MULTIPLICATIVE PERTURBATIONS OF C-0-SEMIGROUPS AND SOME APPLICATIONS TO STEP RESPONSES AND CUMULATIVE OUTPUTS

For a C-o-semigroup T(.), we prove a general multiplicative perturbati on theorem which subsumes many known multiplicative and additive pertu rbation theorems, and provides a general framework for systematic stud y 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 A(s) of the pa ir (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)paral lel to=0(1)(t-->0(+)). Furthermore, C-o-semigroups S(.) which satisfy parallel to S(t)-T(t)parallel to=O(t)(t-->0(+)) are exactly those mixe d-type perturbations caused by Lipschitz continuous step responses. Pe rturbations related to a cumulative output V(.) are also investigated by using a multiplicative perturbation theorem of Desch and Schappache r. 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 b ounded if and only if SV(T(.), t) is sufficiently small for all small t. (C) 1995 Academic Press, Inc.