## The Annals of Applied Probability

### On the stability of planar randomly switched systems

#### 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

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

#### 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

#### References

• [1] Bakhtin, Y. and Hurth, T. (2012). Invariant densities for dynamical systems with random switching. Nonlinearity 25 2937–2952.
• [2] Balde, M., Boscain, U. and Mason, P. (2009). A note on stability conditions for planar switched systems. Internat. J. Control 82 1882–1888.
• [3] Chamayou, J.-F. and Letac, G. (1991). Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theoret. Probab. 4 3–36.
• [4] Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. J. R. Stat. Soc. Ser. B Stat. Methodol. 46 353–388. With discussion.
• [5] Furstenberg, H. (1963). Noncommuting random products. Trans. Amer. Math. Soc. 108 377–428.
• [6] Jacobsen, M. (2006). Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Birkhäuser, Boston, MA.
• [7] Kushner, H. J. (1984). Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory. MIT Press Series in Signal Processing, Optimization, and Control 6. MIT Press, Cambridge, MA.
• [8] Marklof, J., Tourigny, Y. and Wołowski, L. (2008). Explicit invariant measures for products of random matrices. Trans. Amer. Math. Soc. 360 3391–3427.
• [9] Yin, G. G. and Zhu, C. (2010). Hybrid Switching Diffusions: Properties and Applications. Stochastic Modelling and Applied Probability 63. Springer, New York.