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February 2014 On the stability of planar randomly switched systems
Michel Benaïm, Stéphane Le Borgne, Florent Malrieu, Pierre-André Zitt
Ann. Appl. Probab. 24(1): 292-311 (February 2014). DOI: 10.1214/13-AAP924

Abstract

Consider the random process ${(X_{t})}_{t\geq0}$ solution of $\dot{X}_{t}=A_{I_{t}}X_{t}$, where ${(I_{t})}_{t\geq0}$ is a Markov process on $\{0,1\}$, and $A_{0}$ and $A_{1}$ are real Hurwitz matrices on $\mathbb{R}^{2}$. Assuming that there exists $\lambda\in(0,1)$ such that $(1-\lambda)A_{0}+\lambda A_{1}$ has a positive eigenvalue, we establish that $\|X_{t}\|$ may converge to 0 or $+\infty$ depending on the jump rate of the process $I$. An application to product of random matrices is studied. This paper can be viewed as a probabilistic counterpart of the paper [Internat. J. Control 82 (2009) 1882–1888] by Balde, Boscain and Mason.

Citation

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Michel Benaïm. Stéphane Le Borgne. Florent Malrieu. Pierre-André Zitt. "On the stability of planar randomly switched systems." Ann. Appl. Probab. 24 (1) 292 - 311, February 2014. https://doi.org/10.1214/13-AAP924

Information

Published: February 2014
First available in Project Euclid: 9 January 2014

zbMATH: 1288.93090
MathSciNet: MR3161648
Digital Object Identifier: 10.1214/13-AAP924

Subjects:
Primary: 60J75
Secondary: 34D23, 60J57, 93E15

Rights: Copyright © 2014 Institute of Mathematical Statistics

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Vol.24 • No. 1 • February 2014
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