Filippov's and Filippov-Wa.zewski's theorems on closed domains

The celebrated Filippov's theorem implies that, given a trajectory x(1) : [0, + infinity[ \-> R-n of a differential inclusion x(t) is an element of F(t, x) with the set-valued map F measurable in t and k-Lipschitz in x, for any initial condition x(2)(0) is an element of R-n, there exists a trajectory x(2)( . ) starting from x(2)(0) such that \ x(1)(t) - x(2)(t)\ less than or equal to ekt \ x(1)(0) - x(2)(0)\. FilippovWaiewski's theorem establishes the possibility of approximating any trajectory of the convexified differential inclusion x'is an element of (co) overbar F(t, x) by a trajectory of the original inclusion x' EF(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 Theta subset of R-n. The latter is allowed to be non smooth. We impose a generalized Soner type condition on F and Theta, 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.