Equivalence between Differential Inclusions Involving Prox-regular sets and maximal monotone operators

TL;DR

This paper establishes an equivalence between differential inclusions involving prox-regular sets and maximal monotone operators, enabling the use of monotone operator theory for stability and existence analysis of nonsmooth dynamical systems.

Contribution

It introduces a novel equivalence between two classes of differential inclusions, allowing the application of maximal monotone operator theory to prox-regular set dynamics.

Findings

Existence and stability of solutions are proven using the equivalence.

Solutions exhibit continuity and differentiability properties.

The approach is applied to a Luenberger-like observer with exponential convergence.

Abstract

In this paper, we study the existence and the stability in the sense of Lyapunov of solutions for\ differential inclusions governed by the normal cone to a prox-regular set and subject to a Lipschitzian perturbation. We prove that such, apparently, more general nonsmooth dynamics can be indeed remodelled into the classical theory of differential inclusions involving maximal monotone operators. This result is new in the literature and permits us to make use of the rich and abundant achievements in this class of monotone operators to derive the desired existence result and stability analysis, as well as the continuity and differentiability properties of the solutions. This going back and forth between these two models of differential inclusions is made possible thanks to a viability result for maximal monotone operators. As an application, we study a Luenberger-like observer, which is shown to converge exponentially to the actual state when the initial value of the state's estimation remains in a neighborhood of the initial value of the original system.