Local nonsmooth Lyapunov pairs for first-order evolution differential inclusions

TL;DR

This paper develops a theory for Lyapunov stability of first-order differential inclusions with maximally monotone operators having domains with nonempty interior, providing explicit criteria and applications to viability and convexity.

Contribution

It introduces new Lyapunov pair criteria for differential inclusions with maximally monotone operators in Hilbert spaces, extending previous stability results.

Findings

Explicit Lyapunov pair criteria derived

Applications to viability of closed sets demonstrated

Conditions involving continuity and convexity analyzed

Abstract

The general theory of Lyapunov's stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in a previous work. This new contribution focuses on the natural case when the maximally monotone operator governing the given inclusion has a domain with nonempty interior. This setting permits to have nonincreasing Lyapunov functions on the whole trajectory of the solution to the given differential inclusion. It also allows some more explicit criteria for Lyapunov's pairs. Some consequences to the viability of closed sets are given, as well as some useful cases relying on the continuity or/and convexity of the involved functions. Our analysis makes use of standard tools from convex and variational analysis.