On the marginal instability of linear switched systems

Stability properties for continuous-time linear switched systems are determined by the Lyapunov exponent associated with the system, which is the analogous of the joint spectral radius for the discrete-time case. This paper is concerned with the characterizations of stability properties when the Lyapunov exponent is zero. In this case it is well known that the system can be stable as well as unstable, even if it is never asymptotically stable nor it admits a trajectory blowing up exponentially fast. Our main result asserts that a switched system whose Lyapunov exponent is zero may be unstable only if a certain resonance condition is satisfied.