On universal classes of Lyapunov functions for linear switched systems

TL;DR

This paper explores the concept of universality in classes of Lyapunov functions for linear switched systems, establishing conditions for universality and identifying obstructions for certain function classes.

Contribution

It proves that families approximating all convex absolutely homogeneous functions are universal and shows that certain piecewise-polynomial classes cannot be universal.

Findings

Absolutely homogeneous functions can be universal if they approximate all convex absolutely homogeneous functions.

Piecewise-polynomial functions with limited polynomials and degree cannot be universal.

The paper provides conditions and obstructions for universality of Lyapunov function classes.

Abstract

In this paper we discuss the notion of universality for classes of candidate common Lyapunov functions of linear switched systems. On the one hand, we prove that a family of absolutely homogeneous functions is universal as soon as it approximates arbitrarily well every convex absolutely homogeneous function for the $C^0$ topology of the unit sphere. On the other hand, we prove several obstructions for a class to be universal, showing, in particular, that families of piecewise-polynomial continuous functions whose construction involves at most $l$ polynomials of degree at most $m$ (for given positive integers $l,m$) cannot be universal.