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

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 setvalued 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.

Digital implementation of sliding‐mode control via the implicit method: A tutorial

“…is a so-called proximal, (or proximity, or proximation) operator. This is quite classical in Optimization and Convex Analysis, and the associated algorithm is a proximal point algorithm [1,28,21]. Let us now consider (10).…”

“…In this section, the robust control problem is made further complex [85], by considering parameter uncertainties in (21), i.e. :…”

“…Considering maximal monotone operators in (32), allows one to enlarge the set of multivalued controllers u s (t). It encompasses the signum multifunction as in (21).…”

“…The fundamental idea originates from numerical simulation and analysis of mechanical systems with unilateral contact and set-valued friction, in particular the Moreau-Jean algorithm [1,21,53,89]. This is close also to so-called proximal algorithms in Optimization [12,97] as introduced in [88], as well as discretization of differential inclusions with maximal monotone set-valued part [15,16,77].…”

“…We should also mention the analysis of the implicit Euler method for linear complementarity systems and relay systems, with main applications in circuits with nonsmooth components [21,70,22]. In slidingmode control and observation, the implicit approach has been initiated and deeply analysed by two main research groups, in France [2,3,4,7,19,48,50,51,47,83,85,87,101,36,119,49,20], and in Japan [9,64,11,79,56,65,55,125,126,59,62,10,60,123,124].…”