×
Adler, Martin; Bombieri, Miriam; Engel, Klaus-Jochen

On perturbations of generators of \(C_0\)-semigroups. (English) Zbl 1472.47030

Abstr. Appl. Anal. 2014, Article ID 213020, 13 p. (2014).
Summary: We present a perturbation result for generators of \(C_0\)-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.

Cited in 9 Documents

MSC:

47D06 One-parameter semigroups and linear evolution equations
47A55 Perturbation theory of linear operators

Keywords:

perturbation; generators of \(C_0\)-semigroups; Weiss-Staffans perturbation theorem; Desch-Schappacher perturbation theorem
PDF BibTeX XML Cite
\textit{M. Adler} et al., Abstr. Appl. Anal. 2014, Article ID 213020, 13 p. (2014; Zbl 1472.47030)
Full Text: DOI arXiv

References:

[1] Kato, T., Perturbation Theory for Linear Operators. Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, 132 (1966), New York, NY, USA: Springer, New York, NY, USA · Zbl 0148.12601
[2] Engel, K.-J.; Nagel, R., One-Parameter Semigroups for Linear Evolution Equations. One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194 (1999), Springer
[3] Weiss, G., Regular linear systems with feedback, Mathematics of Control, Signals, and Systems, 7, 1, 23-57 (1994) · Zbl 0819.93034
[4] Staffans, O., Well-Posed Linear Systems. Well-Posed Linear Systems, Encyclopedia of Mathematics and its Applications, 103 (2005), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 1057.93001
[5] Greiner, G., Perturbing the boundary conditions of a generator, Houston Journal of Mathematics, 13, 2, 213-229 (1987) · Zbl 0639.47034
[6] Weiss, G., Admissibility of unbounded control operators, SIAM Journal on Control and Optimization, 27, 3, 527-545 (1989) · Zbl 0685.93043
[7] Engel, K.-J., On the characterization of admissible control- and observation operators, Systems & Control Letters, 34, 4, 225-227 (1998) · Zbl 0909.93034
[8] Weiss, G., Admissible observation operators for linear semigroups, Israel Journal of Mathematics, 65, 1, 17-43 (1989) · Zbl 0696.47040
[9] Helton, J. W., Systems with infinite-dimensional state space: the Hilbert space approach, Proceedings of the IEEE, 64, 1, 145-160 (1976)
[10] Weiss, G., Transfer functions of regular linear systems. Part I: characterizations of regularity, Transactions of the American Mathematical Society, 342, 2, 827-854 (1994) · Zbl 0798.93036
[11] Hadd, S., Unbounded perturbations of \(C_0\)-semigroups on Banach spaces and applications, Semigroup Forum, 70, 3, 451-465 (2005) · Zbl 1074.47017
[12] Salamon, D., Infinite-dimensional linear systems with unbounded control and observation: a functional analytic approach, Transactions of the American Mathematical Society, 300, 2, 383-431 (1987) · Zbl 0623.93040
[13] Bombieri, M.; Engel, K., A semigroup characterization of well-posed linear control systems, Semigroup Forum, 88, 2, 366-396 (2014) · Zbl 1291.93155
[14] Staffans, O. J.; Weiss, G., Transfer functions of regular linear systems part III: inversions and duality, Integral Equations and Operator Theory, 49, 4, 517-558 (2004) · Zbl 1052.93032
[15] Weiss, G.; Kappel, F.; Kunisch, K.; Schappacher, W., The representation of regular linear systems on Hilbert spaces, Control and Estimation of Distributed Parameter Systems (Proceedings Vorau, 1988). Control and Estimation of Distributed Parameter Systems (Proceedings Vorau, 1988), International Series of Numerical Mathematics, 91, 401-416 (1989), Basel, Switzerland: Birkhäuser, Basel, Switzerland
[16] Weiss, G., Representation of shift-invariant operators on \(L^{ 2}\) by \(H^\infty\) transfer functions: an elementary proof, a generalization to \(L^{ p}\) , and a counterexample for \(L^\infty \), Mathematics of Control, Signals, and Systems, 4, 2, 193-203 (1991) · Zbl 0724.93021
[17] Arendt, W.; Batty, C. J. K.; Hieber, M.; Neubrander, F., Vector-Valued Laplace Transforms and Cauchy Problems. Vector-Valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96 (2001), Basel, Switzerland: Birkhäuser, Basel, Switzerland · Zbl 0978.34001
[18] Desch, W.; Schappacher, W., Some generation results for perturbed semigroups, Semigroup Theory and Applications (Trieste, 1987). Semigroup Theory and Applications (Trieste, 1987), Lecture Notes in Pure and Applied Mathematics, 116, 125-152 (1989), New York, NY, USA: Dekker, New York, NY, USA
[19] Tucsnak, M.; Weiss, G., Observation and Control for Operator Semigroups. Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts (2009), Birkhäuser · Zbl 1188.93002
[20] Miyadera, I., On perturbation theory for semi-groups of operators, Tohoku Mathematical Journal, 18, 299-310 (1966) · Zbl 0193.10902
[21] Voigt, J., On the perturbation theory for strongly continuous semigroups, Mathematische Annalen, 229, 2, 163-171 (1977) · Zbl 0338.47018
[22] Hadd, S.; Manzo, R.; Rhandi, A., Unbounded perturbations of the generator domain and applications · Zbl 1304.47054
[23] Neidhardt, H., On abstract linear evolution equations, I, Mathematische Nachrichten, 103, 283-298 (1981) · Zbl 0492.47021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.