State Constrained Optimal Control Problems With Control On The Acceleration. Applications To Kinetic Mean Field Games

Relying on the careful study of a related problem in the calculus of variations, we study a class of optimal control problems in which the control lies on the acceleration, with state constraints on the position variable. In dimension one, we find explicit formulas in the special case when the running cost is a power of the acceleration (in absolute value) and the terminal cost is zero. For more general costs or/and higher dimensions, we study the singularities of the value function. We also prove the closedness (in the C 1 topology) of the graph of the multivalued mapping which maps a point in the state space to the set of optimal trajectories which start from this point. A consequence of the latter is the existence, under general assumptions, of relaxed equilibria for a class of kinetic mean field games with state constraints.