二维区域上的Baouendi-Grushin型退化抛物方程的定量唯一延拓性及其在控制问题中的应用

In this project, we shall study the quantitative unique continuation properties for Baouendi-Grushin type degenerate parabolic equation in two dimensional region with their applications in control problems. The quantitative unique continuation properties studied in this project contain mainly the observability inequality and the point-wise interpolation inequality with respect to the time variable. For our purposes, we shall redefine an distance and an integration on 2D real space based on Lie-algebra construction of the Baouendi-Grushin operator. In this framwork, using the Carleman estimate method, we shall show that the desired observability inequality holds when the terminal time is large enough. Meanwhile, in this setting, we shall define a proper frequency function and use the frequency function method to prove that the point-wise interpolation inequality with respect to the time variable holds when the terminal time is large enough. Moreover, we shall give some counterexamples to show that the above two kinds of inequalities do not hold if the terminal time is sufficiently small. We shall also consider the observability inequality on a measurable set for the Baouendi-Grushin type degenerate parabolic equation, the controllability and the time optimal control problem of the controlled Baouendi-Grushin type degenerate parabolic equation as the applications of our main results.

本项目拟就二维区域上的Baouendi-Grushin型退化抛物方程的定量唯一延拓性（主要包括能观性不等式和关于时间变量为点态的插值不等式）及其在控制问题中的应用展开研究和讨论。我们根据Baouendi-Grushin算子的李代数结构重新定义二维实空间的距离和积分。在此框架下，利用Carleman估计方法，证明当终端时间充分大时，所需的能观性不等式成立。同时，通过定义合适的频率函数，利用频率函数方法，证明当终端时间充分大时，关于时间变量为点态的插值不等式成立。另外，我们将列举某些反例，证明当终端时间充分小时，上述两种不等式不成立。作为主要结论的应用，我们着重讨论可测集上的能观性不等式，受控的Baouendi-Grushin型退化抛物方程的能控性以及时间最优控制问题。

本项目研究分布参数系统控制理论中的若干问题。具体地，我们主要研究了抽象抛物方程最优控制问题的最优状态估计问题、变系数抛物方程的三球定理、脉冲控制系统能稳的充分必要条件、时间最优控制问题的切换次数和切换方向和耦合热方程约束能控性的充分必要条件等问题。主要成果如下：一、我们给出了抛物方程控制系统的观测问题，给出了使用观测数据获得的最优状态估计的显式表达式；二、对于变系数的抛物方程，我们获得了显式的三球定理，并将其应用至时间最优控制问题之中；三、在有限维系统的框架下，我们给出了脉冲控制系统能稳的充分必要条件；四、在有限维系统时间最优控制问题的框架下，我们给出了最优控制的切换次数、切换方向等显式估计；五、以耦合热方程的脉冲控制问题为研究对象，我们给出了使得系统约束能控的充分必要条件。这些成果对本学科的发展具有相当的学术和应用价值。