A Characterization of Lyapunov Inequalities for Stability of Switched Systems

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

Research output: Contribution to journalArticlepeer-review

32 Scopus citations


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.

Original languageEnglish (US)
Article number7858590
Pages (from-to)3062-3067
Number of pages6
JournalIEEE Transactions on Automatic Control
Issue number6
StatePublished - Jun 2017

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


  • Linear matrix inequalities
  • lyapunov methods
  • set theory
  • stability
  • switching systems (control)


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