Multiplicative perturbations of \(C_ 0\)-semigroups and some applications to step responses and cumulative outputs
type: Item
instance of: scholarly article
title: Multiplicative perturbations of \(C_ 0\)-semigroups and some applications to step responses and cumulative outputs
zbMATH Open document ID: 0823.47036
author: Sen-Yen Shaw
author: S. I. Piskarev
published in: Journal of Functional Analysis
publication date: 1995-11-05T00:00:00Z
review text: For a \(C_ 0\)-semigroup \(T(\cdot)\), the authors 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 \(SV(U(\cdot), t)\) of \(U(\cdot)\) on \([0, t]\) tends on 0 as \(t\to 0^ +\), then the infinitesimal operator \(A_ s\) of the pair \((T(\cdot), U(\cdot))\), as a mixed-type perturbation of the generator \(A\) of \(T(\cdot)\), generates a \(C_ 0\)-semigroup \(T_ s(\cdot)\) with \(\| T_ s(t)- T(t)\|= o(1)(t\to 0^ +)\). Furthermore, \(C_ 0\)-semigroups \(S(\cdot)\) which satisfy \(\| S(t)- T(t)]= O(t)(t\to 0^ +)\) are exactly those mixed-type perturbations caused by Lipschitz continuous step responses. Perturbations related to a cumulative output \(V(\cdot)\) are also investigated by using a multiplicative perturbation theorem of Desch and Schappacher. In particular, the authors show that bounded additive perturbations of \(A\) are exactly those mixed-type perturbations caused by Lipschitz continuous cumulative outputs. It is also shown that the generator of \(T(\cdot)\) is bounded if and only if \(SV(T(\cdot), t)\) is sufficiently small for all small \(t\).
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 764011
zbMATH Keywords: general multiplicative perturbation theorem
zbMATH Keywords: perturbation associated with a step response
zbMATH Keywords: linear dynamical system
zbMATH Keywords: infinitesimal operator
zbMATH Keywords: mixed- type perturbation
zbMATH Keywords: Lipschitz continuous cumulative outputs
MaRDI profile type: Publication
full work available at URL: jfan.1995.1034
OpenAlex ID: W1986543501
DOI: 10.1006/JFAN.1995.1034
description: scientific article; zbMATH DE number 764011