Cone-Copositive Lyapunov Functions for Complementarity Systems: Converse Result and Polynomial Approximation

TL;DR

This paper introduces cone-copositive Lyapunov functions for stability analysis of complementarity systems, proves their existence for exponentially stable systems, and provides algorithms for their numerical approximation using polynomial and rational functions.

Contribution

It establishes the existence of cone-copositive Lyapunov functions for complementarity systems and develops algorithms for their polynomial and rational approximation.

Findings

Exponentially stable complementarity systems admit differentiable cone-copositive Lyapunov functions.

Algorithms based on linear and semidefinite programming effectively approximate these Lyapunov functions.

Numerical examples demonstrate the practical applicability of the proposed methods.

Abstract

This article establishes the existence of Lyapunov functions for analyzing the stability of a class of state-constrained systems, and it describes algorithms for their numerical computation. The system model consists of a differential equation coupled with a set-valued relation which introduces discontinuities in the vector field at the boundaries of the constraint set. In particular, the set-valued relation is described by the subdifferential of the indicator function of a closed convex cone, which results in a cone-complementarity system. The question of analyzing stability of such systems is addressed by constructing cone-copositive Lyapunov functions. As a first analytical result, we show that exponentially stable complementarity systems always admit a continuously differentiable cone-copositive Lyapunov function. Putting some more structure on the system vector field, such as homogeneity, we can show that the aforementioned functions can be approximated by a rational function of cone-copositive homogeneous polynomials. This later class of functions is seen to be particularly amenable for numerical computation as we provide two types of algorithms for precisely that purpose. These algorithms consist of a hierarchy of either linear or semidefinite optimization problems for computing the desired cone-copositive Lyapunov function. Some examples are given to illustrate our approach.