Lyapunov Theory for Zeno Stability
Andrew Lamperski and Aaron D. Ames
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. The importance of understanding this behavior lies
in the multitude of hybrid models in which it appears, e.g., mechanical systems undergoing impacts.
Without detecting and characterizing this phenomena, it is not possible to fully understand the rich
behavior that these systems can display. 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. These conditions will be applied to Lagrangian hybrid systems, which
model mechanical systems undergoing impacts, yielding simple sufficient conditions for Zeno behavior.
Finally, the results will be applied to robotic bipedal walking.
Zeno behavior occurs in hybrid systems when an execution (or solution) undergoes infinitely
many discrete transitions in a finite amount of time. Prior to the introduction of hybrid systems,
and in contrast with the view of Zeno behavior as a modeling pathology, Zeno phenomena
have long been studied in fields such as nonsmooth mechanics and optimal control. While
domain-specific understanding of Zeno behavior is, in some cases, quite sophisticated, many