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Abstract

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 differential inclusions), every epi-Lipschitzian initial compact set (of the Euclidean space) preserves this regularity for a (possibly short) time, i.e. its reachable set is also epi-Lipschitzian for all small times. 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 (to reachable sets) with respect to both time and base point.