Lyapunov Stability of Differential Inclusions with Lipschitz Cusco Perturbations of Maximal Monotone Operators

Abstract

We give criteria for weak and strong invariant closed sets for differential inclusions given in R n and governed by Lipschitz Cusco perturbations of maximal monotone operators. Correspondingly, we provide different characterizations for the associated strong Lyapunov functions. The resulting conditions only depend on the data of the system.

Locating Theorems of Differential Inclusions Governed by Maximally Monotone Operators

“…Recently, differential inclusions (1.1)-(1.4) have attracted much attention since they are relevant in various areas, including for instance physics, electrical engineering, economics, biology, population dynamics and many others. For phenomena described by (1.1), we refer the reader to [1,2,3,11,10,15,17,24,32,39].…”

“…(ii) If F is Lipschitz Cusco, then thanks to a selection theorem given in [8], Adly, Hantoute and Nguyen [3] rewrited (3.13) as (1.1) and proved that (3.13) has at least one solution.…”

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

“…Time-varying systems. Still dealing with systems possessing continuous solutions, let us cite the so-called Lyapunov pairs (V (t, x), W (x)) (whose definition is very close to dissipation inequality in (D.2) with null input) [ 300,301,21,34,35,38,32,31,30,556,341,376], which are used to prove stability of equilibria, but also existence of solutions and invariance of sets in the FOSwP. The proximal normal cone and the proximal subdifferential (see Sections A.…”