This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce a compact perturbation method which does not depend on the Riesz basis property, but depends only on the continuity of projection with respect to a weaker norm, which is obviously true in many cases of application. Next, in the case of fewer Neumann boundary controls, the non-exact boundary controllability for the initial data with the same level of energy is shown.
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Part of the work was done during the visit of the second author at the Laboratoire International Associé Sino-Français de Mathématiques Appliquées (LIASFMA) and the School of Mathematical Sciences of Fudan University during June–August 2014. He would like to thank their hospitality and support.
The authors are very grateful to Professor Xu Zhang for bringing their attention to the references [11, 12] and for the valuable discussions on several occasions, and would like also to thank the referees for their very valuable comments and remarks, which were greatly appreciated to improve the presentation of the paper.
Dedicated to Professor Haim Brezis on the occasion of his 70th birthday
This work was supported by the National Basic Research Program of China (No. 2013CB834100) and the National Natural Science Foundation of China (No. 11121101).
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Li, T., Rao, B. Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. Chin. Ann. Math. Ser. B 38, 473–488 (2017). https://doi.org/10.1007/s11401-017-1078-5
- Compactness-uniqueness perturbation
- Boundary observability
- Exact boundary controllability
- Non-exact boundary controllability
- Coupled system of wave equations
- Neumann boundary condition
2000 MR Subject Classification