# A characterization of strong iISS for time-varying impulsive systems

## TL;DR

This paper characterizes strong integral input-to-state stability (iISS) for time-varying impulsive systems, linking it to stability under zero input and bounded energy input, extending existing results to impulsive dynamics.

## Contribution

It extends the characterization of iISS to strong stability in time-varying impulsive systems, where previous results were limited to non-impulsive or weak stability cases.

## Key findings

- iISS characterized by 0-GUAS and UBEBS in impulsive systems
- Strong asymptotic stability is key for the characterization
- Results applicable to general time-varying impulsive systems

## Abstract

For general time-varying or switched (nonlinear) systems, converse Lyapunov theorems for stability are not available. In these cases, the integral input-to-state stability (iISS) property is not equivalent to the existence of an iISS-Lyapunov function but can still be characterized as the combination of global uniform asymptotic stability under zero input (0-GUAS) and uniformly bounded energy input-bounded state (UBEBS). For impulsive systems, asymptotic stability can be weak (when the asymptotic decay depends only on elapsed time) or strong (when such a decay depends also on the number of impulses that occurred). This paper shows that the mentioned characterization of iISS remains valid for time-varying impulsive systems, provided that stability is understood in the strong sense.