On the stability of planar randomly switched systems

Consider the random process (Xt)t.0 solution of .Xt=AItXt, where (It)t.0 is a Markov process on {0,1}, and A0 and A1 are real Hurwitz matrices on R2. Assuming that there exists ..(0,1) such that (1..)A0+.A1 has a positive eigenvalue, we establish that .Xt. may converge to 0 or +. 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.