Multiplicative perturbations of \(C_ 0\)-semigroups and some applications to step responses and cumulative outputs
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 \(t\).
general multiplicative perturbation theorem
perturbation associated with a step response
scientific article; zbMATH DE number 764011