Contributions to the theory of mean field games.

Summary 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.