A Note on Marginal Stability of Switched Systems

In the stability analysis of switched dynamical systems, much effort has been devoting to the asymptotic stability and exponential stability, but little focused on marginal stability. In this work, we present criteria for marginal stability and marginal instability of switched systems. We prove that the stability is equivalent to the existence of a common weak Lyapunov function which is generally not continuous. A sufficient condition is also provided for marginal stability in terms of matrix equalities. Finally, we reveal the subtle properties for marginal stability and marginal instability through the largest invariant set contained in a polyhedron.