Non-Standard Zeno-Free Simulation Semantics for Hybrid Dynamical Systems

Geometric-Zeno behavior is one of the most challenging problems in the analysis and simulation of hybrid systems. Geometric-Zeno solutions involve an accumulation of an infinite number of discrete events occurring in a finite amount of time. In models that exhibit geometric-Zeno, discrete events occur at an increasingly smaller distance in time, converging to a limit point according to a geometric series. In practice, simulating models of hybrid systems exhibiting geometric-Zeno is highly challenging, in that the simulation either halts or produces faulty results, as the time interval lengths are decreasing to arbitrary small positive numbers. Although many simulation tools for hybrid systems have been developed in the past years, none of them have a Zeno-free semantic model. All of them produce faulty results when simulating geometric-Zeno models. In this paper, we propose a non-standard Zeno-free mathematical formalism for detecting and eliminating geometric-Zeno during simulation. We derive sufficient conditions for the existence of geometric-Zeno behavior based on the existence of a non-standard contraction map in a complete metric space. We also provide methods for carrying solutions from pre-Zeno to post-Zeno. We illustrate the concepts with examples throughout the paper.