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Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability

Bernard Brogliato 1 Aneel Tanwani 2
1 TRIPOP [2020-....] - Modélisation, simulation et commande des systèmes dynamiques non lisses [2020-....]
Inria Grenoble - Rhône-Alpes, LJK [2020-....] - Laboratoire Jean Kuntzmann [2020-....]
2 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes
Abstract : This survey article addresses the class of continuous-time systems where a system modeled by ordinary differential equations (ODEs) is coupled with a static and time-varying set-valued operator in the feedback. Interconnections of this form model certain classes of nonsmooth systems including sweeping processes, differential inclusions with maximal monotone right-hand side, complementarity systems, differential and evolution variational inequalities, projected dynamical systems, some piecewise linear switching systems. Such mathematical models have seen applications in electrical circuits, mechanical systems, hysteresis effects, and many more. When we impose a passivity assumption on the open-loop system, and regard the set-valued operator in the feedback as maximally monotone, we obtain a set-valued Lur'e dynamical system. In this article we review the mathematical formalisms, their relationships, main application fields, well-posedness (existence, uniqueness, continuous dependence of solutions), and stability of equilibria. An exhaustive bibliography is provided.
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Submitted on : Wednesday, December 4, 2019 - 4:39:49 PM
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Bernard Brogliato, Aneel Tanwani. Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability. SIAM Review, Society for Industrial and Applied Mathematics, 2020, 62 (1), pp.3-129. ⟨10.1137/18M1234795⟩. ⟨hal-02379498⟩

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