Filippov's and Filippov-Wazewski'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 a ny initial condition x(2)(0) is an element of R-n, there exists a trajector y 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 establishe s the possibility of approximating any trajectory of the convexified differ ential 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 conditio n. 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 r esults to infinite horizon constrained problems. Applications to the study of regularity of value functions of optimal control problems with state con straints are discussed as well. (C) 2000 Academic Press.