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Dynamical systems coupled with monotone set-valued operators: formalisms, applications, well-posedness, and stability. (English) Zbl 1450.34002

The paper presents a thorough survey for a class of nonsmooth dynamical systems described by ordinary differential equations interconnected with static or time-dependent set-valued feedbacks. This class contains other fundamental formalisms such as Moreau’s Sweeping Processes, Differential Inclusions, Differential Variational Inequalities, Complementarity Systems, Projected Dynamical Systems, Switching Systems ... These formalisms are related to each other and each has its own advantages as well as drawbacks. Applications can be found in electrical circuits, mechanical systems, sliding mode control and observer, biological systems, neural networks, crowd dynamics, traffic flow networks, economical systems and finance, energy systems, aerosols dynamics…
Well-posedness of the problem is obtained by using regularization or discretization. Stability of equilibria and asymptotic behavior of solutions are also provided by using passivity conditions, Lyapunov-like functions and the Krasovskii-LaSalle invariance principle.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A36 Discontinuous ordinary differential equations
34A60 Ordinary differential inclusions
34D20 Stability of solutions to ordinary differential equations
49J52 Nonsmooth analysis
49J53 Set-valued and variational analysis
93D15 Stabilization of systems by feedback
93D20 Asymptotic stability in control theory

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