## Section: New Results

### On the stability of planar randomly switched systems

Participant : Florent Malrieu.

This is a collaboration with Michel Benaïm (université de Neuchâtel), Stéphane Le Borgne (IRMAR) and Pierre–André Zitt (université de Paris–Est Marne–la–Vallée).

The paper [28] illustrates some surprising instability properties that may occur when stable ODE's are switched using Markov dependent coefficients. Consider the random process $\left({X}_{t}\right)$ solution of $d{X}_{t}/dt=A\left({I}_{t}\right){X}_{t}$ where $\left({I}_{t}\right)$ 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 the norm of ${X}_{t}$ may converge to 0 or infinity, depending on the the jump rate of the process $I$. An application to product of random matrices is studied. This work can be viewed as a probabilistic counterpart of the paper [26] by Baldé, Boscain and Mason.