SIAM Journal on Control and Optimization, Volume 59, Issue 1, Page 131-155, January 2021.
We investigate the null controllability of the wave equation with a Kelvin--Voigt damping on the two-dimensional torus ${\mathbb T} ^2$. We consider a distributed control supported in a moving domain $\omega (t)$ with a uniform motion at a constant velocity $c=(1,\zeta)$. The results we obtain depend strongly on the topological features of the geodesics of ${\mathbb T} ^2$ with constant velocity $c$. When $\zeta\in {\mathbb Q}$, writing $\zeta=p/q$ with $p,q$ relatively prime, we prove that the null controllability holds if roughly the diameter of $\omega (0)$ is larger than $1/p$ and if the control time is larger than $q$. We also prove that for almost every $\zeta \in {\mathbb R}_+\setminus {\mathbb Q}$, and also for some particular values including, e.g., $\zeta=e$, the null controllability holds for any choice of $\omega (0)$ and for a sufficiently large control time. The proofs rely on a delicate construction of the weight function in a Carleman estimate which gets rid of a topological assumption on the control region often encountered in the literature. Diophantine approximations are also needed when $\zeta$ is irrational.

中文翻译：

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二维圆环上结构阻尼波动方程的零可控性

SIAM控制与优化杂志，第59卷，第1期，第131-155页，2021年1月。
我们用二维圆环$ {\ mathbb T} ^ 2 $上的Kelvin-Voigt阻尼研究波动方程的零可控性。我们考虑在运动域$ \ω（t）$中以恒定速度$ c =（1，\ zeta）$匀速运动支持的分布式控制。我们获得的结果在很大程度上取决于恒定速度$ c $的测地线$ {\ mathbb T} ^ 2 $的拓扑特征。当{\ mathbb Q} $中的$ \ zeta \，用$ p，q $书写$ \ zeta = p / q $相对质数时，我们证明如果近似于$ \ omega（0）$的直径，则零可控性成立。大于$ 1 / p $，并且控制时间大于$ q $。我们还证明，对于{\ mathbb R} _ + \ setminus {\ mathbb Q} $中的几乎每个$ \ zeta \，以及某些特定值，例如$ \ zeta = e $，空的可控制性适用于$ \ omega（0）$的任何选择以及足够大的控制时间。证明依赖于Carleman估计中权函数的精细构造，从而摆脱了文献中经常遇到的控制区域的拓扑假设。当$ \ zeta $不合理时，也需要Diophantine近似值。