# Integral Input-to-State Stability of Nonlinear Time-Delay Systems with Delay-Dependent Impulse Effects

## TL;DR

This paper establishes new criteria for integral input-to-state stability of nonlinear time-delay impulsive systems, considering various stability and destabilizing scenarios, using Lyapunov-Krasovskii functionals, with applications to bilinear systems.

## Contribution

It introduces novel iISS criteria for nonlinear impulsive systems with delays, covering cases with destabilizing impulses, unstable dynamics, and stabilizing impulses, expanding existing stability theory.

## Key findings

- Derived iISS criteria for systems with destabilizing impulses
- Proved that frequent impulses can stabilize systems with unstable dynamics
- Demonstrated the criteria on bilinear systems with simulations

## Abstract

This paper studies integral input-to-state stability (iISS) of nonlinear impulsive systems with time-delay in both the continuous dynamics and the impulses. Several iISS results are established by using the method of Lyapunov-Krasovskii functionals. For impulsive systems with iISS continuous dynamics and destabilizing impulses, we derive two iISS criteria that guarantee the uniform iISS of the whole system provided that the time period between two successive impulse moments is appropriately bounded from below. Then we provide an iISS result for systems with unstable continuous dynamics and stabilizing impulses. For this scenario, it is shown that the iISS properties are guaranteed if the impulses occur frequently enough. For impulsive systems with stabilizing impulses and stable continuous dynamics for zero input, we obtain an iISS result which shows that the entire system is uniformly iISS over arbitrary impulse time sequences. As applications, iISS properties of a class of bilinear systems are studied in details with simulations to demonstrate the presented results.