Stability radius and dichotomy radius of infinite-dimensional linear systems under unbounded perturbations via Yosida distance

Abstract

In this paper, by using the concept of Yosida distance between two closed linear operators, we study the stability radius $r\left(A\right)$ of linear systems $u^{\prime }\left(t\right)=Au\left(t\right)$, where $A$ is the generator of an analytic semigroup, under unbounded perturbations in this class of generators. We show that $r\left(A\right)=1/{sup}_{s\in R}{\Vert R(is,A)\Vert }_{L\left(X\right)}$, so extending a classic result by Henrichsen and Pritchard to the infinite-dimensional case. A formula of the dichotomy radius is also established. Finally we give an estimate of the stability radius of general $C_0$-semigroups. Two examples from a parabolic equation are given.