具有端点质量的一维波动方程半离散格式的一致指数稳定性

On uniform exponential stability of semi-discrete scheme for 1-D wave equation with a tip mass

无穷维系统主要由偏微分方程描述,可是大部分用偏微分方程描述的控制系统,无论是单纯的数值实验还是需要应用到实际的问题中去,都需要对方程进行有限数值离散.本文考虑了端点带有质量的波动方程在边界反馈控制下半离散格式的一致指数稳定性.首先,原闭环系统通过降阶法变成低阶的等价系统,通过一种间接Lyapunov函数方法证明了降阶等价的连续系统是一致指数稳定的.其次,对等价系统空间变量离散得到半离散的差分格式.平行于连续系统,间接Lyapunov函数方法证明了半离散系统的一致指数稳定性.数值实验证明了基于降阶法的一致指数稳定性和经典半离散格式的非一致指数稳定性.

Most of the infinite-dimensional systems are described by partial differential equations(PDEs).For PDEs,discretization is most often necessarily for numerical simulation and applications.This paper considers the uniform ex-ponential stability of a semi-discrete model for a 1-D wave equation with tip mass under boundary feedback control.The original closed-loop system is transformedfirstly into a low-order equivalent system by order reduction method and the ex-ponential stability of the transformed system by an indirect Lyapunov method is established.The equivalent system is then discretized into a series of semi-discrete systems in spacial variable.Parallel to the continuous system,the semi-discrete systems are proved to be uniformly exponentially stable by means of the indirect Lyapunov method.Numerical simula-tions illustrate why the classical semi-discrete scheme does not preserve the uniformly exponential stability while the order reduction semi-discrete scheme does.