Robustness of linear systems

Abstract

In this paper we consider linear systems which are stable and examine the robustness of this property. The perturbations are assumed to have unbounded structure and we determine the smallest perturbation which destroys stability. We begin with an abstract analysis of the problem, but in later sections specialize to three types of systems: those whose mild solutions are given in terms of 1. (a) strongly continuous semigroups, 2. (b) resolvent operators-integrodifferential systems, 3. (c) evolution operators-time varying systems. For B a Banach space, L 2( t 0, T; B) denotes the space of all square integrable functions defined on [ t 0, T] with values in B. For B 1, B 2 Banach spaces, [ t 0, T] denotes the space of strongly measurable functions f(·):[ t 0, T]→ L( B 1, B 2) with ess sup ∥ f(·)∥⩽∞. B − 1( t 0, ∞; L( B 1, B 2)) denotes the space of strongly measurable functions f(·): [ t 0, ∞) → L( B 1, B 2) with ∥ f( t)∥ < e − βt k( t) for some β > 0 and k(·) ϵ L 1( t 0, ∞).