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Exponential stability of perturbed superstable systems in Hilbert spaces

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Abstract

We study exponential stability of superstable systems in Hilbert spaces under perturbations. Formulas to calculate or to estimate the exponential growth bound of the perturbed systems are derived via which sufficient conditions on exponential stability are established. The obtained results are applied to a partial differential equation governing the vibration of a smart beam made of self-straining material. Several numerical simulations are given.

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Acknowledgements

We are indebted to the anonymous referee and Professors A. V. Balakrishnan, Ciprian Preda for a lot of helpful comments and suggestions. We are grateful to Prof. Gen Qi Xu for mentioning the essential spectrum approach to perturbation of operator semigroups. Thanks also go to Prof. Bao-Zhu Guo and Dr. Vivek Natarajan for helpful discussions on zeros of the function (4.2).

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, People’s Republic of China

    Jian-Hua Chen

  2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, People’s Republic of China

    Nian-Yu Yi

  3. School of Physics and Optoelectronic Engineering, Foshan University, Foshan, 528000, People’s Republic of China

    Yong-Yao Li

Authors
  1. Jian-Hua Chen
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  2. Nian-Yu Yi
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  3. Yong-Yao Li
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Corresponding author

Correspondence to Jian-Hua Chen.

Additional information

Communicated by Markus Haase.

Chen’s research was supported by the NSF of China grant 11501189 and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. Yi’s research was partially supported by NSFC Project 11201397, and Hunan Provincial NSF Project 2015JJ2145.

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Chen, JH., Yi, NY. & Li, YY. Exponential stability of perturbed superstable systems in Hilbert spaces. Semigroup Forum 96, 126–141 (2018). https://doi.org/10.1007/s00233-017-9865-6

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