Stability of Continuous Time Linear Systems with Bounded Switching Intervals

We address the stability problem for linear switching systems with mode-dependent restrictions on the switching intervals. Their lengths can be bounded both from below (the guaranteed dwell-time) and from above (the maximal time of a single mode). The presence of upper bounds makes this problem quite different from the classical case. For instance, a stable system can now consist of unstable matrices, it may not possess Lyapunov functions, etc. We introduce the concept of a convex Lyapunov multifunction with discrete monotonicity. Its existence, as well as the existence of invariant norms, are proved. We also establish a modified Berger-Wang formula over periodizable switching laws. Those results provide a method of computation of the Lyapunov exponent with an arbitrary precision. Its efficiency is shown in numerical examples.