# An entropy-based bound for the computational complexity of a switched system

## TL;DR

This paper introduces an entropy-based bound for the computational complexity of switched systems, linking the joint spectral radius to system entropy and p-radius, and proposes a reduction method for low-rank matrices.

## Contribution

It provides a new entropy-based guarantee for the sum of squares method's upper bound on the joint spectral radius of constrained switched systems.

## Key findings

- The entropy and p-radius influence the accuracy of stability bounds.
- A reduction method simplifies the computation of the joint spectral radius for low-rank matrices.
- The approach enhances understanding of stability analysis in hybrid systems.

## Abstract

The joint spectral radius (JSR) of a set of matrices characterizes the maximal asymptotic growth rate of an infinite product of matrices of the set. This quantity appears in a number of applications including the stability of switched and hybrid systems. A popular method used for the stability analysis of these systems searches for a Lyapunov function with convex optimization tools. We analyse the accuracy of this method for constrained switched systems, a class of systems that has attracted increasing attention recently. We provide a new guarantee for the upper bound provided by the sum of squares implementation of the method. This guarantee relies on the p-radius of the system and the entropy of the language of allowed switching sequences. We end this paper with a method to reduce the computation of the JSR of low rank matrices to the computation of the constrained JSR of matrices of small dimension.