# On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems

## TL;DR

This paper explores the relationship between deterministic and probabilistic measures of asymptotic behavior in discrete-time linear switched systems, aiming to characterize when these measures are equal.

## Contribution

It provides a characterization of the conditions under which the deterministic joint spectral radius equals certain probabilistic spectral radii.

## Key findings

- Identifies conditions for equality between deterministic and probabilistic spectral radii.
- Analyzes the sets of matrices where such equalities hold.
- Provides insights into the structure of systems with matching spectral radii.

## Abstract

Given a discrete-time linear switched system $\Sigma(\mathcal A)$ associated with a finite set $\mathcal A$ of matrices, we consider the measures of its asymptotic behavior given by, on the one hand, its deterministic joint spectral radius $\rho_{\mathrm d}(\mathcal A)$ and, on the other hand, its probabilistic joint spectral radii $\rho_{\mathrm p}(\nu,P,\mathcal A)$ for Markov random switching signals with transition matrix $P$ and a corresponding invariant probability $\nu$. Note that $\rho_{\mathrm d}(\mathcal A)$ is larger than or equal to $\rho_{\mathrm p}(\nu,P,\mathcal A)$ for every pair $(\nu, P)$. In this paper, we investigate the cases of equality of $\rho_{\mathrm d}(\mathcal A)$ with either a single $\rho_{\mathrm p}(\nu,P,\mathcal A)$ or with the supremum of $\rho_{\mathrm p}(\nu,P,\mathcal A)$ over $(\nu,P)$ and we aim at characterizing the sets $\mathcal A$ for which such equalities may occur.