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

Yacine Chitour (1) , Guilherme Mazanti (1, 2) , Mario Sigalotti (3, 4)

1 L2S - Laboratoire des signaux et systèmes
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
3 CaGE - Control And GEometry
4 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions

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.

Mots clés: Linear switched systems; Discrete time; Joint spectral radius; Markov process

Domaines: Optimisation et contrôle [math.OC]; Systèmes dynamiques [math.DS]; Probabilités [math.PR]

Yacine Chitour, Guilherme Mazanti, Mario Sigalotti. On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems. Linear Algebra and its Applications, 2021, 613, pp.24-45. ⟨10.1016/j.laa.2020.12.013⟩. ⟨hal-01961003v4⟩