Third-order nilpotency, nice reachability and asymptotic stability

Yoav Sharon, Michael Margaliot

Research output: Contribution to journalArticlepeer-review


We consider an affine control system whose vector fields span a third-order nilpotent Lie algebra. We show that the reachable set at time T using measurable controls is equivalent to the reachable set at time T using piecewise-constant controls with no more than four switches. The bound on the number of switches is uniform over any final time T. As a corollary, we derive a new sufficient condition for stability of nonlinear switched systems under arbitrary switching. This provides a partial solution to an open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, in: V. Blondel, A. Megretski (Eds.), Unsolved Problems in Mathematical Systems and Control Theory, Princeton Univ. Press, 2004, pp. 203-207].

Original languageEnglish
Pages (from-to)136-150
Number of pages15
JournalJournal of Differential Equations
Issue number1
StatePublished - 1 Feb 2007


  • Arbitrary switching
  • Bang-bang control
  • Chen series
  • First- and second-order maximum principle
  • Global asymptotic stability
  • Lie bracket
  • Nonlinear switched systems
  • Optimal control
  • P. Hall basis
  • Singular control
  • Sussmann's product expansion


Dive into the research topics of 'Third-order nilpotency, nice reachability and asymptotic stability'. Together they form a unique fingerprint.

Cite this