Geodesic
Discontinuous sweeping process with prox-regular sets
Adly, Samir1  ; Nacry, Florent1  ; Thibault, Lionel2, 3 
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
ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1293-1329

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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.

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DOI : 10.1051/cocv/2016053
Classification : 49J52, 49J53, 34A60
Keywords: Variational analysis, measure differential inclusions, sweeping process, prox-regular set, B.V. solutions, Moreau’s catching-up algorithm, nonlinear differential complementarity systems

Adly, Samir 1 ; Nacry, Florent 1 ; Thibault, Lionel 2, 3

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 
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     title = {Discontinuous sweeping process with prox-regular sets},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1293--1329},
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Adly, Samir; Nacry, Florent; Thibault, Lionel. Discontinuous sweeping process with prox-regular sets. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1293-1329. doi: 10.1051/cocv/2016053

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