Deterministic Mean Field Games with Control on the Acceleration and State Constraints

We consider deterministic mean field games in which the agents control their acceleration and are constrained to remain in a region of $R^n$. We study relaxed equilibria in the Lagrangian setting; they are described by a probability measure on trajectories. The main results of the paper concern the existence of relaxed equilibria under suitable assumptions. A difficulty in the proof of existence comes from the fact that the optimal trajectories of the related optimal control problem do not form a compact set. The proof requires closed graph properties of the map which associates to initial conditions the set of optimal trajectories.