## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 24, Number 1 (2014), 292-311.

### On the stability of planar randomly switched systems

Michel Benaïm, Stéphane Le Borgne, Florent Malrieu, and Pierre-André Zitt

#### 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 24, Number 1 (2014), 292-311.

**Dates**

First available in Project Euclid: 9 January 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1389278726

**Digital Object Identifier**

doi:10.1214/13-AAP924

**Mathematical Reviews number (MathSciNet)**

MR3161648

**Zentralblatt MATH identifier**

1288.93090

**Subjects**

Primary: 60J75: Jump processes

Secondary: 60J57: Multiplicative functionals 93E15: Stochastic stability 34D23: Global stability

**Keywords**

Ergodicity linear differential equations Lyapunov exponent planar switched systems piecewise deterministic Markov process product of random matrices

#### Citation

Benaïm, Michel; Le Borgne, Stéphane; Malrieu, Florent; Zitt, Pierre-André. On the stability of planar randomly switched systems. Ann. Appl. Probab. 24 (2014), no. 1, 292--311. doi:10.1214/13-AAP924. https://projecteuclid.org/euclid.aoap/1389278726