Third-order nilpotency, finite switchings and asymptotic stability

Yoav Sharon, Michael Margaliot*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

Abstract

We show that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields, which span a third-order nilpotent Lie algebra, is globally 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 of finding the "most unstable" trajectory for an associated control system. We use the Agrachev-Gamkrelidze second-order maximum principle to show that there always exists an optimal control that is piecewise constant with no more than four switches. This property is obtained as a special case of a reachability result by piecewise constant controls that is of independent interest. By construction, our criterion also holds for the more general case of differential inclusions.

Original languageEnglish
Title of host publicationProceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
Pages5415-5420
Number of pages6
DOIs
StatePublished - 2005
Event44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05 - Seville, Spain
Duration: 12 Dec 200515 Dec 2005

Publication series

NameProceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
Volume2005

Conference

Conference44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
Country/TerritorySpain
CitySeville
Period12/12/0515/12/05

Keywords

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

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