非线性偏微分方程组的精确能控性和反馈镇定性

We will study simultaneous exact controllability, indirect exact controllability and indirect feedback stabilization of systems coupled by two or more multidimensional(multiple space variables) partial differential equations with nonlinear principal part. Firstly, we will study locally simultaneous (indirect) exact controllability around an equilibrium of the system, and then globally simultaneous(indirect) exact controllability in such a way that the state of the system moves from an equilibrium in one location to an equilibrium in another location. We will use geometric analysis method to study nonlinear hyperbolic-hyperbolic systems and nonlinear Schr?dinger-Schr?dinger systems, use geometric analysis method and “Carleman estimate+energy estimate” to study nonlinear hyperbolic-parabolic systems, and seek the geometrical conditions which make simultaneous (indirect) exact controllability of those three kinds of systems hold. We will use “Carleman estimate+energy estimate” to study simultaneous (indirect) exact controllability of nonlinear parabolic-parabolic systems. We will design dissipative structure on the boundary or interior for a certain equation of the system, and use energy estimate to obtain global solutions and decay speed of the energy.

本项目将研究含有两个或多个高维的(多空间变量)、具有非线性主部的偏微分方程的耦合系统的同步精确能控性、间接精确能控性和间接反馈镇定性。本项目的基本思路是，先研究非线性耦合系统在平衡态附近的局部的同步（间接）精确能控性，然后研究从一个平衡态到另一个平衡态的大范围同步（间接）精确能控性。本项目拟采用几何分析方法研究非线性双曲-双曲系统和非线性薛定谔-薛定谔系统，将几何分析方法和“Carleman 型不等式+通常能量估计”方法相结合起来研究非线性双曲-抛物系统，找到这三类系统的同步精确能控性和间接精确能控性成立的几何条件。本项目拟采用“Carleman 型不等式+通常能量估计”方法研究非线性抛物-抛物系统的同步精确能控性和间接精确能控性。在非线性耦合系统中的某个方程的边界或内部引入耗散结构，用能量估计方法来得到系统的整体解的存在性以及系统能量的衰减速度。

项目负责人利用黎曼几何工具得到了在一定曲率条件下的变系数薛定谔算子的Carleman不等式。利用此不等式，负责人得到了带有低阶项的变系数的薛定谔方程的一个反问题：方程中的势函数可由方程的解在区域的部分边界上的外法向导数唯一确定。由偏微分方程的反问题与能控性的紧密联系，利用本项目的研究成果负责人可以继续开展如下工作：利用变系数薛定谔算子的Carleman不等式研究变系数薛定谔方程组、变系数板方程组、非线性薛定谔方程组、非线性板方程组的同步（间接）精确能控性、反馈镇定性以及反问题等。