Ergodic behavior of control systems and first-order mean field games

The work in this thesis concerns the analysis of first-order mean field game (MFG) systems with control of acceleration and the study of the long time-average behavior of control systems of sub-Riemannian type. More precisely, in the first part we begin by studying the well-posedness of the MFG system associated with a control problem with linear state equation. In particular, via a relaxed approach, we prove the existence and the uniqueness of mild solutions and we also study their regularity. Then, we focus on the MFG system with control of the acceleration, a particular case of the one above, and we investigate the long time-average behavior of solutions showing the convergence to the critical constant. Here, as for the previous analysis, the main issues are the lack of strict convexity and coercivity of the Hamiltonian with respect to the momentum variable. Indeed, for instance, when studying the asymptotic behavior of the control system this lead us to a non existence result of continuous viscosity solutions to the ergodic Hamilton-Jacobi equation. Consequently, it does not allowed us to the define the ergodic MFG system as one would expect. We conclude this first part establishing a connection between the MFG system with control of acceleration and the classical one. To do so, we study the singular perturbation problem for MFG system of acceleration, that is, we analyze the behavior of solutions to the system when the acceleration cost goes to zero. Again, we solve the problem by using variation techniques due to the problems arising from the lack of strict convexity and coercivity of the Hamiltonian with respect to the momentum variable. In the second part, we concentrate the attention to drift-less affine control systems (sub-Riemannian type). Differently from the case of acceleration, we prove that there exists a critical constant and the ergodic Hamilton-Jacobi equation associated with such a constant has continuous viscosity solutions. This is possible appealing to the properties of the sub-Riemannian geometry on the state space. Still using the properties of this geometry we finally define the Lax-Oleinink semigroup and we prove the existence of a fixed point of such semigroup. We conclude this part, and thus this thesis, extending the celebrated Aubry-Mather Theory to the case of sub-Riemannian control system. We first show a variational representation formula for the critical constant and from this we define the Aubry set. By using a dynamical approach we study the analytical and topological properties of such sets as, for instance, horizontal differentiability of the critical solution at any points lying in such a set.