Discontinuous sweeping process with prox-regular sets

Samir Adly (1) , Florent Nacry (2) , Lionel Thibault (3)

1 XLIM-MATHIS - Mathématiques & Sécurité de l'information
2 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations
3 UM - Université de Montpellier

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.

Mots clés: sweeping process; prox-regular set; measure differential inclusions; Variational analysis; nonlinear differential complementarity systems; Moreau’s catching-up algorithm; B.V. solutions

Domaines: Mathématiques [math]

Samir Adly, Florent Nacry, Lionel Thibault. Discontinuous sweeping process with prox-regular sets. ESAIM: Control, Optimisation and Calculus of Variations, 2017, 23 (4), pp.1293-1329. ⟨10.1051/cocv/2016053⟩. ⟨hal-02922729⟩