A Lie-algebraic condition for stability of switched nonlinear systems

Michael Margaliot*, Daniel Liberzon

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review


We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [1]. To prove the result, we consider an optimal control problem which consists in finding the "most unstable" trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.

Original languageEnglish
Article numberFrB01.6
Pages (from-to)4619-4624
Number of pages6
JournalProceedings of the IEEE Conference on Decision and Control
StatePublished - 2004
Event2004 43rd IEEE Conference on Decision and Control (CDC) - Nassau, Bahamas
Duration: 14 Dec 200417 Dec 2004


  • Differential inclusion
  • Global asymptotic stability
  • Lie bracket
  • Maximum principle
  • Optimal control
  • Switched nonlinear system


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