Existence of Solutions to the Nonconvex Sweeping Process

Abstract

In this note, we prove the existence of solutions for the sweeping process problem x′(t)∈−NC(t)(x(t)) a.e., x(t)∈C(t), x(0)=x0∈C(0), where C(.) is an arbitrary Hausdorff–Lipschitzean multifunction, from I=[0, T] onto the set of nonempty closed subsets of Rd. This generalizes a well known result of J. J. Moreau, (1971, in “Sem. d'Analyse Convexe, Montpelier,” Exp. 15) in the convex case.