Beyond Zeno: Get on with It!

In this paper we propose a technique to extend the simulation of a Zeno hybrid system beyond its Zeno time point. A Zeno hybrid system model is a hybrid system with an execution that takes an infinite number of discrete transitions during a finite time interval. We argue that the presence of Zeno behavior indicates that the hybrid system model is incomplete by considering some classical Zeno models that incompletely describe the dynamics of the system being modeled. This motivates the systematic development of a method for completing hybrid system models through the introduction of new post-Zeno states, where the completed hybrid system transitions to these post-Zeno states at the Zeno time point. In practice, simulating a Zeno hybrid system is challenging in that simulation effectively halts near the Zeno time point. Moreover, due to unavoidable numerical errors, it is not practical to exactly simulate a Zeno hybrid system. Therefore, we propose a method for constructing approximations of Zeno models by leveraging the completed hybrid system model. Using these approximation, we can simulate a Zeno hybrid system model beyond its Zeno point and reveal the complete dynamics of the system being modeled.