具有边界切换反馈控制的波方程网络及其稳定性分析

Stability analysis of networks of wave equations with boundary switch feedback controls

中文关键词  波方程  切换反馈控制  达朗贝尔公式  指数稳定性  衰减速率

英文关键词  wave equation  switch feedback control  D’Alembert formula  exponential stability  decay rate

中文摘要

      本文主要就具有两种不同切换反馈控制的网络系统进行探讨. 该系统是由波方程所描述的二叉树结构, 其中, 系统的一端固定, 另外两端分别施加弹性反馈控制α(t)u_2t(x,t),β(t)u_3t(x,t), 并且α(t),β(t)都是以2T为周期的Heaviside型函数. 考虑两种情况下闭环系统的稳定性: 一种是两个切换控制同时开同时关的情况, 一种是两个控制器不同时开关的情况, 即在任意的时间段只有一个控制器在工作. 本文利用达朗贝尔方法给出闭环系统的具体的解的表达式, 然后通过迭代方法对产生的迭代矩阵进行谱分析, 由此得到系统的解的性质, 从而得到了系统的指数稳定性和衰减速率.

英文摘要

      We consider the system with two different kinds of switch controls. The system is described by the wave equations with the structure of binary tree, of which one end is fixed and the other two ends are damped by the velocities feedback controls α(t)u_2t(x; t); β(t)u_3t(x; t) respectively, where the α(t); β(t) are both Heaviside-type functions of periodic 2T. According to two different cases, we analyze the stability of the closed-loop system. The first one is that the two controls are on-off simulatively, the other case is that there is only one controller working at anytime. Using the D’Alembert formula, we get the explicit expression of the solution for the closed loop system. And then, we analyze the spectrum of the iterate matrices, from which we obtained the properties of the solution. Finally, we get the exponential stability and the decay rate.