Well-Posedness of Nonsmooth Lurie Dynamical Systems Involving Maximal Monotone Operators

Abstract

Many physical phenomena can be modeled as a feedback connection of a linear dynamical systems combined with a nonlinear function which satisfies a sector condition. The concept of absolute stability, proposed by Lurie and Postnikov (Appl Math Mech 8(3), 1944) in the early 1940s, constitutes an important tool in the theory of control systems. Lurie dynamical systems have been studied extensively in the literature with nonlinear (but smooth) feedback functions that can be formulated as an ordinary differential equation. Many concrete applications in engineering can be modeled by a set-valued feedback law in order to take into account the nonsmooth relation between the output and the state variables. In this paper, we show the well-posedness of nonsmooth Lurie dynamical systems involving maximal monotone operators. This includes the case where the set-valued law is given by the subdifferential of a convex, proper, and lower semicontinuous function. Some existence and uniqueness results are given depending on the data of the problem and particularly the interplay between the matrix D and the set-valued map \(\mathcal {F}\). We will also give some conditions ensuring that the extended resolvent \((D+\mathcal {F})^{-1}\) is single-valued and Lipschitz continuous. The main tools used are derived from convex analysis and maximal monotone theory.