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Equivalence between observability at the boundary and exponential stabilization for an ACL beam actuated by a voltage source without magnetic effects

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Abstract

In this paper, we consider a one-dimensional dissipative system of a Rayleigh beam coupled with two wave equations, modeling an active constrained layer beam which consists of a stiff layer, a viscoelastic layer and a piezoelectric layer actuated by a voltage source without magnetic effects. We prove the equivalence between observability at the boundary and exponential stability previously proved by Yang and Wang (J Math Anal Appl 448:1204–1227, 2017). This is achieved by using some ingenious calculus techniques and the multiplier method to establish two observable lemmas for the conservative system and the auxiliary system, respectively, so as to overcome the complexity of equations and the difficulties caused by the appearance of the coupling term.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China [grant number 11771216], the Key Research and Development Program of Jiangsu Province (Social Development) [grant number BE2019725] and the Qing Lan Project of Jiangsu Province and Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant No. KYCX20_0945).

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  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Yanning An, Wenjun Liu & Aowen Kong

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  1. Yanning An
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  2. Wenjun Liu
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  3. Aowen Kong
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Correspondence to Wenjun Liu.

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An, Y., Liu, W. & Kong, A. Equivalence between observability at the boundary and exponential stabilization for an ACL beam actuated by a voltage source without magnetic effects. Z. Angew. Math. Phys. 73, 156 (2022). https://doi.org/10.1007/s00033-022-01798-6

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  • DOI: https://doi.org/10.1007/s00033-022-01798-6

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