Input-to-state stability of nonlinear impulsive systems

TL;DR

This paper establishes conditions under which nonlinear impulsive systems are input-to-state stable (ISS), introduces new small-gain theorems for interconnected systems, and extends results to systems in Banach spaces.

Contribution

It provides novel ISS stability criteria for impulsive systems, including generalized dwell-time conditions and small-gain theorems for interconnected systems in Banach spaces.

Findings

ISS stability under fixed dwell-time conditions

Uniform ISS with exponential ISS Lyapunov functions

Construction of ISS Lyapunov functions for interconnected systems

Abstract

We prove that impulsive systems, which possess an ISS Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the exponential one, we provide a stronger result, which guarantees uniform ISS of the whole system over sequences satisfying the generalized average dwell-time condition. Then we prove two small-gain theorems that provide a construction of an ISS Lyapunov function for an interconnection of impulsive systems, if the ISS-Lyapunov functions for subsystems are known. The construction of local ISS Lyapunov functions via linearization method is provided. Relations between small-gain and dwell-time conditions as well as between different types of dwell-time conditions are also investigated. Although our results are novel already in the context of finite-dimensional systems, we prove them for systems based on differential equations in Banach spaces that makes obtained results considerably more general.