ESAIM: COCV, Vol. 13, N°4, pp. 776-792
DOI: 10.1051/cocv:2007040

On the well-posedness and regularity of the wave equation with variable coefficients

Bao-Zhu Guo^{1, 2} and Zhi-Xiong Zhang<sup>1, 3

1</sup>  Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100080, P.R. China; bzguo@iss.ac.cn
²  School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa.
³  Graduate University of Chinese Academy of Sciences, Beijing 100049, P.R. China.

(Received January 12, 2006. Revised May 25, 2006. Published online September 5, 2007.)

Abstract
An open-loop system of a multidimensional wave equation with variable coefficients, partial boundary Dirichlet control and collocated observation is considered. It is shown that the system is well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The Riemannian geometry method is used in the proof of regularity and the feedthrough operator is explicitly computed.

Mathematics Subject Classification. 35J50, 93C20, 93C25

Key words: Wave equation, transfer function, well-posed and regular system, boundary control and observation.


© EDP Sciences, SMAI 2007