Stability of Simultaneously Triangularizable Switched Systems on Hybrid Domains

In this paper, we extend the results of [8, 15, 22] which provide sufficient conditions for the global exponential stability of switched systems under arbitrary switching via the existence of a common quadratic Lyapunov function. In particular, we extend the Lie algebraic results in [15] to switched systems with hybrid non-uniform discrete and continuous domains, a direct unifying generalization of switched systems on R and Z, and extend the results in [8, 22] to a larger class of switched systems, namely those whose subsystem matrices are simultaneously triangularizable. In addition, we explore an easily checkable characterization of our required hypotheses for the theorems. Finally, conditions are provided under which there exists a stabilizing switching pattern for a collection of (not necessarily stable) linear systems that are simultaneously triangularizable and separate criteria are formed which imply the stability of the system under a given switching pattern given a priori.