Robustness of strong stability of semigroups
type: Item
instance of: scholarly article
title: Robustness of strong stability of semigroups
zbMATH Open document ID: 1320.47045
author: Lassi Paunonen
published in: Journal of Differential Equations
publication date: 2014-10-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.5674
review text: Robustness of strong stability of \(C_0\)-semigroups on Hilbert spaces under additive finite-rank perturbation is studied. Throughout this article, it is assumed that \(A\) generates a strongly stable \(C_0\)-semigroup \((T(t))_{t\geq 0}\) on a Hilbert space \(X\) with \(\sigma(A)\cap i\mathbb R=\{i\omega_k\}_{k=1,\dots,N}\) and the norm of its resolvent operator is polynomially bounded near these points. The first main result of this paper shows that, if \(B\in L(\mathbb C^p,X)\) and \(C\in L(X,\mathbb C^p)\) with \({\mathcal R}(B)\subset {\mathcal R}((i\omega_k-A)^\beta)\) and \({\mathcal R}(C^*)\subset {\mathcal R}((-i\omega_k-A^*)^\beta)\) for all \(k+1,\dots,N\) and some \(\beta+\gamma\) sufficiently large, and the graph norms \(\|B\|+\|(i\omega_k-A)^{-\beta}B\|\) and \(\|C\|+\|(-i\omega_k-A^*)^{-\gamma}C^*\|\) are sufficiently small, then the semigroup generated by \(A+BC\) is strongly stable. The second main result of this paper presents a similar perturbation result for polynomially stable \(C_0\)-semigroups. This improves the perturbation results concerning polynomially stability obtained by the author in [J. Funct. Anal. 263, No. 9, 2555--2583 (2012; Zbl 1263.47049); Syst. Control Lett. 62, No. 4, 331--337 (2013; Zbl 1261.93059)].
reviewed by: Birgit Jacob
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47A55
Mathematics Subject Classification ID: 93D09
zbMATH DE Number: 6359818
zbMATH Keywords: strongly continuous semigroups
zbMATH Keywords: strong stability
zbMATH Keywords: polynomial stability
zbMATH Keywords: robustness
MaRDI profile type: Publication
OpenAlex ID: W2049894267
arXiv ID: 1306.5674
cites work: Tauberian Theorems and Stability of One-Parameter Semigroups
cites work: Q2716079
cites work: Polynomial stability of operator semigroups
cites work: Non-uniform stability for bounded semi-groups on Banach spaces
cites work: Optimal polynomial decay of functions and operator semigroups
cites work: Q4678956
cites work: One-Parameter Semigroups for Linear Evolution Equations
cites work: Conditions on the generator of a uniformly bounded \(C_0\)-semigroup
cites work: The functional calculus for sectorial operators
cites work: Asymptotic stability of linear differential equations in Banach spaces
cites work: Decay estimates of functions through singular extensions of vector-valued Laplace transforms
cites work: Robustness of strongly and polynomially stable semigroups
cites work: Robustness of polynomial stability with respect to unbounded perturbations
cites work: Robust output regulation and the preservation of polynomial closed-loop stability
cites work: Q4533160
cites work: Q3417978
DOI: 10.1016/J.JDE.2014.08.011
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description: scientific article; zbMATH DE number 6359818