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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 ${ℝ}^2$. 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.