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In this paper, we investigate conditions for existence of Zeno behaviors in hybrid systems. These are behaviors that occur in a hybrid system when the system undergoes an unbounded number of discrete transitions in a finite and bounded length of time. 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 will be 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. We derive necessary and sufficient conditions for both strong Zenoness and Zenoness, under certain assumptions. Our 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. We also study the relation between the possibility of existence of Zeno behaviors in a system and the problem of existence of non-Zeno safety controllers that prevent the system from entering a suitably defines illegal region of its operating space. In particular, we show that if the system is Zeno (but not strongly Zeno), then a minimally-interventive safety controller may not exist, even if a safety controller does exist, disproving a conjecture made earlier in the literature. We also argue that any attempt of "regularizing" Zeno systems by forcing delays between successive configuration switches will not be fruitful.

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