Lyapunov theory for zeno stability

Andrew Lamperski, Aaron D. Ames

Research output: Contribution to journalArticlepeer-review

60 Scopus citations


Zeno behavior is a dynamic phenomenon unique to hybrid systems in which an infinite number of discrete transitions occurs in a finite amount of time. This behavior commonly arises in mechanical systems undergoing impacts and optimal control problems, but its characterization for general hybrid systems is not completely understood. The goal of this paper is to develop a stability theory for Zeno hybrid systems that parallels classical Lyapunov theory; that is, we present Lyapunov-like sufficient conditions for Zeno behavior obtained by mapping solutions of complex hybrid systems to solutions of simpler Zeno hybrid systems defined on the first quadrant of the plane. These conditions are applied to Lagrangian hybrid systems, which model mechanical systems undergoing impacts, yielding simple sufficient conditions for Zeno behavior. Finally, the results are applied to robotic bipedal walking.

Original languageEnglish (US)
Article number6237496
Pages (from-to)100-112
Number of pages13
JournalIEEE Transactions on Automatic Control
Issue number1
StatePublished - 2013


  • Hybrid systems
  • Lyapunov method
  • mechanical systems
  • stability


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