Degasperis-Procesi方程若干控制问题的研究

The project is concerned with some control problems related to newly proposed integrable shallow water wave equations: the Degasperis-Procesi equation and the two-component Degasperis-Procesi equation. Some important open problems, such as controllability and asymptotical stabilizability under the distributed control and boundary control, will be considered. The returning method and the fixed point principle are applied to establish the controllability of the Degasperis-Procesi equations. Design a feedback control law, by using the fixed point principle, we will establish the global well-posedness of the closed-loop systems. Collect the appreciate Lyapunov function, and by using the Lyapunov stability theory, we will establish the stability of the closed-loop system, and discuss the well-posedness of the initial value problem of the controlled system in the appropriate Banach spaces. Using the finite difference method for numerical simulation, validation of the theoretical results is confirmed. This project has strong application background based on the nonlinear system control theory, launches a new research topic in applied mathematics, nonlinear system control and numerical calculation, and may has potential application in some relative fields.

本项目主要研究一类完全可积的浅水波方程Degasperis-Procesi方程和二元Degasperis-Procesi方程的若干控制问题, 特别是能控性和渐近能稳性问题. 将应用返回（returning）方法和不动点原理建立Degasperis-Procesi方程在分布控制下的能控性. 设计反馈控制律，应用不动点原理建立闭环系统的全局适定性, 设计Lyapunov函数,利用Lyapunov稳定性原理建立闭环系统的稳定性, 并讨论在初始值属于合适的Banach空间时受控系统解的适定性. 利用有限差分方法进行数值模拟, 验证理论结果. 本项目基于具有强烈应用背景的非线性系统, 展开全新的控制理论研究课题, 在应用数学、非线性系统控制和数值计算上给出具有潜在应用的新结果.

本项目主要研究一类完全可积的浅水波方程方程和二元浅水波方程的若干控制问题, 从理论上获得了D-P方程和C-H方程在边界输出反馈控制作用下的能控性。从理论上获得了D-P方程和C-H方程边界输出反馈控制作用下的能稳性，设计出反馈控制器，反馈控制器能使系统的解关于空间变量的一阶偏导数有一个大于0的下确界，系统的解自然单调下降，就可以防止系统的解在有限时间内发生爆破。构造Lyapunov函数，保证系统的解充分接近系统的某个能控的特殊解，而且在边界上快速衰减。最后从理论上得到了D-P方程和C-H方程在Dirichlet边界反馈控制作用下的闭环系统的适定性。考虑系统在某个函数空间内的能量表达式，作为Lyapunov函数，根据Lyapunov稳定性定义设计边界控制器使系统稳定。