# Uniform Input-to-State Stability for Switched and Time-Varying Impulsive Systems

## TL;DR

This paper introduces a Lyapunov-function-based approach to establish uniform input-to-state stability for time-varying impulsive systems, generalizing existing stability criteria across various system classes.

## Contribution

It develops a unified method for ISS analysis of impulsive systems with inputs, extending previous results to broader classes of systems and stability conditions.

## Key findings

- Provides novel sufficient conditions for ISS of impulsive systems.
- Generalizes stability results to time-varying and switched impulsive systems.
- Introduces a reduction technique to scalar systems for stability assessment.

## Abstract

We provide a Lyapunov-function-based method for establishing different types of uniform input-to-state stability (ISS) for time-varying impulsive systems. The method generalizes to impulsive systems with inputs the well-established philosophy of assessing the stability of a system by reducing the problem to that of the stability of a scalar system given by the evolution of the Lyapunov function on the system trajectories. This reduction is performed in such a way so that the resulting scalar system has no inputs. Novel sufficient conditions for ISS are provided, which generalize existing results for time-invariant and time-varying, switched and nonswitched, impulsive and nonimpulsive systems in several directions.