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Journal Articles Linear Algebra and its Applications Year : 2021

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

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.
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Dates and versions

hal-01961003 , version 1 (19-12-2018)
hal-01961003 , version 2 (15-11-2019)
hal-01961003 , version 3 (04-11-2020)
hal-01961003 , version 4 (16-11-2021)

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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⟩
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