Multiplicative Perturbations of C0-Semigroups and Some Applications to Step Responses and Cumulative Outputs
✍ Scribed by S. Piskarev; S.Y. Shaw
- Publisher
- Elsevier Science
- Year
- 1995
- Tongue
- English
- Weight
- 822 KB
- Volume
- 128
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
For a (C_{14})-semigroup (T(\cdot)), 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(\cdot)) of a linear dynamical system. If the semivariation (S V(U(\cdot), t)) of (U(\cdot)) on ([0, t]) tends to 0 as (t \rightarrow 0^{*}), then the infinitesimal operator (A_{8}) of the pair ((T(\cdot), U(\cdot))), as a mixed-type perturbation of the generator (A) of (T(\cdot)). generates a (C_{11})-semigroup (T_{s}(\cdot)) with (\left|T_{s}(t)-T(t)\right|=o(1)\left(t \rightarrow 0^{+}\right)). Furthermore. (C_{(1)})-semigroups (S(\cdot)) which satisfy (|S(t)-T(t)|=O(t)\left(t \rightarrow 0^{+}\right))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 mixedtype perturbations caused by Lipschitz continuous cummulative outputs. It is also shown that the generator of (T(\cdot)) is bounded if and only if (S V(T(\cdot), t)) is sufficiently small for all small (t). ' 1995 Academic Press. Inc.