Geometric Stability Analysis for Differential Inclusions Governed by Maximally Monotone Operators

TL;DR

This paper introduces a geometric framework for analyzing the stability of differential inclusions governed by maximally monotone operators, using regularization and nonsmooth Lyapunov methods to handle complex dynamics.

Contribution

It presents a novel geometric decomposition and a regularized approximation approach that facilitates stability analysis of complex differential inclusions.

Findings

The framework enables stability analysis using nonsmooth Lyapunov methods.

A Lipschitz selection preserves geometric features of the original system.

Examples demonstrate the method's practical applicability.

Abstract

This paper develops a geometric framework for the stability analysis of differential inclusions governed by maximally monotone operators. A key structural decomposition expresses the operator as the sum of a convexified limit mapping and a normal cone. However, the resulting dynamics are often difficult to analyze directly due to the absence of Lipschitz selections and boundedness. To overcome these challenges, we introduce a regularized system based on a fixed Lipschitz approximation of the convexified mapping. From this approximation, we extract a single-valued Lipschitz selection that preserves the essential geometric features of the original system. This framework enables the application of nonsmooth Lyapunov methods and Hamiltonian-based stability criteria. Instead of approximating trajectories, we focus on analyzing a simplified system that faithfully reflects the structure of the original dynamics. Several examples are provided to illustrate the method's practicality and scope.