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Stabilization of the Euler–Bernoulli equation via boundary connection with heat equation

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Abstract

In this paper, we are concerned with the stabilization of a coupled system of Euler–Bernoulli beam or plate with heat equation, where the heat equation (or vice versa the beam equation) is considered as the controller of the whole system. The dissipative damping is produced in the heat equation via the boundary connections only. The one-dimensional problem is thoroughly studied by Riesz basis approach: The closed-loop system is showed to be a Riesz spectral system and the spectrum-determined growth condition holds. As the consequences, the boundary connections with dissipation only in heat equation can stabilize exponentially the whole system, and the solution of the system has the Gevrey regularity. The exponential stability is proved for a two dimensional system with additional dissipation in the boundary of the plate part. The study gives rise to a different design in control of distributed parameter systems through weak connections with subsystems where the controls are imposed.

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Acknowledgments

The authors would like to thank the referees and associate editor for their very helpful suggestions and comments.

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Authors and Affiliations

  1. School of Mathematics, Beijing Institute of Technology, Beijing, 100081, People’s Republic of China

    Qiong Zhang & Jun-Min Wang

  2. Academy of Mathematics and Systems Science, Academia Sinica, Beijing, 100190, People’s Republic of China

    Bao-Zhu Guo

  3. School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

    Bao-Zhu Guo

Authors
  1. Qiong Zhang
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  2. Jun-Min Wang
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  3. Bao-Zhu Guo
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Corresponding author

Correspondence to Jun-Min Wang.

Additional information

This work was carried out with the support of the National Natural Science Foundation of China, the National Basic Research Program of China (2011CB808002), and the National Research Foundation of South Africa.

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Zhang, Q., Wang, JM. & Guo, BZ. Stabilization of the Euler–Bernoulli equation via boundary connection with heat equation. Math. Control Signals Syst. 26, 77–118 (2014). https://doi.org/10.1007/s00498-013-0107-5

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  • DOI: https://doi.org/10.1007/s00498-013-0107-5

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