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Journal of Convex Analysis 17 (2010), No. 3&4, 961--990
Copyright Heldermann Verlag 2010

Existence and Uniqueness of Solutions for Non-Autonomous Complementarity Dynamical Systems

Bernard Brogliato
INRIA - ZIRST Montbonnot, 655 Avenue de l'Europe, 38334 Saint Ismier, France

Lionel Thibault
Université Montpellier II, Dép. de Mathématiques, Case courrier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France

This paper deals with the well-posedness of a class of complementarity dynamical systems. Both the linear and the nonlinear cases are treated, and the systems are non-autonomous. A specific "input-output" property is used to perform a change of state vector which allows one to transform the complementarity dynamics into a perturbed Moreau's sweeping process. Then the results obtained by J. F. Edmond and L. Thibault ["Relaxation of an optimal control problem involving a perturbed sweeping process", Mathematical Programming 104 (2005) 347--373; and "BV solutions of nonconvex sweeping process differential inclusions with perturbation", Journal of Differential Equations 226 (2006) 135--179] and L. Thibault ["Sweeping process with regular and nonregular sets", Journal of Differential Equations, 193 (2003) 1--26] on the well-posedness of the sweeping process are used. Absolutely continuous as well as bounded variation solutions (with state jumps) are examined in this work.

Keywords: Moreau's sweeping process, complementarity system, differential inclusion, existence, uniqueness, prox-regular set, state jumps, bounded variation.

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