Singular Perturbation and Homogenization Problems in Control Theory, Differential Games and fully nonlinear Partial Differential Equations

In this thesis we address different topics related to homogenization of first and second order fully nonlinear PDEs, essentially of Hamilton--Jacobi type, and more generally to singular perturbation in optimal control problems and differential games, in the light of the viscosity solution theory. We take into account a singularly perturbed control systems (i.e. a system where the state variables evolve with two different time scales), both in the deterministic and in the stochastic setting, and the related first and second order Hamilton-Jacobi equations. A first part of the work is devoted to order reduction procedures: the goal of such procedures is to obtain, as the perturbation parameter tends to zero, a system where only the slow variables appear. The construction of the limit dynamics relies on the asymptotic behavior of the fast variables of the original system. We use limiting relaxed controls, i.e. suitably defined Radon probability measures to average the fast part of the controlled dynamics. We give - both in the deterministic and in the stochastic framework - representation formulae for the effective Hamiltonian in terms of limiting relaxed controls. This allow a control interpretation of the limiting dynamics. As an application of these reduction procedures, we study the propagation of fronts moving with normal velocity depending on the position and undergoing fast oscillations. In the second part of the work we study asymptotic controllability properties of a deterministic singularly perturbed systems and of the limit system. We prove first that, under suitable assumptions, the weak lower semilimit of Lyapunov functions of a singularly perturbed system is a lower semicontinuous Lyapunov function for the limiting system. Furthermore, we also prove that the asymptotic controllability to the origin of the (smaller) limit system is enough to infer asymptotic controllability of the slow part of the (larger) perturbed system. More precisely, perturbing a Lyapunov pair for the limit dynamics, we construct a Lyapunov pair for the original system. The third and last part of the thesis concerns homogenization of non-coercive Hamilton-Jacobi equations with oscillating Hamiltonian and initial data. We take into account a rather general class of Hamiltonians convex in some gradient variables and concave with respect to the others. In particular it is shown that for some of these equations homogenization does not take place, in contrast with the usual coercive case. Sufficient conditions for homogenization are provided involving the structure of the running cost and the initial data.