Electron. J. Diff. Equ., Vol. 2015 (2015), No. 31, pp. 1-13.

Exact controllability problem of a wave equation in non-cylindrical domains

Hua Wang, Yijun He, Shengjia Li

Abstract:
Let $\alpha: [0, \infty)\to(0, \infty)$ be a twice continuous differentiable function which satisfies that $\alpha(0)=1$, $\alpha'$ is monotone and $0<c_1\le \alpha'(t)\le c_2<1$ for some constants $c_1$ and $c_2$. The exact controllability of a one-dimensional wave equation in a non-cylindrical domain is proved. This equation characterizes small vibrations of a string with one of its endpoint fixed and the other moving with speed $\alpha'(t)$. By using the Hilbert Uniqueness Method, we obtain the exact controllability results of this equation with Dirichlet boundary control on one endpoint. We also give an estimate on the controllability time that depends only on $c_1$ and $c_2$.

Submitted December 6, 2014. Published January 30, 2015. Math Subject Classifications: 35L05, 93B05.
Key Words: Exact controllability; non-cylindrical domain; Hilbert uniqueness method.

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Hua Wang
School of Mathematical Sciences
Shanxi University
Taiyuan 030006, China
email: 197wang@163.com
Yijun He
School of Mathematical Sciences
Shanxi University
Taiyuan 030006, China
email: heyijun@sxu.edu.cn
Shengjia Li
School of Mathematical Sciences
Shanxi University
Taiyuan 030006, China
email: shjiali@sxu.edu.cn

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