|Publication ahead of print|
|Published online||31 May 2017|
Discontinuous sweeping process with prox-regular sets
1 Laboratoire XLIM, Université de Limoges, 123 Avenue Albert Thomas, 87060 Limoges, Cedex, France.
2 Département de Mathématiques, Université Montpellier, 34095 Montpellier, cedex 5, France.
3 Centro de Modelamiento Matematico, Universidad de Chile, Santiago, Chile.
Received: 6 October 2015
Accepted: 7 July 2016
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.
Mathematics Subject Classification: 49J52 / 49J53 / 34A60
Key words: Variational analysis / measure differential inclusions / sweeping process / prox-regular set / B.V. solutions / Moreau’s catching-up algorithm / nonlinear differential complementarity systems
© EDP Sciences, SMAI 2017
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