一类2×2双曲偏微分系统的PDP边界控制  

The PDP Boundary Control for a Class of 2×2 Hyperbolic Partial Differential System

作　　者：庞玉婷 赵东霞 赵鑫 高彩霞[1] Pang Yuting;Zhao Dongxia;Zhao Xin;Gao Caixia(School of Mathematics,North University of China,Taiyuan 030051;School of Electrical and Power Engineering,Taiyuan University of Technology,Taiyuan 030024)

机构地区：[1]中北大学数学学院,太原030051 [2]太原理工大学电气与动力工程学院,太原030024

出　　处：《数学物理学报（A辑）》2023年第5期1471-1482,共12页Acta Mathematica Scientia

基　　金：山西省基础研究计划资助项目(20210302123046)。

摘　　要：该文研究了具有恒定坡度和底部摩擦的单段明渠系统的指数稳定性,该系统由2×2双曲偏微分方程描述.通过设计位置反馈和延迟位置反馈(简称PDP)边界控制器来解决系统的反馈镇定问题.首先,利用算子半群理论给出系统适定性的证明.然后,通过构造合适的Lyapunov函数分析闭环系统的指数稳定性,并获得了反馈参数和时滞需要满足的充分条件.此外,还利用谱分析方法建立了系统算子特征根和特征向量的渐近表达式.最后,通过一个数值仿真示例来评估PDP控制器性能.This paper studies the exponential stability of a single open-channel system with constant slope and bottom friction,which is described by a 2×2 hyperbolic partial differential equation.The position feedback and delayed position feedback(PDP for short)boundary controller is designed to solve the problem of feedback stabilization.Firstly,the well-posedness of the system is proved by using operator semigroup theory.Then,the exponential stability of the closed-loop system is analyzed by constructing an appropriate Lyapunov function,and sufficient conditions for feedback parameters and time-delay are obtained.In addition,the asymptotic expressions of the eigenvalues and the eigenfunctions of the system operator are given by spectral analysis method.Finally,a numerical example is used to evaluate the performance of the PDP controller.

关 键 词：双曲PDE系统 LYAPUNOV函数 PDP边界控制器 指数稳定性 

分 类 号：O231.4[理学—运筹学与控制论]