KOBEISSI Ziad

This thesis deals with the theory of mean field games (MFG for short). The main part is dedicated to a class of games in which agents may interact through their law of states and controls. we use the terminology mean field games of controls (MFGC for short) to refer to this class of games. First, we assume that the optimal dynamics depends upon the law of controls in a Lipschitz way, with a Lipchitz constant smaller than one. In this case, we give several existence results on the solutions of the MFGC system, and one uniqueness result under a short-time horizon assumption. Second, we introduce a scheme and make simulations for a model of crowd motion. Thrid, under a monotonicity assumption on the interactions through the law of controls, we prove existence and uniqueness of the solution of the MFGC system. Finally, we introduce an algorithm for solving MFG systems of variational type, we use a preconditioned strategy based on a multigrid method.

This thesis deals with the theory of mean-field games. Most of it is devoted to mean-field games in which players can interact through their state and control law. We will use the terminology mean-field control game to designate such games. First, we make a structural assumption, which essentially consists in saying that the optimal dynamics depends on the control law in a lipschitzian way with a constant less than one. In this case, we prove several existence results for solutions to the mean control field game system, and a uniqueness result in short time. In a second step, we set up a numerical scheme and perform simulations for population motion models. In a third step, we show the existence and uniqueness when the control interaction satisfies a monotonicity condition. The last chapter concerns a numerical solution algorithm for mean-field games of variational type and without interaction via the control law. We use a preconditioning strategy by a multigrid method to obtain a fast convergence.