Quadratic Lyapunov theory for dynamic linear switched systems.

Abstract

In this work, a special class of time-varying linear systems in the arbitrary time scale setting is examined by considering the qualitative properties of their solutions. Building on the work of J. DaCunha and A. Ramos, Lyapunov's Second (or Direct) Method is utilized to determine when the solutions to a given switched system are asymptotically stable.

Three major classes of switched systems are analyzed which exhibit a convenient containment scheme so as to recover early results as special cases of later, more general results. The stability of switched systems under both arbitrary and particular switching is considered, in addition to design parameters of the time scale domain which also imply stability. A new approach to Lyapunov theory for time scales is then considered for switched systems which do not necessarily belong to any class of systems, contrasting and generalizing previous results. Finally, extensions of the contained theory are considered and a nontrivial generalization of a major result by D. Liberzon and A. Agrachev is investigated and conjectured.