On the stability of planar randomly switched systems

TL;DR

This paper investigates the stability of planar randomly switched systems driven by a Markov process, revealing conditions under which the system's state norm converges to zero or diverges, with implications for random matrix products.

Contribution

It provides a probabilistic analysis of stability for planar switched systems with Markov switching, extending previous deterministic stability criteria.

Findings

The norm of the system state can tend to zero or infinity depending on the switching rate.

A key condition involves the convex combination of matrices having a positive eigenvalue.

Application to the stability of products of random matrices is demonstrated.

Abstract

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 R2. Assuming that there exists lambda in (0, 1) such that (1 - \lambda)A0 + \lambdaA1 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 paper can be viewed as a probabilistic counterpart of the paper "A note on stability conditions for planar switched systems" by Balde, Boscain and Mason.