Polynomial stability of an abstract system with local damping

Otared Kavian; Qiong Zhang

doi:10.1016/j.jmaa.2022.126133

Polynomial stability of an abstract system with local damping

Otared Kavian, Qiong Zhang^*

^*Corresponding author for this work

School of Mathematics and Statistics

Université Paris–Saclay (site UVSQ)

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper, we consider the polynomial stability for an abstract system of the type u_tt+Lu+Bu_t=0, where L is a self-adjoint operator on a Hilbert space and operator B represents the local damping. By establishing precise estimates on the resolvent, we prove polynomial decay of the corresponding semigroup. The results reveal that the rate of decay depends strongly on the concentration of eigenvalues of operator L and non-degeneration of operator B. Finally, several examples are given as an application of our abstract results.

Original language	English
Article number	126133
Journal	Journal of Mathematical Analysis and Applications
Volume	512
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2022.126133
Publication status	Published - 15 Aug 2022

Keywords

Local damping
Polynomial stability
Semigroup

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10.1016/j.jmaa.2022.126133

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Kavian, O., & Zhang, Q. (2022). Polynomial stability of an abstract system with local damping. Journal of Mathematical Analysis and Applications, 512(2), Article 126133. https://doi.org/10.1016/j.jmaa.2022.126133

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abstract = "In this paper, we consider the polynomial stability for an abstract system of the type utt+Lu+But=0, where L is a self-adjoint operator on a Hilbert space and operator B represents the local damping. By establishing precise estimates on the resolvent, we prove polynomial decay of the corresponding semigroup. The results reveal that the rate of decay depends strongly on the concentration of eigenvalues of operator L and non-degeneration of operator B. Finally, several examples are given as an application of our abstract results.",

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N2 - In this paper, we consider the polynomial stability for an abstract system of the type utt+Lu+But=0, where L is a self-adjoint operator on a Hilbert space and operator B represents the local damping. By establishing precise estimates on the resolvent, we prove polynomial decay of the corresponding semigroup. The results reveal that the rate of decay depends strongly on the concentration of eigenvalues of operator L and non-degeneration of operator B. Finally, several examples are given as an application of our abstract results.

AB - In this paper, we consider the polynomial stability for an abstract system of the type utt+Lu+But=0, where L is a self-adjoint operator on a Hilbert space and operator B represents the local damping. By establishing precise estimates on the resolvent, we prove polynomial decay of the corresponding semigroup. The results reveal that the rate of decay depends strongly on the concentration of eigenvalues of operator L and non-degeneration of operator B. Finally, several examples are given as an application of our abstract results.

KW - Local damping

KW - Polynomial stability

KW - Semigroup

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Polynomial stability of an abstract system with local damping

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