On non convex differential inclusions whose state is constrained in the closure of an open set. Applications to dynamic programming

In this paper the authors consider the problem of finding Filippov estimates when suitable state constraints are assumed to be satisfied by the solutions of a differential inclusion system. This assumption is similar to controllability conditions introduced by Soner in the case of control systems. Two results are presented involving different regularity properties of the boundary of the constraint set. The first result states that continuity of the solution with respect to the initial condition holds when the boundary of the constraint set is assumed to be locally Lipschitz. The second result states that continuity both of the solution and of its time-derivative with respect to the initial condition holds when the boundary of the constraint set is assumed to be smooth in the sense that the signed distance map to this boundary is differentiable with locally Lipschitz gradient on a neighborhood of the boundary. An application is provided to dynamic programming for optimal control problems.