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Lack of uniformly exponential stabilization for isometric $ C_0$-semigroups under compact perturbation of the generators in Banach spaces


Authors: Faming Guo, Ke Guo and Chaolun Zhang
Journal: Proc. Amer. Math. Soc. 135 (2007), 1881-1887
MSC (2000): Primary 47A50, 47A55
DOI: https://doi.org/10.1090/S0002-9939-07-08698-4
Published electronically: February 2, 2007
Addendum: Proc. Amer. Math. Soc. 137 (2009), 2809-2812
MathSciNet review: 2286100
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Abstract: This paper is concerned with non-uniformly exponential stabilization for infinite-dimensional linear systems under compact feedback in Banach spaces. We prove that a compact perturbation of the generator of an isometric $ C_0$-semigroup cannot generate a uniformly exponentially stable $ C_0$-semigroup in a Banach space. Finally, examples are provided to illustrate our result.


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Additional Information

Faming Guo
Affiliation: Department of Applied Mathematics, College of Information Management, Chengdu University of Technology, Chengdu, 610059, People’s Republic of China
Email: guofm@uestc.edu.cn

Ke Guo
Affiliation: Department of Applied Mathematics, College of Information Management, Chengdu University of Technology, Chengdu, 610059, People’s Republic of China

Chaolun Zhang
Affiliation: Institute of Applied Mathematics, Xihua University, Chengdu, 610039, People’s Republic of China

DOI: https://doi.org/10.1090/S0002-9939-07-08698-4
Keywords: Compact perturbation, isometric $C_0$-semigroup, exponential stability.
Received by editor(s): January 14, 2006
Received by editor(s) in revised form: February 23, 2006
Published electronically: February 2, 2007
Additional Notes: This research was supported by the Postdoctoral Science Foundation of China.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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