论文标题
具有非矩形控制集的广义二维Baouendi-Grushin方程的零控制属性
Null-controllability properties of the generalized two-dimensional Baouendi-Grushin equation with non-rectangular control sets
论文作者
论文摘要
我们考虑通用Baouendi -Grushin方程$（\ partial_t- \ partial_x^2 -q（x）^2 \ partial_y^2）F =1_ΩU$上的null可控制性问题。当控制域$ω$是垂直条或$ q（x）= x $时，尖锐的可控性结果已经存在。在本文中，我们为一般$ Q $和非矩形控制区域$ω$的零控制时间的最小时间提供了上限和下限。在某些$ω$的几何形状中，上限和下限是相等的，在这种情况下，我们知道零控制性最小时间的确切值。 我们的证明依赖于几种工具：$ω$是垂直条带和截止符号的截止论点，即无可控性的最小时间； Schrödinger运算符的光谱分析$ - \ partial_x^2 +ν^2 q（x）^2 $当$ \ re（ν）> 0 $，pseudo-differential-type type operators在多项式和runge的定理上用于下限。
We consider the null-controllability problem for the generalized Baouendi-Grushin equation $(\partial_t - \partial_x^2 - q(x)^2\partial_y^2)f = 1_ωu$ on a rectangular domain. Sharp controllability results already exist when the control domain $ω$ is a vertical strip, or when $q(x) = x$. In this article, we provide upper and lower bounds for the minimal time of null-controllability for general $q$ and non-rectangular control region $ω$. In some geometries for $ω$, the upper bound and the lower bound are equal, in which case, we know the exact value of the minimal time of null-controllability. Our proof relies on several tools: known results when $ω$ is a vertical strip and cutoff arguments for the upper bound of the minimal time of null-controllability; spectral analysis of the Schrödinger operator $-\partial_x^2 + ν^2 q(x)^2$ when $\Re(ν)>0$, pseudo-differential-type operators on polynomials and Runge's theorem for the lower bound.
