On the convergence of the optimal value function for singularly perturbed differential inclusions

We consider Mayer type optimization problems for nonlinear singularly perturbed differential inclusions. Especially we are interested in the behaviour of the optimal value function as the perturbation parameter tends to zero. For that purpose we construct a strong limiting system for the slow motion in form of an averaged differential inclusion. We give sufficient conditions under which the value function of the original singularly perturbed problem converges to the value function corresponding to the averaged differential inclusion. These conditions are of controllability respectively stability type and concern only the fast subsystems with fixed slow state