Dissipative Dynamical Systems With Set-Valued Feedback Loops: Well-Posed Set-Valued Lur’e Dynamical Systems

Passivity and dissipativity, in Jan Willems’s sense, are known to be powerful tools for stability analysis feedback control design. For instance, passive systems negative interconnection with slope-restricted, static, single-valued, smooth nonlinearities, as Lur’e systems, have been thoroughly studied automatic control, yielding the so-called absolute problem (with famous Popov circle criteria). On other hand, large classes of nonsmooth (complementarity dynamical relay projected evolution differential variational inequalities, Moreau’s sweeping processes, maximal monotone inclusions), applications circuits, mechanics, economics, interpreted set-valued which nonlinearity is a multivalued mapping. Therefore, closed-loop system inclusion certain type, well posedness must analyzed prerequisite control. This introductory article focuses on such systems. It shown how existence uniqueness solutions these specific inclusions benefit lot from passivity monotonicity (which form incremental passivity) Available results reviewed, many illustrative examples given, some open issues highlighted.