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 ⊂ RN preserves this regularity for a short time, i.e. θF (t, K) is also epi-Lipschitzian for all small t > 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) 7→ TC θF (t,K)(y) ⊂ R N with respect to both arguments.