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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
2 DISCO - Dynamical Interconnected Systems in COmplex Environments
Inria Saclay - Ile de France, L2S - Laboratoire des signaux et systèmes
3 CaGE - Control And GEometry
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
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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Submitted on : Wednesday, November 4, 2020 - 5:16:51 PM
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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, Elsevier, 2021, 613, pp.24-45. ⟨10.1016/j.laa.2020.12.013⟩. ⟨hal-01961003v3⟩

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