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

3 CaGE - Control And GEometry
Inria de Paris, LJLL - 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.
Type de document :
Pré-publication, Document de travail
2018
Domaine :

https://hal.archives-ouvertes.fr/hal-01961003
Contributeur : Mario Sigalotti <>
Soumis le : mercredi 19 décembre 2018 - 16:20:33
Dernière modification le : vendredi 4 janvier 2019 - 17:33:39

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gap-preprint.pdf
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### Identifiants

• HAL Id : hal-01961003, version 1
• 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. 2018. 〈hal-01961003〉

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