On topological entropy and stability of switched linear systems

This paper studies topological entropy and stability properties of switched linear systems. First, we show that the exponential growth rates of solutions of a switched linear system are essentially upper bounded by its topological entropy. Second, we estimate the topological entropy of a switched linear system by decomposing it into a part that is generated by scalar multiples of the identity matrix and a part that has zero entropy, and proving that the overall topological entropy is upper bounded by that of the former. Third, we prove that a switched linear system is globally exponentially stable if its topological entropy remains zero under a destabilizing perturbation. Finally, the entropy estimation via decomposition and the entropy-based stability condition are applied to three classes of switched linear systems to construct novel upper bounds for topological entropy and novel sufficient conditions for global exponential stability.