Further characterizations of integral input-to-state stability for hybrid systems

TL;DR

This paper explores advanced characterizations of integral input-to-state stability (iISS) in hybrid systems, establishing key equivalences with other stability notions and proposing a non-smooth Lyapunov approach.

Contribution

It introduces new equivalences between iISS, 0-UGAS, and UBEBS, along with a non-smooth Lyapunov characterization for hybrid systems.

Findings

Proves the equivalence between 0-UGAS plus UBEBS and iISS.

Provides necessary conditions for 0-UGAS and UBEBS.

Proposes a non-smooth Lyapunov characterization for hybrid systems.

Abstract

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.