Lack of uniform stabilization for noncontractive semigroups under compact perturbation
Author:
R. Triggiani
Journal:
Proc. Amer. Math. Soc. 105 (1989), 375383
MSC:
Primary 47D05
MathSciNet review:
953013
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Abstract: Let , be a strongly continuous semigroup on a Hilbert space (or, more generally, on a reflexive Banach space with the approximating property), with infinitesimal generator . Let: (i) either or be strongly stable, yet (ii) not uniformly stable as . Then, for any compact operator on , the semigroup generated by cannot be uniformly stable as . This result is 'optimal' within the class of compact perturbations . It improves upon a prior result in [G.1] by removing the assumption that be a contraction for positive times. Moreover, it complements a result in [R.1] where was assumed to be a group, contractive for negative times. Our proof is different from both [R.1 and G.1]. Application include physically significant dynamical systems of hyperbolic type in feedback form, where the results of either [R.1 or G.1] are not applicable, as the free dyamics is not a contraction.
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 J. S. Gibson, A note on stabilization of infinite dimensional linear oscillators by compact linear feedback, SIAM J. Control Optim. 18 (1980), 311316. MR 569020 (81f:47040)
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 T. Kato, Perturbation theory of linear operators, SpringerVerlag, 1966. MR 0203473 (34:3324)
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 I. Lasiecka and R. Triggiani, Dirichlet boundary stabilization of the wave equation with damping feedback of finite range, J. Math. Anal. Appl. 97 (1983), 112130. MR 721233 (85c:93071)
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 , Finite rank, relatively bounded perturbations of semigroups generators. I, Wellposedness and boundary feedback hyperbolic dynamics, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. Ser. 12 (4) (1985), MR 848843 (88c:47081a)
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 , Finite rank, relatively bounded perturbations of semigroups generators. II, Spectrum and Riesz basis assignment with applications to feedback systems, Ann. Mat. Pura Appl. 143 (1986), 47100. MR 859597 (88c:47081b)
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 A. Naylor and G. Sell, Linear operator theory, Holt, Rinehart and Wiston, 1971. MR 0461166 (57:1151)
 [R.1]
 D. L. Russell, Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory, J. Math. Anal. Appl. 40 (1972), 336368. MR 0324228 (48:2580)
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 R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl. 52 (1975), 383403. Addendum JMAA 52 (1975), 492493. MR 0445388 (56:3730)
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 , Finite rank, relatively bounded perturbations of semigroups generators. III, A sharp result on the lack of uniform stabilization. Proc. First Conf. on Control and Communication Theory, Washington, D.C., 1987. Edited by N. De Cloris, Optimization Software, Los Angeles. Also, SpringerVerlag Lecture Notes in Control and Information Sciences, vol. 111, Analysis and optimization of systems, A. Bensoussan, J. L. Lions, editors; Proceedings of Conference held at Antibes, France, June 1987.
 [T.3]
 , Pathological asymptotic behavior of control systems in Banach space, J. Math. Anal. Appl. 49 (1975), 411429. MR 0375060 (51:11256)
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DOI:
http://dx.doi.org/10.1090/S00029939198909530130
PII:
S 00029939(1989)09530130
Article copyright:
© Copyright 1989 American Mathematical Society
