Existence of hybrid limit cycles and Zhukovskii stability in hybrid systems

This work pertains to the study of the existence of hybrid limit cycles for a class of hybrid systems. Necessary conditions, particularly, a condition using a forward invariance notion, for existence of hybrid limit cycles are first presented. Due to its usefulness in continuous-time systems with limit cycles, the notion of Zhukovskii stability, typically stated for continuous-time systems, is extended to hybrid systems given by the combination of continuous dynamics on a flow set and discrete dynamics on a jump set. A sufficient condition using incremental graphical stability is proposed for Zhukovskii stability. Examples illustrate the results and provide insight about the existence (or lack of) of hybrid limit cycles when Zhukovskii stability and incremental graphical stability hold.

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