Analysis of Zeno behaviors in a class of hybrid systems

This note investigates conditions for existence of Zeno behaviors (where a system undergoes an unbounded number of discrete transitions in a finite length of time) in a class of hybrid systems. Zeno behavior occurs, for example, when a controller unsuccessfully attempts to satisfy an invariance specification by switching the system among different configurations faster and faster. Two types of Zeno systems are investigated: (1) strongly Zeno systems where all runs of the system are Zeno and (2) (weakly) Zeno systems where only some runs of the system are Zeno. For constant-rate and bounded-rate hybrid systems and some nonlinear generalizations, necessary and sufficient conditions for both Zenoness and strong Zenoness are derived. The analysis is based on studying the trajectory set of a certain "equivalent" continuous-time system that is associated with the dynamic equations of the hybrid system. The relation between the possibility of existence of Zeno behaviors in a system and the problem of existence of non-Zeno safety controllers (that keep the system in a specified region of its operating space) is also examined. It is shown that in certain Zeno systems, a minimally-interventive safety controller may not exist, even if a safety controller exists, disproving a conjecture made earlier in the literature.