Lack of uniform stabilization for noncontractive semigroups under compact perturbation

Author:
R. Triggiani

Journal:
Proc. Amer. Math. Soc. **105** (1989), 375-383

MSC:
Primary 47D05

DOI:
https://doi.org/10.1090/S0002-9939-1989-0953013-0

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.

**[B.1]**A. V. Balakrishnan,*Applied functional analysis*, 2nd ed., Applications of Mathematics, vol. 3, Springer-Verlag, New York-Berlin, 1981. MR**612793****[G.1]**J. S. Gibson,*A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback*, SIAM J. Control Optim.**18**(1980), no. 3, 311–316. MR**569020**, https://doi.org/10.1137/0318022**[K.1]**Tosio Kato,*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473****[L-T.1]**I. Lasiecka and R. Triggiani,*Dirichlet boundary stabilization of the wave equation with damping feedback of finite range*, J. Math. Anal. Appl.**97**(1983), no. 1, 112–130. MR**721233**, https://doi.org/10.1016/0022-247X(83)90241-X**[L-T.2]**I. Lasiecka and R. Triggiani,*Finite rank, relatively bounded perturbations of semigroups generators. I. Well-posedness and boundary feedback hyperbolic dynamics*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**12**(1985), no. 4, 641–668 (1986). MR**848843****[L-T.3]**I. Lasiecka and R. Triggiani,*Finite rank, relatively bounded perturbations of semigroups generators. II. Spectrum and Riesz basis assignment with applications to feedback systems*, Ann. Mat. Pura Appl. (4)**143**(1986), 47–100. MR**859597**, https://doi.org/10.1007/BF01769210**[N-S.1]**Arch W. Naylor and George R. Sell,*Linear operator theory in engineering and science*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR**0461166****[R.1]**David L. Russell,*Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory*, J. Math. Anal. Appl.**40**(1972), 336–368. MR**0324228**, https://doi.org/10.1016/0022-247X(72)90055-8**[T.1]**Roberto Triggiani,*On the stabilizability problem in Banach space*, J. Math. Anal. Appl.**52**(1975), no. 3, 383–403. MR**0445388**, https://doi.org/10.1016/0022-247X(75)90067-0**[T.2]**-,*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, Springer-Verlag 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]**Roberto Triggiani,*Pathological asymptotic behavior of control systems in Banach space*, J. Math. Anal. Appl.**49**(1975), 411–429. MR**0375060**, https://doi.org/10.1016/0022-247X(75)90188-2

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DOI:
https://doi.org/10.1090/S0002-9939-1989-0953013-0

Article copyright:
© Copyright 1989
American Mathematical Society