A Characterization of Lyapunov Inequalities for Stability of Switched Systems

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Raphaël M. Jungers, Amir Ali Ahmadi, Pablo A. Parrilo, Mardavij Roozbehani

Published: 2017, Last Modified: 07 May 2026IEEE Trans. Autom. Control. 2017EveryoneRevisionsBibTeXCC BY-SA 4.0
Abstract: We study stability criteria for discrete-time switched systems and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past 15 years. We prove in this note that a family of language-theoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies stability of a switched system. Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.