Dynamical Systems Coupled with Monotone Set-Valued Operators: Formalisms, Applications, Well-Posedness, and Stability

This survey article addresses the class of continuous-time systems where a system modeled by ordinary differential equations is coupled with a static or 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, and 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.