# A Characterization of Integral ISS for Switched and Time-varying Systems

## TL;DR

This paper develops a new characterization of integral input-to-state stability (iISS) applicable to switched and time-varying systems, especially where traditional Lyapunov methods are unavailable, providing necessary and sufficient conditions.

## Contribution

It introduces a valid characterization of iISS for complex systems where converse Lyapunov theorems do not apply, clarifying the relationship with supply functions and system bounds.

## Key findings

- Provides necessary and sufficient conditions for iISS in switched and time-varying systems.
- Identifies suitable iISS gains related to supply functions.
- Highlights limitations of existing extensions of iISS characterizations.

## Abstract

Most of the existing characterizations of the integral input-to-state stability (iISS) property are not valid for time-varying or switched systems in cases where converse Lyapunov theorems for stability are not available. This note provides a characterization that is valid for switched and time-varying systems, and shows that natural extensions of some of the existing characterizations result in only sufficient but not necessary conditions. The results provided also pinpoint suitable iISS gains and relate these to supply functions and bounds on the function defining the system dynamics.