Locating Theorems of Differential Inclusions Governed by Maximally Monotone Operators

TL;DR

This paper advances the understanding of the long-term behavior of solutions to differential inclusions governed by maximally monotone operators by refining invariance principles and providing sharper tools for locating omega-limit sets.

Contribution

It introduces a refined and extended version of LaSalle's invariance principle tailored for differential inclusions with maximally monotone operators, improving the analysis of asymptotic behavior.

Findings

Refined invariance principle for differential inclusions

Sharper location of omega-limit sets

Use of nonsmooth Lyapunov functions

Abstract

In this paper, we are interested in studying the asymptotic behavior of the solutions of differential inclusions governed by maximally monotone operators. In the case where the LaSalle's invariance principle is inconclusive, we provide a refined version of the invariance principle theorem. This result derives from the problem of locating the $\omega$-limit set of a bounded solution of the dynamic. In addition, we propose an extension of LaSalle's invariance principle, which allows us to give a sharper location of the $\omega$-limit set. The provided results are given in terms of nonsmooth Lyapunov pair-type functions.