Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On well-posedness, regularity and exact controllability for problems of transmission of plate equation with variable coefficients


Authors: Bao-Zhu Guo and Zhi-Chao Shao
Journal: Quart. Appl. Math. 65 (2007), 705-736
MSC (2000): Primary 35L35, 93C20, 93D15, 93B05, 93B07
DOI: https://doi.org/10.1090/S0033-569X-07-01069-9
Published electronically: October 5, 2007
MathSciNet review: 2370357
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Abstract | References | Similar Articles | Additional Information

Abstract: A system of transmission of Euler-Bernoulli plate equation with variable coefficients under Neumann control and collocated observation is studied. Using the multiplier method on a Riemannian manifold, it is shown that the system is well-posed in the sense of D. Salamon. This establishes the equivalence between the exact controllability of an open-loop system and the exponential stability of a closed-loop system under the proportional output feedback. The regularity of the system in the sense of G. Weiss is also proved, and the feedthrough operator is found to be zero. These properties make this PDE system parallel in many ways to the finite-dimensional ones. Finally, the exact controllability of an open-loop system is developed under a uniqueness assumption by establishing the observability inequality for the dual system.


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Additional Information

Bao-Zhu Guo
Affiliation: Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100080, P.R. China and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
Email: bzguo@iss.ac.cn

Zhi-Chao Shao
Affiliation: School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
Email: zcshao@cam.wits.ac.za

DOI: https://doi.org/10.1090/S0033-569X-07-01069-9
Keywords: Euler-Bernoulli plate equation, well-posedness and regularity, boundary control and observation, exact controllability, exact observability, multiplier method on Riemannian manifold.
Received by editor(s): June 15, 2006
Published electronically: October 5, 2007
Additional Notes: This work was carried out with the support of the National Natural Science Foundation of China and the National Research Foundation of South Africa. Zhi-Chao Shao acknowledges the support of the Postdoctoral Fellowship of the Claude Leon Foundation of South Africa.
Article copyright: © Copyright 2007 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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