In this work we present further characterizations of integral input-to-state stability (iISS) for hybrid systems. In particular, the equivalence between 0-input uniform global asymptotic stability (0-UGAS) plus uniform bounded energy bounded state (UBEBS) and iISS is examined. In order to show this equivalence, some necessary conditions for 0-UGAS and UBEBS are provided. In addition, a non-smooth Lyapunov characterization for hybrid systems is proposed and proven. With the help of the aforementioned equivalence, the combination of local iISS and practical iISS, which are defined in this work, is also shown to be equivalent to iISS under one condition on the local and practical quantifiers.

中文翻译：

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混合系统积分输入状态稳定性的进一步表征

在这项工作中，我们进一步描述了混合系统的积分输入状态稳定性（iISS）。特别是，检查了 0 输入均匀全局渐近稳定性 (0-UGAS) 加上均匀有界能量有界态 (UBEBS) 和 iISS 之间的等价性。为了证明这种等价性，提供了 0-UGAS 和 UBEBS 的一些必要条件。此外，还提出并证明了混合系统的非平滑李亚普诺夫表征。借助上述等价性，本文定义的局部 iISS 和实用 iISS 的组合也被证明在局部量词和实用量词的某一条件下与 iISS 等价。