Viscosity Solutions Methods for Singular Perturbations in Deterministic and Stochastic Control

Viscosity solutions methods are used to pass to the limit in some penalization problems for rst order and second order, degenerate parabolic, Hamilton-Jacobi-Bellman equations. This characterizes the limit of the value functions of singularly perturbed optimal control problems for nonlinear deterministic systems and controlled degenerate diiusions, respectively. The results cover also cases where the usual order reduction method does not give the correct limit, and diiusion processes with fast state variables depending nonlinearly on the control. Some connections with ergodic control and periodic homogenization are discussed.