Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems

TL;DR

This paper demonstrates that for any fixed degree, there are stable switched linear systems that do not admit simple polynomial, polytopic, or piecewise quadratic Lyapunov functions, indicating inherent complexity in stability certification.

Contribution

It establishes lower bounds on the complexity of Lyapunov functions needed for stability, linking these bounds to classical problems like the finiteness conjecture and joint spectral radius.

Findings

Existence of stable systems without simple Lyapunov functions of bounded complexity.

Lower bounds imply no finite-size linear or semidefinite programs can universally certify stability.

Connections to classical problems in spectral radius and algebraicity of Lyapunov functions.

Abstract

We show that for any positive integer $d$, there are families of switched linear systems---in fixed dimension and defined by two matrices only---that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree $\leq d$, or (ii) a polytopic Lyapunov function with $\leq d$ facets, or (iii) a piecewise quadratic Lyapunov function with $\leq d$ pieces. This implies that there cannot be an upper bound on the size of the linear and semidefinite programs that search for such stability certificates. Several constructive and non-constructive arguments are presented which connect our problem to known (and rather classical) results in the literature regarding the finiteness conjecture, undecidability, and non-algebraicity of the joint spectral radius. In particular, we show that existence of an extremal piecewise algebraic Lyapunov function implies the finiteness property of the optimal product, generalizing a result of Lagarias and Wang. As a corollary, we prove that the finiteness property holds for sets of matrices with an extremal Lyapunov function belonging to some of the most popular function classes in controls.