Stability of linear systems with bounded switching intervals

Abstract

We address the stability problem for linear switching systems with mode-dependent restrictions on the switching intervals. Their lengths can be bounded as from below (the guaranteed dwell-time) as from above. The upper bounds make this problem quite different from the classical case: a stable system can consist of unstable matrices, it may not possess Lyapunov functions, etc. We introduce the concept of Lyapunov multifunction with discrete monotonicity, which gives upper bounds for the Lyapunov exponent. Its existence as well as the existence of invariant norms are proved. Tight lower bounds are obtained in terms of a modified Berger-Wang formula over periodizable switching laws. Based on those results we develop a method of computation of the Lyapunov exponent with an arbitrary precision and analyse its efficiency in numerical results. The case when some of upper bounds can be cancelled is analysed.