Sufficient conditions for the existence of zeno behavior in a class of nonlinear hybrid systems via constant approximations

Abstract

Abstract-The existence of Zeno behavior in hybrid systems is related to a certain type of equilibria, termed Zeno equilibria, that are invariant under the discrete, but not the continuous, dynamics of a hybrid system. In analogy to the standard procedure of linearizing a vector field at an equilibrium point to determine its stability, in this paper we study the local behavior of a hybrid system near a Zeno equilibrium point by considering the value of the vector field on each domain at this point, i.e., we consider constant approximations of nonlinear hybrid systems. By means of these constant approximations, we are able to derive conditions that simultaneously imply both the existence of Zeno behavior and the local exponential stability of a Zeno equilibrium point. Moreover, since these conditions are in terms of the value of the vector field on each domain at a point, they are remarkably easy to verify.

Stability and Completion of Zeno Equilibria in Lagrangian Hybrid Systems

“…In the context of Lagrangian hybrid systems, Lamperski and Ames [21] provided Lyapunov-like conditions for Zeno behavior near isolated Zeno equilibrium points. Ames et al [2] derived sufficient conditions for Zeno behavior in a special class of hybrid systems, called first quadrant hybrid systems. This idea was then generalized in [22] to set-valued first quadrant hybrid systems, with application to nonisolated Zeno equilibria in Lagrangian hybrid systems.…”

On the existence of Zeno behavior in hybrid systems with non-isolated Zeno equilibria

“…In particular, we study first quadrant hybrid systems with dynamics governed by simple differential inclusions termed first quadrant interval hybrid systems. We find sufficient conditions for Zeno behavior in first quadrant interval hybrid systems extending those in [5].…”

“…These systems are easy to analyze, yet flexible enough to capture important characteristics of nontrivial systems. Note that first quadrant interval hybrid systems are a variant on first quadrant hybrid systems which have been studied in [5], [17].…”

“…That is, we define hybrid systems, executions, and Zeno equilibria. For a more on hybrid systems see [16] and for more on Zeno behavior see [2], [3], [4], [5], [13].…”

“…Recent work [5], provides very simple sufficient conditions for Zeno behavior depending only on the value of the vector fields at the Zeno equilibrium.…”