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 first 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 deterministic systems and for controlled degenerate diffusions. The results apply to cases where the usual order reduction method does not give the correct limit, and to systems with fast state variables depending nonlinearly on the control. Some connections with ergodic control and periodic homogenization and discussed.