弹性弦Dirichlet边界反馈控制的镇定与Riesz基生成  

Dirichlet Boundary Stabilization and Riesz Basis Property of One-Dimensional String Equation

作　　者：谢宇[1] 郭宝珠[1] 

机构地区：[1]中国科学院数学与系统科学研究院,北京100080

出　　处：《数学的实践与认识》2003年第3期99-108,共10页Mathematics in Practice and Theory

基　　金：国家自然科学基金项目资助 (60 174 0 0 8)

摘　　要：本文通过一端固定 ,一端 Dirichlet边界控制的一维波动方程说明系统是 Salamon- W eiss意义下适定和正则的 .由此说明 ,由 J.L.Lions引入的用于研究双曲方程精确可控性的 H ilbert唯一性方法是控制论中著名的对偶原理 .我们讨论了系统的指数镇定及闭环系统的广义本征函数生成 Riesz基和谱确定增长条件 .我们希望通过本文使读者对目前线性偏微分控制理论的一个新动向有一基本的了解 .This paper studies a one-dimensional wave equation with one end fixed and Dirichlet boundary feedback control at another. The system is shown to be well-posed and regular in the class of Salamon-Weiss system theory. This explains rigorously that the Hilbert-Uniqueness-Method introduced by J.L.lions in studying the exact controllability of hyperbolic systems is the well-known Duality-Principle in control theory. The Riesz basis property, spectrum-determined growth condition and exponential stability for the closed-loop system are concluded. Through this example, one can catch a glimpse of a new trend appeared very recently in Partial Differential Equation control theory.

关 键 词：弹性弦 Dirichlet边界 反馈控制 镇定 Riesz基生成 线性偏微分控制系统 HILBERT空间 适定系统 Salamon-Weiss适定 输入 输出 波动方程 直接传输算子 

分 类 号：O231[理学—运筹学与控制论]