G 1 ' G 2 . For w P W , we can write w of the form w " w n `ŵ n , where w n P G 2 and ŵn P G 1 . Hence, for a.e. t P r0, T s, from (4.2.5) and (4.2.7), one gets xB t v n ptq, wy W 1 ,W " Since there exists a constant C independent of n such that }w n

	d dt ˆżΓ	v n wdx ˙" d dt ˆżΓ	v n w n dx ˙" ´xhptq, w n y W 1 ,W	`żΓ	µBv n Bw n dx.
					(4.2.11)

.2.10)

  .2.14) where d 1 , . . . , d N are given real valued C 1 functions defined in r0, T s. For all n ě N , multiplying (4.2.7) by d k ptq, summing for k " 1, . . . , n and integrating over p0, T q leads to Preliminary: a modified heat equation on the network with general Kirchhoff conditions Since the functions of the form (4.2.14) are dense in L 2 p0, T ; V q, (4.2.16) holds for all test function v P L 2 p0, T ; V q. Recalling the isomorphism v P V Þ Ñ vϕ P W (see Remark 4.1.8), we obtain that, for all w P W and ψ P C1

	4.2. c p0, T q,
	´ż T	xB t v, wyψdt	`ż T	ż	µBvBwψdxdt "	ż T	xh, wyψdt.
		0					0		Γ	0
	This implies that, for a.e. t P p0, T q,
		´xB t v, wy `B pv, wq " xh, wy for all w P W.
	Using Theorem 3.1 in [84] (or the same argument as in [49, pages 287-288]), we see that v P
	Cpr0, T s; L 2 ϕ pΓqq, where L 2 ϕ pΓq " tw : Γ Ñ R : below and from above by positive numbers, L 2 ϕ pΓq " L 2 pΓq with equivalent norms. Moreover, ş Γ w 2 ϕdx ă `8u, and since ϕ is bounded from
	max 0ďtďT	}vp¨, tq} L 2
	´ż T	ż		B t v n vϕdxdt	`ż T	ż	µBv n B pvϕq dxdt "	ż T	xh, vϕydt.	(4.2.15)
	0	Γ				0		Γ	0
	Letting n Ñ `8, we obtain from (4.2.13) that
	´ż T	xB t v, vϕydt	`ż T	ż	µBvB pvϕq dxdt "	ż T	xh, vϕydt.	(4.2.16)
		0					0	Γ	0

  Proof of Theorem 4.2.5. It is enough to prove estimate (4.2.17) for v n . Multiplying (4.2.7) by ´d dt y n k , summing for k " 1, . . . , n and using (4.2.5) leads to ż Γ pB t v n q 2 ϕdx ´żΓ µBv n B pB t v n ϕq dx " ´żΓ hB t v n ϕdx,

	hence
	ż

Γ pB t v n q 2 ϕdx ´żΓ µB t pBv n q 2 2 ϕdx ´żΓ µBv n B t v n Bϕdx " ´żΓ hB t v n ϕdx.

  . Taking pm ε ´mq e ´λt ϕ ´1 as a test-function in the versions of (4.3.2) satisfied by m ε and m, subtracting, we obtain that B t ´pm ε ´mq 2 e ´λt ¯`λ 2 pm ε ´mq 2 e ´λt  ϕ ´1dx `żΓ µpB pm ε ´mqq 2 e ´λt ϕ ´1dx `żΓ µ pm ε ´mq B pm ε ´mq e ´λt Bpϕ ´1qdx `żΓ pb ε m ε ´bmq B pm ε ´mq e ´λt ϕ ´1dx `żΓ pb ε m ε ´bmq pm ε ´mq e ´λt Bpϕ ´1qdx " 0. (4.3.4)

			$
			' &	H px, pq	if |p| ď n,
			' %	H ˆx,	p |p|	n ˙if |p| ą n.
	ż	"	1
	Γ		2

  for all ξ P R n . (4.4.24) Looking for u n of the form (4.4.21) and setting Y " py n 1 , . . . , y n n q T , 9 Y " `d dt y n 1 , . . . , d dt y n n ˘T , (4.2.7) implies that we have to solve the following a system of ODEs:

	$
	' &	´Mn 9 Y ptq `BY ptq `HpY qptq " Gptq,	t P r0, T s
		ˆżΓ
	' %	Y pT q "	u

T f 1 ψ 2 dx, ¨¨¨, ż Γ u T f n ψ 2 dx ˙T , (4.4.25)

  and f is bounded, there exists a constant C such that Chapter 4. A Class of Mean Field Games on Networks. Part two: Finite Horizon GamesThe desired estimate on u n is obtained from the previous inequality in a similar way as in the proof of Lemma 4.2.3, by taking λ large enough.Step 2: Uniform estimates of u n in L 8 p0, T ; F q X L 2 p0, T ; H 2 b pΓqq and of B t u n in L 2 pΓ p0, T qq. Multiplying (4.4.23) by B t y n k ptq f k e λt ψ where λ is a positive constant to be chosen later, integrating by part the term containing H and f (all the integration by parts are justified) summing for k " 1, . . . , n and using (4.4.21), we obtain that ´żΓ pB t u n q 2 e λt ψdx `żΓ µBu n B ´Bt u n e λt ψ ¯dx `żΓ B pH px, u n qq B t u n e λt ψdx (4.4.27) iα rH α pν i , u n | Γα pν i , tqq ´f | Γα pν i , tqs B t u n | Γα pν i , tq ψ| Γα pν i q e λt " ż Γ Bf B t u n ψe λt dx. Note that from (4.1.31) and (4.1.32), |B pH px, u n qq| ď C 0 p1 `|u n | `|Bu n |q (4.4.28) so, from Step 1, this function is bounded in L 2 pΓ ˆp0, T qq by a constant. Moreover, Γ µBu n B `Bt u n e λt ψ ˘dx as in the proof of Theorem 4.2.5. Therefore, the only new difficulty with respect to the proof of Theorem 4.2.5 consists of obtaining a bound for the term iα rH α pν i , u n | Γα pν i , tqq ´f | Γα pν i , tqs B t u n | Γα pν i , tq e λt ψ| Γα pν i q .

		ÿ
	´ÿ iPI	αPA i
	ż T	ż					ˆż T	ż	pBf q 2 e λt dxdt	˙1 2 ˆż T	ż	˙1 2 pB t u n q 2 e λt ψdxdt	,	(4.4.29)
	s							s		Γ	s	Γ
	and we can also estimate the term	ş
				ÿ	ÿ
				iPI	αPA i
		´żΓ	"	B t	ˆu2 n 2	e λt ˙´λ 2	u 2 n e λt		ψdx	`żΓ	µ |Bu n | 2 e λt ψdx	´C ż

Γ |u n | p|u n | `|Bu n |q e λt dx ďC ż Γ p|u n | `|Bu n |qe λt dx. (4.4.26) n Γ Bf B t u n ψe λt dxdt ď C n

  u n | Γα pν i , sqq B t u n | Γα pν i , sq " d dt J iα pu n | Γα pν i , sqq. Γα pν i , tqq B t u n | Γα pν i , tq e λt ψ| Γα pν i q ¯dt "n iα ψ| Γα pν i q ˆ´J iα pu n | Γα pν i , T qq e λT `Jiα pu n | Γα pν i , sqq e λs `λ ż T Since H α px, ¨q is sublinear, see (4.1.30), |J iα ppq| is subquadratic, i.e., |J iα ppq| ď Cp1 `p2 q, for a constant C independent of α and i. This implies that ˇˇˇż T s ´niα H α pν i , u n | Γα pν i , tqq B t u n | Γα pν i , tq e λt ψ| Γα pν i q ¯dt ˇˇď C ´eλT `u2 n | Γα pν i , T q e λT `u2 n | Γα pν i , sq e λs ¯`Cλ Γα pν i , tq ˘eλt dt. Γα pν i , tq ˘eλt dt ď Cλe λT . To summarize ˇˇˇż T s ´niα H α pν i , u n | Γα pν i , tqq B t u n | Γα pν i , tq e λt ψ| Γα pν i q ¯dt ˇˇď Γα pν i , sq 2 e λs ¯`C pλq. (4.4.30) 4.4. The Hamilton-Jacobi equation Similarly, using the fact that f P CpΓ ˆr0, T sq and B t f | Γα pν i , ¨q P L 2 p0, T q, and integrating by part, we see that Γα pν i , tq B t u n | Γα pν i , tq e λt dt ˇˇ" ˇˇˇp f | Γα u n q| Γα pν i , T q e λT ´pf | Γα u n q| Γα pν i , tq e λs ´ż T s pλf | Γα pν i , tq `Bt f | Γα pν i , tqq u n | Γα pν i , tq e λt dt ˇˇď C ˆ|u n | Γα pν i , T q |e λT `|u n | Γα pν i , sq |e λs `λ ż T s |u n | Γα pν i , tq| e λt dt Γα pν i , tqq 2 e λt dt. From Step 1 and the assumptions on f , the last three terms in the right hand side of the latter estimate are bounded by a constant depending on λ, but not on n. To summarize, Γα pν i , tq B t u n | Γα pν i , tq e λt dt ˇˇˇď C ´|u n | Γα pν i , T q |e λT `|u n | Γα pν i , sq |e λs ¯`C pλq.(4.4.31) To conclude from (4.4.30) and (4.4.31), we use the following estimates Then proceeding as in the proof of Theorem 4.2.5 and combining (4.4.27), (4.4.28), (4.4.29), (4.4.30) and (4.4.31) with (4.4.32), we find the desired estimates by taking λ large enough.

	ˇˇˇż	T
	s	f | 1
			ż T	ż T
	2 pB t f | ˇˇˇż s u 2 n | Γα pν i , tq e λt dt `1 2 s T
	s	f | $
			' ' &	|u
			'
			'
			%
	We can then write
	´ż T
			ż T
			`1 `u2
			0
	Note that, from Step 1 and the stability of the trace, λ	ş T s `1 `u2
			C ´u2

s ´niα H α pν i , u n | s J iα pu n | Γα pν i , tqq e λt dt ˙. n | n | n | Γα pν i , T q e λT `u2 n | n | Γα pν i , tq| ď C ˆżΓα |u n px, tq| dx `żΓα |Bu n px, tq| dx ˙, u 2 n | Γα pν i , tq ď C ˆżΓα u 2 n px, tq dx `żΓα |u n Bu n px, tq| dx ˙,

(4.4.32)

for t " s and t " T .

  Definition 4.4.15. Let x P Γ α 0 " rν i 0 , ν i 1 s and y P Γ αm " " ν im , ν i m`1 ‰ . We denote the set of paths joining from x to y by Ý Ñ xy. More precisely, if L P Ý Ñ xy, we can write L under the formL " x Ñ ν i 1 Ñ ν i 2 Ñ . . . Ñ ν im Ñ y, with ν i k P V and " ν i k , ν i k`1 ‰ " Γ α k .The integral of a function φ on L is defined by ż

L φ pξq dξ " ż rx,ν i 1 s φ pξq dξ `m ÿ k"1 ż rν i k ,ν i k`1 s φ pξq dξ `żrν im ,

ys φ pξq dξ, (4.4.34) recalling that the integrals on a segment are defined in (4.1.8). Lemma 4.4.16. Let u be the unique solution of (4.4.12) with u T " Bv T . Then for all x, y P Γ and a.e. t P r0, T s, ż L 1 u pζ, tq dζ " ż L 2 u pζ, tq dζ, for all L 1 , L 2 P Ý Ñ xy. This means that the integral of u from x to y does not depend on the path. Hence, for any L P Ý Ñ xy, we can define ż Ý Ñ xy u pζ, tq dζ :" ż L u pζ, tq dζ.

u ˚pxq " lim sup yÑx upyq is the upper-semicontinuous (USC) envelope of u and u˚pxq " lim infyÑx upyq is the lower-semicontinous (LSC) envelope of u.

This chapter is accepted: Manh-Khang Dao, Hamilton-Jacobi equations for optimal control on networks with entry or exit costs, ESAIM: Control, Optimisation and Calculus of Variations.

O * " 0 if x " O.(2.1.1)

this chapter is submitted and a preprint is available: Yves Achdou, Manh-Khang Dao, Olivier Ley and Nicoletta Tchou, A Class of Infinite Horizon Mean Field Games on Networks, hal-01802120.

Freidlin & Sheu proved in[START_REF] Freidlin | Diffusion processes on graphs: stochastic differential equations, large deviation principle[END_REF] that

this chapter is a work in preparation: Yves Achdou, Manh-Khang Dao, Olivier Ley and Nicoletta Tchou, A Class of Finite Horizon Mean Field Games on Networks.

Acknowledgements

Appendix

On the other hand, since x j 0 ,ε,γ P Γ j 0 z tOu, we have λU c px j 0 ,ε,γ q `Hj 0 ˆxj 0 ,ε,γ , ´xj 0 ,ε,γ `δ pεq ε ˙ď 0.

Subtracting the two inequalities and using properties of Hamiltonian H j 0 , let ε tend to 0 then γ tend to 0, we obtain that U c pOq ´Wc pOq ď 0, which is contradictory.

Since for α P A, any smooth function on Γ compactly supported in Γ α zV clearly belongs to DpLq, (3.1.14) implies that m satisfies ´µα B 2 m `B pmaq " 0 (3. 1.15) in the sense of distributions in the edges Γ α zV, α P A. This implies that there exists a real number c α such that ´µα Bm| Γα " ´m| Γα a| Γα `cα . (3.1.16) So m| Γα is C 1 regular, and (3.1. [START_REF] Barles | Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit[END_REF]) is true pointwise. Using this information and (3.1.14), we find that, for all u P DpLq,

This and (3.1.16) imply that

Bu| Γ β pxqdx " 0.

(3.1.17)

For all i P I, it is possible to choose a function u P DpLq such that 1. upν j q " δ i,j for all j P I;

2. B α upν j q " 0 for all j P I and α P A j .

Using such a test-function in (3.1.17) implies that for all i P I, where n iα is defined in (3.1.6). For all i P I and α, β P A i , it is possible to choose a function u P DpLq such that 1. u takes the same value at each vertex of Γ, which implies that ş Γ δ Bu| Γ δ pxqdx " 0 for all δ P A; 2. B α upν i q " 1{p iα , B β upν i q " ´1{p iβ and all the other first order directional derivatives of u at the vertices are 0.

Using such a test-function in (3.1.17) yields m| Γα pν i q γ iα " m| Γ β pν i q γ iβ , for all α, β P A i , ν i P V, in which γ iα " p iα µ α , for all i P I, α P A i . (3.1.19) Next, for i P I, multiplying (3.1.16) at x " ν i by n iα for all α P A i , then summing over all α P A i , we get ÿ αPA i µ α B α m pν i q ´niα ´m| Γα pν i q a| Γα pν i q ´cα ¯" 0, Chapter 3. A Class of Mean Field Games on Networks. Part One: the Ergodic Case and using (3.1.18), we obtain that ÿ αPA i µ α B α m pν i q ´niα a| Γα pν i q m| Γα pν i q " 0, for all i P I. (3.1.20) Summarizing, we get the following boundary value problem for m (recalling that the coefficients n iα are defined in (3.1.6)):

´µα B 2 m `B pmaq " 0,

x P pΓ α zVq , α P A, ÿ αPA i µ α B α m pν i q ´niα a| Γα pν i qm| Γα pν i q " 0, ν i P V, m| Γα pν i q γ iα " m| Γ β pν i q γ iβ , α, β P A i , ν i P V.

(3.1.21)

Formal derivation of the MFG system on Γ

Consider a continuum of indistinguishable agents moving on the network Γ. The set of Borel probability measure on Γ is denoted by P pΓq. Under suitable assumptions, the theory of MFGs asserts that the distribution of states is absolutely continuous with respect to Lebesgue measure on Γ. Hereafter, m stands for the density of the distribution of states: m ě 0 and ş Γ mpxqdx " 1. The state of a representative agent at time t is a time-continuous controlled stochastic process X t in Γ, as defined in Section 3.1.2, where the control is the drift a t , supposed to be of the form a t " apX t q. The function X Þ Ñ apXq is the feedback.

For a representative agent, the optimal control problem is of the form:

L pX s , a s q `V rm p¨, sq pX s qs ds

where E x stands for the expectation conditioned by the event X 0 " x. The functions and operators involved in (3.1.22) will be described below. Let us assume that there is an optimal feedback law, i.e. a function a ‹ defined on Γ which is sufficiently regular in the edges of the network, such that the optimal control at time t is given by a ‹ t " a ‹ pX t q. Then, almost surely if X t P Γ α zV, dπ ´1 α pX t q " a ‹ α pπ ´1 α pX t qqdt `a2µ α dW t .

An informal way to describe the behavior of the process at the vertices is as follows: if X t hits ν i P V, then it enters Γ α , α P A i with probability p iα ą 0.

Let us discuss the ingredients in (3.1.22): the running cost depends separately on the control and on the distribution of states. The contribution of the distribution of states involves the coupling cost operator, which can either be nonlocal, i.e. V : P pΓq Ñ C 2 pΓq, or local, i.e. V rmspxq " F pmpxqq assuming that m is absolutely continuous with respect to the Lebesgue measure, where F : R `Ñ R is a continuous function.

The contribution of the control involves the Lagrangian L, i.e. a real valued function defined on pY αPA Γ α zVq ˆR. If x P Γ α zV and a P R, Lpx, aq " L α pπ ´1 α pxq, aq, where L α is a continuous real valued function defined on r0, α s ˆR. We assume that lim |a|Ñ8 inf yPΓα Lαpy,aq |a| " `8. Further assumptions on L and V will be made below.

Under suitable assumptions, the Ito calculus recalled in Section 3.1.2 and the dynamic programming principle lead to the following ergodic Hamilton-Jacobi equation on Γ, more precisely

Introduction and main results

the following boundary value problem:

´µα B 2 v `H px, Bvq `ρ " V rms pxq, x P pΓ α zVq , α P A, ÿ αPA i γ iα µ α B α v pν i q " 0, ν i P V, v| Γα pν i q " v| Γ β pν i q , α, β P A i , ν i P V, ż Γ vpxqdx " 0.

(3. 1.23) We refer to [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF] Lasry | Mean field games[END_REF] for the interpretation of the value function v and the ergodic cost ρ.

Let us comment the different equations in (3.1.23):

1. The Hamiltonian H is a real valued function defined on pY αPA Γ α zVq ˆR. For x P Γ α zV and p P R, H px, pq " sup a ´ap ´Lα `π´1 α pxq, a ˘( ,

The Hamiltonian is supposed to be C 1 and coercive with respect to p uniformly in x.

2. The second equation in (3.1.23) is a Kirchhoff transmission condition (or Neumann boundary condition if ν i P BΓ); it is the consequence of the assumption on the behavior of X s at vertices. It involves the positive constants γ iα defined in (3.1.19).

3. The third condition means in particular that v is continuous at the vertices.

4. The fourth equation is a normalization condition.

If (3.1.23) has a smooth solution, then it provides a feedback law for the optimal control problem, i.e. a ‹ pxq " ´Bp H px, Bv pxqq .

At the MFG equilibrium, m is the density of the invariant measure associated with the optimal feedback law, so, according to Section 3.1.

At a vertex ν i , i P I, the transmission conditions for both v and m consist of d ν i " 7pA i q linear relations, which is the appropriate number of relations to have a well posed problem. If ν i P BΓ, there is of course only one Neumann like condition for v and for m.

Remark 3.1.6. In [START_REF] Camilli | Stationary mean field games systems defined on networks[END_REF], the authors assume that γ iα " γ iβ for all i P I, α, β P A i . Therefore, the density m does not have jumps across the transition vertices.

Chapter 3. A Class of Mean Field Games on Networks. Part One: the Ergodic Case

Assumptions and main results

Assumptions

Let pµ α q αPA be a family of positive numbers, and for each i P I let pγ iα q αPA i be a family of positive numbers such that ř αPA i γ iα µ α " 1. Consider the Hamiltonian H : Γ ˆR Ñ R, with H| Γα : Γ α ˆR Ñ R. We assume that, for some positive constants C 0 , C 1 , C 2 and q P p1, 2s,

for all m which are absolutely continuous with respect to the Lebesgue measure and such that d m pxq " m pxq dx. We shall also suppose that F is bounded from below, i.e., there exists a positive constant M such that F prq ě ´M, for all r P r0, `8q .

(3.1.31)

Function spaces related to the Kirchhoff conditions

Let us introduce two function spaces on Γ, which will be the key ingredients in order to build weak solutions of (3.1.24).

Definition 3.1.8. We define two Sobolev spaces: " w : Γ Ñ R : w α P H 1 p0, α q for all α P A, and

which is also a Hilbert space, endowed with the norm }w} W " ´řαPA }w α } 2

Definition 3.1.9. Let the function ψ P W and φ P P CpΓq be defined as follows:

ψ is constant on the edges Γ α which touch the boundary of Γ.

(3.1.34)

Preliminary: A class of linear boundary value problems

φ is constant on the edges Γ α which touch the boundary of Γ.

(3.1.35)

Note that both functions ψ, φ are positive and bounded. We set ψ " max Γ ψ, ψ " min Γ ψ, φ " max Γ φ, φ " min Γ φ.

Remark 3.1.10. One can see that v P V Þ ÝÑ vψ is an isomorphism from V onto W and w P W Þ ÝÑ wφ is the inverse isomorphism.

Definition 3.1.11. Let the function space W Ă W be defined as follows:

(note that v is continuous at the vertices from the definition of C 2 pΓq), and m satisfies ÿ αPA ż Γα rµ α BmBu `B pmB p H px, Bvqq us dx " 0, for all u P V, (3.1.38) where V is defined in (3.1.32).

We are ready to state the main result:

The proof, omitted in what follows, is similar to that of Theorem 3.4.1 below.

Preliminary: A class of linear boundary value problems

This section contains elementary results on the solvability of some linear boundary value problems on Γ. To the best of our knowledge, these results are not available in the literature.

Chapter 3. A Class of Mean Field Games on Networks. Part One: the Ergodic Case

A first class of problems

We recall that the constants µ α and γ i,α are defined in Section 3.1.2. Let λ be a positive number. We start with very simple linear boundary value problems, in which the only difficulty is the Kirchhoff condition:

where f P W Let us first study the homogeneous case, i.e. f " 0.

Lemma 3.2.5. The following linear boundary value problem

has a unique solution: v " 0.

Proof. Let I i :" tk P I : k " i; ν k P Γ α for some α P A i u be the set of indices of the vertices which are connected to ν i . By Remark 3.1.1, it is not restrictive to assume (in the remainder of the proof) that for all k P I i , Γ α " Γ α ik " rν i , ν k s is oriented from ν i to ν k .

For k P I i , Γ α " rν i , ν k s, using the parametrization (3.

It follows that B α v pν i q " ? λξ α " ? λ sinh `?λ α ˘"v pν k q ´v pν i q cosh `?λ α ˘‰. Hence, the transmission condition in (3.2.3) becomes: for all i P I, 0 "

Therefore, we obtain a system of linear equations of the form M U " 0 with M " pM ij q 1ďi,jďN , N " 7pIq, and U " pv pν 1 q , . . . , v pν N qq T , where M is defined by

For all i P I, since cosh `?λ α ik ˘ą 1 for all k P I i , the sum of the entries on each row is positive and M is diagonal dominant. Hence, M is invertible and the system has a unique solution U " 0. Finally, by solving the ODE in each branch Γ β with v β p0q " v β p β q " 0, we obtain that v " 0 on Γ.

Let us now study the non-homogeneous problems (3.

is also continuous by the open mapping theorem. This yields that `Id ´λ0 `I ˝Rλ 0 ˘˘´1 | V ˝Rλ 0 is continuous from W 1 into V . Hence, there exists a positive constant C such that for any f P W 1 , there exists a unique weak solution v of (3.2.1) and }v} V ď C }f } W 1 . The proof of Lemma 3.2.6 is complete.

The Kolmogorov equation

This subsection is devoted to the following boundary value problem including a Kolmogorov equation Remark 3.2.8. By using the fact that Γ α are line segments, i.e. one dimensional and solving the ODE, we see that if v is a weak solution of (3.2.11), then v P C 2 pΓq.

The uniqueness of solutions of (3.2.11), up to the addition of constants, is obtained by using a maximum principle: Lemma 3.2.9. For b P P C pΓq, the set of solutions of (3.2.11) is the set of constant functions on Γ.

Proof of Lemma 3.2.9. First of all, any constant function on Γ is a solution of (3.2.11). Now let v be a solution of (3.2.11) then v P C 2 pΓq by Remark 3.2.8. Assume that the maximum of v over Γ is achieved in Γ α ; by the maximum principle, it is achieved at some endpoint ν i of Γ α . Without loss of generality, using Remark 3.1.1, we can assume that π β pν i q " 0 for all β P A i . We have B β v pν i q ě 0 for all β P A i because ν i is the maximum point of v. Since all the coefficients γ iβ , µ β are positive, by the Kirchhoff condition if ν i is a transition vertex, or by Chapter 3. A Class of Mean Field Games on Networks. Part One: the Ergodic Case the Neumann boundary condition if ν i is a boundary vertex, we infer that B β v pν i q " 0 for all β P A i . This implies that Bv β is a solution of the first order linear homogeneous ODE $ & % ´µβ u 1 `bβ u " 0, on r0, β s , u p0q " 0. Therefore, Bv β " 0 and v is constant on Γ β for all β P A i . We can propagate this argument, starting from the vertices connected to ν i . Since the network Γ is connected and v is continuous, we obtain that v is constant on Γ. Using similar arguments as in the proof of existence in Theorem 3.2.13, we see that for λ 0 large enough and all g P L 2 pΓq, there exists a unique solution v P V of (3.2.23). Moreover, there exists a constant C such that }v} V ď C }g} L 2 pΓq for all g P L 2 pΓq. This allows us to define a continuous operator

The dual Fokker-Planck equation

Then we define Rλ 0 " I ˝Tλ 0 where I is the injection from V in L 2 pΓq, which is compact. Therefore, Rλ 0 is a compact operator from L 2 pΓq into L 2 pΓq. For any g P L 2 pΓq, set v " T λ 0 g which is the unique solution of (3.2.23). Noticing that T λ 0 pv, wq " A λ 0 pw, vq for all v P V, w P W , we obtain that

Thus R ‹ λ 0 " Rλ 0 . But ker `Id ´λ0 Rλ 0 ˘is the set of solutions of (3.2.11), which, from Lemma 3.2.9, consists of constant functions on Γ. This implies that dim ker `Id ´λ0 R ‹ λ 0 ˘" 1 and then that dim ker pId ´λ0

Finally, since the solutions m of (3.2.12) are in ker pId ´λ0 R λ 0 q and satisfy in addition the normalization condition ş Γ mdx " 1, we obtain the desired uniqueness property in Theorem 3.2.13.

Hamilton-Jacobi equation and the ergodic problem

The Hamilton-Jacobi equation

This section is devoted to the following boundary value problem including a Hamilton-Jacobi equation:

where λ is a positive constant and the Hamiltonian H : Γ ˆR Ñ R is defined in Section 3.1, except that, in (3.

where C 2 is a positive constant. There exists a classical solution v of (3.3.1). Moreover, if H α is locally Lipschitz with respect to both variables for all α P A, then the solution v belongs to C 2,1 pΓq.

Remark 3.3.3. Assume (3.3.2) and that v P H 2 pΓq Ă V is a weak solution of (3.3.1). From the compact embedding of H 2 pr0, α sq into C 1,σ pr0, α sq for all σ P p0, 1{2q, we get v P C 1,σ pΓq. Therefore, from the PDE in (3.3.1)

It follows that v is a classical solution of (3.3.1).

Remark 3.3.4. Assume now that H is locally Lipschitz continuous and that v P H 2 pΓq Ă V is a weak solution of (3.3.1). From Remark 3.3.3, v P C 1,σ pΓq for σ P p0, 1{2q and the function ´λv α ´Hα p¨, Bv α q belongs to C 0,σ pr0, α sq. Then, from the first line of (3.3.1), v P C 2,σ pΓq. This implies that Bv α P Lipr0, α s and using the PDE again, we see that v P C 2,1 pΓq.

Let us start with the case when H is a bounded Hamiltonian.

3.4)

There exists a classical solution v of (3.3.1). Moreover, if H α is locally Lipschitz in r0, α s ˆR for all α P A then the solution v belongs to C 2,1 pΓq.

Proof of Lemma 3.3.5. For any u P V , from Lemma 3.2.6, the following boundary value problem:

has a unique weak solution v P V . This allows us to define the map T

Moreover, from Lemma 3.2.6, there exists a constant C such that

where |Γ| " Σ αPA α . Therefore, from the PDE in (3.3.5),

where µ :" min αPA µ α . From (3.3.6) and (3.3.7), T pV q is a bounded subset of H 2 pΓq defined in Definition 3.1.3. From the compact embedding of H 2 pΓq into V , we deduce that T pV q is a compact subset of V . Next, we claim that T is continuous from

Chapter 3. A Class of Mean Field Games on Networks. Part One: the Ergodic Case we need to prove that v n Ñ v in V . Since tv n u is uniformly bounded in H 2 pΓq, then, up to the extraction of a subsequence, v n Ñ p v in C 1,σ pΓq for some σ P p0, 1{2q. From (3.3.8), we have that Bu n Ñ Bu in L 2 pΓ α q for all α P A. This yields that, up to another extraction of a subsequence, Bu n Ñ Bu almost everywhere in Γ α . Thus H px, Bu n q Ñ H px, Buq in L 2 pΓ α q by Lebesgue dominated convergence theorem. Hence, p v is a weak solution of (3.3.5). Since the latter is unique, p v " v and we can conclude that the whole sequence v n converges to v. The claim is proved.

From Schauder fixed point theorem, see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Corollary 11.2], T admits a fixed point which is a weak solution of (3.3.1). Moreover, recalling that v P H 2 pΓq, we obtain that v is a classical solution of (3.3.1) from Remark 3.3.3.

Assume now that H is locally Lipschitz. Since v α P H 2 p0, α q for all α P A, we may use Remark 3.3.4 and obtain that v P C 2,1 pΓq.

Proof of Lemma 3.3.6. The proof is reminiscent of an argument in [START_REF] Camilli | The vanishing viscosity limit for Hamilton-Jacobi equations on networks[END_REF]. Suppose by contradiction that δ :" max Γ tu ´vu ą 0. Let x 0 P Γ α be a maximum point of u ´v. It suffices to consider the case when x 0 P V, since if x 0 P ΓzV, then

and we obtain a contradiction with the first line of (3.3.9). Now consider the case when x 0 " ν i P V; from Remark 3.1.1, we can assume without restriction that π α p0q " ν i . Since u ´v achieves its maximum over Γ at ν i , we obtain that

From Kirchhoff conditions in (3.3.9), this implies that

It follows that Bv α p0q " Bu α p0q. Using the first line of (3.3.9), we get that

`λ pu α p0q ´vα p0qq ą 0.

Therefore, u α ´vα is locally strictly convex in r0, α s near 0 and its first order derivative vanishes at 0. This contradicts the fact that ν i is the maximum point of u ´v.

We now turn to Proposition 3.3.2.

Proof of Proposition 3.3.2. We adapt the classical proof of Boccardo, Murat and Puel in [START_REF] Boccardo | Existence de solutions faibles pour des équations elliptiques quasi-linéaires à croissance quadratique[END_REF].

First of all, we truncate the Hamiltonian as follows: Thus B 2 v n is uniformly bounded in L 1 pΓq. This and the previous estimate on }Bv n } L 2 pΓq yield that Bv n is uniformly bounded in L 8 pΓq, from the continuous embedding of W 1,1 p0, α q into C pr0, α sq. Therefore, from (3.3.10), we get that B 2 v n is uniformly bounded in L 8 pΓq. This implies in particular that v n is uniformly bounded in W 2,8 pΓq. Hence, for any σ P p0, 1q, up to the extraction of a subsequence, there exists v P V such that v n Ñ v in C 1,σ pΓq. This yields that H n px, Bv n q Ñ H px, Bvq for all x P Γ. By Lebesgue's Dominated Convergence Theorem, we obtain that v is a weak solution of (3.3.1), and since v P C 1,σ pΓq, by Remark 3.3.3, v is a classical solution of (3.3.1).

Assume now that H is locally Lipschitz. We may use Remark 3.3.4 and obtain that v P C 2,1 pΓq. The proof is complete.

The ergodic problem

For f P P C pΓq, we wish to prove the existence of pv, ρq P C 2 pΓq ˆR such that

Hence, from (3.3.19), B 2 u λ is bounded in L 1 pΓq uniformly with respect to λ P p0, 1s. From the continuous embedding of W 1,1 p0, α q in Cpr0, α sq, we infer that Bu λ | Γα is bounded in CpΓ α q uniformly with respect to λ P p0, 1s. From the equation (3.3.18) and (3.3.17), this implies that u λ is bounded in C 2 pΓq uniformly with respect to λ P p0, 1s.

After the extraction of a subsequence, we may assume that when λ Ñ 0 `, the sequence u λ converges to some function v P C 1,1 pΓq and that λ min v λ converges to some constant ρ. Notice that v still satisfies the Kirchhoff conditions since Bu λ | Γα pν i q Ñ Bv| Γα pν i q as λ Ñ 0 `. Passing to the limit in (3.3.18), we get that the couple pv, ρq satisfies (3.3.12) in the weak sense, then in the classical sense by using an argument similar to Remark 3.3.3. Adding a constant to v, we also get (3.3.13).

Furthermore, if for some σ P p0, 1q, f | Γα P C 0,σ pΓ α q for all α P A, a bootstrap argument using the Lipschitz continuity of H on the bounded subsets of Γ ˆR shows that u λ is bounded in C 2,σ pΓq uniformly with respect to λ P p0, 1s. After a further extraction of a subsequence if necessary, we obtain (3.3.15).

Proof of uniqueness in Theorem 3.3.7. Assume that there exist two solutions pv, ρq and pṽ, ρq of (3.3.12)- (3.3.13). First of all, we claim that ρ " ρ. By symmetry, it suffices to prove that ρ ě ρ. Let x 0 be a maximum point of e :" ṽ ´v. Using similar arguments as in the proof of Lemma 3.3.6, with λv and λu respectively replaced by ρ and ρ, we get ρ ě ρ and the claim is proved.

We now prove the uniqueness of v. Since H α belongs to C 1 pΓ α ˆRq for all α P A, then e is a solution of

´"ż 1 0 B p H α py, θBv α `p1 ´θq Bṽ α q dθ  Be α " 0, in p0, α q, with the same transition and boundary condition as in (3.3.12). By Lemma 3.2.9, e is a constant function on Γ. Moreover, from (3.3.13) , we know that ş Γ edx " 0. This yields that e " 0 on Γ. Hence, (3.3.12)-(3.3.13) has a unique solution.

Remark 3.3.8. Since there exists a unique solution of (3.3.12)-(3.3.13), we conclude that the whole sequence pu λ , λv λ q in the proof of Theorem 3.3.7 converges to pv, ρq as λ Ñ 0.

Proof of the main result

We first prove Theorem 3.1.13 when F is bounded.

(3.4.1) Multiplying the equation for v by m and integrating over Γ α , we get ż Γα µ α BvBm `rH px, Bv 1 q ´H px, Bv 2 q `ρ ´pF pm 1 q ´F pm 2 qqs mdx ´rµ α m α Bv α s α 0 " 0.

(3.4.2) Multiplying the equation for m by v and integrating over Γ α , we get

´

Subtracting (3.4.2) to (3.4.3), summing over α P A, assembling the terms corresponding to a same vertex ν i and taking into account the transmission and the normalization condition for v and m, we obtain 0 "

Since F is strictly monotone then the first sum is non-negative. Moreover, by the convexity of H and the positivity of m 1 , m 2 , the last two sums are non-negative. Therefore, we have that m 1 " m 2 . From Theorem 3.3.7, we finally obtain v 1 " v 2 and ρ 1 " ρ 2 .

Proof of Theorem 3.1.13 for a general coupling F . We only need to modify the proof of existence.

We now truncate the coupling function as follows: The assumption that all the edges have unit length is not restrictive, because we can always rescale the constants µ α and the piecewise continuous function a.

The goal is to derive the boundary value problem satisfied by the law of the stochastic process X t . Since the derivation here is formal, we assume that the law of the stochastic process X t is a measure which is absolutely continuous with respect to the Lebesgue measure on Γ and regular enough so that the following computations make sense. Let mpx, tq be its density. We have where n iα is defined in (4.1.6). We choose first, for every α P A, a smooth function u which is compactly supported in pΓ α zVq ˆr0, T s. Hence u| Γ β pν i , tq " 0 and B β upν i , tq " 0 for all i P I, β P A i . Notice that up¨, tq P D. It follows that m satisfies `Bt m ´µα B 2 m `B pmaq ˘px, tq " 0, for x P Γ α zV, t P p0, T q, α P A.

(4.1.18)

For a smooth function χ : r0, T s Ñ R compactly supported in p0, T q, we may choose for every i P I, a smooth function u such that upν j , tq " χptqδ i,j for all t P r0, T s, j P I and B α upν j , tq " 0 for all t P r0, T s, j P I and α P A j , we infer a condition for m at the vertices, ÿ αPA i n iα a| Γα pν i , tqm| Γα pν i , tq ´µα B α mpν i , tq " 0 for all i P I, t P p0, T q. This condition is called a transmission condition if ν i is a transition vertex and reduces to a Robin boundary condition when ν i is a boundary vertex. Finally, for a smooth function χ : r0, T s Ñ R compactly supported in p0, T q, for every transition vertex ν i P VzBΓ and α, β P A i , we choose u such that

• The directional derivatives of u at the vertices ν " ν i are 0.

Using such a test-function in (

px, tq P pΓ α zVq ˆp0, T q, α P A, ÿ

Formal derivation of the MFG system on Γ

Consider a continuum of indistinguishable agents moving on the network Γ. Under suitable assumptions, the theory of MFGs asserts that the distribution of states is absolutely continuous with respect to Lebesgue measure on Γ. Hereafter, m stands for the density of the distribution of states: m ě 0 and ş Γ mpx, tqdx " 1 for t P r0, T s. The state of a representative agent at time t is a time-continuous controlled stochastic process X t in Γ, as defined in Section 4.1.2, where the control is the drift a t , supposed to be of the form a t " apX t , tq.

For a representative agent, the optimal control problem is of the form

where E xt stands for the expectation conditioned by the event X t " x. The functions and operators involved in (4.1.22) will be described below. Let us assume that there is an optimal feedback law, i.e. a function a ‹ defined on Γ ˆr0, T s which is sufficiently regular in the edges of the network, such that the optimal control at time t is given by a ‹ t " a ‹ pX t , tq. Then, almost surely if

An informal way to describe the behavior of the process at the vertices is as follows: if X t hits ν i P V, then it enters Γ α , α P A i with probability p iα ą 0, (p iα was introduced in Section 4.1.2).

Let us discuss the ingredients in (4. 1.22). The running cost depends separately on the control and on the distribution of states. The contribution of the distribution of states involves the Chapter 4. A Class of Mean Field Games on Networks. Part two: Finite Horizon Games coupling cost operator V , which may be either nonlocal and regularizing, i.e., V : P pΓq Ñ C 2 pΓq for example, or local, i.e. V rmspxq " F pmpxqq where F : R `Ñ R is a continuous function.

The contribution of the control involves the Lagrangian L, i.e., a real valued function defined on pY αPA Γ α zVq ˆR. If x P Γ α zV and a P R, Lpx, aq " L α pπ ´1 α pxq, aq, where L α is a continuous real valued function defined on r0, α s ˆR. We assume that lim |a|Ñ8 inf yPΓα Lαpy,aq |a| " `8. The last one is the terminal cost v T . Further assumptions on L, V and v T will be made below.

Under suitable assumptions, Ito calculus as in [START_REF] Freidlin | Diffusion processes on graphs and the averaging principle[END_REF][START_REF] Freidlin | Diffusion processes on graphs: stochastic differential equations, large deviation principle[END_REF] and the dynamic programming principle lead to the following HJB equation on Γ, more precisely the following boundary value problem:

for all ν i P V, t P p0, T q α, β P A i , ÿ

We refer to [START_REF] Lasry | Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF] Lasry | Mean field games[END_REF] Note that such an assumption is satisfied by local operators of the form V rmspxq " F pmpxqq where F is a Lipschitz continuous function.

(Initial and terminal data) v T P H 1 pΓq and m 0 P L 2 pΓq X M.

The above set of assumptions, referred to as (H), will be the running assumptions hereafter. We will use the following notation: µ :" min αPA µ α ą 0 and µ :" max αPA µ α . We will also say that the coupling V is strictly increasing if, for any m 1 , m 2 P M X L 2 pΓq, ż Γ pm 1 ´m2 qpV rm 1 s ´V rm 2 sqdx ě 0 and equality implies m 1 " m 2 .

Stronger assumptions on the coupling operator

We will sometimes need to strengthen the assumptions on the coupling operator, namely that V has the following smoothing properties:

V maps the topological dual of W to H 1 b pΓq; more precisely, V defines a Lipschitz map from W 1 to H 1 b pΓq. Note that such an assumption is not satisfied by local operators.

Definition of solutions and main result

Definition 4.1.10. (solutions of the MFG system) A weak solution of the Mean field games system (4.1.24) is a pair pv, mq such that v P L 2 `0, T ; H 2 pΓq ˘X Cpr0, T s; V q, B t v P L 2 `0, T ; L 2 pΓq ˘, (ii) (Uniqueness) If V is strictly increasing, then the solution is unique.

(iii) (Regularity) If V satisfies furthermore the stronger assumptions made in Section 4.1.4 and if v T P C 2`η pΓq X D for some η P p0, 1q (D is given in (4.1.13)), then v P C 2,1 pΓ ˆr0, T sq. Moreover, if for all α P A, B p H α px, pq is a Lipschitz function defined in Γ α ˆR, and if m 0 P W , then m P Cpr0, T s; W q X W 1,2 p0, T ; L 2 pΓqq X L 2 p0, T ; H 2 b pΓqq.

Preliminary: a modified heat equation on the network with general Kirchhoff conditions

This section contains results on the solvability of some linear boundary value problems with terminal condition, that will be useful in what follows. Consider

where h P L 2 p0, T ; W 1 q and v T P L 2 pΓq. Definition 4.2.1. If v T P L 2 pΓq and h P L 2 p0, T ; W 1 q, a weak solution of (4.2.1) is a function v P L 2 p0, T ; V q X Cpr0, T s; L 2 pΓqq such that B t v P L 2 p0, T ; W 1 q and $ & % ´xB t v ptq , wy W 1 ,W `B pv p¨, tq , wq " xhptq, wy W 1 ,W for all w P W and a.e. t P p0, T q, v px, T q " v T pxq, (4.2.2) where B : V ˆW Ñ R is the bilinear form defined as follows: B pv, wq :"

We use the Galerkin's method (see [START_REF] Evans | Partial differential equations[END_REF]), i.e., we construct solutions of some finite-dimensional approximations to (4.2.1).

Recall that ϕ has been defined in Definition 4.1.7. We notice first that the symmetric bilinear form p Bpu, vq :" ş Γ µϕBuBv is such that pu, vq Þ Ñ pu, vq L 2 pΓq `p Bpu, vq is an inner product in V equivalent to the standard inner product in V , namely pu, vq V " pu, vq L 2 pΓq `şΓ BuBv. Therefore, by standard Fredholm's theory, there exist • a non decreasing sequence of nonnegative real numbers pλ k q 8

k"1 , that tends to `8 as k Ñ 8,

• a Hilbert basis pv k q 8

k"1 of L 2 pΓq , which is also a a total sequence of V (and orthogonal if V is endowed with the scalar product pu, vq L 2 pΓq `p Bpu, vq), Chapter 4. A Class of Mean Field Games on Networks. Part two: Finite Horizon Games

Note also that v k is a weak solution of

Finally, by Remark 4.1.8, the sequence pϕv k q 8 k"1 is a total family in W (but is not orthogonal if W is endowed with the standard inner product). Lemma 4.2.2. For any positive integer n, there exist n absolutely continuous functions y n k : r0, T s Ñ R , k " 1, . . . , n, and a function v n : r0, T s Ñ L 2 pΓq of the form

)

Since ϕ is positive and pv k q 8 k"1 is a Hilbert basis of L 2 pΓq, we can check that M n is a positive definite matrix and there exist two constants c, C independent of n such that

Then, choosing λ " 2µ 2 ||Bϕ|| 2 L 8 pΓq {pϕ 2 µq, we obtain that

for some constant C depending only on Γ, µ, T and ϕ. Estimate of B 2 v n in L 2 pΓ ˆp0, T qq. Finally, using the PDE in (4.2.1), we can see that B 2 v n belongs to L 2 pΓ ˆp0, T qq and is bounded by C ´}h} L 2 pΓˆp0,T qq `}v T } V ¯, hence v n is bounded in L 2 `0, T ; H 2 pΓq ˘by the same quantity. The Kirchhoff conditions (which boil down to Neumann conditions at BΓ) are therefore satisfied in a strong sense for almost all t. Using Theorem 3.1 in [START_REF] Lions | Problèmes aux limites non homogènes et applications[END_REF] (or a similar argument as [START_REF] Evans | Partial differential equations[END_REF] pages 287-288), we see that v in Cpr0, T s; V q. The assumptions on the coefficents b ε and b imply in fact that b ε Ñ b in L p pΓ ˆp0, T qq for all 1 ď p ă 8. On the other hand, we know that m P L q pΓ ˆp0, T qq for all 1 ď q ă 8. From the latter observation with p " q " 4, we see that the quantity ş T 0 ş Γ ´|b ε ´b| 2 m 2 ¯e´λt ϕ ´1dxdt tends to 0 as ε Ñ 0 uniformly in λ ą 0. We write

The Fokker-Planck equation

Choosing λ large enough and integrating the latter inequality from 0 to t P r0, T s, we obtain

Subtracting the two versions of (4.3.2) and using the latter estimate also yields

which achieves the proof.

The Hamilton-Jacobi equation

This section is devoted to the following boundary value problem including a Hamilton-Jacobi equation

where f P L 2 pΓ ˆp0, T qq, v T P V and the Hamiltonian H : Γ ˆR Ñ R satisfies the running assumptions (H). We start by proving existence and uniqueness of a weak solution for (4.4.1). Next, further regularity for the solution will be obtained under stronger assumptions. Uniqueness is a direct consequence of the following proposition.

Existence and uniqueness for the Hamilton-Jacobi equation

for all w P W, w ě 0, a.a. t P p0, T q , v px, T q ď v T pxq ď v px, T q for a.a. x P Γ.

Proof of Proposition 4.4.3. Setting v " v ´v, we have, for all w P W such that w ě 0 and for a.a t P p0, T q: ż Γ ´Bt vw `µBvBw `pH px, Bvq ´H px, Bvqq wdx ď 0, and v px, T q ď 0 for all x P Γ. Set v `" v 1 tvą0u and w " v `eλt ϕ. We have

`żΓ rH px, Bvq ´H px, Bvqs v `ϕe λt dx ď 0.

Integrating from 0 to T , we get

From (4.1.31), |H px, Bvq ´H px, Bvq| ď C 0 |Bv|. Hence, since v `pT q " 0 and |Bv|v `" |Bv `|v àlmost everywhere, we get

For λ large enough, the first term in the left hand side is not smaller than the second term. This implies that v `" 0. Now we prove Theorem 4.4.3. We start with a bounded Hamiltonian H. 3 when H is bounded by C H . Take v P L 2 p0, T ; V q and f P L 2 pΓ ˆp0, T qq. From Theorem 4.2.4 and Theorem 4.2.5 with h " f ´H px, Bvq and v T P V , the following boundary value problem

has a unique weak solution v P L 2 `0, T ; H 2 pΓq ˘X Cpr0, T s; V q X W 1,2 `0, T ; L 2 pΓq ˘. This allows us to define the map T :

Using again Theorem 4.2.5, we have that T is continuous from L 2 p0, T ; V q to L 2 p0, T ; V q. Moreover, there exists a constant C depending only on C H ,Γ,pµ α q αPA , f , T , ϕ and v T such that

Therefore, from Aubin-Lions theorem (see Lemma 4.6.1), we obtain that T `L2 p0, T ; V q ˘is relatively compact in L 2 p0, T ; V q. By Schauder fixed point theorem, see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Corollary 11.2], the operator T admits a fixed point which is a weak solution of (4.4.1).

Proof of existence in

Regularity for the Hamilton-Jacobi equation

In this section, we prove further regularity for the solution of (4.4.1).

Theorem 4.4.5. We suppose that the assumptions of Theorem 4.4.2 hold and that, in addition, v T P H 2 pΓq satisfies the Kirchhoff conditions given by the third equation in (4.4.1), f P P CpΓ r0, T sq X L 2 p0, T ; H 1 b pΩqq and B t f P L 2 p0, T ; H 1 b pΓqq. Then, the unique solution v of (4.4.1) satisfies v P L 2 `0, T ; H 3 pΓq ˘and B t v P L 2 `0, T ; H 1 pΓq ˘. Moreover, there exists a constant C depending only on }v T } H 2 pΓq , pµ α q αPA , H and f such that

If, in addition, there exists η P p0, 1q such that v T P C 2`η pΓq then there exists τ P p0, 1q such v P C 2`τ,1`τ 2 pΓ ˆr0, T sq, and v is a classical solution of (4.4.1).

The main idea to prove Theorem 4.4.5 is to differentiate (4.4.1) with respect to the space variable and to prove some regularity properties for the derived equation. Let us explain formally our method. Assuming the solution v of (4.4.1) is in C 2,1 pΓ ˆp0, T qq and taking the spacederivative of (4.4.1) on pΓ α zVq ˆp0, T q, we have ´Bt Bv ´µα B 3 v `B pHpx, Bvqq " Bf.

Therefore, u " Bv satisfies the following PDE ´Bt u ´µα B 2 u `B pH px, uqq " Bf, with terminal condition upx, T q " Bv T pxq. From the Kirchhoff conditions in (4.4.1) and Remark 4.1.1, we obtain a condition for u of Dirichlet type, namely ÿ

Note that the latter condition is an homogeneous Dirichlet condition at the boundary vertices of Γ. Now, by extending continuously the PDEs in (4.4.1) until the vertex ν i in the branchs Γ α and Γ β , α, β P A i , and using the continuity condition in (4.4.1), one gets

This gives a second transmission condition for u at ν i P VzBΓ of Robin type, namely

which is equivalent to

Hence, we shall study the following nonlinear boundary value problem for u " Bv,

where Bf P L 2 pΓ ˆp0, T qq and u T P F defined in (4.4.13) below. Theorem 4.4.5 will follow by choosing u T " Bv T .

In order to define the weak solutions of (4.4.12), we need the following subspaces of H 1 b pΓq. Definition 4.4.6. We define the Sobolev spaces F :"

ψ is constant on the edges Γ α which touch the boundary of Γ.

(4.4.15)

Note that ψ is positive and bounded. The map f Þ ÝÑ f ψ is an isomorphism from F onto E.

Definition 4.4.8. A weak solution of (4.4.12) is a function u P L 2 p0, T ; F q such that B t u P L 2 p0, T ; E 1 q, u p¨, T q " u T and We use the Galerkin's method to construct solutions of certain finite-dimension approximations to (4.4.12).

We notice first that the symmetric bilinear form q Bpu, vq :" ş Γ µψ ´1BuBv is such that pu, vq Þ Ñ pu, vq L 2 pΓq `q Bpu, vq is an inner product in E equivalent to the standard inner product in E, namely pu, vq E " pu, vq L 2 pΓq `şΓ BuBv. Therefore, by standard Fredholm's theory, there exist • a non decreasing sequence of nonnegative real numbers pλ k q 8

k"1 , that tends to `8 as k Ñ 8

• A Hilbert basis pe k q 8

k"1 of L 2 pΓq, which is also a a total sequence of E (and orthogonal if E is endowed with the scalar product pu, vq L 2 pΓq `q Bpu, vq), such that q Bpe k , eq " λ k pe k , eq L 2 pΓq , for all e P E. (4.4.19) Note that ż

Note also that e k is a weak solution of

(4. 4.20) which implies that e k | Γα P C 2 pΓ α q for all α P A.

Finally, the sequence pf k q 8 k"1 given by f k " ψ ´1e k is a total family in F (but is not orthogonal). such that for all k " 1, . . . , n,

where we have set i m`1 " i 0 . Now using (4. where the last identity comes from the assumption that v T P V (the continuity of v T ).

Lemma 4.4.17. If u T " Bv T P F , then the weak solution u of (4.4.12) satisfies u " Bv where v is the unique solution of (4.4.1).

Proof of Lemma 4.4.17. For simplicity, we write the proof in the case when BΓ " H. The proof is similar in the other case.

Let us fix some vertex ν k P BΓ. From standard regularity results for Hamilton-Jacobi equation with homogeneous Neumann condition, we know that that there exists ω, a closed neighborhood of tν k u in Γ made of a single straigt line segment and containing no other vertices of Γ than ν k , such that v| ωˆp0,T q P L 2 p0, T ; H 3 pωqq X Cpr0, T s; H 2 pωq X W 1,2 p0, T ; H 1 pωqq. Hence, v satisfies the Hamilton-Jacobi equation at almost every point of ω ˆp0, T q. Moreover the equation

holds for almost every t P p0, T q and in L 2 p0, T q.

For every x P Γ and t P r0, T s, we define v px, tq " v pν k , tq `żÝÑ ν k x u pζ, tq dζ. We claim that v is a solution of (4.4.1). First, v p¨, tq is continuous on Γ. Indeed, vpy, tq ´vpx, tq " ş Ý Ñ xy upζ, tqdζ. On the other hand, u P Cpr0, T s; F q Ă L 8 pΓ ˆr0, T sq. It follows that |vpy, tq ´vpx, tq| ď ||u|| L 8 pΓˆr0,T sq distpx, yq which implies that v p¨, tq is continuous on Γ.

Next, from the terminal conditions for u,

where the last identity follows from the continuity of v T on Γ.

Let us check the Kirchhoff condition for v. Take ν i P V and α P A i . From (4.1.7), for a.e. t P p0, T q, B α vpν i , tq " n iα Bv| Γα pν i , tq and from (4.4.37), Bv| Γα pν i , tq " u| Γα pν i , tq. Since up¨, tq P F , we get

which is exactly the Kirchhoff condition for v at ν i .

There remains to prove v solves the Hamilton-Jacobi equation in ΓzV: Take x P Γ α zV for some α P A and consider a path ÝÑ ν k x Q L " ν i 0 Ñ ¨¨¨Ñ ν im Ñ x, where i 0 " k and ν im P Γ α . Let ν i m`1 be the other endpoint of Γ α . We proceed as in the proof of Lemma 4.4.16: the following conditions 1. e| Γα " 0 on each edge Γ α not contained in L 2. for all j " 0, . . . m, e| Γ j " 1 i j ăi j`1 ´1i j ąi j`1 if Γ j is the edge joining ν i j and ν i j`1 define a unique piecewise constant function e which takes at most two values on L, namely ˘1. Note that e does not belong to E because epν k q " 0, but that e satisfies ř αPA i n iα e| Γα pν i q " 0 for all ν i P VzBΓ.

Using this function, a similar computation as in the proof of Lemma 4.4.16 implies that, for almost every t P p0, T q, B t vpx, tq ´Bt vpν k , tq " ´µα Bu| Γα px, tq `H px, u| Γα px, tqq ´f px, tq

Then, using (4.4.36) and the fact that Bv " u, the latter identity yields that for almost every px, tq P p0, T q ˆΓ, B t vpx, tq `µα B 2 v px, tq ´H px, Bv px, tqq `f px, tq " 0.

We have proven that v is a solution of (4.4.1). Since v is the unique solution of (4.4.1), we conclude that v " v and Bv " u.

We are now ready to give the proof of Theorem 4.4.5.

Proof of Theorem 4.4.5. Since Bv " u by Lemma 4.4.17 and u satisfies (4.4.18) by Theorem 4.4.11, we obtain that v P L 2 `0, T ; H 3 pΓq ˘and B t v P L 2 `0, T ; H 1 pΓq ˘and (4.4.9) holds.

Therefore, using an interpolation result combined with Sobolev embeddings, see [START_REF] Amann | Compact embeddings of vector-valued Sobolev and Besov spaces[END_REF] or Lemma 4.6.2 in the Appendix, v P C 1`σ,σ{2 pΓ ˆr0, T sq for some 0 ă σ ă 1. Finally, we know that since f P W 1,2 p0, T, H 1 b pΓqq, f | Γαˆr0,T s P C η,η pΓ α ˆr0, T sq for all η P p0, 1{2q. If f P C η, η 2 pΓ α ˆr0, T sq for some η P p0, 1{2q, we claim that v P C 2,1 pΓ ˆr0, T sq. This is a direct consequence of a theorem of Von Below, see the main theorem in [START_REF] Below | Classical solvability of linear parabolic equations on networks[END_REF], for the (modified) heat equation ´Bt w ´µα B 2 w " gpx, tq, (4.4.38) with the same Kirchhoff conditions as in (4.4.1): Note that if the terminal Cauchy condition for w is wp¨, t " T q " v T and if g " f ´Hpx, Bvq, then w " v. Now g " f ´Hpx, Bvq P C τ, τ 2 pΓ α ˆr0, T sq, where 1{2 ą τ " minpσ, ηq ą 0. Using the result in [START_REF] Below | Classical solvability of linear parabolic equations on networks[END_REF], we obtain that v " w P C 2`τ,1`τ {2 pΓ α ˆr0, T sq, then that v is a classical solution of (4.4.1).

Existence, uniqueness and regularity for the MFG system (Proof of Theorem 4.1.11)

Proof of existence in Theorem 4. 1.11. First of all, given m 0 and v T , let us construct the map T from L 2 p0, T ; V q to itself as follows.

Given v P L 2 p0, T ; V q, we first define m as the weak solution of (4.3.1) with initial data m 0 and b " H p px, Bvq. We know that m P L 2 p0, T ; W q X Cpr0, T s; L 2 pΓqq X W 1,2 p0, T ; V 1 q.

We claim that if v n Ñ v in L 2 p0, T ; V q then H p p¨, Bv n q tends to H p p¨, Bvq in L 2 pΓ ˆp0, T qq. To prove the claim, we argue by contradiction: assume that there exist a positive number and a subsequence v φpnq such that }H p p¨, Bv φpnq q ´Hp p¨, Bvq} L 2 pΓˆp0,T qq ą . Then since Bv φpnq tends to Bv in L 2 pΓ ˆp0, T qq, we can extract another subsequence v ψpnq from v φpnq such that Bv ψpnq tends to Bv almost every where in Γ ˆp0, T q. From the continuity of H p , we deduce that H p p¨, Bv ψpnq q tends to H p p¨, Bvq almost everywhere in Γ ˆp0, T q. Since there exists a positive constant C 0 such that }H p p¨, Bv ψpnq q} 8 ď C 0 , }H p p¨, Bvq} 8 ď C 0 , Lebesgue dominated convergence theorem ensures that H p p¨, Bv ψpnq q tends to H p p¨, Bvq in L 2 pΓ ˆp0, T qq, which is the desired contradiction.

To summarize, H p p¨, Bv n q tends to H p p¨, Bvq in L 2 pΓ ˆp0, T qq on the one hand, and for a positive constant C 0 , }H p p¨, Bv n q} 8 ď C 0 , }H p p¨, Bvq} 8 ď C 0 . Using Lemma 4.3.4, we see that m n , the weak solution of (4.3.1) with initial data m 0 and b " H p px, Bv n q converges to m in L 2 p0, T ; W q X L 8 `0, T ; L 2 pΓq ˘X W 1,2 p0, T ; V 1 q. Hence, the map v Þ Ñ m is continuous from L 2 p0, T ; V q to L 2 p0, T ; W q X L 8 `0, T ; L 2 pΓq ˘X W 1,2 p0, T ; V 1 q. Moreover, the a priori estimate (4.3.3) holds uniformly with respect to v.

Then, knowing m, we construct T pvq " r v as the unique weak solution of (4.4.1) with f px, tq " V rmp¨, tqspxq. Note that m Þ Ñ f is continuous and locally bounded from L 2 pΓˆp0, T qq to L 2 pΓp 0, T qq. Then Lemma 4.4.4 ensures that the map m Ñ ṽ is continuous from L 2 pΓ ˆp0, T qq to L 2 `0, T ; H 2 pΓq ˘XL 8 p0, T ; V qXW 1,2 p0, T ; L 2 pΓqq. From Aubin-Lions theorem, see Lemma 4.6.1, m Ñ ṽ maps bounded sets of L 2 pΓ ˆp0, T qq to relatively compact sets of L 2 p0, T ; V q.

Therefore, the map T : v Þ Ñ ṽ is continuous from L 2 p0, T ; V q to L 2 p0, T ; V q and has a relatively compact image. Finally, we apply Schauder fixed point theorem [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]Corollary 11.2] and conclude that the map T admits a fixed point v. We know that v P L 2 `0, T ; H 2 pΓq ˘X L 8 p0, T ; V q X W 1,2 p0, T ; L 2 pΓqq and m P L 2 p0, T ; W q X L 8 `0, T ; L 2 pΓq ˘X W 1,2 p0, T ; V 1 pΓqq.

Hence, there exists a weak solution pv, mq to the mean field games system (4.1.24). Theorem 4.1.11. We assume that there exist two solutions pv 1 , m 1 q and Chapter 4. A Class of Mean Field Games on Networks. Part two: Finite Horizon Games pv 2 , m 2 q of (4.1.24). We set v " v 1 ´v2 and m " m 1 ´m2 and write the system for v, m

Proof of uniqueness in

Γα m 1 rH px, Bv 2 q ´H px, Bv 1 q ´Bp H px, Bv 1 q Bvs dx `ÿ αPA ż Γα m 2 rH px, Bv 1 q ´H px, Bv 2 q `Bp H px, Bv 1 q Bvs dx " 0.

Since V is strictly monotone, the first sum is nonnegative. Moreover, ż T 0 ż Γ B t pm vq dxdt " ż Γ rmpx, T q vpx, T q ´mpx, 0q vpx, 0qsdx " 0, since vpx, T q " 0 and mpx, 0q " 0. From the convexity of H and the fact that m 1 , m 2 are nonnegative, the last two sums are nonnegative. Therefore, all the terms are zero and thanks again to the fact that V is strictly increasing, we obtain m 1 " m 2 . From Lemma 4.4.2, we finally obtain v 1 " v 2 .

Proof of regularity in Theorem 4.1.11. We make the stronger assumptions written in Section 4.1.4 on the coupling operator V . We know that V rms P W 1,2 p0, T ; H 1 b pΓqq X P CpΓ ˆr0, T sq. Assuming also that v T P V and Bv T P F , we can apply the regularity result in Theorem 4.4.5: v P L 2 `0, T ; H 3 pΓq ˘X W 1,2 `0, T ; H 1 pΓq ˘.

Moreover, since V rms P W 1,2 p0, T, H 1 b pΓqq, we know that pV rmsq| Γαˆr0,T s P C σ,σ{2 pΓ α ˆr0, T sq for all 0 ă σ ă 1{2. If v T P C 2`η X D for some η P p0, 1q (D is defined in (4.1.13)), then from Theorem 4.4.5, v P C 2`τ,1`τ {2 pΓ ˆr0, T sq for some τ P p0, 1q and the boundary value problem for v is satisfied in a classical sense.

In turn, if for all α P A, B p H α px, pq is a Lipschitz function defined in Γ α ˆR, and if m 0 P W , then we can use the latter regularity of v and arguments similar to those contained in the proof of Theorem 4.2.5 and prove that m P Cpr0, T s; W q X W 1,2 p0, T ; L 2 pΓqq X L 2 p0, T ; H 2 b pΓqq.

4.6. Appendix: Some continuous and compact embeddings 4.6 Appendix: Some continuous and compact embeddings Lemma 4.6.1. (Aubin-Lions Lemma, see [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites non linéaires[END_REF]) Let X 0 ,X and X 1 be function spaces, (X 0 and X 1 are reflexive). Suppose that X 0 is compactly embedded in X and that X is continuously embedded in X 1 . Consider some real numbers 1 ă p, q ă `8. Then the following set tv : p0, T q Þ Ñ X 0 : v P L p p0, T ; X 0 q , B t v P L q p0, T ; X 1 qu is compactly embedded in L p p0, T ; Xq.

Lemma 4.6.2. (Amann, see [START_REF] Amann | Compact embeddings of vector-valued Sobolev and Besov spaces[END_REF]) Let φ : ra, bs ˆr0, T s Ñ R such that φ P L 2 p0, T ; H 2 pa, bqq and B t φ P L 2 p0, T ; L 2 pa, bqq. Then φ P C s p0, T ; H 1 pa, bqq for some s P p0, 1{2q.

This result is a consequence of the general result [11, Theorem 1.1] taking into account [START_REF] Amann | Compact embeddings of vector-valued Sobolev and Besov spaces[END_REF]Remark 7.4]. More precisely, we have E 1 :" H 2 pa, bq compact ãÑ E :" H 1 pa, bq ãÑ E 0 :" L 2 pa, bq.

Let r 0 " r 1 " r " 2, σ 0 " 0, σ 1 " 2 and σ " 1. For any ν P p0, 1q, we define 1 r ν " 1 r 0 `1 ´ν r 1 , σ ν :" p1 ´νqs 0 `νs 1 .

This implies that r ν " 2 and σ ν " 2ν. Therefore, if ν P p1{2, 1q, then the following inequality is satisfied σ ´1{r ă σ ν ´1{r ν ă σ 1 ´1{r 1 .

Hence, we infer from [START_REF] Amann | Compact embeddings of vector-valued Sobolev and Besov spaces[END_REF]Remark 7.4] E 1 ãÑ pE 0 , E 1 q ν,1 ãÑ pE 0 .E 1 q ν,rν " W σν ,rν pa, bq ãÑ E, where pE 0 , E 1 q ν,1 , pE 0 .E 1 q ν,rν are interpolation spaces. This is precisely the assumption allowing to apply [11, Theorem 1.1], which gives the result of Lemma 4.6.2.

Titre : Équation de Hamilton-Jacobi et jeux à champ moyen sur les réseaux

Mots clés : Problèmes de contrôle optimal, équation de Hamilton-Jacobi, jeux à champ moyen, réseaux Résumé : Cette thèse porte d'une part sur l'éude d'éuations de Hamilton-Jacobi-Bellman (HJB) associées à des problèmes de contrôle optimal et d'autre part de jeux à champ moyen (MFG) avec la particularitéqu'on se place sur des réseaux, pour lesquels on autorise différentes dynamiques et coûts dans chaque arête.

Dans la première partie, on considère un problème de contrôle optimal sur les réseaux dans lesquels on rajoute des coûts d'entré (ou de sortie) aux sommets conduisant à une éentuelle discontinuitéde la fonction valeur. Celle-ci est caractéisée comme l'unique solution de viscosité d'une eq ation Hamilton-Jacobi (HJ) pour laquelle une condition de jonction adéquate est établie.

La deuxième partie concerne les MFG stochastiques sur les réseaux dans le cas ergodique. Ils sont décrits par un système couplant une équation de HJB et une équation de Fokker-Planck, dont les inconnues sont: la densité m qui est en général discontinue aux sommets et satisfait deux conditions de transmission aux sommets; la fonction valeur v qui est continue et satisfait des conditions de Kirchhoff aux sommets et enfin la constante ergodique ρ. L'existence et l'unicitésont prouvés pour des Hamiltoniens sous-quadratiques et des couplages bornés inférieurement généraux.

Enfin, dans une dernière partie, nous étudions le même problème non stationnaire pour des Hamiltoniens sous-linéaires et un couplage régularisant. La principale difficulté supplémentaire par rapport au cas stationnaire est d'établir la réularite. Notre approche consiste à éudier la solution de l'equ tion de HJ déivé pour gagner de la régularité sur la solution de l'equ tion initiale.

Title : Hamilton-Jacobi equations and Mean field games on networks

Keywords : Optimal control problems, Hamilton-Jacobi equation, Mean Field Games, networks

Abstract:

The dissertation focuses on the study of Hamilton-Jacobi-Bellman (HJB) equations associated with optimal control problems and mean field games (MFG) problems in the case when the state space is a network. Different dynamics and running costs are allowed in each edge of the network.

In the first part, we consider an optimal control on networks in which there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. The value function is characterized as the unique viscosity solution of a Hamilton-Jacobi (HJ) equation for which an adequate junction condition is established.

The second part is about stochastic MFG for which the state space is a network in the ergodic case. They are described by a system coupling a HJB equation and a Fokker-Planck equation, whose unknowns are: the density m which is in general discontinuous at the vertices and satisfies dual transmission condition; the value function v is continuous and satisfies general Kirchhoff conditions at the vertices and the ergodic constant ρ. Existence and uniqueness are proven for subquadratic Hamiltonian and the general coupling term which is bounded from below.

Finally, in the last part, we study non-stationary stochastic MFG on networks. The transition conditions for v and m are similar to the ones given in second part. We prove the existence and uniqueness of a weak solution for sublinear Hamiltonian and bounded non-local regularizing coupling term. The main additional difficulty compared to the stationary case is to establish the regularity of the system. Our approach is to study the solution of the derived HJ equation to gain regularity over the initial equation.