一类 2$\times$2 双曲偏微分系统的 PDP 边界控制

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 1471-1482.

一类 2$\times$2 双曲偏微分系统的 PDP 边界控制

庞玉婷¹(),赵东霞^1,^*(),赵鑫²(),高彩霞¹()

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

收稿日期:2022-03-18 修回日期:2023-03-08 出版日期:2023-10-26 发布日期:2023-08-09
通讯作者: 赵东霞 E-mail:2116786325@qq.com;zhaodongxia6@sina.com;1808642517@qq.com;1519546532@qq.com
作者简介:庞玉婷，Email: 2116786325@qq.com;|赵鑫，Email: 1808642517@qq.com;|高彩霞，Email: 1519546532@qq.com
基金资助:
山西省基础研究计划资助项目(20210302123046)

The PDP Boundary Control for a Class of 2$\times$2 Hyperbolic Partial Differential System

Pang Yuting¹(),Zhao Dongxia^1,^*(),Zhao Xin²(),Gao Caixia¹()

¹School of Mathematics, North University of China, Taiyuan 030051
²School of Electrical and Power Engineering, Taiyuan University of Technology, Taiyuan 030024

Received:2022-03-18 Revised:2023-03-08 Online:2023-10-26 Published:2023-08-09
Contact: Dongxia Zhao E-mail:2116786325@qq.com;zhaodongxia6@sina.com;1808642517@qq.com;1519546532@qq.com
Supported by:
Fundamental Research Program of Shanxi Province(20210302123046)

摘要/Abstract

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$2\times2$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.

关键词: 双曲PDE系统, Lyapunov函数, PDP边界控制器, 指数稳定性

Abstract:

This paper studies the exponential stability of a single open-channel system with constant slope and bottom friction, which is described by a $2\times2$ 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.

Key words: Hyperbolic PDE system, Lyapunov function, PDP boundary controller, Exponential stability

中图分类号:

O231.4

庞玉婷,赵东霞,赵鑫,高彩霞. 一类 2$\times$2 双曲偏微分系统的 PDP 边界控制[J]. 数学物理学报, 2023, 43(5): 1471-1482.

Pang Yuting,Zhao Dongxia,Zhao Xin,Gao Caixia. The PDP Boundary Control for a Class of 2$\times$2 Hyperbolic Partial Differential System[J]. Acta mathematica scientia,Series A, 2023, 43(5): 1471-1482.

图/表 4

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