具有内奇异位的二维Grushin方程的近似能控性
Approximate controllability for a 2D Grushin equation with potential having an internal singularity
摘要：
研究了具有平方反比势的矩形$(x,y) \in (-1,1) \times (0,1)$上Grushin方程的逼近能控性.该模型对应于Boscain和Laurent研究的与$\mathbb{R}^2$上Grushin度量相关的Laplace-Beltrami算子的热方程。该算子在$\{ x=0 \}$线上既是退化的又是奇异的。通过伴随系统的唯一延拓研究了其近似能控性。对于所研究的奇异性范围，证明了无论退化程度如何，近似能控性都成立。由于系统内部的平方反比奇异性，研究其适定性是本文的一个重点。设计了一个扩展的奇异算子，通过奇异性施加适当的传输条件。然后，唯一的延续依赖于在一个变量中的2D解的傅立叶分解和由傅立叶分量求解的1D热方程的Carleman估计。Carleman估计使用适当的Hardy不等式。
英文摘要：
This paper is dedicated to approximate controllability for Grushin equation on the rectangle $(x,y) \in (-1,1) \times (0,1)$ with an inverse square potential. This model corresponds to the heat equation for the Laplace-Beltrami operator associated to the Grushin metric on $\mathbb{R}^2$, studied by Boscain and Laurent. The operator is both degenerate and singular on the line $\{ x=0 \}$. The approximate controllability is studied through unique continuation of the adjoint system. For the range of singularity under study, approximate controllability is proved to hold whatever the degeneracy is. Due to the internal inverse square singularity, a key point in this work is the study of well-posedness. An extension of the singular operator is designed imposing suitable transmission conditions through the singularity. Then, unique continuation relies on the Fourier decomposition of the 2D solution in one variable and Carleman estimates for the 1D heat equation solved by the Fourier components. The Carleman estimate uses a suitable Hardy inequality.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
