Multiplicative Perturbations of C0-Semigroups and Some Applications to Step Responses and Cumulative Outputs

Abstract

For a C0-semigroup T(·), we prove a general multiplicative perturbation theorem which subsumes many known multiplicative and additive perturbation theorems, and provides a general framework for systematic study of the perturbation associated with a step response U(·) of a linear dynamical system. If the semivariation SV(U(·), t) of U(·) on [0, t] tends to 0 as t→0+, then the infinitesimal operator As of the pair (T(·), U(·)), as a mixed-type perturbation of the generator A of T(·), generates a C-o-semigroup T-s(.) with parallel to T-s(t)-T(t)parallel to=0(1)(t→0(+)). Furthermore, C0-semigroups S(.) which satisfy ||S(t)-T(t)||=O(t)(t→0(+)) are exactly those mixed-type perturbations caused by Lipschitz continuous step responses. Perturbations related to a cumulative output V(·) are also investigated by using a multiplicative perturbation theorem of Desch and Schappacher. In particular, we show that bounded additive perturbations of A are exactly those mixed-type perturbations caused by Lipschitz continuous cummulative outputs. It is also shown that the generator of T(·) is bounded if and only if SV(T(·), t) is sufficiently small for all small t.