Epi-Lipschitzian reachable sets of differential inclusions

The reachable sets of a differential inclusion have nonsmooth topological boundaries in general. The main result of this paper is that under the well-known assumptions of Filippov’s existence theorem (about solutions of differential inclusions), every epi-Lipschitzian initial compact set K⊂RNK⊂RN preserves this regularity for a short time, i.e. ϑF(t,K)ϑF(t,K) is also epi-Lipschitzian for all small t>0t>0.The proof is based on Rockafellar’s geometric characterization of epi-Lipschitzian sets and uses a new result about the “inner semicontinuity” of Clarke tangent cone (t,y)↦TϑF(t,K)C(y)⊂RN with respect to both arguments.