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

• Main Authors: Hua Wang, Yijun He, Shengjia Li
• Format: Article
• Language: English
• Published: Texas State University 2015-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: Exact controllability; non-cylindrical domain; Hilbert uniqueness method
• Online Access: http://ejde.math.txstate.edu/Volumes/2015/31/abstr.html

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 $c_2$.