Abstract
We present a perturbation result for generators ofC0-semigroups which can be considered as an operator theoretic version of the Weiss-Staffans perturbation theorem for abstract linear systems. The results are illustrated by applications to the Desch-Schappacher and the Miyadera-Voigt perturbation theorems and to unbounded perturbations of the boundary conditions of a generator.
Highlights
In his classic [1] “Perturbation Theory for Linear Operators”, Kato addresses, among others, the following general problem.Given operators A and P on a Banach space X, how should one define their “sum” A + P and which properties of A are preserved under the perturbation by P?In the present paper we study this problem in the context of operator semigroups
The norm condition (12) in the previous definition combined with the denseness of D(A) ⊂ X implies that there exists an observability map satisfying Ct0 ≤ M such that
For further reference we summarize some of the previous notions in a single notation
Summary
In his classic [1] “Perturbation Theory for Linear Operators”, Kato addresses, among others, the following general problem. The bounded perturbation theorem ([2, Section III.1]), the Desch-Schappacher ([2, Section III.3.a]), and the Miyadera-Voigt theorems ([2, Section III.3.c]) give some well-known answers in these cases It seems that the Weiss-Staffans theorem on the wellposedness of perturbed linear systems (cf [3, Theorems 6.1 and 7.2] and [4, Sections 7.1 and 7.4]) is a general result in this direction. Since here T and TK are C0-semigroups with generators A and AK, respectively, this result implicitly contains a perturbation theorem for generators of C0-semigroups To apply this theorem to a perturbed operator AP as appearing in (2) one first has to construct an abstract linear system with appropriate generating operators and a suitable admissible feedback operator incorporating the unperturbed generator A and the perturbation P. (We assume that the observation and control spaces coincide This is no restriction of generality and somewhat simplifies the presentation.) On these spaces consider the following operators:.
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