Analyzing Feedback Interconnections of Maximal Monotone Systems Using Dissipativity Approach

Abstract

We consider interconnections of two dynamical systems in feedback configuration. The dynamics of the individual systems are modeled by a differential inclusion, and the corresponding set-valued mapping is (anti-) maximal monotone with respect to the state of the system for each fixed value of the external signal that defines the interconnection. We provide conditions on these mappings under which the dynamics of the resulting interconnected system are (anti-) maximal monotone. An interpretation of our main result is provided: firstly, by considering dynamical systems defined by the gradient of a saddle function, and secondly, by considering an interconnection of incrementally passive systems. In the same spirit, when we associate more structure to the individual systems by considering linear complementarity systems, we allow for more flexibility in describing the interconnections and derive more specific sufficient conditions in terms of system matrices that result in the overall system being described by (anti-) maximal monotone operator.