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

中文翻译：

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具有集值反馈环的耗散动力系统：适定集值 Lur 动力系统


从 Jan Willems 的意义上来说，无源性和耗散性被认为是稳定性分析和反馈控制设计的强大工具。例如，具有斜率限制、静态、单值、平滑非线性的负反馈互连无源系统，称为 Lur'e 系统，已在自动控制中得到深入研究，产生了所谓的绝对稳定性问题（著名的波波夫和圆准则）。另一方面，大类非光滑系统（互补动力系统、中继系统、投影动力系统、演化和微分变分不等式、莫罗的扫掠过程和最大单调微分包含）在电路、力学和经济学中的应用是解释为集值 Lur'e 系统，其中反馈非线性是多值映射。因此，闭环系统是某种类型的微分包含体，必须对其适定性进行分析，作为稳定性和控制的前提。这篇介绍性文章重点讨论这种定值反馈系统的适定性。它显示了这些特定微分包含的解的存在性和唯一性如何从系统的被动性和反馈集值映射的最大单调性（这是增量被动性的一种形式）中受益匪浅。回顾了现有的结果，给出了许多说明性示例，并强调了一些悬而未决的问题。