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

3 DISCO - Dynamical Interconnected Systems in COmplex Environments
L2S - Laboratoire des signaux et systèmes, Inria Saclay - Ile de France
4 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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Cited literature [27 references]

https://hal.archives-ouvertes.fr/hal-01961003
Contributor : Guilherme Mazanti <>
Submitted on : Friday, November 15, 2019 - 6:36:19 PM
Last modification on : Wednesday, December 4, 2019 - 12:18:02 PM

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### Identifiers

• HAL Id : hal-01961003, version 2
• ARXIV : 1812.08399

### Citation

Yacine Chitour, Guilherme Mazanti, Mario Sigalotti. On the gap between deterministic and probabilistic joint spectral radii for discrete-time linear systems. 2019. ⟨hal-01961003v2⟩

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