ON THE STABILITY OF PLANAR RANDOMLY SWITCHED SYSTEMS

Consider the random process (X-t)(t >= 0) solution of (X) over dot(t) = A(It) X-t, where (I-t)(t >= 0) 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 is an element of (0, 1) such that (1 - lambda)A(0) + lambda A(1) has a positive eigenvalue, we establish that parallel to X-t parallel to may converge to 0 or +infinity 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.