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: 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.