Prof Lionel Thibault

Prof Dan Goreac

Prof Alain Rapaport

Prof Alberto Seeger

Prof. Mircea Prof Sofonea

Briec Walter

Asma Mansouri

Sara Dridi

Ahlem Benraouda

Hadjer Hechaichi

Amira Amel

Rayan Djazia

Fatma

Warda

Aicha Zahira

Wissam

Their insight, feedback, and comments were influential to improve the quality of the dis-

Quelques aspects sur les processus de rafle maine de la non convexité. Dans le cadre mathématique, le problème a connu plein de développements numériques de simulation; particulièrement la simulation de la dynamique de corps rigides en interaction. La théorie de contrôle optimal a aussi une partie significative des revues. Comme nous avons mentionné auparavant, le manuscrit est consacré à l'étude de l'existence de solutions pour des processus de rafle. Nous utiliserons des outils d'analyse non lisses divers et des différents concepts et notions de solutions. Autrement dit, des différents types des procédures de sélection des trajectoires d'inclusions différentielles.

La première partie de notre étude concerne la notion de tube-solutions qui a été présentée et étudiée à fond par M. Frigon (voir, [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF][START_REF] Frigon | Boundary and periodic value problems for systems of nonlinear second order differential equations[END_REF][START_REF] Frigon | Théorèmes d'existence de solutions d'inclusions différentielles[END_REF][START_REF] Frigon | Existence Results for Some Initial and Boundary Value Problems Without Growth Restriction[END_REF][START_REF] Frigon | Nonlinear first order initial and periodic problems in Banach spaces[END_REF]). Cette notion est prouvée être un outil important dans l'étude de beaucoup de classes d'inclusions différentielles. Nous soulignons que ce type de solutions étend des conditions d'Hartman [START_REF] Hartman | On boundary value problems for systems of ordinary, nonlinear, second order differential equations[END_REF] et la notion célèbre de sous et sur-solutions (voir, [START_REF] Dragoni | Il problema dei valori ai limiti studiato in grande per gli integrali di una equazione differenziale del secondo ordine[END_REF][START_REF] Perron | Ein neuer Existenzbeweis für die Integrale der Differentialgleichung y = f (x, y)[END_REF][START_REF] Pouso | Upper and lower solutions for first order discontinuous BVPs[END_REF]). Ce dernier est un outil crucial pour examiner des résultats d'existence pour les équations différentielles ordinaires. L'avantage de cette méthode se trouve dans le fait qu'elle assure qu'une solution du problème considéré se trouve dans un tube. En somme, nous avons des informations sur la présence et l'emplacement des solutions. Cette notion a trouvé beaucoup d'applications comme, par exemple, pour obtenir des résultats de multiplicité.

La deuxième partie de notre étude est consacré aux solutions monotones. Le problème de trouver de solutions monotones résulte de questions de viabilité, où nous choisissons les trajectoires qui sont viables dans le sens qu'elles satisfont toujours des contraintes données. Aubin [START_REF] Aubin | Monotone evolution of resource allocations[END_REF] a étudié ce problème en adaptant des méthodes présentées dans Aubin, Cellina et Nohel [START_REF] Aubin | Monotone trajectories of multivalued dynamical systems[END_REF] et l'a, ensuite, examiné à fond dans [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Aubin | Viability theory[END_REF][START_REF] Clarke | Monotone invariant solutions to differential inclusions[END_REF] et les références là-dedans. On considère la monotonie comme une procédure de sélection qui choisit parmi toutes les trajectoires de l'inclusion différentielle, celles qui améliorent l'état du système comme le temps s'écoule. Les motivations et les applications possibles de ce concept sont dans: la théorie des jeux, l'économie et le contrôle optimal où il est possible de choisir des trajectoires en minimisant ou maximisant une fonction définie sur un ensemble de trajectoires.

Processus de rafle

Dans cette thèse, nous nous intéressons à une classe d'inclusions différentielles appelée le processus de rafle. Soient H un espace de Hilbert et C(t) une application multivoque définie sur l'intervalle J de R qui contient son origine t 0 et dont les xvi Quelques aspects sur les inégalités variationnelles différentielles valeurs sont convexes fermées non vides. Un processus de rafle est le problème d'évolution défini par:

(1.1) -x (t) ∈ ∂I C(t) (x(t)), dans J, x(t 0 ) = x 0 , où x 0 ∈ C(t 0 ) est une condition initiale et ∂I C(t) (u) est le sous-différentiel de la fonction convexe u → I C(t) (u) l'indicatrice de l'ensemble C(t) (i.e. I C(t) (u) = 0 si u ∈ C(t) et +∞ sinon).

Ce problème abstrait a été introduit par Jean Jaques Moreau au Séminaire de l'Analyse Convexe de l'Université de Montpellier dans les années soixante-dix [START_REF] Moreau | Evolution problem associated with a moving convex set in a Hilbert space[END_REF][START_REF] Moreau | Liaisons unilatérales sans frottement et chocs inélastiques[END_REF] en vue de l'analyse des systèmes plastiques. Il considérait le problème d'évolution suivant: C(t) est un ensemble de déplacement et x(t) un point dans cet ensemble (x(t) ∈ C(t) puisque ∂I C(t) (•) est vide en tout point hors C(t)). Lorsque x(t) est un point intérieur, il reste au repos jusqu'à ce qu'il soit attrapé par le bord de C(t) qui se déplace, il part dans la direction normale à celui-ci, comme poussé par la frontière, afin de rester dans C(t). Le problème (1.1) s'écrit aussi -x (t) ∈ N C(t) (x(t)).

Rappelons que l'ensemble ∂I C(t) (u) est égale au cône des vecteurs normaux à C(t), dans le sens sortant, au point u (si u est un point intérieur de C(t), l'ensemble est réduit à {0}; vide si et seulement si u ∈ C(t)). Quelques aspects sur les processus de rafle où x(t) est le courant par l'indicateur/résistance, i(t) est une source de courant et ν(t) est la tension à travers la diode D.

Aperçu de la thèse

Dans cette thèse, en employant des outils de l'analyse non lisse, multivoque et variationnelle, nous avons étudié certaines inclusions différentielles, particulièrement, nous nous sommes concentrés sur les processus de rafle. En somme, une inclusion différentielle impliquant les cônes normaux qui modèlent l'évolution de trajectoires contraintes par des ensembles convexes et variables dans le temps (voir la section précédente). Ces nouveaux résultats sont présentés dans les deux derniers chapitres. Le premier résultat en constitue le quatrième chapitre:

Chapitre 4: Existence de tube-solutions pour des processus de rafle non convexes.

Nous établissons quelques résultats d'existence de tube-solutions de processus de rafle perturbés avec des ensembles prox-réguliers. Notre approche emploie des propriétés appartenant à la classe d'ensembles prox-réguliers ainsi que des techniques de théorèmes de point fixes.

Soient un réel T > 0 et deux applications multivoques C : [0, T ] ⇒ R n et F : [0, T ] × R n ⇒ R n à valeurs fermées non vides. Supposons que tous les ensembles C(t) soient ρ-prox-réguliers pour une même constante ρ ≥ 0. Dans ce chapitre, nous considérons l'inclusion différentielle (1.2)    x (t) ∈ -N C(t) (x(t)) + F (t, x(t)), p.p. t ∈ [0, T ], x(0) = x 0 ∈ C(0), x(t) ∈ C(t), ∀t ∈ [0, T ]; où N C(t) (x(t)) désigne le cône normal de Clarke et x 0 ∈ C(0).

Ensemble prox-régulier

Les ensembles uniformément prox-régulier sont initialement définis par Federer [START_REF] Malinvaud | Leçons de théorie microéconomique[END_REF] dans R n sous le nom "positively reached sets". Puis, le concept de prox-régularité a été considéré par plusieurs auteurs sous différents noms. Finalement, Poliquin, Rockafellar et Thibault ont introduit la dénomination de ce concept pour les ensembles aux espaces d'Hilbert. Cette classe d'ensembles est plus générale et comprend xviii Quelques aspects sur les inégalités variationnelles différentielles les ensembles convexes. Il se trouve que cette classe partage avec cela beaucoup de propriétés excellentes concernant les applications dans l'optimisation, la théorie du contrôle, etc.

On dit qu'un sous-ensemble C de R n est prox-régulier en x ∈ C pour v ∈ N C (x), si C est localement fermé à x et il existe ε > 0 et ρ > 0 tels que pour tous x ∈ C et v ∈ N C (x) avec x -x < ε et v -v < ε, alors x est l'unique point de {x ∈ C | x -x < ε} la plus proche à x + ρ -1 v.

On dit qu'un vecteur v de R n est un vecteur normal proximal de C à x ∈ C si et seulement s'il existe deux constantes ρ > 0 et ζ > 0 tel que

v, x -x ≤ ρ 2 x -x 2 , pour tout x ∈ C ∩ B(x, ζ).
Le cône de tous les vecteurs normaux proximaux de C en x est appelée le cône proximal normal de C en x et noté par N P C (x). Le cône proximal est un opérateur important dans la suite. Il décrit l'ensemble des bonnes directions selon lesquelles on peut projeter (dans le sens suivant): Proposition 1.3.1. Pour tout x ∈ C, on a:

N P C (x) = {v ∈ R n : ∃ρ > 0 pour tout x ∈ proj C (x + ρv)},
où proj C est la projection sur l'ensemble C.

la projection proj C est continue en tout point à distance inférieure strictement à ρ. D'un point de vue géométrique: il est possible de rouler une boule de rayon ρ continûment sur toute la frontière de C. Cette représentation géométrique conduit au concept des ensembles (uniformément) ρ-prox-régulier.

Généralement, nous avons N P C (x) ⊂ N C (x), où N C (x) est le cône normal au sens de l'Analyse Convexe. Cependant, pour un ensemble ρ-prox-régulier C on a N P C (x) = N C (x).

Définition 1.3.1. Pour ρ > 0, l'ensemble C est dit ρ-prox-régulier si pour tout

x ∈ C et v ∈ N C (x) avec v < 1, v, x -x ≤ ρ 2
x -x 2 pour tout x ∈ C, cette condition signifie que x est l'unique point de C le plus proche à x + ρ -1 v.

Une propriété principale des ensembles prox-réguliers est décrite par l'hypomonotonie du cône proximal normal.

xix Quelques aspects sur les processus de rafle Proposition 1.3.2. Si C est un ensemble ρ-prox-régulier, l'opérateur multivoque N C (•) est hypomonotone: pour tout x 1 , x 2 ∈ C, u 1 ∈ N C (x 1 ) et u 2 ∈ N C (x 2 )

x 1 -x 2 , u 1 -u 2 ≥ -ρ x 1 -x 2 2 .
Pour plus d'informations voir [START_REF] Poliquin | Prox-regular functions in variational analysis[END_REF][START_REF] Poliquin | Local differentiability of distance functions[END_REF].

Résultats principaux

Considérons l'inclusion différentielle (1.2). Parmi les travaux faites auparavant dans le cadre non convexe, Colombo et Monteiro Marques [START_REF] Colombo | Sweeping by continuous proxregular set[END_REF] ont prouvé l'existence locale et globale ainsi que l'unicité des solutions d'un système sans perturbation, quand les ensembles C(t) sont prox-réguliers contenus dans l'intérieur d'un sousensemble approprié se déplaçant continûment. Thibault [START_REF] Thibault | Sweeping process with regular and nonregular sets[END_REF] a montré l'existence d'une solution de (1.2) quand les ensembles C(t) sont non convexe et variants de manière absolument continue et F est une application multivoque scalairement semicontinue supérieure. Nombre d'autres travaux peuvent être trouvés dans plusieurs références, voir [START_REF] Bounkhel | Nonconvex sweeping process and prox-regularity in Hilbert space[END_REF][START_REF] Castaing | Evolution equations governed by the sweeping process[END_REF][START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF]. Une solution classique du problème (1.2) est une application absolument continue x(•) sur [0, T ] telle que x(0) = x 0 , x(t) ∈ C(t) pour tout t ∈ [0, T ] et telle que l'inclusion donnée dans (1.2) soit vérifiée pour presque tout t ∈ [0, T ], i.e., une application x ∈ A(x 0 ) telle que

A(x 0 ) = {x ∈ W 1,p ([0, T ], R n ), tel que x(0) = x 0 }, où W 1,p ([0, T ], R n ) = {x ∈ C[0, T ] : tel que x est absolument continue et x ∈ L p ([0, T ], R n )}.
Dans notre cas, cette solution se trouve dans un tube appelé le tube-solutions. Pour cela, nous introduisons la notion de L 2 -tube-solution pour (1.2). Cette notion a été introduite par Frigon [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF] dans le cas général avec des termes maximaux monotones. Tout d'abord, supposons le suivant:

(P) Soit ρ ≥ 0, C : [0, T ] ⇒ R n est une application multivoque continue à valeurs ρ-prox-réguliers. 4. x(0) -α(0) ≤ β(0).

Définition 1.3.2. Soient α ∈ W 1,2 ([0, T ], R n ) et β ∈ W
On note par

B β (α) = {x ∈ C([0, T ], R n ) : x(t) -α(t) ≤ β(t), ∀ t ∈ [0, T ]}.
Les hypothèses suivantes vont intervenir dans les prochains théorèmes;

(H) F : [0, T ] × R n ⇒ R n est une application multivoque à valeurs compactes convexes tels que (S -L 2 ) Il existe un couple (α,

(i) t → F (t, x) est mesurable pour tout x ∈ R n , (ii)
β) ∈ W 1,2 ([0, T ], R n ) × W 1,2 ([0, T ], [0, ∞)) un L 2 - tube-solution de (1.2).
Par la suite, on donne les deux résultats principaux d'existence de cette partie.

Théorème 1.3.3. Supposons que les hypothèses (P), (H), (H k ), et (S -L 2 ) soient satisfaites. Alors, le problème (1.2) a une solution dans W 1,2 ([0, T ], R n ) ∩ B β (α).

xxi Quelques aspects sur les processus de rafle Théorème 1.3.4. Supposons que les hypothèses (P), (H), (H k ), et (S -L 2 ) soient satisfaites. Alors, le problème (1.2) a une solution dans W 1,2 ([0, T ], R n ) ∩ B β (α).

Afin de prouver l'existence d'une solution du problème (1.2), nous allons modifier la fonction F en utilisant le L 2 -tube-solution. Pour ces opérateurs modifiés nous allons associer deux problèmes et pour lesquels nous allons prouver l'existence d'une solution. Finalement, nous déduisons que cette solution est en effet la solution de notre problème d'origine (1.2).

Maintenant, nous présentons quelques opérateurs inspirés par ceux présentés dans le papier de M. Frigon [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF] nous les avons adaptés au cadre prox-régulier.

Considérons (α, β) et δ donné comme dans l'hypothèse (S -L 2 ). Définissons

F : [0, T ] × R n ⇒ R n par F = F ∩ G, où F (t, x) = F (t, x t ), si x -α(t) > β(t), F (t, x), si x -α(t) ≤ β(t); G(t, x) =                α (t) + δ(t), si β(t) = 0, R n , si x -α(t) ≤ β(t) et β(t) > 0, {z : x -α(t), z -δ(t) -α (t) ≤ β (t) x -α(t) -ρ x -α(t) 2 }, sinon; avec x t = x, si x -α(t) ≤ β(t), α(t) + β(t)
x-α(t) (x -α(t)), si x -α(t) > β(t). De même, nous définissons

F : [0, T ] × R n ⇒ R n par F = F ∩ G, où G(t, x) =          α (t) + δ(t), si β(t) = 0, R n , si x -α(t) < β(t), {z : x -α(t), z -δ(t) -α (t) ≤ β (t) x -α(t) -ρ x -α(t) 2 }, sinon. Proposition 1.3.5. Supposons que les hypothèses (H), (H k ) et (S -L 2 ) soient satisfaites. Soit F : C([0, T ], R n ) ⇒ L 2 ([0, T ], R n ) définie par F(x) = {ν ∈ L 2 ([0, T ], R n ) : ν(t) ∈ F (t, x(t)) + x(t) t , p.p. t ∈ [0, T ]} xxii
Quelques aspects sur les inégalités variationnelles différentielles avec L 2 ([0, T ], R n ) muni de la topologie faible. Alors, F est semi-continue supérieurement à valeurs compactes convexes. De plus, il existe l ∈

L 2 ([0, T ], [0, ∞)) tel que, pour tout x ∈ C([0, T ], R n ) et tout ν ∈ F(x), ν(t) ≤ l(t) p.p. t ∈ [0, T ]. Proposition 1.3.6. Supposons que les hypothèses (H), (H k ) et (S -L 2 ) soient satisfaites. Alors, il existe une fonction continue f : C([0, T ], R n ) → L 1 ([0, T ], R n ) tel que f (x)(t) ∈ F (t, x(t)) + x(t) t , p.p. t ∈ [0, T ]. De plus, il existe l ∈ L 2 ([0, T ], [0, ∞)) tel que, pour tout x ∈ C([0, T ], R n ), f (x)(t) ≤ l(t) p.p. t ∈ [0, T ]. Alors, f (C([0, T ], R n )) ⊂ {ν ∈ L 2 ([0, T ], R n ) : ν(t) ≤ l(t) p.p. t ∈ [0, T ]}.
Maintenant, considérons les opérateurs modifiés suivants: pour tout (t,

x) ∈ [0, T ] × R n , F ρ : [0, T ] × R n ⇒ R n donné par F ρ (t, x) = F (t, x) + ρx.
De même, nous définissons

F ρ : [0, T ] × R n ⇒ R n par F ρ (t, x) = F (t, x) + ρx. Proposition 1.3.7. Supposons que les hypothèses (H), (H k ) et (S -L 2 ) soient satisfaites. Soit F ρ : C([0, T ], R n ) ⇒ L 2 ([0, T ], R n ) définie par F ρ (x) = {ν ∈ L 2 ([0, T ], R n ) : ν(t) ∈ F ρ (t, x(t)) + x(t) t p.p. t ∈ [0, T ]} avec L 2 ([0, T ], R n ) muni de la topologie faible. Alors, F ρ est semi-continue supérieu- rement à valeurs compactes convexes. De plus, il existe l ∈ L 2 ([0, T ], [0, ∞)) tel que, pour tout x ∈ C([0, T ], R n ) et tout ν ∈ F ρ (x), ν(t) ≤ l(t) p.p. t ∈ [0, T ]. Proposition 1.3.8. Supposons que les hypothèses (H), (H k ) et (S -L 2 ) soient satisfaites. Alors, il existe une fonction continue f ρ : C([0, T ], R n ) → L 1 ([0, T ], R n ) tel que f ρ (x)(t) ∈ F ρ (t, x(t)) + x(t) t , p.p. t ∈ [0, T ]. De plus, il existe l ∈ L 2 ([0, T ], [0, ∞)) tel que, pour tout x ∈ C([0, T ], R n ), f ρ (x)(t) ≤ l(t) p.p. t ∈ [0, T ]. Alors, f ρ (C([0, T ], R n )) ⊂ {ν ∈ L 2 ([0, T ], R n ) : ν(t) ≤ l(t), p.p. t ∈ [0, T ]}.
xxiii Quelques aspects sur les processus de rafle Notre méthode nécessite de réinterpréter le cône normal à des ensembles variant dans le temps en tant qu'opérateur maximal monotone dans L 2 . Pour cela, nous introduisons l'opérateur

N C(•) : dom(N C(•) ) ⊂ L 2 ([0, T ], H) ⇒ L 2 ([0, T ], H) défini par N C(•) (x) = {ν ∈ L 2 ([0, T ], H) : ν(t) ∈ N C(t) (x(t)), p.p. t ∈ [0, T ]}, où t ⇒ C(t) est une application multivoque à valeurs non vides convexes. L'opéra- teur N C(•) est maximal monotone avec dom(N C(•) ) = {x ∈ L 2 ([0, T ], H) : x(t) ∈ C(t) p.p. t ∈ [0, T ] et ∃ ν ∈ L 2 ([0, T ], H) | ν(t) ∈ N C(t) (x(t)), p.p. t ∈ [0, T ]}.
Proposition 1.3.9. Supposons que l'hypothèse (P) soit satisfaite. Alors, les opérateurs

N C(•) + ρ id et N C(•) + ρ id sont maximaux monotones.
Pour établir nos résultats d'existence, laissez-nous définir

A + : dom( A + ) ⊂ L 2 ([0, T ], R n ) ⇒ L 2 ([0, T ], R n ) par A + (x) = L(x) + (N C(•) + ρ id)(x), où dom( A + ) = A(x 0 ) ∩ dom(N C(•) ) et L(x) = x .
Proposition 1.3.10. Supposons que l'hypothèse (P) soit satisfaite. Alors, l'opérateur multivoque A + est maximal monotone.

Vu que A + est maximal monotone, alors id + A + est surjective et inversible. Ceci nous permet de définir

P + (x) = (id + A + ) -1 (x) ∈ A(x 0 ), où x ∈ L 2 ([0, T ], R n ). P + est appelé le résolvant de A + tel que λ = 1.
Proposition 1.3.11. Supposons que l'hypothèse (P) soit satisfaite. Alors, l'opérateur P + :

L 2 ([0, T ], R n ) -→ W 1,2 ([0, T ], R n ) est complètement continu, quand W 1,2 ([0, T ], R n ) muni de la topologie de C([0, T ], R n ) et continu quand L 2 ([0, T ], R n ) muni de la topologie faible.
Pour montrer l'existence de nos solutions, nous introduisons deux inclusions différentielles modifiées par les opérateurs qu'on a présentés plus tôt. Considérons les problèmes suivants; xxiv Quelques aspects sur les inégalités variationnelles différentielles

(1.3)    x (t) + x(t) ∈ -(N C(t) + ρ id)(x(t)) + F ρ (t, x(t)) + x(t) t , p.p. t ∈ [0, T ], x(0) = x 0 ∈ C(0); x(t) ∈ C(t), ∀t ∈ [0, T ]; et (1.4)    x (t) + x(t) ∈ -(N C(t) + ρ id)(x(t)) + F ρ (t, x(t)) + x(t) t , p.p. t ∈ [0, T ], x(0) = x 0 ∈ C(0); x(t) ∈ C(t), ∀t ∈ [0, T ].
Toutes solutions de (1.

Autres résultats

Considérons le problème

(1.5)    x (t) ∈ N C(t) (x(t)) + F (t, x(t)), p.p. t ∈ [0, T ], x(0) = x 0 ∈ C(0); x(t) ∈ C(t). ∀t ∈ [0, T ].
Nous pouvons fournir des résultats d'existence semblables à ceux obtenus pour le problème (1.2). À ce but, laissez-nous présenter une notion adaptée de L 2 -solutiontube pour ce problème.

Définition 1.3.3. Soient α ∈ W 1,2 ([0, T ], R n ) et β ∈ W 1,2 ([0, T ], R). On dit que (α, β) est un L 2 -tube-solution de (1.5), s'il existe δ ∈ L 2 ([0, T ], R n ) tels que 1. δ(t) ∈ N C(t) (α(t)), p.p. t ∈ [0, T ]; 2. pour tout t ∈ [0, T ] et x ∈ R n tel que x -α(t) = β(t), il existe ν ∈ F (t, x) tel que x -α(t), ν -δ(t) -α (t) ≥ β(t)β (t) + ρβ 2 (t) ; 3. α (t) ∈ -δ(t) + F (t, α(t)), p.p. sur {t ∈ [0, T ] : β(t) = 0}; 4. x(0) -α(0) ≤ β(0).
xxv Quelques aspects sur les processus de rafle Rappelons que

B β (α) = {x ∈ C([0, T ], R n ) : x(t) -α(t) ≤ β(t), ∀ t ∈ [0, T ]}.
Considérons l'hypothèse modifiée suivante:

(S -L 2 ) * Il existe un couple (α, β) ∈ W 1,2 ([0, T ], R n ) × W 1,2 ([0, T ], [0, ∞)) un L 2 -
tube-solution de (1.5).

([0, T ], R n ) ∩ B β (α).
Considérons les problèmes supplémentaires :

   x (t) -x(t) ∈ (N C(t) + ρ id)(x(t)) + F * ρ (t, x(t)) -x(t) t , p.p. t ∈ [0, T ], x(0) = x 0 ∈ C(0); x(t) ∈ C(t). ∀t ∈ [0, T ]; et    x (t) -x(t) ∈ (N C(t) + ρ id)(x(t)) + F * ρ (t, x(t)) -x(t) t , p.p. t ∈ [0, T ], x(0) = x 0 ∈ C(0); x(t) ∈ C(t). ∀t ∈ [0, T ]; où F * ρ (t, x) = F * (t, x) -ρx et F * ρ = F * (t, x) -ρx pour tout (t, x) ∈ [0, T ] × R n , avec F * = F ∩ G * et F * = F ∩ G * , tels que G * (t, x) =                α (t) + δ(t), si β(t) = 0, R n , si x -α(t) ≤ β(t) et β(t) > 0, {z : x -α(t), z + δ(t) -α (t) ≥ β (t) x -α(t) + ρ x -α(t)
G * (t, x) =          α (t) + δ(t), si β(t) = 0, R n , si x -α(t) < β(t), {z : x -α(t), z + δ(t) -α (t) ≥ β (t) x -α(t) + ρ x -α(t) 2 }, sinon.
Le deuxième résultat en constitue le cinquième chapitre:

Chapitre 5: Existence de solutions monotones par rapport à un préordre et applications.

Dans la deuxième partie (chapitre 5), nous fournissons un résultat d'existence de solutions bi-monotones pour un système d'inclusions différentielles associées aux sous-différentiels de fonctions convexes-concaves. Nos solutions sont monotones par rapport à un préordre, i.e., une relation binaire réflexive et transitive. La limite quand le temps tend vers l'infini d'une solution de notre système est un point-selle pour une fonction convexe-concave. Les résultats obtenus peuvent être appliqués à deux problèmes. D'une part, à un jeu de deux joueurs avec un payement collectif, et d'autre part, à un problème de production pour une entreprise qui veut maximiser son profit (voir section 5.6). Commençons par présenter le problème suivant; Soient K p ⊂ R n et K q ⊂ R m deux sous-ensembles compacts convexes. Soit Γ : K p × K q → R + une fonction convexe-concave. Soient P : K p ⇒ K p et Q : K q ⇒ K q deux préordres à valeurs compactes convexes non vides. Considérons le système mixte

(1.6)    x (t) ∈ proj T P (x(t)) (x(t)) (-∂ x Γ(x(t), y(t)), p.p. t ∈ [0, +∞), y (t) ∈ proj T Q(y(t)) (y(t)) (∂ + y Γ(x(t), y(t)), p.p. t ∈ [0, +∞), x(0) = x 0 , y(0) = y 0 , où x 0 et y 0 sont donnés (donc x 0 ∈ P (x 0 ) et y 0 ∈ Q(y 0 ) compte tenu de la réflexivité des applications multivoques P et Q en tant que préordres). Définition 1.3.4. Une application multivoque R : R n ⇒ R n est un préordre s'il satisfait les conditions (i) x ∈ R(x), pour tout x ∈ C (réflexivité); (ii) z ∈ R(y), y ∈ R(x) ⇒ z ∈ R(x) (transitivité).

De plus, xxvii

Quelques aspects sur les processus de rafle

• ∂ u Γ : K p × K q ⇒ R
n est une application multivoque semi-continue supérieurement appelée le sous-différentiel de la fonction convexe Γ(•, v) par rapport à u définie par

∂ u Γ(u, v) = {u * ∈ R n | Γ(u , v) ≥ Γ(u, v) + u * , u -u , u ∈ R n }. • ∂ + v Γ : K p × K q ⇒ R
n est une application multivoque semi-continue supérieurement appelée le sous-différentiel concave de la fonction concave Γ(u, •) par rapport à v définie par

∂ + v Γ(u, v) = {v * ∈ R n | Γ(u, v ) ≤ Γ(u, v) + v * , v -v , v ∈ R n }.
• T C (u) est le cône tangent à l'ensemble convexe C en u défini par

T C (u) = cl {λ(u -v) | v ∈ C et λ > 0} .
• proj note la projection habituel au sens de l'analyse convexe.

La trajectoire x(•) est monotone par rapport au préordre P (•), la contrainte variable et se déplace sur la direction de descente maximale de la fonction Γ(x(•), •) qui est compatible avec P (•). De plus, la trajectoire y(•) est monotone par rapport au préordre Q(•) et se déplace sur la direction de montée minimale de la fonction Γ(•, y(•)) qui est compatible avec Q(•). Définition 1.3.5. On dit qu'une trajectoire x(•) définie sur un intervalle [0, T [ est monotone si et seulement si ∀t, s ≥ 0, t ≥ s, on a x(t) ∈ P (x(s)).

Définition 1.3.6 (Solutions P × Q-monotones). On dit que les trajectoires x : [0, ∞) → K p et y : [0, ∞) → K q commençant à x 0 ∈ K p et y 0 ∈ K q respectivement, sont P × Q-monotones si 1. x(•) est monotone par rapport à P (•), 2. y(•) est monotone par rapport à Q(•).
x (t) = proj T P (x(t)) (x(t)) (-∇w(x(t))),
x(0) = x 0 ; où, ∇w(•) est le gradient de la fonction convexe w(•) de classe C 1 , (cf. [START_REF] Malinvaud | Leçons de théorie microéconomique[END_REF]). Dans le temps initial t = 0, la société a un faisceau d'entrée x 0 et elle doit gérer le flux de données pour faire face à la demande surgissant sur le marché. De plus, elle vise à trouver, parmi les courants de données qui garantiraient l'élargissement de sa part de marché, ceux qui font la fonction de coût diminuant.

Notre argument est construit sur quelques résultats présentés par Falcone et Siconolfi dans leur papier [START_REF] Falcone | Maximum descent monotone solutions of an ordinary differential equation with a discontinuous right-hand side[END_REF], par exemple, nous utiliserons le même préordre P (•) qu'ils ont défini pour leur modèle économique. Cependant, nos résultats concernant l'existence d'un point-selle sont nouveaux.

Résultats principaux

Les hypothèses suivantes vont intervenir dans les prochains théorèmes;

Soient K p ⊂ R n et K q ⊂ R m deux
ensembles compacts convexes non vides. Soit Γ : K p × K q → R + une fonction qui satisfait les hypothèses suivantes: (H 1 ) Pour tout y ∈ K q fixé, la fonction x → Γ(x, y) est convexe et semi-continue inférieurement.

(H De même laissez-nous définir le préordre Q : K q ⇒ K q par:

Q(y) = {s ∈ K q : max z∈Fq(s) h q (z) ≥ max z∈Fq(y)
h q (z)}, ∀y ∈ K q , où h q et F q satisfont les hypothèses suivantes;

(H 1 q ) F q : K q ⇒ R n + est une application multivoque, telle que (a) F q (x) est un ensemble convexe compact, ∀x ∈ K q ;

(b) F q (•) est concave;

(c) F q (•) est continue.

(H 2 q ) h q : R n + → R + est une fonction à valeurs réelles, telle que (a) h q (•) est continue;

(b) h q (•) est strictement concave;

(c) si y 1 ≥ y 2 , alors h q (y 1 ) ≥ h q (y 2 ), ∀y 1 , y 2 ∈ K q .

Proposition 1.3.16. Supposons que les hypothèses (H 1 q ) et (H 2 q ) soient satisfaites. Alors, Q(•) est un préordre continu à valeurs non vides compactes convexes. Γ(x, y).

Le résultat suivant est nécessaire pour montrer le théorème ci-dessus. Notons que (i) Si π(-ψ(x, y)) = 0, alors pour chaque y ∈ K q il n'y a aucune direction de descente de Γ(•, y) dans T (x).

T P (x) (x) = T (x), et T Q(y) (y) = T (y). N P (x) (x) = N (x), et N Q(y) (y) = N (y). π(f (x, y)) = proj T (x) (f (x, y)), et π (f (x, y)) = proj T (y) (f (x, y)), avec f pour être choisi. Proposition 1.3.18. Soient x ∈ K p et y ∈ K q . Soient ψ(x,
xxxi Quelques aspects sur les processus de rafle (ii) Si π(-ψ(x, y)) = 0, alors ψ(x, y), π(-ψ(x, y)) < 0, et pour tout u ∈ T (x), tel que u = π(-ψ(x, y)) , on ait ψ(x, y), π(-ψ(x, y)) < ψ(x, y), u .

(iii) Si π (ϕ(x, y)) = 0, alors pour chaque x ∈ K p il n'y a aucune direction de montée de Γ(x, •) dans T (y).

(iv) Si π (ϕ(x, y)) = 0, alors ϕ(x, y), π (ϕ(x, y)) > 0, et pour tout v ∈ T (y), tel que u = π (ϕ(x, y)) , on ait ϕ(x, y), π (ϕ(x, y)) > ϕ(x, y), v .

Cas particuliers

Commençons par le cas convexe. Considérons l'inclusion différentielle

(1.7) x (t) ∈ proj T P (x(t)) (x(t)) (-∂w(x(t))), p.p. dans [0, ∞), x(0) = x 0 ,
où ∂w(•) est le sous-différentiel de la fonction convexe w(•) donné par

∂w(u) = {u * ∈ R n | w(v) -w(u) ≥ u * , v -u , ∀v ∈ R n }, ∀u ∈ R n .
La fonction w(•) doit vérifier l'hypothèse suivante;

A0 w : K → R + est
P (x) (x) (f (x)) = π(f (x)), et T P (x) (x) = T (x), et N P (x) (x) = N (x). Proposition 1.3.19. Soient x ∈ K et ψ(x) une sélection mesurable dans ∂w(x). Alors, d dt w(x(t)) = ψ(x(t)), x (t) , p.p. t ≥ 0.
De plus, (i) si π(-ψ(x)) = 0, il n'y a aucune direction de descente de w(•) dans T (x);

(ii) si π(-ψ(x)) = 0, alors ψ(x), π(-ψ(x)) < 0, et pour chaque u ∈ T (x), tel que u = π(-ψ(x)) , on ait ψ(x), π(-ψ(x)) < ψ(x), u . w(x).

Continuons avec le cas concave. Considérons l'inclusion différentielle (1.9)

x (t) ∈ proj T Q(x(t)) (x(t)) (∂ + w(x(t))), p.p. dans [0, ∞), x(0) = x 0 , où ∂ + w(•) est le sous-différentiel concave de la fonction w(•) donné par ∂ + w(u) = {u * ∈ R n | w(v) -w(u) ≤ u * , v -u , v ∈ C}, ∀u ∈ R n .
Dans ce cas, la fonction w(•) doit vérifier l'hypothèse suivante; xxxiii Quelques aspects sur les processus de rafle (A0) + w : K → R + est une fonction concave semi-continue supérieurement avec un domaine non vide.

De plus, soit K un sous-ensemble compact convexe de R m + . Le préordre Q sur K est donné par :

Q(x) = {y ∈ K : max z∈F (y) h(z) ≥ max z∈F (x) h(z)}, ∀x ∈ K.
Dans ce cas, les fonctions F et h satisfont les mêmes hypothèses que les deux fonctions F q et h q respectivement. Alors, le préordre Q est continu à valeurs non vides compactes convexes. Le problème d'évolution dans ce cas est le suivant: Nous cherchons des trajectoires x : [0, ∞) → K de (1.9) commençant à x 0 ∈ K, telles que

(i) x(•) est monotone par rapport à Q(•), (ii) w(x(t)) est une fonction croissante de t ∈ [0, +∞).
Dans ce cas, nous considérons les notations suivantes; De plus, (i) si π (ϕ(x)) = 0, il n'y a aucune direction de descente de w(•) dans T (x);

proj T Q(x) (x) (f (x)) = π (f (x)), et T Q(y) (y) = T (y), et N Q(y) (y) = N (y).
(ii) si π (ϕ(x)) = 0, alors ϕ(x), π (ϕ(x)) > 0, et pour chaque u ∈ T (x), tel que u = π (ϕ(x)) , on ait ϕ(x), π (ϕ(x)) > ϕ(x), u . w(x).

Finalement, nous concluons par le cas prox-régulier. Considérons l'inclusion différentielle (1.7) mais avec le sous-différentiel proximal de la fonction prox-régulier w(•). Pour cela, nous donnons quelques définitions. Soient f : R n → R ∪{+∞} une fonction et x ∈ dom f . Définition 1.3.8. On définit le sous-différentiel proximal de f en x, noté ∂ P f (x), comme l'ensemble convexe formé des vecteurs x * de R n pour lesquels il existe deux constantes c > 0 et ε > 0 telles que

x * , x -x ≤ f (x ) -f (x) + c 2 x -x 2 , pour tout x ∈ B(x, ε). Définition 1.3.9. On dit que f est prox-régulier à x pour v ∈ ∂f (x) s'il existe certains c > 0 et > 0 tel que pour tout (x, v) ∈ gph ∂f avec x -x < , |f (x) - f (x)| < et v -v < on a f (x ) -f (x) ≥ v, x -x - c 2 x -x 2 , pour tout x ∈ B(x, ).
Si les propriétés ci-dessus sont satisfaites pour tous les vecteurs v ∈ ∂f (x), la fonction f est dite prox-régulier à x. Quand f est prox-régulier à chaque point de E ∩dom ∂f , nous dirons que f est prox-régulier sur l'ensemble E.

Proposition 1.3.23. Une fonction f est dite c-prox-régulier sur un voisinage de

x 0 dans R n si et seulement s'il existe > 0 et c > 0 pour chaque (x, v) ∈ gph ∂f avec x -x 0 < , on a f (x ) -f (x) ≥ v, x -x - c 2 x -x 2 , pour tout x ∈ B(x 0 , ).
Par la suite, nous considérons les hypothèses suivantes; soit c ≥ 0.

(A0) c w : K → R + est une fonction semi-continue inférieurement et c-prox-régulier. w(x).

(A2)

xxxvi

INTRODUCTION

In this chapter, we present some results on sweeping processes, which cover two different areas. On the one hand, we study the existence of solutions-tube for nonconvex sweeping processes. On the other hand, we investigate the existence of monotone solutions with respect to a preorder for a mixed system and we provide some applications. This work is based on [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Cornet | Existence of slow solutions for a class of differential inclusions[END_REF][START_REF] Falcone | Maximum descent monotone solutions of an ordinary differential equation with a discontinuous right-hand side[END_REF][START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF][START_REF] Frigon | Boundary and periodic value problems for systems of nonlinear second order differential equations[END_REF][START_REF] Frigon | Théorèmes d'existence de solutions d'inclusions différentielles[END_REF][START_REF] Henry | An existence theorem for a class of differential equations with multivalued right-hand side[END_REF][START_REF] Poliquin | Prox-regular functions in variational analysis[END_REF].

Motivation

The sweeping process, motivated by plasticity theory, was introduced in the seventies by J.J. Moreau. It is considered to be an evolution problem conditioned by inequality constraints. The classical theory of the sweeping process establishes the existence and uniqueness of Lipschitzian solutions for a given moving set. Even though the theory originated in mechanics, nowadays, this problem is an object of mathematical research since it is essential not only in mechanics, but also in economics, electrical engineering, biology, and so on. The theory is actually crucial in many branches of science with various applications; in particular, it is vital to understanding quasistatic elastoplasticity, crowd motion, magnetic hysteresis, social economic modeling, and many other processes. Several works have already dealt with significant components of the subject. Some have concentrated on convexity and similar developments in the area of nonconvexity. Others have focused on developing numerical methods for the study of systems composed of interacting rigid bodies. Optimal control theory also constitutes a significant part of the work that has been done. As we mentioned previously, 1 Some results on sweeping processes this manuscript is dedicated to studying the existence of solutions for sweeping processes. We will use various non-smooth analysis tools and different concepts and notions of solutions. In other words, we will be investigating different types of selection procedures for the trajectories of differential inclusions.

The first part of our study is concerned with the notion of the solution-tube, which was introduced and thoroughly studied by M. Frigon (see, [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF][START_REF] Frigon | Boundary and periodic value problems for systems of nonlinear second order differential equations[END_REF][START_REF] Frigon | Théorèmes d'existence de solutions d'inclusions différentielles[END_REF][START_REF] Frigon | Existence Results for Some Initial and Boundary Value Problems Without Growth Restriction[END_REF][START_REF] Frigon | Nonlinear first order initial and periodic problems in Banach spaces[END_REF]). This notion has proved to be an important tool in the study of many classes of differential inclusions. We emphasize that this type of solution extends the Hartman condition [START_REF] Hartman | On boundary value problems for systems of ordinary, nonlinear, second order differential equations[END_REF] and the well-known notion of upper and lower solutions (see, [START_REF] Dragoni | Il problema dei valori ai limiti studiato in grande per gli integrali di una equazione differenziale del secondo ordine[END_REF][START_REF] Perron | Ein neuer Existenzbeweis für die Integrale der Differentialgleichung y = f (x, y)[END_REF][START_REF] Pouso | Upper and lower solutions for first order discontinuous BVPs[END_REF]). It worth pointing out that the latter is a crucial tool with which to investigate existence results for differential equations. The advantage of using a solution-tube lies in the fact that it allows one to ensure the existence of a solution of the considered problem lying in a tube, i.e., we have information pertaining to the presence and location of the solutions. This notion has found many applications in, for instance, obtaining multiplicity results.

Our study is carried out in the non-convex framework, which to our knowledge has not been considered before. The class of prox-regular sets and functions is an important tool in non-smooth analysis. Rockafellar introduced in [START_REF] Rockafellar | Favorable classes of Lipschitz continuous functions in subgradient optimization[END_REF] the class of lower-C 2 functions in the finite-dimensional context which is of great importance in optimization. Later, Poliquin and Rockafellar in [START_REF] Poliquin | Prox-regular functions in variational analysis[END_REF] identified, in the finitedimensional setting, a large class of prox-regular functions, which includes lower-C 2 and strongly amenable functions. Via the Mordukhovich (limiting) normal cone they also defined the concept of prox-regularity for sets. Thereafter, Poliquin et al. studied the prox-regularity of sets in [START_REF] Poliquin | Local differentiability of distance functions[END_REF] in any Hilbert space. In [START_REF] Bernard | Uniform prox-regularity of functions and epigraphs in Hilbert spaces[END_REF], Bernard and Thibault studied the local uniform prox-regularity of functions in any Hilbert space, and they made a connection between the lower-C 2 property and prox-regularity of epigraphs. This class of functions covers all lower semi-continuous, proper, convex functions, lower-C 2 functions, and strongly amenable functions, i.e., a large core of functions of interest in variational analysis and optimization.

The second part of our study deals with the study of monotone solutions. The problem of finding monotone and feasible solutions arise from viability issues, in which we select trajectories which are viable in the sense that they always satisfy given constraints. Aubin [START_REF] Aubin | Monotone evolution of resource allocations[END_REF] studied this issue by adapting methods introduced in Aubin, Cellina, and Nohel [START_REF] Aubin | Monotone trajectories of multivalued dynamical systems[END_REF] and then thoroughly investigated in [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Aubin | Viability theory[END_REF][START_REF] Clarke | Monotone invariant solutions to differential inclusions[END_REF] and the references therein. Monotonicity is considered to be a selection procedure which selects among all the trajectories of the differential inclusion those trajectories that improve the state of the system as time elapses. Motivations and possible applications can be found in game theory, economics, and optimal control, in which it is possible to select trajectories by minimizing or maximizing a function defined on a set of trajectories (see, section 5.3.2).

Objective and Outlines

In this thesis, by employing tools from non-smooth, multivalued and variational analysis, we have studied differential variational inequalities; in particular, we have focused on the so-called Moreau sweeping process, i.e., a differential inclusion involving normal cones that model the evolution of state trajectories constrained by convex and time varying sets.

On the one hand, motivated by the works of M. Frigon, we investigate the existence of a solution-tube for a time-dependent perturbed sweeping process with (uniformly) ρ-prox-regular sets. Note that our approach does not use viability results as in [START_REF] Aubin | Set-valued analysis[END_REF]. Instead, we use a method inspired by [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF]. Our approach employs properties belonging to the class of prox-regular sets and functions as well as fixed point theory techniques. It is known that the normal cone associated with a ρ-proxregular set is hypomonotone. However, we apply properties of monotone operators by constructing a new operator using the relation between prox-regular and convex functions. Also, when ρ = 0, we recover the global results of the convex case.

On the other hand, we investigate the existence of monotone solutions with respect to a preorder. From this standpoint, we present an existence theorem for a mixed system of projected differential inclusions associated to subdifferentials of convex-concave functions. Our solutions are monotone with respect to a preorder. This latter set of solutions is a continuous multivalued map with convex compact values. Moreover, the trajectory solution of our system converges to a limit point, which is a saddle-point for a convex-concave function. The result obtained can be applied in two problems. First, in a game involving two players with a collective payoff, and, second, in a production problem for a company which wishes to maximize its profit. Finally, we note that the preorder has prox-regular values when we use proxregular functions. In this case, we use a different approach to obtain the existence of monotone solutions.

The manuscript is divided into five chapters. In chapter 2, we give a brief introduction and motivation for studying sweeping processes. Chapter 3 contains a summary of basic facts needed throughout the work. These include information on convex, multivalued, and non-smooth analyses. Chapter 4 deals with a timedependent sweeping process. Here, our goal is to prove the existence of solutions which are in a tube called solutions-tube. Our approach is based on fixed points theory. Chapter 5 is devoted to the study of a mixed system with a non-continuous right-hand side, in which we investigate the existence of monotone solutions as well as the presence of a minimax. We provide some applications to motivate our investigations in this chapter.

In the following we present a short overview of the thesis:

Chapter 3 -Mathematical background. This chapter is dedicated to preliminary results for a few specific topics which we will need in later chapters. This results contain reminders for convex, multivalued, and non-smooth analyses. We also review some of the standard facts concerning convex sets and functions, with special attention given to minimax theory. Next, the chapter contains some results on fixed point theory, which is a fundamental technique used in our proofs, solutions-tube and monotone solutions. These types of solutions are of great interest in a number of applications, see [START_REF] Frigon | Boundary and periodic value problems for systems of nonlinear second order differential equations[END_REF][START_REF] Frigon | Théorèmes d'existence de solutions d'inclusions différentielles[END_REF][START_REF] Frigon | Existence Results for Some Initial and Boundary Value Problems Without Growth Restriction[END_REF][START_REF] Frigon | Multiplicity results for systems of first order differential inclusions[END_REF] for solutions-tube and [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Aubin | Monotone trajectories of multivalued dynamical systems[END_REF][START_REF] Aubin | Viability theory[END_REF][START_REF] Clarke | Monotone invariant solutions to differential inclusions[END_REF] for monotone solutions.

Chapter 4 -Existence of solutions-tube for non-convex sweeping processes. We establish some existence results of solutions-tube for non-convex sweeping processes. Our approach employs properties belonging to the class of prox-regular sets and functions as well as fixed point theory techniques.

In this chapter, we will be concerned, for any fixed x 0 ∈ C(0), with the differential inclusion

   x (t) ∈ -N C(t) (x(t)) + F (t, x(t)), a.e. t ∈ [0, T ], x(0) = x 0 ∈ C(0), x(t) ∈ C(t), ∀t ∈ [0, T ].
where T ≥ 0 and for each t

∈ [0, T ], C(t) is a nonempty closed uniformly prox-regular subset of R n . Moreover, N C(t) (x(t)) is the normal cone to C(t) at x(t) ∈ C(t) for all t ∈ [0, T ], and F : [0, T ] × R n ⇒ R n is a multivalued map with compact values which satisfies different semi-continuity conditions.
Chapter 5 -Existence of monotone solutions with respect to a preorder and applications. In this chapter, we present an existence theorem for P (•)× Q(•)-monotone solutions of differential inclusions with a discontinuous righthand side. The monotonicity that we consider is with respect to a preorder.

Moreover, we show that the limit points (x, ỹ) of a solution is a saddle-point for the convex-concave function Γ(•, •) in P (x) × Q(ỹ). The obtained results can be used for studying a game involving two players with a collective pay-off. Another application is in a production problem of a company which wishes to maximize its profit. Finally, some particular cases are also considered: with a function w(•) which is supposed convex, concave, then prox-regular. We will be concerned with the following problem:

Let K p ⊂ R n , K q ⊂ R
m be two convex compact sets. Let x 0 be any fixed point in K p , and let y 0 be any fixed point in K q . We investigate the existence of P (•) × Q(•)-monotone solutions of the following mixed system

   x (t) ∈ proj T P (x(t)) (x(t)) (-∂ x Γ(x(t), y(t)), a.e. t ∈ [0, +∞), y (t) ∈ proj T Q(y(t)) (y(t)) (∂ + y Γ(x(t), y(t)), a.e. t ∈ [0, +∞), x(0) = x 0 , y(0) = y 0 .
Here, Γ : K p ×K q → R + is a convex-concave function, P : K p ⇒ K p a preorder on K p , and Q : K q ⇒ K q a preorder on K q . Moreover,

• ∂ u Γ(u, v) is the subdifferential of the convex function Γ(•, v) with respect to u, • ∂ + v Γ(u, v)
is the superdifferential of the concave function Γ(u, •) with respect to v,

• T C denote the tangent cone to the convex set C in R n , and

• proj denote the usual projection on a closed convex set.

CHAPTER 3

MATHEMATICAL BACKGROUND

Introduction

This chapter is dedicated to preliminary results for a few specific topics which we will need in later chapters. We provide some reminders of theoretical backgrounds of analyses as, for example, some points on convex analysis; in particular, information on convex functions and sets. We also present some differential properties of convex functions and a short review of saddle-functions and minimax problems. This is followed by standard facts on multivalued maps and some semi-continuity concepts which are similar to upper and lower ones for real-valued functions if inclusion reinstates order. We recall some results on fixed point theory which is a fundamental technique used in our proofs. An introduction to differential inclusions is presented as well as some existence results, namely, the solutions-tube and monotone solutions.

In the last section of this part, we look more closely at basic notions of nonsmooth analysis, for instance, we present some basic definitions and facts on tangent and normal cones. Then, we give a short review on the properties of prox-regular sets and functions as well as some points on the differential properties of prox-regular functions. Finally, we give a brief exposition of sweeping processes.

Convex analysis

In this section, we present some properties of convex sets and functions as well as the differential and subdifferential properties of convex functions. The concept Some results on sweeping processes of convexity has far-reaching consequences in variational analysis in the study of maximization and minimization problems. In addition, we recall some results on saddle-functions and minimax problems. The tools presented in this section are essential for the rest of the work.

Convex sets and functions

Definition 3.2.1. We say that

• a subset C of R n is convex if for every x 0 ∈ C and x 1 ∈ C one has (1 -λ)x 0 + λx 1 ∈ C for all λ ∈ [0, 1], • a function f : R n → R on a convex set C is a convex function if for every x 0 ∈ C and x 1 ∈ C one has f ((1 -λ)x 0 + λx 1 ) ≤ (1 -λ)f (x 0 ) + λf (x 1 ) for all λ ∈ [0, 1],
and f is strictly convex if this inequality is strict for points x 0 = x 1 with f (x 0 ), f (x 1 ) finite, and λ ∈ (0, 1).

Note that if f is convex then -f is concave, consequently, it satisfy the opposite inequalities under similar hypotheses.

Let us recall some notions relative to convex functions. Let f : R n → R ∪{+∞} be a convex function; the effective domain of f , which will be denoted by dom f , is the convex set defined by

dom f = {x ∈ R n | f (x) < +∞}.
f is said to be proper if its effective domain is nonempty.

Example 1.

1. The empty set is convex, and so is every singleton C = {x}.

2. Note that the supermum

f (x) = sup{f i (x) | i ∈ I},
of a family of convex functions is a convex function.

Chapter 3 Mathematical backgrounds

There are several useful correspondences between convex sets and functions. The simplest associates with each set C in R n the indicator function I C (x) of C, where

I C (x) = 0 if x ∈ C, +∞ if x ∈ C.
Obviously dom I C = C, and I C is proper and convex if and only if C is nonempty and convex.

The sets

lev ≤α = {x ∈ R n | f (x) ≤ α}, lev ≥α = {x ∈ R n | f (x) ≥ α},
are called respectively the lower and upper level sets.

Proposition 3.2.2. For a convex function f : R n → R all level sets of type lev ≤α f are convex.

Level sets of the type lev ≥α f are convex when, instead, f is concave.

Definition 3.2.3. Let f : R n → R be a function. We say that f is

• lower semi-continuous at x if lim inf x→x f (x) ≥ f (x), or equivalently lim inf x→x f (x) = f (x), • upper semi-continuous at x if lim sup x→x f (x) ≤ f (x), or equivalently lim sup x→x f (x) = f (x),
• continuous if and only if it is both lower and upper semi-continuous

lim x→x f (x) = f (x) ⇐⇒ lim inf x→x f (x) = lim sup x→x f (x).
In the following we recall some results on projections onto convex sets. We devote special attention to the case when these sets are closed convex cones. Recall that, a subset

T of R n is a cone if ∀λ ≤ 0, ∀u ∈ T, λu ∈ T.

Its Polar cone is given by

T -= {p ∈ R n | p, x ≤ 0 for all x ∈ T }.
Corollary 3.2.4 (cf. [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF]). Suppose C is a nonempty, closed, convex subset of R n . For all x in R n , there exists a unique element in C, denoted by proj C (x), satisfying

(i) x -proj C (x) ≤ x -v for all v ∈ C.
Or equivalently

(ii) x -proj C (x), v -proj C (x) ≤ 0 for all v ∈ C. Definition 3.2.5. The map proj C is called the projector onto C. It is non-expansive proj C (x) -proj C (y) ≤ x -y .

And monotone proj

C (x) -proj C (y), x -y ≥ 0. Proposition 3.2.6. If T is a closed convex cone of R n , then proj T is characterized by      i) proj T (x) ∈ T, ii) ∀ v ∈ T, proj T (x) -x, v ≤ 0, iii) proj T (x) -x, proj T (x) = 0. Moreover, it satisfies, i) proj T (x) ≤ x , ii) (id -proj T )(x) ≤ x .
Furthermore, if N = T -denote the polar cone, id -proj T = proj N is the projector on N , and we have

if x = v T + v N where v T ∈ T, v N ∈ N and v T , v N = 0. Then v T = proj T (x) and v N = proj N (x).
Definition 3.2.7. For a proper, lower semi-continuous function f : R n → R and parameter value λ > 0, the Moreau envelope function e λ f and proximal mapping P λ f are defined by

e λ f (x) = inf x ∈R n f (x ) + 1 2λ x -x 2 , P λ f (x) = arg min x ∈R n f (x ) + 1 2λ x -x 2 .

Figure .3.1 : Approximation by a Moreau envelope function

Generally, e λ f approximates f from bellow in the manner represented in Figure .3.1 . The mapping e λ f approximates f better and better when λ is smaller and smaller, indeed, e λ f increases pointwise to f as λ ↓ 0.

Note that P λ f is a multivalued mapping, this concept will be studied in the next section. Moreover, if f is taken to be the indicator function I C , then P λ f coincides with the projection mapping proj C .

Normal convex integrand

Let T > 0, a convex integrand on L p ([0, T ]) is an extended-real-valued functional of the form

I f (u) = T 0 f (t, u(t))dt, for u ∈ L p ([0, T ]),
where

f is a function from [0, T ] × R n to R ∪{+∞}, such that f (t, x) is a convex function of x ∈ R n for each t ∈ [0, T ]. Such a functional is, as the name implies, convex on L p ([0, T ]), if in a rather general sense it is well-defined. Definition 3.2.8. A convex integrand is called normal if f (t, x
) is proper and lower semi-continuous in x for each t, and if further there exists a countable collection U of measurable functions u from [0, T ] to R n having the following properties:

(a) for each u ∈ U , f (t, u(t)) is measurable in t, (b) for each t, U t ∩ dom f t is dense in dom f t , where U t = {u(t) | u ∈ U }.
Some results on sweeping processes Lemma 3.2.9.

[77] Suppose f a convex integrand such that f (t, x) is measurable in t for each fixed x, and such that, for each t, f (t, x) is lower semi-continuous in x and has interior points in its effective domain. Then f is a normal convex integrand.

Conjugate functions

Let f : R n → R be a proper function. The function f * : R n → R associating to every

x * ∈ R n : f * (x * ) = sup x { x * , x -f (x)},
is called the conjugate function of f . The conjugate function f * is a lower semi-continuous convex function.

We always have the following inequality

∀x, x * ∈ R n , x * , x ≤ f (x) + f * (x * ),
called the Fenchel inequality .

Differentiability and subdifferentiability

In this subsection, we provide some useful differential properties of convex functions.

Directional derivative

Let f : R n → R ∪{+∞} be a function. The directional derivative of f at x with respect to a vector v is defined to be the limit

f (x; v) = lim λ↓0 f (x + λv) -f (x) λ ,
if it exists (+∞ and -∞ being allowed as limits). If f (x; v) exists for all directions v, we say that f is differentiable at x. If moreover

f (x; v) = ∇f (x), v ,
is continuous and linear, we say that f is Gâteaux differentiable, where ∇f (x) is the gradient of f at x. Convex functions are not differentiable in the usual sense in general. Nevertheless, the directional derivative always exists. Theorem 3.2.10 (cf. [START_REF] Rockafellar | Convex analysis[END_REF]). Let f be a convex function, and let x be a point where

f is finite. Then f (x; v) exists for any v ∈ R n . Moreover, v → f (x; v) is a convex non-decreasing function. Subdifferential Let f : R n → R ∪{+∞} be a convex function. The set ∂f (x) = {x * ∈ R n | f (y) -f (x) ≥ x * , y -x , ∀y ∈ R n },
is called the subdifferential of f at x. The subdifferential ∂f (x) is a multivalued map with convex values. The domain of the subdifferential ∂f is defined by

dom(∂f ) = {x ∈ R n : ∂f (x) = ∅}.
In general, the subdifferential may be empty. If ∂f (x) is not empty, then f is said to be subdifferentiable at x.

Example 1 (Subdifferential of the square of the norm). Let u ∈ R n . For

f (u) = 1 2 u -b 2 ,
with fixed b ∈ R n we have:

u * ∈ ∂f (u) ⇐⇒ u * , u -b = u * u -b and u * = u -b .
Therefore, ∂f (u) = {u -b}.

The directional derivatives of any proper convex function f , not necessarily differentiable, can be described in terms of subgradient vectors. Theorem 3.2.11. Let f be a convex function, and let x be a point where f is finite. Then x * is a subgradient of f at x if and only if

f (x; v) ≥ x * , v , ∀v ∈ R n .
Theorem 3.2.12. Let f be a convex and lower semi-continuous with Int(dom f ) nonempty. Then ∂f (•) is an u.s.c. map on Int(dom f ) with nonempty bounded values. Proposition 3.2.13. Let f be a proper convex function. The following propositions are equivalent

     a) x ∈ R n minimizes f on R n , b) 0 ∈ ∂f (x), c) for all v ∈ R n , 0 ≤ f (x; v).

Some results on sweeping processes

For proofs and more information see [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Rockafellar | Convex analysis[END_REF].

Theorem 3.2.14 (cf. [START_REF] Rockafellar | variational Analysis[END_REF]). For any proper, convex function f : R n → R, the mapping ∂f : R n ⇒ R n is monotone. Indeed, a proper, lower semi-continuous function f is convex if and only if ∂f is monotone, in which case ∂f is maximal monotone.

Note that the Fréchet derivative of Moreau envelope e λ f is nothing but the Yosida approximation (see, the last part of section 3.3.1) of index λ of the maximal monotone operator ∂f ∂(e λ ) = (∂f ) λ .

Theorem 3.2.15 (cf. [START_REF] Rockafellar | variational Analysis[END_REF]). For any proper, lower semi-continuous, convex function f : R n → R, the Yosida regularization of the subgradient mapping ∂f for any λ > 0 is the gradient mapping ∇e λ f associated with the Moreau envelope e λ f : one has

∇e λ f = (λ id +(∂f ) -1 ) -1 .
This mapping is maximal monotone, single-valued and Lipschitz continuous globally on R n with constant λ -1 . Consequently, e λ f is of class C 1+ (i.e., is differentiable with ∇e λ f strictly continuous).

For more information about the relationship between maximal monotone maps and proper, lower semi-continuous, convex functions, we refer the reader to [START_REF] Rockafellar | variational Analysis[END_REF].

Concave functions Definition 3.2.16. Let C ⊂ R n be a convex set. We say that a function f :

C → R is concave on C if f (λx + (1 -λ)y) ≥ λf (x) + (1 -λ)f (y), ∀x, y ∈ C with x = y, ∀ λ ∈ [0, 1]. Definition 3.2.17. A vector x * is a supergradient of f at the point x if f (y) ≤ f (x) + x * , y -x , ∀ y ∈ C.
The set of all supergradients of f at x is called the superdifferential of f at x ∈ C, and is denoted by ∂ + f (x). i.e.,

∂ + f (x) = {x * ∈ R n | f (y) -f (x) ≤ x * , y -x , y ∈ C}.
Chapter 3 Mathematical backgrounds Lemma 3.2.18. (cf. [START_REF] Rockafellar | Convex analysis[END_REF]) Let f be a concave function on the convex set C. Then

x * ∈ ∂ + f (x) ⇐⇒ x * , v ≥ f (x; v), ∀v ∈ R n .
We point out that for each direction v, f (x; v) is a lower semi-continuous function of x, when f is concave on the convex set C (see, [START_REF] Fenchel | Convex cones, sets and functions[END_REF]). Moreover, ∂ + f is upper semi-continuous at x. Indeed, let x k be a sequence converging to x and a sequence

x * k converging to x * such that x * k ∈ ∂ + f (x k ). By Lemma 3.2.18, x * k , v ≥ f (x k ; v) for any v, which implies x * , v ≥ f (x; v)
by the lower semi-continuity of f . Since v is arbitrary, then by Lemma 3.2.18 one has x * ∈ ∂ + f (x).

Saddle-functions and minimax theory

Let C and D be subsets of R n and R m respectively, and let K be a function from C × D to R. We say that K is a convex-concave function if K(x, y) is a convex function of x ∈ C for each y ∈ D and a concave function of y ∈ D for each x ∈ C. Concave-convex functions are defined similarly. We speak of both kinds of functions as saddle-functions.

Given any convex-concave function K on R n × R m , we define

dom 1 K = {x | K(x, y) < +∞, ∀y}, dom 2 K = {y | K(x, y) > -∞, ∀x}.
Observe that dom 1 K is the intersection of the effective domains of the convex functions K(x, •) as x ranges over R n , while dom 2 K is the intersection of the effective domains of the concave functions K(•, y) as y ranges over R m . In particular, dom

1 K is a convex set in R n and dom 2 K is a convex set in R m . The product set dom K = dom 1 K × dom 2 K, is called the effective domain of K. Since -∞ < K(x, y) < +∞. When x ∈ dom 1 K and y ∈ dom 2 K, K is finite on its domain. If dom K = ∅, K is said to be proper.
Let us recall some results about directional derivatives and subgradients of saddle-functions. Let K be a saddle-function on R n × R m , and let (x, y) be a point where K is finite. The (one-sided) directional derivative of K at (x, y) with respect to (u, v) is defined by

K (x, y; u, v) = lim λ↓0 [K(x + λu, y + λv) -K(x, y)]/λ, if this limit exists. The directional derivatives K (x, y; u, 0) = lim λ↓0 [K(x + λu, y) -K(x, y)]/λ, K (x, y; 0, v) = lim λ↓0 [K(x, y + λv) -K(x, y)]/λ, certainly exist.
Given any convex-concave function K on R n × R m and (x, y) in R n × R m , we define ∂ x K(x, y) to be the set of all subgradients of the convex function K(•, y) at x, i.e., the set of all vectors x * ∈ R n such that

K(x , y) ≥ K(x, y) + x * , x -x , x ∈ R n .
Similarly, we define ∂ + y K(x, y) to be the set of all supergradients of the concave function K(x, •) at y, i.e., the set of all vectors y * ∈ R m such that K(x, y ) ≤ K(x, y) + y * , y -y , y ∈ R m . Moreover, if (x, y) is an interior point of dom K, then ∂ x K(x, y) and ∂ + y K(x, y) are nonempty bounded convex sets and

K (x, y; x , 0) = sup{ x * , x | x * ∈ ∂ x K(x, y)}, K (x, y; 0, y ) = inf{ y * , y | y * ∈ ∂ + y K(x, y)}.
Theorem 3.2.19 (cf. [START_REF] Rockafellar | Convex analysis[END_REF]). Let C × D be an open convex set in R n × R m , and let K be a convex-concave function finite on C × D. Then, for each u, K (x, y; u, 0) is an upper semi-continuous function of (x, y) on C × D, and for each v, K (x, y; 0, v) is a lower semi-continuous function on C × D. Moreover, given any (x, y) in on C × D and any ε > 0, there exists δ > 0 such that ∂K(x 0 , y 0 ) ⊂ ∂K(x, y) + εB, ∀x 0 ∈ B(x, δ), ∀y 0 ∈ B(y, δ).

Chapter 3 Mathematical backgrounds

Minimax problems

Minimax theory treats a class of extremum problems which involve, not simply minimization or maximization, but a combination of both. Let C and D be arbitrary non-empty sets, and K be a function from C ×D to R. For each x ∈ C, one can take the supremum of K(x, y) over y ∈ D and then take the infimum of this supremum as a function on C. The equality so obtained is

inf x∈C sup y∈D k(x, y).
On the other hand, for each y ∈ D one can take the infimum of K(x, y) over x ∈ C and then take the supremum of this infimum as a function on D. This forms

sup y∈D inf x∈C K(x, y).
If the "sup inf"and "inf sup"are equal, the common value is called the minimax or the saddle-value of K with respect to minimizing over C and maximizing over D.

In general, the "sup inf"and the "inf sup"might not be equal, but a certain inequality is at least satisfied. If K is any function from a non-empty product set C × D to R, then sup

y∈D inf x∈C K(x, y) ≤ inf x∈C sup y∈D k(x, y).
However, we are more interested in the concept of saddle-point, which is given by the following definition, we refer to [START_REF] Rockafellar | Convex analysis[END_REF] for more details.

Definition 3.2.20 (Saddle point). Let C and D be nonempty sets, and

K : C×D → R be a function. A point (x, ỹ) ∈ C × D is said to be a saddle point of K on C × D if (*) K(x, y) ≤ K(x, ỹ) ≤ K(x, ỹ), for all (x, y) ∈ C × D.
The condition (*) is equivalent to

min x∈C max y∈D K(x, y) = max y∈D min x∈C K(x, y).
In general, minimax problems for closed proper saddle-functions on R n × R m correspond to minimax problems for certain finite saddle-functions on convex product sets.

The following result is given by K. Fan [START_REF] Fan | Minimax theorems[END_REF] for convex-concave functions defined on compact sets. Some results on sweeping processes Theorem 3.2.21 (Theorem 1, [START_REF] Fan | Minimax theorems[END_REF]). Let C ⊂ R n and D ⊂ R m be two nonempty compact sets. Let K : C ×D → R be a function satisfying the following assumptions:

Multivalued analysis

In this section, we have compiled some basic facts on multivalued maps which are needed for the study of differential inclusions. For instance, we present various continuity properties (upper and lower semi-continuity, Lipschitz continuity). We make some reminders on the selection problem, that is, the question of constructing single-valued maps with graphs contained in the graph of a given multivalued map.

Then, we recall some results on fixed point theory. Moreover, we provide an introduction to differential inclusions which are a generalization of the concept of ordinary differential equations by replacing single-valued functions by multivalued ones. In addition, we call back some existence results. Since we will be concerned with the existence of solutions, we summarize without proofs some results on the presence of two types of solutions for differential inclusions, namely, solutions-tube and monotone solutions. For more informations see [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Yu | Setvalued maps[END_REF][START_REF] Deimling | Multivalued Differential Equations[END_REF][START_REF] Himmelberg | Measurable relations[END_REF][START_REF] Hu | Handbook of Multivalued Analysis[END_REF] and the references are therein.

Multivalued maps

The investigation of differential equations with multivalued right-hand side requires some basic knowledge of the general theory of multivalued maps. In this subsection, we present a review of some of the standard facts on multivalued maps and most useful semi-continuity concepts. Set X and Y to be two topological spaces.

Definition 3.3.1. A multivalued map F from X to Y is a map that associates with any x ∈ X a set F (x) ⊂ Y .
The graph of F is, by definition, the set

gph F = {(x, y) ∈ X × Y | y ∈ F (x)}. Chapter 3 Mathematical backgrounds The set dom(F ) = {x ∈ X | F (x) = ∅},
is called the domain of F . When dom(F ) = X, we say that the map F is strict. We say that a map F is proper if its domain is nonempty.

In this thesis, we shall assume that the images F (x) are nonempty for all x ∈ X.

The inverse map F -1 : Y ⇒ X is given by

F -1 (y) = {x ∈ X | (x, y) ∈ gph F }.
The range R(F ) is, by definition, the set

R(F ) = ∪ x∈X F (x).
We shall say that a multivalued map is compact (bounded ) if its range R(F ) is a compact (respectively bounded) subset of Y .

A multivalued map F from X to Y is said to be concave, if for every

x 1 , x 2 ∈ X and λ ∈ [0, 1] we have λF (x 1 ) + (1 -λ)F (x 2 ) ⊆ F (λx 1 + (1 -λ)x 2 ).
Note that F is concave if and only if gph F is a convex subset of X × Y . Moreover if F is concave, then F (x) is convex, for every x ∈ dom(F ).

Example 2.

1. Let f : X → Y be a single-valued map. Then its inverse f -1 : Y ⇒ X is usually multivalued.

2. Let V : X → Y be a function. Define the multivalued map

V + (x) = V (x) + R + , V (x) < +∞, ∅, V (x) = +∞.
The domain of V + coincides with the set of the points x such that V (x) < ∞, and gph V + is the epigraph of V .

Concepts of continuity of multivalued maps

Let F : X ⇒ Y be a multivalued map, and let C be a subset of Y . We have two types of inverse images of C under the action of F , namely

F + (C) = {x ∈ X : F (x) ⊆ C} (the strong inverse image of C),
Some results on sweeping processes and

F -(C) = {x ∈ X : F (x) ∩ C = ∅} (the weak inverse image of C).
Evidently we have

F + (C) ⊆ F -(C) ⊆ X.
Definition 3.3.2. We say that F : X ⇒ Y is upper semi-continuous (u.s.c.) at x ∈ X if and only if for any neighborhood U of F (x),

∃r > 0 such that F (x ) ⊂ U, ∀x ∈ B(x, r).
It is said to be upper semi-continuous if and only if it is so at every x ∈ X.

Proposition 3.3.3. Given a multivalued mapping F : X ⇒ Y , the following statements are equivalent.

(a) F is u.s.c. We say that a multivalued

(b) For every C ⊆ Y closed, F -(C) is closed in X.
F is closed, if gph F ⊆ X × Y is closed. Moreover, if F is u.s.c. and K ⊆ X is a compact, then F (K) ⊆ Y is compact. Definition 3.3.6. We say that F : X ⇒ Y is lower semi-continuous (l.s.c.) at x ∈ X if for all U ⊆ Y open such that F (x) ∩ U = ∅, ∃r > 0 such that F (x ) ∩ U = ∅, ∀x ∈ B(x, r).
It is said to be lower semi-continuous if and only if it is so at every x ∈ X. Proposition 3.3.7. Given a multivalued mapping F : X ⇒ Y , the following statements are equivalent.

(a) F is l.s.c.

(b) For every C ⊆ Y closed, F + (C) is closed in X.
(c) For any y ∈ F (x) and any sequence of elements x n ∈ dom(F ) converging to x, there exists a sequence of elements y n ∈ F (x n ) converging to y.

Example 3.

1. The multivalued map F 1 defined by

F 1 (x) = [-1, +1], if x = 0, {0}, if x = 0;
is l.s.c. at zero but not u.s.c. at zero.

2. The multivalued map F 2 defined by

F 2 (x) = {0}, if x = 0, [-1, +1], if x = 0;
is u.s.c. at zero but not l.s.c. at zero.

Note that if

F : R ⇒ R is defined by F (x) = [ψ(x), ϕ(x)] = {y ∈ R : ψ(x) ≤ y ≤ ϕ(x)},
and ψ is lower semi-continuous, ϕ is upper semi-continuous, then F is u.s.c. Whereas if ψ is upper semi-continuous and ϕ is lower semi-continuous, then F is l.s.c. Definition 3.3.8. We say that F : X ⇒ Y is continuous at x ∈ X, if it is both u.s.c. and l.s.c. at x. If that is true at every x ∈ X, then we say that F is continuous. Proposition 3.3.9. Given a multivalued mapping F : X ⇒ Y , the following statements are equivalent.

Measurability of multivalued maps

Let (Ω, Σ) be a measurable space and (X, d) a metric space, where d denote the distance function.

Definition 3.3.11. Let (X, d) be a metric space and A ⊆ X. For every x ∈ X, we define

d(x, A) = inf a∈A d(x, a).
As usual, we adopt the convention that inf ∅ = +∞. Definition 3.3.12. Let F : Ω ⇒ X be a multivalued map.

(a) We say that F is measurable, if for all U ⊆ X open, we have F -(U ) = {ω ∈ Ω : F (ω) ∩ U = ∅} ∈ Σ. (b) We say that f is graph measurable, if gph F = {(ω, x) ∈ Ω × X : x ∈ F (ω)} ∈ Σ × B(X)
: [0, T ] × R n ⇒ R n is integrably bounded if there exists h ∈ L 1 ([0, T ], R
) such that for almost every t ∈ [0, T ] and every x ∈ R n , one has y ≤ h(t) for all y ∈ F (t, x).

Maximal monotone maps

In this part, H denotes a Hilbert space.

Definition 3.3.17. A multivalued map A :

H ⇒ H is called monotone if and only if v 1 -v 2 , x 1 -x 2 ≥ 0, ∀x 1 , x 2 ∈ dom(A), ∀v i ∈ A(x i ), i = 1, 2.
A monotone multivalued map is maximal if there is no other monotone multivalued map à whose graph contains strictly the graph of A.

An equivalent formulation is the following: A is maximal if

v 1 -v 2 , x 1 -x 2 ≥ 0, ∀x 1 ∈ dom(A), ∀v 1 ∈ A(x 1 ) ⇒ x 2 ∈ dom(A) and v 2 ∈ A(x 2 ).
Example 2. Let f be a non-decreasing application from R into R, the operator

f : x ∈ R → [f (x-), f (x+)],
is maximal monotone in R. All the maximal monotone operators of R are of this type.

Let us recall some results concerning monotone maps. More properties and the proofs can be found for instance in [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF][START_REF] Hu | Handbook of Multivalued Analysis[END_REF][START_REF] Rockafellar | variational Analysis[END_REF][START_REF] Zeidler | Nonlinear Monotone Operators. II/B: Nonlinear monotone operators[END_REF]. Lemma 3.3.18 (cf. [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF]). A multivalued monotone map A : dom(A) ⊂ H ⇒ H is locally bounded at every point in the interior of its domain.

Lemma 3.3.19 (cf. [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF]). Let A : dom(A) ⊂ H ⇒ H be a multivalued maximal monotone operator. Then A has closed, convex values, and gph(A) := {(x, x * ) :

x * ∈ A(x)} is sequentially closed in (H, τ s ) × (H, τ w ) and in (H, τ w ) × (H, τ s ), where τ s and τ w denote, respectively, the strong and the weak topologies of H. Lemma 3.3.20 (cf. [START_REF] Aubin | Differential Inclusions[END_REF]). Let A : dom(A) ⊂ H ⇒ H be a multivalued monotone operator. It is maximal if and only if id +A is surjective. Lemma 3.3.21 (cf. [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF]). Let A : dom(A) ⊂ H ⇒ H be a multivalued maximal monotone operator and S : H → H a single valued Lipschitzian monotone operator. Then A + S is maximal monotone. Lemma 3.3.22 (cf. [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF]). Let A : dom(A) ⊂ H ⇒ H and S : dom(S) ⊂ H ⇒ H be two multivalued maximal monotone operators such that dom(A) ∩ dom(S) = ∅. Then ν ∈ Im(id +A λ + S) if and only if {A λ (x λ )} is bounded as λ → 0 + , where ν = (id +A λ + S)(x λ ). Moreover, we have that

x λ → x and A λ (x λ ) → µ ∈ A(x) ∩ {ν -x -S(x)}.

Some results on sweeping processes

A maximal monotone map A can be approximated in some sense by a singlevalued Lipschitzian map denoted by A λ defined from H to H and called the Yosida approximation of A. For λ > 0, it is given by

J λ = (id +λA) -1 the resolvent of A, A λ = 1
λ (id -J λ ) the Yosida approximation of A. Recall that J λ and A λ are single valued, J λ is non-expansive, A λ is monotone and Lipschitzian with constant 1/λ, and therefore maximal monotone. Moreover,

dom(J λ ) = dom(A λ ) = H.
Furthermore, for all λ > 0,

       A λ (x) ∈ A(J λ (x)), for every x ∈ dom(A), A λ (x) ≤ inf{ ν : ν ∈ A(x)}, J λ (x) → x, as λ → 0, A λ (x) → ν 0 ∈ A(x), as λ → 0,
where ν 0 is the element of minimal norm in A(x).

The selection problem

Given two sets X, Y and a multivalued map F : X ⇒ Y , a selection of F is a singlevalued map f : X → Y such that f (x) ∈ F (x) for all x ∈ X. When X, Y both have topological structure, it is natural to look for continuous selections. If X = Ω has a measure-theoretic structure, then we seek for measurable selections. So, our main preoccupation in this section is to provide some known selections theorems.

Example 3. An u.s.c. multivalued map need not have a continuous selection. We consider the u.s.c. multivalued map F : R ⇒ R defined by

F (x) =      -1, if x < 0, [0, 1], if x = 0, 1, if x > 0.
It is clear that F cannot have a continuous selection. Note that F (x) = ∂ϕ(x) where ϕ(x) = |x| (the subdifferential in the sense of convex analysis, see section 3.2.10).

However, an u.s.c. mapping can be approached by a continuous single-valued almost selection that called an approximate selection.

Chapter 3 Mathematical backgrounds Definition 3.3.23. Let f : R n → R be a given function, and let x be a given point in R n . The function f is said to be locally Lipschitz in x if there exist a scalar κ and a positive number ε such that

f (x 1 ) -f (x 2 ) ≤ κ x 1 -x 2 , for all x 1 , x 2 ∈ B(x, ε).
Theorem 3.3.24 (The approximate selection theorem [START_REF] Aubin | Differential Inclusions[END_REF]). Let X be a metric space, Y a Banach space, and F a map from X into the convex subsets of Y be u.s.c. Then for every ε > 0 there exists a locally Lipschitzian map f ε from X to Y such that its range is contained in the convex hull of the range of F and gph(f ε ) ⊂ gph(F ) + εB. Now we give the celebrated Michael's selection theorem.

Theorem 3.3.25 ( Michael's selection theorem [START_REF] Aubin | Differential Inclusions[END_REF]). Let X be a metric space, Y a Banach space, and F from X into the closed convex subsets of Y be l.s.c. Then there exists f : X → Y , a continuous selection from F .

Next, we turn our attention to measurable selections of measurable multivalued maps. The first result in this direction is the so-called Kuratowski, Ryll-Nardzewski selection theorem. The technique of its proof is similar to the one given for Michael's selection theorem, see [START_REF] Kuratowski | A general theorem on selectors[END_REF]. Definition 3.3.26. A Hausdorff topological space (X, τ ) (τ being the Hausdorff topology of X), is said to be a Polish space, if it is separable and there exists a metric on X for which the topology τ is complete. Theorem 3.3.27 ( Kuratowski, Ryll-Nardzewski selection theorem [START_REF] Kuratowski | A general theorem on selectors[END_REF]). If (Ω, Σ) is a measurable space, Y is a Polish space, and F : Ω ⇒ Y is measurable. Then F admits a measurable selection.

The following result is due to Fryskowski and Bressan-Colombo for maps taking values in a separable L 1 space with decomposable values.

Let (Ω, Σ, µ) be a measurable space, where µ is a nonatomic probability measure on Ω, Y a Banach space, and L 1 (Ω, Y ) denotes the Banach space of µ-integrable functions from Ω to Y .

Definition 3.3.28. A set K ⊆ L 1 (Ω; Y ) is decomposable if u.χ A + v.χ Ω\A ∈ K whenever u, v ∈ K, A ∈ Ω.

Some results on sweeping processes

The collection of all nonempty decomposable subsets of L 1 (Ω, Y ) is denoted by D(L 1 (Ω; Y )). Theorem 3.3.29 ( Fryskowski, Bressan-Colombo selection theorem [START_REF] Bressan | Extensions and selections of maps with decomposable values[END_REF]). Let X be a separable metric space, and let F : X ⇒ D(L 1 (Ω, Y )) be a l.s.c. multivalued map with closed decomposable values. Then F has a continuous selection.

The last two theorems will be used in Chapter 4. This chapter deals with the existence of solutions-tube of a sweeping process in the non-convex framework.

Fixed point theory

Here, we shall focus our attention on only two results in this direction, the Schauder's and Kakutani's fixed point theorems. It is important to emphasize that Schauder's fixed point theorem is given for single-valued maps. Moreover, Kakutani's fixed point theorem is a generalization of this latter for multivalued maps. Theorem 3.3.30 ( The Schauder's fixed point theorem [START_REF] Granas | Fixed Point Theory[END_REF]). Let K be a nonempty closed convex bounded subset of a Banach space X and let A : K → X be a compact operator such that A(K) ⊂ K. Then A has at least one fixed point. Theorem 3.3.31 ( Kakutani's fixed point theorem [START_REF] Aubin | Differential Inclusions[END_REF]). Let K be a compact convex subset of a Banach space X and let F from K into its compact convex subsets be an u.s.c. multivalued mapping. Then F has a fixed point, i.e., ∃ x such that x ∈ F (x).

Introduction to differential inclusions

In 1890, Peano [START_REF] Peano | Démonstration de l'intégrabilité des équations différentielles ordinaires[END_REF] showed that for T > 0, the Cauchy problem

(3.1) x (t) = f (t, x(t)), t ∈ [0, T ], x(0) = x 0 ,
where f : [0, T ] × R n → R is continuous, has local solutions although the uniqueness does not hold in general. There is a great variety of motivations that led mathematicians to study dynamical systems having velocities not uniquely determined by the state of the system, but depending loosely upon it. In other words, to replace (3.1) by differential inclusions

x ∈ F (x),
where F is a multivalued map that associates to the state x of the system the set of feasible velocities. These motivations arise from its potential applications in different fields, besides mathematical and physical motivations, social and biological sciences provide many problems governed by differential inclusions.

In what follows we shall give some existence results to differential inclusions of the form

(3.2) x (t) ∈ F (t, x(t)), x(0) = x 0 .
Its important for a differential inclusion to make the following assumption: a solution x(•) has to be, at least an absolutely continuous function.

Example 4. The Cauchy problem

x ∈      1, x < 0, [-1, 1], x = 0, t ∈ [0, T ], -1,
x > 0.

x(0) = x 0 , has no continuously differentiable solution whenever x 0 = 0 and T > x 0 .

Definition 3.3.32. A function x : [α, β] → R n is called absolutely continuous if for any ε > 0, there exists δ > 0 such that, for any countable collection of disjoint subintervals

[α k , β k ] of [α, β] such that (β k -α k ) < δ, we have |x(β k ) -x(α k )| < ε.
An absolutely continuous function has a finite derivative x except at most on a set of measure zero, it has the corresponding integral equation:

x(β) -x(α) = β α x (s)ds.
The conditions to be imposed on the multivalued mapping F in order to have solutions for a differential inclusion are of two kinds: regularity conditions on the map and topological or geometric conditions on the sets F (x). Various combinations are possible.

Some results on sweeping processes

Recall that, The set of solutions of (3.2) is the map x 0 → S(x 0 ) defined from a subset Ω of R n into the subsets of C([0, T ], R n ), that associates to a point x 0 the set of solutions to the differential inclusion (3.2). The map S is u.s.c. with compact values when the mapping F is u.s.c., but S(x 0 ) need not be closed if F is only l.s.c. Moreover, S(x 0 ) is connected in both situations, but need not be convex.

In this subsection, we choose to present three kinds of differential inclusions: with u.s.c. maps and with l.s.c. or continuous ones.

Case of upper semi-continuous maps

The most interesting result in the upper semi-continuous case is analogous to Peano's theorem; this latter stats the existence of a local solution to a differential inclusion for upper semi-continuous maps with compact convex values. We recall that if K is a closed convex subset, m(K) = proj K (0) is the element of K with the minimal norm.

Where proj denote the usual projection on convex sets. Definition 3.3.33. We say that a solution to the differential inclusion (3.2) is a slow solution if, for almost all t ∈ [0, T ], x (t) = m(F (t, x(t))). Theorem 3.3.34. Let X be a Hilbert space, Ω ⊂ R ×X be an open set containing (0, x 0 ). Let F be an u.s.c. map from Ω into the nonempty closed convex subsets of X. We assume that (x, t) → m(F (t, x)) is locally compact. Then there exists T > 0 and an absolutely continuous function x(•) defined on [0, T ], a solution to the differential inclusion x (t) ∈ F (t, x(t)), x(0) = x 0 .

Lemma (ii) for any x ∈ R n there exists a measurable function t → f (t, x) satisfying f (t, x) ∈ F (t, x);

(iii) there exists a function b(•) ∈ L 1 ([0, T ]; R n ) such that f (t, x) ≤ b(t), ∀t ∈ [0, T ].
Then for any x 0 ∈ R n there exists a solution x(•) to differential inclusion (3.2).

Case of lower semi-continuous maps

If F is a l.s.c. map with closed convex values, Michael's Selection Theorem guarantees the existence of continuous selections.

Theorem 3.3.37. Let F be a l.s.c. multivalued mapping, from some open region Ω ⊂ R × R n into the nonempty, closed and convex subsets of R n . Let (0, x 0 ) ∈ Ω.

Then there exists some interval I = (ω -, ω + ), ω -< 0 < ω + and at least one continuously differentiable function x : I → R n , a solution to the Cauchy problem for the differential inclusion

x (t) ∈ F (t, x(t)), x(0) = x 0 .
Moreover, either ω + = +∞ or the solution x(t) tends to the boundary of Ω as t → ω + , and analogously for ω -.

Case of continuous maps

When F is a continuous multivalued mapping, the problem (3.2) has also a slow solution on a maximal interval of existence.

Theorem 3.3.38. Let F be a continuous map with closed convex values defined on an open subset Ω ⊂ R × R n that contains (0, x 0 ). Then there exists some interval (ω -, ω + ), ω -< 0 < ω + , on which a slow solution issued from x 0 does exist. Moreover, ω + = +∞ or the slow solution tends to the boundary of Ω when t tends to ω + .

Some references for these and other results regarding the existence of solutions for differential inclusions are [START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Aubin | Set-valued analysis[END_REF][START_REF] Deimling | Multivalued Differential Equations[END_REF].

Existence of solutions-tube

Motivated by the works of M. Frigon we shall investigate in chapter 4 the existence of solutions-tube for a time-dependent sweeping process. To this aim, we present in this section the notion of solutions-tube introduced by Frigon in [START_REF] Frigon | Théorèmes d'existence de solutions d'inclusions différentielles[END_REF]. Note that, this concept is a generalization of the notion of upper and lower solutions in the case of real valued functions, described for instance in [START_REF] Agarwal | A note on upper and lower solutions for first order inclusions of upper semicontinuous or lower semicontinuous type[END_REF].

We shall consider the following differential inclusion

(3.3) x (t) ∈ F (t, x(t)), a.e. t ∈ [0, T ], x(0) = x 0 , where F : [0, T ] × R ⇒ R. Definition 3.3.39. A function β ∈ W 1,1 ([0, T ]) is said to be an upper solution for (3.3) if (a) For almost every t ∈ [0, T ] there exists v ∈ F (t, β(t)) with v ≥ β (t), i.e., F (t, β(t)) ∩ [β (t), ∞) = ∅, (b) and β(0) ≥ x 0 .
Similarly, a function α ∈ W 1,1 ([0, T ]) is said to be a lower solution for (3.3) if (a) For almost every t ∈ [0, T ] there exists v ∈ F (t, α(t)) with v ≤ α (t), i.e.,

F (t, α(t)) ∩ (-∞, α (t)] = ∅, (b) and α(0) ≤ x 0 . Theorem 3.3.40. Suppose F : [0, T ] × R ⇒ K where K is a nonempty, convex, compact subset of R, satisfies the following conditions: (i) t → F (t, x) is measurable for every x ∈ R, (ii) x → F (t, x) is u.s.c. for a.e. t ∈ [0, T ],
(iii) for each r > 0, there exists h r ∈ L 1 [0, T ] with |F (t, x)| ≤ h r (t) for a.e. t ∈ [0, T ] and x ∈ R with |x| ≤ r.

Chapter 3 Mathematical backgrounds

Moreover, assume there exists α, β ∈ W

: [0, T ] × R n ⇒ R n .
In what follows, we introduce the notion of solutions-tube of (3.3), as well as some results concerning the existence of solutions of (3.3) in the case where this problem has a solution-tube.

Definition 3.3.42. Let (v, r) ∈ W 2,1 ([0, T ], R n ) × W 2,1 ([0, T ], R + ).
We say that (v, r) is a solution-tube of (3.3) if the following conditions hold:

1. for a.e. t ∈ [0, T ] and for any x ∈ R n such that x -v(t) = r(t), there exists y ∈ F (t, x) such that

x -v(t), y -v (t) ≤ r(t)r (t);

2. v (t) ∈ F (t, v(t)) for a.e. on {t ∈ [0, T ] : r(t) = 0};

3. x 0 -v(0) ≤ r(0).

Remark 3.3.43. In the scalar case, if α ≤ β ∈ W 2,1 ([0, T ], R) are respectively, lower and upper solutions of (3.3), then

v = α + β 2 and r = β -α 2 ,
is a solution-tube of (3.3).

Lemma 3.3.44. Let F : [0, T ] × R n ⇒ R n be a multivalued mapping. Assume there exist v ∈ W 1,1 ([0, T ], R n ) and r ∈ W 1,1 ([0, T ], R + ) such that
Some results on sweeping processes

(i) x 0 -v(0) ≤ r(0); (ii) F (t, x) ⊂ {y ∈ R n : x -v(t), y -v (t) < x -v(t) r (t)} a.e. t ∈ [0, T ] and for all x ∈ R n such that x -v(t) > r(t).
Then, any solution

x ∈ W 1,1 ([0, T ], R n ) of (3.3) satisfies x(t) -v(t) ≤ r(t) for all t ∈ [0, T ]. Theorem 3.3.45. Let F : [0, T ] × R n ⇒ R n be
v, r) ∈ W 1,1 ([0, T ], R n )× W 1,1 ([0, T ], R + ) a solution-tube of (3.3). Then, problem (3.3) has a solution x ∈ W 1,1 ([0, T ], R n ) satisfies x(t) -v(t) ≤ r(t) for all t ∈ [0, T ].
For proofs and for more informations see [START_REF] Frigon | Boundary and periodic value problems for systems of nonlinear second order differential equations[END_REF][START_REF] Frigon | Théorèmes d'existence de solutions d'inclusions différentielles[END_REF][START_REF] Frigon | Existence Results for Some Initial and Boundary Value Problems Without Growth Restriction[END_REF][START_REF] Frigon | Multiplicity results for systems of first order differential inclusions[END_REF].

Existence of monotone solutions

Let K be a subset of a Hilbert space X and F (x) a multivalued mapping from K into X with nonempty compact values. Let be a preorder defined on K, i.e., a binary relation x y which is (i) ∀x ∈ K, x x (reflexive), (ii) if y x and z y, then z x (transitive).

Let [0, T ] be any finite interval (T > 0). We say that an absolutely continuous function x from [0, T ] into X is a monotone trajectory for F starting at

x 0 ∈ K if        (i) x (t) ∈ F (x(t)) for almost all t in [0, T ], (ii) x(0) = x 0 , (iii) x(t) ∈ K for all t ∈ [0, T ], (iv) if t ≥ s then x(s) x(t).
The preorder is characterized by the multivalued mapping P : K ⇒ K defined by P (x) = {y ∈ K such that y x}. P (x) is the set of elements y better than x.

Definition 3.3.46. a multivalued map P which satisfies the following properties:

(i) x ∈ P (x), for any x ∈ K (reflexivity); (ii) z ∈ P (y), y ∈ P (x) ⇒ z ∈ P (x) (transitivity), is called a preorder.

Chapter 3 Mathematical backgrounds

So, a preorder is defined by y x if y belongs to P (x).

Definition 3.3.47. We say that a trajectory x(•) defined on [0, T [ is monotone if and only if ∀t, s ≥ 0, t ≥ s, we have that x(t) ∈ P (x(s)).

In this section, we recall the following results from Haddad [START_REF] Haddad | Monotone trajectories of differential inclusions and functional differential inclusions with memory[END_REF] and Aubin and Cellina [START_REF] Aubin | Differential Inclusions[END_REF] respectively. Note that T P (x) (x) denotes the tangent cone to P (x) at x, see Section 3.4.1.

Theorem 3. 3.48 ([47]). Let K be a nonempty, compact convex subset of R n . Let F be an u.s.c. map from K to R n with compact convex values, and let P : K ⇒ K be a continuous preorder with compact convex values satisfying

F (x) ∩ T P (x) (x) = ∅, for all x ∈ K.
Then, for all x 0 in K, there exists a P -monotone trajectory x : [0, +∞) → R n which is a solution to the differential inclusion

x (t) ∈ F (x(t)), x(0) = x 0 .
Theorem 3.3.49 ([2], Theorem 4.2.3, page 182). Let K be a nonempty compact subset of R n , F be an u.s.c. from K to R n with compact convex values, and let P : K ⇒ K be a l.s.c. preorder with a closed graph. Assume that the following condition is satisfied

F (x) ∩ T P (x) (x) = ∅, for all x ∈ K.
Then, for all x 0 in K, there exists a P -monotone trajectory x : [0, +∞) → R n which is a solution to the differential inclusion

x (t) ∈ F (x(t)), x(0) = x 0 .

Non-smooth analysis

In this section, we recall some basic definitions and properties on the tangent and normal cones. This latter is an essential tool in the study of Moreau sweeping processes. We also present several properties of prox-regular functions and sets which are useful for generalizing convex differential inclusions to non-convex ones. We end the section by a brief introduction to Moreau sweeping processes.

Normal cones and tangent cones

We recall the notions of tangent and normal cones (see, for instance [START_REF] Aubin | Differential Inclusions[END_REF]).

Definition 3.4.1. Let C be a closed set of R n . For x ∈ C, we define by

T C (x) = {v ∈ R n | lim inf h→0 + d C (x + hv)/h = 0},
the tangent cone to C at x and by (id +λ∂I C ) -1 (x) = proj C (x), and the Yosida approximation

N C (x) = T C (x) -= {p ∈ R n | p, v ≤ 0 for all v ∈ T C (x)
∂I C (x) =      ∅, if x / ∈ C, {0}, if x ∈ Int(C), The outward normal cone of C, if x ∈ ∂C.
(∂I C ) λ (x) = 1 λ (x -proj C (x)) is the subdifferential of the function (I C ) λ (x) = 1 2λ x -proj C (x) 2 .
Let us recall some well-known properties of tangent and normal cones associated with convex sets (see, [START_REF] Aubin | Differential Inclusions[END_REF]). Proposition 3.4.4. Let C be a convex subset of R n and x ∈ C, then

T C (x) = cl{λ(y -x)| y ∈ C and λ > 0}, N C (x) = {p ∈ R n | p, x ≥ p, y for all y ∈ C}.
Thus, T C (x) and N C (x) are closed convex cones. Theorem 3.4.5. Let C be a closed convex subset of R n . Then

(i) x → N C (x) has a closed graph, (ii) x → T C (x) is lower semi-continuous.
Moreover, let x 0 ∈ C ⊂ R n , and suppose there exists r > 0 such that C ∩ B(x 0 , r) is compact. Then, the normal cone N C (•) is closed at x 0 . Corollary 3.4.6. The normal cone N C of a nonempty, closed, convex subset C of R n is maximal monotone.

Prox-regular functions and sets

The class of (uniformly) ρ-prox-regular functions was introduced and studied thoroughly in [START_REF] Bernard | Uniform prox-regularity of functions and epigraphs in Hilbert spaces[END_REF][START_REF] Poliquin | Prox-regular functions in variational analysis[END_REF][START_REF] Poliquin | Local differentiability of distance functions[END_REF] and such locally Lipschitz functions can be characterized via the prox-regularity of the epigraph. This class of functions is large enough. From the geometrical point of view, this type contains any convex proper and lower semicontinuous function, a lower-C 2 function and strongly amenable function such that it is possible to draw a parabola under their epigraphs with a curvative bounded from below by ρ. The class of prox-regular functions is much larger than the ρ-proxregular one.

To be beyond the convexity, the class of prox-regular sets appears under different names in the literature (weakly convex sets). First by Federer [START_REF] Federer | Curvature measures[END_REF], where he introduced these sets in R n as the "sets with positive reach." This notion extends the Steiner polynomial formula to a much larger class of subsets of R n than those of convex sets and shares with it many excellent properties concerning the applications in optimization, control theory, and others. The concept of a prox-regular set C in a Hilbert space at a point x ∈ C is somehow related to the hypo-monotonicity of some truncation of the proximal normal cone mapping N C (•) around this point x.

Prox-regular functions

Let f : R n → R ∪ {+∞} be a function and let x ∈ dom f . The proximal subdifferential ∂ P f (x) and the Fréchet subdifferential ∂ F f (x) are defined as follows.

A vector x * is in ∂ P f (x) if there exists constant real numbers c > 0 and ε > 0 such that for all x ∈ B(x, ε) (see for instance [START_REF] Clarke | Nonsmooth analysis and control theory[END_REF])

x * , x -x ≤ f (x ) -f (x) + c 2 x -x 2 .
In a similar way, a vector x * in ∂ F f (x) if for each real ε > 0 there exists some ζ > 0 such that for all x ∈ B(x, ζ) (see for instance [START_REF] Mordukhovich | Nonsmooth sequential analysis in Asplund spaces[END_REF])

x * , x -x ≤ f (x ) -f (x) + ε x -x 2 . If f (x) is not finite one define ∂ P f (x) = ∂ F f (x) = ∅.
Considering the weak sequential outer limit lim sup

x→ f x ∂ P f (x) = {w = lim xn : xn ∈ ∂ P f (x n ), x n → f x},
where x n → f x means that x n -x → 0 and f (x n ) → f (x), moreover (see for example [START_REF] Ioffe | Proximal analysis and approximate subdifferentials[END_REF]) lim sup

x→ f x ∂ P f (x) = lim sup x→ f x ∂ F f (x)
. This set is generally called the limiting subdifferential of f at x and it will be denoted by ∂f (x). Definition 3.4.7 ([70]). We say that f is prox-regular at x for v ∈ ∂f (x) if there exist some ρ > 0 and > 0 such that for all (x, v)

∈ gph ∂f with x -x < , |f (x) -f (x)| < and v -v < we have f (x ) ≥ f (x) + v, x -x - ρ 2 x -x 2 , for all x ∈ B(x, ).
If the property holds for all vectors v ∈ ∂f (x), the function f is said to be proxregular at x. When f is prox-regular at each point of E ∩ dom ∂f , we will say that f is prox-regular on the set E. Proposition 3.4.8 (cf. [START_REF] Bernard | Uniform prox-regularity of functions and epigraphs in Hilbert spaces[END_REF]). A function f is (uniformly) ρ-prox-regular on some neighborhood of x 0 ∈ R n if and only if there exist > 0 and ρ > 0, for any (x, v) ∈ gph ∂f with x -x 0 < , one has

f (x ) ≥ f (x) + v, x -x - ρ 2
x -x 2 , for all x ∈ B(x 0 , ).

Definition 3.4.9 (subdifferentially continuous functions). A function f : R → R is subdifferentially continuous at x for v, where v ∈ ∂f (x), if for every δ > 0 there exists > 0 such that f (x) -f (x) < δ whenever x -x < and v -v < with v ∈ ∂f (x).

A proper, convex function is prox-regular and subdifferentially continuous.

f (x) = 1, if x < 1, x -2, if x ≥ 1.
f is lower semi-continuous and prox-regular everywhere. The graph of ∂f has a peculiarity at (x, v) = (0, 1). As (x, v) → (x, v) in gph ∂f along this branch we have

f (x) = 1, so f (x) f (x) = -1.
In what follows we suppose that f is a subdifferentially continuous functions.

Theorem 3.4.10 (cf. [START_REF] Poliquin | Prox-regular functions in variational analysis[END_REF]). When f is lower semi-continuous, the following are equivalent.

(a) The function f is ρ-prox-regular on K.

Some results on sweeping processes (b) The vector v is a proximal subgradient to f at x, and there exist ρ > 0 such that ∂f + ρ id is monotone, i.e.,

v 2 -v 1 , x 2 -x 1 ≥ -ρ x 2 -x 1 2 , whenever i = 1, 2 one has x i ∈ K and v i ∈ ∂f (x i ). Definition 3.4.11. A multivalued mapping A : R n ⇒ R n is ρ-hypomonotone on a set E if A + ρ id is monotone on E. If E = R n we will say that A is ρ-hypomonotone.
It is important to emphasize that, a proper lower semi-continuous function, f is ρ-prox-regular if and only if it is expressible as the difference between a finite convex function and a positive multiple of 1 2 • 2 in the sense of the next Lemma. Also, the class of uniformly prox-regular functions coincides with the class of locally Lipschitz weakly convex functions. Lemma 3.4.12 (cf. [START_REF] Bernard | Uniform prox-regularity of functions and epigraphs in Hilbert spaces[END_REF]). Let f be a proper, lower semi-continuous function. Then the following assertions are equivalent:

(a) ∂f is ρ-hypomonotone, (b) f = g -ρ 2
• 2 with g proper, lower semi-continuous and convex. We recall some results on Moreau envelopes and proximal mappings. For their proofs and for more information you can see [START_REF] Rockafellar | variational Analysis[END_REF].

Theorem 3.4.13. Let f : R n → R be lower semi-continuous, proper, and convex. Then the following properties hold for every λ > 0.

(a) The proximal mapping P λ f is single valued and continuous.

(b) The envelope function e λ f is convex and continuously differentiable, the gradient being

∇e λ f (x) = λ -1 [x -P λ f (x)].
Proposition 3.4.14. For a proper, lower semi-continuous function f : R n → R and any λ > 0, the proximal mapping P λ f is monotone. If f is also convex, then P λ f is maximal monotone and non-expansive, hence single valued. More generally in the possible absence of convexity, for a given λ > 0, the following conditions are equivalent:

(a) P λ f is maximal monotone, (b) f + λ -1 j is convex for j = 1 2 • 2 ,
(c) id +λ∂f is monotone.

Chapter 3 Mathematical backgrounds

Prox-regular sets

Proximal regularity can be defined for sets as well as for functions.

Definition 3.4.15 ([70]). A set C ⊂ R n is prox-regular at x for v, where x ∈ C and v ∈ N C (x), if C is locally closed at x and there exist > 0 and ρ > 0 such that whenever x ∈ C and v ∈ N C (x) with x -x < and v -v < , then x is the unique nearest point of {x | x -x < } to x + ρ -1 v.

The proximal subdifferential ∂ P I C (x) of the indicator function is the proximal normal cone N P C (x) of C at x. This means that a vector v ∈ R n is a proximal normal vector of C at x ∈ C if and only if there are constant real numbers ρ > 0 and

ζ > 0 such that v, x -x ≤ ρ 2 x -x 2 for all x ∈ C ∩ B(x, ζ).
The proximal cone is given by the equality (see [START_REF] Clarke | Nonsmooth analysis and control theory[END_REF])

N P C (x) = {v ∈ R n : ∃ρ > 0 such that x ∈ proj C (x + ρv)}.
The uniformity of the positive constant ρ for the unit proximal normal vectors to C leads to the concept of uniformly ρ-prox-regular sets. In general, we only have

N P C (x) ⊂ N C (x)
and the inclusion can be strict. However, for a ρ-prox-regular set C, we have N P C (x) = N C (x), see [START_REF] Poliquin | Local differentiability of distance functions[END_REF]. Uniform prox-regularity defined in [START_REF] Poliquin | Local differentiability of distance functions[END_REF] is as follows.

Definition 3.4.16 (cf. [START_REF] Poliquin | Local differentiability of distance functions[END_REF]). A closed set C is ρ-prox-regular with constant ρ > 0 if whenever

x ∈ C and v ∈ N C (x) with v < 1, then v, x -x ≤ ρ 2 x -x 2 , for all x ∈ C.
This condition means that x is the unique nearest point of C to x + ρ -1 v.

Example 4.

1. Convex sets: a nonempty closed set of a real Hilbert space is 0-prox-regular if and only if it is convex.

Complement of an open ball:

Let H be a real Hilbert space with H = {0}.

Then, the set C = H\B(0, ρ) is ρ-prox-regular. (a) The set C is ρ-prox-regular, (b) there exist ρ > 0 such that N C + ρ id is monotone, i.e., whenever for i = 1, 2 one has

x i ∈ C and v i ∈ N C (x i ) v 2 -v 1 , x 2 -x 1 ≥ -ρ x 2 -x 1 2 .
The following proposition gives a precise connection between the prox-regularity of functions and sets. Proposition 3.4.18 (cf. [START_REF] Poliquin | Prox-regular functions in variational analysis[END_REF]). A closed set C is ρ-prox-regular if and only if its indicator function I C is ρ-prox-regular on C.

Sweeping processes

In this section, we shall illustrate the sweeping process. This latter is a timedependent evolution inclusion modeling the displacement of a point constrained to belong to a moving set C(t), which velocity in the opposite of the normal cone to C(t). Jean Jacques Moreau introduced the sweeping process at the Convex Analysis Seminar of the University of Montpellier in the seventies [START_REF] Moreau | Evolution problem associated with a moving convex set in a Hilbert space[END_REF][START_REF] Moreau | Liaisons unilatérales sans frottement et chocs inélastiques[END_REF]. Its simplest form is

(3.4) -x (t) ∈ N C(t) (x(t)).
The problem consists in finding an absolutely continuous R n -valued mapping x satisfying the inclusion .3.1 . Where the set N C(t) (x(t)) is the normal cone to the nonempty, closed convex set C(t) at the point x(t).

The evolution process given by the sweeping process may be illustrated in a mechanical language. It is evident if C(t) possesses a nonempty interior: The moving point x → x(t) remains at rest as long as it happens to lie in the interior; when x(t) lies on the boundary of the moving set, there exists a possibly non zero element p(t) ∈ N C(t) (x(t)) such that -x (t) = p(t) and it can only proceed in a normal inward direction, as it pushed by this boundary. Consequently, the only way to make the point x(•) move is to "sweep" it with the set C(•).

Another mechanical interpretation is the following (see, Moreau [START_REF] Moreau | Evolution problem associated with a moving convex set in a Hilbert space[END_REF]): Given a Hilbert space H od dimension 2. Take H × R as the physical three-dimensional space, with R corresponding to a vertical axis oriented downward. Take the set S = {(x, t) ∈ H × R : u ∈ C(t)} as a solid cavity, supposed to have a smooth boundary, and the curve γ = {(x, t) ∈ H × R : u = x(t)}, i.e., the graph of the unknown function x, as a tiny stationary water stream falling down the cavity. Condition (3.4) is equivalent to the statements:

(i) any arc of this stream which happens to be loose from the cavity wall is rectilinear and vertical;

(ii) when water is running over the wall, it describes a line orthogonal to the level curves of the wall surface, i.e., a line of steepest descent; this agrees with hydrodynamics under the simplifying assumption that inertia may be neglected compared to friction and gravity;

(iii) the minus sign in (3.4) precisely expresses that such a flow in contact with the wall may take place only on the part of this wall exposed upward. If it crosses the rim of a possible overhang, water will get loose from the surface and fall vertically down as formulated in (i).

Example 6 ([18]

). Consider the mechanical system illustrated in Figure .3.5 . The linear spring with stiffness k > 0 is acted upon by two forces that balance: the elastic force k(x -x e (t)) with x e (t) a forced displacement, and the friction force -µmg sgn(x ), where µ > 0 is the coefficient of friction, g is the gravity acceleration. The equilibrium of the massless spring states that k(x -x e (t)) ∈ -µmg sgn(x ). This is rewritten equivalently as The first-order sweeping process is extended to its perturbed version as

x ∈ -N - µmg k +xe(t), µmg k +xe(t) (x(t)).
(3.5) x (t) ∈ -N C(t) (x(t)) + F (t, x(t)),
where F is a multivalued mapping. In many works F is considered as a singlevalued vector field f . Applications are in crowed motion [START_REF] Maury | A discrete contact model for crowd motion[END_REF][START_REF] Venel | A numerical scheme for a class of sweeping processes[END_REF], mechanics [START_REF] Moreau | Evolution problem associated with a moving convex set in a Hilbert space[END_REF][START_REF] Moreau | Liaisons unilatérales sans frottement et chocs inélastiques[END_REF], electronic circuit [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF][START_REF] Brogliato | Nonsmooth mechanics. Models, dynamics and control[END_REF].

Example 7 ([18]

). Consider the circuit in Figure .3.6 . Its dynamics are given by

-x (t) - R L x(t) ∈ N [i(t),+∞] (x(t)).
Where x(t) is the current through the indicator/resistor, i(t) is a current source, and ν(t) is the voltage across the diode D.

Résumé du chapitre

Ce chapitre est consacré aux résultats préliminaires de quelques sujets spécifiques dont nous avons besoin dans les chapitres postérieurs. Ainsi, nous faisons quelques rappels théoriques d'analyse comme, par exemple, quelques points sur l'analyse convexe, particulièrement, nous présentons quelques propriétés d'ensemble et de fonctions convexes ainsi que des propriétés différentielles et sous-différentielles de fonctions convexes. Par la suite, nous fournissons quelques outils sur les fonctions convexes-concaves ainsi qu'une introduction aux problèmes de minimax.

Ceci est suivi par quelques rappels sur les applications multivoques qui sont nécessaires pour l'étude d'inclusions différentielles. Cela est suivi par, quelques concepts de semi-continuités diverses (la semi-continuité supérieure, inférieure et la continuité Lipschitzienne). Nous rappelons quelques résultats sur la théorie de point fixe qui est une technique fondamentale dans nos preuves. Une introduction aux inclusions différentielles est présentée, aussi bien que, quelques résultats d'existence, essentiellement, l'existence de tube-solutions et solutions monotones.

Dans la dernière section de cette chapitre, nous regardons de près quelques notions de base d'analyse non lisse, par exemple, nous présentons quelques définitions et propriétés des cônes tangents et normaux. Par la suite, nous donnons une revue court sur les propriétés d'ensemble et fonctions prox-réguliers, ainsi que, quelques points sur les propriétés différentielles de fonctions prox-régulières. Finalement, nous fournissons une exposition brève des processus de rafle.

CHAPTER 4 EXISTENCE OF SOLUTIONS-TUBE FOR NON-CONVEX SWEEPING PROCESSES

Introduction

We establish some existence results of solutions-tube for non-convex sweeping processes. Our approach employs properties belonging to the class of prox-regular sets and functions as well as fixed point theory techniques.

In this chapter, we will be concerned, for any fixed x 0 ∈ C(0), with the differential inclusion (4.1)

   x (t) ∈ -N C(t) (x(t)) + F (t, x(t)), a.e. t ∈ [0, T ], x(0) = x 0 ∈ C(0), x(t) ∈ C(t), ∀t ∈ [0, T ],
here, T > 0 is a non-negative real number and for each t

∈ [0, T ], C(t) is a nonempty closed subset of R n with N C(t) (x(t)) is the normal cone to C(t) at x(t) for all t ∈ [0, T ], and F : [0, T ] × R n ⇒ R n is a multivalued map.
The sweeping process has been studied by several authors under various assumptions and has found a great number of applications. It has proved useful in many different areas. In particular, it has been played an important role in mechanics (see for example [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems-Shocks and Dry Friction[END_REF][START_REF] Moreau | Liaisons unilatérales sans frottement et chocs inélastiques[END_REF]). In the non-convex setting, Colombo and Monteiro Marques [START_REF] Colombo | Sweeping by continuous proxregular set[END_REF] proved local and global existence as well as the uniqueness of solutions of a system without perturbation, when the sets C(t) are prox-regular contained in the Some results on sweeping processes interior of a suitable subset moving continuously. Thibault [START_REF] Thibault | Sweeping process with regular and nonregular sets[END_REF] showed the existence of a solution of (4.1) when the sets C(t) are non-convex moving continuously and F is a scalarly upper semi-continuous multivalued mapping. For more information on the perturbed time-dependent sweeping process, we refer to [START_REF] Bounkhel | Nonconvex sweeping process and prox-regularity in Hilbert space[END_REF][START_REF] Castaing | Evolution equations governed by the sweeping process[END_REF][START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF]. The statedependent sweeping process (depending on time and solution) when C(t) ≡ C(t, x) is a convex set was studied in [START_REF] Kaadoud | Résolution de problèmes de rafle et application à un problème de frottement. (French) [ Solution of sweeping problems and application to a friction problem[END_REF], see also [START_REF] Chemetov | Non-convex quasi-variational differential inclusions[END_REF][START_REF] Haddad | Reduction of state dependent sweeping process to unconstrained differential inclusion[END_REF] in the non-convex case. In [START_REF] Tolstonogov | Convex-valued selectors of a Nemytskii operator with nonconvex values and their applications[END_REF], Tolstonogov and Timoshin obtained the existence of a solution for a mapping F having different semi-continuity types at different points of its domain.

Moreover, numerical methods are developed in [START_REF] Brogliato | Numerical simulation of finite dimensional multibody nonsmooth mechanical systems[END_REF] for the study of systems composed of interacting rigid bodies. One of the ways uses Moreau's sweeping process, which is similar to a reflected problem. Finally, let us mention that some optimal control problems governed by variational inequalities have also been studied in several works. We refer to [START_REF] Bergounioux | Pontryagin maximum principle for optimal control of variational inequalities[END_REF][START_REF] Bonnans | Control problems with mixed constraints and application to an optimal investment problem[END_REF][START_REF] Goreac | Some Applications of Linear Programming Formulations in Stochastic Control[END_REF][START_REF] Serea | Reflected differential games[END_REF][START_REF] Serea | On reflecting boundary problem for optimal control[END_REF][START_REF] Serea | Optimality conditions for reflecting boundary control problems[END_REF] and the references therein.

In this chapter we are interested in finding L p -solution-tube to the problem (4.1) when the sets C(t) are non-convex. This type of solution extends the Hartman condition [START_REF] Hartman | On boundary value problems for systems of ordinary, nonlinear, second order differential equations[END_REF] and the notion of upper-lower solutions, was introduced by Frigon in [START_REF] Frigon | Théorèmes d'existence de solutions d'inclusions différentielles[END_REF] for a differential inclusion in the following form

x (t) ∈ F (t, x(t)), x(0) = x(1).
For more details, we refer the reader to [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF][START_REF] Frigon | Boundary and periodic value problems for systems of nonlinear second order differential equations[END_REF][START_REF] Frigon | Existence Results for Some Initial and Boundary Value Problems Without Growth Restriction[END_REF][START_REF] Frigon | Nonlinear first order initial and periodic problems in Banach spaces[END_REF]. An important tool to prove existence results for differential equations with initial or boundary conditions is the well-known method of lower and upper solutions described for instance in [START_REF] Dragoni | Il problema dei valori ai limiti studiato in grande per gli integrali di una equazione differenziale del secondo ordine[END_REF][START_REF] Perron | Ein neuer Existenzbeweis für die Integrale der Differentialgleichung y = f (x, y)[END_REF][START_REF] Pouso | Upper and lower solutions for first order discontinuous BVPs[END_REF]. A critical feature of the solution-tube method is the fact that it allows one to ensure the existence of a solution of the considered problem lying in a tube, i.e., we have information pertaining to the presence and location of the solutions. Another compelling property is the fact that it allows for obtaining multiplicity results, i. e. The existence of several solutions for differential inclusions located in disjoint tubes. Note that our approach does not use viability results as in [START_REF] Aubin | Set-valued analysis[END_REF]. Instead, we use a method inspired by [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF]. Fixed point theory are used to establish existence results for systems of first-order differential inclusions with maximal monotone terms satisfying the periodic boundary condition.

In our case, we deal with multivalued first order differential inclusion with a hypomonotone term. Additionally, this term is depending on time (time-dependent sweeping process) with initial value condition. The multivalued perturbation F satisfies a lower or an upper semi-continuity condition. We obtain our main results using Schauder's and Kakutani's fixed points (see section 3.3.3).

For more applications of the solution-tube method see e. g. [START_REF] Zhang | Semilinear second order differential inclusions with nonlinear boundary values[END_REF] for priori bounds for the solutions of semilinear second order differential inclusions or [START_REF] Frigon | Systems of first order inclusions on time scales[END_REF][START_REF] Frigon | Multiplicity results for systems of first order differential inclusions[END_REF] for multiplicity results of systems of first-order differential inclusions.

The chapter is organized as follows: In Section 4.2 we present without proofs the main existence results under various assumptions. In Section 4.3 we introduce some operators which will be needed for the proof of our main results. Supposing that the sets C(•) are uniformly prox-regular we establish existence of solutions for (4.1) in Section 4.4. The last section is dedicated to further results.

Main results

The aim of this section is to provide without proofs the main theorems of existence of solutions for the problem (4.1), i.e., an absolutely continuous function x ∈ A(x 0 ) satisfying (4.1), where

A(x 0 ) = {x ∈ W 1,p ([0, T ], R n ), such that x(0) = x 0 }, with W 1,p ([0, T ], R n ) = {x ∈ C[0, T ] : such that x is absolutely continuous and x ∈ L p ([0, T ], R n )}.
We adapt the notion of L 2 -solution-tube of the problem (4.1) introduced in the paper of M. Frigon [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF] for the periodic case to the prox-regular framework. First we shall make the following assumption: let ρ ≥ 0.

(P) The multivalued mapping C : [0, T ] ⇒ R n is continuous and has (uniform) ρ-prox-regular values.

Definition 4.2.1. Let α ∈ W 1,2 ([0, T ], R n ), and β ∈ W 1,2 ([0, T ], R). We say that (α, β) is an L 2 -solution-tube of (4.1) if there exists δ ∈ L 2 ([0, T ], R n ) such that 1. δ(t) ∈ N C(t) (α(t)) a.e. t ∈ [0, T ];
2. for a.e. t ∈ [0, T ], and every

x ∈ R n such that x -α(t) = β(t), there exists ν ∈ F (t, x) such that x -α(t), ν -δ(t) -α (t) ≤ β(t)β (t) -ρβ 2 (t) ; 3. α (t) ∈ -δ(t) + F (t, α(t)) a.e. on {t ∈ [0, T ] : β(t) = 0}; 4. x(0) -α(0) ≤ β(0).

Some results on sweeping processes

We denote by

B β (α) = {x ∈ C([0, T ], R n ) : x(t) -α(t) ≤ β(t) ∀ t ∈ [0, T ]}. Remark 4.2.2. Note that ∂I C(t) = N C(t) . Since δ(t) ∈ N C(t) (α(t))
, then one has

I C(t) (x) -I C(t) (α(t)) + ρ 2 x -α(t) 2 ≥ x -α(t), δ(t) .
Let (α, β) an L 2 -solution-tube of (4.1), then, for a.e. t ∈ [0, T ], and every x ∈ R n such that x -α(t) = β(t), there exists ν ∈ F (t, x) such that

I C(t) (x) -I C(t) (α(t)) + ρ 2 x -α(t) 2 + β(t)β (t) -ρβ 2 (t) ≥ x -α(t), ν -α (t) ,
when x ∈ C(t), we have

β(t)β (t) - ρ 2 β 2 (t) ≥ x -α(t), ν -α (t) .
To establish our main existence results, we make several assumptions:

(H) F : [0, T ] × R n ⇒ R n is a multivalued map with compact convex values such that (i) t → F (t, x) is measurable for all x ∈ R n , (ii) x → F (t, x) is u.s.c. a.e. t ∈ [0, T ]. (H) F : [0, T ] × R n ⇒ R n is a multivalued map with compact values such that (i) x → F (t, x) is l.s.c. a.e. t ∈ [0, T ], (ii) (t, x) → F (t, x) is L ⊗ B-measurable.
Here [0, T ] × R n is endowed with the σ-algebra generated by subsets C × D, where C ⊂ [0, T ] and D ⊂ R n are, respectively, Lebesgue and Borel measurable.

(H k ) For every k ≥ 0, there exists

l k ∈ L 2 ([0, T ]) such that max{ µ : µ ∈ F (t, x), x ≤ k} ≤ l k (t) a.e. t ∈ [0, T ]. (S -L 2 ) There exists (α, β) ∈ W 1,2 ([0, T ], R n ) × W 1,2 ([0, T ], [0, ∞)) an L 2 -solution- tube of (4.1).
Chapter 4 Existence of solutions-tube for non-convex sweeping processes Let us state the main existence results in the following. Note that the proofs are postponed in the next section.

Preliminary tools

We begin by constructing several operators which will be used in the proofs. Let t ⇒ C(t) be a continuous multivalued map with C(t) closed, convex and nonempty for all t ∈ [0, T ]. We associate with

N C(•) the operator N C(•) : dom(N C(•) ) ⊂ L 2 ([0, T ], R n ) ⇒ L 2 ([0, T ], R n ) defined by N C(•) (x) = {ν ∈ L 2 ([0, T ], R n ) : ν(t) ∈ N C(t) (x(t)) a.e. t ∈ [0, T ]}. The mapping N C(•) is a maximal monotone operator with dom(N C(•) ) = {x ∈ L 2 ([0, T ], R n ) : x(t) ∈ C(t) a.e. t ∈ [0, T ] and ∃ ν ∈ L 2 ([0, T ], R n ) | ν(t) ∈ N C(t) (x(t)) a.e. t ∈ [0, T ]}.
To verify the maximal monotonicity of the operator N C(•) one can use the function

f (t, x) : [0, T ] × R n → R ∪ {+∞} defined as f (t, x) := I C(t) (x),
where I C(t) is the indicator function (0 on C(t) and +∞ otherwise). Then we easily verify that (a) for each t ∈ [0, T ], f (t, •) is proper, convex and lower semicontinuous (because C(t) are closed, convex and non-empty), and (b) the function (t, x) → f (t, x) is measurable, as a consequence of the continuity of the mapping t → C(t).

It follows that f is a convex normal integrand by Lemma 3.2.9 [START_REF] Rockafellar | Integrals which are convex functionals[END_REF]. We define the function

I f : L 2 ([0, T ], R n ) → R ∪ {+∞} as I f (x(•)) := T 0 I C(t) (x(t))dt.
By Michael's continuous selection Theorem 3.3.25 [START_REF] Aubin | Differential Inclusions[END_REF] (C(•) being continuous and consequently lower semi-continuous), there exists a continuous selection x 0 (•) of C(•), which then belongs to L 2 ([0, T ], R n ). Hence, I f defines a proper, lower semicontinuous and convex function on L 2 ([0, T ], R n ). Consequently, its Fenchel subdifferential mapping ∂I f : L 2 ([0, T ], R n ) → L 2 ([0, T ], R n ) induces a maximal monotone operator by Theorem 3.2.14. We conclude by observing that we have the following equality ∂I f = N C(•) (x(•)) which ensures the maximal monotonicity of N C(•) .

Moreover, we present some mappings inspired by the operators introduced in the paper of M. Frigon [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF]. We adapted them to the prox-regular framework.

Consider (α, β) and δ given as in (S -L 2 ) (see Definition 4.2.1). Define

F : [0, T ] × R n ⇒ R n by F = F ∩ G,
where

F (t, x) = F (t, x t ), if x -α(t) > β(t), F (t, x), if x -α(t) ≤ β(t); G(t, x) =                α (t) + δ(t), if β(t) = 0, R n , if x -α(t) ≤ β(t) and β(t) > 0, {z : x -α(t), z -δ(t) -α (t) ≤ β (t) x -α(t) -ρ x -α(t) 2 }, otherwise; with (4.2) x t = x, if x -α(t) ≤ β(t), α(t) + β(t)
x-α(t) (x -α(t)), if x -α(t) > β(t).

Similarly, we define

F : [0, T ] × R n ⇒ R n by F = F ∩ G,
where

G(t, x) =          α (t) + δ(t), if β(t) = 0, R n , if x -α(t) < β(t), {z : x -α(t), z -δ(t) -α (t) ≤ β (t) x -α(t) -ρ x -α(t) 2 }, otherwise. Proposition 4.3.1. Assume (H), (H k ) and (S -L 2 ) hold. Let F : C([0, T ], R n ) ⇒ L 2 ([0, T ], R n ) be defined by F(x) = {ν ∈ L 2 ([0, T ], R n ) : ν(t) ∈ F (t, x(t)) + x(t) t a.e. t ∈ [0, T ]}
with L 2 ([0, T ], R n ) supplied with the weak topology. Then F is u.s.c with compact convex values. Furthermore, there exists l ∈ L 2 ([0, T ], [0, ∞)) such that, for all x ∈ C([0, T ], R n ) and every ν ∈ F(x),

ν(t) ≤ l(t) a.e. t ∈ [0, T ].
Proof. First, since the map F has nonempty, compact convex values. Then, it follows from (S -L 2 ) that F (t, x) is non-empty and closed convex, for every x ∈ R n and almost all t ∈ [0, T ]. Furthermore, it is easy to verify that t → F (t, x) is measurable for all x ∈ R n and that x → F (t, x) is u.s.c. a.e. t ∈ [0, T ]. Therefore, for all x ∈ C([0, T ], R n ), t → F (t, x(t)) is measurable, and consequently has a measurable selection by the Kuratowski, Ryll-Nardzewski selection theorem (see, [START_REF] Kuratowski | A general theorem on selectors[END_REF]).

Second, there exists l ∈ L 2 ([0, T ], [0, ∞)) such that for all x ∈ C([0, T ], R n ) and all ν ∈ F(x),

(4.3) ν(t) ≤ l(t) a.e. t ∈ [0, T ].
Indeed, the previous estimation follows from (H k ) and the fact that

x(t) t ≤ α 0 + β 0 .
Therefore, F has nonempty values. Third, it is easy to adapt the proof given in [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF] to show that the multivalued map F has closed convex, bounded values in L 2 ([0, T ], R n ) endowed with the strong topology, and hence has nonempty, closed convex values in the weak topology. Therefore, F has compact convex values in the weak topology by (4.3).

Moreover, F is u.s.c. Indeed, let {x n } a sequence in {x ∈ C([0, T ], R n ) : F(x) ∩ U = ∅} converging to x 0 , such that U be a weakly closed subset of L 2 ([0, T ], R n ). Owing to (4.3), a sequence ν n ∈ F(x n ) ∩ U has a weakly convergent subsequence that we denote also {ν n }; and we denote by ν its weak limit, observe that ν ∈ U , since U is weakly closed. Further, there exists µ n ∈ co{ν n , ν n+1 , . . . } such that the sequence {µ n } converges strongly to ν ∈ L 2 ([0, T ], R n ), with µ n (t) → ν(t) almost everywhere in [0, T ]. Moreover, since F has convex values and is u.s.c., we have

ν(t) ⊂ n≥1 co m≥n ν n (t) ⊂ n≥1 co m≥n F (t, x n (t)) + x(t) t ⊂ F (t, x 0 (t)) + x(t) t a.e. t ∈ [0, T ]. Therefore, ν ∈ F(x 0 ) ∩ U , hence the set {x ∈ C([0, T ], R n ) : F(x) ∩ U = ∅} is closed in L 2 ([0, T ], R n ).
C([0, T ], R n ) → L 1 ([0, T ], R n ), a continuous single valued map, such that f (x)(t) ∈ F (t, x(t)) + x(t) t a.e. t ∈ [0, T ].
Further, there exists l ∈ L 2 ([0, T ], [0, ∞)) such that,

for every x ∈ C([0, T ], R n ), f (x)(t) ≤ l(t) a.e. t ∈ [0, T ], hence f (C([0, T ], R n )) ⊂ {ν ∈ L 2 ([0, T ], R n ) : ν(t) ≤ l(t) a.e. t ∈ [0, T ]}.
Proof. Note that F has nonempty values because of (S -L 2 ). Moreover, we can prove that (t, x) → F (t, x) is L ⊗ B-measurable, and x → F (t, x) is l.s.c. a.e. t ∈ [0, T ].

Indeed, let U ⊂ R n be open, and denote by O = {x ∈ R n : F (t, x) ∩ U = ∅}. We have several cases. If β(t) = 0, F (t, x) = α (t) + δ(t) for all x ∈ R n , and hence O is open. In the case β(t) > 0, we consider x ∈ O. Whenever x -α(t) < β(t), there exists ζ > 0 such that u -α(t) < β(t) for all u ∈ B(x, ζ). Then F (t, u) = F (t, u) for all u ∈ B(x, ζ). Consequently, the lower semi-continuity of F with respect to its second variable implies that there exists a neighborhood of x in O and therefore O is open. Moreover, if x -α(t) ≥ β(t), there exists

z 0 ∈ U ∩ F (t, x t ) ∩ {z : x -α(t), z -δ(t) -α (t) ≤ β (t) x -α(t) -ρ x -α(t) 2 }.
Set ε > 0 such that B(z 0 , ε) ⊂ U and fix µ = -ε(x -α(t))/(2 x -α(t) ). For y ∈ S n-1 and λ ≥ 0.

x + λy -α(t),

z 0 + µ -δ(t) -α (t) = x -α(t), z 0 -δ(t) -α (t) + λ y, z 0 + µ -δ(t) -α (t) + x -α(t), µ ≤ x -α(t) β (t) -ρ x -α(t) 2 + λ z 0 + µ -δ(t) -α (t) - ε 2 x -α(t)
On the other hand, for λ is sub-unitary, we have

x + λy -α(t) 2 ≤ x -α(t) 2 + λ(1 + 2 x -α(t) ).
So, we have x + λy -α(t), z 0 + µ -δ(t) -α (t)

≤ x + λy -α(t) β (t) -ρ x + λy -α(t) 2 +λ(ρ(1 + 2 x -α(t) ) + |β (t)| + z 0 + µ -δ(t) -α (t) ) - ε 2 x -α(t) .
Some results on sweeping processes Now let us consider the following modified operators: for all (t, x) ∈ [0, T ] × R n ,

F ρ : [0, T ] × R n ⇒ R n given by F ρ (t, x) = F (t, x) + ρx.
Similarly, we define

F ρ : [0, T ] × R n ⇒ R n by F ρ (t, x) = F (t, x) + ρx.
The last two propositions enable us to formulate:

Proposition 4.3.3. Assume (H), (H k ), and (S -L 2 ). Let F ρ : C([0, T ], R n ) ⇒ L 2 ([0, T ], R n ) be defined by F ρ (x) = {ν ∈ L 2 ([0, T ], R n ) : ν(t) ∈ F ρ (t,
: C([0, T ], R n ) → L 1 ([0, T ], R n ) such that f ρ (x)(t) ∈ F ρ (t, x(t)) + x(t) t , a.e. t ∈ [0, T ].
Further, there exists l ∈ L 2 ([0, T ], [0, ∞)) such that,

for every x ∈ C([0, T ], R n ), f ρ (x)(t) ≤ l(t), a.e. t ∈ [0, T ], hence f ρ (C([0, T ], R n )) ⊂ ν ∈ L 2 ([0, T ], R n ) : ν(t) ≤ l(t), a.e. t ∈ [0, T ] .
Remark 4.3.5. Note that under the assumption (H k ), the operators F ρ and F ρ are bounded by

l ρ k ∈ L 2 ([0, T ]) such that l ρ k = l k + ρk.
In particular, for the operator N K + ρ id we have (N K + ρid) λ (x) ≤ ρ x , for every x ∈ K with K is a prox-regular subset of R n .

z λ = x 0 + ∇e λ g(x 0 ), such that g = I C(•) + ρ 2 • 2 .
Using the facts that x λ (0) = x 0 , (N C(•) + ρ id) λ is maximal monotone and

(N C(t) + ρ id) λ (x 0 ) = ∇e λ g(x 0 ),
we have that

T 0 l(t) -z λ , x λ (t) -x 0 dt = T 0 x λ (t) -x 0 + x λ (t), x λ (t) -x 0 dt + T 0 (N C(t) + ρ id) λ (x λ (t)) -(N C(t) + ρ id) λ (x 0 ), x λ (t) -x 0 dt ≥ T 0 x λ (t) -x 0 2 dt + 1 2 ( x λ (T ) -x 0 2 -x λ (0) -x 0 2 ) ≥ x λ -x 0 2 L 2 .
By the Cauchy-Schwarz inequality we have

x λ -x 0 2 L 2 ≤ l(t) -z λ , x λ (t) -x 0 L 2 ≤ l -z λ L 2 x λ -x 0 L 2 .
Therefore,

x λ -x 0 L 2 ≤ l -z λ L 2 ≤ l L 2 + (1 + ρ) x 0 . Consequently, {x λ } is bounded in L 2 ([0, T ], R n
) by some constant denoted by c. We have

(N C(•) + ρ id) λ (x λ ) L 2 ≤ inf{ ν L 2 : ν ∈ (N C(•) + ρ id)(x λ )} ≤ ρ x λ L 2 ≤ ρc. Consequently, {(N C(t) + ρ id) λ (x λ )} is bounded in L 2 ([0, T ], R n ).
Then, we conclude that A + is maximal by applying Lemma 3.3.22.

Since A + is maximal monotone, id + A + is surjective and invertible. This enables us to define P + (x) = (id + A + ) -1 (x) ∈ A(x 0 ).

P + is called the resolvent of A + for λ = 1, which is defined for x ∈ L 2 ([0, T ], R n ).
Proposition 4.3.8. Assume that (P) holds. Then, the operator

P + : L 2 ([0, T ], R n ) → W 1,2 ([0, T ], R n ), is completely continuous, when W 1,2 ([0, T ], R n ) is supplied with the topology of C([0, T ], R n ),
and is continuous when L 2 ([0, T ], R n ) is supplied with the weak topology.

Proof. Since L 2 ([0, T ], R n ) is a Hilbert space, then, it is sufficient for us to show that,

if ν n ν weakly in L 2 ([0, T ], R n ), u n = P + (ν n ) → u = P + (ν) in C([0, T ], R n ). Since P + : L 2 ([0, T ], R n ) → L 2 ([0, T ], R n ) is non-expansive, we deduce that {u n } is bounded in L 2 ([0, T ], R n ) by a constant m ≥ 0. Let us show now that u n → u in C([0, T ], R n ). For λ > 0, let u λ n be the unique solution of ν n = u λ n + (u λ n ) + (N C(t) + ρ id) λ (u λ n ).
Then, by Lemma 3.3.22 we have that u λ n → u n , and

(N C(t) + ρ id) λ (u λ n ) → µ n ∈ (N C(•) + ρ id)(u n ) ∩ {ν n -u n -u n } in L 2 ([0, T ], R n ).
Moreover, we have

µ n ≤ lim λ→0 + (N C(t) + ρ id) λ (u λ n ) L 2 ≤ lim λ→0 + inf{ z L 2 : z ∈ (N C(•) + ρ id)(u λ n )} ≤ lim λ→0 + ρ u λ n ≤ ρm.
Then, the sequence

{µ n } is bounded in L 2 ([0, T ], R n ), and hence {u n } is bounded in W 1,2 ([0, T ], R n ).
Hence, there are subsequences that we denote also by {u n } and {µ n } such that µ n µ weakly in L 2 ([0, T ], R n ), and u n → u strongly in C([0, T ], R n ) and hence in L 2 ([0, T ], R n ), and weakly in W 1,2 ([0, T ], R n ). Now, we have that A + is maximal monotone, so, we deduce that (u,

u + µ) ∈ gph( A + ), which is closed in (L 2 ([0, T ], R n ), τ s ) × (L 2 ([0, T ], R n ), τ w ). It follows that u n = P + (ν n ) → u = P + (ν) strongly in C([0, T ], R n ). Remark 4.3.9. Let l ∈ L 2 ([0, T ], [0, ∞)) and O = {ν ∈ L 2 ([0, T ], R n ) : ν(t) ≤ l(t)
a.e. t ∈ [0, T ]} supplied with the topology of L 1 ([0, T ], R n ). Note that we can show that P + : O → C([0, T ], R n ) is continuous and compact following the same arguments as in the proof of the previous proposition.

Proofs of the main results

In this section we suppose that C(•) : [0, T ] ⇒ R n be a continuous set-valued map, with nonempty closed values and C(t) is ρ-prox-regular for every t ∈ [0, T ]. In this case the normal cone N C(•) : gph C(•) ⇒ R n has a closed graph.

We consider the following problems (4.4)

   x (t) + x(t) ∈ -(N C(t) + ρ id)(x(t)) + F ρ (t, x(t)) + x(t) t , a.e. t ∈ [0, T ], x(0) = x 0 ∈ C(0); x(t) ∈ C(t), ∀t ∈ [0, T ]; and (4.5)
   x (t) + x(t) ∈ -(N C(t) + ρ id)(x(t)) + F ρ (t, x(t)) + x(t) t , a.e. t ∈ [0, T ], x(0) = x 0 ∈ C(0); x(t) ∈ C(t), ∀t ∈ [0, T ].
solutions of (4.4) and (4.5) are in B β (α). Proof. We follow some arguments presented in [START_REF] Frigon | Nonlinear first order initial and periodic problems in Banach spaces[END_REF]. First, we assume that x is a solution of (4.4). Then we know that there exists ν,

δ x ∈ L 2 ([0, T ], R n ) such that δ x (t) ∈ N C(t) (x(t)), ν(t) ∈ F ρ (t, x(t)),
hence take ν(t) = z(t) + ρx(t) such that z(t) ∈ F (t, x(t)), and

x (t) + x(t) = -δ x (t) -ρx(t) + ν(t) + x(t) t a.e. t ∈ [0, T ]. It remains to show that x(t) -α(t) ≤ β(t) for all t ∈ [0, T ]. Recall that x(t) t = α(t) + β(t) x-α(t) (x -α(t)).
Now, we suppose that there exists t 1 ∈ (0, T ] with x(t 1 ) -α(t 1 ) > β(t 1 ). The condition (4) in Definition 4.2.1 implies that there exists t 0 ∈ [0, t 1 ) such that x(t 0 ) -α(t 0 ) = β(t 0 ) and x(t) -α(t) > β(t) > 0 for t ∈ (t 0 , t 1 ). From condition (2) in the same definition we deduce that a.e. on (t 0 , t 1 ) we have the following estimation:

x(t) -α(t), x (t) -α (t) x(t) -α(t) = x(t) -α(t), -δ x (t) -ρx(t) + ν(t) + x(t) t -x(t) -α (t) x(t) -α(t) = x(t) -α(t), -δ x (t) + z(t) -x(t) + α(t) + β(t) x(t)-α(t) (x(t) -α(t)) -α (t) x(t) -α(t) = β(t) -x(t) -α(t) + x(t) -α(t), z(t) -δ(t) -α (t) x(t) -α(t) - x(t) -α(t), δ x (t) -δ(t) x(t) -α(t) ≤ β (t) -ρ x(t) -α(t) + ρ x(t) -α(t) ≤ β (t).
because, the operator N C(•) + ρ id is maximal monotone, and

δ(t) ∈ N C(t) (α(t)), z(t) = ν(t) -ρx(t) and z(t) ∈ G(t, x(t)).
Also, for x(t) -α(t) > β(t) = 0 and t ∈ (t 0 , t 1 ), we deduce that z(t) = v (t) + δ(t), x (t) = α (t) + x(t) t -x(t) and

x(t) -α(t), x (t) -α (t) x(t) -α(t) = -x(t) -α(t) < 0 = β (t).
Thus, the function x -α -β is decreasing on (t 0 , t 1 ), which is a contradiction.

We continue with the presentation of the proofs of the existence theorems.

Proof of Theorem 4.2.3. Note that Proposition 4.3.3 implies that the operator

F ρ : C([0, T ], R n ) ⇒ L 2 ([0, T ], R n ) is u.s.c
∈ L 2 ([0, T ], [0, ∞)) such that ν(t) ≤ l(t) a.e. t ∈ [0, T ] for every ν ∈ F ρ (x) and x ∈ C([0, T ], R n ).
Let us give some properties of the operator

P + • F ρ : C([0, T ], R n ) ⇒ C([0, T ], R n ).
β(t) = 0}; 4. x(0) -α(0) ≤ β(0).
We also denote

B β (α) = {x ∈ C([0, T ], R n ) : x(t) -α(t) ≤ β(t) ∀ t ∈ [0, T ]}.
In this part we need to consider the following assumption:

(S -L 2 ) * there exists (α, β) ∈ W 1,2 ([0, T ], R n ) × W 1,2 ([0, T ], [0, ∞)) an L 2 -solution- tube of (4.6).
We provide similar existence results to those obtained in the previous section. The proofs are analogous to those introduced in section 4.4 when we consider the supplementary problems:

   x (t) -x(t) ∈ (N C(t) + ρ id)(x(t)) + F * ρ (t, x(t)) -x(t) t , a.e. t ∈ [0, T ], x(0) = x 0 ∈ C(0); x(t) ∈ C(t). ∀t ∈ [0, T ]; and    x (t) -x(t) ∈ (N C(t) + ρ id)(x(t)) + F * ρ (t, x(t)) -x(t) t , a.e. t ∈ [0, T ], x(0) = x 0 ∈ C(0); x(t) ∈ C(t). ∀t ∈ [0, T ];
where

F * ρ (t, x) = F * (t, x) -ρx and F * ρ = F * (t, x) -ρx for all (t, x) ∈ [0, T ] × R n , with F * = F ∩ G * and F * = F ∩ G * , such that G * (t, x) =                α (t) + δ(t), if β(t) = 0, R n , if x -α(t) ≤ β(t) and β(t) > 0, {z : x -α(t), z + δ(t) -α (t) ≥ β (t) x -α(t) + ρ x -α(t) 2 }, otherwise; and G * (t, x) =          α (t) + δ(t), if β(t) = 0, R n , if x -α(t) < β(t), {z : x -α(t), z + δ(t) -α (t) ≥ β (t) x -α(t) + ρ x -α(t) 2 }, otherwise.

Résumé du chapitre

Nous établissons quelques résultats d'existence de tube-solutions pour des processus de rafle non convexes. Notre approche emploie des propriétés appartenant à la classe d'ensembles prox-réguliers ainsi que des techniques de théorie de point fixe.

Dans ce chapitre, nous considérons l'inclusion différentielle

   x (t) ∈ -N C(t) (x(t)) + F (t, x(t)), p.p. t ∈ [0, T ], x(0) = x 0 ∈ C(0), x(t) ∈ C(t), ∀t ∈ [0, T ], où T > 0 est un nombre réel, C : [0, T ] ⇒ R n et F : [0, T ] × R n ⇒ R n deux

CHAPTER 5 EXISTENCE OF MONOTONE SOLUTIONS WITH RESPECT TO A PREORDER AND APPLICATIONS

Introduction

In this chapter, we study the existence of P (•)×Q(•)-monotone solutions of a system of differential inclusions with a discontinuous right-hand side. The monotonicity that we consider is with respect to a preorder. Moreover, we show that a limit point (x, ỹ) of a solution is a saddle-point for the convex-concave function Γ(•, •) in P (x) × Q(ỹ). The obtained results can be used for studying a game involving two players with a collective pay-off. Another application is in a production problem of a company which wishes to maximize its profit. Finally, some particular cases are also considered: with a function w(•) which is supposed convex, concave, then prox-regular. We start by presenting the problem in the following:

Let K p ⊂ R n and K q ⊂ R m be two convex compact sets. Let Γ : K p × K q → R + is a convex-concave function. Let P : K p ⇒ K p and Q : K q ⇒ K q two preorders. We consider the following mixed system (5.1)

   x (t) ∈ proj T P (x(t)) (x(t)) (-∂ x Γ(x(t), y(t)), a.e. t ∈ [0, +∞), y (t) ∈ proj T Q(y(t)) (y(t)) (∂ + y Γ(x(t), y(t)), a.e. t ∈ [0, +∞), x(0) = x 0 , y(0) = y 0 ;
where x 0 and y 0 are given (therefore x 0 ∈ P (x 0 ) and y 0 ∈ Q(y 0 ) as a result of the reflexivity of the multivalued maps P and Q as preorders). Moreover,

• ∂ u Γ(u, v) is the subdifferential of the convex function Γ(•, v) with respect to u, • ∂ + v Γ(u, v)
is the superdifferential of the concave function Γ(u, •) with respect to v,

• T C denote the tangent cone to the convex set C in R n , and

• proj denote the usual projection on a closed convex set.

We point out that, we face here at least tow difficulties given by the fact that the multivalued map x → proj T P (x) (x) (-∂ x Γ(x, y)) has no continuity properties nor convex values.

The trajectories x(•) are monotone with respect to the preorder P (•), the variable constraint, and move along the maximum descent direction of the function Γ(x(•), •) which is consistent with P (•). Moreover, the trajectories y(•) are monotone with respect to the preorder Q(•), and move along the minimum ascent direction of the function Γ(•, y(•)) which is consistent with Q(•).

The problem of finding monotone and feasible solutions arises from viability issues, in which we select trajectories which are viable in the sense that they always satisfy a given constraint. Aubin [START_REF] Aubin | Monotone evolution of resource allocations[END_REF] studied this issue by adapting methods introduced in Aubin, Cellina, Nohel [START_REF] Aubin | Monotone trajectories of multivalued dynamical systems[END_REF] and thoroughly investigated in [START_REF] Aubin | Viability theory[END_REF][START_REF] Aubin | Differential Inclusions[END_REF][START_REF] Clarke | Monotone invariant solutions to differential inclusions[END_REF] and the references therein.

In order to study the planning procedures in mathematical economy, Henry [START_REF] Henry | An existence theorem for a class of differential equations with multivalued right-hand side[END_REF] introduced in the 70s the differential inclusion (*)

x (t) ∈ proj T C (x(t)) F (x(t)), x(0) = x 0 ∈ C,
where C is a nonempty closed convex subset of R n and F : C → R n is an upper semicontinuous multivalued mapping. Later on, this inclusion has been associated to the existence of a minimal norm absolutely continuous solution for the following problem

x (t) ∈ -N C (x(t)) + F (x(t)), x(0) = x 0 ∈ C,
by Cornet [START_REF] Cornet | Existence of slow solutions for a class of differential inclusions[END_REF].

In [START_REF] Falcone | Maximum descent monotone solutions of an ordinary differential equation with a discontinuous right-hand side[END_REF], Falcone and Siconolfi studied an economic model of a firm, this latter transforms m resources into n different consumption goods and controls its product activity by means of the input flow. This model leads to a mathematical problem of the following type

x (t) = proj T P (x(t)) (x(t)) (-∇w(x(t))),
x(0) = x 0 , applications where, ∇w(•) is the gradient of the C 1 convex function w(•) (cf. [START_REF] Malinvaud | Leçons de théorie microéconomique[END_REF]). At the initial time t = 0, the firm has an input bundle x 0 , and it has to manage the data flow to achieve the demand arising on the market. Moreover, it aims to find, among the data currents which would guarantee the maintenance of its market share, those along which the cost function decrease.

As we mentioned before, we are looking for monotone solutions with respect to a preorder. One of the reasons is that the advantage of the monotonicity feature lies in the fact that it allows one to select among the trajectories of the differential inclusions in (5.1) those trajectories that improve the state of the system when the time elapses. From this standpoint, we prove that a trajectory solution of our system converges to a limit point which is a saddle-point for the convex-concave function Γ. Our argument is built on some notions presented by Falcone and Siconolfi in their paper, for instance, we shall use the same preorder P (•) they defined for their economic model. However, our results concerning the existence of a saddle-point are new and incorporate a more general framework. The obtained results can be applied in two problems their utility found at least two applications: first, in a game involving two players with a collective pay-off, second, in a production problem for a company which wishes to maximize its profit.

Moreover, we discuss the existence of monotone solutions to differential inclusions of the type (*) with subdifferential operators of convex, concave, and prox-regular functions. We note that the preorder has prox-regular values when we use proxregular functions. In this case, we use a different approach to obtain our existence results.

The chapter is organized as follows. The first section provides a presentation of the problem. The second section provides a detailed presentation of the main result without proofs. In section 5.3 we give the proof of the main results, and we investigate the existence of a saddle-point. In section 5.4, we present some particular cases. In section 5.5, we generalize the results of section 5.4 to the prox-regular case. In the last section, we provide two applications.

Main results

Let K p ⊂ R n and K q ⊂ R m be two nonempty, compact, convex sets. Let Γ : K p × K q → R + be a function satisfying the following assumptions:

(H 1 ) For every fixed y ∈ K q , the function x → Γ(x, y) is convex and lower semicontinuous. applications (b) h q (•) is strictly concave;

(c) if y 1 ≥ y 2 , then h q (y 1 ) ≥ h q (y 2 ), ∀y 1 , y 2 ∈ K q .

We have the following result.

Proposition 5.2.2. Assume that assumptions (H 1 q ) and (H 2 q ) are satisfied. Then, Q(•) is a continuous preorder with nonempty compact convex values.

Proof. First, we show that Q(•) has a closed graph. Let {x k } ⊆ K q and {y k } ⊆ Q(K q ) be two sequences, such that

y k → y, with y k ∈ Q(x k ), x k → x.
We want to prove that y ∈ Q(x). Since y k ∈ Q(x k ), we denote by {z k } and {s k } two sequences such that

h q (z k ) = max z∈Fq(y k ) h q (z) ≥ max s∈Fq(x k ) h q (s) = h q (s k ).
Since F q (•) is u.s.c. F q (K q ) is compact, we can find two convergent subsequences of {z k } and {s k }. We conclude the proof by the continuity of F q (•) and h q (•). From the previous results and the compactness of K q it follows that Q(•) is u.s.c.

Next, let us show that Q(•) is l.s.c. For this aim, let x ∈ K q and {x k } ⊆ K q be a sequence. If

x k → x and y ∈ Q(x),

we shall consider the sequence

y k = proj Q(x k ) y,
observe that y k ∈ Q(x k ). We want to prove that y k → y in K q . Arguing by contradiction, suppose that there exists a subsequence, which still denoted by y k , such that y k -y > η > 0, for some η > 0 and ∀k ≥ 1.

This implies that x k ∈ Q(y).

Let us now define the sequence

ω k = τ k x k + (1 -τ k )y, where τ k ∈ [0, 1] and lim k→+∞ τ k = 0.
By the concavity assumption on h q (•) and F q (•), and since x k ∈ Q(y), we obtain max

z∈Fq(ω k ) h q (z) ≥ max z∈τ k Fq(x k )+(1-τ k )Fq(y) h q (z), > τ k max z∈Fq(x k ) h q (z) + (1 -τ k ) max z∈Fq(y) h q (z), > max z∈Fq(y)
h q (z) = h q (z).

Since ω k → y, there exists N ∈ N, for all k ≥ N such that ω k -y < η.

By the fact that Q(y) is closed, x ∈ Q(y) and since y ∈ Q(x), we have

Q(x) = Q(y).
Moreover, by the continuity assumption on h q (•) and F q (•), we have that

h q (z k ) = max z∈Fq(x k ) h q (z) → h q (z).
Therefore, we can find a constant M ∈ N such that

h q (z M ) = max z∈Fq(x M ) h q (z) ≤ max z∈Fq(ω N ) h q (z).
Consequently, from the definition of Q(•), we have that ω N ∈ Q(x M ), which contradicts the fact that y M -y ≤ z -y , ∀ z ∈ Q(x M ).

Furthermore, observe that Q(x) = {y ∈ K | F q (y) ∩ lev ≥cx h q = ∅} = F -1 q (lev ≥cx h q ).

where lev ≥cx h q = {x ∈ R n + | h q (x) ≥ c x } is the level set of h q . Since h q (•) is concave, the set lev ≥cx h q is convex. By the concavity of F q (•), we conclude that Q(•) has convex values. Finally, the fact that Q(•) has bounded images is immediate since Q(•) is u.s.c., and K q is compact. Definition 5.2.3. (P ×Q-monotone solutions) We say that trajectories x : [0, ∞) → K p , starting at x 0 ∈ K p and y : [0, ∞) → K q , starting at y 0 ∈ K q , are P × Qmonotone, if 1. x(•) is monotone with respect to P (•), Chapter 5 Existence of monotone solutions with respect to a preorder and applications 2. y(•) is monotone with respect to Q(•).

Observe that for every t > 0, and h > 0 one has Γ(x(t + h), y(t)) ≤ Γ(x(t), y(t)) ≤ Γ(x(t), y(t + h)).

We shall show that the solutions x(•) and y(•) of the mixed system (5.1) converge to the points x and ỹ which verify Γ(x, ỹ) = min Γ(x, y).

We finish this section by presenting our main result.

Theorem 5.2.4 (Main result). Let P (•) and Q(•) be two continuous preorders with convex compact values defined on compact convex subsets K p and K q respectively, and let Γ : K p ×K q → R + be a function satisfying (H 1 ) and (H 2 ). Then, there exists a P (•) × Q(•)-monotone solution (x(•), y(•)) of (5.1) for any initial points x 0 ∈ K p and y 0 ∈ K q . Moreover, let x(•) and y(•) be solutions of the mixed problem (5.1), and let x = lim tn→+∞ x(t n ) and ỹ = lim tn→+∞ y(t n ).

Then, we have Γ(x, y).

Proof of the main result

An existence result

Let K be a convex compact set of R n , and let x 0 be a fixed point in K. In this section, we shall consider the following differential inclusion (5.3) x (t) ∈ proj T R(x(t)) (x(t)) (V (x(t))), a.e. in [0, ∞), x(0) = x 0 , here, V : K ⇒ R n is an u.s.c. multivalued mapping, and R(•) is a preorder defined on K.

We shall present an abstract theorem concerning the existence of monotone solutions to problem (5.3) which will be used in the proof of our main result. We shall use the following notations: N (x) = N R(x) (x), T (x) = T R(x) (x), and π(V (x)) = proj T R(x) (x) (V (x)). applications Now let us show that H V is closed at all x ∈ E. Let {x n }, {v n }, {p n }, be sequences such that x n ∈ E, v n ∈ V (x n ), and p n ∈ N (x n ). Additionally p n , y n ≥ 0, for all n, {x n } converge to x, and {v n -p n } converge to some ỹ in R n . Without loss of generality, we can assume that the sequences {v n } and {p n } converges to some elements ṽ and p in R n . Note that N (•) is closed at x, by virtue of the lower semicontinuity of the preorder R(•) (see, Proposition 5.3.2). Therefore, p ∈ N (x). Since the multivalued mapping V is u.s.c. at x, then ṽ ∈ V (x). Consequently, we have ỹ = ṽ -p and p, ỹ ≥ 0, hence, ỹ ∈ H V (x). Thus, H V is u.s.c. with nonempty, convex, compact values.

We continue by showing that π(V (x)) = H V (x) ∩ T (x). By inequality (ii) in Corollary 3.2.4, we have that v -π(v) ∈ N T (x) (π(v)), for any v ∈ V (x).

Since N T (x) (µ) ⊂ N (x), ∀ µ ∈ T (x), we have that v -π(v) ∈ N (x). Moreover, we have π(v) ∈ T (x), and consequently π(v) ∈ H V (x) ∩ T (x) for any v ∈ V (x). Therefore π(V (x)) ⊂ H V (x) ∩ T (x).

At this point we proved that the multivalued map H V is u.s.c. with nonempty convex compact values and verifies the tangential condition

H V (x) ∩ T (x) = ∅, ∀x ∈ K.
The assumptions of Theorem 3.3.48 are satisfied by the set K, and by the multivalued map H V from K to R n . Thus, for any x 0 in K, there exists a R-monotone trajectory x : [0, +∞) → R n which is a solution of the Cauchy problem x (t) ∈ H V (x(t)), a.e. in [0, ∞), x(0) = x 0 .

Moreover, for almost every t ∈ [0, ∞), the derivative x (t) of a monotone solution x(•) to (5.16) We continue by showing that π(V (x)) ⊃ H V (x) ∩ T (x) for all x ∈ K. Let v -p ∈ H V (x) ∩ T (x), for some v ∈ V (x). Consequently, we have the equality

H V (x) ∩ T (x) = π(V (x)), ∀x ∈ K.
which gives the desired results.

Remark 5.3.3. A solution x(t) of (5.3) has a right-derivative x + (t) everywhere, and

x + (t) ∈ proj T R(x(t)) (x(t)) (V (x(t))), ∀t ∈ [0, ∞). applications which contradicts the fact that ϕ(x, ỹ), π (ϕ(x, ỹ)) > 0. On the other hand, let us suppose that π(-ψ(x, ỹ)) = 0. Then, for h > 0, we have (5.9) Γ(x, ỹ) ≤ Γ(x(h), ỹ).

Suppose that ψ(x, ỹ), π(-ψ(x, ỹ)) < 0, then there exist ε > 0 and η < 0 such that ψ(u, z), π(-ψ(u, z)) < η < 0, ∀u ∈ B(x, ε), ∀z ∈ B(ỹ, ε).

Otherwise, there will be two sequences {u n }, and {z n }, with u n → x, and z n → ỹ, such that ψ(u n , z n ), π(-ψ(u n , z n )) → 0, that is, π(-ψ(u n , z n )) → 0.

Let ψ n be a sequence of continuous approximate selections of ∂ x Γ that converges to ψ. From Corollary 3.2.4 we have -ψ n (u n , z n ) -π(-ψ n (u n , z n )), µ -π(-ψ n (u n , z n )) ≤ 0, ∀µ ∈ T (u n ).

Therefore -ψ(u n , z n ) -π(-ψ(u n , z n )) ∈ N (u n ).

Recall that by Proposition 5.3.2, the normal cone N (•) has a closed graph. Then, we have -ψ(x, ỹ) ∈ N (x), which contradicts the fact that ψ(x, ỹ), π(-ψ(x, ỹ)) < 0.

Moreover, since d dt Γ(x(t), ỹ) = ψ(x(t), ỹ), x (t) ,

for T > T we have Γ(x(T + h), ỹ) = Γ(x(T ), ỹ) + T +h T ψ(x(s), ỹ), π(-ψ(x(s), ỹ)) ds < Γ(x(T ), ỹ) + ηh < Γ(x, ỹ), which contradicts (5.9). Finally, we finish the proof by applying Theorem 3.2.21.

Particular cases

The convex case

This part deals with the existence of P -monotone solutions of the differential inclusion (5.10) x (t) ∈ proj T P (x(t)) (x(t)) (-∂w(x(t))), a.e. in [0, ∞), x(0) = x 0 , where, ∂w(•) is the subdifferential of the function w(•), which satisfies the following assumption: A0 w : K → R + is a proper, lower semicontinuous, convex function.

Existence of monotone solutions

Let K be a convex compact set of R m + . We define the following preorder P : K ⇒ K We shall make the following assumptions:

A1 Let F : K ⇒ R n + be a multivalued mapping. Such that w(x(t)).

We recall that the projection on a convex set is a non-expansive map, hence we have that π(ψ(x)) ∈ B(0, α), where α = max x∈K ψ(x) .

Note that π(-ψ(x)) is bounded. Therefore, for any ε > 0, there exist ζ > 0, and T > 0 such that, if x(t) ∈ B(x, ε/2) then x(t + τ ) ∈ B(x, ε), for any |τ | ≤ ζ, ∀t ≥ T.

We shall prove that, if ψ(x), π(-ψ(x)) < 0, then, there exist ε > 0 and η < 0 such that ψ(z), π(-ψ(z)) < η < 0, ∀z ∈ B(x, ε).

Otherwise, there will be a sequence z n , with z n → x, such that ψ(z n ), π(-ψ(z n )) → 0, that is, π(-ψ(z n )) → 0.

Once again, let ψ n to be a sequence of continuous maps that converges to some measurable selection in ∂w. According to Corollary 3. w(x(t)).

Note that, π(ϕ(x)) is bounded. Consequently, for any ε > 0, we can find ζ > 0, and T > 0 such that if x(t) ∈ B(x, ε/2) then x(t + τ ) ∈ B(x, ε), for any |τ | ≤ ζ, ∀t ≥ T.

We would like to show that, if ϕ(x), π(ϕ(x)) > 0, then, there exist ε > 0 and η > 0 such that ϕ(z), π(ϕ(z)) > η > 0, ∀z ∈ B(x, ε).

Otherwise, there will be a sequence z n converging to x, such that ϕ(z n ), π(ϕ(z n )) -→ 0.

Let ϕ n be a sequence of continuous maps that converges to some measurable selection denoted ϕ in ∂ + w. From Corollary 3.2.4 statement (ii), we have

ϕ n (z n ) -π(ϕ n (z n )) ∈ N (z n ).
Consequently, it follows that ϕ(x) ∈ N (x), and this contradicts the fact that ϕ(x), π(ϕ(x)) > 0.

We proceed by supposing that π(ϕ(x)) = 0. By equality (5.18), there exists T > T such that w(x(T + ζ)) = w(x(T )) + T +ζ T ϕ(x(s)), π(ϕ(x(s))) ds > w(x(T )) + ηζ > w(x), which contradicts (5.19). applications

We would like to prove that y ∈ P (x). Since y k ∈ P (x k ), we denote by {z k } and {s k } two sequences such that h(z k ) = min z∈F (y k ) h(z) ≤ min s∈F (x k ) h(s) = h(s k ).

Since F (•) is u.s.c., F (K) is compact, we can find two convergent subsequences of {z k } and {s k }. We conclude the proof by the continuity of F (•) and h(•). From the previous results and the compactness of K it follows that P (•) is u.s.c.

Next, let us show that P (•) is l.s.c. For this aim, let x ∈ K and {x k } ⊆ K be a sequence. If

x k → x and y ∈ P (x).

we shall consider the sequence y k = proj P (x k ) y.

Observe that y k ∈ P (x k ). We want to prove that y k → y in K. Arguing by contradiction, suppose that there exists a subsequence still denoted by y k , such that y k -y > η > 0, for some η > 0 and ∀k ≥ 1.

This implies that x k ∈ P (y).

Let us define the sequence h(z),

≤ h(τ k z 1 + (1 -τ k )z 2 ), ≤ τ k h(z 1 ) + (1 -τ k )h(z 2 ) + τ k (1 -τ k ) c 2 z 1 -z 2 2 , ≤ min z∈F (y) h(z) + τ k (1 -τ k ) c 2 z 1 -z 2 2 .
Taking τ k → 0, one has applications Then

v 1 -v 2 , ȳ1 -ȳ2 -z 1 -z 2 , z1 -z2 ≥ 0.
Moreover, Θ is r-prox-regular, so N Θ (z) is r-hypomonotone. Furthermore, F is locally Lipschitz with constant k > 0. Therefore,

v 1 -v 2 , ȳ1 -ȳ2 ≥ z 1 -z 2 , z1 -z2 ≥ - r 2 z1 -z2 2 ≥ - rk 2 2 ȳ1 -ȳ2 2 .
Consequently, P (x) is prox-regular. Finally, the boundness of images is immediate from the fact that P (•) is u.s.c., and K is compact.. We shall continue by studying the limit points of a solution of the problem (5.10) under assumption (A0) c . Since w(•) is c-prox-regular, we have where, ψ(•) is a measurable selection in ∂ P w(•). Indeed, for ε > 0, we let ψ ε be a sequence of continuous maps that converges to some measurable selection denoted ψ of ∂ P w. For h > 0 we have w(x(t + h)) -w(x(t)) ≥ ψ ε (x(t)), x(t + h) -x(t) -c 2 x(t + h) -x(t) 2 + ε, and w(x(t)) -w(x(t + h)) ≥ ψ ε (x(t + h)), x(t) -x(t + h) -c 2 x(t) -x(t + h) 2 + ε.

By dividing the two sides of the second inequality by h > 0, and then taking ε goes to 0, we obtain (5.20).

We can formulate now our result.

Figure 1 . 1 :

 11 Figure 1.1: Circuit avec une diode optimale et une source de courant

  x → F (t, x) est semi-continue supérieurement, p.p. t ∈ [0, T ]. (H) F : [0, T ] × R n ⇒ R n estune application multivoque à valeurs compactes tels que(i) x → F (t, x) est semi-continue inférieurement, p.p. t ∈ [0, T ], (ii) (t, x) → F (t, x) est L ⊗ B-mesurable. où [0, T ] × R n munide la σ-algèbre engendrée par les sous-ensembles C × D, avec C ⊂ [0, T ] et D ⊂ R n sont, respectivement, Lebesgue et Borel mesurables. (H k ) Pour tout k ≥ 0, il existe l k ∈ L 2 ([0, T ]) tel que max { µ : µ ∈ F (t, x), x ≤ k} ≤ l k (t), p.p. t ∈ [0, T ].

Dans [ 33 ]

 33 , Falcone et Siconolfi ont étudié un modèle économique d'une société, cette dernière transforme m ressources en n biens de consommations et contrôle son xxviii Quelques aspects sur les inégalités variationnelles différentielles activité de production au moyen du flux d'entrée. Ce modèle mène à un problème mathématique du type suivant

  Nous montrerons que les trajectoires x(•) et y(•) du système mixte (1.6) convergent vers les points limites x et ỹ qui vérifient Γ(x, ỹ) = min x∈P (x) max y∈Q(ỹ) Γ(x, y) = max y∈Q(ỹ) min x∈P (x) Γ(x, y). Pour faciliter la lecture, rappelons la définition d'un point-selle. xxx Quelques aspects sur les inégalités variationnelles différentielles Définition 1.3.7. Soient C, D deux ensembles non vides, et G : C × D → R une fonction. un point (x, ỹ) ∈ C × D est un point-selle de G sur C × D si (*) G(x, y) ≤ G(x, ỹ) ≤ G(x, ỹ), pour chaque (x, y) ∈ C × D.

Proposition 1 . 3 . 21 .

 1321 Soient x ∈ K et ϕ(x) une sélection mesurable dans ∂ + w(x). Alors, d dt w(x(t)) = ϕ(x(t)), x (t) , p.p. t ≥ 0.

  (a) for every v ∈ D the function u → K(u, v) is convex and lower semi-continuous, (b) for every u ∈ C the function v → K(u, v) is concave and upper semi-continuous.

Proposition 3 . 3 . 4 .

 334 The graph of an u.s.c. multivalued map with closed values from X to Y is closed.Even for single-valued maps, the converse of Proposition 3.3.4 is not in general true.Proposition 3.3.5. If F : X ⇒ Y has a closed graph and it is locally compact, then F is u.s.c.

  (a) F is continuous. (b) For every C ⊆ Y closed, F + (C) and F -(C) are both closed in X. Definition 3.3.10. A mapping F : X ⇒ Y is Lipschitz continuous on U , a subset of X, if it has nonempty closed values on U and there exists κ > 0, a Lipschitz constant, such that y -y ≤ κ x -x , ∀x, x ∈ U, ∀y ∈ F (x), ∀y ∈ F (x ).

Definition 3 . 3 . 14 .

 3314 with B(X) being the Borel σ-field of X. Proposition 3.3.13. If F : Ω ⇒ Y and for every C ⊆ X we have F -(C) ∈ Σ. Then F is measurable. We say that A ⊆ [0, T ] × R n is L ⊗ B measurable if A belongs to the σ-algebra generated by all sets of the form N × D where N is Lebesgue measurable in [0, T ] and D is Borel measurable in R n . Definition 3.3.15. Let X and Y be two Hausdorff topological spaces. A function ϕ : Ω × X → Y is said to be a Carathéodory function, if (a) For every x ∈ X, ω -→ ϕ(ω, x) is (Σ, B(Y ))-measurable and (B(Y ) is the Borel σ-field of Y ). (b) For every ω ∈ Ω, x -→ ϕ(ω, x) is continuous. Definition 3.3.16. a multivalued mapping F

  Figure .3.2 : Tangent cones (in clear blue) and normal cones (in pink) to different type of sets

Chapter 3 3 . 4 . 3 .

 3343 Therefore, ∂I C (x) = N C (x) and dom(N C ) = C. Mathematical backgrounds Remark The resolvent (id +λ∂I C ) -1 of ∂I C is the projection on C. i.e.

Figure . 3 . 3 :

 33 Figure .3.3 : A prox-regular function which fail to be subdifferentially continuous

Figure . 3 . 4 :

 34 Figure .3.4 : Two sets which fail to be uniformly prox-regular

Figure . 3 . 5 :

 35 Figure .3.5 : Mass and spring with coulomb's friction

Figure . 3 . 6 :

 36 Figure .3.6 : Circuit with ideal diode and current source

Theorem 4 . 2 . 3 .

 423 Assume (P), (H), (H k ) and (S -L 2 ) hold. Then the problem (4.1) has a solution in W 1,2 ([0, T ], R n ) ∩ B β (α).

Theorem 4 . 2 . 4 .

 424 Assume (P), (H), (H k ) and (S -L 2 ) hold. Then the problem (4.1) has a solution in W 1,2 ([0, T ], R n ) ∩ B β (α).

Proposition 4 . 3 . 2 .

 432 Assume (H), (H k ) and (S -L 2 ) hold. Then there exists f :

Proposition 4 . 3 . 6 .

 436 Assume that (P) holds. Then, the operators N C(•) + ρ id and N C(•) + ρ id are maximal monotone. (N C(•) + ρ id) λ is maximal monotone, and hence id +L + (N C(•) + ρ id) λ is surjective by Lemmas 3.3.20 and 3.3.21. Then, for l ∈ L 2 ([0, T ], R n ), there exists x λ ∈ A(x 0 ) such that (id +L + (N C(t) + ρ id) λ )x λ = l for all t ∈ [0, T ]. Let x 0 ∈ C(0), and denote

Proposition 4 . 4 . 1 .

 441 Let (S -L 2 ) and (P) be satisfied. Then every solution x ∈ A(x 0 ) of (4.4) or (4.5) belongs to B β (α).

Theorem 4 . 5 . 2 .

 452 Assume (H), (H k ), (P), and (S -L 2 ) * . Then the problem (4.6)has a solution x ∈ W 1,2 ([0, T ], R n ) ∩ B β (α).

Theorem 4 . 5 . 3 .

 453 Assume (H), (H k ), (P), and (S -L 2 ) * . Then the problem (4.6) has a solution x ∈ W 1,2 ([0, T ], R n ) ∩ B β (α).

  Hence, there exists p∈ N T (x) (π(v)) ⊂ N (x), such that π(v) = v -p. Let us suppose that p, v -p < 0.Then, we have the following inequality p,1 2 π(v) -(v -p) = p, -1 2 (v-p) > 0, which contradicts the fact that p ∈ N T (x) (v -p).

Then, we have

  p, µ -(v -p) = p, µ -p, v -p ≤ 0, ∀µ ∈ T (x), ∀v ∈ V (x); Therefore, p ∈ N T (x) (v -p),and hence, v -p = π(v), ∀v ∈ V (x).

  , y(t)) = ϕ(x, y(t)), y (t) , for T > T we haveΓ(x, y(T + h )) = Γ(x, y(T )) + T +h T ϕ(x, y(s)), π (ϕ(x, y(s))) ds > Γ(x, y(T )) + ηh > Γ(x, ỹ),which contradicts (5.8).

(5. 11 )

 11 P (x) = {y ∈ K : min z∈F (y) h(z) ≤ min z∈F (x) h(z)}, ∀x ∈ K.

  (a) F (x) is a convex compact set, ∀x ∈ K; (b) F (•) is concave; (c) F (•) is a continuous. A2 Let h : R n + → R + be a single-valued function. Such that applications Moreover, let x(•) be a solution of the Cauchy problem (5.10), and let First, we note that the existence of solutions of problem (5.10) follows from Proposition 5.4.1, Assumption A0, and Theorem 5.3.1. Second, we point out that by Proposition 5.4.3 a point x verifies (5.15) if and only if π(-ψ(x)) = 0. We shall show that the limit points represent the minimum of the function w(•) with respect to the preorder P (•). Since x(•) is monotone with respect to w(•), we have that (5.16) w(x) = min t∈[0,+∞)

  2.4 statement (ii), we have-ψ n (z n ) -π(-ψ n (z n )) ∈ N (z n ). applicationsWe shall show that the limit points satisfies this condition. Since x(•) is monotone with respect to w(•), one has(5.19) w(x) = max t∈[0,+∞)

ω k = τ k x k + ( 1 -

 1 τ k )y, where τ k ∈ [0, 1] and lim k→+∞ τ k = 0.Since h(•) is uniformly prox-regular with constant c > 0, the function h = h+ c 2 • 2 is convex. By the convexity of h (•) and the concavity of F (•), we obtain forz 1 ∈ F (x k ), z 2 ∈ F (y) min z∈F (ω k ) h(z) ≤ min z∈τ k F (x k )+(1-τ k )F (y)

v 1 ∈

 1 D * F (ȳ 1 , z1 )(z 1 ) ⇔ (v 1 , -z 1 ) ∈ N gph(F ) (ȳ 1 , z1 ), and for z 2 ∈ N Θ (z 2 ), z2 ∈ F (ȳ 2 ) ∩ Θ v 2 ∈ D * F (ȳ 2 , z2 )(z 2 ) ⇔ (v 2 , -z 2 ) ∈ N gph(F ) (ȳ 2 , z2 ). Since gph(F ) is convex we have (v 1 , -z 1 ) -(v 2 , -z 2 ), (ȳ 1 , z1 ) -(ȳ 2 , z2 ) ≥ 0.

  t)) = ψ(x), x (t) a.e. t ≥ 0.

  3) et(1.4) se trouve dans le tube B β (α).

Proposition 1.3.12. Supposons que les hypothèses (S -L 2 ) et (P) soient satisfaites. Alors, chaque solution x ∈ A(x 0 ) de (1.3) ou (1.4) appartient à B β (α).

  Théorème 1.3.13. Supposons que les hypothèses (P), (H), (H k ), et (S -L 2 )

* soient satisfaites. Alors, le problème (1.5) a une solution dans

W 1,2 ([0, T ], R n ) ∩ B β (α).

Théorème 1.3.14. Supposons que les hypothèses (P), (H), (H k ), et (S -L 2 ) * soient satisfaites. Alors, le problème (1.5) a une solution dans W 1,2

  ≥ x 2 , alors h p (x 1 ) ≤ h p (x 2 ), ∀x 1 , x 2 ∈ K p .

	Quelques aspects sur les processus de rafle
	(H 2 p ) h p : R n + → R + est une fonction à valeurs réelles, telle que
	(a) h p (•) est continue;
	(b) h p (•) est strictement convexe;
	(c) si x 1 Proposition 1.3.15. Supposons que les hypothèses (H 1 p ) et (H 2 p ) soient satisfaites.
	Alors, P (•) est un préordre continu à valeurs non vides compactes convexes.
				semi-continue
	supérieurement.
	Définissons l'application multivoque P : K p ⇒ K p par
	P (x) = {s ∈ K p :	min z∈Fp(s)	h p (z) ≤ min z∈Fp(x)	h p (z)} ∀x ∈ K p ,
	où h p et F p satisfont les hypothèses suivantes;
	(H 1 p ) F p : K p ⇒ R n + est une application multivoque, telle que
	(a) F p (x) est un ensemble convexe compact, ∀x ∈ K p ;
	(b) F p (•) est concave;
	(c) F p (•) est continue.
			xxix

2

) Pour tout x ∈ K p fixé, la fonction y → Γ(x, y) est concave et

  K p × K q → R + une fonction qui satisfaisait les hypothèses (H 1 ) et (H 2 ).

	Alors, notre résultat d'existence est donné dans le théorème suivant;
	Théorème 1.3.17. Soient P (•) et Q(•) deux préordres continus à valeurs compactes
	convexes définis sur les sous-ensembles compacts convexes K p et K q respectivement.
	Soit Γ : Alors, il existe une solution P (•) × Q(•)-monotone (x(•), y(•)) de (1.6) pour tous
	points initiaux x 0 ∈ K p et y 0 ∈ K q .
	De plus, soient x(•) et y(•) solutions du problème mixte (1.6) et soient
	x = lim tn→+∞	x(t n ) et ỹ = lim tn→+∞	y(t n );
	alors,
	Γ(x, ỹ) = min	max	Γ(x, y) = max	min
	x∈P (x)	y∈Q(ỹ)	y∈Q(ỹ)	x∈P (x)

  1,1 [0, T ] respectively lower and upper solutions of (3.3) with α(t) ≤ β(t) for t ∈ [0, T ]. Then (3.3) has a solution x ∈ W 1,1 [0, T ]

	with
	α(t) ≤ x(t) ≤ β(t),	for t ∈ [0, T ].
	Theorem 3.3.41. Suppose F : [0, T ] × R ⇒ K where K is a nonempty compact
	subset of R, satisfies the following conditions:
	(i) (t, x) → F (t, x) is L ⊗ B measurable,
	(ii) x → F (t, x) is l.s.c. for a.e. t ∈ [0, T ].

Moreover, assume there exists α, β ∈ W 1,1 [0, T ] respectively lower and upper solutions of

(3.3)

with α(t) ≤ β(t) for t ∈ [0, T ]. Then (3.3) has a solution x ∈ W 1,1 [0, T ] with α(t) ≤ x(t) ≤ β(t), for t ∈ [0, T ].

Take the differential inclusion (3.3) with F

  a Carathéodory multivalued mapping with nonempty, closed, convex values. Assume there exists (

  x(t)) + x(t) t , a.e. t ∈ [0, T ]} where L 2 ([0, T ], R n ) supplied with the weak topology. Then F ρ is u.s.c. with compact convex values. Furthermore, there exists l ∈ L 2 ([0, T ], [0, ∞)) such that, for every x ∈ C([0, T ], R n ) and every ν ∈ F Proposition 4.3.4. Assume that (H), (H k ) and (S -L 2 ) hold. Then there exists a continuous single valued map f ρ

ρ (x), ν(t) ≤ l(t) a.e. t ∈ [0, T ].

  with closed convex, and bounded values whenever L 2 ([0, T ], R n ) is supplied with the weak topology. Recall that, there exists l

  2. for a.e. t ∈ [0, T ], and every x ∈ R n such that x -α(t) = β(t), there exists ν ∈ F (t, x) such that

	x -α(t), ν + δ(t) -α (t) ≥ β(t)β (t) + ρβ 2 (t);
	3. α (t) ∈ δ(t) + F (t, α(t)) a.e. on {t ∈ [0, T ] :

  applications multivoques. L'application F satisfait différents types de semi-continuité et l'ensemble C(t) est fermé non vide et uniformément prox-régulier. Dans ce contexte, N C(t) (x(t)) est le cône normal à C(t) au point x(t).

  exists, and satisfiesx (t) = lim h→0 + [x(t + h) -x(t)]/h.Note that for h sufficiently small, x(t + h) stays in R(x(t)) for all t + h ≥ t (see, Definition 3.3.47), and we can deduce that x (t) ∈ T (x(t)) almost everywhere in [0, ∞). Then

x (t) ∈ H V (x(t)) ∩ T (x(t)),

a.e. in [0, ∞).

Acknowledgments

The process of earning a doctorate and writing a dissertation is long, arduous, and it is certainly not done singlehandedly. Now it is time to express my gratitude to all the people that supported me during what ended up being the toughest times of my life.

Chapter 4 Existence of solutions-tube for non-convex sweeping processes Choose 0 < λ 0 < ε x -α(t) /2(ρ(1 + 2 x -α(t) ) + |β (t)| + z 0 + µ -δ(t) -α (t) ). hence, for all u ∈ B(x, λ 0 ),

Further, the lower semi-continuity of F with respect to its second variable ensure that there exists λ 1 > 0 such that B(x, λ 1 ) ⊂ {u : F (t, u t ) ∩ U = ∅}. Hence, for λ 2 = min{λ 0 , λ 1 }, B(x, λ 2 ) ⊂ {u ∈ R n : U ∩F (t, u t )∩{z : u-α(t), z -δ(t)-α (t) ≤ β (t) u-α(t)ρ u -α(t) 2 } = ∅} ⊂ O. Consequently, O is an open set. Therefore, x → F (t, x) is l.s.c.

Moreover, following the same arguments as in [START_REF] Frigon | Systems of first order differential inclusions with maximal monotone terms[END_REF] we can prove that assumptions (H k ) and (S -L 2 ) guarantee that, there exists l ∈ L 2 ([0, T ], R n ) such that, for every x ∈ C([0, T ], R n ), F (t, x(t)) + x(t) t ⊂ B(0, l(t)) a.e. t ∈ [0, T ], since x(t) t ≤ α 0 + β 0 . Furthermore, we conclude that, for all x ∈ C([0, T ], R n ), the map t → F (t, x(t)) is measurable with closed nonempty values; (see, [START_REF] Hiai | Integrals, conditional expectations, and martingales of multivalued functions[END_REF]). By the Kuratowski Ryll-Nardzewski selection theorem [START_REF] Kuratowski | A general theorem on selectors[END_REF], this map has a measurable selection; so, the map F : C([0, T ], R n ) → L 2 ([0, T ], R n ) defined by F(x) = {ν ∈ L 2 ([0, T ], R n ) : ν(t) ∈ F (t, x(t)) + x(t) t a.e. t ∈ [0, T ]} has bounded nonempty values. We can observe that F has closed, decomposable values, i.e. for every ν, z ∈ F and every measurable set Ω ⊂ [0, T ], νχ Ω +zχ Ω c ∈ F(x).

Now let {x n } be a sequence in {x ∈ C([0, T ], R n ) : F(x) ⊂ E} converging to x 0 , such that E ⊂ L 2 ([0, T ], R n ) is closed. Let ν ∈ F(x 0 ). For every n ∈ N, there exists ν n ∈ F(x n ) such that ν n (t) -ν(t) = d(ν(t), F (t, x n (t)) + x(t) t ).

By the lower semi-continuity of x → F (t, x) + x(t) t we have d(ν(t), F (t, x n (t)) + x(t) t ) → 0 a.e. t ∈ [0, T ].

We apply the Lebesgue dominated convergence theorem, hence ν n → ν in L 2 ([0, T ], R n ) and ν ∈ E. So, F(x 0 ) ⊂ E. consequently, F is l.s.c.

Finally, Fryskowski, Bressan-Colombo selection theorem [START_REF] Bressan | Extensions and selections of maps with decomposable values[END_REF] and [START_REF] Fryszkowski | Continuous selections of Aumann integrals[END_REF] allows us to conclude the proof.

Proof. Let us consider the indicator function I C(•) . Under (P), I C(•) is proper, lower semi-continuous. Take g = I C(•) + ρ 2 • 2 on one hand. We obtain that ∂g = ∂I C(•) + ρ id = N C(•) + ρ id. The operator N C(•) is ρ-hypomonotone, and hence by Lemma 3.4.12 the function g is proper, lower semi-continuous and convex. Consequently, Theorem 3.2.14 implies that ∂g = N C(•) + ρ id is maximal monotone.

Moreover, by a simple modification of the function in the proof of the maximality of N C(•) and by taking

we obtain that N C(•) + ρ id is maximal monotone.

In order to establish our existence results, let us define

where dom(

Proposition 4.3.7. Assume that (P) holds. Then, the operator A + is a multivalued maximal monotone operator.

Proof. We first show that A + is monotone. It is easy to verify that, D( A + ) = ∅. So, take x, x ∈ dom( A + ) and µ ∈ (N

Consider x, x ∈ A(x 0 ) then x(0) = x(0) and we have

On the other hand, we want to show that A + is maximal which is equivalent to showing that id + A + is surjective. We use the fact that id +L is invertible and hence surjective, then by Lemma 3.3.20, L is maximal monotone. Since for λ > 0, (N C(•) + ρ id) λ is single valued, monotone and Lipschitzian, L + Some results on sweeping processes

Let us give some properties of the operator

Proposition 4.3.8 and Remark 4.3.9 imply that P + • f ρ is a single-valued continuous compact operator. By the Schauder's fixed point theorem, there exists a fixed point x to the operator P + • f ρ , and hence a solution to the problem (4.5). Finally, Proposition 4.4.1 ensures that this solution is in B β (α).

Remark 4.4.2. Note that the convex case (i.e., when the sets C(t) are convexes) is a particular case of the ρ-prox-regular one with ρ = 0.

Further results

Now we consider the problem:

We can provide existence results similar to those obtained for the problem (4.1). To this aim, let us introduce an adapted notion of L 2 -solution-tube for this problem.

(H 2 ) For every fixed x ∈ K p , the function y → Γ(x, y) is concave and upper semicontinuous.

Let us define the multivalued mapping P : K p ⇒ K p by:

Suppose that the following assumptions hold:

We have the following result.

Proposition 5.2.1 (cf. [START_REF] Falcone | Maximum descent monotone solutions of an ordinary differential equation with a discontinuous right-hand side[END_REF]). Assume that assumptions (H 1 p ) and (H 2 p ) are satisfied. Then, P (•) is a continuous preorder with nonempty compact convex values.

Similarly, let us define the multivalued mapping Q : K q ⇒ K q by:

As previously, we suppose that the following assumptions hold:

(H 2 q ) h q : R m + → R + is a single-valued function satisfying;

(a) h q (•) is continuous; Some results on sweeping processes Theorem 5.3.1 (Existence result). Let R(•) be a continuous preorder with convex compact values defined on a compact convex subset K in R n , and let V : K → R n be u.s.c. Then, for any initial point x 0 ∈ K there exists a R-monotone solution x(t) of (5.3).

Proposition 5.3.2. Let R(•) be a continuous preorder. Then for every x ∈ C, the multivalued mapping N R(x) (x) has a closed graph.

Proof. Let z n and x n be two sequences such that

. Then, there is y ∈ R(x) such that z, y > z, x ; so, by the lower semicontinuity of R(•), there exists a sequence {y n }, convergent to y, such that y n ∈ R(x n ). Thus, we have z n , y n → z, y , and definitively

which is a contradiction.

We are now ready to give the proof of Theorem 5.3.1.

Proof of Theorem 5.3.1. Let us consider the multivalued mapping

We shall prove that H V (•) is u.s.c., with convex compact values. Since V is u.s.c. at x 0 , then for all ε > 0, there exists r > 0 and κ > 0 such that

Observe that, for all v ∈ V (x), y ∈ H V (x), and all p ∈ N (x) we have p, y ≥ 0 therefore p, v -p 2 ≥ 0.

Clearly, H V (x) is nonempty, compact, for all x fixed in E.

Chapter 5 Existence of monotone solutions with respect to a preorder and applications

Proof of Theorem 5.2.4

In this section, we shall give the proof of our main result: the existence of P (•) × Q(•)-monotone solutions to problem (5.1). For doing this we need the following proposition describing the monotonicity of the function Γ(•, •). We denote by

, and T Q(y) (y) = T (y).

with f to be singled out.

Proposition 5.3.4. Let x ∈ K p and y ∈ K q . Let ψ(x, y) be a measurable selection in ∂ x Γ(x, y), and let ϕ(x, y) be a measurable selection in ∂ + y Γ(x, y). The following statements hold

(i) If π(-ψ(x, y)) = 0, then for every y ∈ K q there are no descent directions of Γ(•, y) in T (x).

(ii) If π(-ψ(x, y)) = 0, then ψ(x, y), π(-ψ(x, y)) < 0, and for any u ∈ T (x), such that u = π(-ψ(x, y)) , one has that ψ(x, y), π(-ψ(x, y)) < ψ(x, y), u .

(iii) If π (ϕ(x, y)) = 0, then for every x ∈ K p there are no ascent directions of Γ(x, •) in T (y).

and for any v ∈ T (y), such that u = π (ϕ(x, y)) , one has that ϕ(x, y), π (ϕ(x, y)) > ϕ(x, y), v .

Some results on sweeping processes

Proof. First of all, for every ε > 0 let ψ ε and ϕ ε be two sequences of continuous single-valued maps approaching ∂ x Γ and ∂ + y Γ respectively in the sense of the approximate selection Theorem 3.3.24. For h > 0, and t ≥ 0, we have that

We begin by dividing the two sides by h > 0, taking in account that ψ ε and ϕ ε are continuous. Thereafter, ψ ε converges to some measurable selection denoted ψ in ∂ x Γ, and ϕ ε converges to some measurable selection denoted ϕ in ∂ + y Γ. Consequently, we obtain (5.4).

Second, let us verify the statement (i). To this aim, for u ∈ T (x), we have that

Therefore, if π(-ψ(x, y)) = 0, we obtain ψ(x, y), u > 0. Consequently, 0 ∈ ∂ x Γ(x, y).

In addition, from (5.5) and Proposition 3.2.6 we have that

Moreover,

Since u = π(-ψ(x, y)) , we obtain -ψ(x, y), u ≤ -ψ(x, y), π(-ψ(x, y)) ,

Chapter 5 Existence of monotone solutions with respect to a preorder and applications and where the equality holds if and only if u = π(-ψ(x, y)). Thus, the statement (ii) holds. Third, let v ∈ T (y), we have

Therefore, if π (ϕ(x, y)) = 0, we obtain ϕ(x, y), v < 0. Consequently, 0 ∈ ∂ + y Γ(x, y), and the statement (iii) holds.

Finally, let us verify the statement (iv). For that reason, let v ∈ T (y). From (5.6) and Proposition 3.2.6 we have that

Where the equality holds if and only if v = π (ϕ(x, y)).

We continue with the proof of the main result.

Proof of Theorem 5.2.4. First, let us note that the existence of solutions of the mixed problem (5.1) follows from Proposition 5.2.1 and 5.2.2, Assumption (H 1 ) and (H 2 ), and the existence Theorem 5.3.1. Indeed, we consider that

Second, observe that points x and ỹ verify (5.2) if and only if

So we shall show that the limit points verifies the condition (5.7). For all t > 0, and h > 0, we have that

Some results on sweeping processes Therefore,

Recall that (x, ỹ) ∈ P (x) × Q(ỹ), consequently, for h > 0, and h > 0 we have that

Recall that, π(ϕ(•, •)) and π (-ψ(•, •)) are bounded, so for any ε > 0, we can find ζ > 0, and

On one hand, let us suppose that π (ϕ(x, ỹ)) = 0. Then, for h > 0, one has

We want to show that if ϕ(x, ỹ), π (ϕ(x, ỹ)) > 0, there exist ε > 0 and η > 0 such that ϕ(u, z), π (ϕ(u, z)) > η > 0, ∀z ∈ B(ỹ, ε), ∀u ∈ B(x, ε).

Otherwise, there will be two sequences {u n }, and {z n }, with u n → x, and z n → ỹ, such that

Let ϕ n be a sequence of continuous approximate selections of ∂ + y Γ that converges to ϕ. From Corollary 3.2.4 we have that

Therefore, we have

Note that by Proposition 5.3.2, the normal cone N (•) has a closed graph. Then, ϕ(x, ỹ) ∈ N (ỹ), Chapter 5 Existence of monotone solutions with respect to a preorder and applications

Let us introduce the evolution problem. We are looking for trajectories x : [0, ∞) → K, starting at x 0 ∈ K, such that:

We have the following result.

Proposition 5.4.1 (cf. [START_REF] Falcone | Maximum descent monotone solutions of an ordinary differential equation with a discontinuous right-hand side[END_REF]). Assume that assumptions A1 and A2 are satisfied. Then, the preorder P (•) defined by (5.11) is continuous with nonempty compact convex values.

Remark 5.4.2. Proposition 5.4.1 shows that the preorder P (x) is a closed convex set. Consequently T P (x) (x) the tangent cone on P (x) at x is l.s.c., closed convex cone, and this determines the discontinuity of the right-hand side of (5.10).

We continue by showing that, for all ψ(x) ∈ ∂w(x), π(-ψ(x)) guarantees the maximum descent of the function w(•) among the vectors of the tangent cone T (x). Recall that, T (x) = T P (x) (x), and N (x) = N P (x) (x)

Proposition 5.4.3. Let x ∈ K, and let ψ(x) be a measurable selection in ∂w(x).

Then, we have

Moreover, (i) if π(-ψ(x)) = 0, then there are no descent directions of w(•) in T (x);

Moreover, for any u ∈ T (x), such that

we have ψ(x), π(-ψ(x)) < ψ(x), u .

Proof. First, let us show that equality (5.12) holds. To this end, for every ε > 0, we let ψ ε be a sequence of continuous maps approaching ∂w in the sense of the approximate selection Theorem 3.3.24. For every h > 0, and t ≥ 0, we obtain

We begin by dividing the two sides by h > 0, taking in account that ψ ε are continuous. Thereafter, ψ ε converges to some measurable selection denoted ψ in ∂w, and consequently, we obtain (5.12). Second, for any u ∈ T (x) we have,

Let π(-ψ(x)) = 0 we obtain that ψ(x), u > 0, and we conclude that 0 ∈ ∂w(x). Then, the statement (i) holds. Statement (ii) follows from Proposition 3.2.6. In fact we have

We can conclude the proof using equality (5.13). Indeed, from (5.14) we have

Here the equality holds if and only if u = π(-ψ(x)).

Now we study the limit points of a solution of (5.10). More precisely, we prove that the limit points of a solution of (5.10) is a minimum of the function w(•) with respect to the preorder P (•). Theorem 5.4.4. Let P (•) be a continuous preorder with convex compact values defined on a compact convex subset K, and let w : K → R + be a function satisfying A0. Then, for any initial point x 0 ∈ K there exists a P -monotone solution x(•) of (5.10). Some results on sweeping processes Proposition 5.3.2 implies that the normal cone N (•) has a closed graph. Then, -ψ(x) ∈ N (x), which contradicts the fact that ψ(x), π(-ψ(x)) < 0.

We shall continue by supposing that π(-ψ(x)) = 0. By equality (5.12) in Proposition 5.4.3, there exists T > T such that

which contradicts (5.16). This ends the proof.

The concave case

Let us consider the following differential inclusion (5.17)

where, ∂ + w(•) is the superdifferential of the function w(•), which satisfies the following assumption:

(A0) + w : K → R + is a proper, concave, and upper semicontinuous function.

Existence of monotone solutions

Let K be a convex compact set of R m + . We define the following preorder

We shall make the following assumption:

Chapter 5 Existence of monotone solutions with respect to a preorder and applications

Let us introduce the evolution problem. We are looking for trajectories x : [0, ∞) → K to problem (5.17), starting at x 0 ∈ K, such that: (i) x(•) is monotone with respect to Q(•) in the sense of Definition 3.3.47, (ii) w(x(t)) is an increasing function of t ∈ [0, +∞). Proposition 5.4.5. Assume that assumptions A1 and (A2) + are satisfied. Then, Q(•) is a continuous preorder with nonempty compact convex values.

In the following proposition we shall use the following notations:

Proposition 5.4.6. Let x ∈ K, and let ϕ(x) be a measurable selection in ∂ + w(x). Then

Moreover, (i) if π(ϕ(x)) = 0, then there are no ascent directions of w(•) in T (x),

(ii) if π(ϕ(x)) = 0, then ϕ(x), π(ϕ(x)) > 0, and for any u ∈ T (x), such that u = π(ϕ(x)) , one has ϕ(x), π(ϕ(x)) > ϕ(x), u .

Proof. The proof of Proposition 5.4.6 is similar to the one given for Proposition 5.4.3 with appropriate adjustments. Since the mapping ∂ + w is u.s.c., then for ε > 0 we take ϕ ε a sequence of continuous maps approaching ∂ + w in the sense of the approximate selection Theorem 3.3.24. For h > 0, and t ≥ 0, we obtain

We begin by dividing the two sides by h > 0, taking in account that ϕ ε are continuous. Thereafter, ϕ ε converges to some measurable selection denoted ϕ in ∂ + w, and consequently, we obtain (5.18).

Some results on sweeping processes

Let ϕ ∈ ∂ + w, for any u ∈ T (x) we have

The statement (i) follows from Lemma 3.2.18. The first inequality of statement (ii) follows from Proposition 3.2.6. Indeed we have

For any u ∈ T (x) we have

and since u = π(ϕ(x)) , we obtain

where the equality holds if and only if u = π(ϕ(x)).

We turn now to study the limit points of a solution of (5.17). We shall prove that a limit point of a solution of (5.17) is the maximum of the function w(•) with respect to the preorder Q(•).

Theorem 5.4.7. Let Q(•) be a continuous preorder with convex compact values defined on a compact convex subset K, and let w : K → R + be a function satisfying (A0) + . Then, for any initial point x 0 ∈ K there exists a Q-monotone solution x(•) of (5.17).

Moreover, if x(•) is a solution of the Cauchy problem (5.17), such that

then, we have

w(x).

Proof. First, we point out that, the existence of Q-monotone solutions to problem (5.17) follows from Proposition 5.4.5, Assumption (A0) + , and Theorem 5.3.1. Second, we observe that from Proposition 5.4.6, x is a maximum of w on Q(x) if and only if π(ϕ(x)) = 0.

Some results on sweeping processes

Further results: the prox-regular case

First of all, we provide some results on the coderivative of a multivalued mapping which will be used later. R m ⇒ R n defined by:

Definition 5.5.2. For a multivalued mapping S : R n ⇒ R m and any x and ȳ ∈ S(x) such that gph(S) is locally closed at (x, ȳ), we say that S is metrically regular if there exists U a neighborhood of x, W a neighborhood of ȳ and a positive constant κ, such that d(x, S -1 (y)) ≤ κd(y, S(x)) when x ∈ U, y ∈ W.

Theorem 5.5.3. (cf. [START_REF] Mordukhovich | Variational Analysis and Generalized Differentiation I. Basic Theory[END_REF]) Let x ∈ S -1 (Θ), where S : R n ⇒ R m is a closed graph multivalued mapping, Θ ⊂ R m is a closed set and S(•) ∩ Θ is locally compact at x. Assume that S is metrically regular around (x, ȳ) for every ȳ ∈ S(x) ∩ Θ. Then one has

We consider the following assumptions: let c ≥ 0.

(A0) c w : K → R + is a proper, lower semicontinuous and c-prox-regular function. Proof. Let us first show that P (•) has a closed graph. Let {x k } ⊆ K and {y k } ⊆ P (K) be two sequences, such that

Some results on sweeping processes min

Since ω k → y, there exists N ∈ N; so, for all k ≥ N such that ω k -y < η.

Note that P (y) is closed, so we have that x ∈ P (y). Moreover, y ∈ P (x) hence we conclude that P (x) = P (y).

By the continuity assumption on h(•) and F (•), we have that

Therefore, we can find a constant M ∈ N such that

This implies that ω N ∈ P (x M ) because of the definition of P (•), which contradicts the fact that y M -y ≤ z -y , ∀ z ∈ P (x M ). Now we discuss the regularity of the images P (x). It is easy to see that

where

x } is the level set of h. Since h is lower semicontinuous and c-prox-regular, the set lev ≤cx h is uniformly prox-regular with some constant r > 0 (see, [START_REF] Bounkhel | Subdifferential stability of the distance function and its applications to nonconvex economies and equilibrium[END_REF]).

Moreover, F is concave, hence gph(F ) is convex. On one hand, F has compact valued and convex-graph. Then, by Corollary 9.34 and Theorem 9.30 in [START_REF] Rockafellar | variational Analysis[END_REF] the mapping F is locally Lipschitz continuous. On the other hand, F is continuous and convex-graph. Then, by Theorem 9.48 in [START_REF] Rockafellar | variational Analysis[END_REF] the mapping F is metrically regular around (ȳ, z) for every ȳ ∈ F -1 (z). By Theorem 5.5.3, we have that for all ȳ ∈ F -1 (Θ)

where Θ = lev ≤cx h. Let us show that N F -1 (Θ) (ȳ) is hypomonotone. To this aim, let x ∈ K, let ȳ1 , ȳ2 ∈ P (x), and let v 1 ∈ N P (x) (ȳ 1 ), v 2 ∈ N P (x) (ȳ 2 ), we have, for

Some results on sweeping processes Theorem 5.5.5. Let P (•) be a continuous preorder with prox-regular compact values defined on a compact convex subset K, and let w : K → R + be a function satisfying (A0) c . Then, for any initial point x 0 ∈ K there exists a P -monotone solution x(•) of (5.10). Moreover, let x(•) be a solution of the Cauchy problem (5.10), and let

then, we have

w(x).

Proof. On one hand, P (x) is prox-regular, the tangent cone T (x) is a closed convex cone, hence proj T (x) is well-defined (see, Proposition 3. The second part of the theorem follows from equality (5.20) and Proposition 5.4.3.

Applications

Example 5. Game. We consider a two players p and q game with a collective payoff. The loss function h p : R n + → R + represents the negative gain of player p, with the preorder P : K p → K p given by

Similarly, the gain function h q : R n + → R + represents the positive gain of player q, with the preorder Q : K q → K q given by Q(y) = {s ∈ K q : max z∈F (s) h q (z) ≥ max z∈F (y) h q (z)}, ∀y ∈ K q .

The sets K p and K q are the sets of strategies of player p and player q respectively. x(t) is a strategy of player p in K p and y(t) is a strategy of player q in K q .

Chapter 5 Existence of monotone solutions with respect to a preorder and applications Player p seeks to minimize h p and player q seeks to maximize h q . Therefore, P (x) and Q(y) represent the sets of optimal strategies of players p and q respectively. So, our problem here is to prove that (x * , y * ) is the maximum for the collective pay-off function given by Γ(x, y), i.e., Γ(x * , y * ) = max (x,y)∈P (x * )×Q(y * ) Γ(x, y).

it is sufficient to maximize a collective well-being with Γ(x, y) = r(y) -w(x), such that r is the revenue and w is the cost function.

Example 6. Production. We consider a firm which produces n consumption goods using m inputs which it buys at constant prices. We consider resources and materials as well as energy, labor, ..., etc, as inputs. The firm is assumed to be small that no issue of scarcity will arise: at any time it can choose any input in the regular set K of available resources, where K ⊆ R m is a convex compact set. Set F : K ⇒ R n + as the production function of the firm, the production technologies of the firm are represented by the set F (x) the production output, for a given input bundle x. The firm can produce different outputs with the same input x. The outputs are considered as consumption goods: none of them will come back in the production cycle as an input. The production function F satisfies the following assumptions • F (x) is a convex compact set, for all input x which implies that the firm can only produce bounded outputs with bounded resources.

• F (•) is concave which means that the combination of resources enriches the production possibility set.

• F (•) is continuous, because we suppose that technology is fixed, and no technological renewal is allowed.

A loss function h p : R n + → R + represents the revealed preferences of consumers relative to different product bundles which are something the firm can know from past experience and market researches: h(•) represents the firm's knowledge of the market. h(•) is continuous, strictly convex and a decreasing function. The consumers will always prefer abundance to the scarcity of consumption goods. The aim of the firm is to derive its production activity by the input in order to make the loss function decrease; So, we define the preorder P by

Some results on sweeping processes P (•) is continuous with compact convex values.

Similarly, let h q : R n + → R + be the gain function. The firm derives its production activity from the input in order to make the gain function increase. Therefore, we define the preorder Q by

We shall also consider the profit of the firm Γ : K p × K q → R + given by Γ(x, y) = r(y) -w(x) for all input-output vector (x, y).

The profit maximization is the process by which the firm determines the price and output level that returns the greatest profit. Therefore, to find (x * , y * ) such that Γ(x * , y * ) = r(y * ) -w(x * ) = max (x,y)∈P (x * )×Q(y * ) r(y) -w(x).

Résumé du chapitre

Dans ce chapitre, nous fournissons un résultat d'existence des solutions bi-monotones pour un système d'inclusions différentielles associées aux sous-différentiels des fonctions convexes-concaves. Nous solutions sont monotones par rapport à un préordre, i.e., une relation binaire réflexive et transitive. La limite quand le temps tend vers l'infini d'une solution de notre système est un point-selle pour une fonction convexeconcave. Les résultats obtenus peuvent être appliqués à deux problèmes. D'une part, à un problème du jeu de deux joueurs avec un payement collectif et, d'autre part, à un problème de production pour une entreprise qui veut maximiser son profit. Nous considérons le problème suivant: Soient K p ⊂ R n et K q ⊂ R m deux sous-ensembles compacts convexes. Soit Γ : K p × K q → R + une fonction convexe-concave. Soient P : K p ⇒ K p et Q : K q ⇒ K q deux préordres à valeurs compactes convexes non vides. Considérons le système mixte    x (t) ∈ proj T P (x(t)) (x(t)) (-∂ x Γ(x(t), y(t)), p.p. t ∈ [0, +∞), y (t) ∈ proj T Q(y(t)) (y(t)) (∂ + y Γ(x(t), y(t)), p.p. t ∈ [0, +∞), x(0) = x 0 , y(0) = y 0 , où x 0 et y 0 sont donnés (donc x 0 ∈ P (x 0 ) et y 0 ∈ Q(y 0 ) compte tenu de la réflexivité des applications multivoques P et Q en tant que préordres).

Résumé

Dans cette thèse, on s'intéresse à l'étude d'existence de solutions pour les processus de rafle. Ce problème prend la forme d'une inclusion différentielle contrainte avec des cônes normaux qui apparaissent naturellement dans nombreuses applications telles que le mouvement de foule, l'élastoplasticité, les mécaniques, les circuits électroniques, etc.

L'objectif de ce travail est de rapprocher deux importantes classes d'inclusions différentielles. D'une part, nous établissons quelques résultats d'existence de tubesolutions de processus de rafle perturbés avec des ensembles uniformément proxréguliers. D'autre part, nous présentons des résultats d'existence de solutions bimonotones par rapport à un préordre pour un système d'inclusions différentielles projetées associées aux sous-différentiels de fonctions convexes-concaves. De plus, nous montrons l'existence d'un point-selle pour notre système et nous fournissons deux exemples d'applications.

Mots clés: Inclusion différentielle, processus de rafle, ensembles prox-réguliers, tube-solutions, solutions monotones, point-selle, régularisation de Moreau-Yosida.

Abstract

In this thesis, we were interested in the study of the existence of solutions for sweeping processes. This problem takes the form of a constrained differential inclusion involving normal cones which appears naturally in many applications such as crowd motion, elastoplasticity, mechanics, electrical circuit, etc.

The aim of this work is to bring together two classes of differential inclusions. On one hand, we establish some existence results of solutions-tube for perturbed sweeping processes with uniformly prox-regular sets. On the other hand, we present existence results of bi-monotone solutions with respect to a preorder for a system of projected differential inclusions associated to subdifferentials of convex-concave functions. In addition, we show that our system has a saddle-point and we provide two examples of applications.

Key words: Differential inclusion, sweeping process, prox-regular sets, solutionstube, monotone solutions, saddle-points, Moreau-Yosida regularization.