These De L'

Universite De

Ecole Doctorale

M Mamadou

Lamine Diagne

Directeurs De

M Bernhard Maschke

M Françoise Couenne

Jury M Yann

Le Gorrec

Ensmm De Besançon

M Didier

M Michaël

Di Loreto

Mme Françoise Couenne

Claude Universite

Bernard -Lyon

M François-Noël

M Le Professeur Hamda

Ben Hadid

Mme Le Professeur

Germain Gillet

M Alain Helleu

Composantes Sante

J Etienne

D Bourgeois

F De Marchi

F Fleury

H Parrot Département

M N Siauve

S Akkouche

A Goldman

H Ben Hadid

Mme S Fleck

M C Collignon

M P Fournier

M G Pignault

C Coulet

R Bernard

Modelling and control of systems of conservation laws with a moving interface: an application to an extrusion process

To my mother, Fatoumata Sylla Résumé en français L'étude de la commande des systèmes de lois de conservation couplés par une interface mobile reste très peu développée dans la littérature. Ces problèmes d'interface apparaissent dans la modélisation d'un grand nombre de systèmes physiques donnant lieu à des phénomènes de changement de phases, de croissance de cristaux, d'évolution biologique (modèle d'évolution de tumeur) etc. Dans cette thèse, nous nous intéressons aux systèmes hyperboliques couplés à travers une interface mobile. Cette étude est développée autour d'un modèle bi-zone d'un procédé d'extrusion. D'autre part nous proposons une analyse des structures d'interface en utilisant le formalisme Hamiltonien à port. Les contributions de cette thèse sont évoquées ci dessous : le Chapitre 3 est dédié à la modélisation des procédés d'extrusion. Le modèle est basée sur la description des phénomènes de transport de masse et de chaleur dans une extrudeuse. Nous obtenons par les bilans de matière et d'énergie des équations de transport couplées à travers une interface mobile. Deux relations de couplage à l'interface sont étudiées : l'une repose sur l'hypothèse de continuité de pression et l'autre sur la continuité du ux de quantité de mouvement. Des simulations numériques sont proposées pour illustrer la cohérence du modèle. le Chapitre 4 traite de l'analyse mathématique des équations de transport dénies sur des domaines temps-variants complémentaires. Nous utilisons le théorème du point xe de Banach pour prouver l'existence et l'unicité de la solution pour le modèle de l'extrudeuse. La méthode de résolution repose sur un changement de coordonnées qui permet de dénir le système sur un domaine xe. Cette normalisation fait apparaître des termes de convection ctifs dans les coecients multiplicatifs de l'opérateur diérentiel de transport. Nous obtenons nalement l'existence, l'unicité et la régularité de la solution faible du problem de Cauchy. le Chapitre 5 concerne l'étude d'un système hyperbolique couplé à une interface mobile.

Il s'agit des équations issues du bilan de masse dans une extrudeuse sous certaines hypothèses simplicatrices. Nous représentons les equations aux dérivées partielles et ordinaires couplées par un système à retards particulier : le retard est sur l'entrée et dépend de l'état. Cette représentation est obtenue en résolvant l'équation de transport de masse par la méthode des caractéristiques. La stabilisation de l'interface autour d'une position d'équilibre s'eectue au moyen d'un prédicteur. le Chapitre 6 est une extension de l'analyse des problèmes d'interface à une classe particulière de systèmes. Nous considérons deux systèmes de deux lois de conservation décrites sous le formalisme Hamiltonien à port et dénis sur des domaines complémentaires variant dans le temps. Nous proposons une formulation basée sur l'utilisation de la fonction couleur ou fonction caractéristique du domaine pour étendre les variables sur l'ensemble du domaine et dénissons un système Hamiltonien augmenté des fonctions couleur. Dans un deuxième temps nous considérons un système à interface mobile et nous montrons que le système augmenté d'une entrée dénie comme étant la vitesse de l'interface avec une sortie conjuguée reste un système Hamiltonien à port.

Modélisation d'un procédé d'extrusion pour la commande

Les procédés d'extrusion sont destinés à la fabrication de produits nis ou semi-nis par utilisation de vis sans n en rotation dans un fourreau cylindrique disposant d'un système de chauage comme le montre le schéma de la gure 4.4.1.

La matière tout au long de son parcours subit certaines transformations (réaction chimique, cuisson, homogénéisation...). Elle est ainsi transportée vers le dispositif de mise en forme, la lière, par lequel la matière est expulsée par des petits orices sous l'action de fortes pressions. D'un point de vue industriel, on cherche à obtenir à la sortie de la machine un débit régulier, avec un matériau homogène ayant les propriétés d'usage désirées, à température contrôlée, et des conditions de production satisfaisantes (débit maximal, consommation énergétique limitée). Dans ce chapitre nous élaborons un modèle dynamique qui pourra être utilisé pour la commande.

Ces procédés sont complexes et composés de phénomènes hydrodynamiques, thermiques et physico-chimiques fortement couplés. Une littérature importante existe sur la modélisation en régime stationnaire des écoulements de la matière dans une extrudeuse [START_REF] Vergnes | Global computer software for polymer ows in corotating twin screw extruders[END_REF][START_REF] Fukuoka | Numerical analysis of a reactive extrusion process. part ii: simulations and verications for the twin screw extrusion[END_REF][START_REF] Lili | Numerical simulation of reactive extrusion processes for activated anionic polymerization[END_REF]. Des modèles dynamiques ont aussi été développés. On trouve des modèles issus de l'identication paramétrique [START_REF] Kulshreshtha | Simulating the performance of a control system for food extruders using model-based set-point adjustment[END_REF][START_REF] Haley | On-line system identication and control design of an extrusion cooking process: Part I. system identication[END_REF][START_REF] Wang | Dual-target predictive control and application in food extrusion[END_REF][START_REF] Wang | Continuous-time model predictive control of food extruder[END_REF] développés pour la commande PID ou prédictive.

Des modèles dynamiques en dimension nie [START_REF] Choulak | Generic dynamic model for simulation and control of reactive extrusion[END_REF] et innie [START_REF] Kulshrestha | An unsteady state model for twin screw extruders[END_REF][START_REF] Kim | Isothermal transient startup for starved ow modular co-rotating twin screw extruder[END_REF][START_REF] Kim | Non-isothermal transient startup for starved ow modular co-rotating twin screw extruder[END_REF][START_REF] Janssen | A dynamic model for multiple steady states in reactive extrusion[END_REF][START_REF] Janssen | A dynamic model accounting for oscillating behavior in extrusion reaction[END_REF] ont été développés. Citons le modèle de dimension nie d'extrusion réactive proposé par [START_REF] Choulak | Generic dynamic model for simulation and control of reactive extrusion[END_REF] prédisant le régime dynamique sous diérentes conditions opératoires (débit d'alimentation, concentration à l'entrée des diérentes espèces, puissance de chaue des fourreaux) et basé sur une représentation par une cascade de RCP A (réacteurs continus parfaitement agités) avec reux. En dimension innie, les modèles 1D proposés dans [START_REF] Kulshrestha | An unsteady state model for twin screw extruders[END_REF] et [START_REF] Li | Modelling extrusion cooking[END_REF] mettent en évidence les couplages des phénomènes de transport de la matière et d'échange de chaleur.

Ces modèles de connaissance, en dimension innie, restent toujours inexploités pour des objectifs de commande. La diculté pour la commande de ces procédés à partir de ces modèles résulte de l'existence d'une interface mobile entre deux zones aux propriétés d'écoulement diérentes.

Il existe une littérature très fournie concernant la modélisation et la simulation de systèmes avec interface mobile. Un problème très souvent traité et référencé, comme le problème de Stefan, étudie le déplacement de l'interface entre deux zones spatiales ayant des dynamiques diérentes sous l'inuence de phénomènes diusifs et/ou convectifs. Il concerne un vaste champ d'applications [START_REF] Gupta | The classical Stefan problem. Basic concepts, Modelling and Analysis[END_REF] comme par exemple, le gonement de nanocapsules [START_REF] Bouchemal | Polyamides nanocapsules : Modeling and wall thickness estimation[END_REF], la lyophilisation [START_REF] Velardi | Development of simplied models for the freeze-drying process and investigation of the optimal operating conditions[END_REF], la cuisson [START_REF] Purlis | A moving boundary problem in a food material undergoing volumechange -simulation of bread baking[END_REF].

La commande de systèmes à frontière mobile a aussi été étudiée dans [START_REF] Armaou | Robust control of parabolic pde systems with timedependent spatial domains[END_REF] pour un modèle parabolique semi-linéaire et dans [START_REF] Daraoui | Model predictive control during the primary drying stage of lyophilisation[END_REF] pour un modèle parabolique non linéaire. Dans cette seconde publication, le problème de commande est étudié à partir d'un modèle linéarisé de dimension nie. Une discussion concernant les problèmes de commande d'écoulements réactifs à frontière mobile est aussi proposée dans [START_REF] Petit | Control problems for one-dimensional uids and reactive uids with moving interfaces[END_REF].

Dans ce chapitre nous proposons un modèle bi-zone d'extrudeuse composé de deux systèmes d'équations d'évolution de type hyperbolique couplés par une interface mobile dont la dynamique est donnée par une EDO. Le modèle 1D bi-zone est déduit des bilans de matière et d'énergie exprimées en termes de taux de remplissage et d'un taux d'humidité ainsi que de la température, les variables de commande étant la vitesse de rotation des vis, la température du fourreau et le débit d'alimentation. Les relations d'interface sont déduites des hypothèses de continuité de ux de quantité de mouvement ou de pression l'interface et permettent de calculer la pression en amont de la lière.

Modèle bi-zone d'une extrudeuse

Nous présentons ici un modèle idéalisé des phénomènes de transfert de matière et de chaleur dans une extrudeuse corotative bi-vis dont le prol de vis est supposé uniforme. De plus nous supposons l'existence de deux zones (représentées sur la gure 4.4.5), une zone de convoyage et une zone de pompage suivant que la matière remplisse complètement ou non le volume ecace de l'extrudeuse [START_REF] Kulshrestha | An unsteady state model for twin screw extruders[END_REF] ; [START_REF] Li | Modelling extrusion cooking[END_REF]. La zone partiellement remplie, notée PFZ, est une zone de transport avec une grande surface libre permettant l'introduction aisée de réactifs : la matière est chauée et transportée vers la zone de fusion. Cette région est caractérisée par des débits de reux nuls (la matière est transportée en aval sous l'eet de la rotation de la vis), une pression constante égale à la pression atmosphérique et de faibles échanges thermiques entre la matière et le fourreau.

Dans la zone entièrement remplie, notée FFZ, la matière est chauée, mélangée, mise sous pression et convectée vers la lière avec des échanges thermiques importants entre le fourreau et la matière. Un gradient de pression se crée à partir de la lière provoquant la circulation d'un débit de reux. De forts auto-échauements par dissipation visqueuse sont aussi présents.

Ces deux zones sont couplées à travers une interface mobile notée l(t) qui sépare la zone où l'extrudeuse est complètement remplie (taux de remplissage égal à 1) de celle où elle est partiellement remplie (taux de remplissage < 1). Cet interface caractérise donc la discontinuité du taux de remplissage.

Nous considérons que l'extrudat est un mélange contenant de l'eau, ce qui est le cas par exemple de l'extrusion alimentaire qui donne lieu à l'élaboration de produits complexes associant céréales et matières grasses et pour laquelle la teneur en eau est un facteur important. De plus nous faisons les hypothèses suivantes : l'écoulement est uni-dimensionnel ; c'est l'écoulement longitudinal qui détermine le débit de l'extrudeuse, la capacité calorique C, la viscosité η et la masse volumique ρ 0 sont supposées constantes et les pertes de chaleur sont négligeables sur l'axe des vis. Le modèle repose sur l'écriture des bilans de masse totale, de l'eau et du bilan d'énergie pour chaque zone. Le bilan de quantité de mouvement est écrit et réduit à un modèle unidimensionnel stationnaire [START_REF] Booy | Isothermal ow of viscous liquids in corotating twin screw devices[END_REF]. Les phénomènes de transfert de chaleur et de matière sont décrits selon les zones (P F Z ou F F Z) par les variables suivantes : le taux de remplissage f (x, t) : il est déni comme le rapport entre la densité linéique de volume occupé par la matière et la densité linéique de volume disponible ou ecace (dans la direction de l'écoulement de coordonnée x) ; le taux d'humidité M (x, t) : c'est le rapport de densité linéique de volume d'eau présente dans la matière sur la densité de volume disponible ; la température T (x, t) ; la pression P (x, t).

x ∈ [0, L] est la variable spatiale et t ∈ R + la variable temporelle.

1.1.1.2 Modèle de la zone partiellement remplie ou P F Z Dans cette zone le taux de remplissage f p (x, t) varie dans l'intervalle [0, 1[. L'expression des bilans de matière et d'énergie aboutit aux équations d'évolution suivantes sur le taux de remplissage f p (x, t), le taux d'humidité M p (x, t) et la température T p (x, t) :

∂ ∂t ⎛ ⎜ ⎜ ⎝ f p (x, t) M p (x, t) T p (x, t) ⎞ ⎟ ⎟ ⎠ = -ξN (t)I 3 ∂ ∂x ⎛ ⎜ ⎜ ⎝ f p (x, t) M p (x, t) T p (x, t) ⎞ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎝ 0 0 Ω p (f p , N, T p , T Fp ) ⎞ ⎟ ⎟ ⎠
(1.1.1) I 3 est la matrice identité de dimension 3. Le premier terme de ces équations d'évolution consiste simplement en trois équations de transport à la vitesse v (t) = ξN (t) uniforme et proportionnelle à la vitesse de rotation des vis N (t) et au pas de vis ξ. Le terme source Ω intervient dans le bilan d'énergie uniquement et est déni par la fonction:

Ω p (f, N, T, T F ) = μηN 2 (t) f p (x, t)ρ 0 V ef f C + S ech α ρ 0 V ef f C (T F -T p ) (1.1.2)
Son premier terme correspond à la dissipation visqueuse due à la rotation et est proportionnelle au carré de la vitesse de rotation des vis N (t) et le second terme représente les échanges de chaleur entre le fourreau et la matière et est proportionnel à la diérence de température (T Fp -T p ) entre le fourreau et la matière dans l'extrudeuse. Les paramètres sont la capacité calorique C J kg -1 K -1 , la masse volumique ρ 0 kg m -3 , η P a s -1 le coecient de viscosité, μ J kg -1 K -1 le facteur de dissipation visqueuse de la matière, V ef f = S ef f ξ est le volume ecace de l'extrudeuse (où S ef f est la section ecace de l'extrudeuse et ξ son pas de vis) et S ech m 2 est la surface d'échange, α Jm -2 s -1 K -1 le coecient d'échange thermique entre la matière et fourreau. Rappelons nalement que l'on suppose l'équilibre de pression avec la pression atmosphérique: P (x, t) = P 0 .

1.1.1.3 Modèle de la zone entièrement remplie ou F F Z Par dénition de la zone entièrement remplie, le taux de remplissage atteint sa borne supérieure et vérie: f f (x, t) = 1. C'est donc la variable conjuguée, la pression qui varie dans cette zone. De l'hypothèse de la stationnarité de l'écoulement, on déduit l'expression du gradient de pression :

∂P (x, t) ∂x = η V ef f N (t)ρ 0 -F d Bρ 0 (1.1.3)
qui est proportionnel à la diérence entre la capacité maximale de pompage (V ef f N (t)ρ 0 ) et le débit net en sortie de lière F d :

F d = K d η
ΔP avec ΔP = (P (L, t) -P 0 ) (1. 1.4) où K d caractérise la géométrie de la lière et dépendant aussi des caractéristiques géométriques de la vis B.

Les bilans de matière et d'énergie impliquent les équations d'évolution suivantes sur le taux d'humidité M f (x, t) et la température T f (x, t) dans la zone entièrement remplie :

∂ ∂t M f (x, t) T f (x, t) = - F d ξ ρ 0 V ef f I 2 ∂ ∂x M f (x, t) T f (x, t) + 0 Ω f (N, T f , T F f ) (1.1.5)
De nouveau le premier terme de ces équations d'évolution consiste simplement en deux équations de transport à la vitesse v (t) = F d ξ ρ 0 V ef f et le second Ω 2 représente la source d'énergie thermique due à dissipation visqueuse et aux échanges de chaleur entre le fourreau et la matière.

Ω f (N, T, T F ) = μηN 2 (t) ρ 0 V ef f C + S ech α ρ 0 V ef f C (T F -T f ) (1.1.6)
1.1.2 Dynamique de l'interface mobile l(t) et relations d'interface

1.1.2.

Relations d'interface

Nous faisons dans la suite l'hypothèse que la température et le taux d'humidité sont continus à l'interface :

T p (l -, t) = T f (l + , t) M p (l -, t) = M f (l + , t) où l ± (t) = lim ε→0 l(t) ± ε
La troisième relation d'interface est basée sur deux hypothèses qui permettent de résoudre l'équation 1.1.3 et de calculer la pression dans la zone entièrement remplie.

Cas 1 : la continuité du ux de quantité de mouvement F (l -, t)v(l -, t) + P (l -, t)f p (l -, t)S ef f = F (l + , t)v(l + , t) + P (l + , t)S ef f (1.1.7) avec :

F (l -, t) = ρ 0 N (t)V ef f f p (l -, t) F (l + , t) = F d (t) v(l -, t) = ξN (t) v(l + , t) = ξF d (t)
ρ 0 V ef f P (l -, t) = P 0 L' intégration de l'équation 1.1.3 donne :

P (l + , t) = P (L, t) + η V ef f Nρ 0 -F d Bρ 0 (l + -L) (1.1.8)
Des équations 1.1.4, 1.1.7 et 1.1.8 nous déduisons la pression à l'interface :

P (L, t) = P 0 + -[1 + K d Bρ 0 (L -l + )] + √ Δ 2K 2 d η 2 ρ 0 S 2 ef f
(1.1.9) avec Δ = 1 + K d Bρ 0 (Ll + ) 2 + Ω f p (l -, t), N(t), l +

(1. 1.10) et

Ω = 2K d ηS ef f 2 ηV ef f N (t) Bρ 0 (L -l + ) +ξ 2 N 2 (t)f p (l -, t) -(1 -f p (l -, t)) P 0 ρ 0 (1.1.11)
Le modèle suppose l'existence de deux zones P F Z et F F Z, ce qui n'est vrai que si ΔP = P (L, t) -P 0 est positif. De ce fait Ω doit être strictement positif. Cette contrainte sur la fonction Ω (f p (l -, t), N(t), l + ) permet de déduire pour une vitesse de rotation donnée, la valeur minimum du taux de remplissage à la frontière pour que l'hypothèse sur l'existence des zones P F Z et F F Z soit satisfaite.

Cas 2 : la continuité de pression

⎧ ⎨ ⎩ P (L, t) = P 0 + ηV ef f N (t)ρ 0 (L-l(t)) (Bρ 0 +K d (L-l(t))) F d (t) = K d V ef f N (t)ρ 0 (L-l(t)) (Bρ 0 +K d (L-l(t)))
(1.1.12)

Dans ce cas de gure le débit en sortie de lière est positif si (Ll(t)) > 0 ce qui est nécessairement vrai. Cette relation d'interface n'exige aucune contrainte sur le taux de remplissage à la frontière.

1.1.2.2 Bilan de masse et dynamique de l'interface La P F Z et la F F Z sont séparées par une interface mobile l(t). L'équation de la frontière s'obtient par un bilan de matière totale sur la zone entièrement remplie :

dl(t) dt = F d -F (l(t)) ρ 0 S ef f (1 -f p (l -, t))
(1. 1.13) F (l(t)) représente le débit entrant dans la zone entièrement remplie et F d est le débit en sortie de lière. Le terme ρ 0 S ef f (1 -f p (l -, t)) montre que l'évolution de la frontière dépend du remplissage du volume non occupé par la matière ou volume disponible. Nous rappelons que ce modèle n'est valable que pour f p (l -) < 1.

Conditions frontières et conditions initiales

Les conditions limites en x = 0 et en x = L sont établies sous l'hypothèse de la continuité du débit de matière, de la température et du ux de quantité de mouvement. Le débit, la température et le taux d'humidité sont supposés continus : le débit est égal au débit d'alimentation en matière F in (t), la température T p (0, t) et le taux d'humidité M p (0, t) sont égaux à ceux de la matière mise dans le circuit d'alimentation soit T in (t) et M in (t) :

f p (0, t) =
F in (t) ρ 0 NV ef f (1.1.14) T p (0, t) = T in (t) (1.1.15) M p (0, t) = M in (t) (1.1.16) 1.1.3.2 Conditions initiales À t = 0 nous supposons l'existence des deux zones d'où les conditions initiales suivantes : pour la P F Z :

f p (x, 0) = f p0 (1.1.17) M p (x, 0) = M p0 (1.1.

18)

T p (x, 0) = T p0 (1.1.19) pour la F F Z :

M f (x, 0) = M f 0 (1.1.20) T f (x, 0) = T f 0 (1.1.21)
1.2 Etude du problème de Cauchy associé au modèle de l'extrudeuse Le problème d'existence de solution pour les modèles de système à frontières mobiles est largement étudié dans le cas de modèles paraboliques [START_REF] Muntean | Well-posedness of a moving-boundary problem with two moving reaction strips[END_REF] et des modèles hyperboliques [START_REF] Raul Borsche | On the coupling of systems of hyperbolic conservation laws with ordinary dierential equations[END_REF] ; [START_REF] Coutand | Well-posedness in smooth function spaces for movingboundary 1-d compressible euler equations in physical vacuum[END_REF] où des EDPs sont dénies sur la première zone spatiale, une EDO sur l'interface et un modèle algébrique sur la seconde zone spatiale. Le modèle d'extrudeuse considéré ici se distingue par le fait qu'il est composé de deux systèmes d'équations d'évolution de type hyperbolique couplés par une interface mobile. Nous proposons d'étudier le modèle associé au procédé d'extrusion par le théorème du point xe de Banach. Dans un premier temps nous représentons le système dans le domaine (0, 1) en utilisant les changements de coordonnées appropriés pour chaque zone. L'argument du point xe permet ensuite de dénir les conditions d'existence de solution pour ce problème.

Modèle adimensionnel à domaine xe (0, 1)

Dans cette section nous présentons deux changement de variables qui permettent d'écrire les équations d'évolution sur le domaine xe (0, 1). Le domaine total (0, L) est transformé en l'intervalle (0, 1) et la position de l'interface mobile est ramenée à la valeur xe 1. Un terme de convection ctive fonction de l(t) apparaît alors dans les équations d'évolution.

Le modèle normalisé est constitué: d'une équation aux dérivées ordinaires décrivant la dynamique dde l'interface ⎧ ⎨ ⎩ l(t) = F (l(t), N(t), f p (t, 1)) , t ∈ (0, T )

l(0) = l 0 ,
(1.2.1) d'une équation de transport de masse:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t f p (t, x) + α p (t, x)∂ x f p (t,
x) = 0, (t, x) ∈ (0, T ) × (0, 1)

f p (0, x) = f 0 p (x), x ∈ (0, 1) f p (t, 0) = F in (t) θ(N (t)) , t ∈ (0, T ) (1.2.2)
de deux equations de transport du taux d'humidité:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t M p (t, x) + α p (t, x)∂ x M p (t, x) = 0, (t, x) ∈ (0, T ) × (0, 1) ∂ t M f (t, x) + α f (t, x)∂ x M f (t, x) = 0, (t, x) ∈ (0, T ) × (0, 1) M p (0, x) = M 0 p (x), M f (0, x) = M 0 f (x), x ∈ (0, 1) M p (t, 0) = M in (t), M f (t, 0) = M p (t, 1), t ∈ (0, T ) (1.2.3) et de deux équations de transport de la température ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t T p (t, x) + α p (t, x)∂ x T p (t, x) = Ω p , (t, x) ∈ (0, T ) × (0, 1) ∂ t T f (t, x) + α f (t, x)∂ x T f (t, x) = Ω f , (t, x) ∈ (0, T ) × (0, 1) T p (0, x) = T 0 p (x), T f (0, x) = T 0 f (x), x ∈ (0, 1) T p (t, 0) = T in (t), T f (t, 0) = T p (t, 1), t ∈ (0, T ) (1.2.4)
(l, f p , M p , M f , T p , T f ) ∈ W 1,∞ (0, T ) × W 1,∞ (Q) × (C 0 ([0, T ]; L 2 (0, 1))) 4 , tel que pour tout τ ∈ [0, T ], toute fonction test ϕ 1 , ϕ 2 = (ϕ 21 , ϕ 22 ), ϕ 3 = (ϕ 31 , ϕ 32 ) ∈ C 1 ([0, T ] × [0, 1]) tel que ϕ i (τ, x) = 0, ∀x ∈ [0, 1], i = 1, 2, 3, (1.2.5) ϕ i (t, 1) = 0, ∀t ∈ [0, T ], i = 1, 2, 3, (1.2.6) nous avons l(t) = l 0 + ¢ t 0 F (l(s), N(s), f p (s, 1)) ds, t ∈ (0, τ), (1.2.7) ¢ τ 0 ¢ 1 0 f p (∂ t ϕ 1 + ∂ x (α p ϕ 1 )) dx dt + ¢ 1 0 f 0 p (x)ϕ 1 (0, x) dx + ¢ τ 0 F in (t) θ(N (t)) α p (t, 0)ϕ 1 (t, 0) dt = 0, (1.2.8) ¢ τ 0 ¢ 1 0 M p (∂ t ϕ 21 + ∂ x (α p ϕ 21 )) dx dt + ¢ τ 0 ¢ 1 0 M f (∂ t ϕ 22 + ∂ x (α f ϕ 22 )) dx dt + ¢ τ 0 α p (t, 0)M in (t)ϕ 21 (t, 0) dt + ¢ τ 0 α f (t, 0)M p (t, 1)ϕ 22 (t, 0) dt + ¢ 1 0 M 0 p (x)ϕ 21 (0, x) dx + ¢ 1 0 M 0 f (x)ϕ 22 (0, x) dx = 0, (1.2.9) ¢ τ 0 ¢ 1 0 T p (∂ t ϕ 31 + ∂ x (α p ϕ 31 )) dx dt + ¢ τ 0 ¢ 1 0 T f (∂ t ϕ 32 + ∂ x (α f ϕ 32 )) dx dt + ¢ τ 0 ¢ 1 0 Ω p ϕ 31 dx dt + ¢ τ 0 ¢ 1 0 Ω f ϕ 32 dx dt + ¢ τ 0 α p (t, 0)T in (t)ϕ 31 (t, 0) dt + ¢ τ 0 α f (t, 0)T p (t, 1)ϕ 32 (t, 0) dt + ¢ 1 0 T 0 p (x)ϕ 31 (0, x) dx + ¢ 1 0 T 0 f (x)ϕ 32 (0, x) dx = 0. (1.2.10)
Nous proposons ces deux théorèmes:

Theorème 2. Soient T > 0 et (l e , N e , f pe ) un équilibre constant,

F (l e , N e , f pe ) = 0 (1.2.11) avec 0 < f pe < 1, 0 < l e < L.
Sous l'hypothèse de compatibilité en (0, 0)

F in (0) θ(N (0)) = f 0 p (0).
(1.2.12)

il existe ε 0 (dépendant de T ) tel que pour tout ε ∈ (0, ε 0 ], si

f 0 p (•)-f pe W 1,∞ + F in (•) θ(N (•)) -f pe W 1,∞ + N (•)-N e W
(l, f p , M p , M f , T p , T f ) ∈ W 1,∞ (0, T ) × W 1,∞ (Q) × (C 0 ([0, T ]; L 2 (0, 1)))
f p -f pe W 1,∞ + l -l e W 1,∞ C ε 0 • ε, (1.2.14) M p C 0 ([0,T ];L 2 (0,1)) C ε 0 • M 0 p L 2 + M in L 2 , (1.2.15) T p C 0 ([0,T ];L 2 (0,1)) C ε 0 • T 0 p L 2 + T in L 2 , (1.2.16) M f C 0 ([0,T ];L 2 (0,1)) C ε 0 • M 0 p L 2 + M in L 2 + M 0 f L 2 ,
(1.2.17)

T f C 0 ([0,T ];L 2 (0,1)) C ε 0 • T 0 p L 2 + T in L 2 + T 0 f L 2 , (1.2.18)
où C ε 0 est une constante dépendant de ε 0 mais indépendant de ε.

Theorème 3. Sous les hypothèses du Théorème 2, supposons que f 0

p (•) ∈ H 2 (0, 1), F in (•) θ(N (•)) ∈
H 2 (0, T ), et la condition de compatibilité suivante en (0, 0) vériée:

f 0 px (0) + l(0) ζN (0) • F in (0)θ(N (0)) -F in (0)θ (N (0))N (0) θ 2 (N (0)) = 0. (1.2.19) alors il existe ε 0 (dépendant de T ) tel que pour tout ε ∈ (0, ε 0 ], si f 0 p (•)-f pe H 2 (0,1) + F in (•) θ(N (•)) -f pe H 2 (0,T ) + N (•)-N e W
(l, f p , M p , M f , T p , T f ) ∈ W 1,∞ (0, T ) × C 0 ([0, T ]; H 2 (0, 1)) ×(C 0 ([0, T ]; L 2 (0, 1))) 4 avec : f p -f pe C 0 ([0,T ];H 2 (0,1)) C ε 0 • ε, (1.2.21)
où C ε 0 est une constante dépendant de ε 0 mais indépendant de ε.

Stabilisation de l'interface par une approche de commande prédictive

Cette partie traite de la stabilisation de la position de l'interface par une régulation du débit d'alimentation en matière dans une extrudeuse. Ce débit entrant agit sur l'équation de transport du taux de remplissage dans la zone partiellement remplie dénie sur un domaine variable dans le temps. Il apparaît dans ce contexte un retard du au temps de propagation de l'entrée et ce retard est fonction du taux de remplissage à l'interface f p (l(t), t). Le problème de commande est représenté par la stabilisation un système à retard sur l'entrée. Les systèmes à retard sur l'entrée ne sont pas fréquemment utilisés pour les problèmes de contrôle en génie des procédés même si les modèles dynamiques de diérents procédés laissent apparaître des délais de propagation signicatifs [START_REF] Christodes | Feedback control of nonlinear dierential dierence equation systems[END_REF][START_REF] Antoniades | Robust control of nonlinear time-delay systems[END_REF][START_REF] Shu Xia | Distributed control of plant-wide chemical processes with uncertain time-delays[END_REF][START_REF] Xu | Distributed control of plant-wide chemical processes with uncertain time-delays[END_REF][START_REF] Bués | Delay model for a cycling transport through porous medium[END_REF]. Généralement, les problèmes de régulation des systèmes d'équation de transport se ramènent à des problèmes de stabilisation de systèmes à retards. En eet, une résolution de des equations aux dérivées partielles par la méthode des caractéristiques permet de transformer les EDPs en des systèmes d'équations aux dérivées ordinaires comportant un retard [START_REF] Gundepudi | Velocity control of partial dierential equation systems with single characteristic variable[END_REF][START_REF] Shang | Feedback control of hyperbolic distributed parameter systems[END_REF][START_REF] Aguilar-Garnica | Robust continuous velocity control of convective spatially distributed systems[END_REF][START_REF] Aguilar-Garnica | PI controller design for a class of distributed parameter systems[END_REF]. Dans cette section nous utilisons cette méthode des caractéristiques pour transformer le système original en un système à retard sur l'entrée équivalent.

Les équations de conservation de masse dans une extrudeuse

La modélisation de l'extrudeuse fait apparaître deux zones géométriques dans lesquelles la matière est transportée à des vitesse diérentes. Ces deux zones sont séparées par une interface mobile dont la dynamique est dictée par une équation aux dérivées ordinaires obtenue par un bilan de masse dans la zone entièrement remplie. Ici, nous rappelons les équations issues du bilan de masse dans les zones partiellement et entièrement remplies. La zone partiellement remplie est dénie dans le domaine spatial [0, l(t)[ et la zone entièrement remplie dans ]l(t), L] (L représente la longueur de l'extrudeuse et l(t) la position de l'interface).

En représentant la variation de la masse dans la zone partiellement remplie par le taux de remplissage f p (x, t), le bilan de masse est représenté par une équation de transport dont la vélocité est déterminée par la vitesse de rotation N (t) et le pas ξ de la vis. Sous l'hypothèse d'une continuité de débit à l'entrée {x = 0}, la condition frontière associée au taux de remplissage se déduit de la vitesse de convection, du volume disponible V ef f , de la masse volumique ρ 0 , et du débit d'alimentation F in (t) :

⎧ ⎨ ⎩ ∂fp ∂t (x, t) = -ξN (t) ∂fp ∂x (x, t) x ∈ [0, l(t)[ f p (0, t) = f in p (t) = F in (t) ρ 0 ξN (t)V ef f (1.3.1)
Dans la zone entièrement remplie la vitesse de convection est proportionnelle au débit de sortie de lière F d (t). Le bilan de masse dans cette zone permet d'obtenir l'équation d'évolution de la position de l' interface l(t) :

⎧ ⎨ ⎩ dl(t) dt = F d (t)-ρ 0 N (t)V ef f fp(l(t),t)) ρ 0 S ef f (1-fp(l(t),t)) l(0) = l 0 0 < l 0 < L, (1.3.2)
Le débit F d (t) est une fonction des caractéristiques géométriques de la lière K d , de la viscosité η et de la pression en L, P (L, t).

⎧ ⎨ ⎩ F d (t) = K d η ΔP (t) ΔP (t) = (P (L, t) -P 0 ), (1.3.3) P 0 est la pression atmosphérique.
Dans cette partie nous considérons que la relation de couplage est dénie par une continuité de la pression à l'interface. Alors:

ΔP (t) = ηV ef f N (t)ρ 0 (L -l(t)) (Bρ 0 + K d (L -l(t))) (1.3.4)

Résolution de l'équation de transport

Considérons l'équation de transport (1.3.1) avec la condition frontière associée f p (0, t). La dérivée totale de la variable f p s'écrit:

df p (x(τ ), t) dτ = dt dτ ∂f p (x(τ ), t) ∂t + dx(τ ) dτ ∂f p (x(τ ), t) ∂x
Les solutions de l'équation (1.3.1), le long des caractéristiques sont données par:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ dt dτ = 1 dx(τ ) dτ = ξN (t) dfp(x(τ ),t) dτ = 0 (1.3.5)
Comme N (t) est une fonction de t, le système (1.3.5) ne peut être résolu explicitement. En intégrant les equations caractéristiques:

x 1 x 2 x 3 = t x f p ∈ R 3 nous obtenons: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x 1 (τ, s) = τ + s x 2 (τ, s) = ξ ¡ τ +s s N (τ )dτ x 3 (s) = f p (s) , (1.3.6)
N (s) = N (τ + s) - x 2 ξ (1.3.7)
Finalement:

s = N -1 N (t) - x ξ (1.3.8)
Toute fonction constante le long des caractéristiques s'exprime de la manière suivante:

f p (x, t) = f p N -1 N (t) - x ξ (1.3.9)
La solution satisfaisant la condition frontière associée à l'équation de transport (1.3.1) est donnée par:

f p (x, t) = f in p N -1 N (t) - x ξ (1.3.10)
Cette solution peut être exprimée en fonction du débit d'alimentation en matière:

f in p N -1 N (t) - x ξ = F in N -1 N (t) -x ξ ρ 0 N (t)V ef f , (1.3.11)
1.3.3 Système à retard sur l'entrée dépendant de l'état Dans la suite nous considérons que la vitesse de rotation de la vis est constante N (t) = N 0 . Le problème de commande consiste à stabiliser l'interface en agissant sur le débit d'alimentation en matière (1.3.11). Dans ce cas de gure, la solution de l'équation de transport s'écrit :

⎧ ⎨ ⎩ f p (x, t) = f in p (t -x N 0 ξ ) f p (l(t), t) = f in p (t -l(t) N 0 ξ ), (1.3.12)
et l'équation de la dynamique d'interface devient :

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ dl(t) dt = F d (t)-ρ 0 N 0 V ef f f in p t- l(t) N 0 ξ ρ 0 S ef f 1-f in p t- l(t) N 0 ξ F d (t) = K d η ΔP (t) (1.3.13)
L'équation (1.3.13) dénit un système à retard sur l'entrée dépendant de l'état et peut s'écrire sous la forme abstraite:

⎧ ⎨ ⎩ dl(t) dt = f (l(t), U(t -D(l(t))) U (t) = f in p (t) (1.3.14)
où f est une fonction non-linéaire et D(l(t)) la fonction retard dépendant de l'état et agissant sur l'entrée U (t). Le problème de stabilisation est résolu en utilisant le contrôleur prédictif proposé par [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF].

Systèmes Hamiltoniens à ports couplés par une interface mobile

De nombreux systèmes physiques décrits par des équations aux dérivées partielles admettent une formulation Hamiltonienne à ports est sont appelés systèmes Hamiltoniens à ports frontière [START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF][START_REF] Maschke | Advanced Topics in Control Systems Theory[END_REF][START_REF] Macchelli | Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach, chapter Innite-dimensional Port-Hamiltonian Systems[END_REF]. Ces systèmes ont été étudiés sous diérents aspects incluant l'existence de solution et la commande dans le cas linéaire [START_REF] Macchelli | Modeling and control of the Timoshenko beam. the Distributed Port Hamiltonian approach[END_REF][START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric dierential operators[END_REF][START_REF] Villegas | A Port-Hamiltonian Approach to Distributed Parameter Systems[END_REF][START_REF] Zwart | Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain[END_REF][START_REF] Villegas | Stability and stabilization of a class of boundary control systems[END_REF][START_REF] Jacob | Linear Port-Hamiltonian Systems on Innite-dimensional Spaces[END_REF][START_REF] Ortega | On compensation of wave reexions in transmission lines and applications to the overvoltage problem in AC motor drives[END_REF][START_REF] Macchelli | Control by interconnection of mixed Port Hamiltonian systems[END_REF].

Ici nous explorons le couplage de deux systèmes Hamiltoniens à ports à travers une interface mobile. Cette classe de systèmes apparaît dans diérent contexte liée aux procédés impliquant des changements de phase, des phénomène d'évaporation etc. Dans cette étude nous considérons l'exemple de deux gaz séparés par un piston de masse nulle. L'approche proposée est basée sur les travaux de [START_REF] Godlewski | The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. the scalar case[END_REF][START_REF] Godlewski | The numerical interface coupling of nonlinear systems of conservation laws: II. the case of systems[END_REF][START_REF] Ambroso | A Relaxation Method for the Coupling of Systems of Conservation Laws[END_REF][START_REF] Boutin | Dafermos regularization for interface coupling of conservation laws[END_REF] concernant les systèmes hyperboliques couplés par une interface xe. Les auteurs proposent d'augmenter le système par une fonction couleur qui est dénie comme étant la fonction caractéristique du domaine.

Denition d'un système Hamiltonien à ports pour deux lois de conservation

Dans un premier temps nous rappelons la dénition d'un système Hamiltonien à ports frontière en considérant un système de deux lois de conservation déni par :

∂ t x + ∂ z N (x) = 0 (1.4.1)
Nous rappelons que le système de deux lois de conservation (1.4.1) peut être exprimé sous la forme:

∂ t x(z, t) = J δ x H (1.4.2)
Où δ x H est la dérivée variationnelle d'une fonction Hamiltonienne correspondant à l'énergie totale du système. En eet le ux N (x) est donné par la relation suivante :

N (x) = 0 -1 -1 0 δ x 1 H δ x 2 H, (1.4.3)
et l'opérateur canonique Hamiltonien par l'égalité ci-dessous :

J = 0 -∂ z ∂ * z 0 (1.4.4)
L'équation (1.4.1) augmentée des variables de ports

f ∂ e ∂ = δ x 2 H δ x 1 H a,b = 0 1 1 0 δ x 1 H δ x 2 H a,b (1.4.5)
Dénit un système Hamiltonien à ports frontière.

∂ z x i + ∂ z N i x i (1.4.6)
Nous dénissons les fonctions couleurs qui sont des fonctions caractéristiques du domaine par:

c 0 (z, t) = 1 z ∈ [a, 0[ 0 z ∈ [0, b] and c0 (z, t) = 1 z ∈ ]0, b] 0 z ∈ [a, 0] (1.4.7)
L'augmentation du système Hamiltonien par des variables couleurs satisfaisant les lois de conservation

∂ t c = ∂ t c = 0
(1.4.8) avec les conditions initiales c 0 and c0 et des conditions aux bords compatibles permet de dénir un système Hamiltonien sur le domaine total [a , b]. Nous dénissons le vecteur d'état étendu :

x = x T , c, c T (1.4.9) et la fonction Hamiltonienne H (x, c, c) = ¡ b a H (x, c, c) dz avec une densité d'énergie H (x, c, c) = c H -(x) + c H + (x).
(1.4.10) La dérivée variationnelle s'écrit :

δ xH (x) = ⎛ ⎜ ⎜ ⎝ δ x H (x, c, c) δ c H (x, c, c) δ cH (x, c, c) ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ c δ x H -(x) + c δ x H + (x) H -(x) H + (x) ⎞ ⎟ ⎟ ⎠ (1.4.11)
Considérons le système (1.4.12) :

∂ t x = J a δ xH (x) + Ie I (1.4.12)
avec une application d'entrée dénie par:

I T = 0 -1 0 0 (1.4.13)
et l'opérateur augmenté:

J a = ⎛ ⎜ ⎜ ⎝ 0 d -d * 0 0 2 0 2 0 2 ⎞ ⎟ ⎟ ⎠ (1.4.14)
L'opérateur d est un opérateur diérentiel non-linéaire , modulé par c(z, t) and c(z, t) et déni par:

d = -[∂ z c. + ∂ c.]
(1.4.15)

Son dual formel est:

d * = -d + [(∂ z c) -(∂ z c)]
(1.4.

D I = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ f f I f ∂ ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ ẽ e I e ∂ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ∈ F × E / f f I = J a I -I T 0 ẽ e I and f ∂ e ∂ = 0 1 (c + c) 0 e 1 e 2 a,b (1.4.17) avec les variables de ux f = (f 1 , f 2 , f c , f c) T et les variables d'eort ẽ = (e 1 , e
B = F × E avec F = L 2 ((a, b) , R) 5 × R 2 et E = dom d * × dom d × L 2 ((a, b) , R) 3 × R 2 muni du produit bi-linéaire ⎛ ⎜ ⎜ ⎝ f f I f ∂ ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ ẽ e I e ∂ ⎞ ⎟ ⎟ ⎠ = ¡ b a ẽT f dz + e T ∂ Σf ∂ + ¡ b a e I T f I dz (1.4.18) avec e T ∂ Σf ∂ = e ∂ (a)f ∂ (a) -e ∂ (b)f ∂ (b)
, dénit une structure de Dirac.

Interconnexion de deux systèmes Hamiltoniens à ports à travers une interface mobile

Pour une interface dont la position varie en fonction du temps nous dénissons les fonctions couleur suivantes:

c l(t) (z, t) = 1 z ∈ [a, l (t) [ 0 z ∈ [l (t) , b] (1.4.19) et cl(t) (z, t) = 1 z ∈]l (t) , b] 0 z ∈ [a, l (t)]
(1.4.20)

Ces fonctions couleurs obéissent à des equations de transport dont la vélocité est donnée par la vitesse de l'interface l (t) :

∂ t c(z, t) = -l (t) ∂ z c(z, t) and ∂ t c(z, t) = -l (t) ∂ z c(z, t) (1.4.21)
avec les conditions initiales:

c(z, 0) = c l(0) (z, t) and c(z, 0) = cl(0) (z, t) (1.4.22)
Le système Hamiltonien associé aux deux systèmes de lois de conservation couplés par une interface mobile peut se dénir en considérant un système étendu et augmenté des fonctions couleurs.

∂ t ⎛ ⎜ ⎜ ⎝ x c c ⎞ ⎟ ⎟ ⎠ = J a ⎛ ⎜ ⎜ ⎝ δ x H (x, c, c) δ c H (x, c, c) δ cH (x, c, c) ⎞ ⎟ ⎟ ⎠ + Ie I + l (t) ⎛ ⎜ ⎜ ⎝ c x c x -1 0 0 -1 ⎞ ⎟ ⎟ ⎠ ∂ z c c (1.4.23)
l'application d'entrée à l'interface I est dénie par (1.4.13).

L'application d'entrée associé à la dynamique de l'interface est dénie par (1.4.24)

G (x, c, c) = ⎛ ⎜ ⎜ ⎝ c x c x -1 0 0 -1 ⎞ ⎟ ⎟ ⎠ ∂ z c c (1.4.24)
et sa sortie conjuguée e l est dénie par la relation:

e l = ¢ b a δ xH (x) T G (x, c, c) dz
où:

e l = G T x, c, c , δ xH (x) = ¢ b a δ xH (x) T G (x, c, c) dz (1.4.25)
Proposition 2. L'ensemble des relations D M associées au système de deux lois de conser- vation dénis par les variables

D M = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ f f I e l f ∂ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ẽ e I ˙l e ∂ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ∈ F × E / ⎛ ⎜ ⎜ ⎝ f f I -e l ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ J a I G(x, c, c) -I T 0 0 -G T (x, c, c) 0 0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ ẽ e I l ⎞ ⎟ ⎟ ⎠ and f ∂ e ∂ = 0 1 (c + c) 0 e 1 e 2 a,b ⎫ ⎬ ⎭ (1.4.26) avec les variables de ux f = (f 1 , f 2 , f c , f c) T et les variables d'eort ẽ = (e 1 , e
G T par (6.3.11) et l'espace produit B = F × E avec F = L 2 ((a, b) , R) 5 × R × R 9 et E = dom d * × dom d × L 2 ((a, b) , R) 3 × R × R 9 muni du produit bi-linéaire ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ f f I e l f ∂ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ẽ e I ˙l e ∂ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ¡ b a ẽT f dz + ¡ b a e I T f I dz + e T ∂ Σf ∂ -e l l (

Motivation and previous work

Balance equations provide the foundation for much of the physical-based modelling in uid dynamics. They are also the starting point for developing qualitative understanding of phenomenological observations in uid mechanics, heat transfer, mass transfer, and reaction engineering. Describing physical phenomena using variables that inuence these behavior is the main issue of modelling and a large class of physical systems are characterized by a spatiotemporal dynamics. Physical systems obey to physical laws expressing conservation of certain quantities (mass, momentum, energy) and can be represented by partial dierential equations. These innite dimensional systems which are generally deduced from the conservation laws appear in modelling of transport and diusion phenomena. In this context, the analysis of coupled PDEs through an interface is not current in the literature which concern to the study of these innite dimensional systems. The lack of physical model which express clearly the interface structure and the coupling conditions can be considered as the real challenge in modelling point of view. It is clear that, in addition to the denition of each model of conservation laws, an interfacial model must be constructed in order to precise the nature of the informations that are exchanged at the coupling region. This interfacial model may be formulated for instance, when imposing the continuity of a given set of variables or a given linear or nonlinear function of variables while preserving physically a coherent description of processes. In this aspect, lot of studies which are related to coupled hyperbolic systems of conservation laws through an xed interface exist in the literature. In [START_REF] Boutin | Etude Mathématique et Numérique des Equations Hyperboliques Nonlinéaires: couplage de modèles et chocs non classiques[END_REF], the coupling problem is considered in the space domain R which is separated by a thin interface I located at {0}. The system is described on each side of the interface by 1D hyperbolic systems of conservation law with dierent equations of state.

One may consider two cases of coupling formulation:

The rst one which is conservative, is called a ux coupling methods in the sense that the solution performs continuous ow at the interface. This approach is considered 2. INTRODUCTION when the conservation laws are described by discontinuous coecients in the domains R -, * and R +, * [START_REF] Karlsen | On the existence and compactness of a twodimensional resonant system of conservation laws[END_REF][START_REF] Helluy | Relaxation models of phase transition ows[END_REF][START_REF] Godlewski | The Riemann problem for a simple model of phase transition[END_REF][START_REF] Burger | Conservation laws with discontinuous ux : a short introduction[END_REF][START_REF] Bachmann | Existence and uniqueness of entropy solution of scalar conservation laws with a ux function involving discontinuous coecients[END_REF].

The second approach is related to state coupling methods which consist to impose the continuity of the state variables at the interface. Then, the relation between the traces of the solution on both sides of the interface can be interpreted in terms of two sets of boundary conditions. The left trace of the rst conservation laws which are dened in R -, * serves as boundary condition for the second one. Also, right trace of the second conservation laws which are dened in R +, * stand as a boundary condition for the rst one [START_REF] Ambroso | Extension of interface coupling to general Lagrangian systems[END_REF][START_REF] Ambroso | Coupling of general Lagrangian systems[END_REF][START_REF] Ambroso | Relaxation methods and coupling procedures[END_REF][START_REF] Godlewski | The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. the scalar case[END_REF][START_REF] Godlewski | The numerical interface coupling of nonlinear systems of conservation laws: II. the case of systems[END_REF].

For physical problems the requirement of continuity of all the unknowns variables of the system is quite impossible. For example, considering a ow in tubes of dierent sections, a pressure drop is observed at the discontinuities of the medium [START_REF] Ambroso | Interface model coupling via prescribed local ux balance[END_REF]. An alternative model suggested by this observation is the state coupling methods: the interface requires continuity of a set of variables which are nonlinear transformation of the state variables, selected according to the physical constraints.

The mathematical and numerical studies of this class of systems calls for interesting questions in the last decades. Such xed interface problems typically appear in nuclear reactor and steam generators at the phase change. They also concern the uid dynamic modelling in porous and non-porous media, and are very frequent in networks trac ow modelling (supply chains connecting several suppliers, a vertical manhole with two horizontal tubes) [START_REF] Raul Borsche | On the coupling of systems of hyperbolic conservation laws with ordinary dierential equations[END_REF].

However, control techniques which are dedicated to coupled innite dimensional systems by a xed interface is very dicult to implement. Networks of ow transportation is the major example which is explored in control theory concerning these systems. For an irrigation canal which is connected to secondary canals and reservoirs, the control is located at the interfaces which interconnect dierent sub-systems. The objectives of such control strategies is usually the regulation of water levels close to a given reference set-point [START_REF] Begovich | Predictive control with constraints of a multi-pool irrigation canal prototype[END_REF][START_REF] Akouz | Predictive control applied to asce canal 2[END_REF][START_REF] Ruiz | Predictive control in irrigation canal operation[END_REF][START_REF] Malaterre | Multivariable predictive control of irrigation canals. design and evaluation on a 2-pool model[END_REF], with input values being minimized, or changes in inputs being minimized [START_REF] Begovich | Predictive control with constraints of a multi-pool irrigation canal prototype[END_REF][START_REF] Wahlin | Automatic downstream water-level feedback control of branching canal networks: Simulation results[END_REF]. The inputs are here acting at the interface which is a gate modelling the pressure discontinuity in dierent region of the network. So the regulation is performed by opening gates at these points using model predictive controllers, feedback or feedforward controllers. Usually, due to the complexity and size of water networks, control of such systems is not performed in a centralized way. Distributed control composed by local controllers which are designed in such way that they take into account the eects of local actions on the overall system performance are developed. Then, control strategy is based on a potential cooperation, negotiation and communication between many controllers as in all transportation networks systems [START_REF] Negenborn | A distributed model predictive control approach for the control of irrigation canals[END_REF].

This thesis is devoted to the analysis of PDEs which are coupled through a moving interface by extrusion process modelling. The interface dynamic is given by an ordinary dierential equation and the PDEs are dened in two complementary time-varying domains. Such sys-tems arises in many applications like multi-phase ow, crystal growth, freeze drying process modelling [START_REF] Dieulot | An arrangement of ideal zones with shifting boundaries as a way to model mixing processes in unsteady stirring conditions in agitated vessels[END_REF][START_REF] Dunbar | Motion planning for a nonlinear stefan problem[END_REF][START_REF] Gupta | The classical Stefan problem. Basic concepts, Modelling and Analysis[END_REF][START_REF] Purlis | A moving boundary problem in a food material undergoing volumechange -simulation of bread baking[END_REF][START_REF] Velardi | Development of simplied models for the freeze-drying process and investigation of the optimal operating conditions[END_REF][START_REF] Bouchemal | Polyamides nanocapsules : Modeling and wall thickness estimation[END_REF][START_REF] Daraoui | Model predictive control during the primary drying stage of lyophilisation[END_REF], mixing systems (model of torus reactor including a well-mixed zone and a transport zone), Diesel Oxidation Catalyst [START_REF] Petit | Control problems for one-dimensional uids and reactive uids with moving interfaces[END_REF]. For moving interface problems, one should also dene the transmission conditions at the interface: the coupling aspect is formulated using a continuity argument as it will be shown in Chapter 6. As for xed interface, there exists innite manner for modelling the interconnection between the two sub-systems of conservation laws based on conservation of a physical quantity. The main dierence between xed and moving interface problem appears in the analysis of interfacial topological changes which are introduced by the interface motion. In the case of extrusion processes, the system should take into account thermal and friction eect as well as other factors which derive from modelling aspect. The coupled system may be also exposed to oscillations at the moving junction which can generate numerical instability in computations.

Over the last two decades, there has been an ongoing quest for new computational methods to solve multi-phase ows and applications include the study of condensation and several phase change processes. This thrust has been motivated in part by the energy industry because those processes allow uids to store and release large amounts of heat energy. The main challenges for a direct numerical simulation come from the fact that the interface location must be calculated as part of the process solution and discontinuities in materials properties across the interface must be preserved. The rst attempt at advanced numerical simulations deals with the simulation of a two dimensional thin lm boiling using moving triangular grids [START_REF] Welch | Local simulation of two-phase ows including interface tracking with mass transfer[END_REF]. The others approaches as front tracking [START_REF] Daly | Numerical study of density current surges[END_REF][START_REF] Tryggvason | A front-tracking method for the computations of multiphase ow[END_REF] allow to track explicitly the interface by a moving front on a xed mesh. Also, simulations are carried out using a level set approach [START_REF] Osher | Fronts propagating with curvature-dependent speed : algorithms based on Hamilton-Jacobi formulations[END_REF] as well as volume of uid approach [START_REF] Harlow | Numerical calculation of time-dependent viscous incompressible ow of uid with free surfaces[END_REF], [START_REF] Daly | Numerical study of two uid Rayleigh-Taylor instability[END_REF]. These numerical methods will be discussed succinctly in Chapter 3 which is dedicated to the modelling and the numerical computation of the extrusion process model. Note that the advantage of the level set method is that topological changes are handled naturally without special treatments.

Mathematical analysis of such systems is generally performed using coordinates change which allows to treat the problem in a xed domain. In Chapter 4 we propose such kind of transformations in order to deal with the cauchy problem of the extruder process model.

Over the last few years, the problem of controlling coupled PDEs with time-varying domains has become subject to very great interest. In [START_REF] Petit | Control problems for one-dimensional uids and reactive uids with moving interfaces[END_REF] a boundary control problem is formulated for the outlet temperature control in Diesel Oxidation Catalyst process. As it is described in [START_REF] Petit | Control problems for one-dimensional uids and reactive uids with moving interfaces[END_REF] the distributed reactive gaseous system takes into account a moving interface model which separates the upstream reactive zone from the downstream transport zone. The author of [START_REF] Petit | Control problems for one-dimensional uids and reactive uids with moving interfaces[END_REF] points out that the mobile interface dynamic in this process stand as a function of measured disturbances and the control variable. Also, the control problem is tackled using Laplace transformation to build a transfer function as a modied Bessel function which can be translated in the time domain. Then, the input/output relation obeys a straightforward open-loop control law: given histories for the output temperature, one can simply determine 2. INTRODUCTION the corresponding inlet temperature histories. For nonlinear Stefan problem control strategy based on the resolution of the inverse problem is proposed in [START_REF] Petit | Control problems for one-dimensional uids and reactive uids with moving interfaces[END_REF]. In many cases concerning control of quasi-linear parabolic PDEs with moving interface, standard Galerkin's method is used to derive an approximate nite dimensional one, for applying classical control strategy as feedback [START_REF] El-Farra | Constrained nonlinear control of parabolic PDE systems[END_REF][START_REF] Armaou | Nonlinear feedback control of parabolic partial dierential equation systems with time-dependent spatial domains[END_REF][START_REF] Armaou | Robust control of parabolic pde systems with timedependent spatial domains[END_REF]. In Chapter 5 we suggest to treat moving interface problem using delay system framework.

Organization and contribution of the thesis

The thesis is divided into four chapters: Chapter 3 is devoted to the modelling of an extrusion process as coupled transport equations trough moving interface. The model derives from mass and energy balance in an extrusion process and is formulated as two non-autonomous systems of transport equations coupled by a moving interface which is described by an ordinary dierential equation. Two coupling relations are proposed for mass balance equation. The rst one states the continuity of pressure and the second one is based on momentum ux continuity at the mobile interface. The analysis of these two coupling relations is performed in Chapter 5 Chapter 4 is dedicated to a mathematical analysis of transport equations which are dened in complementary time-varying domains. We illustrate how xed point argument and contraction mapping principle can be used in order to state the well-posedness of such coupled hyperbolic Cauchy problem using the extrusion process model. Our strategy is based on a change of coordinates which allows to express the system on a xed domain. This normalization generates ctives convection terms depending on the interface dynamic in the transport equations. We obtain the existence, uniqueness and regularity of the weak solution for Cauchy problem.

In Chapter 5, an analysis of 1D hyperbolic partial dierential equation with moving interface as a delay system is proposed. The model derives from the mass balance of an extrusion process that describes the mass transport by hyperbolic partial dierential equation coupled with an ordinary dierential equation. The ordinary dierential equation represents the interface motion. Solving the transport equation by the method of characteristics, we obtain a state-dependent-input-delay control problem. The stabilization of the system around an equilibrium is obtained using a state predictor.

Chapter 6 is an extension of our analysis on moving interface problem to a particular class of conservations laws. We consider the port-Hamiltonian formulation of systems of two conservation laws dened on two complementary intervals of some interval of the real line and coupled by a moving interface. We recall rst how two port Hamiltonian systems coupled by an interface may be expressed as a port Hamiltonian systems augmented with two variables being the characteristic functions of the two spatial domains.

Then, we consider the case of a moving interface and show that it may be expressed as the previous port Hamiltonian system augmented with an input, being the velocity of the interface and dene a conjugated output variable. We give some structure to the interface relations dening the dynamics of the displacement of the interface and with the simple example of two gases coupled by a moving piston we discuss the passivity property.

The thesis concludes with some nal remarks and perspectives on future work in Chapter 7. Extrusion processes modelling for control

Publications

Introduction

Extruders are designed to process highly viscous materials. They are mainly used in the chemical industries for polymer processing as well as in the food industries. An extruder is made of a barrel, the temperature of which is regulated. One or two Archimedean screws are rotating inside the barrel. The extruder is equipped with a die where the material comes out of the process (Fig. 3.1.1). The extruder is of particular interest due to its modular geometry allowing the control of capacities of mixing along the machine. Another interesting property of the extruder is that the ll ratio along the axial direction of the screws can be less than one in some part of the system according to the screw conguration and the operating conditions. For modelling purposes, the main phenomenon is obviously the uid ow which may be considered as highly viscous Newtonian or non-Newtonian uid ows interacting with heat transfer and possibly chemical reactions. These processes occur within a complex nonstationary volume delimited by the barrel and the rotating screw. Another point is that the uid viscosity may signicantly change due to composition and temperature changes.

The most important part of the extruder is the screw conguration which modulates extensively the mechanical energy. Direct pitch screws promote the transport by having conveying eect and reverse pitch screws which restrict the convection phenomena are useful for melting eect.

Single-screw extruders are limited in their ability to transport high viscous material and have poor mixing capability. To improve the mixing eect distributive mixing elements called mixers blocks are often incorporated in the screw design. Generally the mixing section is a restrictive region where the material is subjected to high stresses that for a short time. A typical single-screw extruder consists of three zones: feeding zone which is essentially a transport zone, mixing zone and nishing zone (cooking zone for food processing).

Twin-screw extruders provide a high degree of heat transfer (Fig. 3.1.2). This type of extruders can be classied on the basis of direction of screw rotation in counter-rotating and co-rotating twin-screw extruders. Counter-rotating screws move in opposing directions and co-rotating screws move in the same direction. Counter-rotating screws are not widely used in food industry although they are excellent conveyors. They are good in processing relatively non viscous materials requiring low speed and long resident time. The main advantages of a co-rotating conguration which are commonly used in the snack food industry are the self-cleanning capacity, the excellent mixing quality and the very high degassing rate.

Both counter-and co-rotating screws can be either intermeshing or non-intermeshing. In an intermeshing conguration one screw penetrates the channel of the second screw, producing a positive pumping action, ecient mixing and self-cleaning capacity. Non-intermeshing screws do not engage or interfere with each other and depend on friction as single-screw extruders. They are not designed for mixing and are similar to single-screw extruders, even if they have higher capacities.

In the literature there exists no result about the control of an innite dimensional model of an extruder. In this chapter, the proposed model oers the perspective to develop control methods for these processes using dynamic physical models.

The studies performed on the automated control of extruders are based on experimental data from pilot-scale units to develop transfer functions for controller design. The ultimate control systems involved manipulation of screw speed, feed rate, inlet solvent fraction, and barrel temperature. The models were developed to control motor torque, solvent content, product temperature, specic mechanical energy, viscosity, residence time and die ow, etc. As an example, the screw speed and temperature of the mixing zone determine the amount of cooking, moisture, extent of mixing, and exiting temperature of the nal product. The transport phenomena occurring in the extruder lead to dierent models taking into account the delay corresponding to the time each particle spends in the barrel section (residence time). This suggests that the transport phenomena occurring in the extruder can be included in a specic delay system framework which oers many control strategies in order to obtain desired properties of extruded material. A number of control methodologies have been proposed to regulate the product quality for those processes. A simple methodology can be a single-input single-output (SISO) feedback control linking the output at the die to the input parameter such as the screw speed. In [START_REF] Kulshreshtha | Simulating the performance of a control system for food extruders using model-based set-point adjustment[END_REF][START_REF] Hamane | Design of decoupling system for eliminating disturbutance on extruder temperature control system[END_REF][START_REF] Kulshreshtha | Simulating the performance of a control system for food extruders using model-based set-point adjustment[END_REF][START_REF] Mcafee | A novel approach to dynamic modeling of polymer extrusion for improved process control[END_REF][START_REF] Elsey | Modeling and control of a food extrusion process[END_REF][START_REF] Moreira | Feedforward control model for a twin-screw food extruder[END_REF][START_REF] Singh | Modeling and process control of twin-screw cooking food extruder[END_REF][START_REF] Broadhead | Closed loop viscosity control of reactive extrusion with an in line rheometer[END_REF] proportional integral derivative P ID feedback controllers are performed through dierent models which are deduced from experimental data under specic operating conditions for food and polymer extrusion. In context of food extrusion a multi-variables predictive control which allows to track both input and output for a given set point is proposed by [START_REF] Wang | Dual-target predictive control and application in food extrusion[END_REF] as a Dual Target Predictive Controller (DT P C). The predictive capabilities of (DT P C) are provided by an on-line system identication. Predictive controllers are also implemented for many applications as in [START_REF] Nield | Control of a LPDE reactive extrusion process[END_REF][START_REF] Tan | Self-tuning predictive control of processing temperature for food extrusion[END_REF][START_REF] Haley | On-line system identication and control design of an extrusion cooking process: Part I. system identication[END_REF][START_REF] Haley | On-line system identication and control design of an extrusion cooking process: Part II. system model predictive and inferential control design[END_REF].

As far as modelling issues are concerned, one can nd many papers devoted to steady-state modelling for design purpose, [START_REF] Vergnes | Modeling of reactive systems in twin-screw extrusion: challenges and applications[END_REF], but works devoted to dynamic modelling of extruders are less common, [START_REF] Kim | Isothermal transient startup for starved ow modular co-rotating twin screw extruder[END_REF][START_REF] Kim | Non-isothermal transient startup for starved ow modular co-rotating twin screw extruder[END_REF][START_REF] Janssen | A dynamic model for multiple steady states in reactive extrusion[END_REF][START_REF] Janssen | A dynamic model accounting for oscillating behavior in extrusion reaction[END_REF][START_REF] Choulak | Generic dynamic model for simulation and control of reactive extrusion[END_REF]. For control purposes, one can mainly cites the following works: [START_REF] Choulak | Generic dynamic model for simulation and control of reactive extrusion[END_REF] proposes a nite dimensional model based on a ow representation made of a series of Continuous Stirred Tank Reactors (CSTR) with variable back-ows. This approach allows simulating the lling process of the machine as well as all the other state-variables. This model has been validated in the case of the polymerization of ε-Caprolactone.

An innite dimensional model has been developed in [START_REF] Kulshrestha | An unsteady state model for twin screw extruders[END_REF][START_REF] Li | Modelling extrusion cooking[END_REF][START_REF] Li | Modelling extrusion cooking[END_REF] within the context of food processing. In this case, the Fully and Partially Filled Zones are dened in advance. These two zones are modelled by transport equations with dierent velocities. A moving interface obeying an ordinary dierential equation interconnects the two innite dimensional systems. Those models remained undeveloped for control because the authors do not explicitly express all constraints, particularly at the interface. The same methods are used in [START_REF] Prat | Solid-liquid transport in a modied corotating twin-screw extruder-dynamic simulator and experimental validations[END_REF] to describe the extraction of solvent in a vegetable matter. The authors represent the liquid accumulation in porous matter. They consider otherwise, transport (Partially Filled) and pressing (Fully Filled) Zones without discussing the interconnection through the interface which separate the two zones. Transport equations are also used in [START_REF] Leong | Soft computing applications in dynamic model identication of polymer extrusion process[END_REF] for modelling the solid fusion by extrusion under stationary conditions.

In this chapter, a non-linear innite dimensional 1D model of an extruder is developed. The model is inspired by [START_REF] Kulshrestha | An unsteady state model for twin screw extruders[END_REF][START_REF] Li | Modelling extrusion cooking[END_REF][START_REF] Li | Modelling extrusion cooking[END_REF] and is focused on the analysis of interface relations as transmission conditions. The obtained model is simulated using nite volume method adapted to a moving interface system by extending the variables with a color function. The color function is associated to the time varying domains as a characteristic function which also obey to a transport equation. The velocity of transport is here given by the mobile interface motion.

Organization of the chapter

The organization of this chapter is as follows: rst in Section 2.2, we present a geometric decomposition of the extruder by dening a Partially and a Fully Filled Zone. Then, we obtain a bi-zone model of the process by dening two zones which are coupled by a moving interface. In Section 2.3, we propose a model of the Partially Filled Zone by considering extrusion of homogeneous material and Section 2.4 is dedicated to the modelling of Fully Filled Zone. In section 2.5 we present the dynamics of the moving interface and two classes of coupling relations. Section 2.6 is devoted to the discussion on numerical computation and the simulation of the interface dynamics, lling ratio and temperature variables.

Contributions

The main contribution of this chapter is the formulation of a dynamic input/output 1D model of an extrusion process. The modelling objective is to analyze evolution of temperature, die pressure and lling ratio under some operating conditions. The model is based on structural decomposition of the extruder in two zones. These zones which are Partially and Fully Filled arise from the geometric structure of the machine. Expressing conservation of mass and energy in these two zones, one obtains a system of two non-autonomous transport equations which are interconnected by an ordinary dierential equation on the moving interface. The specicity of the interface relations is discussed using continuity of pressure and continuity of momentum ux. Numerical method which is inspired from the height function method is performed using color function. The color function which stand as characteristic function of the complementary time-varying domains allows to extend the denition of variables in the total domain. Simulations are presented to illustrate the coherence of this physical model relating heat and mass transport in nonhomogeneous media.

Bi-zone model of an extruder 3.2.1 Zones dened by the pressure gradient

The complexity of screw conguration in an extruder makes dicult the design a nonisothermal ow model [START_REF] Booy | Isothermal ow of viscous liquids in corotating twin screw devices[END_REF]. In [START_REF] Booy | Geometry of fully wiped twin-screw equipment[END_REF], an analysis of the ow in the channel of co-rotating twin-screws with the same speed is developed and the authors show how a reasonable ow analysis can be made by writing a single screw extrusion process as an equivalent model. In an extruder, the net ow at the die exit is mainly due to the ow of material in the longitudinal direction if one neglects the clearance between the screw and the barrel and the vibrations. Therefore, the ow dynamic which is dominated by the convection eect in the direction of screw axis is sucient to represent the material ow. This means that the transverse ow corresponding to a recirculation of the material in the plane perpendicular to the screw channel is neglected. From a macroscopic point of view, a 1D model describes clearly the material transport in an extruder. So, the material is driven from the feed to the die by the pumping eect of the screw rotation. It becomes clear that the geometric structure of the die inuences the material transport along the extruder. Therefore, the material is accumulated behind the die and lls completely the available volume in this region. This spatial domain where the extruder is completely lled is called the Fully Filled Zone (FFZ). The ow in this Fully Filled Zone depends on the pumping capacity of the screw and also on the pressure ow. The pressure gradient which appears due to the die restriction is given by Navier-Stokes equation which provides a mathematical model of the uid motion. The extruder which is initially empty, may also comprise a spatial domain that is not completely lled by the material. This region which corresponds to a conveying region is called Partially Filled Zone (PFZ). In this domain, there is no pressure build-up. This means that the pressure gradient is zero and the pressure inside the barrel is approximatively equal to the atmospheric pressure. The transport velocity of the material can be controlled by the screw speed and the feed rate. In an extrusion process with a complex geometry of the screws, there exist alternating Partially and Fully Filled Zones. In the remainder of the thesis, we shall assume that the geometry of the equivalent screw is uniform and that there exist only one Partially Filled Zone and one Fully Filled Zone. These two zones are coupled by an interface which characterizes the spatial domains where the pressure gradient is null or not (and accordingly where the extruder is Partially or Fully Filled). The moving interface is assumed to be thin, i.e. reduced to a point. In the sequel the spatial domain of the extruder will be taken to be the real interval [0, L] where L > 0 is the length of the extruder. Denoting by l(t) ∈ [0, L], the position of the thin interface, the domain of the Partially Filled Zone is [0, l(t)[ and the Fully Filled Zone is dened in interval ]l(t), L]. The interface is moving according to the volume of material which is accumulated in the Fully Filled region. It is clear, that the interface which separates these two zones is a function of the dierence between feed and die rates. In conclusion we shall consider three interdependent dynamics which described the evolution of the material in the Partially and Fully Filled Zones (PFZ and FFZ) and the evolution of the interface position. We shall assume that the extruded melt is composed of some species blended with water and that it forms an incompressible homogeneous mixture. For example, in food extrusion processes the water content is an important factor which characterizes the quality of extruded material. We also make the following assumptions: the twin-screw mechanism is approximated by a single screw and the relative motion of the screws is not taken into account.

the ow is one-dimensional and strictly convective in the down barrel direction. the heat capacity C p and the density ρ 0 are assumed to be constant, the screw pitch ξ is uniform along the extruder, there is no reaction which occurs along the process, the extruder is divided in time-varying spatial zones where the material lls completely or not the eective volume V ef f [START_REF] Kulshrestha | An unsteady state model for twin screw extruders[END_REF][START_REF] Li | Modelling extrusion cooking[END_REF].

there exists an interface l(t) between the Partially Filled Zone and the Fully Filled Zone corresponding to discontinuity of the lled volume (or lled volume fraction also called lling ratio);

the Partially Filled Zone is submitted to atmospheric pressure P 0 .

Denition of the variables

let us remind that extrusion processes are used for the production of various products which may vary in their physical structures, composition and functional properties. Depending on the desired material and the desired properties of the product, the extrusion process can combine dierent functions: transport of material, mixing, compression, shear, chemical reactions, heating, cooling, cooking, drying and formatting. Each of these functions can be exploited to various degrees by the appropriate choice of the control parameters and the design of the machine. For a food extrusion process, the transformation mechanisms that occur lead to consider the following variables:

1. The lling ratio: This rate is dened as the ratio between the volume occupied by the material (V o ) and the free volume or eective volume between the barrel and the screw.

Eective volume (V ef f ) is the total volume that can be occupied by material.

(3.2.1)

f (x, t) = V o (x, t) V ef f (x, t) ∈ (R + × R + )
Remark 1. In the Partially Filled Zone, the partial lling characteristic implies that the volume occupied by material is less than the total available volume:

0 < f p (x, t) < 1.
The FFZ has an occupied volume equal to the free volume and f f (x, t) = 1 .

2. The moisture content is the fraction of water stored in the material which is conveyed.

This rate is dened as the ratio between the volume of water V e and the eective volume

V ef f : (3.2.2) M (x, t) = V e (x, t) V ef f ∀(x, t) ∈ (R + × R + )
Remark 2. The moisture content is always less than 1 for both two zones and its evolution aects the viscosity η(x, t) along the extruder. This rate determines the quality of the product see e.g [START_REF] Haley | On-line system identication and control design of an extrusion cooking process: Part I. system identication[END_REF], where the author considers the moisture content to evaluate volume expansion during cooking popcorn process.

3. The temperature: T (x, t) depends on exchanges of heat between the barrel and the extruded material and also on the viscous dissipation phenomena due to the transformation of mechanical energy into heat.

4. The Pressure: in the Partially Filled Zone the pressure is supposed to be equal to air pressure P 0 whereas in the completely lled zone, P (x, t) depends on the reux ow due to geometric characteristics of the die.

5. The viscosity η(x, t) stands as a function of temperature and moisture depending on the considered zone.

For the coupled system the lling ratio f p , the temperature (T p and T f ), the moisture (M p and M f ), viscosity (η p and η f ), and pressure P are the variables. The screw speed N (t), barrel temperature T b and feed rate F in (t) are the input variables.

Model of the Partially Filled Zone

Mass balance in the Partially Filled Zone

The mass balance equations in the PFZ, is written on the spatial domain [0, l(t)[, in terms of the lling ratio f p (x, t) (the lled volume fraction which may be related to the total mass density) [START_REF] Kulshrestha | An unsteady state model for twin screw extruders[END_REF] where l(t) is the position of the thin interface which separates the PFZ and FFZ zones. A local analysis of the ow in one component of the screw is performed to obtain the ow dynamics in the Partially Filled Zone.

We have assumed that the extruded material is composed by a homogeneous mixture and no chemical transformation is operating in the process. So, no species are produced or destroyed and the variation of the amount of material in time is derived as the change of inow and outow through the cross section S ef f dening an elementary volume in this area.

This cross section is the eective section between the screw and the barrel. The variation in the elementary volume between x 2 and x 1 with (0 < x 1 < x 2 < l(t)) in cartesian coordinates

x, y, z (Fig. 3.3.1).

¢ x 2 x 1 ¢ zmaxfp(x,t) 0 ρ 0 y max dzdx = F (x 1 , t) -F (x 2 , t) (3.3.1)
F (x 1 ) and F (x 2 ) are the inow and outow at x 1 and x 2 respectively. The right term of equation (3.3.1) can be written in an integral form using the divergence operator (in the 1D case it reduces to the dierential operator ∂ x ):

F (x 1 , t) -F (x 2 , t) = ¢ x 2 x 1 ∂ x F (x, t)dx (3.3.2)
Developing equation (3.3.1) and using Leibniz formula we obtain the following equality:

¢ x 2 x 1 ρ 0 y max z max ∂ t f p (x, t) + ¢ zmaxf (x,t) 0 d dt (ρ 0 y max z max )dz dx = - ¢ x 2 x 1 ∂ x F (x, t)dx
ρ 0 , y max , and z max are constant and the product y max z max corresponds to the eective section S ef f (Fig. 3.3.1). Thus, equation (3.3.1) implies that: The eective volume for a screw element is given by the following relation:

ρ 0 S ef f ∂ t f p (x, t) = -∂ x F (x, t) (3.
V ef f = ξS ef f (3.3.4)
F (x, t) is the material ow at the spatial coordinates x for a given time t. It could be expressed as a function of the lling ratio f p (x, t) and the screw speed N (t):

F (x, t) = N (t)ρ 0 V ef f f p (x, t) (3.3.5)
Using the denition of the eective volume (3.3.4) and the ow of matter (3.3.5) in this situation, we obtain the following transport equation which described the transport of the homogeneous material in the down barrel direction. The transport velocity which depends on the screw pitch ξ is the transversal component of the rotational velocity N (t).

∂ t f p (x, t) = -ξN (t)∂ x f p (x, t) (t, x) ∈ (R + , [0, l(t)[) (3.3.6) (3.3.7)
f p (x, t) lling ratio in Partially Filled Zone;

N (t) screw speed ;
ξ uniform pitch of the screw.

Moisture balance in the Partially Filled Zone

The moisture content in the Partially Filled Zone M p (x, t) is the fraction of water in the transported homogeneous mixture. It represents the amount of water stored in the processing material and can be assimilated to the fraction of water in the eective volume. The moisture content has the same behavior as the homogeneous mixture because the mass density is supposed to be constant. Therefore, its evolution is also described by a transport equation as previously (3.3.9):

∂ t M p (x, t) = -ξN (t)∂ x M p (x, t) (t, x) ∈ (R + , [0, l(t)[) (3.3.8) (3.3.9)

Energy balance in the Partially Filled Zone

The energy balance allows to describe the evolution of the temperature along the extruder.

There are two ma jor eects which aect temperature behavior of the melt:

1. The mechanical energy provided by the rotation of the screw, which results in deformations of the extruded material. Usually, a drive unit with variable speed causes rotation of the screw. This mechanical power generates heat under the shearing eect:

it is the power generated by viscous dissipation. We would like to mention that heat produced by the internal friction of screws aects the viscosity η(x, t) and consequently modies the behavior of matter. The shearing eect is particularly important in the channel and in dierent geometric area of screw circuit. Especially, the area between the top of screws thread and the inner surface of the barrel. The thermal power Q c generated by viscous dissipation per unit occupied volume V o is a function of the viscosity η(x, t), a geometrical constant C (depending on shear in dierent areas of screw circuit), the screw speed N (t) (which is related to mechanical energy) and viscous dissipation factor γ. As in [START_REF] Kulshrestha | An unsteady state model for twin screw extruders[END_REF] the γ parameter is approximated by a linear function in the lling ratio (ϑ is a given constant):

⎧ ⎨ ⎩ Q c = γ(x,t)Cη(x,t)N 2 (t) f (x,t)V ef f γ(x, t) = ϑ + (1 -ϑ)f p (x, t) (3.3.10)
In the Partially Filled Zone the viscosity η p (x, t) for an unplasticized material is given by equation (3.3.11) as in [START_REF] Li | Modelling extrusion cooking[END_REF]. The viscosity decays when the moisture content increases.

Given the initial viscosity η 0 p , the power law is given by the following equation (β p is a constant depending on rheological properties of the extruded material):

η p (x, t) = η 0 p e -βpMp(x,t) (3.3.11) 2.
The heat ow is supplied by the temperature of the barrel surrounding the screw.

This barrel is equipped with a heating system composed by electric circuits. Also, the circulation of a thermal uid (hot water, steam, oil or uids special) in channels and a system of cooling by circulation of uid (air, water or oil) inuence the barrel temperature T b (x, t). A heat ow φ e is exchanged between the barrel and the mixture through a jacket surface S e in accordance with a coecient depending on the properties of the extruded material.

φ e = αS e (T b (x, t) -T p (x, t)) V ef f (3.3.12)
T b is the barrel temperature;

T p is the temperature in the PFZ; α coecient of exchange;

S e surface of exchange.

The rst laws of thermodynamics law states that the variation of energy per unit of time for a closed system is equal to the power of the external forces plus the external heat exchanged between the system and its environment. Dening u(x, t) as the internal energy per unit of volume and h(x, t) as the enthalpy, we obtain the following relation:

u(x, t) = h(x, t) -P (x, t) (3.3.13) P (x, t)
is the pressure which is supposed to be uniform. The variation of internal energy u(x, t) per unit of volume depends on the enthalpy inow and outow F (x, t) and sources terms such as viscous dissipation Q c (x, t) and convective heat from the barrel to the mass φ e .

Its expression is given by the following relation:

ρ 0 S ef f ∂ t (uf p ) dx = [F h] x x+dx -ρ 0 S ef f P 0 ∂ x f p dx + (Q c + φ e )S ef f f p dx (3.3.14)
Using (3.3.13) and dening the ux F (x, t) as the enthalpy ux which crosses the section S ef f in the Partially Filled Zone with transport velocity ξN (t), (3.3.14) is rewritten as:

ρ 0 ∂ t [(h -P )f p ] = -ρ 0 ξN (t)∂ x (hf p ) -ρ 0 P 0 ∂ t f p + Q c f p + φ e f p (3.3.15)
Since the pressure in the Partially Filled Zone is equivalent to the air pressure P 0 , (3.3.15) becomes:

∂ t h = -ξN (t)∂ x h + Q c ρ 0 + φ e ρ 0 (3.3.16)
The relation between the enthalpy and the temperature T p of the homogeneous mixture with constant mass density is given by :

∂h = C p ∂T p (3.3.17)
C p is a the specic heat capacity. Combining (3.3.10), (3.3.11), (3.3.12), (3.3.16) and (3.3.17), one obtains the following nonlinear time varying transport equation with source terms:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∂ t T p (x, t) = -ξN (t)∂ x T p (x, t) + ληp(x,t)N 2 (t)
φfp(x,t)

+ θ φ (T b -T p (x, t)) θ = αS e λ = γC φ = ρ 0 C p V ef f η p (x, t) = η 0 p e -βpMp(x,t) (t, x) ∈ (R + , [0, l(t)[) (3.3.18)
γ and C are supposed to be constant.

Model of the Fully Filled Zone

The momentum balance of the Fully Filled Zone

In this section, the characteristics of the ow in the Fully Filled Zone are analyzed. The main aspect is that the speed of transportation is essentially determined by the outlet ow rate depending on the pressure at the die. In the Fully Filled Zone the pressure is generated to overcome die section oering ow restriction and the location of the peak pressure coincides with the beginning of the restriction. The length of the Fully Filled Zone is a function of the pressure needed for the melt to ow across the die. In this zone the ow has two components: the rst component is generated by the pumping eect of the screw. In most cases, this is a forward ow, but it could be a backward ow if a reverse screw element is used.

the second component is the pressure ow, which could be a backward ow (called leakage backow) if there is a forward pressure build-up, but could also be a forward ow if there is a forward pressure drop.

The Fully Filled Zone is dened in the spatial domain ]l(t), L], where L is the length of the extruder. By denition the mass balance states that mass accumulated in the Fully Filled Zone is the dierence between the inow and the outow mass as previously. This is a consequence of constant density, and it means that the net forward mass F d (t) is constant across this zone.

The available volume is completely lled and the lling ratio have no dynamic: f f = 1.

The momentum balance equation in the Fully Filled Zone is expressed in term of pressure gradient and yields the following equation for the pressure:

∂ x P (x, t) = η f (x, t) ρ 0 V ef f N (t) -F d (t) Bρ 0 ; (t, x) ∈ (R + , ]l(t), L] (3.4.1)
B is a coecient of pressure ow and η f (x, t) is the melt viscosity in the F F Z [START_REF] Li | Modelling extrusion cooking[END_REF].

η f (x, t) is a decreasing exponential function depending on the moisture content M f (x, t) and the temperature T f (x, t). δ and β f are moisture content and temperature constants of viscosity respectively.

η f (x, t) = η 0 f e -β f M f (x,t) e -δT f (x,t) (3.4.2)
Remark 3. The pressure gradient dened by the equation (3.4.1) appears in the Fully Filled

Zone. The discontinuity of the pressure (the Partially Filled Zone is submitted to atmospheric pressure P 0 ) determines the position of the interface which separates the two zones. Thus, the interface l(t) arises from a change of causality between the lling ratio and the pressure from the Partially Filled Zone to the Fully Filled Zone. Recall that the lling ratio is related to the fraction of volume which is occupied by the extruded material and the pressure corresponds to its conjugate variable.

Integrating the equation (3.4.1), we can derive the value of the pressure at the die exit

P (L, t). P (L, t) -P ((l(t), t) = ¢ L l(t) η f (x, t) ρ 0 V ef f N (t) -F d (t) Bρ 0 dx (3.4.3)
Remark 4. It becomes clear that the coupling relations at the interface depend on the possible solutions of equation (3.4.3). This equation interconnects the two zones and the pressure at the interface position P (l(t), t) should be expressed by taking into account the inuence of the Partially Filled Zone. This aspect will be analyzed under a simplied assumption in the section 3.5.2 dedicated to the interface relations. In ours sense, the formulation of an analytical solution of the integral (3.4.3) with respect to the distributed viscosity η f (x, t) is a dicult point for the construction of explicit coupling relation by (3.4.3).

The net forward ow is determined by the die conductance K d and viscosity η d by the following relation

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ F d (t) = K d η d ΔP (t) ΔP (t) = (P (L, t) -P 0 ) η d = η f (L, t) (3.4.4) η d is equal to η f (x, t) at the die entry.
Remark 5. For a Newtonian uid which ows in a tube of length L and radius R (Fig. 3

.4.1)

K d is given by the following equation: The moisture content M f (x, t) is also dened as the fraction of water in the extruded material, see the Partially Filled Zone. This variable obeys a similar transport equation (3.3.9) with a dierent velocity. The ow in this zone is the net forward ow and for an uniform screw the velocity v(t) of the convection is proportional to F d (t):

K d = π 8 R 4 L (3.
v(t) = 1 S ef f F d (t) ρ 0 = ξF d ρ 0 V ef f (3.4.6)
Then the moisture content balance equation stands as:

∂ t M f (x, t) = - ξF d (t) ρ 0 V ef f ∂ x M f (x, t) (t, x) ∈ (R + , ]l(t), L]
(3.4.7)

Energy balance in the Fully Filled Zone

In this the Fully Filled Zone, the thermal phenomena are identical to those which occur in the Partially Filled Zone. As for moisture content equation (3.4.7), the heat transport velocity is given by equation (3.4.6). Viscous dissipation and heat exchange have to be considered as source terms. The viscous heat generation is much important in this zone due to the mixing eect. One should point out that the permanent contact between the melt and the barrel increases the heat exchange with the mixture. Here, the viscosity depends on both temperature and moisture evolutions as it is shown by equation (3.4.2). Recalling, equation (3.3.14)(with f p = 1), and the melt viscosity (3.4.2) the following equations are obtained to describe the temperature evolution T f (x, t):

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ∂ t T f (x, t) = -ξF d (t) ρ 0 V ef f ∂ x T f (x, t) + λη f (x,t)N 2 (t) φ + θ φ (T b -T f (x, t)) θ = αS e λ = γC φ = ρ 0 C f V ef f η f (x, t) = η 0 f e -β f M f (x,t) e -δT f (x,t) (t, x) ∈ (R + , ]l(t), L]) (3.4.8)
As for the Partially Filled Zone, the coecients γ and C are supposed to be constant.

Description of the interface 3.5.1 Dynamics of the moving interface

The interface that separates the Partially Filled Zone to the Fully Filled Zone is located at the position l (t) where the pressure changes from air pressure to greater than air pressure and the pressure gradient changes from zero to positive value (and the lling ratio changes from values strictly less than one to one) . The position of the interface will, in general, change due to the dierence between the incoming and outcoming mass ow rates. For a nonuniform screw prole, interfaces are determined from the screw geometry as the end point of the restrictive sections [START_REF] Li | Modelling extrusion cooking[END_REF]. However, the location of the interfaces is in general, unknown and must be calculated along with the solution of the model equations. In this section we calculate the equation associated with the interface l(t) for a uniform screw prole. The variation of the position of the interface is derived from the local mass balance at the interface. We consider a xed domain [l -, l + ] strictly containing the position of the interface that is l -< l (t) < l + for some time interval and dene all functions f p (x, t) and F (x, t) as left continuous.

Then, the mass balance is written as follows:

d dt ¢ l + l - ¢ zmaxf (x,t) 0 ρ 0 y max dzdx = F (l -) -F (l + ) = ΔF (3.5.1)
Remind that the lling ratio f (x, t) is :

f p (x, t) in the Partially Filled Zone (x ∈ [l -, l(t)[), f f (x, t) = 1 in Fully Filled Zone (x ∈]l(t), l + ]).
Thus equation (3.5.1) is rewritten as:

¢ zmaxfp(x,t) 0 d dt ¢ l(t) l - ρ 0 y max dxdz + d dt ¢ l + l(t) ρ 0 y max dxdz = ΔF (3.5.2)
Developing this integral (3.5.2) and using f (l + , t) = f f = 1, we deduce the following relation:

dl(t) dt (ρ 0 S ef f f p (l -, t) -1) + ¢ l + l - d dt ¢ zmaxfp(x,t) 0 ρ 0 y max dzdx = ΔF (3.5.3) = ¢ l + l - ρ 0 y max z max ∂ t f (x, t)dx (3.5.4) = ¢ l(t) l - ρ 0 y max z max ∂f p (x, t) ∂t dx + ¢ l + l(t) ρ 0 y max z max ∂f f (x, t) ∂t dx (3.5.5)
let us remind that the lling ratio is dened by f p (x, t) in the P F Z and is equal to 1 in the F F Z, then (3.5.3) is rewritten as:

dl(t) dt ρ 0 S ef f f p (l -, t) -1 + ¢ l(t) l - ρ 0 S ef f ∂f (x, t) ∂t dx = ΔF
Using the transport equation of the lling ratio (3.3.7), one obtains equation (3.5.6):

dl(t) dt ρ 0 S ef f f p (l -, t) -1 + ¢ l(t) l - ξN (t)ρ 0 S ef f ∂f p (x, t) ∂x dx = ΔF (3.5.6)
Which can be rewritten as:

dl(t) dt ρ 0 S ef f f p (l -, t) -1 + ξN (t)ρ 0 S ef f f p (l(t)) -f p (l -, t) = ΔF (3.5.7)
By considering the rst order approximation of the lling ratio f p (l -, t) and the ow rate F (l -, t) in (3.5.7) we nd (3.5.8):

dl(t) dt f p (l(t), t) + l(t) -l -∂f p (l(t), t) ∂x -1 + ξN (t) l(t) -l -∂f p (l(t), t) ∂x = 1 ρ 0 S ef f F (l(t), t) -l(t) -l -∂F (l(t), t) ∂x -F (l + , t) (3.5.8)
When l -→ l (t), the equation (3.5.8) becomes:

dl(t) dt = F (l + ) -F (l(t), t) ρ 0 S ef f (1 -f p (l(t), t) (3.5.9)
In the Fully Filled Zone the net ow rate is uniform and equal to the net ow at the die F d (t) which is a function of the pressure gradient (3.4.4). The ow rate F (l(t), t) is dened by the maximum pumping capacity of the screw which is reduced to the eective fraction of convected material at the left limit of the interface.

F (l(t), t) = ρ 0 ξN (t)V ef f f p (l(t), t) (3.5.10)
Finally the ordinary dierential equation of the moving interface is:

dl(t) dt = F d (t) -ρ 0 ξN (t)V ef f f p (l(t), t) ρ 0 S ef f (1 -f p (l(t), t)) (3.5.11)
Remark 6. The term (1f p (l(t), t)) shows that the extruded material can only ll the fraction of volume which is unoccupied.

Secondly, the condition 0 < f p (l(t), t) < 1 is necessary to have a well-posed dierential equation.

Interface relations

Throughout this chapter we build a simplied model of the extrusion process based on the conservation of mass, energy and momentum. The model is described by considering two zones which are linked by a moving interface. It shows that the lling ratio is discontinuous from the PFZ to the FFZ: see Remark 4. Then, the dynamics of temperature and moisture are not inuenced the net ow. The system is now "decoupled" because the interface dynamics (3.5.11) depends on the lling ratio at the interface f p (l(t), t).

0 < f p (x, t) <

Coupling relations for moisture content and temperature evolutions

The temperature and the moisture content are supposed to be continuous at the interface.

This assumption is physically coherent under a constant viscosity. In our perspective, the strong assumption concerning the viscosity is considered because we focus our analysis on the particular structure of the system consisting of transport equations related by a moving interface. and the position of the interface l(t). Recall that the value of the lling ratio determines the interface position l(t) (3.5.11).

⎧ ⎨ ⎩ T p (l -, t) = T f (l + , t) M p (l -, t) = M f (l + ,
⎧ ⎨ ⎩ P (L, t) = P 0 + ηV ef f N (t)ρ 0 (L-l(t)) (Bρ 0 +K d (L-l(t))) F d (t) = K d V ef f N (t)ρ 0 (L-l(t)) (Bρ 0 +K d (L-l(t))) (3.5.13)
recall B and K d are some geometric parameters of the screw and the die respectively.

Continuity of the ux of momentum ux

The second assumption is based on the continuity of momentum ux which is given by the following relation:

F (l -, t)v(l -, t) + P 0 f p (l -, t)S ef f = F d (t)v(l + , t) + P (l + , t)S ef f , (3.5.14)
where

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ F (l -, t) = ρ 0 N (t)V ef f f (l -, t) v(l -, t) = ξN (t) v(l + , t) = ξF d (t) ρ 0 V ef f . ( 3
ΔP (L, t) = ν -[1 + K d Bρ 0 (L -l(t))] + Δ(l(t), N(t), f p (l(t), t)) , (3.5.16)
where ν =

η 2 f ρ 0 S 2 ef f 2K 2 d and ΔP (L, t) = P (L, t) -P 0 , Δ(l(t), N(t), f p (l(t), t)) = [1 + K d Bρ 0 (L -l(t))] 2 + Ω f p (l(t), t), N(t), l + (3.5.17)
and

Ω = 2ρ 0 ν η f V ef f N (t) Bρ 0 (L -l(t)) +ξ 2 N 2 (t)f p (l(t), t) -(1 -f p (l(t), t)) P 0 ρ 0
The equation (3.5.16) restricts the solution of the system because it induces a positivity constraint. Indeed, the die ow rate which depends on the pressure P (L, t) should be positive and this condition is satised if P (L, t) > P 0 . Then, it becomes necessary to dene a critical value of the lling ratio f pc (l(t), t) as:

f pc (l(t), t)) > P 0 - ηV ef f N (t) B (L -l(t)) ρ 0 ξ 2 N 2 (t) + P 0 (3.5.18)

Control problem associated to the extrusion process model

Here we recall the equations which dene the model of both Partially and Fully Filled Zones in the extruder. An analysis of the global system leads to a control problem for the design of extruded material. The control variables associated to this extrusion process model are:

the boundary conditions at x = 0 for lling ratio, moisture content and temperature which are dened by:

f in p (t) = f p (0, t) = F in (t) ρ 0 N (t)V ef f , M p (0, t) = M in (t), T p (0, t) = T in (t), (3.6.1)
respectively. F in (t) is the feed rate, T b (x, t), the barrel temperature and N (t) the screw speed.

Under the assumption of constant viscosity η, the control objective is to design the temperature T f (L, t) and the die pressure P (L, t). Under the hypothesis of constant viscosity, the moisture content dynamics can be neglected in the control point of view because its evolution inuences essentially the viscosity design as it is shown by equations (3.3.11) and (3.4.2).

In the both following equations (3.6.2) and (3.6.4), ξ, θ, φ and λ are parameters depending on the screw geometry, the barrel characteristics and the material properties (see Section 3.3.3). We mention that at the initial time t = 0, we assume the existence of both Partially and Fully Filled Zones in the spatial domains [0, l 0 ) and (l 0 , L] respectively. L is the length of the extruder, l 0 being the initial position of the moving interface. Thus, the system dened in the coupled domains [0, l(t)) and (l(t), L] is given by the following equations:

1. For the Partially Filled Zone, lling ratio, moisture content and temperature dynamics are given by equations (3.3.7), (3.3.9) and (3.3.18) respectively on the spatial domain

[0, l(t)): ∂ t ⎛ ⎜ ⎜ ⎝ f p M p T p ⎞ ⎟ ⎟ ⎠ (x, t) = -ξN (t)∂ x ⎛ ⎜ ⎜ ⎝ f p M p T p ⎞ ⎟ ⎟ ⎠ (x, t) + ⎛ ⎜ ⎜ ⎝ 0 0 Ω p (f p , N, T p , T b ) ⎞ ⎟ ⎟ ⎠ (3.6.2)
where the source term (3.3.18) is given by the equation :

Ω p (f p , N, T p , T b ) = ληN 2 (t) φf p (x, t) + θ φ (T b (x, t) -T p (x, t)). (3.6.3) 2.
For the Fully Filled Zone, the moisture content and the temperature are given by equations (3.4.7) and (3.4.8) respectively, on the spatial domain (l(t), L]:

∂ t M f T f (x, t) = - ξF d (t) ρ 0 V ef f ∂ x M f T f (x, t) + 0 Ω f (N, T f , T b ) (3.6.4) Ω f (N, T f , T b )
is the source term which is given by the following equality:

Ω f (N, T f , T b ) = ληN 2 (t) φ + θ φ (T b (x, t) -T f (x, t)) (3.6.5)
Recall that the die net ow F d (t) is a function of the pressure at the die exit (3.4.4):

F d (t) = K d η ΔP (t) ΔP (t) = (P (L, t) -P 0 ) (3.6.6)
3. For interface coupling relations the temperature and the moisture content are supposed to be continuous at the interface. We consider two coupling conditions concerning the pressure (or the lling ratio) which is a discontinuous variable:

the rst one derives from the continuity of momentum ux at the interface l(t) (for the parameter see equations (3.5.17) and (3.5.18)):

ΔP (L, t) = ν -[1 + K d Bρ 0 (L -l(t))] + Δ(l(t), N(t), f p (l(t), t)) ,(3.6.7)
where Δ is the dierence between the pressure at L and the atmospheric pressure, L being the extruder length (P (L, t) -P 0 ). This coupling relation generates a constraint on the lling ratio (ΔP (t) should be positive (in some sense the square roots term on Δ should be positive). Thus :

f pc (l(t), t)) > P 0 - ηV ef f N (t) B (L -l(t)) ρ 0 ξ 2 N 2 (t) + P 0 , ( 3.6.8)
Where f pc (l(t), t)) represents the minimal value of the lling ratio for a coherent physical model.

the second one is constructed under the hypothesis of the continuity of pressure at interface l(t). Thus:

⎧ ⎨ ⎩ ΔP (t) = ηV ef f N (t)ρ 0 (L-l(t)) (Bρ 0 +K d (L-l(t))) F d (t) = K d V ef f N (t)ρ 0 (L-l(t)) (Bρ 0 +K d (L-l(t))) .
(3.6.9)

4. The dynamics of the moving interface l(t) is deduced from a local mass balance and it follows :

⎧ ⎨ ⎩ dl(t) dt = F d (t)-ρ 0 V ef f N (t)(1-fp(t,l(t))) ρ 0 S ef f (1-fp(l(t),t)) l(0) = l 0
F in = 135/3600kg.s -1
Filling ratio

f in p = F in ρ 0 V ef f N 0 = 0.3148 Barrel temperature T b = 330K Inlet temperature T in = 293K Extruder length L = 2m Table 3.1: Input values N 0 , f in p , T in , T b
The initial conditions are dened for the lling ratio, the interface position and the temperature along the extruder as f 0 p , l 0 , T 0 p and T 0 f respectively. Initial lling ratio

f 0 p = 0.3

Initial interface position

l 0 = 1m
Initial temperature along the extruder

T 0 = T 0 p = T 0 f = 293K Table 3.2: Initial conditions f 0 p , l 0 , T 0
Now, considering the initial and boundary data which are dened in (Tab. 3.1) and (Tab. 3.2), we dene the following the simulated system composed by: 1. The lling ratio and the temperature dynamics in the Partially Filled Zone x ∈ [0, l(t)):

∂ t f p T p (x, t) = -ξN 0 ∂ x f p T p (x, t) + 0 Ω p (f p , T p ) , (3.7.1)
where the source term Ω p (f p , T p ) is given by the equation:

Ω p (f p , T p ) = ληN 2 0 φf p (x, t) + θ φ (T b -T p (x, t)). (3.7.2)
2. The temperature dynamics in the Fully Filled Zone x ∈ (l(t), L]:

∂ t T f (x, t) = - ξF d (t) ρ 0 V ef f ∂ x T f (x, t) + Ω f (T f ), (3.7.3)
where

Ω f (T f ) = ληN 2 0 (t) φ + θ φ (T b -T f (x, t)). (3.7.4) 3.
Interface relations which consist of continuity of temperature (T p (l -, t) = T f (l + , t)) and pressure at the coupling point l(t). As for (3.7.5), we nd:

⎧ ⎨ ⎩ P (L, t) = P 0 + ηV ef f N 0 ρ 0 (L-l(t)) (Bρ 0 +K d (L-l(t))) F d (t) = K d V ef f N 0 ρ 0 (L-l(t)) (Bρ 0 +K d (L-l(t))) .
(3.7.5)

4. The moving interface whose dynamics is given by the following equation:

dl(t) dt = F d (t) -ρ 0 ξN 0 V ef f f p (l(t), t) ρ 0 S ef f (1 -f p (l(t), t)) (3.7.6)
Remark 7. In the case of continuity of momentum ux at the interface, the initial condition must be chosen with respect to the minimal value of the lling ratio f pc (3.6.8).

Numerical computation by nite volume method

There is a fairly extensive literature on numerical analysis of partial dierential equations with time-varying domains. Most of the numerical works have been focused on multi-phase ow problems including the inter-phase coupling terms which require specic treatment by numerical algorithms. A very important feature of multi-phase ow simulations is the interface between materials and phases, and it is often crucial to track such interfaces at each step of the computation. There exist two classes of numerical representations which allow to perform numerical experiments on moving interface problems:

For Lagrangian representation, the mesh deforms with the material and automatically maintain interfaces. These representations fail if the mesh is excessively deformed by topological changes because the mesh reconstitution for each step is strongly based on the interface form.

For Eulerian representation, the material moves through a stationary mesh and requires special procedures to track precisely the interfaces position in the ow.

In uid dynamics, however, both Lagrangian and Eulerian coordinates have been used with considerable success. Because each coordinates representation has advantages and disadvantages, the choice of which representation to use depends on the characteristics of the problem to be solved. The two representations dier, however, in the manner in which the uid elements are moved to their next positions after their new velocities have been computed. In the Lagrangian case the grid simply moves with the computed element velocities, while in an Eulerian calculation it is necessary to compute the ow of uid through the mesh.

By denition the interface is a boundary which separates two dierent physical regions. In many cases, the numerical analysis is based on the use of marker-cells, volume of uids, height functions or Level Set methods which allow to keeping track of the interface position [START_REF] Harlow | Numerical calculation of time-dependent viscous incompressible ow of uid with free surfaces[END_REF][START_REF] Daly | Numerical study of two uid Rayleigh-Taylor instability[END_REF][START_REF] Daly | Numerical study of density current surges[END_REF][START_REF] Tryggvason | A front-tracking method for the computations of multiphase ow[END_REF]. Also, there exist many algorithms which are developed to rebuild the interface along the numerical calculation specially for volume of uid methods one can mention works of [START_REF] Noh | Slic (simple line interface calculation)[END_REF][START_REF] Parker | Two and three dimensional eulerian simulation of uid ow with material interfaces[END_REF][START_REF] Saltzman | A 3-D adaptive mesh renement algorithm for multimaterial gas dynamics[END_REF]. Here we present a brief description of the height function method. The idea behind the height function method is to dene the interface distance from a reference line as a function of position along the reference line. For example, in a rectangular mesh of cells of width δx and height δy one might dene the vertical height (h) of the interface above the bottom of the mesh in each column of cells. This would approximate a curve h = f (x, t) by assigning values of h to discrete values of x. This representation is extremely ecient, requiring only a one-dimensional storage array to record the surface height values. Likewise, the evolution of the surface only requires the updating of the one-dimensional array. Then, the time evolution of the height function is governed by a kinematic equation expressing the fact that the surface must move with the uid:

∂ t h + u∂ x h = v (3.7.7)
where (u, v) are the uid velocity components in the (x, y) coordinates direction.

Numerical analysis of the extruder model which is described by coupled hyperbolic systems in complementary time-varying domains through an interface is very similar to uid-structure coupling problems. Thus the height function methods presented previously may be relevant in this case. Indeed, we consider a xed mesh on which a color function whose value are 1 or 0 according to the Partially Filled Zone and the Fully Filled Zone. It is important to note that the position of the interface can be calculated explicitly by integrating the color function over the whole domain [0, L] if appropriates initial conditions are chosen. Here we dene the color function as the characteristic function of the domain:

c(x, t) = 1 x ∈ [0, l (t) [ 0 x ∈ [l (t) , L] (3.7.8)
We mention that the characteristic function (3.7.8) is the solution of the transport equation whose velocity corresponds to the dynamics of the interface.

∂ ∂t c(x, t) = -l (t) ∂ ∂x c(x, t)
(3.7.9) with the following initial conditions:

c(x, 0) = 1 x ∈ [0, l 0 [ 0 x ∈ [l 0 , L]
(3.7.10) l 0 being the position of the interface at time t = 0.

All the states and ux variables which describe the extruder model in PFZ and FFZ are extended over the whole domain [0, L], using color function (3.7.8) and its complement as follows:

N (x, t) = c(x, t)N p (x, t) + (1 -c(x, t))N f (x, t) (3.7.11)
The nite volume method which is generally well adapted to the numerical calculation of the transport equations is used to simulated the system. The numerical scheme is applied on a xed mesh considering the extended variables which arise from a linear combination (3.7.11).

Simulations results

The interface position l(t) Recall that the extruder length is L = 2m. The gure (Fig. 3.7.1) shows the evolution of the interface l(t) from the initial position l 0 = 1m to l(t 1 ) = 1.760m, t 1 = 350s. The equilibrium position of the interface l e = 1.826m is given by:

l e = L - Bρ 0 f p in K d (1 -f p in ) , (3.7.12)
where f p in is the inlet lling ratio deduced from the value of the feed rate F in (see Tab. 3.1). We remark that at time t = 350s the interface converges to the equilibrium with an error equal to 6.6%

The lling ratio evolution at dierent spatial coordinates First, at time t = 0, f p (t) has a constant initial value which is f 0 p = 0.3 (Tab. 3.2). Recall that the extruder length is L = 2m. Consequently, the coordinates x 1 , belongs to the Partially Filled Zone and the positions x 2 and x 3 belong to the Fully Filled Zone.

Secondly, the interface moves increasingly starting from the initial value l 0 = 1m to its nal value l(t 1 ) = 1.760m (t 1 = 350s) (Fig. 3 Finally at positions x 1 x 2 x 3 , the nal value of the lling ratio is f p = 0.1749. We observe that delay of propagation of the boundary input f p in also appears in the simulation (Fig. 3.7.2): the input acts on the initial value after a delay which depends on the speed and the pitch of the screw which are the constant "parameters" N 0 and ξ, respectively.

The temperature in Partially and Fully Filled Zones Recall that the barrel temperature is xed at T b = 330K, the initial value of temperature along the extruder is T 0 = 293K and the inlet temperature is T in = 293K(continuity of temperature at the entry of the extruder). Using the previous arguments we observe that at the initial time t 0 : the spatial coordinate x 1 = L/4 = 0.5m belongs to the Partially Filled Zone, and the positions x 2 = 3L/4 = 1.5m and x 3 = L = 2m belong to the Fully Filled Zone

The gure (Fig. 3.7.3) shows that: at the spatial coordinate x 3 = L = 2m which belongs to the Fully Filled Zone, the temperature reaches its nal value T (L, t 1 ) = 355K, (t 1 = 350s); at the spatial coordinate x 3 = L = 2m which belongs to the Partially Filled Zone the temperature increases from the initial value and reaches its nal value

T (x 3 , t 1 ) = 361K.
The evolution of the temperature at the spatial coordinate x 2 = 3L/4 = 1.5m is particular due to the change of zone.

Remark 8. At time t 1 = 350s the value of the temperature in the Partially Filled Zone is superior to its value in the Fully Filled Zone and both temperatures that we observe in (Fig. 3.7.3) are greater than the barrel temperature. The extruders includes generally a cooling system which regulate the heat ux between the barrel and the material. Normally, in a real extruder, the shear is more important in the Fully Filled Zone and we should observe the opposite phenomenon. The strong assumption of constant viscosity which mitigates the eect of the power generated by viscous dissipation.

The evolution of temperature prole at dierent times The gure (Fig. 3.7.4) is a screenshot of the temperature along the extruder at dierent times t 1 , t 2 , t 3 and t 4 . It allows to observe the evolution of temperature proles depending on the interface position. For each captured time, the " dot" shows the position of the interface. The peak from which the temperature starts to decrease rapidly corresponds to the position of the interface at the time when the temperature prole is captured.

Conclusion

In this chapter the model of an extrusion process has been presented. The model is derived from balance laws expressing conservation of mass, moisture content and energy. Firstly, we describe the ow in a nonhomogeneous medium which corresponds to the geometric structure of the extruder. We explain how a gradient of pressure is build-up in some Fully Filled Zone due to the accumulation of material behind the die. Then, by a structural decomposition of the extruder, we obtain transport equations in Partially and Fully Filled Zone in the time-varying domains [0, l(t)[ and ]l(t), L] respectively. The obtained model is composed by nonlinear hyperbolic systems with source terms and a mobile interface l(t) stands as an interconnection component. The main diculty in such system arises from the denition of the coupling conditions between the two zones. We propose two coupling relations at the moving interface by means of state and ux coupling methods. The rst transmission condition is based on a pressure continuity hypothesis and the second one is obtained assuming the continuity of momentum ux. An analysis of the proposed coupling relations shows how the system is constrained in the case of the continuity of momentum ux. Another interesting point in this chapter concerns the brief analysis of numerical methods concerning uids dynamics. The color function which is proposed in [START_REF] Boutin | Etude Mathématique et Numérique des Equations Hyperboliques Nonlinéaires: couplage de modèles et chocs non classiques[END_REF] permits to extend the denition of variables of each zone in the whole domain [0, L] and a nite volume method is performed for simulations. The results illustrate the dynamics of moving interface, lling ratio and temperature along the process. The position of the moving interface appears clearly in the simulation as a point of discontinuity. In the next chapter, an mathematical analysis will be performed in order to prove the well-posedness of the Cauchy problem of the extruder process model. The Cauchy problem for coupled hyperbolic systems through a moving interface

Introduction

The mathematical analysis of mobile interfaces in the context of moving boundary problems have been an active subject in the last decades and their mathematical understanding continues to be an important interdisciplinary tool for the scientic applications. Such class of partial dierential equations arises in many applications devoted to modelling of biological systems and reaction diusion processes. In this context, [START_REF] Choi | Global existence of solutions to a coupled parabolichyperbolic system with moving boundary[END_REF] study the global existence of solutions to a coupled parabolic-hyperbolic system with moving boundary representing cell mobility. A similar type of nonlinear moving-boundary problem consisting of a hyperbolic equation and a parabolic equation for modelling blood ow through viscoelastic arteries [START_REF] Canic | Existence of a Unique Solution to a Nonlinear Moving-Boundary Problem of Mixed Type Arising in Modelling Blood Flow[END_REF] and tumor growth [START_REF] Chen | A two-phase free boundary problem with discontinuous velocity: Application to tumor model[END_REF] is study in terms of well-posedness. General reaction-absorptiondiusion processes involves also moving boundary as it is stated in [START_REF] Jr Robertt | Moving boundaries in reaction-diusion systems with absorption[END_REF]. In order to prove local existence and uniqueness of a solution for such problem, the key of the resolution is to transform the system with moving boundary into a system dened on a xed domain. Then, the results of many studies dedicated to hyperbolic system of conservation laws may be useful for establishing the existence, uniqueness, regularity and continuous dependence of solutions. The well-posedness study of hyperbolic systems of conservation laws which are dened on xed domains is proposed in [START_REF] Bianchini | Vanishing viscosity solutions of nonlinear hyperbolic systems[END_REF][START_REF] Li | Semi-global C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF][START_REF] Wang | Exact controllability for nonautonomous rst order quasilinear hyperbolic systems[END_REF][START_REF] Coron | Analysis of a conservation law modeling a highly re-entrant manufacturing system[END_REF][START_REF] Shang | Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system[END_REF] (and the references therein) in the content of weak solutions. In this chapter we apply the method proposed in [START_REF] Shang | Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system[END_REF][START_REF] Coron | Analysis of a conservation law modeling a highly re-entrant manufacturing system[END_REF] to prove the existence and the uniqueness of solutions for the extrusion process model after a change of coordinates which allows to express the system in a xed domain (0, 1). In [START_REF] Shang | Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system[END_REF] a scalar conservation law that models a highly re-entrant manufacturing system is studying as a generalization of [START_REF] Coron | Analysis of a conservation law modeling a highly re-entrant manufacturing system[END_REF]. The proof of the existence and uniqueness of the weak solution of the corresponding Cauchy problem with initial and boundary data is given using a xed point argument. We mention that rst results concerning the well-posedness of extrusion processes as transport equations coupled via complementary time varying domains is proposed in [START_REF] Diagne | Modélisation et commande d'un système d'équations aux dérivées partielles à frontière mobile : application au procédé d'extrusion[END_REF].

In [START_REF] Diagne | Modélisation et commande d'un système d'équations aux dérivées partielles à frontière mobile : application au procédé d'extrusion[END_REF] The mathematical analysis is performed for the linearized model of the extruder and the well-posedness is proved using perturbation theory on the linear operator [START_REF] Kato | Pertubation Theory for Linear Operators[END_REF]. Here we are interested in the nonlinear model of the extruder which is described by conservation laws of Chapter 3.

We consider the cauchy problem for extruder model composed of two systems of conservation laws together with an ordinary dierential equation. Using Banach xed point theorem, we prove the well-posedness directly on the nonlinear model. Recall that the model is based on the decomposition of the extruder in Partially and Fully Filled Zones. Let us recall the transport equations of mass, moisture content and energy in an extruder. In this chapter we dene ζ as the screw pitch and ξ(t) as the characteristic curves.

1. For the Partially Filled Zone, we consider the lling ratio f p , the moisture content M p and the temperature T p as the state variables.

The transport equations which are associated to these variables are dened at time t ∈ [0, T ], and position x ∈ [0, l(t)) where l(t) represents the moving interface between the two zones (3.3.7), (3.3.9), (3.3.18). The lling ratio f p and the moisture M p are strictly positive functions which are less than one according to the modelling assumptions. The source term (3.3.18) is given by the equation : The denition of the variables above (4.1.3) can be found in section 3.3.3.

∂ t ⎛ ⎜ ⎜ ⎝ f p M p T p ⎞ ⎟ ⎟ ⎠ = -α p (N )∂ x ⎛ ⎜ ⎜ ⎝ f p M p T p ⎞ ⎟ ⎟ ⎠ + ⎛ ⎜ ⎜ ⎝ 0 0 Ω p (f p , N, T p , T b ) ⎞ ⎟ ⎟ ⎠
Ω p (f p , N, T p , T b ) = C 1 (T p -T b ) + g 1 (N, f p ),

2.

For the Fully Filled Zone, we consider the moisture M f and temperature T f as states variables. The transport equations which are associated to these variables are dened at time t ∈ [0, T ], and position x ∈ (l(t), L] where L is the length of the extruder (3.4.7), (3.4.8). The moisture content is the fraction of water which takes values in ]0, 1].

∂ t M f T f = -α f (N, l, f p ) ∂ x M f T f + 0 Ω f (N, T f , T b ) (4.1.6)
where

α f (N, l, f p ) = ζF d (t) ρ 0 V ef f (4.1.7)
The source term is given by the equation:

Ω f (N, T f , T b ) = C 1 (T f -T b ) + g 2 (N ), (4.1.8)
with

g 2 := ληN 2 (t) ρ 0 V ef f c p
The denition of the variables above (4.1.8) can be found in section 3.3.3. Assuming the continuity of momentum ux at the interface l(t), the net ow F d (t) is a function of l(t), f p (t, l(t)) and N (t) and under hypothesis of constant viscosity η, we nd:

ΔP (L, t) = ν -[1 + K d Bρ 0 (L -l(t))] + Δ(l(t), N(t), f p (l(t), t)) ,(4.1.9)
where ν = η 2

f ρ 0 S 2 ef f 2K 2 d and ΔP (L, t) = P (L, t) -P 0 , Δ(l(t), N(t), f p (l(t), t)) = [1 + K d Bρ 0 (L -l(t))] 2 + Ω f p (l(t), t), N(t), l + (4.1.10)
and

Ω = 2ρ 0 ν η f V ef f N (t) Bρ 0 (L -l(t)) +ζ 2 N 2 (t)f p (l(t), t) -(1 -f p (l(t), t)) P 0 ρ 0 F d (t) = K d η (P (L, t) -P 0 ), (4.1.11)
3. The moving interface l(t) dynamic is deduced from a total mass balance and is:

⎧ ⎨ ⎩ l(t) = F (l(t), N(t), f p (t, l(t))) , 0 < t < T, l(0) = l 0 , (4.1.12)
where

F (l(t), N(t), f p (t, l(t)) = F d (t) -ρ 0 V ef f N (t)(1 -f p (t, l(t))) ρ 0 S ef f (1 -f p (t, l(t))
) .

(4.1.13)

Recall that the Partially and Fully Filled Zones exist if the positivity constraint is satised:

=⇒ f pc (t, l(t)) > 1 ζ 2 N 2 (t) + P 0 ρ 0 P 0 ρ 0 -K 2 (L -l(t)) (4.1.14)
For the system dened by (4.1.1), (4.1.6) (4.1.9) and (4.1.12) , we dene:

the initial conditions as:

(f p (0, x), M p (0, x), T p (0, x)) T = (f 0 p (x), M 0 p (x), T 0 p (x)) T , 0 x l 0 , (4.1.15) (M f (0, x), T f (0, x)) T = (M 0 f (x), T 0 f (x)) T , l 0 x L.
(4.1.16) Here we assume that f 0 p satised:

0 < γ 0 f 0 p L ∞ γ 1 < 1.
(4.1.17)

the boundary conditions as:

f p (t, 0), M p (t, 0), T p (t, 0) T = F in (t) θ(N (t)) , M in (t), T in (t)
T , t ∈ (0, T ), (4.1.18) where

θ(N (t)) = ρ 0 V ef f N (t).
F in (t) is the feed rate. F in (t) and N (t) are known functions of time t. and the interface coupling relation for temperature and moisture as:

M p (t, l(t)), T p (t, l(t)) T = M f (t, l(t)), T f (t, l(t)) T . ( 4

Organization of the chapter

The organization of this chapter is as follows/ First in Section 3.2, using a change of variables, we transform the time-varying spatial domain problem into a xed domain problem. In Section 3.3 we present the main results on the existence and uniqueness of solutions in W 1,∞ (Theorem 3) and H 2 (Theorem 4) spaces. In Section 3.4 we give the proof of Theorem 3 and nally in Section 3.5 we give an idea of the proof of Theorem 4.

Contributions

In this chapter, we prove the existence, uniqueness and regularity of the weak solution to the free boundary Cauchy problem (4.1.1), (4.1.6), (4.1.12), (4.1.15), (4.1.16), (4.1.18), (4.1.19). To tackle this free boundary problem, we make a change of variables on the spatial variables and transform the time-varying spatial domain into a xed domain. Here, we use Banach xed point theorem to prove the existence of weak solutions in W 1,∞ and H 2 .

Model expressed in xed spatial domain

The main diculty of the well-posedness study arises from the moving domains [0, l(t)[ and ]l(t), L], l(t) being the interface dynamics. In order to deal with this system of balance equations in xed domains, a classical change of spatial variables is performed for the two zones leading to two systems of conservation laws with additive ctitious convection terms in the transport velocities. Now the Partially and Fully Filled Zone equations are dened on

Q := (0, T ) × (0, 1).
1. For Partially Filled Zone (PFZ), after the change of variables from (0, l(t)) to the interval (0, 1), see [START_REF] Diagne | Well posedness of the model of an extruder in innite dimension[END_REF], via

y(t, x) := x l(t) ,
we normalize the system (4.1.1) to a new system dened on Q := (0, T ) × (0, 1). For the sake of simplicity, we still denote the space variable y as x and unknown the functions as (f p , M p , T p ), the velocity term as α p and the source term as Ω p . We have

∂ t ⎛ ⎜ ⎜ ⎝ f p (t, x) M p (t, x) T p (t, x) ⎞ ⎟ ⎟ ⎠ + α p (t, x)∂ x ⎛ ⎜ ⎜ ⎝ f p (t, x) M p (t, x) T p (t, x) ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ 0 0 Ω p (f p , N, T p , T b ) ⎞ ⎟ ⎟ ⎠ (4.2.1)
with

α p (t, x) = 1 l(t) ζN (t) -x l(t) = 1 l(t) ζN (l(t)) -xF (l(t), N(t), f p (t, 1)) . (4.2.2)
The source term is given by the same relation (4.1.3):

Ω p (f p , N, T p , T b ) = C 1 (T p -T b ) + g 1 (N, f p ), (4.2.3)
2. For the Fully Filled Zone (FFZ), after the change of variables from (l(t), L) onto the interval (0, 1)

y(t, x) := x -l(t) L -l(t) ,
system (4.1.6) can be normalized to a new system dened on Q := (0, T ) × (0, 1). For the sake of simplicity, we still denote the space variable y as x and unknown functions as (M f , T f ), the velocity term as α f and the source term as Ω f .

∂ t M f (t, x) T f (t, x) + α f (t, x)∂ x M f (t, x) T f (t, x) = 0 Ω f (N, T f , T b ) , (4.2.4) with α f (t, x) = 1 L -l(t) ζF d (t) ρ 0 V ef f + (x -1) l(t) , = 1 L -l(t) ζF d (t) ρ 0 V ef f + (x -1)F (l(t), N(t), f p (t, 1)) , ( 4.2.5)
where

F d (t) := F d (l(t), N(t), f p (t, 1))
is dened by (4.1.11). The source term is also given by the same relation (4.1.8). which are the die net ow rate and the lling ratio at the interface, respectively.

Ω f (N, T f , T b ) = C 1 (T f -T b ) + g 2 (N ), (4.2
f p (t, 0), M p (t, 0), T p (t, 0) T = F in (t) θ(N (t)) , M in (t), T in (t) T , t ∈ (0, T ) (4.2.7) M f (t, 0), T f (t, 0) T = M p (t,
Consequently, in both Partially and Fully Filled Zones the net ow rate F d (t), lling ratio at the interface f p (l(t), t) and the screw speed N (t) must be considered in the convection terms: the interface dynamic is a nonlinear function of these functions.

In summary, we consider a coupled system composed of an Ordirany Dierential Equation (ODE) for moving interface

⎧ ⎨ ⎩ l(t) = F (l(t), N(t), f p (t, 1)) , t ∈ (0, T ) l(0) = l 0 , ( 4.2.9)
a transport equation for lling ratio

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t f p (t, x) + α p (t, x)∂ x f p (t, x) = 0, (t, x) ∈ (0, T ) × (0, 1) f p (0, x) = f 0 p (x), x ∈ (0, 1) f p (t, 0) = F in (t) θ(N (t)) , t ∈ (0, T ) (4.2.10)
two transport equations for the moisture contains

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t M p (t, x) + α p (t, x)∂ x M p (t, x) = 0, (t, x) ∈ (0, T ) × (0, 1) ∂ t M f (t, x) + α f (t, x)∂ x M f (t, x) = 0, (t, x) ∈ (0, T ) × (0, 1) M p (0, x) = M 0 p (x), M f (0, x) = M 0 f (x), x ∈ (0, 1) M p (t, 0) = M in (t), M f (t, 0) = M p (t, 1), t ∈ (0, T ) (4.2.11)
and two transport equations for the temperature

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t T p (t, x) + α p (t, x)∂ x T p (t, x) = Ω p , (t, x) ∈ (0, T ) × (0, 1) ∂ t T f (t, x) + α f (t, x)∂ x T f (t, x) = Ω f , (t, x) ∈ (0, T ) × (0, 1) T p (0, x) = T 0 p (x), T f (0, x) = T 0 f (x), x ∈ (0, 1) T p (t, 0) = T in (t), T f (t, 0) = T p (t, 1), t ∈ (0, T ) (4.2.12)
In the whole chapter, unless otherwise specied, we always assume that l 0 ∈ (0, L),

f 0 p ∈ W 1,∞ (0, 1), M 0 p , T 0 p , M 0 f , T 0 f ∈ L 2 (0, 1), M in , T in ∈ L 2 (0, T ), F in , N ∈ W 1,∞ (0, T ) and T b ∈ L ∞ (Q).
For the sake of simplicity, we denote from now on

f L ∞ ( f W 1,∞ , f L 2 , resp.) as the L ∞ (W 1,∞ , L 2 ,resp.
) norm of the function f with respect to its unknowns.

F W 1,∞ : = |α| 1 ess sup le-ε 1 <x 1 <le+ε 1 Ne-ε 1 <x 2 <Ne+ε 1 fpe-ε 1 <x 3 <fpe+ε 1 |D α F (x 1 , x 2 , x 3 )|.

Main Results

We recall from [40, Section 2.1], the usual denition of a weak solution to Cauchy problem (4.2.9)-(4.2.12). Denition 1. Let T > 0 be given. A weak solution of Cauchy problem (4.2.9)-(4.2.12) is 4 , such that for every τ ∈ [0, T ], every test function

a vector function (l, f p , M p , M f , T p , T f ) ∈ W 1,∞ (0, T ) × W 1,∞ (Q) × (C 0 ([0, T ]; L 2 (0, 1)))
ϕ 1 , ϕ 2 = (ϕ 21 , ϕ 22 ), ϕ 3 = (ϕ 31 , ϕ 32 ) ∈ C 1 ([0, T ] × [0, 1]) such that ϕ i (τ, x) = 0, ∀x ∈ [0, 1], i = 1, 2, 3, (4.3.1)
ϕ i (t, 1) = 0, ∀t ∈ [0, T ], i = 1, 2, 3, (4.3.2)
one has

l(t) = l 0 + ¢ t 0 F (l(s), N(s), f p (s, 1)) ds, t ∈ (0, τ), (4.3.3) ¢ τ 0 ¢ 1 0 f p (∂ t ϕ 1 + ∂ x (α p ϕ 1 )) dx dt + ¢ 1 0 f 0 p (x)ϕ 1 (0, x) dx + ¢ τ 0 F in (t) θ(N (t)) α p (t, 0)ϕ 1 (t, 0) dt = 0, (4.3.4) ¢ τ 0 ¢ 1 0 M p (∂ t ϕ 21 + ∂ x (α p ϕ 21 )) dx dt + ¢ τ 0 ¢ 1 0 M f (∂ t ϕ 22 + ∂ x (α f ϕ 22 )) dx dt + ¢ τ 0 α p (t, 0)M in (t)ϕ 21 (t, 0) dt + ¢ τ 0 α f (t, 0)M p (t, 1)ϕ 22 (t, 0) dt + ¢ 1 0 M 0 p (x)ϕ 21 (0, x) dx + ¢ 1 0 M 0 f (x)ϕ 22 (0, x) dx = 0, (4.3.5) ¢ τ 0 ¢ 1 0 T p (∂ t ϕ 31 + ∂ x (α p ϕ 31 )) dx dt + ¢ τ 0 ¢ 1 0 T f (∂ t ϕ 32 + ∂ x (α f ϕ 32 )) dx dt + ¢ τ 0 ¢ 1 0 Ω p ϕ 31 dx dt + ¢ τ 0 ¢ 1 0 Ω f ϕ 32 dx dt + ¢ τ 0 α p (t, 0)T in (t)ϕ 31 (t, 0) dt + ¢ τ 0 α f (t, 0)T p (t, 1)ϕ 32 (t, 0) dt + ¢ 1 0 T 0 p (x)ϕ 31 (0, x) dx + ¢ 1 0 T 0 f (x)ϕ 32 (0, x) dx = 0. (4.3.6)
We have the following two theorems Theorem 3. Let T > 0. Let (l e , N e , f pe ) be a constant equilibrium,

F (l e , N e , f pe ) = 0 (4.3.7)
with 0 < f pe < 1, 0 < l e < L. Assume that the compatibility condition at (0, 0) holds

F in (0) θ(N (0)) = f 0 p (0). (4.3.8)
Then, there exists ε 0 (depending on T ) such that for any ε ∈ (0, ε 0 ], if

f 0 p (•)-f pe W 1,∞ + F in (•) θ(N (•)) -f pe W 1,∞ + N (•)-N e W 1,∞ +|l 0 -l e | ε. (4.3.9)
Cauchy problem (4.2.9)-(4.2.12) admits a unique solution (l,

f p , M p , M f , T p , T f ) ∈ W 1,∞ (0, T )× W 1,∞ (Q) × (C 0 ([0, T ]; L 2 (0, 1))) 4
, and the following estimates hold

f p -f pe W 1,∞ + l -l e W 1,∞ C ε 0 • ε, (4.3.10) M p C 0 ([0,T ];L 2 (0,1)) C ε 0 • M 0 p L 2 + M in L 2 , (4.3.11) T p C 0 ([0,T ];L 2 (0,1)) C ε 0 • T 0 p L 2 + T in L 2 , (4.3.12) M f C 0 ([0,T ];L 2 (0,1)) C ε 0 • M 0 p L 2 + M in L 2 + M 0 f L 2 , (4.3.13) T f C 0 ([0,T ];L 2 (0,1)) C ε 0 • T 0 p L 2 + T in L 2 + T 0 f L 2 , (4.3.14)
where C ε 0 is a constant depending on ε 0 , but independent of ε.

Theorem 4. Under the assumptions of Theorem 3, we assume further that f 0

p (•) ∈ H 2 (0, 1), F in (•) θ(N (•))
∈ H 2 (0, T ), and the compatibility condition at (0, 0) holds

f 0 px (0) + l(0) ζN (0) • F in (0)θ(N (0)) -F in (0)θ (N (0))N (0) θ 2 (N (0)) = 0. (4.3.15)
Then, there exists ε 0 (depending on T ) such that for any ε ∈ (0, ε 0 ], if Cauchy problem (4.2.9)-(4.2.12) has a unique solution (l,

f 0 p (•)-f pe H 2 (0,1) + F in (•) θ(N (•)) -f pe H 2 (0,T ) + N (•)-N e W 1,∞ +|l 0 -l e | ε,
f p , M p , M f , T p , T f ) ∈ W 1,∞ (0, T ) × C 0 ([0, T ]; H 2 (0, 1)) ×(C 0 ([0, T ]; L 2 (0, 1)
)) 4 with the additional estimate

f p -f pe C 0 ([0,T ];H 2 (0,1)) C ε 0 • ε, (4.3.17)
where C ε 0 is a constant depending on ε 0 , but independent of ε.

Remark 10. The solution we obtained in Theorem 3 or in Theorem 4 is called semi-global solution since it exists on any preassigned time interval [0, T ] if (l, f p ) has some kind of smallness (depending on T ), see [START_REF] Li | Semi-global C 1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF][START_REF] Wang | Exact controllability for nonautonomous rst order quasilinear hyperbolic systems[END_REF]. On the other hand, we do not require any smallness on M p , M f , T p and T f since their equations are linear.

Remark 11. We have the hidden regularity that

(M p , M f , T p , T f ) ∈ (C 0 ([0, 1]; L 2 (0, T ))) 4
both in Theorem 3 and in Theorem 4. Similarly, we also have

f p ∈ C 0 ([0, 1]; H 2 (0, T )) in Theorem 4.
For the proof of Remark 11, one can refer to [START_REF] Coron | Analysis of a conservation law modeling a highly re-entrant manufacturing system[END_REF][START_REF] Shang | Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system[END_REF].

Proof of Theorem 1

Sketch of the prove of Theorem 3:

1. In Section 4.4.1 we apply contraction mapping principle to prove the existence and uniqueness of l(t) and f p (t,

> 0 suitably small, ∃ δ 1 = δ 1 (ε 1 , f 0 p -f pe W 1,∞ , |l 0 -l e |), ∀ε ∈ (0, ε 1 ], ∀f 0 p ∈ W 1,∞ (0, 1), ∀F in , N ∈ W 1,∞ (0, T ), ∀l 0 ∈ (0, L) with f 0 p (•)-f pe W 1,∞ + F in (•) θ(N (•)) -f pe W 1,∞ + N (•)-N e W 1,∞ +|l 0 -l e | ε, (4.4.1)
Cauchy problem (4.2.9)-(4.2.10) admits a unique local solution on [0, δ 1 ], which satises the estimates: Step 1. Fixed point argument We choose ε 1 such that 0 < ε 1 < min{l e , Ll e }. We denote

f p (t, •) -f pe W 1,∞ C ε 1 • ε, ∀t ∈ [0, δ 1 ], (4.4.2) |l(t) -l e | C ε 1 • ε, ∀t ∈ [0, δ 1 ]. ( 4
F W 1,∞ : = |α| 1 ess sup le-ε 1 <x 1 <le+ε 1 Ne-ε 1 <x 2 <Ne+ε 1 fpe-ε 1 <x 3 <fpe+ε 1 |D α F (x 1 , x 2 , x 3 )|, (4.4.5)
Ψ(t) : = (l(t), f p (t, 1)), t ∈ [0, T ].
For any given δ > 0 small enough (which will be dened later), let us dene a domain candidate as a closed subset of C 0 ([0, δ]) with respect to C 0 norm.

Ω δ := Ψ ∈ C 0 ([0, δ]) : Ψ C 0 ([0,δ]) : = max{ l C 0 ([0,δ]) , f p (•, 1) C 0 ([0,δ]) } L + 1} , ( 4.4.6)
Let us dene a map F := (F 1 , F 2 ), where F :

Ω δ → C 0 ([0, δ]), Ψ → F(Ψ) as F 1 (l)(t) = l 0 + ¢ t 0 F l(s), N(s), f p (s, 1) ds, ( 4.4.7)
F 2 (f p )(t, 1) = f 0 p (ξ 2 (0)), From (4.2.2), we solve the linear ODE (4.4.9)

ξ 2 (s) = e ¡ t s F (l(σ),N (σ),fp(σ,1)) l(σ) dσ - ¢ t s ζN (σ) l(σ) • e - ¡ s σ F (l(s),N (s),fp(s,1)) l(s)
ds dσ, 0 s t. (4.4.10) We solve the linear ODE (4.4.13)

It is obvious that F maps into Ω δ itself if 0 < δ < min{T 1 , T },
ξ(s) = - ¢ 0 s ζN (σ) l(σ) • e - ¡ s σ F (l(s),N (s),fp(s,1)) l(s)
ds dσ, 0 s t. Now we prove that, if δ is small enough, F is a contraction mapping on Ω δ with respect to the C 0 norm. Let Ψ = (l, f p ), Ψ = ( l, fp ) ∈ Ω δ , for any xed t ∈ [0, δ], we have

|F 1 ( l)(t) -F 1 (l)(t)| = ¢ t 0 F l(s), N(s), fp (s, 1)) ds - ¢ t 0 F l(s), N(s), f p (s, 1)) ds δ F W 1,∞ l -l C 0 ([0,δ]) + fp (•, 1) -f p (•, 1) C 0 ([0,δ]) . ( 4
ds dσ f 0 p W 1,∞ e ¡ t 0 F ( l(σ),N (σ), fp(σ,1)) l(σ)
dσe

¡ t 0 F (l(σ),N (σ),fp(σ,1)) l(σ) dσ + ¢ t 0 ζN (σ) l(σ) • e - ¡ 0 σ F ( l(s),N (s), fp(s,1)) l(s) ds - ζN (σ) l(σ) • e - ¡ 0 σ F (l(s),N (s),fp(s,1)) l(s) ds dσ f 0 p W 1,∞ e δ F W 1,∞ le-ε 1 ¢ t 0 F ( l, N, fp )l -F (N, l, f p ) l(σ) ll dσ + ¢ t 0 ζN (σ)(l(σ) -l(σ)) l(σ)l(σ) e ¡ σ 0 F ( l(s),N (s), fp(s,1)) l(s) ds dσ + ¢ t 0 ζN (σ) l(σ) e δ F W 1,∞ le-ε 1 ¢ σ 0 F ( l, N, fp )l -F (N, l, f p ) l ll ds dσ f 0 p W 1,∞ e δ F W 1,∞ le-ε 1 ¢ t 0 F ( l, N, fp ) l -l -l F ( l, N, fp ) -F (N, l, f p ) ll dσ + ¢ t 0 ζN (σ)(l(σ) -l(σ)) l(σ)l(σ) e ¡ σ 0 F ( l(s),N (s), fp(s,1)) l(s) ds dσ + ¢ t 0 ζN (σ) l(σ) e δ F W 1,∞ le-ε 1 ¢ σ 0 F ( l, N, fp ) l -l -l F ( l, N, fp ) -F (N, l, f p ) ll ds dσ .
(4.4.17)

Since t ∈ [0, δ], l < l eε 1 , l < l eε 1 and noting (4.4.5), from (4.4.17), we obtain

|F 2 ( fp )(t, 1) -F 2 (f p )(t, 1)| f 0 p W 1,∞ δe δ F W 1,∞ le-ε 1 F W 1,∞ l e -ε 1 ( l -l C 0 ([0,δ]) + fp -f p C 0 ([0,δ]) ) + F W 1,∞ (l e -ε 1 ) 2 l -l C 0 ([0,δ]) + δ 2 ζ N L ∞ l e -ε 1 e δ F W 1,∞ le-ε 1 F W 1,∞ l e -ε 1 ( l -l C 0 ([0,δ]) + fp -f p C 0 ([0,δ]) ) + F W 1,∞ (l e -ε 1 ) 2 l -l C 0 ([0,δ]) + δζ N L ∞ (l e -ε 1 ) 2 e δ F W 1,∞ le-ε 1 l -l C 0 ([0,δ]) .
(4.4.18)

Hence, from (4.4.15) and (4.4.18), we have

F( Ψ) -F(Ψ) C 0 ([0,δ]) = max F 1 ( l)(•) -F 1 (l)(•) C 0 ([0,δ]) , F 2 ( fp )(•, 1) -F 2 (f p )(•, 1) C 0 ([0,δ]) δ max{ l -l C 0 ([0,δ]) , fp -f p C 0 ([0,δ]) } • 2 F W 1,∞ + f 0 p W 1,∞ • e δ F W 1,∞ le-ε 1 • 2 F W 1,∞ l e -ε 1 + F W 1,∞ (l e -ε 1 ) 2 1 + δζ N L ∞ l e -ε 1 + ζ N L ∞ (l e -ε 1 ) 2 δC ε 1 • max l -l C 0 ([0,δ]) , fp -f p C 0 ([0,δ]) . ( 4.4.19)
We choose δ 1 (depending on ε 1 ,

f 0 p W 1,∞ , F W 1,∞ , N L ∞ ) be small enough such that F( Ψ) -F(Ψ) C 0 ([0,δ 1 ]) 1 2 Ψ -Ψ C 0 ([0,δ 1 ]) .
(4.4.20)

By means of the contraction mapping principle, there exists a unique xed point Step 2. Construction of a weak solution By xed point argument, we get the existence and uniqueness of local solution l(t) and f p (t, 1). Next, we construct local solution to (4.2.10) by characteristic method.

Ψ = F(Ψ) in Ω δ 1 . t x 1 s x s x 2 s x t t 1 x 0 x 3 s
For any xed t ∈ [0, δ 1 ], when 0 x ξ(t), we dene the characteristic curve ξ 1 (s) as (see Fig 4 For any given l ∈ Ω δ 1 ,M , when ξ(t) x 1, we dene the characteristic curve ξ 3 (s) as (see

ξ 3 (s) = xe ¡ t s F (l(σ),N (σ),fp(σ,1)) l(σ) dσ - ¢ t s ζN (σ) l(σ) • e - ¡ s σ F (l(s),N (s),fp(s,1)) l(s)
ds dσ. x = ξ 3 (0). We have

∂ t ∂t = -α p (t, x)l( t) ζN ( t) • e ¡ t t F (l(σ),N (σ),fp(σ,1)) l(σ) dσ , (4.4.25) ∂ t ∂x = -l( t) ζN ( t) • e ¡ t t F (l(σ),N (σ),fp(σ,1)) l(σ)
dσ . Now, we construct a solution for the lling ratio f p for PFZ zone as Let τ ∈ [0, δ 1 ] and any ϕ 1 ∈ C 1 ([0, τ] × [0, 1]) with ϕ 1 (τ, x) ≡ 0 and ϕ 1 (t, 1) ≡ 0 be given, we have

f p (t, x) = ⎧ ⎪ ⎨ ⎪ ⎩ F in ( t) θ(N ( t)) , 0 x ξ(t), f 0 p (x), ξ(t) x 1, (4.4
A := ¢ τ 0 ¢ 1 0 f p (t, x)(ϕ 1t (t, x) + (α p ϕ 1 (t, x)) x ) dx dt = ¢ τ 0 ¢ ξ(t) 0 F in ( t) θ(N ( t)) (ϕ 1t (t, x) + (α p ϕ 1 (t, x)) x ) dx dt + ¢ τ 0 ¢ 1 ξ(t) f 0 p (x)(ϕ 1t (t, x) + (α p ϕ 1 (t, x)) x ) dx dt.
By (4.4.26) and (4.4.28), we obtain

A = ¢ τ 0 ¢ t 0 F in ( t) θ(N ( t)) • ζN ( t) l( t) e - ¡ t t F (s) l(s) ds (ϕ 1t (t, ξ 1 (t)) + (α p (t, ξ 1 (t))ϕ 1 (t, ξ 1 (t))) x ) d t dt + ¢ τ 0 ¢ 1- ¡ t 0 αp(s,ξ 2 (s)) ds 0 f 0 p (x) • e - ¡ t 0 F (s) l(s) ds (ϕ 1t (t, ξ 3 (t)) + (α p (t, ξ 3 (t))ϕ 1 (t, ξ 3 (t))) x ) dx dt = ¢ τ 0 ¢ t 0 F in ( t) θ(N ( t)) • ζN ( t) l( t) e - ¡ t t F (s) l(s) ds ( dϕ 1 (t, ξ 1 (t)) dt - F (t) l(t) ϕ 1 (t, ξ 1 (t))) d t dt + ¢ τ 0 ¢ 1- ¡ t 0 αp(s,ξ 2 (s)) ds 0 f 0 p (x) • e - ¡ t 0 F (s) l(s) ds ( dϕ 1 (t, ξ 3 (t)) dt - F (t) l(t) ϕ 1 (t, ξ 3 (t))) dx dt = ¢ τ 0 ¢ t 0 F in ( t) θ(N ( t)) • ζN ( t) l( t) • d ϕ 1 (t, ξ 1 (t))e - ¡ t t F (s) l(s) ds dt d t dt + ¢ τ 0 ¢ 1- ¡ t 0 αp(s,ξ 2 (s)) ds 0 f 0 p (x) • d ϕ 1 (t, ξ 3 (t))e - ¡ t 0 F (s) l(s) ds dt dx dt
By changing the order of integral, we have

A = ¢ τ 0 ¢ τ t F in ( t) θ(N ( t)) • ζN ( t) l( t) • d ϕ 1 (t, ξ 1 (t))e - ¡ t t F (s) l(s) ds dt dt d t + ¢ f (τ ) 0 ¢ τ 0 + ¢ 1 f (τ ) ¢ f -1 (x) 0 f 0 p (x) • d ϕ 1 (t, ξ 3 (t))e - ¡ t 0 F (s) l(s) ds dt dt dx,
where

f (t) = 1 - ¢ t 0 α p (s, ξ 2 (s)) ds
represents the coordinate that the characteristic curve ξ 2 intersects with x-axis and f -1 (x) represents the time when the characteristic curve starting from (0, x) arrives at the boundary x = 1. Consequently, for any β ∈ [f (τ ), 1], ξ 2 and ξ 3 are identical to each other since they pass through the same point (0, β), so we get

ξ 3 (f -1 (x)) = ξ 2 (f -1 (x)) = 1.

Finally, we get

A = - ¢ τ 0 F in ( t) θ(N ( t)) • ζN ( t) l( t) ϕ 1 ( t, 0) d t - ¢ f (τ ) 0 f 0 p (x)ϕ 1 (0, x) dx - ¢ 1 f (τ ) f 0 p (x)ϕ 1 (0, x) dx = - ¢ τ 0 F in ( t) θ(N ( t)) • ζN ( t) l( t) ϕ 1 ( t, 0) d t - ¢ 1 0 f 0 p (x)ϕ 1 (0, x) dx.
This proves that f p given by (4.4.29) satises (4.3.4) for every τ ∈ [0, δ 1 ].

Step 3. Uniqueness of solution. Since the xed point iteration yields the uniqueness of the xed point (l(t), f p (t, 1)) that determines the characteristic curves ξ 1 and ξ 3 . Then (4.2.10) becomes a linear forward Cauchy problem, it follows easily the uniqueness of f p (t, x).

Step 4. Priori estimate From (4.4.1) and the expression of the solution for the lling ratio (4.4.29), it is easy to check that, for any t ∈ [0, δ 1 ] From now on, we denote by C ε 1 various constants that may depend on ε 1 .

f p (t, •) -f pe L ∞ ε.
| l(t)| = | l(t) -le | =|F (l(t), N(t), f p (t, 1)) -F (l e , N e , f pe ) | F W 1,∞ |l(t) -l e | + F W 1,∞ |N (t) -N e | + F W 1,∞ |f p (t, 1) -f pe |.
| l(t)| C ε 1 |l(t) -l e | + C ε 1 ε. (4.4.32)
By (4.4.1) and using Grownwall's inequality, we get (4.4.3). Furthermore, With the existence of the interface l(t) and f p (t, 1), t ∈ [0, T ], using characteristic method, we construct a solution M p as

∂f p (t, x) ∂x L ∞ ∂ F in ( t) θ(N ( t)) ∂x L ∞ + ∂f 0 p (x) ∂x L ∞ F in (•) θ(N (•)) -f pe W 1,∞ • ∂ t ∂x L ∞ + f 0 p (•) -f pe W 1,∞ • ∂ x ∂x L ∞ .
f p (δ 1 , •) -f pe W 1,∞ (0,1) C ε 1 • ε ε 1 , (4.4.34) |l(δ 1 ) -l e | C ε 1 • ε ε 1 .
M p (t, x) = ⎧ ⎨ ⎩ M 0 p (x), ξ(t) x 1, t ∈ [0, T ],
M in ( t), else, Next we prove that M p ∈ C 0 ([0, T ]; L 2 (0, 1)). For any given 0 t 1 < t 2 δ, we prove that

M p (t 2 , x) -M p (t 1 , x) L 2 (0,1) → 0, as |t 2 -t 1 | → 0. (4.4.37)
We estimate

¡ ξ(t 1 ) 0 |M p (t 2 , x)-M p (t 1 , x)| 2 dx, ¡ ξ(t 2 ) ξ(t 1 ) |M p (t 2 , x)-M p (t 1 , x)| 2 dx, ¡ 1 ξ(t 2 ) |M p (t 2 , x)- M p (t
|M p (t 2 , x) -M p (t 1 , x)| = |M in ( t2 ) -M in ( t1 )|, (4.4.38)
where t1 and t2 are dened by

0 = xe ¡ t 1 t1 F (l(σ),N (σ),fp(σ,1)) l(σ) dσ - ¢ t 1 t1 ζN (σ) l(σ) • e - ¡ t1 σ F (l(s),N (s),fp(s,1)) l(s)
ds dσ, (4.4.39)

0 = xe ¡ t 2 t2 F (l(σ),N (σ),fp(σ,1)) l(σ) dσ - ¢ t 2 t2 ζN (σ) l(σ) • e - ¡ t2 σ F (l(s),N (s),fp(s,1)) l(s)
ds dσ.

(4.4.40)

For the given Hence, we have

M in ∈ L 2 (0, T ), we let M n in ∈ C 1 ([0, T ]) be such that M n in → M in in L 2 (0, T ).

We have that for almost every

x ∈ [0, ξ(t 1 )] |M p (t 2 , x) -M p (t 1 , x)| |M in ( t1 ) -M n in ( t1 )| + |M n in ( t1 ) -M n in ( t2 )| + |M n in ( t2 ) -M in ( t2 )|.
| t2 -t1 | 1 D ¢ t2 t1 ζN (σ) l(σ) • e - ¡ t1 σ F (l(s),N (s),fp(s,1)) l(s) ds dσ =e ( ¡ t2 t1 - ¡ t 2 t 1
)

F (l(σ),N (σ),fp(σ,1)) l(σ) dσ ¢ t 1 t2 ζN (σ) l(σ) • e - ¡ t2 σ F (l(s),N (s),fp(s,1)) l(s) ds dσ - ¢ t 1 t2 ζN (σ) l(σ) • e - ¡ t1 σ F (l(s),N (s),fp(s,1)) l(s) ds dσ + e ( ¡ t2 t1 - ¡ t 2 t 1
)

F (l(σ),N (σ),fp(σ,1)) l(σ) dσ ¢ t 2 t 1 ζN (σ) l(σ) • e - ¡ t2 σ F (l(s),N (s),fp(s,1)) l(s) ds dσ =(e ( ¡ t2 t1 - ¡ t 2 t 1
)

F (l(σ),N (σ),fp(σ,1)) l(σ)
dσ -1)

¢ t 1 t2 ζN (σ) l(σ) • e - ¡ t2 σ F (l(s),N (s),fp(s,1)) l(s)
ds dσ

+ ¢ t 1 t2 ζN (σ) l(σ) • (e - ¡ t2 σ F (l(s),N (s),fp(s,1)) l(s) ds -e - ¡ t1 σ F (l(s),N (s),fp(s,1)) l(s) ds ) dσ + e ( ¡ t2 t1 - ¡ t 2 t 1
)

F (l(σ),N (σ),fp(σ,1)) l(σ) dσ ¢ t 2 t 1 ζN (σ) l(σ) • e - ¡ t2 σ F (l(s),N (s),fp(s,1)) l(s)
ds dσ, (4.4.43) where

D := ζ(N e -ε 1 ) l e + ε 1 . (4.4.44) We get | t2 -t1 | C|t 2 -t 1 |.
¢ ξ(t 1 ) 0 |M p (t 2 , x) -M p (t 1 , x)| 2 dx C M n in -M in 2 L 2 (0,T ) + C n |t 2 -t 1 | 2 . ( 4
¡ t 1 0 F (l(σ),N (σ),fp(σ,1)) l(σ) dσ - ¢ t 1 0 ζN (σ) l(σ) • e - ¡ 0 σ F (l(s),N (s),fp(s,1)) l(s)
ds dσ.

¢ ξ(t 2 ) ξ(t 1 ) |M p (t 2 , x) -M p (t 1 , x)| 2 dx 2 ¢ ξ(t 2 ) ξ(t 1 ) |M p (t 2 , x)| 2 dx + 2 ¢ ξ(t 2 ) ξ(t 1 ) |M p (t 1 , x)| 2 dx = 2 ¢ Γ -1 2 (0) 0 |M in (t)| 2 • l(t) ζN (t) • e ¡ t 2 t F (l(σ),N (σ),fp(σ,1)) l(σ) dσ dt + 2 ¢ Γ 1 (0) 0 |M 0 p (x)| 2 • e ¡ t 2 0 F (l(σ),N (σ),fp(σ,1)) l(σ)
dσ dx, (4.4.49) where Γ 1 and Γ 2 are dened by

Γ 1 (s) = ξ(t 2 )e ¡ t 1 s F (N (σ),l(σ),fp(σ,1)) l(σ) dσ - ¢ t 1 s ζN (σ) l(σ) • e - ¡ s σ F (N (
|Γ 1 (0)| = ¢ t 2 0 ζN (σ) l(σ) • e ¡ t 1 0 - ¡ t 2 σ F (N (s),l(s),fp(s,1)) l(s) ds dσ - ¢ t 1 0 ζN (σ) l(σ) • e - ¡ 0 σ F (N (s),l(s),fp(s,1)) l(s) ds dσ ¢ t 2 0 ζN (σ) l(σ) e ¡ t 1 0 - ¡ t 2 σ F (N (s),l(s),fp(s,1)) l(s) ds -e - ¡ 0 σ F (N (s),l(s),fp(s,1)) l(s) ds dσ + ¢ t 2 t 1 ζN (σ) l(σ) • e - ¡ 0 σ F (N (s),l(s),fp(s,1)) l(s) ds dσ C ¢ t 2 0 ¢ t 1 0 - ¢ t 2 σ - ¢ σ 0 F (N (s), l(s), f p (s, 1)) l(s) ds dσ + C|t 2 -t 1 | C|t 2 -t 1 |. ( 4.4.52)
From (4.4.51), we get

0 = ξ(t 1 )e ¢ t 2 Γ -1 2 (0) F (N (σ), l(σ), f p (σ, 1)) l(σ) dσ - ¢ t 2 Γ -1 2 (0) ζN (σ) l(σ) • e - ¡ Γ -1 2 (0) σ F (N (s),l(s),fp(s,1)) l(s)
ds dσ.

(4.4.53) Hence,

¢ ξ(t 2 ) ξ(t 1 ) |M p (t 2 , x) -M p (t 1 , x)| 2 dx C|t 2 -t 1 |.
|M p (t 2 , x) -M p (t 1 , x)| = |M 0 p (x 2 ) -M 0 p (x 1 )|, (4.4.55)
where x1 is dened by (4.4.48) and x2 is dened by

x2 = xe ¡ t 2 0 F (N (σ),l(σ),fp(σ,1)) l(σ) dσ - ¢ t 2 0 ζN (σ) l(σ) • e - ¡ 0 σ F (N (s),l(s),fp(s,1)) l(s)
ds dσ. (4.4.57) For the given M 0 p ∈ L 2 (0, 1), we let M 0n p ∈ C 1 ([0, 1]) be such that M 0n p → M 0 p in L 2 (0, 1). From (4.4.55), we get

|M p (t 2 , x) -M p (t 1 , x)| |M 0 p (x 2 ) -M 0n p (x 2 )| + |M 0n p (x 2 ) -M 0n p (x 1 )| + |M 0n p (x 1 ) -M 0 p (x 1 )|. ( 4
|M p (t 2 , x) -M p (t 1 , x)| 2 dx C M n in -M in 2 L 2 (0,T ) + C M 0n p -M 0 p 2 L 2 (0,1) + C n |t 2 -t 1 | 2 + C|t 2 -t 1 | 2 . ( 4
ξ(s) = - ¢ 0 s 1 L -l(σ) ζF d (σ) ρ 0 V ef f -F (N (σ), l(σ), f p (σ, 1)) • e ¡ s σ F (N (s),l(s),fp(s,1)) L-l(s)
ds dσ. (4.4.61) For any xed (t, x), when 0 x ξ(t) we dene the characteristic curve ξ1 (s) (see Fig 4 .4.5)

ξ1 (s) = xe - ¡ t s F (N (σ),l(σ),fp(σ,1)) L-l(σ) dσ - ¢ t s 1 L -l(σ) ζF d (σ) ρ 0 V ef f -F (N (σ), l(σ), f p (σ, 1)) • e ¡ s σ F (N (s),l(s),fp(s,1)) L-l(s)
ds dσ.

We dene t as ξ1 ( t) = 0.

ξ3 (s) = xe - ¡ t s F (N (σ),l(σ),fp(σ,1)) L-l(σ) dσ - ¢ t s 1 L -l(σ) ζF d (σ) ρ 0 V ef f -F (N (σ), l(σ), f p (σ, 1)) • e ¡ s σ F (N (s),l(s),fp(s,1)) L-l(s)
ds dσ.

We dene x as

x := ξ3 (0).

(4.4.63)

Now, we construct a solution With the existence of the interface l(t) and f p (t, 1), t ∈ [0, δ 1 ], using characteristic method, we construct a solution T p as

M f (t, x) = ⎧ ⎨ ⎩ M 0 f (x), ξ(t) x 1, t ∈ [0, T ], M p ( t,
¢ 1 0 M 2 p (t, x)dx ¢ 1 0 (M 0 p (x)) 2 dx + ¢ 1 0 (M in ( t)) 2 dx = ¢ 1 0 (M 0 p (x)) 2 ∂x ∂ x dx + ¢ T 0 (M in ( t)) 2 ∂x ∂ t d t. ( 4
T p (t, x) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ T 0 p (x)e C 1 t + ¢ t 0 e C 1 (t-s) g 1 (N (s), f p (s, ξ 2 (s))) -C 1 T b (x, ξ 2 (s)) ds, ξ(t) x 1, T in ( t)e C 1 t + ¢ t t e C 1 (t-s) g 1 (N (s), f p (s, ξ 1 (s))) -C 1 T b (x, ξ 1 (s)
) ds, else, (4.4.67) with t and x dened by (4.4.22) and (4.4.24).

With the existence of the solution for temperature equation for PFZ zone, we use the interface relation in (4.2.12) to solve the FFZ zone. We construct a solution T f as

T f (t, x) = ⎧ ⎪ ⎨ ⎪ ⎩ T 0 f (x)e C 1 t + ¡ t 0 e C 1 (t-s) g 2 (N (s)) -C 1 T b (s, ξ2 (s)) ds, ξ(t) x 1, T p ( t, 1)e C 1 t + ¡ t t e C 1 (t-s) g 2 (N (s)) -C 1 T b (s,

Proof of Theorem 2

In order to conclude Theorem 4, it is sucient to prove that f pxx ∈ C 0 ([0, T ]; L 2 (0, 1)).

Derive the system of f pxx and apply L 2 well-posedness for linear system.

From (4.2.10) and take derivative with respect to x in (4.2.10), we have

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ t f px (t, x) + α p (t, x)∂ x f px (t, x) = -α px f px (t, x), (t, x) ∈ Q f px (0, x) = f 0 px (x), x ∈ (0, 1)
f px (t, 0) = -l(t) ζN (t) • F in (t)θ(N (t)) -F in (t)θ (N (t))N (t) θ 2 (N (t)) , t ∈ (0, T ) (4.5.1)
Similarly to the proof of Theorem 3, when t ∈ [0, δ], with δ dened by (4.4.11), we have

f px (t, x) = ⎧ ⎪ ⎨ ⎪ ⎩ -l( t) ζN ( t) • F in ( t)θ(N ( t)) -F in ( t)θ (N ( t))N ( t) θ 2 (N ( t)) • e ¡ t t F (l(s),N (s),fp(s,1)) l(s) ds , 0 x ξ(t), f 0 px (x) • e ¡ t 0 F (l(s),N (s),fp(s,1)) l(s)
ds , ξ(t) x 1, (4.5.2) where t and x are dened by (5.2.14) and (4.4.24).

Since H 2 can be embedded in W 1,∞ , in order to prove Theorem 4, we only have to prove f pxx (t, •) ∈ L 2 (0, 1) for any given t ∈ [0, δ]. >From (4.5.2), we have

¢ 1 0 |f pxx (t, x)| 2 dx 2 ¢ 1 0 d d t l( t) ζN ( t) • d d t F in ( t) θ(N ( t)) • e ¡ t t F (l(s),N (s),fp(s,1)) l(s) ds ∂ t ∂x 2 dx (4.5.3) + 2 ¢ 1 0 d dx f 0 px (x) • e ¡ t 0 F (l(s),N (s),fp(s,1)) l(s) ds ∂ x ∂x 2 dx.
d dx f 0 px (x) • e ¡ t 0 F (l(s),N (s),fp(s,1)) l(s) ds ∂ x ∂x 2 dx =2e 4 ¡ t 0 F (l(s),N (s),fp(s,1)) l(s) ds ¢ 1 0 |f 0 pxx (x)| 2 dx =2e 3 ¡ t 0 F (l(s),N (s),fp(s,1)) l(s) ds ¢ 1 0 |f 0 pxx (x)| 2 dx 2e 3δ F W 1,∞ le-ε f 0 p (t, •)
¢ δ 0 d d t l( t) ζN ( t) • d d t F in ( t) θ(N ( t)) • e ¡ t t F (l(s),N (s),fp(s,1)) l(s) ds 2 d t = 6(l e + ε) ζ(N e -ε) ¢ δ 0 d d t l( t) ζN ( t) • d d t F in ( t) θ(N ( t)) • e ¡ t t F (l(s),N (s),fp(s,1)) l(s) ds 2 d t + ¢ δ 0 l( t) ζN ( t) • d 2 d t2 F in ( t) θ(N ( t)) • e ¡ t t F (l(s),N (s),fp(s,1)) l(s) ds 2 d t + ¢ δ 0 l( t) ζN ( t) • d d t F in ( t) θ(N ( t)) • e ¡ t t F (l(s),N (s),fp(s,1)) l(s) ds • F (l( t), N( t), f p ( t, 1)) l( t) 2 d t 6(l e + ε) ζ(N e -ε) 2 l 2 W 1,∞ N 2 1 + (l e + ε) 2 N 2 W 1,∞ ζ 2 N 4 0 • e 2δ F W 1,∞ le-ε • F in (•) θ(N (•)) 2 H 2 (0,T ) + (l e + ε) 2 ζ 2 N 2 0 • e 2δ F W 1,∞ le-ε • F in (•) θ(N (•)) 2 H 2 (0,T ) + (l e + ε) 2 ζ 2 N 2 0 • F 2 W 1,∞ (l e -ε) 2 • e 2δ F W 1,∞ le-ε • F in (•) θ(N (•)) 2 H 2 (0,T ) = 6(l e + ε) ζ 3 N 3 0 • e 2δ F W 1,∞ le-ε • F in (•) θ(N (•)) 2 H 2 (0,T ) 2 l 2 W 1,∞ N 2 1 + (l e + ε) 2 N 2 W 1,∞ N 2 0 + (l e + ε) 2 1 + F 2 W 1,∞
(l eε) 2 .

(4.5.6) Combining (4.5.3), (4.5.4), (4.5.5) and (4.5.6), we get f pxx (t, •) ∈ L 2 (0, 1) for any given t ∈ [0, δ]. The proof is complete.

Conclusion

In this chapter we perform an mathematical analysis of coupled transport equations which are dened in complementary time varying domains. The goal of the analysis is to prove the well-posedness of this class of system. The Partially Filled Zone equations are characterized by a time-varying velocity which is a function of the screw speed N (t). In the Fully Filled Zone the convection velocity is strongly related to the interface properties by its dependence on die net ow rate F d (t). We approach this problem by performing a change of coordinates which consists to express the system in a xed domain (0, 1). This transformation generates ctive convective terms in the transport operator. By contraction mapping principle (Banach xed point theorem), we prove the existence and uniqueness of l(t) and f p (t; 1) for t small. With the existence of l(t) and f p (t; 1), we then construct a local solution to Cauchy problem. By an iterative method we extend the results to the existence and the uniqueness of semi-global solution. The solution we obtained in Theorem 3 and Theorem 4 is called semi-global solution since it exists on any preassigned time interval [0; T ] if the initial and boundary data has some kind of smallness (depending on T). We point out that the assumption of constant viscosity allows to release the strong coupling relation between the equation of e mass, moisture and energy balances. The coupled system which is composed by the transport equation of the lling ratio and the interface motion equation is in a sense a free boundary problem which can be treated independently. Here the free boundary dynamic is a function of the dierence of pressure ΔP (t) and it evolves independently from the energy and moisture balances. Ours future works in the context of well-posedness and moving interface problems will deal with this extrusion process model with distributed viscosity and pressure continuity relation at the interface.

Chapter 5

Mass transport equation with moving interface and its control as an input delay system

Introduction

In this part we address the problem of the stabilization of the position of the moving interface given by the equation (3.3.7) thanks to the inlet ow rate acting on the boundary of the transport equation of the Partially Filled Zone (3.5.11). The control input does not appears directly in (3.3.7) but it appears with a delay depending on the position of the moving interface through the lling ratio f p (l(t), t). So we propose to tackle the problem as a time input delay system control problem.

So far the control oriented models of extruders are issued from some black box linear model identication around some operating conditions. Following the objectives of the control and the choice of the manipulated variables these models can be time delay linear MIMO or SISO models [START_REF] Moreira | Feedforward control model for a twin-screw food extruder[END_REF][START_REF] Nield | Control of a LPDE reactive extrusion process[END_REF][START_REF] Wang | Dual-target predictive control and application in food extrusion[END_REF][START_REF] Previdi | Design of a feedback control system for real-time control of ow in a single-screw extruder[END_REF][START_REF] Garge | Inference-based scheme for controlling product end-use properties in reactive extrusion processes[END_REF]. These time delays are due to time delays associated with the o line measurements of key product properties, non physical time delays due to approximation of nonlinear dynamics and in some case due to the transport of the matter in the extruder [START_REF] Moreira | Feedforward control model for a twin-screw food extruder[END_REF]. These time delays are always assumed to be constant. In some sense the basic nature of the extruders: the two-zone model is never exploited. We leave the reader refer to the introduction of Chapter 3 to have more details about the control strategy used for these extrusion processes. Time delay systems are not commonly used in chemical engineering even if the dynamic models of many chemical engineering processes involve signicant time delays due to transportation lag such as the example treated in [START_REF] Christodes | Feedback control of nonlinear dierential dierence equation systems[END_REF] of two chemical reactors represented as continuous stirred tank reactors with recycle loop. Others examples can be found in [START_REF] Antoniades | Robust control of nonlinear time-delay systems[END_REF][START_REF] Shu Xia | Distributed control of plant-wide chemical processes with uncertain time-delays[END_REF][START_REF] Xu | Distributed control of plant-wide chemical processes with uncertain time-delays[END_REF][START_REF] Bués | Delay model for a cycling transport through porous medium[END_REF].

For spatially distributed parameter systems, some authors treats the regulation problem of convection governed dynamical processes described by linear rst order partial dierential equations by using the method of characteristics(MC) in order to transform the PDEs into a set of time delayed ordinary dierential equations [START_REF] Gundepudi | Velocity control of partial dierential equation systems with single characteristic variable[END_REF][START_REF] Shang | Feedback control of hyperbolic distributed parameter systems[END_REF][START_REF] Aguilar-Garnica | Robust continuous velocity control of convective spatially distributed systems[END_REF][START_REF] Aguilar-Garnica | PI controller design for a class of distributed parameter systems[END_REF].

Research on time delay systems initially focused on open-loop stable linear systems with a single manipulated input delay which are described by transfer function models, in which the presence of the time delay prevents the use of large controller gains. To overcome the destabilizing eect of the high gain, Smith [START_REF] Smith | Closer control of loops with dead time[END_REF] proposed in 1957 a control structure, known as the Smith predictor that permits to improve the control performance; it completely eliminates the time delay from the characteristic equation of the closed-loop system, allowing the use of larger controller gains.

Various controllers have been developed to address time-delay induced performance and stability issues as described in the survey paper [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]. Delay systems can be controlled in dierent manners using methods based on an algebraic Riccati approach (where stability conditions are expressed by the solvability of a Riccati equation or by the feasibility of a linear matrix inequality) [START_REF] Mao | D-stability for linear continuous-time systems with multiple time delays[END_REF][START_REF] Yu | An LMI approach to robust H∞ ltering for uncertain systems with time-varying distributed delays[END_REF], sliding mode controllers [START_REF] El-Khazali | Robust stabilization of uncertain time-delay systems using variable structure output feedback control[END_REF][START_REF] Gouaisbaut | A sliding mode control for linear systems with input and state delays[END_REF][START_REF] Gouaisbaut | Robust control of delay systems: a sliding mode control design via lmi[END_REF][START_REF] Zhao | Integral sliding mode control for a class nonlinear systems[END_REF][START_REF] Song | Model reference adaptive integral-type sliding mode control design for a class of uncertain systems[END_REF][START_REF] Jafarov | Robust delay-dependent stabilization of uncertain time-delay systems by variable structure control[END_REF][START_REF] Mirkin | Continuous model reference adaptive control with sliding mode for a class of nonlinear plants with unknown state delay[END_REF] or variants of the Smith predictor [START_REF] Smith | Closer control of loops with dead time[END_REF][START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF][START_REF] Artstein | Linear systems with delayed controls: A reduction[END_REF][START_REF] Evesque | Adaptive control scheme for uncertain time-delay systems[END_REF][START_REF] Hua | Time-delay compensation control of networked control systems using time-stamp based state prediction[END_REF][START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF].

For the problem under consideration we will have to deal with the class of the input delay systems more precisely the class of state dependent input delay systems. The approach chosen for the synthesis of the control is based on the works proposed in [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF] for state dependent input delay systems using a predictor.

Organization of the chapter

This chapter is organized as follow: in section 5.2 the system formed by the transport PDE coupled with the ODE of the moving boundary is transformed into a state dependent input delay system. In section 5.3 we propose a rapid overview of the predictor based controllers for input delay systems. The design of the controller is described in section 5.4. Simulation results are proposed in section 5.5. Finally concluding remarks are established in section 5.6.

Contribution

This chapter is dedicated to the control of 1D hyperbolic partial dierential equation with moving interface as a delay system. The model derives from the mass balance of the extrusion process that describes the strong coupling between the mass transport equation and an ordinary dierential equation which represents the interface motion. Solving the transport equation of lling ratio by the method of characteristics , we obtain an state-dependent-inputdelay control problem. The stabilization of the system around an equilibrium is done by using a state predictor.

From transport equation to delay system for the extrusion process

The main idea in this section is to dene a delay system from the model given by equations (3.5.11) and (3.3.7). Our aim is to solve the hyperbolic partial dierential equation by the method of characteristics and to express the solution as a function of the boundary inow data. The delay represents the propagation time of the matter from the boundary {x = 0} to the interface {x = l(t)} due to the screw speed . The solution is injected in the equation (5.2.2) in order to obtain a delay system. This computation leads to a nonlinear scalar state dependent input system depending on the interface position l(t).

Mass conservation equations in an extrusion process

In Chapter 3 we have presented the model of the extrusion process considering an homogeneous melt which is convected from the feed to the die by screw rotation.

The modelling of the extruder makes appear two geometric zones in which the matter ows at dierent velocities. These two zones are separated by an moving interface whose dynamics is given by an ordinary dierential equation expressing the mass balance in the Fully Filled Zone.

In this section we recall the mass transport model of homogeneous material in the PFZ given by equation (3.3.7) as well as the total mass balance in the FFZ gives rise to the moving interface dynamics (3.5.11).

The PFZ is dened in [0, l(t)[ and the FFZ in ]l(t), L] (where L represents the extruder length and l(t) the interface position).

Considering the lling ratio f p (x, t) as the distributed variable, the PFZ mass balance leads to a transport equation whose velocity is determined by the screw speed N (t) and the equivalent pitch ξ of the screw. Under the ow continuity assumption at {x = 0}, one can express the boundary lling ratio in terms of the PFZ convective velocity ξN (t), the available volume V ef f , the material feed rate F in (t) and the mass density ρ 0 .

⎧ ⎨ ⎩ ∂fp ∂t (x, t) = -ξN (t) ∂fp ∂x (x, t) x ∈ [0, l(t)[ f p (0, t) = f in p (t) = F in (t) ρ 0 ξN (t)V ef f (5.2.1)
In the FFZ, the velocity of the convection is determined by the net ow at the die F d (t).

The FFZ mass balance is associated to the motion of the moving interface l(t) between there two zones:

⎧ ⎨ ⎩ dl(t) dt = F d (t)-ρ 0 N (t)V ef f fp(l(t),t)) ρ 0 S ef f (1-fp(l(t),t)) l(0) = l 0 0 < l 0 < L, (5.2.2)
where L is the extruder length. F d (t), the net ow at the die is a function of the geometric characteristics of the die K d , the constant viscosity η and the pressure build-up in this zone P (x, t). (5.2.3) where P 0 is the air pressure.

⎧ ⎨ ⎩ F d (t) = K d η ΔP (t) ΔP (t) = (P (L, t) -P 0 ),

Recalling the formulation of coupling relations at l(t)

As it is described in Chapter 3 the coupling relation between the two zones (P F Zl(t) -F F Z) can be formulated in two ways:

1. The rst one is a ux coupling relation (3.5.14) and derives from the continuity of the momentum ux and allows to express the die pressure P (L, t) as a function of N (t), l(t), and f p (l(t), t):

ΔP (t) = -[1 + K d Bρ 0 (L -l(t))] + √ Δ 2K 2 d η 2 ρ 0 S 2 ef f (5.2.4) with Δ = 1 + K d Bρ 0 (L -l(t)) 2 + Ω 3 (f p (l(t), t), N(t), l(t))
and

Ω 3 = 2K d ηS ef f 2 ηV ef f N (t) Bρ 0 (L -l(t)) +(ξ 2 N 2 (t) + P 0 ρ 0 )f p (l(t), t) - P 0 ρ 0 (5.2.5)
2. the second one is a state coupling relation and derives from the continuity of the pressure at the interface l(t) (P (l -, t) = P (l + , t) = P 0 ), then:

ΔP (t) = ηV ef f N (t)ρ 0 (L -l(t)) (Bρ 0 + K d (L -l(t))) (5.2.6)
Recall that the main assumption of this modelling point of view is the existence of two zones PFZ and FFZ to make appear the interface dynamics. This property is preserved if and only if ΔP (t) is strictly positive. It means that the output ow at the die F d (t) must be a positive quantity. This constraint implies by the following inequalities:

For continuous momentum ux at l(t)

Ω 3 (f p (l(t), t), N(t), l(t)) > 0 (5.2.7)
Then a critical value of the lling ratio at the interface coupling is dened as f pc (l(t), t)). Its expression is recalled here (3.5.18):

f pc (l(t), t)) > P 0 - ηV ef f N (t) B (L -l(t)) ρ 0 ξ 2 N 2 (t) + P 0 (5.2.8)
For a given function N (t), one can dene the set (f p (l(t), t), l(t)) which satises the constraint (5.2.8).

For continuous pressure at l(t) there is no constraint because the positivity condition on ΔP (L, t) is always satised (L > l(t)).

Transport equation solution

Consider the rst order linear transport equation (5.2.1) in the two variables t and x with the boundary condition f p (0, t).

The total derivative with respect to some variable τ can be written as:

df p (x(τ ), t) dτ = dt dτ ∂f p (x(τ ), t) ∂t + dx(τ ) dτ ∂f p (x(τ ), t) ∂x
Dening the solution along the characteristic curves, we have:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ dt dτ = 1 dx(τ ) dτ = ξN (t) dfp(x(τ ),t) dτ = 0
(5.2.9)

The last equation is deduced from equation (5.2.1) and f p (x(τ ), t) is a constant function along the characteristics.

Since N (t) is an unknown function of t, the system (5.2.9) cannot be explicitly solved. Integrating the characteristics equations

x 1 x 2 x 3 = t x f p ∈ R 3
we obtain:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x 1 (τ, s) = τ + s x 2 (τ, s) = ξ ¡ τ +s s N (τ )dτ x 3 (s) = f p (s) (5.2.10)
Where s is the constant of integration. Let us introduce N (t) as the primitive function of N (t). It is obvious that:

N (s) = N (τ + s) - x 2 ξ
(5.2.11)

Using equation (5.2.10) the following equality is deduced:

s = N -1 N (t) - x ξ
(5.2.12)

Any function which is constant along the characteristic curves can be expressed as:

f p (x, t) = f p N -1 N (t) - x ξ (5.2.13)
To nd the solution that satises the prescribed boundary conditions (5.2.1), we substitute the general solution formula and obtain:

f p (x, t) = f in p N -1 N (t) - x ξ (5.2.14)
Recall that we assume the ow continuity at {x = 0}. Thus, the input lling ratio

f in p (t)
which is the boundary condition can be deduced from the feed rate F in (t), we obtain:

f in p N -1 N (t) - x ξ = F in N -1 N (t) -x ξ ρ 0 N (t)V ef f , (5.2.15)
Where ρ 0 is the mass density and V ef f the eective volume.

State dependent input delay system for the extruder model

In the following we assume that the screw speed is constant N (t) = N 0 . The control problem consists to stabilize the interface position by a feedback on the "abstract" input variable f in p which depends on the feed rate (5.2.15). We nd: 5.2.16) and this simplication results to the following system:

⎧ ⎨ ⎩ f p (x, t) = f in p (t -x N 0 ξ ) f p (l(t), t) = f in p (t -l(t) N 0 ξ ), (
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ dl(t) dt = F d (t)-ρ 0 N 0 V ef f f in p t- l(t) N 0 ξ ρ 0 S ef f 1-f in p t- l(t) N 0 ξ F d (t) = K d η ΔP (t) (5.2.17)
Now, it becomes clear that equation (5.2.17) denes a nonlinear state dependent input delay system which can be written in the abstract form:

⎧ ⎨ ⎩ dl(t) dt = f (l(t), U(t -D(l(t))) U (t) = f in p (t) (5.2.18)
Where f is the nonlinear function and D(l(t)) the delay function depending in the state variable and acting on the input function U (t). As it will be presented in section 5.3.3, [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF] solve the regional stabilization of this class of problem using a predictor with an horizon depending on the future value of the state. The stability around the equilibrium is ensured for small variations of the initial conditions of the state and the input when the delay is large. The section 5.3 is dedicated to the general review of predictor-based controllers in the cases of constant input delay, varying input delay and nally state dependent input delay.

Generality on predictor-based controllers

Predictor-based controllers for constant input delay systems

As already mentioned the predictor-based controllers were rst introduced for linear system by Smith [START_REF] Smith | Closer control of loops with dead time[END_REF] in the frequency domain and later in the state space representation by [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF][START_REF] Artstein | Linear systems with delayed controls: A reduction[END_REF]. These latter references use the terms nite spectrum assignment (FSA) and model reduction respectively to represent the modied Smith predictor. These extensions oer more possibilities on stability issues since they work for unstable systems contrary to the Smith predictor.

We present rst the philosophy of such modied Smith predictors.

We consider a linear system with constant input time-delay D as (5.3.1):

Ẋ(t) = AX(t) + BU (t -D) X ∈ R n U ∈ R m (5.3.1)
The important point here is that the knowledge of an initial data as X(t 0 ) and the future inputs U is not sucient to compute the future dynamics of the state X(t). One should complete the system by dening the input U (t) over the history window [t 0 -D, t 0 ]. Supposing that this system (5.3.1) is controllable and introducing K ∈ R m×n as vector gain which stabilizes the free-delay system Ẋ(t) = AX(t) + BU (t) (by static state feedback), one can build the predictor which guaranteed the stability of the delayed system. A modication of the classical Smith predictor scheme allows to compensate the delay D by introducing control law which makes the system free of delay. This control law is dened by the following equation:

U (t) = KX(t + D) X ∈ R n X ∈ R m (5.3.2)
This alternative feedback allows to transform the delayed system (5.3.1) to an free-delay system after a dead-time D as it is represented here:

Ẋ(t) = (A + BK)X(t) (5.3.3)
This equivalence is achieved if one can predict the value of the state at time (t + D). This prediction is obtained by integrating (5.3.1) from an initial condition given by the current state vector X(t):

The delay compensation is then guaranteed by the predictor-based control law: (5.3.11) This control law is implicit but it is computable.

⎧ ⎨ ⎩ U (t) = κ(P(t)) P(t) = X(t) + ¡ t t-D f (P(τ ), U(τ ))dτ

Predictor design for time-varying delay

The study of time-varying delay system is performed for state dependent delay and few results are proposed for the input delay cases [START_REF] Bresch-Pietri | Invoking halanay inequality to conclude on closed-loop stability of processes with input-varying delay[END_REF]. [START_REF] Nihtila | Adaptive control of a continuous-time system with time-varying input delay[END_REF] proposes a parameter-adaptive design in very particular cases of scalar system which a known time varying input delay. In [START_REF] Nihtila | Finite pole assignment for systems with time-varying input delays[END_REF], a general result based on an explicit state-feedback control law for linear time-invariant (LTI) plants with time-varying input delays was presented. But no analysis of stability or feasibility of controller are performed. In the context of Network Controlled Systems (NCS), [START_REF] Witrant | Optimal control design for the stabilization of network controlled systems[END_REF][START_REF] Witrant | Remote stabilization via time-varying communication network delays: Application to tcp networks[END_REF][START_REF] Witrant | Remote output stabilization under two channels time-varying delays[END_REF] propose the control design of a particular case of linear system with time varying input delay. The authors consider an internal delay dynamics representing the transmission channel model and implement a predictor-based controller.

There exist few papers that deal with the compensation of time-varying input delay in nonlinear systems [START_REF] Karafyllis | Finite-time global stabilization by means of time-varying distributed delay feedback[END_REF][START_REF] Krstic | Lyapunov stability of linear predictor feedback for time-varying input delay[END_REF][START_REF] Georges | A nonlinear state feedback design for nonlinear systems with input delay[END_REF]. In [START_REF] Georges | A nonlinear state feedback design for nonlinear systems with input delay[END_REF], the control design approach is based on state prediction computation and the overall method is illustrated on an example of water ow control in openchannel systems.

The key idea in this case is that the state prediction is calculated over a time-varying window starting with the current state as an initial condition and taking into account the future values of the delay.

As in [START_REF] Nihtila | Finite pole assignment for systems with time-varying input delays[END_REF][START_REF] Krstic | Lyapunov stability of linear predictor feedback for time-varying input delay[END_REF] we consider the following linear system:

Ẋ(t) = AX(t) + BU (φ(t)) X ∈ R n (5.3.12)
U is the input and φ(t) a continuous function which contains the delay. The predictor based controller is build under the following assumptions:

φ(t) is a dierentiable function that satises φ(t) is strictly upper-bounded.
with these assumptions that guarantee the causality of the system (5.3.12) and the invertibility of φ(t), the predictor-based controller is generated by introducing the closed loop system :

⎧ ⎨ ⎩ Ẋ(t) = AX(t) + BU (φ(t)) U (φ(t)) = KX(t) ∀φ(t) ≥ 0 (5.3.13)
Consequently, the system (5.3.12) is rewritten as:

Ẋ(t) = (A + BK)X(t) ∀φ(t) ≥ 0 (5.3.14)
where the gain vector K is chosen to ensure the convergence of the closed loop system. The control law can be written using the inverse of φ(t) as:

U (t) = KX(φ -1 (t)) ∀t ≥ 0 (5.3.15)
Using integration formula on equation (5.3.12) we obtain for φ(t) ≤ θ ≤ t: (5.3.16) or equivalently by introducing the change of coordinates μ = φ(τ ):

X(φ -1 (t)) = e A(φ -1 (t)-t) X(t) + ¢ φ -1 (t) t e A(φ -1 (t)-τ ) BU (φ(τ ))dτ
X(φ -1 (t)) = e A(φ -1 (t)-t) X(t) + ¢ t φ(t) e A(φ -1 (t)-φ -1 (μ)) B U (μ) φ [φ -1 (μ)] dμ
(5.3.17)

Replacing (5.3.17) in the control law (5.3.15), it follows that :

U (t) = K e A(φ -1 (t)-t) X(t) + ¢ t φ(t) e A(φ -1 (t)-φ -1 (μ)) B U (μ) φ [φ -1 (μ)] dμ (5.3.18)
Equation (5.3.18) is the predictor feedback of the time-varying input delay (5.3.12) with the predicted state given by equation (5.3.17):

P (t) = X(φ -1 (t)).
Remark 12. In can be noted that the exponential stability of the feedback system with the predictor controller proposed in [START_REF] Nihtila | Finite pole assignment for systems with time-varying input delays[END_REF] is proven in [START_REF] Krstic | Lyapunov stability of linear predictor feedback for time-varying input delay[END_REF]. The proof lays on the construction of a Lyapunov functional using a backstepping transformation with time varying kernels, and transforming the actuator state into a transport partial dierential equation with a convection speed coecient that varies with both space and time.

Prediction of the state of a time varying delay system is very complicated. It necessitates future values of the delay for the horizon [t, φ -1 (t)] that may be not available. Some methodology to compute this horizon is proposed in [START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF] when the delay evolution is known.

Predictor design for state-dependent input delay system

The rst results on the stabilization of a state-dependent input delay linear and nonlinear systems are proposed in [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF]. These papers are an extension of the works on time varying delay systems (see section 5.3.2).

The predictor-based compensator is designed for a particular class of nonlinear systems which are, in the absence of the input delay, either forward complete and globally or locally stabilizable (by a possible time-varying control law). Again the controller uses predictions of future values of the state on appropriate prediction horizons that depend on the current values of the state. Only a regional stability result, is obtained even for the case of forwardcomplete systems. Also, an estimate of the region of attraction for a control scheme based on the construction of a strict, time-varying Lyapunov functional is determined.

The global result is achieved for forward-complete systems under a restrictive Lyapunovlike condition, which has to be veried. This feasibility condition F c (5.3.31), is that the delay rate must be bounded by unity irrespective of the values of the state and input. This condition corresponds to the upper bounded condition established for time varying input delay system. In this context, the control signal never reaches the plant if the delay rate is larger than one.

We present the predictor for the state dependent input delay nonlinear system [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF].

Consider the system which is governed by the following nonlinear equation:

Ẋ(t) = f (X(t), U(t -D(X(t))) with f (0, 0) = 0 (5.3.19)
where

X ∈ R n ; U : [t 0 -D(X(t 0 )), ∞[→ R n ; t > t 0 > 0 D ∈ C 1 (R n ; R + ). f : (R n ; R + ) → R n
is assumed to be locally Lipschitz and the following holds:

|f (X(t), ω)| ≤ α 1 (|X| + |ω|) (5.3.20)
Where

α 1 is a class K ∞ function * .
The state predictor is given by:

P(t) = X(t + D(P(t)))
(

Indeed the predictor P(t) is associated to the value of the state at the time when the applied control reaches the system. This time is itself depending on the state at this time, so P(t). The existence of this implicit relation makes the problem completely dierent from the case of system with independent time varying delay function as it is exposed previously.

The key of the resolution in [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF], is based on the transformations of the time variable t → t + D(P(t)) and t → t -D(X(t)). The predictor-based controller is given by the following relation: (5.3.22) where for all t -D(X(t)) < τ < t and t > t 0 :

U (t) = κ(σ(t), P(t))
P(τ ) = X(t) + ¢ τ t-D(X(t)) f (P(μ), U(μ)) 1 -∇D(P(μ))f (P(μ), U(μ)) dμ (5.3.23)
with the prediction time

σ(μ) = μ + D(P(μ)) (5.3.24)
The initial predictor P (τ ) is given by (5.3.23) for t = t 0 and τ ∈ [t 0 -D(X(t 0 )); t 0 ]. Analyzing the initial predictor for τ = t 0 -D(X(t 0 ), one deduce that:

P(t 0 -D(X(t 0 )) = X(t 0 ) (5.3.25) * a continuous function α: [0, a) → [0, ∞) is said to belong to class K∞ if a = ∞ and α(r) → ∞ as r → ∞
For nonlinear state dependent-input delay systems (5.3.19) which are more restrictive in the sense that the free-delay system is only locally stable, the problem is more complicated [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF].

The regional stability of the delay system (5.3.19) is achieved if the control reaches the system while the state is within the region of attraction of the delay-free closed-loop system and the solution remains at any time within the controller's feasibility region, so that the control signal is never retarded to earlier values, and the delay remains compensated for all subsequent times. This requirement corresponds to set of initial conditions from which all of the solutions belong to this the region of attraction.

It is important to note that in both cases the feasibility condition (5.3.31) must be checked.

Remark 13. For the linear state dependent input delay system the same analysis can be performed. Let us consider the plant:

Ẋ(t) = AX(t) + BU (t -D(X(t)) X ∈ R n (5.3.34)
As soon as the pair (A, B) is stabilizable the free delay system Ẋ(t) = AX(t)+BU (t) satises all the required condition to synthesize the controller for the linear state dependent input delay system. The controller for the time delay system is given by

U (t) = KP(t) (5.3.35) P(τ ) = X(t) + ¢ τ t-D(X(t)) AP(μ) + BU (μ) 1 -∇D(P(μ))(AP(μ) + BU (μ)) dμ for all t -D(X(t)) < τ < t:
The proposed predictor-based controller is perfectly adapted to linear systems, treating them as a special case of the design for nonlinear systems, for which the exponential stability is proved.

Stabilization of the position of the interface

In this section we shall show that a linear feedback stabilizes indeed the dynamics of the interface position considered as the nonlinear control system (5.2.17) with state-dependent delay on the input. The main feature of our problem in this context arises from the physical constraint: the input function corresponds to a lling ratio which takes value in [0, 1[ ; the denominator term of (5.2.17) has an singularity in the term

1 -f in p t -l(t) N 0 ξ .
This problem of singularity is not considered in [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF]. It is clear that the free-delay system which is related to equation (5.2.17) is only locally stable. For the stability issue one should choose carefully the initial condition to guarantee that the delay system stays in the attraction region when the control reaches the system. This basin of attraction must be dened precisely for a stabilization problem around an equilibrium. The net ow at the die is a function of the gradient of pressure ΔP (L, t) and the interface dynamics depends strongly on this parameter. Consequently, the interface dynamics varies according to the coupling relation that is imposed at the interface. Then, the study of the reachable equilibria appears as an important aspect of the stability analysis. In order to formulate the control problem, we start by an analysis of the static models associated to the two interface relations consisting of momentum ux continuity or pressure continuity.

Static models associated with the interface relations

The static model of the position of the interface strongly depends on the considered interface relations which are presented below.

1. Firstly, assuming the interface relations (5.2.4) stating the continuity of momentum ux at the interface, the equilibrium (f pe , l e ) is dened by

⎧ ⎨ ⎩ F de -ρ 0 N 0 V ef f f pe = 0 l e = L - B P 0 (η S ef f ξ Ne) + B ρ 0 K d fpe (1-fpe) -B ρ ξ N 0 fpe (η S ef f ) (5.4.1)
In order to satisfy the positivity of the die net ow F de , at the boundary the critical value of lling ratio at the equilibrium must satisfy:

f pce > 1 (ξ 2 N 2 e + P 0 ρ 0 ) P 0 ρ 0 - ηV ef f N e Bρ 0 (L -l e ) (5.4.2)
The graph of equilibria (f pe , l e ) is then shown on the gure (Fig. 5.4.1). With this interface relation the physically reachable interface position lies in the interval [0, 1.2] meters for a extruder length of L = 2m.

at l = 0m: this case corresponds to a full of matter extruder. The inlet lling ration is f pe = 0.68. Above this critical inlet ow rate the extruder has only the FFZ zone.

at l = 1.2m the inlet ow rate is close to zero.

2. Secondly, assuming the interface relation (5.2.6) stating the continuity of pressure at the interface, the equilibrium (f pe , l e ) is dened by : The graph of equilibria (f pe , l e ) is then shown on the gure 5.4.2. (the total length of the extruder is 2 meters) than with the assumption of continuity of the momentum ux. In this second case, the validity of the equilibrium model is restricted to (f pe , l e ) ∈ [0, 0.67] × [0, 1.2] . This conclusion was predictable because the constraint related to the pressure continuity relation states that the interface position is located in the interval ]0, L[. This aspect guarantees the existence of Partially and Fully Filled Zones and is considered as a modelling assumption.

⎧ ⎨ ⎩ F de -ρ 0 N 0 V ef f f pe = 0 l e = L -Bρ 0 fpe K d (1-fpe) (5.4.3)

Feedback stabilization of the position of the interface

Due to the fact that the pressure continuity relation cover a larger domain than momentum ux coupling condition, we restrict the study to the case with the assumption of continuity of pressure. Then, in the following stability analysis, the gradient of pressure is given by equation (5.2.6).

Feedback stabilization of the nonlinear without delay

As in [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF], we study the stabilization of the free-delay system of equation (5.2.17) around the equilibrium. To satised the property which consists to dene the stabilization problem at the origin (F(0, 0) = 0) we rewrite the system by including the equilibrium point as follows:

dl(t) dt = F d (t) -F de -ρ 0 N 0 V ef f f in p (t) -f pe ρ 0 S ef f 1 -f in p (t) (5.4.5)
with a linear abstract control on the incremental system : δf in p = -K δl.

We mention that the control is implementable if the real input and output variables are considered . As is stated previously the control is directly deduced from the feed rate F i n(t) by the linear relation:

f in p (t) = F in (t) ρ 0 N 0 V ef f .
(5.4.6)

Secondly, there is a dependence between δl and δP (L, t) which can be measure by a pressure jauge. The relation between δl(t) and δP (L, t) arise from the linearized model of pressure gradient equation assuming the pressure continuity at the interface:

⎧ ⎨ ⎩ ΔP (t) = P L, t) -P 0 P (L, t) = P 0 + ηV ef f N 0 ρ 0 (L-l(t)) (Bρ 0 +K d (L-l(t))) (5.4.7)
and:

δP (L, t) = Cδl,
(5.4.8) Where C is an constant depending on physical parameters of the system.

The linearized model of the system (5.4.5) around the equilibrium of the open loop system without delay (5.4.5) is given by:

d(δl) dt = α l δl + α f δf in p (5.4.9)
with:

α l = - (ξN 0 K d Bρ 0 ) (Bρ 0 + K d (L -l e )) 2 (1 -f pe )
(5.4.10) and:

α f = - ξN 0 (1 -f pe ) (5.4.11)
It is clear that the stability of the linearized system is achieved if:

⎧ ⎨ ⎩ K < K max K max = α l α f = K d Bρ 0 (Bρ 0 +K d (L-le)) 2
(5.4.12) By implementing the gain K to the nonlinear system, the equation (5.4.5) becomes:

⎧ ⎨ ⎩ dδl(t) dt = -ϕ k (δl(t)) -ϕ k (δl(t)) = F d (δl(t))-F de +ρ 0 N 0 V ef f Kδl(t) ρ 0 S ef f (1-fpe+Kδl(t)) (5.4.13)
The stability depends on the sign of ϕ k (δl(t)). The physical properties of the process introduce a constraint on the denominator of equation (5.4.13):

⎧ ⎨ ⎩ 0 < (1 -f pe + Kδl(t)) < 1 - 1-fpe K < δl(t) < fpe K (5.4.14)
Due to the positivity of the denominator of equation (5.4.13), the positivity of ϕ k (δl(t))

is achieved if ∀δl(t) = 0:

δl(t) (F d (δl(t)) -F de + ρ 0 N 0 V ef f Kδl(t)) < 0 (5.4.15)
The above expression will vary depending on the selected hypothesis to dene the interface coupling relation. In the case treated in this section, we obtain the following results:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ϕ k (δl(t)) > 0 ⇔ K 2 d η + K C e -K d η δl(t) C e < 0 C e = Bρ 0 + K d η (L -l e ) (5.4.16)
Which is equivalent to:

δl(t) > K d KC e - ηC e K d (5.4.17)
Finally combining equations (5.4.14) and (5.4.17), we can determine the basin of attraction under the assumption of continuous pressure at the interface. The obtained relation proves that the stability region is modulated by the feedback gain K:

sup - 1 -f pe K ; K d KC 2 e - ηC e K d < δl(t) < f pe K (5.4.18)

Control of the system with input delay

Now we recall the delay system which relates the coupling of the P DE and the boundary equation.

dl(t) dt = F d (t) -F de -ρ 0 N 0 V ef f f in p t -l(t) N 0 ξ -f pe ρ 0 S ef f 1 -f in p t -l(t) N 0 ξ (5.4.19)
We design a predictor controller which is developed in section 5.3.3 and physical parameters are chosen to satisfy both assumptions as forward completeness and the delay function

D(l(t))
is also C 1 . The feasibility condition F c which is dened in (5.3.31) is also veried. The control law is the linear feedback given in the above section by the gain K:

⎧ ⎨ ⎩ U (t) = δf p (l(t), t) = -Kδl(t) = -K(P(t) -l e ) D(l(t)) = l(t) ξN 0 (5.4.20)
Where for all t -D(l(t)) ≤ τ ≤ t

P(τ ) = l(t) + ¢ τ t-D(l(t)) F (P(μ), U(μ)) dμ 1 -∇D(P(μ))F (P(μ), U(μ)) (5.4.21)
The initial predictor P(μ) with μ ∈ [-D(l 0 ), 0] is given by (5.4.21) at t = 0. Applying this controller to the system ensures the stabilization of the desired equilibrium position l e .

Simulation results

The Numerical implementation of the predictor (5.4.21) is performed by approximating the distributed delay by a sum of point-wise delays using a numerical quadrature rule. In this way one ends up with a sequence of control-laws of the form:

P(t) = l(t) + N i=1 h(t) F (P i-1 , U i-1 ) 1 -∇D(P i-1 )F (P i-1 , U i-1 ) h(t) = D(l(t)) N (5.5.1)
For th numerical integration , the values of the right-hand side of (5.5.1) involve earlier values of P and the values of the input U . As we stated previously, the main diculty is the practical implementation of the integral term, which needs to be calculated on-line [START_REF] Mondié | A safe implementation for nnite spectrum assignment : robustness analysis[END_REF]. An overview of stability results on the implementation of distributed delay control laws is given in [START_REF] Michiels | The eect of approximating distributed delay control laws on stability[END_REF][START_REF] Mirkin | On the approximation of distributed-delay control laws[END_REF][START_REF] Fattouh | Pulse controller design for linear systems with delayed state and control[END_REF][START_REF] Van Assche | Some problems arising in the implementation of distributed-delay control laws[END_REF]. [START_REF] Bidard | Stabilisation des systémes commandés par réseaux[END_REF] mentions that the problems raised by the implementation of a control law including time varying delay is much more complicated. In this case, the discretization leads to a discrete controller with variable dimension. The resulting loop system therefore has a variable number of poles and zeros (for linear system), which makes dicult the study of the correlation between the numerical instabilities and the sampling period or the method of discretization.

We use a semi-discretization scheme based on the nite volume method for the simulation of the transport equation (5.2.1) with the controller. Time integration is performed with ODE45 routine of MATLAB. The gure (Fig. 5.5.1) shows the convergence of the moving interface from an initial position (l 0 = 1m) to the equilibrium (l e = 1.742m). It remains some static error that can be corrected by using some integrator (l(t 1 ) = 1.760m, t 1 = 450s). One conclude that the system can be stabilized using the state predictor 5.4.21 and the linear gain K max = 0.8383. Remark 14. An interesting future concerning this application is the robustness analysis of the controller. Recall that the model is based on the strong assumption of constant viscosity which is very restrictive in industrial context. In some sense a perturbation of the delay term may be interpreted as a variation of the viscosity or of some others physical variables. This type of extension will allows to consider more realistic extrusion process with application in industry. We suggest to follow the recent work of [START_REF] Bekiaris-Liberis | Robustness of nonlinear predictor feedback laws to time-and state-dependent delay perturbations[END_REF] to pursuit this study on robustness.

Conclusion

The main idea in our approach relies on the development of a bi-zone model separated by a mobile interface l(t). The model describes the mass balance in the Partially and Fully Filled Zones of an extruder as a transport equation coupled with an ordinary dierential equation. For such systems we show that the coupling conditions are very important since they determine the reachable set of equilibrium. We illustrated this aspect by considering two coupling relations: The rst one is derived from the continuity of pressure and the second one from the continuity of momentum ux.

The coupled mass transport and interface equations are turned in a state-dependent input delay system using the explicit solutions which are obtained by the method of characteristic. Here, the main objective is the stabilization of the moving interface around a given equilibrium. This control problem is equivalent to the regulation of net ow at the die F d (t) if the continuity of pressure at the interface l(t) is assumed. Using the control proposed in [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF] we show that the desired equilibrium can be reached with reasonable static error.

Ours analysis must be extended for the control of the energy balance equations which describes heat transport phenomena of the extrusion process as in [START_REF] Diagne | Well posedness of the model of an extruder in innite dimension[END_REF]. The associated transport equations are described by (5.6.1,5.6.2) and the source term Ω p,f groups the heat produced by viscous dissipation that is proportional to N 2 (t)) and the heat exchange with the barrel

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ For x ∈ [0; l(t)[ ∂ ∂t T p (x, t) = -ξN 0 ∂ ∂x T p (x, t) + Ω p (T p , T b ) For x ∈]l(t); L] ∂ ∂t T f (x, t) = -F d (t)ξ ρ 0 V ef f ∂ ∂x T f (x, t) + Ω f (T f , T b )
(5.6.1)

Ω p,f = ⎧ ⎨ ⎩ γηN 2 0 f (x,t)ρ 0 V ef f C + Seα ρ 0 V ef f C (T b -T p ) γηN 2 0 f (x,t)ρ 0 V ef f C + Seα ρ 0 V ef f C (T b -T f ) (5.6.2)
The simulations presented in Figure ( T p (l -, t) = T f (l + , t) Port Hamiltonian formulation of a system of two conservation laws with a moving interface port Hamiltonian systems [START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF][START_REF] Maschke | Advanced Topics in Control Systems Theory[END_REF][START_REF] Macchelli | Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach, chapter Innite-dimensional Port-Hamiltonian Systems[END_REF]. This structure has led to various methods of analysis concerning the existence of solutions, their well-posedness and control in the linear case [START_REF] Macchelli | Modeling and control of the Timoshenko beam. the Distributed Port Hamiltonian approach[END_REF][START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric dierential operators[END_REF][START_REF] Villegas | A Port-Hamiltonian Approach to Distributed Parameter Systems[END_REF][START_REF] Zwart | Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain[END_REF][START_REF] Villegas | Stability and stabilization of a class of boundary control systems[END_REF][START_REF] Jacob | Linear Port-Hamiltonian Systems on Innite-dimensional Spaces[END_REF][START_REF] Ortega | On compensation of wave reexions in transmission lines and applications to the overvoltage problem in AC motor drives[END_REF][START_REF] Macchelli | Control by interconnection of mixed Port Hamiltonian systems[END_REF].

In this chapter we shall investigate wether the port-Hamiltonian formulation may be extended to systems of conservation laws coupled by a moving interface. Such systems occur in various cases when the system is heterogeneous in the considered spatial domain, leading to several phases. The most simple example (which we shall also consider here) consists in two uids which are separated by some moving wall. This wall separates two phases, the two uids which might have dierent properties and induces discontinuities of some variables at the interface. These discontinuities are part of the model of the interface dened by a set of interface relations. The wall separating the two uid may permit, or not, a mass ow or a pressure discontinuity for instance. The interfaces may also separate bubbles in a uid in dierent chemical processes or dene the boundary of polymer nanoparticles in a uid. This interface might also separate the phase in evaporation processes. It might also arise from a constraint on some variables such as volume and lead to a change of causality and order in the spatial domain as for the extrusion process model.

More precisely we are inspired by a classical approach developed for xed interfaces, consisting in augmenting the system of conservation laws of the physical model with color functions [START_REF] Godlewski | The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. the scalar case[END_REF][START_REF] Godlewski | The numerical interface coupling of nonlinear systems of conservation laws: II. the case of systems[END_REF][START_REF] Ambroso | A Relaxation Method for the Coupling of Systems of Conservation Laws[END_REF][START_REF] Boutin | Dafermos regularization for interface coupling of conservation laws[END_REF]. In rst instance we shall show that this system of conservation laws may be formulated as port-Hamiltonian system with a pair of port variables associated to the interface. In second instance we shall generalize the previous approach to moving interfaces and show that it may again be formulated as a port-Hamiltonian system by adding a second pair of port variables corresponding to the motion of the interface. In this model the velocity of the interface appears like an input and the interface relations dening the dynamics of the displacement of the interface are then dened as an external port-based model. In the whole chapter the spatial domain is an interval of the real line and we shall only consider systems of two conservation laws. The sketch of this chapter is the following. In a rst part we consider two Hamiltonian systems consisting of two conservation laws coupled by a xed interface. We rst recall the denition of Stokes-Dirac structures and the boundary port-Hamiltonian formulation of a system of two conservation laws with ux variables deriving from a Hamiltonian. Then, we recall the extended system obtained by introducing color functions, associated with the characteristic functions of the spatial domains dened by the interface. We then dene a Dirac structure and the port Hamiltonian formulation of the systems of conservation laws coupled by a xed interface. In the second part we consider a moving interface and generalize the formulation of the coupled system of conservation laws before formulating it in the port-Hamiltonian framework. Finally we introduce the interface relations dening the dynamics of the interface as a two-port element and illustrate this on the simple example of two gases coupled by a piston.

Organization of the chapter

This chapter is organized as follows. In section 5.2.1 we introduce the notion of a p-system in the port-Hamiltonian framework and present the construction of the associated Dirac structure by adding boundary ow and eort. In section 5.2.2 we study the interconnection of two p-systems through a xed interface and also discuss the transmission condition at the interface in order to dene a Dirac structure arising from the composition of the two complementary domains associated to two p-systems of conservation laws. In section 5.2.3 a new formulation is proposed by augmenting the system with color functions which correspond to characteristic functions of those complementary domains. These color functions are useful to extend the denition of the eort and ow variables in the total domain by introducing a modulated dierential operator. The total Hamiltonian functional is also dened as an extension of the two Hamiltonian functionals by these color functions. Introducing external interface eort and ow variable the associated Dirac structure is constructed. In section 5.3 we discuss the moving interface problem in the extended formulation of coupled systems using color functions and modulated dierential operator. The main idea is to represent the interface dynamic as an external input variable. We suggest a general structure of interface models based on the continuity of one eort variable. Then the coupling conditions at the moving interface guaranteed the existence of a Dirac structure. Finally we illustrate this by an example of two isentropic gases (modelled by a systems of two boundary port Hamiltonian systems) coupled at their interface by a piston in motion.

Contributions

In this chapter we consider the port-Hamiltonian formulation of a system of two conservation laws dened on two complementary intervals of some interval of the real line and coupled by a moving interface. We recall rst how two port Hamiltonian systems coupled by an interface may be expressed as a port Hamiltonian system augmented with two variables being the characteristic functions of the two spatial domains. Then we consider the case of a moving interface and show that it may be expressed as the preceding port Hamiltonian system augmented with an input, being the velocity of the interface and dene a conjugated output variable. We then give some structure to the interface relations dening the dynamics of the displacement of the interface and end with the simple example of two gases coupled by a moving piston.

6.2 Two port Hamiltonian systems coupled by an interface 6.2.1 Port Hamiltonian system of two conservation laws Let us recall briey the port Hamiltonian formulation of a system of two conservation laws according to [START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF][START_REF] Maschke | Advanced Topics in Control Systems Theory[END_REF][START_REF] Macchelli | Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach, chapter Innite-dimensional Port-Hamiltonian Systems[END_REF] which represents a model of vibrating strings or Lagrangian gas dynamics (the p-system).

We consider systems of two conservation laws:

∂ t x + ∂ z N (x) = 0 (6.2.1)
dened on the spatial domain Z = [a, b], with time t ∈ R + and with the 2-dimensional state vector x (z, t) =

x 1 (z, t)

x 2 (z, t)

. The ux variable is dened by

N (x) = 0 -1 -1 0 δ x 1 H δ x 2 H (6.2.2)
with respect to the Hamiltonian functional H (x) = ¡ b a H (x) dz with Hamiltonian density function H (x) (where δ x H denotes the variational derivative of H with respect to x ). Then the system of conservation laws may be rewritten as the Hamiltonian system ∂ t x = J δ x H (6.2.3) generated by the Hamiltonian functional H (x) and dened with respect to the dierential operator

J = 0 -∂ z ∂ * z 0 (6.2.4)
where ∂ * z is the formal adjoint of the operator ∂ z . Indeed if the ux variables (6.2.2) satisfy the boundary conditions given by δ

x 1 H (a) = δ x 1 H (b) = δ x 2 H (a) = δ x 2 H (b) = 0, then ∂ * z = -∂ z
and the equation (6.2.3) is precisely (6.2.1) (6.2.2). Under the same conditions, the operator J is skew-symmetric. Furthermore as it a matrix dierential operator with constant coecients, it satises the Jacobi identities and is a Hamiltonian operator, dening a Poisson bracket on the functionals of the state variables [START_REF] Olver | Applications of Lie Groups to Dierential Equations[END_REF].

However for control purposes, it should be assumed that this is not satised in order to allow for energy exchange of the system with its environment. Therefore the Hamiltonian system is augmented with the boundary port variables

f ∂ e ∂ = δ x 2 H δ x 1 H a,b = 0 1 1 0 δ x 1 H δ x 2 H a,b (6.2.5)
and is thereby extended to a boundary port-Hamiltonian system [START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF][START_REF] Maschke | Advanced Topics in Control Systems Theory[END_REF][START_REF] Macchelli | Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach, chapter Innite-dimensional Port-Hamiltonian Systems[END_REF]. This port-

Hamiltonian system is dened with respect to a Dirac structure which extend the Hamiltonian operator (6.2.4).

Let us recall the denition of a Dirac structure which will be extensively used in this chapter.

Denition 5. [START_REF] Courant | Dirac manifolds[END_REF] Consider two real vector spaces, F the space of ow variables and E the space of eort variables, together with a pairing, that is, a bilinear operation

F × E : → R (f, e) → e, f (6.2.6)
which induces the symmetric bilinear form , on the bond space B = F × E dened as

(f 1 , e 1 ), (f 2 , e 2 ) :=< e 1 , f 2 > + < e 2 , f 1 >, (f i , e i ) ∈ F × E (6.2.7)
A Dirac structure is a linear subspace D ⊂ F × E which is isotropic and co-isotropic that is satises D = D ⊥ , with ⊥ denoting the orthogonal complement with respect to the bilinear form , .

Particular Dirac structures, called Stokes-Dirac structures, are associated with Hamiltonian dierential operators [START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF][START_REF] Macchelli | Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach, chapter Innite-dimensional Port-Hamiltonian Systems[END_REF][START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric dierential operators[END_REF]; here we recall the particular case of the Stokes-Dirac structure associated with the Hamiltonian operator J dened in (6.2.4).

Proposition 6. The linear subspace of the bond space B = F × E , product space of the space of ow variables F and eort variables E where

F = E = L 2 (a, b) , R 2 × R 2 dened by D = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ f 1 f 2 f ∂ ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ e 1 e 2 e ∂ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ∈ F × E | e 1 e 2 ∈ H 1 (a, b) , R 2 2 , f 1 f 2 = J e 1 e 2 and f ∂ e ∂ = 0 1 1 0 e 1 e 2 a,b ⎫ ⎬ ⎭ (6.2.8)
is a Dirac structure and is called Stokes-Dirac structure, with respect to the pairing

⎛ ⎜ ⎜ ⎝ f 1 f 2 f ∂ ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ e 1 e 2 e ∂ ⎞ ⎟ ⎟ ⎠ = ¢ b a (f 1 e 1 + f 2 e 2 ) dz + e ∂ Σ f ∂ with Σ = diag (-1 , 1) . (6.2.9)
In the same way as Hamiltonian systems are dened with respect to Poisson brackets, port-Hamiltonian systems are dened with respect to Dirac structures [START_REF] Van Der Schaft | The Hamiltonian formulation of energy conserving physical systems with external ports[END_REF][START_REF]Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach[END_REF]. Again we refer with [START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF][START_REF] Macchelli | Modeling and Control of Complex Physical Systems -The Port-Hamiltonian Approach, chapter Innite-dimensional Port-Hamiltonian Systems[END_REF][START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric dierential operators[END_REF] for the general denition of boundary port-Hamiltonian dened respect to a Stokes-Dirac structure and will only recall the denition for the case of a system of two conservation laws.

Proposition 7. The system consisting of two conservation laws (6.2.3) augmented with the port variables (6.2.5) is equivalent to

⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ ∂ t x 1 ∂ t x 2 f ∂ ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ δ x 1 H δ x 2 H e ∂ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ∈ D (6.2.10)
and denes a boundary port-Hamiltonian system.

As a consequence of the properties of the Stokes-Dirac structure [START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF], the Hamiltonian function obeys the following balance equation

d dt H = -e ∂ Σ f ∂ 6.2.

Interconnection of port Hamiltonian systems through an interface

In this section we consider two systems of conservation laws which are dened in the spatial domains [a, 0[ and ]0, b] respectively. We denote their state variables and Hamiltonian with exponentand + depending on which half real line they are dened. The two systems are coupled through interface relations coupling the boundary values of the two systems at z = 0, the interface of the two systems. Considering two systems of conservation laws coupled through an interface the most currently considered interface relations are [START_REF] Godlewski | The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. the scalar case[END_REF][START_REF] Boutin | Dafermos regularization for interface coupling of conservation laws[END_REF][START_REF] Ambroso | A Relaxation Method for the Coupling of Systems of Conservation Laws[END_REF] the continuity of the ux variable or the continuity of some other functions of the state variables, called privileged variables [27, p. 5].

For boundary port Hamiltonian systems it is natural to express the interface relations using the boundary port variables (6.2.5) which are in fact, in the case of the canonical systems of two conservation laws, actually the ux variables at the boundary. In this case the space of ow and eort variables are dened as the product space of the ow and eort spaces on each domain:

F = E = L 2 (a, 0) , R 2 × R 2 × L 2 (0, b) , R 2 × R 2
and the Dirac structure is dened with respect to the product operator of (6.2.4)

J 0 2 0 2 J
Considering for instance the interface relations being the balance equation

δ x + 1 H + + δ x - 1
H -= 0 and the continuity equation

δ x + 2 H + = δ x - 2
H -, and writing them in vector notation, one obtains the linear relation between the conjugated power variables In the sequel of the paper, we shall consider the following interface relations where the pair of interface port variables (f I , e I ) is introduced

⎛ ⎝ δ x + 1 H + δ x + 2 H + ⎞ ⎠ 0 + = 0 1 -1 0 ⎛ ⎝ δ x - 1 H - δ x - 2 H - ⎞ ⎠ 0 - (6.
f I = δ x + 2 H + = δ x - 2
H - (6.2.12)

0 = δ x + 1 H + + δ x - 1 H -+ e I (6.2.13)
The equation (6.2.12) is again a continuity equation and the equation (6.2.13) is a balance equation with an external term e I . This are commonly considered interface relations [START_REF] Godlewski | The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. the scalar case[END_REF][START_REF] Boutin | Dafermos regularization for interface coupling of conservation laws[END_REF][START_REF] Ambroso | A Relaxation Method for the Coupling of Systems of Conservation Laws[END_REF] consisting of the continuity equation of one of the ux variable (then called privileged variable) and the introduction of a source term at the interface, in the balance equation of the other ux variable [START_REF] Boutin | Etude Mathématique et Numérique des Equations Hyperboliques Nonlinéaires: couplage de modèles et chocs non classiques[END_REF]. Denoting e + i = δ x + i H + and e - i = δ x - i H -with i = 1, 2 , the interface relations (6.2.12) (6.2.13) dene the linear relations between the conjugated power variables

⎛ ⎜ ⎜ ⎝ e - 2 e + 1 f I ⎞ ⎟ ⎟ ⎠ 0 + = ⎛ ⎜ ⎜ ⎝ 0 1 0 -1 0 -1 0 1 0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ e - 1 e + 2 e I ⎞ ⎟ ⎟ ⎠ 0 - (6.2.14)
with respect to a nonlinear matrix and therefore dene a Dirac structure. The interface relations may of course be much more general using nonlinear functions of the ux variables at the interface, port variables coupled to a dynamical system: if the interface relations dene a Dirac structure coupled to a dissipative port Hamiltonian systems then by composition of Dirac structures, a dissipative port Hamiltonian system is obtained on the product space of the state space of the subsystems [START_REF] Cervera | Interconnection of port-Hamiltonian systems and composition of Dirac structures[END_REF] as for instance in [START_REF] Ortega | On compensation of wave reexions in transmission lines and applications to the overvoltage problem in AC motor drives[END_REF][START_REF] Macchelli | Control by interconnection of mixed Port Hamiltonian systems[END_REF]. However in the sequel we shall depart from this procedure of composition of boundary port Hamiltonian systems. Indeed, as a consequence of considering moving interfaces, timevarying spatial domains have to be considered. These do not appear explicitly as variables in the denition of boundary port Hamiltonian systems. This is the reason why, in the remaining of the paper, we shall use additional state variables, the characteristic functions of the time-varying spatial domains of each subsystem. There is an alternative approach [START_REF] Godlewski | The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. the scalar case[END_REF][START_REF] Boutin | Dafermos regularization for interface coupling of conservation laws[END_REF][START_REF] Ambroso | A Relaxation Method for the Coupling of Systems of Conservation Laws[END_REF] where instead of considering the product spaces of the variables dened in the dierent spatial domains, the state variables of the coupled systems are dened on the composed spatial domain, the interval [a, b]. The interface at z = 0 becomes then an interior point of the spatial domain. However some external variables may still be associated with the interface. Following [START_REF] Abgrall | Computations of compressible multiuids[END_REF][START_REF] Boutin | Dafermos Regularization for Interface Coupling of Conservation Laws[END_REF][START_REF] Ambroso | Coupling Two Scalar Conservation Laws via Dafermos Self-Similar Regularization[END_REF] we shall introduce additional state variables called color functions which are actually the characteristic functions of the domains of the two systems

c 0 (z, t) = 1 z ∈ [a, 0[ 0 z ∈ [0, b] and c0 (z, t) = 1 z ∈ ]0, b] 0 z ∈ [a, 0] (6.2.15)
Hence the state variables of the coupled system may be expressed as the sum of prolongations of the variables of each subsystem to the total spatial domain Z = [a, b] by

x(z, t) = x -(z, t) + x + (z, t) (6.2.16) x -(z, t) = c 0 (z, t)x(z, t) x + (z, t) = c0 (z, t)x(z, t) (6.2.17)
The ux variable of the two conservation laws, becomes

N (x, c 0 , c0 ) = c 0 N -(x) + c0 N + (x) (6.2.18)
The global ux can be written as (6.2.20) where it should be noticed that N -(x) and N + (x) in (6.2.18), (6.2.19), (6.2.20), are dierent ux functions. Now the system is given by the conservation laws dened on the total spatial domain 6.2.3.2 Conservation laws and interface relations as a single system of balance equations

c 0 N (x, c 0 , c0 ) = c 0 N -(x) (6.2.19) c0 N (x, c 0 , c0 ) = c0 N + (x),
We shall now consider the two systems of Hamiltonian conservation laws coupled by the interface relations dened in (6.2.12) and (6.2.13). As a consequence of these relations, considering the denition of the ux variables (6.2.2), it appears that the ux variable N 1 satises a continuity equation at the interface whereas the ux variable N 2 satises a balance equation at the interface.

In the rst instance, let us consider the conservation law of the state variable

x 1 which may be written (on the whole domain [a, b])

∂ t x 1 = -∂ z c 0 N - 1 (x) + c0 N + 1 (x) = -∂ z (c 0 N 1 (x, c 0 , c0 ) + c N 1 (x, c 0 , c0 )) = -[∂ z c 0 . + ∂ z c0 .] d 0 N 1 (x, c 0 , c0 ) (6.2.21)
where the operator

d 0 = -[∂ z c 0 . + ∂ z c0 .] (6.2.22)
acts as the dierential operator -∂ z on each sub-domain (according to the system (6.2.3) (6.2.4)). Indeed (6.

(a ≤ a ≤ b < 0 ) d dt ¢ b a x 1 (z, t) = -N - 1 (a , t) + N - 1 (b , t) or on an interval [a , b ] on the positive real line ( 0 < a < b ≤ b ) d dt ¢ b a x 1 (z, t) = -N + 1 (a , t) + N + 1 (b , t)
Now let us consider the formulation of the conservation law of x 1 on an arbitrary interval [a , b ] containing the interface (with a ≤ a < 0 < b ≤ b). The assumption the continuity of the ux variable N 1 , implies the following consequence on the conservation law of the

variable x 1 d dt ¡ b a x 1 (z, t) dz = d dt ¡ 0 a x 1 (z, t) dz + d dt ¡ b 0 x 1 (z, t) dz = ¡ b a d 0 N 1 (x, c 0 , c0 ) dz = ¡ b a -[∂ z c 0 . + ∂ z c0 .] N 1 (x, c 0 , c0 ) dz = -N - 1 (a , t) + N - 1 (0 -, t) -N + 1 (0 -, t) + N + 1 (b , t) = -N - 1 (a , t) + N + 1 (b , t) = -N 1 (a , t) + N 1 (b , t)
= -e 2 (a , t) + e 2 (b , t) (6.2.23) In the second instance, let us consider the conservation law of the state variable x 2 and remind that , at the interface, the associated ux variable N 2 is supposed to satisfy the balance equation (6.2.13) with the source term e I δ (z) (a Dirac distribution), localized at the interface. But rstly we have to calculate the dual operator, denoted by d * 0 , to the operator d 0 dened in (6.2.22), in order to be able to express the power pairing. Therefore consider two eort variables e 1 and e 2 and compute

¡ b a e 1 (d 0 e 2 ) dz = - ¡ b a (e 1 [∂ z c 0 . + ∂ z c0 .] e 2 ) dz = - ¡ b a (e 1 [∂ z (c 0 e 2 ) + ∂ z (c 0 e 2 )]) dz = -[(c 0 + c0 )e 1 e 2 ] b a + ¡ b a (c 0 e 2 + c0 e 2 ) (∂ z e 1 ) dz = -[(c 0 + c0 )e 1 e 2 ] b a + ¡ b a e 2 [∂ z c 0 . + ∂ z c0 .] e 1 dz - ¡ b a e 2 [(∂ z c 0 ) + (∂ z c0 )] e 1 dz
Hence the dual operator is dened by

d * 0 = [∂ z c 0 . + ∂ z c0 .] -[(∂ z c 0 ) + (∂ z c0 )] = -d 0 + [(∂ z c 0 ) + (∂ z c0 )] (6.2.24)
Using this dual operator the conservation law of the variable x 2 becomes

∂ t x 2 = -d * 0 N 2 -e I δ (z) (6.2.25)
where δ (z) denote the Dirac mass. Indeed, using similar calculation as in the preceding paragraph, one shows that (6.

¡ b a x 2 (z, t) = ¡ b a {-d * 0 N 2 (x, c 0 , c0 ) -e I (t) δ (z)} dz = ¡ b a {(d 0 -[(∂ z c 0 ) -(∂ z c0 )]) N 2 (x, c 0 , c0 ) -e I (t) δ (z)} dz = ¡ b a d 0 N 2 (x, c 0 , c0 ) dz + ¡ b a {([(∂ z c 0 ) -(∂ z c0 )]) N 2 (x, c 0 , c0 ) -e I (t) δ (z)} dz = -N - 2 (a , t) + N - 2 (0 -, t) -N + 2 (0 -, t) + N + 2 (b , t) -N - 2 (0 -, t) + N + 2 (0 -, t) - ¡ b a e I (t) δ (z) dz = -N - 2 (a , t) + N + 2 (b , t) + e I (t) = -e 1 (a , t) + e 1 (b , t) -e I (t) (6.2.26)
On this balance equation, it appears clearly that the ux variables at the interface N -

Hamiltonian system extended with color functions

In the preceding paragraph we have formulated the dynamical equations of the system with an interface, as the system of balance equations (6.2.21) and (6.2.25) using the dual dierential operators (6.2.22) and (6.2.24) which depend on the characteristic functions of the two domains separated by the interface. Following [61][28][10], we shall introduce explicitly these functions as variables of the system; they are then called color functions and will be denoted by c(z, t) and c(z, t). Noticing that the spatial domains separated by the xed interface are constant, hence also their characteristic functions c 0 and c0 dened in (6.2.15), it is clear that they satisfy the trivial conservation laws

∂ t c = ∂ t c = 0 (6.2.27)
with initial conditions being precisely c 0 and c0 and compatible boundary conditions.

In the sequel we shall dene an extended Hamiltonian system composed of the two balance equations (6.2.21) and (6.2.25) with closure equations (6.2.2) indexed by + andfor each spatial subdomain and augmented with the trivial conservation laws (6.2.27). Therefore dene the Hamiltonian functional H

(x, c, c) = ¡ b a H (x, c, c) dz with density H (x, c, c) = c H -(x) + c H + (x)
δ xH (x) = ⎛ ⎜ ⎜ ⎝ δ x H (x, c, c) δ c H (x, c, c) δ cH (x, c, c) ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ c δ x H -(x) + c δ x H + (x) H -(x) H + (x) ⎞ ⎟ ⎟ ⎠ (6.2.30)
Note that the rst row corresponds precisely to the denition of the ux variable (6.2.18) for the particular solution c 0 and c0

δ x H (x, c, c) = c δ x H -(x) + c δ x H + (x) = c N -(x) + c N + (x) = N (x, c, c)
This allows to augment the Hamiltonian system (6.2.3) with the trivial conservation laws of the color functions (6.2.27) obtaining the Hamiltonian system:

∂ t x = J a δ xH (x) + Ie I (6.2.31) I T = 0 -1 0 0 (6.2.32)
with respect to the operator

J a = ⎛ ⎜ ⎜ ⎝ 0 d -d * 0 0 2 0 2 0 2 ⎞ ⎟ ⎟ ⎠ (6.2.33)
where operator d is the nonlinear dierential operator, modulated by c(z, t) and c(z, t) dened by

d = -[∂ z c. + ∂ c.] (6.2.34)
and its formal dual In order to take account of the energy exchange at the boundary {a, b} and dening a conjugated ow variable f I to the interface source term e I at the interface, the Hamiltonian system (6.2.31) will now be extended to a port Hamiltonian systems with boundary and distributed ports. In the begin, the operator J a dened in (6.2.33) and the input map at the interface dened by (6.2.32) are extended to a Stokes-Dirac structure using a similar procedure as in [START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF]. Proposition 8. The set of relations D I associated with a system of two conservation laws dened on the variables

d * = -d + [(∂ z c) -(∂ z c)] (6
D I = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ f f I f ∂ ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ ẽ e I e ∂ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ∈ F × E / f f I = J a I -I T 0 ẽ e I and f ∂ e ∂ = 0 1 (c + c) 0 e 1 e 2 a,
= F × E with F = L 2 ((a, b) , R) 5 × R 2 and E = dom d * × dom d × L 2 ((a, b) , R) 3 × R 2 endowed with the pairing ⎛ ⎜ ⎜ ⎝ f f I f ∂ ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ ẽ e I e ∂ ⎞ ⎟ ⎟ ⎠ = ¡ b a ẽT f dz + e T ∂ Σf ∂ + ¡ b a e I T f I dz (6.2.38) with e T ∂ Σf ∂ = e ∂ (a)f ∂ (a) -e ∂ (b)f ∂ (b)
, denes a Dirac structure.

One may notice immediately that the pair of port variables (e I , f I ) at the interface are distributed variables. Let us prove that the set (6.3.12) is indeed a Dirac structure. Note that we shall use the following notation

f = ⎛ ⎜ ⎜ ⎝ f f I f ∂ ⎞ ⎟ ⎟ ⎠ ; ē = ⎛ ⎜ ⎜ ⎝ ẽ e I e ∂ ⎞ ⎟ ⎟ ⎠ (6.2.39)
Proof. Let us rst show the isotropy condition

D I ⊂ D ⊥ I : ( f 1 , ē1 ), ( f 2 , ē2 ) = 0 ∀( f 1 , ē1 ), ( f 2 , ē2 ) ⊂ D I (6.2.40)
with respect to the bilinear product associated with the pairing (6.2.38) and denoted in the sequel by P and using the identity (6.2.36), one obtains that P = 0 which proves the isotropy condition.

P = ( f 1 , ē1 ), ( f 2 , ē2 ) = f 1 , ē2 + f 2 , ē1 = ¡ b a ẽ2T f 1 dz + ¡ b a ẽ1T f 2 dz + ¡ b a e 1 I T f 2 I dz + ¡ b a e 2 I T f 1 I dz +e 1 ∂ T Σf 2 ∂ + e 2 ∂ T Σf 1 ∂ ( 6
Let us now prove the co-isotropy condition Secondly, let us choose e 1 2 = 0 , e 1 I = 0 and e 1 1 (a) = e 1 2 (b) = 0, then, using the denition of f 2 of the constitutive relations of D I and (6.2.36) with zero boundary conditions, the bilinear product becomes

D ⊥ I ⊂ D I . This amounts to prove that if ( f 2 , ē2 ) ∈ B satises ∀( f 1 , ē1 ) ∈ D I ; ( f 1 , ē1 ), ( f 2 , ē2 ) = 0 then ( f 2 , ē2 ) ∈ D I . Therefore
P = ¢ b a e 2 2 -d * e 1 1 + e 1 1 f 2 1 dz = ¢ b a e 1 1 -de 2 2 + f 2 1 dz (6.2.43)
The condition that P vanishes for any e 1 1 hence implies that :

f 2 1 = de 2 2 .
Thirdly, let us choose e 1 1 = 0 , e 1 I = 0 and e 1 1 (a) = e 1 2 (b) = 0, then, using the denition of The condition that P vanishes for any e 1 2 hence implies that :

f 2 2 = -d * e 2 1 -e 2 I .
Fourthly, let us choose ẽ1 1 = 0 , ẽ1 2 = 0, then, using the denition of f I of the constitutive relations of D I , the bilinear product becomes

P = ¢ b a -e 1 I e 2 2 + f 2 I e 1 I dz = ¢ b a e 1 I -e 2 2 + f 2 I dz (6.2.45)
The condition that P vanishes for any e 1 I hence implies that :

f 2 I = e 2 2 .
Fifth, let us choose e 1 I = 0, then, using the constitutive relations of D I , the previously established relations on f 2 1 , f 2 2 and f 2 I , the relation (6.2.36), the bilinear product becomes

P = ¡ b a e 2 1 de 1 2 + e 2 2 (-d * ) e 1 1 dz + ¡ b a e 1 1 de 2 2 + e 1 2 (-d * ) e 2 1 dz + (c + c) e 1 1 a,b T Σf 2 ∂ + e 2 ∂ T Σ e 1 2 a,b = -(c + c)e 1 1 e 2 2 b a -(c + c)e 2 1 e 1 2 b a (c + c) e 1 1 a,b T Σf 2 ∂ + e 2 ∂ T Σ e 1 2 a,b
The condition that P vanishes for any e 1 1 (a) and e 1 2 (b) hence implies the boundary port variables and

f 2 ∂ e 2 ∂ = 0 1 (c + c) 0 e 2 1 e 2 2 a,b
Comparing the constitutive relations of the Dirac structure D I dened in (6.3.12) with the augmented Hamiltonian system (6.2.31), one may easily see that it may be endowed with a port Hamiltonian structure. Corollary 9. The augmented Hamiltonian systems (6.2.31) with the conjugated ow variable f I = e 2 may be dened as a boundary port-Hamiltonian system with respect to the Dirac structure D I by :

⎛ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎝ ∂ t x f I f ∂ ⎞ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎝ δ xH (x) e I e ∂ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ ∈ D I ,
where the state vector x, dened in (6.2.29), is the vector of conserved quantities of the conservation laws augmented by the color functions of each spatial subdomain, the Hamiltonian H (x), dened in (6.2.28), is the sum of the Hamiltonian of each spatial subdomain, the pair of port variables (f I , e I ) are associated with the interface and the pair of port variables As a consequence of the port-Hamiltonian structure, the augmented Hamiltonian system (6.2.31) with the conjugated ow variable f I = e 2 , satises the following power balance equation:

d dt H(x) = e T ∂ Σ f ∂ + ¡ b a e T I f I dz (6.2.46)
Furthermore, if the Hamiltonians H + (x) and H + (x) are bounded from below,the augmented system has passivity properties. Indeed, although the Hamiltonian of the augmented system (6.2.31) is linear in the two color functions, they are invariants of the system hence, restricted to the invariant submanifold of invariance, it is indeed bounded from below.

Observe that, as we consider general color functions which are not Heaviside functions, the interface is now only dened by the continuity and discontinuity assumption on the two eort variables. The port variables (f I , e I ) are distributed variables on the complete domain (a, b) and hence are not the restriction at the point interface of the eort variables as for instance in (6.2.11). If the color functions are restricted to Heaviside functions then the power inow at the interface appearing in the power balance equation ( 6 involving the same variables as in (6.2.14).

Port Hamiltonian systems coupled through a moving interface

In this section we shall consider the problem of two systems of two conservation laws which are coupled through a moving interface. We shall denote by l (t) the time-varying position of the interface in the interval ]a, b[ and its velocity by l (t) = dl dt . In rst instance we shall show how the formulation as a port-Hamiltonian systems of corollary 9 may be extended to a moving interface. To this end we shall consider the velocity l (t) of displacement of the interface as an input. After formulating the conservation laws in the case of a moving interface, we show that they lead to a Port Hamiltonian system obtained by completing the system of corollary 9 with an input relation and a conjugated port variable to l (t). Secondly, we shall dene a class of models associated with interface relations. We conclude with the simple example of two gases in interaction through a piston.

Balance equations with moving interface

c l(t) (z, t) = 1 z ∈ [a, l (t) [ 0 z ∈ [l (t) , b] (6.3.1) and cl(t) (z, t) = 1 z ∈]l (t) , b] 0 z ∈ [a, l (t)] (6.3.2)
It may be noted immediately that these color functions, depending on the position of the interface l (t), also are the solutions of some conservation laws, namely the transport equations depending on the velocity l (t) of the interface

∂ t c(z, t) = -l (t) ∂ z c(z, t) and ∂ t c(z, t) = -l (t) ∂ z c(z, t) (6.3.3)
with initial conditions c(z, 0) = c l(0) (z, t) and c(z, 0) = cl(0) (z, t) (6.3.4) and associated boundary conditions.

The state variables, the ux variables and the energy function may be dened according to the denitions (6.2.16), (6.2.18) and (6.2.28). Now due to the moving interface, the balance equations on interval [a , b ] with a ≤ a < l (t) < b ≤ b containing the interface, will include an additional term, depending on the velocity l (t) of the interface.

Assuming again the continuity of the eort e 2 at the interface at l(t) (or equivalently of the ux variable N 1 ), the conservation law associated to the variable x 1 becomes d dt

¡ b a x 1 (z, t) dz = d dt ¡ l(t) a x 1 (z, t) dz + d dt ¡ b l(t) x 1 (z, t) dz = ¡ l(t) a ∂ t x 1 (z, t) dz + ¡ b l(t) ∂ t x 1 (z, t) dz + l (t) [x - 1 (l(t), t) -x + 1 (l(t), t)] = ¡ b a d 0 N 1 (x, c l , cl ) dz + l (t) [x - 1 (l(t), t) -x + 1 (l(t), t)] = -N 1 (a , t) + N 1 (b , t) + l (t) [x - 1 (l(t), t) -x + 1 (l(t), t)] = -e 2 (a , t) + e 2 (b , t) + l (t) [x - 1 (l(t), t) -x + 1 (l(t), t)] (6.3.5)
and its local formulation becomes

∂ t x 1 = d 0 N 1 (x, c l , cl ) + l (t) [c x 1 ∂ z c + c x 1 ∂ z c] (6.3.6)
Allowing again a discontinuity of the eort variable e 1 (or equivalently of the ux variable N 2 ), the conservation law associated to the variable x 2 becomes, in a similar way as above

d dt ¡ b a x 2 (z, t) = ¡ b a {-d * 0 N 2 (x, c l , cl ) -e I } dz + l (t) [x - 2 (l(t), t) -x + 2 (l(t), t)] = ¡ b a {(d 0 -[(∂ z c l ) -(∂ z cl )]) N 2 (x, c l , cl ) -e I } dz + l (t) [x - 2 (l(t), t) -x + 2 (l(t), t)] = -N - 2 (a , t) + N + 2 (b , t) + e I (l(t)) + l (t) [x - 2 (l(t), t) -x + 2 (l(t), t)] = -e 2 (a , t) + e 2 (b , t) -e I (l(t)) + l (t) [x - 2 (l(t), t) -x + 2 (l(t), t)] (6.3.7)
and its local formulation becomes

∂ t x 2 = -d * 0 N 2 (x, c l , cl ) -e I + l (t) [c x 2 ∂ z c + c x 2 ∂ z c] ( 6
∂ t ⎛ ⎜ ⎜ ⎝ x c c ⎞ ⎟ ⎟ ⎠ = J a ⎛ ⎜ ⎜ ⎝ δ x H (x, c, c) δ c H (x, c, c) δ cH (x, c, c) ⎞ ⎟ ⎟ ⎠ + Ie I + l (t) ⎛ ⎜ ⎜ ⎝ c x c x -1 0 0 -1 ⎞ ⎟ ⎟ ⎠ ∂ z c c (6.3.9)
with I dened in (6.2.32).

This denes an input map associated with the input l (t), velocity of the interface, as follows Let us now prove the co-isotropy condition D ⊥ M ⊂ D M . This amounts to proving that if ( f 2 , ê2 ) ∈ B satises ∀( f 1 , ê1 ) ∈ D M ; ( f 1 , ê1 ), ( f 2 , ê2 ) = 0 then ( f 2 , ê2 ) ∈ D M . Therefore let us compute the bilinear product (6.3.14), assuming that ( f 1 , ê1 ) ∈ D M . Remind that, from the denition of D M , the variables ẽ1 , e 1 I and l1 may be chosen freely.

G (x, c, c) = ⎛ ⎜ ⎜ ⎝ c x c x -1 0 0 -1 ⎞ ⎟ ⎟ ⎠ ∂ z c c ( 6
D M = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ f f I e l f ∂ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ẽ e I ˙l e ∂ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ∈ F × E / ⎛ ⎜ ⎜ ⎝ f f I -e l ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎝ J a I G(x, c, c) -I T 0 0 -G T (x, c, c) 0 0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ ẽ e I l ⎞ ⎟ ⎟ ⎠ and f ∂ e ∂ = 0 1 (c + c) 0 e 1 e 2
B = F × E with F = L 2 ((a, b) , R) 5 × R × R 9 and E = dom d * × dom d × L 2 ((a, b) , R) 3 × R × R 9 endowed with the pairing ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ f f I e l f ∂ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
In a rst instance, choose l1 = 0. Then the power product becomes The condition that P e vanishes for any e 1 1 hence implies that :

f 2 1 = de 2 2 + (cx 1 ∂ z c + cx 1 ∂ z c).
Thirdly, let us choose e 1 1 = 0, e 1 I = 0 and e 1 1 (a) = e 1 2 (b) = 0, then, using the preceding derived equalities and the denition of f 1 of the constitutive relations of D M and (6. The condition that P e vanishes for any e 1 2 hence implies that :

f 2 2 = -d * e 2 1 -e 2 I + (cx 2 ∂ z c + cx 2 ∂ z c) l2 .
Fourthly, let us choose ẽ1 1 = 0, ẽ1 2 = 0, then, using the denition of f I of the constitutive relations of D M , the bilinear product becomes The condition that P e vanishes for any e 1 I hence implies that:

f 2 I = e 2 2 .
Fifth, let us choose e 1 I = 0, then, using the constitutive relations of D I , the previously established relations on f 2 1 , f 2 2 and f 2 I , the relation (6.2.36) , the bilinear product becomes .

In a second instance, choose l1 = 0. Then, using the previously derived equalities, the power product becomes Note that with the choice of the two additional port variables associated with the color functions, the power balance on the eort and ow variables associated with the color functions do not contribute to the total power balance (corresponding, for physical systems, to the energy balance equation).

The port-Hamiltonian formulation of the system of two conservation laws with a moving interface with velocity l may be formulated as a port-Hamiltonian system. Corollary 11. The augmented Hamiltonian systems (6.3.9) with the conjugated interface ow variable f I = e 2 and conjugated variable e l to the interface velocity, dened in (6. where the state vector x, dened in (6.2.29), is the vector of conserved quantities of the conservation laws augmented by the color functions of each spatial subdomain, the Hamiltonian H (x), dened in (6.

Model of the interface's displacement

In the preceding section we have dened the dynamic model of a system of two conservation laws coupled by a moving interface with velocity l considered as an input variable and the interface relations (6.2.12) and (6.2.13). The port Hamiltonian model with the moving interface, admit as port variables, the port variables (f I , e I ) associated with the ux variables at the interface and the port variables l, e l associated with the displacement of the interface. In this section we shall discuss possible closure relations which could be imposed of these two pairs of port variables at the interface and illustrate it on a very simple example: two gases with a piston at the interface.

In a rst instance one should observe that the dynamics of displacement of the interface is necessarily nite-dimensional while the port variables (f I , e I ) are distributed. Coming back to the motivating example of a thin interface, that is located at some point l (t) which was the departure for the denition of the model in the section 6.

v
where p (v) = -δ v U (v) is the pressure. The interface relation (6.2.12) corresponds to the continuity of the eort variable e 2 = v at the interface, which is the usual hypothesis that there is no cavitation at the piston and that the velocities of the uids on both sides of the piston are equal to the velocity of the piston. And the interface relation (6.2.13) corresponds to the balance of forces exerted on the piston by the pressures e 1 = -p (v) of the gases from both gases and the external force f I .

The system of the two gases with a moving interface is then formulated by the corollary 11

with the color functions being the characteristic functions of each subdomain. As the piston is considered as a thin interface, we use the relation (6.3.29). In order to complete the interface relations we shall assume that the piston has no mass but is subject to friction with coecient ν and an linear elastic force with stiness k. In this case the dynamics of the piston is dened Finally this port Hamiltonian formulation might open the way to the analysis of the wellposedness of these systems (in the continuation of [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric dierential operators[END_REF][START_REF] Zwart | Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain[END_REF]) as well as their passivity-based control which will be the aim of future work.

Chapter 7

Conclusion 7.1 General conclusion

This thesis is focused on modelling, control, and mathematical analysis of 1D transport equations coupled on two complementary time-varying domains. Along the manuscript, we show how a rigorous description of the interface is a major factor for such class of systems. The lack of physical model to express clearly the interface structure and the coupling conditions can be considered as the main challenge in modelling point of view. These interfaces which appear in many problems as the extrusion process and the Stefan problems are sources of real diculties due to the discontinuity of the variables and the interface motion in many physical systems. This discontinuity is expressed in the extrusion modelling by a change of causality between the lling ratio in the Partially Filled Zone and the gradient of pressure which is the associated conjugate variable in the Fully Filled Zone. Many studies of mathematical interest are performed concerning the xed interface is the context of existence and regularity of solutions and well-balanced numerical computation as it is discussed along this thesis.

In Chapter 3 the modelling of an extrusion process has been studied. The proposed model is based on mass and energy balance of an homogeneous material which is heated and transported along the extruder. The rheological behavior of the transported material is related essentially to the viscosity which strongly depends on dynamic variables as the material temperature and moisture content. The distributed viscosity prevents an explicit formulation of interface relations and represents in that sense a very interesting problem for control and mathematical analysis.

In our study we consider a simplied model with constant viscosity. Then, the interface coupling conditions can be explicitly formulated by assuming the continuity of variables as pressure or momentum ux. By numerical computations based on the nite volume method, we simulate the evolution of pressure, die net ow, lling ratio and temperature along the extruder. The tracking of the interface is achieved using an extension of the variables in the whole coupled complementary time-varying domains by means of color functions as charac- In Chapter 5, we deal with the control of the extrusion process. We only consider the homogeneous mass balance equation as an independent system. One should point out that the system can be decoupled only if the viscosity is constant. Indeed, for the process the dynamics of moisture content which is continuous at the interface does not aect the design of the desired properties of the material at the die and the temperature also evolves independently. The evolution of the moisture is in this case determined by an input value (boundary condition) which is transported along the extruder with a delay depending on the velocity of the transport in the partially and Fully Filled Zone. The lling ratio transport equation and the interface dynamics are combined to obtain a delay system framework by solving the partial dierential equation using the method of characteristics. Assuming that the rotation speed of the screw is constant and dening the feed rate as an input variable we obtain a state-dependent input delay system and construct an predictor based controller to stabilize the interface around a given equilibrium. Consequently, we stabilize the net ow at the die. Comparing the interface coupling relations, we show that the admissible set of equilibrium for lling ratio and interface position is more large if we consider continuity of pressure. We conclude that the momentum ux continuity represents a restrictive coupling relation and we perform the control analysis by considering pressure continuity at the interface. The stabilization of the interface leads to the stability of the temperature as it is shown by simulations.

In Chapter 6 we consider coupled port-Hamiltonian systems through a mobile interface.

The motivation of the study in this context is to classify the interface relations for coupled port-Hamiltonian systems in complementary time varying domains. By augmenting the system with the colors functions, we dene a global port-Hamiltonian system on the total domain. The transport operator is then given by a dierential operator which is modulated by the distributed color functions. We show that the extended system is also a Dirac structure by adding two pairs of port-variables which are associated to interface. The rst pair corresponds to the discontinuity assumption on one of the ux variables (the other one being continuous) and the second pair being the velocity of interface and its conjugated variable. The velocity of the color functions transport equations corresponds in this case to the interface velocity.

It is important to note that the eorts and ux variables which are associated to the color functions dynamics do not appear in the power balance equation.

Future research

Coupled partial dierential equations through mobile interface is an emergent subject in control theory. In that perspective, this thesis stands as an exploration which suggests many open problems. In the context of extrusion process modelling, the problem which is related to distributed viscosity is a major interest for various applications. One should also mention that a more complex geometry of screw prole may lead to a succession of partially and fully lled zones which makes arise several mobile interfaces. Another challenge is to deal with reactive extrusion which consisted to a mixture of dierent materials which dierent densities. Wellposedness issues of such system for distributed viscosity can be explored in future works. The extension of the predictor based controller by introducing a time-varying screw speed is a real challenge, for the delay system control. One should mention that the extension of the controller to the temperature design should be performed to guarantee the stability of the whole system. By studying moving interface problem in port-Hamiltonian framework, we make explicit the pairs of conjugated variables needed to express the interface relations. This might be a powerful insight in the various suggested interface relations in the literature and towards a passivity-based denition and classication of these interface relations.

  vi comments help me to improve signicantly the nal version of the manuscript. Professor Jean-Michel Coron has encouraged my collaboration with Peipei Shang and Zhiqiang Wang and gave me the opportunity to learn advanced technics in mathematical analysis. Along this years, I had very interesting discussions with Professor Hans Zwart on Hamiltonian formulation of moving interface problems and his advices gave me large perspectives in this subject. I acknowledge the director of the Laboratoire d'Automatique et de Génie des Procédés (LAGEP), Professor Hatem Fessi for receiving me in his lab these last four years. I have only great thoughts towards the group Dynamique des procédés et commande à base de lois de conservation in which I had the pleasure to work in LAGEP. I would like to thank each and every person of the group, specially Professor Christian Jallut for interesting discussions in extrusion processes modelling. I am very thankful to the personnel of LAGEP, in particular Nadia Chapel, Jean Pierre Valour and Olivier Garrigues. I would like to thank the French Ministry of Higher Education and Research for the grant supporting my Ph.D. work and the French National Research Agency sponsored project HAMECMOPSYS (ANR-11-BS03-0002) for its partial nancial support. Also, I like to thank Weijun, Yongxin, Goma, Soane, Naveed, Mariem, Kun, Lamice, Hoang, Redha with whom I shared very good time. I am grateful to my friends Amadou Bah, Silcarneyni Gueye, Abdou Ndiaye and Samba Cor Thiandoum who encouraged me during these years. Finally, I would like to express my deepest appreciation to my parents ABdou Diagne and Fatoumata Sylla, my brother Papa Ibrahima Diagne and my nephew Souleymane Sy Savané for their love and constant support.

  Chapter 1

Figure 1 . 1 . 1 :

 111 Figure 1.1.1: Description schématique d'une extrudeuse

Figure 1 . 1 . 2 :

 112 Figure 1.1.2: Schéma des deux zones de convoyage et pompage

1. 1 . 1 . 1

 111 Bilans de matière et d'énergie dans les deux zones

1. 1 . 3 . 1

 131 Frontière en x = 0

  1.4.27) avec e T ∂ Σf ∂ = e ∂ (a)f ∂ (a)e ∂ (b)f ∂ (b), dénit une structure de Dirac.

Figure 3 . 1 . 1 :

 311 Figure 3.1.1: Schematic description of an extruder

Figure 3 .

 3 Figure 3.2.1: Bi-zone model of an extruder

Figure 3 . 3 . 1 :

 331 Figure 3.3.1: Cartesian representation of the eective volume V ef f

3 . 3 )

 33 Now extending the local formulation of the mass balance to a global one in the Partially Filled Zone we consider the transport phenomena for one screw element of pitch ξ. In this case, the inow is due to the convection of the extruded material from the upstream channel to the current channel and the outow is determined by the transport from the current channel to the downstream channel (Fig.3.3.2):

Figure 3 . 3 . 2 :

 332 Figure 3.3.2: Description of the ow in a twin-screw extruder[START_REF] Chapet | Techniques de l'ingénieur Plasturgie : procédés d'extrusion[END_REF]

4 . 5 )Figure 3 . 4 . 1 :

 45341 Figure 3.4.1: Poiseuille ow in a tube[START_REF] Choulak | Modélisation et commande d'un procédé d'extrusion reactive[END_REF]

1

 1 in the Partially Filled Zone (P F Z) f f = 1 in the Partially Filled Zone(FFZ) It thus becomes necessary to describe the coupling between the two zones by physical coupling relations ensuring consistency of the model. As stated in the introduction, these relations are established under some continuity assumptions of state or ux variables or some functions of state variables. In this section we discuss coupling relations based on the continuity of the state variables or momentum ux. The interconnection is performed under the assumption of constant viscosity η along the extruder. This hypothesis allows to solve explicitly the equation for the pressure gradient which is a function of the distributed viscosity (3.4.1) and(3.4.3)

7

 7 Open loop simulations of mass and energy balance equations 3.7.1 Description of the simulated extrusion process model In this part, we perform simulations for the mass and energy balance of the extrusion process model. We impose constant screw speed N 0 , feed rate F in , inlet temperature T in and barrel temperature T b . We point out that F in and T in are the boundary conditions associated to the lling ratio and the temperature transport equations. Screw speed N 0 = 325/60rpm.s -1 Feed rate

Figure 3 . 7 . 1 :

 371 Figure 3.7.1: Moving interface position l(t)

Figure 3 . 7 . 2 :

 372 Figure 3.7.2: Filling ratio evolution

  .7.1). Then, the coordinates x 2 and x 3 which initially belong to Fully Filled Zone passe into the Partially Filled Zone due to the evolution of the interface position. The discontinuity of the lling ratio appears clearly in the simulation results (Fig. 3.7.2) and the value of the lling ratio at position x 2 and x 3 at times when occurs the change zone is determined by the transport in the Partially Filled Zone velocity ξN 0 and the inlet lling ratio f p in .

Figure 3 .

 3 Figure 3.7.3: Temperature evolution

Figure 3 . 7 . 4 :

 374 Figure 3.7.4: Temperature and interface position

( 4 .

 4 1.1) where α p (N ) = ζN.

ληN 2

 2 (t) φf p (x, t).

  .1.19) The aim of this part is to analyze the well-posedness of the system which is dened by equations (4.1.1) (4.1.6) and (4.1.12) with associated initial and boundary data (4.1.15), (4.1.16) and 4.1.18) respectively. The interface relations are described by (4.1.19) and the momentum ux continuity equations (4.1.9), (4.1.10) and (4.1.11).

( 4 . 4 . 8 ) where ξ 2 (dξ 2

 44822 see Fig 4.4.1) represents the characteristic curve passing through (t, 1) (s) ds = α p N (s), ξ 2 (s), l(s), f p (s, 1) , 0 s t ξ 2 (t) = 1.

( 4 . 4 . 12 )

 4412 Here ξ represents the characteristic curve passing through (0, 0) (see Fig 4.4.1) ⎧ ⎨ ⎩ dξ ds = α p N (s), ξ(s), l(s), f p (s, 1) , 0 s t, ξ(0) = 0.

( 4 .

 4 4.13)

F

  (l(s),N (s),fp(s,1)) l(s)

Figure 4 . 4 . 1 :

 441 Figure 4.4.1: The characteristics ξ, ξ 1 , ξ 2 and ξ 3 .

  .

F

  (l(s),N (s),fp(s,1)) l(s) ds dσ.

( 4 . 4 . 21 )

 4421 We dene t as (see Fig 4.4.1) ξ 1 ( t) = 0.

Fig 4 . 4

 44 Fig 4.4.1)

( 4 . 4 . 23 )

 4423 We dene x as (see Fig 4.4.1).

  4.1), (4.4.30) and (4.4.31), we get

  (4.4.33) By (4.4.26), (4.4.28) and noting (4.4.1), we obtain (4.4.2).

Figure 4 . 4 . 2 :

 442 Figure 4.4.2: x ∈ [0, ξ(t 1 )].

Figure 4 . 4 . 3 :

 443 Figure 4.4.3: x ∈ [ξ(t 1 ), ξ(t 2 )].

Figure 4 . 4 . 4 :

 444 Figure 4.4.4: x ∈ [ξ(t 2 ), 1].

( 4 . 1 ] + 1

 411 4.35) Take (l(δ 1 ), f p (δ 1 , 1)) as new initial condition, noting(4.4.34) and (4.4.35), after repeating the process in the proof of Lemma 4.4.1, we could solve Cauchy problem (4.2.9) to (4.2.10) on [δ 1 , 2δ 1 ]. For xed T > 0, by at most [ T δ steps, we reduce the value of ε to ε 0 and apply Lemma 4.4.1 successively, then we extend local solution to large time interval, nally to [0, T ].

4. 4 . 3

 43 Solving the moisture equation (4.2.11) for PFZ zone and FFZ zone

( 4 .

 4 4.36) with t and x dened by (4.4.22) and (4.4.24).

F)FF

  (l(s),N (s),fp(s,1)) l(s) ds dσ =e ( (l(σ),N (σ),fp(σ,1)) l(σ) (l(s),N (s),fp(s,1)) l(s) ds dσ.

( 4 .

 4 4.42)

( 4 .

 4 4.45) Hence, from (4.4.38), (4.4.41) and (4.4.45), we get

( 4 . 4 . 54 )

 4454 For almost every x ∈ [ξ(t 2 ), 1], by denition (4.4.36) of M p , we have (seeFig 4.4.4)

( 4 . 4 . 56 )

 4456 It is easy to get|x 2 -x1 | C|t 2t 1 |.

( 4 . 4 . 62 )

 4462 When ξ(t) x 1, we dene the characteristic curve ξ3 (t) (seeFig 4.4.5)

Figure 4 . 4 . 5 :

 445 Figure 4.4.5: The characteristics ξ, ξ1 , ξ2 and ξ3 .

4. 4 . 4

 44 Solving the temperature equation (4.2.12) for PFZ zone and FFZ zone

Figure 5 . 4 . 1 :

 541 Figure 5.4.1: Static model with continuity of momentum ux at the interface

Figure 5 . 4 . 2 :

 542 Figure 5.4.2: Static model with continuity of pressure at the interface

Figure 5 . 5 . 1 :

 551 Figure 5.5.1: Moving interface stability

Figure 5 . 5 . 3 :

 553 Figure 5.5.3: Filling ratio evolution at dierent positions

(

  proportional to (T b -T p,f )). The temperature T p (x, t) describes the temperature prole in the Partially Filled Zone and T f (x, t) is related to the Fully Filled Zone. The moving interface stability results in the convergence of the velocity associated to the transport equation of the Fully Filled Zone which is a function of F d (t). The inuence of the interface stability on the the temperature behavior is shown in Figure (Fig. 5.6.1) and Figure 5.6.2).

Fig. 5 .

 5 6.1) and (Fig. 5.6.2) are performed assuming the continuity of temperature at the interface l(t):

Figure 5 . 6 . 1 :Figure 5 . 6 . 2 :

 561562 Figure 5.6.1: Temperature dynamics at dierent times

6. 1

 1 Introduction It has been shown that a large class of physical distributed parameter systems with boundary external variables, admit a port Hamiltonian formulation and they are called boundary

2 . 11 )

 211 they may immediately be interpreted as dening a Dirac structure on the boundary port variables at the interface. Then, by composition of Dirac structures, the two boundary port Hamiltonian systems can be composed to a single boundary port Hamiltonian system with boundary port variables being δ x -H -(a) and δ x + H + (b), according to (6.2.5)[START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF].

6. 2 . 3

 23 Augmenting the port Hamiltonian systems with color functions 6.2.3.1 Prolongation of the variables on the domain [a, b]

  2.21) corresponds to the local formulation of the conservation laws on arbitrary domain [a , b ] with either on an interval [a , b ] on the negative real line

2 . 25 )

 225 corresponds to the local formulation of the conservation laws on arbitrary interval [a , b ] on the negative real line (a ≤ a ≤ b < 0 ) or on the positive real line ( 0 < a < b ≤ b ). On these intervals the operator -d * 0 acts as the dierential operator -∂ * z according to the Hamiltonian system (6.2.3) (6.2.4). On an arbitrary interval [a , b ] containing the interface (with a ≤ a < 0 < b ≤ b), the balance equation on the variable x 2 is d dt

.2. 35 )

 35 Furthermore the two operators satises, for any two eort variables e 1 and e 2 which do not vanish at the boundary ¢ b a e 1 (d e 2 ) dz = ¢ b a e 2 (d * e 1 ) dz -[(c + c)e 1 e 2 ] b a (6.2.36) 6.2.3.4 Extension to a Boundary Port Hamiltonian system arising from a skewadjoint operator

x 1 x 2

 2 dened on a spatial domain [a, b] z with an interface at the point z = 0 which imposes the continuity of the eort variable e 2 and allows for the discontinuity of the eort variable e 1 is dened by

.2. 41 ) 1 I and e 2 I

 4112 Using the constitutive relations of the set D I , the power product becomes that the terms in the interface variables e

1 IFirstly, let us choose e 1 1 = 0, e 1 2 = 0, e 1 I = 0 and e 1 c = 0 .

 112110 let us compute the bilinear product (6.2.41), assuming that ( f 1 , ē1 ) ∈ D I . One computes from the denition of D I , the variables ẽ1 and e may be chosen freely. The the bilinear product reduces to: P = ¡ b a e 1 c f 2 c dz and the condition that it vanishes for any e 1 c , implies the relation: f 2 c = 0. By symmetry one obtains f 2 c = 0.

  (f ∂ , e ∂ ) are associated with the boundary of the spatial domain [a, b].

  .2.46) depends only on the values of the eort variables at the interface ¢ b a e T I f I dz = -e 1 0 -e 2 0 + + e 1 0 -e 2 0 +

  For a time-varying position l (t) of the interface the spatial domains of the two subsystems are the intervals [a, l(t)[ and ]l(t), b]. The two color functions, the characteristic functions of the domains, depend now on the position of the interface :

.3. 10 )

 10 One may dene then the conjugated output e l by e l = ¢ b a δ xH (x) T G (x, c, c) dz which may also be dened as the pairinge l = G T x, c, c , δ xH (x) = ¢ b a δ xH (x) T G (x, c, c) dz(6.3.11) This leads to a Dirac structure associated with systems of conservation laws with a moving interface as follows.Proposition 10. The set of relations D M associated with a system of two conservation laws dened with the variables x 1 x 2 on the spatial domain [a, b] z and an interface moving with velocity l which imposes the continuity of the eort variable e 2 and allows for the discontinuity of the eort variable e 1 is dened by

  T f I dz + e T ∂ Σf ∂e l l (6.3.13) with e T ∂ Σf ∂ = e ∂ (a)f ∂ (a)e ∂ (b)f ∂ (b), denes a Dirac structure.Proof. Let us rst show the isotropy condition D M ⊂ D ⊥ (6.3.13) by ê, f . Then the isotropy condition is written( f 1 , ê1 ), ( f 2 , ê2 ) = ê1 , f 2 + ê2 , f 1 = 0 ∀(( f 1 , ê1 ), ( f 2 , ê2 ) ⊂ D Mwith respect to the bilinear product associated with the pairing (6.3.13) or in an equivalent way ê, f = 0 ∀( f, ê) ∈ D M Let us hence compute the pairing T f I dz + e T ∂ Σf ∂e l l = ¡ b a ẽT (J a ẽ + I e I + G l) dz + ¡ b a e I T -I T ẽ dz + e ∂ T Σf ∂ the denition of the boundary port-variables f ∂ and e ∂ .

2 ∂ + e 2 ∂T Σ e 1 2G

 221 T (x, c, c) ẽ1 dz l2e 2 l l1(6.3.18)

( 1 1∂ z c e 1 c dz and the condition that it vanishes for any e 1 c

 11 ∂ z c) e 1 c + (∂ z c) e 1 c f 2 c dz (6.3.21) -¢ b a G T (x, c, c) ẽ1 dz l2 (6.3.22)Then, rstly, let us choose e = 0, e 1 2 = 0, e 1 I = 0 and e 1 c = 0. Then, the bilinear product reduces to: , implies the relation:

f 2 c 1 2 = 0 , e 1 I = 0 and e 1 1 (cx 1 ∂f 2 1 -

 211111 + l2 ∂ z c = 0. By symmetry one obtains f 2 c + l2 ∂ z c = 0.Secondly, let us choose e (a) = e 1 2 (b) = 0, then, using the preceding equalities, the denition of f 2 of the constitutive relations of D M and (6.2.36) with zero boundary conditions, the bilinear product becomes z c + cx 1 ∂ z c) (cx 1 ∂ z c + cx 1 ∂ z c) dz(6.3.23)

(cx 2 ∂e 2 I

 22 2.36) with zero boundary conditions, the bilinear product becomes z c + cx 2 ∂ z c) -(cx 2 ∂ z c + cx 2 ∂ z c) l2 dz(6.3.24)

e 1 2 ( 2 ∂ + e 2 ∂T Σ e 1 2 2 ∂ + e 2 ∂T Σ e 1 2

 2221221 -d * ) e 2 1 dz + (c + c) e 1 1 a,b T Σf a,b = -(c + c)e 1 a,bThe condition that P e vanishes for any e 1 1 (a) and e 1 2 (b) hence implies

  The condition that P e vanishes for any l1 hence implies e 2 l = ¡ b a G T ẽ2 dz .

  3.11), may be dened as a boundary port-Hamiltonian system with respect to the Dirac structure D M by

  2.28), is the sum of the Hamiltonian of each spatial subdomain, the pair of port variables (f I , e I ) are associated with the ux variables at the interface, the pair of port variables (l, e l ) are associated with the displacement of the interface and the pair of port variables (f ∂ , e ∂ ) are associated with the boundary of the spatial domain [a, b]. The pairs of port variables (f c ∂ , e c ∂ ) balance the power product associated with the transport equations of the color functions. Computing the balance equation for the Hamiltonian we nd d dt H(x) = e ∂ Σ f ∂ + ¡ b a e T I f I dz + l e l (6.3.28) It may be observed that, when restricting to the particular solution of the color functions (6.3.2) obtained with initial conditions (6.3.4), one can relate these port variables to interfacial eort and ux variables in a similar way as for a xed interface. In this case the balance equation of the Hamiltonian is given by dH dt = e ∂ Σ f ∂ +e 1 (l -) e 2 (l + )e 1 (l -) e 2 (l + ) -l e l with output conjugated to the velocity of the interface being the discontinuity of energy density at the interface e l = -H -(l) + H + (l)

3 . 1 , 2 2.

 312 the port variables (f I , e I ) may be related to a nite-dimensional pair of variables (φ I , I ) ∈ R 2 with the following adjoint relationsφ I e I = ¡ b a δ (zl) f I dz I δ (zl) (6.3.29)which preserves the power product :φ I I = ¡ b a e I f I dz .It should be noted that one could also dene a thick interface by choosing another kernel than δ (zl), with positive values and nite support.In a second instance, one has complete the interface relation with the dynamics of the position of the interface l (t) for instance in terms of a port Hamiltonian system with state variables including l (t) and the port variables (φ I , I ) and l, e l . In this case by interconnection of port Hamiltonian systems through a Dirac structure one may conclude that the complete system is again port Hamiltonian and use its properties for the proof of well-posedness and passivity-based control design.Example 12. Let us conclude this paragraph with the example of two isentropic gases (modelled by a systems of two boundary port Hamiltonian systems[START_REF] Van Der Schaft | Hamiltonian formulation of distributed parameter systems with boundary energy ow[END_REF]) coupled at their interface by some piston in motion. The port Hamiltonian model of the gases is given precisely by the proposition 7 with state variables being the specic volume x 1 (t, z) = v (t, z) , the velocity x 2 (t, z) = v (t, z) , Hamiltonian is the sum of the internal energy density U (v) and the kinetic energy density : H(v, v) = U (v) + v The variational derivative of the Hamiltonian is then

  as a simple integrator dl dt = φ I = v and the conjugated eort variable is the sum of all forces applying on the piston i = -k lν φ I It may be interpreted as a nite-dimensional port Hamiltonian system with state variable l , structure matrix being zero, Hamiltonian function 1 2 k l 2 port-variables (φ I , I ) and dissipative term. Finally the model has to be coupled with the pair of port variables l, e l . One relation is triviall = φ I = vThe second one is less trivial and involves the eort variable e l which is, when the color func- tions are the characteristic functions of both subdomains, the dierence of the Hamiltonian density function at the interface: e l = (-H -(l) + H + (l)) . The most simple way of dening some relation is impose the continuity of the Hamiltonian density (which plays then the role of a privileged variable) which indeed complete the boundary conditions at the interface e l = 0As a consequence using the total energy of the conservation laws and the interface modelH tot (v, v, l) = ¡ b a U (v) + v 2 2dz + 1 2 k l 2 one obtains the power balance equationdH tot dt = -ν v 2v -(a)p -(v) (a) + v + (b)p + (v) (b)6.4 ConclusionIn this chapter we have suggested port Hamiltonian formulation of a system of two conservation laws (on a 1-dimensional spatial domain) coupled by a moving interface. We have rstly augmented the system of conservation laws with two transport equations of the characteristic functions of the subdomains dened by the interface. Then we have derived a the port Hamiltonian formulation of this augmented system with, in addition to the boundary port variables at the boundary of the total domain two pairs of port-variables associated with the interface. The rst pair corresponds to a particular choice of interface relation corresponding to a continuity and a balance equation on the ux variables at the interface and the second pair is dened by the velocity of interface and its conjugated variable. Finally we have illustrated this model with the example of two gases coupled by a moving piston. This is the rst step towards considering the coupling through an interface of Hamiltonian systems composed of an arbitrary number of conservation laws. However the most interesting feature of this formulation is that it makes explicit the pairs of conjugated variables needed to express the interface relations when derived from a port Hamiltonian formulation. This might be a powerful insight in the various suggested interface relations in the literature and towards a passivity-based denition and classication of these interface relations.

126 7 .

 7 CONCLUSION teristic functions of the two moving intervals.In Chapter 4 a mathematical analysis of the extrusion process model has been performed.It has been shown that the Cauchy problem admits a unique solution if initial and boundary data are closed to the equilibrium. Our strategy is based on the transformation of moving domains in xed one using changes of coordinates. Contraction mapping principle (Banach xed point argument) allows to dene the conditions such that the uniqueness and existence of solution are guaranteed. We rst prove the existence of solution for small time and extend ours analysis to semi-global time by iterations.

  Dénition 1. . Soit T > 0, une solution faible du problème de Cauchy (1.2.1)-(1.2.4) est un vecteur

	1.2.2 Principaux résultats
	Nous rappelons [40, Section 2.1], la dénition de la solution faible du problème de Cauchy:
	(??)-(1.2.4).

  1,∞ +|l 0 -l e | ε,

	(1.2.20)
	le problème de Cauchy (1.2.1)-(1.2.4) admet une solution unique

  1. RÉSUMÉ EN FRANÇAIS s étant la constante d'intégration. Soit N (t) la fonction primitive de N (t), nous en dé-

duisons :

  1.4.2 Interconnexion de deux systèmes Hamiltoniens à ports à travers une interface xeNous considérons des systèmes Hamiltoniens à ports de deux lois de conservations dénis sur les domaines spatiaux [a , 0[ et ]0 , b] dénotés par les symboles {-; +} :

  Continuity of pressure at the interface The assumption of the continuity

	(3.5.12)
	t)
	3.5.2.2 Coupling relations for total mass balance equation
	The coupling relation for the homogeneous mixture mass balance are established consid-
	ering two alternatives.
	3.5.2.2.1

of pressure at the interface is a state coupling relation. Knowing that the pressure in the Partially Filled Zone is equal to atmospheric pressure P 0 we nd the equality: P (l + , t) = P 0 .

By integrating the equation of the pressure gradient (3.4.1), we express the die pressure P (L, t)

or the net ow rate F d (t) as a function which is strictly determined by the screw speed N (t)

  .6) From (4.1.18) and (4.1.19), we rewrite the interface relations as the boundary conditions associated to the coupled equations in xed domains:

  a local solution to Cauchy problem (4.2.9)-(4.2.10). Next, we prove the solution we constructed is indeed the unique weak solution to Cauchy problem (4.2.9)-(4.2.10). Finally, we derive the estimates (4.4.2)-(4.4.3) of the local solution.

.

4.3)

where C ε 1 is a constant depending on ε 1 , but independent of ε.

Proof of Lemma 4.4.1: We rst apply contraction mapping principle to prove the existence and uniqueness of l(t) and f p (t, 1) for t small. With the existence of l(t) and f p (t, 1), we then construct

  Lemma 4.4.1, we takeε 2 ε 1 such that C ε 1 • ε 2 ε 1 . Then for any ε ∈ (0, ε 2 ]and any initial-boundary data such that (4.4.1) holds, noting(4.4.19), Cauchy problem (4.2.9)-(4.2.10) admits a solution on [0, δ 1 ]. Furthermore,

	4.4.2 Semi-global solution to Cauchy problem (4.2.9)-(4.2.10)
	By (4.4.2) and (4.4.3) in

  .4.46) Here and hereafter, we denote by C dierent constants which do not depend on x, t 1 , t 2 and n. We denote by C n various constants which do not depend on x, t 1 and t 2 , but may depend on n.

	(4.4.47)
	where t2 is dened by (4.4.40) and x1 are dened by
	x1 = xe

For almost every x ∈ [ξ(t 1 ), ξ(t 2 )], by denition (4.4.36) of M p , we have (see Fig 4.4.3)

|M p (t 2 , x) -M p (t 1 , x)| = |M in ( t2 ) -M 0 p (x 1 )|,

  + C n |t 2t 1 | 2 .

	Noting (4.4.57), we have
	¢ 1 ξ(t 2 )	|M p (t 2 , x) -M p (t 1 , x)| 2 dx C M 0n p -M 0 p	2 L 2 (0,1) (4.4.59)
	Combining (4.4.46), (4.4.54) and (4.4.59), we get
	¢ 1
	0
			.4.58)

  .4.60) Therefore by Lebesque dominated convergence theorem, letting n large enough and then |t 2 -t 1 | small enough, the right hand side of (4.4.60) can be arbitrarily small. Thus, we get(4.4.37). This proves that the function M p dened by (4.4.36) belongs to C 0 ([0, T ]; L 2 (0, 1)).With the existence of the solution M p (t,x) for moisture equation for PFZ zone, we use the interface relation in(4.2.11) to solve the FFZ zone.For any s ∈ [0, T ], we dene the characteristic curve ξ(s) (seeFig 4.4.5)

  By the uniqueness of (l(t), f p (t, 1)) which determines the characteristic curves ξ 1 , ξ 3 , ξ1 and ξ3 , we conclude the uniqueness of M p and M f .

							.4.65)
	By (4.4.26) and (4.4.28), we have
	∂x ∂ x	< 1,	∂x ∂ t	<	ζ(N e + ε 0 ) l e -ε 0	.	(4.4.66)
	Combining (4.4.65) and (4.4.66), we obtain (4.3.11). Similarly, we can obtain (4.3.13).
		t
		t	x 1 s	x s
			x 3 s
					x 2 s
		0	x		1

  ξ1 (s)) ds, else,(4.4.68) where t and x are dened by(4.4.62) and(4.4.63).Similarly as we did for M p and M f , one can easily conclude that (4.4.67) and (4.4.68) give the unique solution of (4.2.12), together with the estimates (4.3.12) and (4.3.14).

  f 1 of the constitutive relations of D I and (6.2.36) with zero boundary conditions, the bilinear product becomes

	P = =	¢ b a ¢ b a	e 2 1 de 1 2 + e 1 2 f 2 2 + e 1 2 e 2 I dz e 1 2 d * e 2 1 + f 2 2 + e 2 I dz	(6.2.44)

  the ow variable f = (f 1 , f 2 , f c , f c) T and the eort variable ẽ = (e 1 , e 2 , e c , e c) T associated with the extended state (6.2.29), the dierential operator J a dened in (6.2.33), the operators d , resp. d * dened in (6.2.34), resp. (6.2.35), the column vector I dened in (6.2.32), the input map G dened in (6.3.10) and its adjoint G T in (6.3.11) and bond space

		(6.3.12)
		⎫ ⎬
	a,b	⎭

with

RÉSUMÉ EN FRANÇAIS

1.1. MODÉLISATION D'UN PROCÉDÉ D'EXTRUSION POUR LA COMMANDE

1.3. STABILISATION DE L'INTERFACE PAR UNE APPROCHE DE COMMANDE PRÉDICTIVE

EXTRUSION PROCESSES MODELLING FOR CONTROL

WELL-POSEDNESS OF THE EXTRUSION PROCESS MODEL

CONTROL OF MASS TRANSPORT AS AN INPUT DELAY SYSTEM

(0 -, t) and N + 2 (0 -, t) are eliminated according to the balance equation (6.2.13).

Acknowledgements I would like to express my deepest appreciation to all those who provided me the possibility to complete this report. I am grateful to my thesis directors Professor Bernhard Maschke and Doctor Françoise Couenne. Without their encouragement and guidance this thesis would not have materialized. I would like to thank my committee members, Professor Jean-Michel Coron, Doctor Michael Di Loreto, Professor Didier Georges, Professor Yann Le Gorrec, Professor Nicolas Petit and Professor Hans Zwart for their detailed revision of the thesis manuscript. Their 4.7 Acknowledgements This work is done in collaboration with: Peipei SHANG, Department of Mathematics, Tongji University, Shanghai 200092, China, Zhiqiang WANG, School of Mathematical Sciences, Fudan University, Shanghai 200433, China. The authors would like to thank Professor Jean-Michel Coron (Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie 4, place Jussieu 75005 Paris France) for his helpful comments and constant support.

8. NOMENCLATURE Bibliography

X(t + D) = e AD X(t) -¢ t t-D

e A(t-τ ) BU (τ )dτ ∀ t > t 0 (5.3.4) It follows that the corresponding feedback law is given by the relation 5.3.4:

e A(t-τ ) BU (τ ) dτ ∀ t > t 0 (5.3.5) Knowing the input along the delay interval [t -D; t], it becomes easy to compute the integral term ¡ t t-D e A(t-τ ) BU (τ )dτ and to design this innite dimensional feedback law (5.3.5).

The numerical approximation of the integral term should be done carefully to guarantee the stability when the controller is implemented [START_REF] Mondié | Finite spectrum assignment of unstable time-delay systems with a safe implementation[END_REF].

So the feedback law (5.3.5) for the system (5.3.1) determines the so called FSA method since the spectrum of the closed loop system coincides with the spectrum of the matrix A + BK.

Considering the predicted state determined by equation (5.3.4):

One easily deduces the predictor that obeys to an free-delay system with its associated feedback as:

This framework corresponds to the reduction model approach.

The predictor (5.3.6) can be extended to nonlinear systems. We present here the result as proposed in [START_REF] Krstic | On compensating long actuator delays in nonlinear control[END_REF][START_REF] Krstic | Delay-Adaptive Full-State Predictor Feedback[END_REF]. We consider systems of the form:

In [START_REF] Krstic | Delay-Adaptive Full-State Predictor Feedback[END_REF] the author dened the predictor for nonlinear forward-complete systems, namely, systems such that have bounded solutions and a suitable continuous gain function for any bounded input function U (t).

The construction of the feedback predictor-based control law for nonlinear system with constant input delay follows the same method as previously. The controlled free-delay nonlinear system (5.3.9) is supposed to be globally asymptotically stable by a feedback law given by equation (5.3.10):

(5.3.9) U (t) = κ(X(t)) (5.3.10) This condition means that at the initial time the predictor ts perfectly with the initial state argument.

The fact that P(t) as given in (5.3.23) is the σ(t)-t = D(P(t)) time units ahead predictor of X(t) can be seen performing the following coordinates change:

(5.3.26) Equation (5.3.25) is rewritten as:

Applying φ -1 to the equation above one obtains:

Then for t > t 0 , the following equality holds:

Considering the change of coordinates (5.3.26), it is obvious that σ(t) = t + D(X(σ(t))) and dierentiating this equality it follows that:

(

It is now clear that the gradient-of-delay term in the denominator of (5.3.23) is the result of a change in the time variable, which allows the predictor to be dened using an integral from the known delayed time φ(t) = t -D(X(t)) until present time , rather than an integral from the present time until the unknown prediction time σ(t) = t + D(P(t)) as it is mentioned previously.

The denominator of the predictor equation (5.3.23) should be positive and this condition stands as the feasibility condition:

Because of this feasibility condition the stabilizing results are not global in general.

In [START_REF] Bekiaris-Liberis | Compensation of time-varying input delay for nonlinear systems[END_REF][START_REF] Bekiaris-Liberis | Compensation of state-dependent input delay for nonlinear systems[END_REF], the local stability of the nonlinear state dependent-input delay system (5.3.19) is guaranteed under assumptions of global stabilizability and forward completeness of the free-delay system Ẋ = f (X, ω). Recall that the forward completeness assumption means that for every initial condition and every locally bounded input signal, the solution of the system Ẋ = f (X, ω) is dened for all t ≥ t 0 . Mathematically, this means that there exists a smooth positive dene function R(X) and K ∞ functions α 2 ,α 3 and α 4 such that:

Chapter 8

Nomenclature

Mass ow feed rate

Mass ow rate at the die f p (-) Barrel temperature α (Jm -2 s -1 K -1 ) Heat exchange coecient μ(J kg -1 K -1 )

Viscous heat generation parameter