Filippov's and Filippov–Ważewski's Theorems on Closed Domains

Abstract

The celebrated Filippov's theorem implies that, given a trajectory x1:[0, +∞[↦Rn of a differential inclusion x′∈F(t, x) with the set-valued map F measurable in t and k-Lipschitz in x, for any initial condition x2(0)∈Rn, there exists a trajectory x2(·) starting from x2(0) such that |x1(t)−x2(t)|⩽ekt|x1(0)−x2(0)|. Filippov– Ważewski's theorem establishes the possibility of approximating any trajectory of the convexified differential inclusion x′∈coF(t, x) by a trajectory of the original inclusion x′∈F(t, x) starting from the same initial condition. In the present paper we extend both theorems to the case when the state variable x is constrained to the closure of an open subset Θ⊂Rn. The latter is allowed to be non smooth. We impose a generalized Soner type condition on F and Θ, yielding extensions of the above classical results to infinite horizon constrained problems. Applications to the study of regularity of value functions of optimal control problems with state constraints are discussed as well.