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Table of contents
1 Introduction
1.1 La théorie des jeux à champ moyen
1.1.1 Le jeu à N joueurs
1.1.2 Le système de jeu à champ moyen
1.1.3 Jeux à champ moyen de contrôle
1.2 Organisation de la thèse
1.2.1 À propos des solutions classiques du système de jeu à champ moyen de contrôle
1.2.2 Approximation par la méthode des différences finies du système de jeu à champ moyen de contrôle
1.2.3 Jeux à champ moyen avec interactions monotones par la loi des états et des contrôles des joueurs
1.2.4 Un algorithme primal-dual pour des jeux à champ moyen dynamique du second-ordre et à couplage local
1.2.5 Perspectives et travaux futurs
2 On Classical Solutions to the Mean Field Game System of Controls
2.1 Introduction
2.2 Notations and assumptions
2.2.1 Notations and definitions
2.2.2 Assumptions
2.2.3 Main results
2.3 The fixed point relation in and the proof of Lemma 2.2.3
2.4 A priori estimates and the proof of Lemma 2.2.4
2.4.1 A priori estimates on u
2.4.2 A priori estimates on m
2.4.3 A priori estimates on derivatives of u
2.5 Existence and uniqueness results under additional assumptions
2.5.1 Solving the MFGC systems for M < 1
2.5.2 Existence results when q0 q0
2.5.3 Existence results which do not need the assumption q0 < q0
2.5.4 Existence and uniqueness results with a short-time horizon assumption
2.6 Applications
2.6.1 Exhaustible ressource model with nonpositively correlated ressources
2.6.2 Price impact models with bid and ask prices
2.6.3 First-order flocking model with velocity as controls
2.6.4 A model of crowd motion
3 Mean Field Games of Controls : Finite Difference Approximations
3.1 Introduction
3.1.1 A brief discussion on the mathematical analysis of (3.1.1)
3.1.2 A more detailed description of the considered class of MFGCs
3.1.3 Organization of the paper
3.2 Finite difference methods
3.2.1 Notations and definitions
3.2.2 The scheme
3.2.3 Solving the discrete version of the Hamilton-Jacobi-Bellman equation
3.2.4 Solving the discrete version of the Fokker-Planck-Kolmogorov equation
3.3 Newton algorithms for solving the whole system (3.2.8)-(3.2.17)
3.3.1 The coupling cost and the average drift
3.3.2 The linearized operators
3.3.3 The algorithm for solving (3.2.8)-(3.2.17)
3.4 Numerical simulations
3.4.1 First example
3.4.2 Second example
4 Mean Field Games with monotonous interactions through the law of states and controls of the agents
4.1 Introduction
4.2 Assumptions
4.2.1 Notations
4.2.2 Hypotheses
4.2.3 Main results
4.2.4 Properties of the Lagrangian and the Hamiltonian in (4.2.3) and (4.2.5)103
4.3 Applications
4.3.1 Exhaustible ressource model
4.3.2 A model of crowd motion
4.4 The fixed point (4.2.5c) and the proof of Lemma 4.2.4
4.4.1 Leray-Schauder Theorem for solving the fixed point in
4.4.2 The continuity of the fixed point in time
4.5 A priori estimates for the solutions to (4.2.5)
4.6 Existence and Uniqueness Results
4.6.1 Proof of Theorem 4.2.6: existence of solutions to (4.2.5)
4.6.2 Proof of Theorem 4.2.7: passing from the torus to Rd
4.6.3 Proof of Theorem 4.2.8: uniqueness of the solutions to (4.2.3) and (4.2.5)
4.6.4 Theorems 4.2.2 and 4.2.3: existence and uniqueness of the solution to (4.1.6)
5 On the implementation of a primal-dual algorithm for second order timedependent mean field games with local couplings
5.1 Introduction
5.2 Preliminaries and the finite difference scheme
5.3 The finite dimensional variational problem and the discrete MFG system .
5.4 A primal-dual algorithm
5.5 Preconditioning strategies
5.5.1 Multigrid preconditioner
5.5.2 Numerical Tests
