Comparison theorem for infinite-dimensional linear impulsive systems

TL;DR

This paper establishes a comparison theorem for infinite-dimensional linear impulsive systems, linking their stability to simpler systems with constant dwell-times, using commutator identities in Banach spaces.

Contribution

It introduces a novel comparison theorem for infinite-dimensional impulsive systems under averaged dwell-time conditions, utilizing commutator identities for stability analysis.

Findings

The theorem reduces stability analysis to simpler systems with constant dwell-times.

An example demonstrates the theorem's application to unstable parabolic systems.

The approach applies to systems with analytic semigroup generators.

Abstract

We consider a linear impulsive system in an infinite-dimensional Banach space. It is assumed that the moments of impulsive action satisfy the averaged dwell-time condition and the linear operator on the right side of the differential equation generates an analytic semigroup in the state space. Using commutator identities, we prove a comparison theorem that reduces the problem of asymptotic stability of the original system to the study of a simpler system with constant dwell-times. An illustrative example of a linear impulsive system of parabolic type in which the continuous and discrete dynamics are both unstable is given.