Perturbed state-dependent sweeping processes

Abstract

In this paper, we consider a class of first-order differential inclusion, the right-hand side of the problem is governed by the so-called nonconvex state-dependent sweeping process. The perturbation, known as the external forces applied on the system, is general and takes the form of a sum of a single-valued mapping depends on velocity and a set-valued mapping depends on state. We study the existence of solutions and the compactness of the attainable set, where the set-valued mapping is an upper semi-continuous with convex values. Then, we treat the autonomous problem under assumptions that do not require the convexity of the values and that weaken the assumption on the upper semi-continuity. The results obtained are applied to the study of a control problem.