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 (Xt) solution of dXt/dt=A(It)Xt where (It) is a Markov process on {0,1} and A0 and A1 are real Hurwitz matrices on 2. Assuming that there exists λ(0,1) such that (1-λ)A0+λA1 has a positive eigenvalue, we establish that the norm of Xt 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.