Contributions to the theory of mean field games.

Summary 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.