论文标题
随机Grushin方程的无效可控性和逆源问题，具有边界退化和奇异性
Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity
论文作者
论文摘要
在本文中，我们考虑了带有边界堕落和奇异性的随机格鲁什方程的无效可控性和逆源问题。我们构建了两个特殊的权重函数，以通过加权身份方法为整个随机Grushin操作员建立两个Carleman估计值。一种是用于具有奇异重量函数的后退随机格鲁什方程。然后，我们将其应用于任何$ t $的随机grushin方程的零可控性，以及任何退化$γ> 0 $，当我们的控制域触摸脱落线$ \ {x = 0 \} $时。为了同时研究确定两种来源的逆源问题，我们证明了另一个卡尔曼估计值，这是针对具有常规重量功能的正向随机Grushin方程。基于此Carleman估计，我们获得了逆源问题的唯一性。
In this paper, we consider a null controllability and an inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity. We construct two special weight functions to establish two Carleman estimates for the whole stochastic Grushin operator with singular potential by a weighted identity method. One is for the backward stochastic Grushin equation with singular weight function. We then apply it to prove the null controllability for stochastic Grushin equation for any $T$ and any degeneracy $γ>0$, when our control domain touches the degeneracy line $\{x=0\}$. In order to study the inverse source problem of determining two kinds of sources simultaneously, we prove the other Carleman estimate, which is for the forward stochastic Grushin equation with regular weight function. Based on this Carleman estimate, we obtain the uniqueness of the inverse source problem.
