A New Measure of Robust Stablity for Linear Ordinary Impulsive Differential Equations

Abstract

A new measure of robust stability for linear ordinary impulsive differential equations with periodic structure is introduced, based on the impulse extension concept. This new stability measure reflects the sensitivity of the model to uncertainty in what we see as the fundamental hypothesis of impulsive models: that the impulse effect occurs quicky enough that its duration can be entirely neglected. The measure, that we call the time-scale tolerance, $\mathcal{E}_{t}$, has the property that, if the vector of durations of impulse effect, a, satisfies $\vert \vert a\vert \vert < \mathcal{E}_{t}$, then both the impulsive model and a family of continuous impulse extension equations (a specific functional differential equation) to which it is related, will all be asymptotically stable. We review linear impulse extension equations, state theorems that describe the convergence of their solutions to the associated impulsive solutions, and introduce all the machinery necessary in the development of the time-scale tolerance, stating theoretical results on its existence and how it can be computed in practice. We conclude with two illustrative examples and a discussion of the limitations of the techniques presented, as well as elaborate on the ways they can be improved.