Discontinuous sweeping process with prox-regular sets

¹ Laboratoire XLIM, Université de Limoges, 123 Avenue Albert Thomas, 87060 Limoges, Cedex, France ;
samir.adly@unilim.fr
² Département de Mathématiques, Université Montpellier, 34095 Montpellier, cedex 5, France
lionel.thibault@univ-montp2.fr
³ Centro de Modelamiento Matematico, Universidad de Chile, Santiago, Chile

Received: 6 October 2015
Accepted: 7 July 2016

Abstract

In this paper, we study the well−posedness (in the sense of existence and uniqueness of a solution) of a discontinuous sweeping process involving prox-regular sets in Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure and the perturbation is assumed to satisfy a Lipschitz property. The existence of a solution with bounded variation is achieved thanks to the Moreau’s catching-up algorithm adapted to this kind of problem. Various properties and estimates of jumps of the solution are also provided. We give sufficient conditions to ensure the uniform prox-regularity when the moving set is described by inequality constraints. As an application, we consider a nonlinear differential complementarity system which is a combination of an ordinary differential equation with a nonlinear complementarily condition. Such problems appear in many areas such as nonsmooth mechanics, nonregular electrical circuits and control systems.