Well-posedness and regularity for non-uniform Schrödinger and Euler-Bernoulli equations with boundary control and observation

Authors:
Bao-Zhu Guo and Zhi-Chao Shao

Journal:
Quart. Appl. Math. **70** (2012), 111-132

MSC (2000):
Primary 35L35, 93C20, 93D15

DOI:
https://doi.org/10.1090/S0033-569X-2011-01243-0

Published electronically:
August 26, 2011

MathSciNet review:
2920619

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Abstract | References | Similar Articles | Additional Information

Abstract: The open-loop systems of a Schrödinger equation and an Euler-Bernoulli equation with variable coefficients and boundary controls and collocated observations are considered. It is shown, with the help of a multiplier method on a Riemannian manifold, that both systems are well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The feed-through operators are found to be zero. The results imply particularly that the exact controlability of each open-loop system is equivalent to the exponential stability of the associated closed-loop system under the output proportional feedback.

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Additional Information

**Bao-Zhu Guo**

Affiliation:
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, People’s Republic of China, Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, People’s Republic of 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 Information Technology and Management and China Service Industry Research Center, University of International Business and Economics, Beijing 100029, People’s Republic of China

Email:
zcshao@amss.ac.cn

DOI:
https://doi.org/10.1090/S0033-569X-2011-01243-0

Keywords:
Schrödinger equation,
Euler-Bernoulli equation,
well-posed and regular system,
variable coefficients,
boundary control and observation.

Received by editor(s):
August 9, 2010

Published electronically:
August 26, 2011

Additional Notes:
This work was carried out with the support of the National Natural Science Foundation of China, the National Research Foundation of South Africa, and the National Basic Research Program of China (973 Program: 2011CB808002).

Zhi-Chao Shao also acknowledges the support of the Scientific Research Foundation for the Returned Overseas Scholar by Education Ministry of China and the program for Innovative Research Team in UIBE

Article copyright:
© Copyright 2011
Brown University