耦合二阶ODE波系统的输出反馈镇定

Output Feedback Stabilization of a Coupled Second Order ODE-Wave System

摘要: 本文考虑耦合二阶ODE波系统的边界输出反馈镇定问题。我们首先通过原系统的输出设计ODE波系统的观测器，并证明观测器的有效性。然后基于文献中给出的状态反馈控制器，我们设计出输出反馈控制器。利用算子半群方法和Back-stepping变换，证明了闭环系统在状态空间中具有唯一解，且闭环系统的解是渐近稳定的。

Abstract: In this paper, we are concerned with boundary output feedback stabilization of a coupled second order ODE-wave system. We first design an observer for ODE-wave system by output of original system and the effectiveness of observer is proved. Then an output feedback controller is proposed based on a state feedback controller in. Operator semi-group method and back-stepping transformation are adopted to prove that the resulting closed-loop system admits a unique solution in state space and the solution of closed-loop system is asymptotically stable.

1. 引言

在航天工业、土木工程领域、工业流水线等方面，一维波动方程或弦方程可以模拟许多物理现象。因此，该模型在理论和实际应用领域都受到了广泛的关注。一般情况下，振动过大往往会降低系统的性能，因此系统的镇定变得非常重要。鉴于工程中易实现性，边界控制器是工程师首选的。在过去的二十年中，文献中出现了一系列关于一维波方程边界镇定的著作( [2] )，其中耗散理论对反馈控制律的设计起着重要的作用。当系统不稳定( [3] )或反稳定( [4] )时，Back-stepping方法可以用来实现镇定、观测器构造、输出调节等目的。

近年来，随着研究的深入，文献中出现很多PDE-ODE耦合系统相关的镇定工作。在 [5] 中，作者考虑了耦合ODE-弦系统的镇定问题，它模拟了塔吊上的平台和缆索的动力学行为。在 [6] 和 [7] 中Krstic等分别考虑ODE-弦和ODE-双曲型方程耦合系统。文献 [6] 的结果被推广到一个内部反阻尼ODE-系统和连通的诺伊曼系统中；另一个推广可以在文献 [8] 中找到，其中是应用于两个边值耦合的ODE-弦方程。上述的PDE-ODE耦合工作仅考虑了边界耦合系统。对于内部点耦合系统， [9] 对内部点耦合的二阶ODE-热系统设计了状态反馈控制器。类似的结果可以在 [1] 中找到，其所考察的是一维波系统。

在本文中，我们考虑以下耦合的二阶ODE-波系统：

$\{\begin{array}{l}{y}^{″}\left(t\right)=a^{2}y\left(t\right)+bu\left(x_0,t\right),t>0,\\ u_{tt}\left(x,t\right)=u_{xx}\left(x,t\right),x\in \left(0,1\right),t>0,\\ u\left(0,t\right)=0,u_x\left(1,t\right)=U\left(t\right),t\ge 0,\\ y_{out}\left(t\right)=\left\{y\left(t\right),u_t\left(1,t\right)\right\},t\ge 0,\\ y\left(0\right)=y_0,y^{\prime }\left(0\right)=y_1,\\ u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \left(0,1\right),\end{array}$ (1.1)

其中 $x_0\in \left(0,1\right)$ 是一个给定的中间点， $a>0$ 和 $b\ne 0$ 是常数， $y\left(t\right)$ 和 $u_t\left(1,t\right)$ 是测量值。设 $X\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right)\right)}^{{\rm T}}={\left(y\left(t\right),y^{\prime }\left(t\right)\right)}^{{\rm T}}$ ，那么我们可以把系统(1.1)写为如下形式：

$\{\begin{array}{l}\stackrel{\dot{}}{X}\left(t\right)=AX\left(t\right)+Bu\left(x_0,t\right),t>0,\\ u_{tt}\left(x,t\right)=u_{xx}\left(x,t\right),x\in \left(0,1\right),t>0,\\ u\left(0,t\right)=0,u_x\left(1,t\right)=U\left(t\right),t\ge 0,\\ y_{out}\left(t\right)=\left\{x_1\left(t\right),u_t\left(1,t\right)\right\},t\ge 0,\\ X\left(0\right)=X_0,u\left(x,0\right)=u_0\left(x\right),u_t\left(x,0\right)=u_1\left(x\right),x\in \left(0,1\right),\end{array}$ (1.2)

其中

$A=\left(\begin{array}{l}01\\ a^{2}0\end{array}\right),B=\left(\begin{array}{l}0\\ b\end{array}\right),X_0=\left(\begin{array}{l}{y}_0\\ y_1\end{array}\right).$

本文的目的是设计一个输出反馈控制器来使整个闭环系统稳定，而状态反馈定律在 [1] 中已经设计出来。由文献 [1] 的引理2.1我们可以知道存在一个矩阵K使得 $A+BK$ 是Hurwitz的。

文章结构如下：在第二部分中，我们通过系统的输出给出控制器的设计，通过Back-stepping变换，我们证明了闭环系统具有唯一的解；第三部分，我们用Back-stepping逆变换证明了闭环系统的指数稳定性。

2. 观测器和控制器设计

首先，我们设计系统(1.2)的观测器如下：

$\{\begin{array}{l}{\stackrel{\dot{}}{\hat{x}}}_1\left(t\right)={\hat{x}}_2\left(t\right)+q_1\left({\hat{x}}_1\left(t\right)-x_1\left(t\right)\right),t>0,\\ {\stackrel{\dot{}}{\hat{x}}}_2\left(t\right)=a^2{\hat{x}}_1\left(t\right)+q_2\left({\hat{x}}_1\left(t\right)-x_1\left(t\right)\right)+b\hat{u}\left(x_0,t\right),t>0,\\ {\hat{u}}_{tt}\left(x,t\right)={\hat{u}}_{xx}\left(x,t\right),t>0,\\ \hat{u}\left(0,t\right)=0,t\ge 0,\\ {\hat{u}}_x\left(1,t\right)=U\left(t\right)-k\left({\hat{u}}_t\left(1,t\right)-u_t\left(1,t\right)\right),t\ge 0,\\ {\hat{x}}_1\left(0\right)={\hat{x}}_{10},{\hat{x}}_2\left(0\right)={\hat{x}}_{20},\\ \hat{u}\left(x,0\right)={\hat{u}}_0\left(x\right),{\hat{u}}_t\left(x,0\right)={\hat{u}}_1\left(x\right),\end{array}$ (2.1)

其中 $k>0$ ， $q_1$ ， $q_2$ 是待定的设计参数。令 $\tilde{x}=\hat{x}-x$ 和 $\tilde{u}=\hat{u}-u$ 代表误差，则 $\tilde{x}$ 和 $\tilde{u}$ 满足

$\{\begin{array}{l}{\stackrel{\dot{}}{\tilde{x}}}_1\left(t\right)={\tilde{x}}_2\left(t\right)+q_1{\tilde{x}}_1\left(t\right),t>0,\\ {\stackrel{\dot{}}{\tilde{x}}}_2\left(t\right)=a^2{\tilde{x}}_1\left(t\right)+q_2{\tilde{x}}_1\left(t\right)+b\tilde{u}\left(x_0,t\right),t>0,\\ {\tilde{u}}_{tt}\left(x,t\right)={\tilde{u}}_{xx}\left(x,t\right),t>0,\\ \tilde{u}\left(0,t\right)=0,t\ge 0,\\ {\tilde{u}}_x\left(1,t\right)=-k{\tilde{u}}_t\left(1,t\right),t\ge 0,\\ {\tilde{x}}_1\left(0\right)={\hat{x}}_{10}-x_{10},{\tilde{x}}_2\left(0\right)={\hat{x}}_{20}-x_{20},\\ \tilde{u}\left(x,0\right)={\hat{u}}_0\left(x\right)-u_0\left(x\right),{\tilde{u}}_t\left(x,0\right)={\tilde{u}}_1\left(x\right)-u_1\left(x\right).\end{array}$ (2.2)

我们定义状态空间：

${\rm H}=\left\{{\left(f,g\right)}^{{\rm T}}\in H^1\left(0,1\right)\times L^2\left(0,1\right)|f\left(0\right)=0\right\}$

内积诱导的范数如下：

${\Vert {\left(f,g\right)}^{{\rm T}}\Vert }_{{\rm H}}^2={\displaystyle {\int }_0^1\left({\left|f^{\prime }\left(x\right)\right|}^2+{\left|g\left(x\right)\right|}^2\right)\text{d}x}.$

众所周知， $\tilde{u}$ -部分在状态空间中具有唯一的指数稳定解。即在此范数意义下，存在常数 $M>0$ ， $\delta >0$ ，有

${\Vert {\left(\tilde{u}\left(\cdot ,t\right),{\tilde{u}}_t\left(\cdot ,t\right)\right)}^{{\rm T}}\Vert }_{{\rm H}}\le M{\text{e}}^{-\delta t}{\Vert {\left(\tilde{u}\left(\cdot ,0\right),{\tilde{u}}_t\left(\cdot ,0\right)\right)}^{{\rm T}}\Vert }_{{\rm H}}.$ (2.3)

利用庞加莱不等式，我们有

$\left|\tilde{u}\left(x_0,t\right)\right|\le {\Vert {\left(\tilde{u}\left(\cdot ,t\right),{\tilde{u}}_t\left(\cdot ,t\right)\right)}^{{\rm T}}\Vert }_{{\rm H}}\le M{\text{e}}^{-\delta t}{\Vert {\left(\tilde{u}\left(\cdot ,0\right),{\tilde{u}}_t\left(\cdot ,0\right)\right)}^{{\rm T}}\Vert }_{{\rm H}}.$ (2.4)

对于系统(2.2)中的ODE-部分，这里存在唯一解

${\left({\tilde{x}}_1,{\tilde{x}}_2\right)}^{{\rm T}}={\text{e}}^{A_{1}t}{\left({\tilde{x}}_1\left(0\right),{\tilde{x}}_2\left(0\right)\right)}^{{\rm T}}+{\displaystyle {\int }_0^t{\text{e}}^{A_1\left(t-s\right)}B\tilde{u}\left(x_0,s\right)\text{d}s},$

其中

$A_1=\left(\begin{array}{l}{q}_{1}1\\ a^2+q_{2}0\end{array}\right),$ (2.5)

$q_1<0$ ， $q_2<-a^2$ 是可调常数。很明显，对于这样选择的参数， $A_1$ 是Hurwitz的。从而 ${\left({\tilde{x}}_1,{\tilde{x}}_2\right)}^{{\rm T}}$ 是指数稳定的。因此我们有如下引理：

引理2.1：假设 $q_1<0$ ， $q_2<-a^2$ ， $k>0$ ，那么对任意的初值 ${\left({\tilde{x}}_{10},{\tilde{x}}_{20},{\tilde{u}}_0,{\tilde{u}}_1\right)}^{{\rm T}}\in {\text{C}}^2\times {\rm H}$ ，系统(2.2)都有唯一的弱解 ${\left({\tilde{x}}_1\left(t\right),{\tilde{x}}_2\left(t\right),\tilde{u}\left(\cdot ,t\right),{\tilde{u}}_t\left(\cdot ,t\right)\right)}^{{\rm T}}\in C\left(0,\infty ;{\text{C}}^2\times {\rm H}\right)$ ，并且系统(2.2)的解是指数稳定的。

受文献 [1] 的状态反馈控制器的启发，我们设计输出反馈控制器如下：

$\begin{array}{l}U\left(t\right)={\displaystyle {\int }_0^1\left[c{l}_{yy}\left(1,y\right)+k_x\left(1,y\right)\right]\hat{u}\left(y,t\right)\text{d}y}-c{l}_y\left(1,1\right)\hat{u}\left(1,t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[{p^{\prime }}_1\left(1\right)q_1\left(y\right)+cp\left(1\right)q\left(y\right)\right]{\hat{u}}_t\left(y,t\right)\text{d}y}+{\varphi }^{\prime }\left(1\right)\hat{X}\left(t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[p^{\prime }\left(1\right)q\left(y\right)+c{p}_1\left(1\right){q^{″}}_1\left(y\right)\right]\hat{u}\left(y,t\right)\text{d}y}+c\varphi \left(1\right)A\hat{X}\left(t\right)\\ +{\displaystyle {\int }_0^1\left[l_x\left(1,y\right)+ck\left(1,y\right)\right]{\hat{u}}_t\left(y,t\right)\text{d}y}+c{u}_t\left(1,t\right),\end{array}$ (2.6)

其中 $c>0$ 是给定常数， $k\left(x,y\right)$ ， $l\left(x,y\right)$ ， $p\left(x\right)$ ， $p_1\left(x\right)$ ， $q\left(y\right)$ ， $q_1\left(y\right)$ 和 $\varphi \left(x\right)$ 满足

$\{\begin{array}{l}k\left(x,y\right)=-\xi \eta \left(1-{\text{e}}^{2\lambda x_0}\right)\left({\text{e}}^{\lambda \left(x-y\right)}-{\text{e}}^{-\lambda \left(x-y\right)}\right)),\\ l\left(x,y\right)=-{\xi }_1{\eta }_1\left(1-{\text{e}}^{2{\lambda }_1{x}_0}\right)\left({\text{e}}^{{\lambda }_1\left(x-y\right)}-{\text{e}}^{-{\lambda }_1\left(x-y\right)}\right),\\ p\left(x\right)=\xi {\text{e}}^{\lambda x}-\xi {\text{e}}^{-\lambda x},\\ q\left(y\right)=\eta {\text{e}}^{\lambda y}-\eta {\text{e}}^{-\lambda \left(2x_0-y\right)},\\ p_1\left(x\right)={\xi }_1{\text{e}}^{{\lambda }_{1}x}-{\xi }_1{\text{e}}^{-{\lambda }_{1}x},\\ q_1\left(y\right)={\eta }_1{\text{e}}^{{\lambda }_{1}y}-{\eta }_1{\text{e}}^{-{\lambda }_1\left(2x_0-y\right)},\\ \varphi \left(x\right)=\frac{{\text{e}}^{{\lambda }_{1}x}-{\text{e}}^{-{\lambda }_{1}x}}{{\text{e}}^{{\lambda }_1{x}_0}-{\text{e}}^{-{\lambda }_1{x}_0}}K,\\ \xi \eta =\frac{KAB}{2\lambda {\text{e}}^{\lambda x_0}\left({\text{e}}^{\lambda x_0}-{\text{e}}^{-\lambda x_0}\right)},\\ {\xi }_1{\eta }_1=\frac{KB}{2{\lambda }_1{\text{e}}^{{\lambda }_1{x}_0}\left({\text{e}}^{{\lambda }_1{x}_0}-{\text{e}}^{-{\lambda }_1{x}_0}\right)}\\ \lambda ={\lambda }_1=a.\end{array}$ (2.7)

这些函数在文献 [1] 中方程(16)~(20)已给出。

在控制器(2.6)之下，我们得到系统(1.2)的闭环系统为：

$\{\begin{array}{l}\stackrel{\dot{}}{X}\left(t\right)=AX\left(t\right)+Bu\left(x_0,t\right),t>0,\\ u_{tt}\left(x,t\right)=u_{xx}\left(x,t\right),x\in \left(0,1\right),t>0,\\ u\left(0,t\right)=0,t>0,\\ u_x\left(1,t\right)={\displaystyle {\int }_0^1\left[c{l}_{yy}\left(1,y\right)+k_x\left(1,y\right)\right]\hat{u}\left(y,t\right)\text{d}y}-c{l}_y\left(1,1\right)\hat{u}\left(1,t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[{p^{\prime }}_1\left(1\right)q_1\left(y\right)+cp\left(1\right)q\left(y\right)\right]{\hat{u}}_t\left(y,t\right)\text{d}y}+{\varphi }^{\prime }\left(1\right)\hat{X}\left(t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[p^{\prime }\left(1\right)q\left(y\right)+c{p}_1\left(1\right){q^{″}}_1\left(y\right)\right]\hat{u}\left(y,t\right)\text{d}y}+c\varphi \left(1\right)A\hat{X}\left(t\right)\\ +{\displaystyle {\int }_0^1\left[l_x\left(1,y\right)+ck\left(1,y\right)\right]{\hat{u}}_t\left(y,t\right)\text{d}y}+c{u}_t\left(1,t\right),t>0,\\ {\stackrel{\dot{}}{\hat{x}}}_1\left(t\right)={\hat{x}}_2\left(t\right)+q_1\left({\hat{x}}_1\left(t\right)-x_1\left(t\right)\right),t>0,\\ {\stackrel{\dot{}}{\hat{x}}}_2\left(t\right)=a^2{\hat{x}}_1\left(t\right)+q_2\left({\hat{x}}_1\left(t\right)-x_1\left(t\right)\right)+b\hat{u}\left(x_0,t\right),t>0,\\ {\hat{u}}_{tt}\left(x,t\right)={\hat{u}}_{xx}\left(x,t\right),t>0,\\ \hat{u}\left(0,t\right)=0,t\ge 0,\\ {\hat{u}}_x\left(1,t\right)=u_x\left(1,t\right)-k\left({\hat{u}}_t\left(1,t\right)-u_t\left(1,t\right)\right),t\ge 0,\end{array}$ (2.8)

这里为了简便我们省略初值。

3. 闭环系统适定性与稳定性

引入误差变量 $\tilde{u}=\hat{u}-u$ 和 $\tilde{X}\left(t\right)={\left({\tilde{x}}_1\left(t\right),{\tilde{x}}_2\left(t\right)\right)}^{{\rm T}}$ ，我们可以重写系统(2.8)如下：

$\{\begin{array}{l}\stackrel{\dot{}}{X}\left(t\right)=AX\left(t\right)+Bu\left(x_0,t\right),t>0,\\ u_{tt}\left(x,t\right)=u_{xx}\left(x,t\right),x\in \left(0,1\right),t>0,\\ u\left(0,t\right)=0,t>0,\\ u_x\left(1,t\right)={\displaystyle {\int }_0^1\left[c{l}_{yy}\left(1,y\right)+k_x\left(1,y\right)\right]u\left(y,t\right)\text{d}y}-c{l}_y\left(1,1\right)u\left(1,t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[{p^{\prime }}_1\left(1\right)q_1\left(y\right)+cp\left(1\right)q\left(y\right)\right]u_t\left(y,t\right)\text{d}y}+{\varphi }^{\prime }\left(1\right)X\left(t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[p^{\prime }\left(1\right)q\left(y\right)+c{p}_1\left(1\right){q^{″}}_1\left(y\right)\right]u\left(y,t\right)\text{d}y}+c\varphi \left(1\right)AX\left(t\right)\\ +{\displaystyle {\int }_0^1\left[l_x\left(1,y\right)+ck\left(1,y\right)\right]u_t\left(y,t\right)\text{d}y}+c{u}_t\left(1,t\right)\\ +{\displaystyle {\int }_0^1\left[c{l}_{yy}\left(1,y\right)+k_x\left(1,y\right)\right]\tilde{u}\left(y,t\right)\text{d}y}-c{l}_y\left(1,1\right)\tilde{u}\left(1,t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[{p^{\prime }}_1\left(1\right)q_1\left(y\right)+cp\left(1\right)q\left(y\right)\right]{\tilde{u}}_t\left(y,t\right)\text{d}y}+{\varphi }^{\prime }\left(1\right)\tilde{X}\left(t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[p^{\prime }\left(1\right)q\left(y\right)+c{p}_1\left(1\right){q^{″}}_1\left(y\right)\right]\tilde{u}\left(y,t\right)\text{d}y}+c\varphi \left(1\right)A\tilde{X}\left(t\right)\\ +{\displaystyle {\int }_0^1\left[l_x\left(1,y\right)+ck\left(1,y\right)\right]{\tilde{u}}_t\left(y,t\right)\text{d}y},t>0,\\ {\stackrel{\dot{}}{\tilde{x}}}_1\left(t\right)={\tilde{x}}_2\left(t\right)+q_1{\tilde{x}}_1\left(t\right),t>0,\\ {\stackrel{\dot{}}{\tilde{x}}}_2\left(t\right)=a^2{\tilde{x}}_1\left(t\right)+q_2{\tilde{x}}_1\left(t\right)+b\tilde{u}\left(x_0,t\right),t>0,\\ {\tilde{u}}_{tt}\left(x,t\right)={\tilde{u}}_{xx}\left(x,t\right),t>0,\\ \tilde{u}\left(0,t\right)=0,t\ge 0,\\ {\tilde{u}}_x\left(1,t\right)=-k{\tilde{u}}_t\left(1,t\right),t\ge 0.\end{array}$ (3.1)

由引理2.1可知， ${\left(\tilde{X},\tilde{u},{\tilde{u}}_t\right)}^{{\rm T}}$ 是指数稳定的。对于 ${\left(X,u,u_t\right)}^{{\rm T}}$ ，我们引用 [1] 中的Back-stepping变换(3)

$\begin{array}{c}w\left(x,t\right)=u\left(x,t\right)-{\displaystyle {\int }_0^{x}k\left(x,y\right)u\left(y,t\right)\text{d}y}-{\displaystyle {\int }_0^{x}l\left(x,y\right)u_t\left(y,t\right)\text{d}y}\\ -p\left(x\right){\displaystyle {\int }_0^{x_0}q\left(y\right)u\left(y,t\right)\text{d}y}-p_1\left(x\right){\displaystyle {\int }_0^{x_0}{q}_1\left(y\right)u_t\left(y,t\right)\text{d}y}-\varphi \left(x\right)X\left(x\right).\end{array}$ (3.2)

于是系统(3.1)中 ${\left(X,u,u_t\right)}^{{\rm T}}$ -部分满足

$\{\begin{array}{l}\stackrel{\dot{}}{X}\left(t\right)=\left(A+BK\right)X\left(t\right)+Bw\left(x_0,t\right),t>0,\\ w_{tt}\left(x,t\right)=w_{xx}\left(x,t\right),x\in \left(0,1\right),t>0,\\ w\left(0,t\right)=0,t>0,\\ w_x\left(1,t\right)=-c{w}_t\left(1,t\right)+f\left(t\right),t>0,\\ w\left(x,0\right)=w_0\left(x\right),w_t\left(x,0\right)=w_1\left(x\right),x\in \left(0,1\right),\\ X\left(0\right)=X_0,\end{array}$ (3.3)

其中

$\begin{array}{c}f\left(t\right)={\displaystyle {\int }_0^1\left[c{l}_{yy}\left(1,y\right)+k_x\left(1,y\right)\right]\tilde{u}\left(y,t\right)\text{d}y}-c{l}_y\left(1,1\right)\tilde{u}\left(1,t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[{p^{\prime }}_1\left(1\right)q_1\left(y\right)+cp\left(1\right)q\left(y\right)\right]{\tilde{u}}_t\left(y,t\right)\text{d}y}+{\varphi }^{\prime }\left(1\right)\tilde{X}\left(t\right)\\ +{\displaystyle {\int }_0^{x_0}\left[p^{\prime }\left(1\right)q\left(y\right)+c{p}_1\left(1\right){q^{″}}_1\left(y\right)\right]\tilde{u}\left(y,t\right)\text{d}y}+c\varphi \left(1\right)A\tilde{X}\left(t\right)\\ +{\displaystyle {\int }_0^1\left[l_x\left(1,y\right)+ck\left(1,y\right)\right]{\tilde{u}}_t\left(y,t\right)\text{d}y}.\end{array}$ (3.4)

由引理2.1知 $f\left(t\right)$ 指数衰减。也就是说，存在常数 $M_1>0$ ， ${\delta }_1>0$ ，其中 ${\delta }_1$ 是独立于初值，使得

$\left|f\left(t\right)\right|\le M_1{\text{e}}^{-{\delta }_{1}t},t\ge 0.$

我们在能量空间H中考虑系统(3.3)的PDE部分

$\{\begin{array}{l}{w}_{tt}\left(x,t\right)=w_{xx}\left(x,t\right),x\in \left(0,1\right),t>0,\\ w\left(0,t\right)=0,t>0,\\ w_x\left(1,t\right)=-c{w}_t\left(1,t\right)+f\left(t\right),t>0,\\ w\left(x,0\right)=w_0\left(x\right),w_t\left(x,0\right)=w_1\left(x\right),x\in \left(0,1\right),\\ X\left(0\right)=X_0.\end{array}$ (3.5)

定义系统(3.5)的算子： ${\rm A}:D\left({\rm A}\right)\to {\rm H}$

$\{\begin{array}{l}{\rm A}{\left(f,g\right)}^{{\rm T}}={\left(g,f^{″}\right)}^{{\rm T}},\forall {\left(f,g\right)}^{{\rm T}}\in D\left({\rm A}\right),\\ D\left({\rm A}\right)=\left\{{\left(f,g\right)}^{{\rm T}}\in H^2\left(0,1\right)\times H^1\left(0,1\right)|{\rm A}{\left(f,g\right)}^{{\rm T}}\in {\rm H},f^{\prime }\left(1\right)=-cg\left(1\right)\right\}.\end{array}$ (3.6)

简单计算表明A的对偶算子为

$\{\begin{array}{l}{\text{A}}^{\ast }\left(\begin{array}{l}\varphi \\ \psi \end{array}\right)=\left(\begin{array}{l}-{\psi }_1\\ -{\varphi }^{″}\end{array}\right),\forall \left(\begin{array}{l}\varphi \\ \psi \end{array}\right)\in D({\text{A}}^{\ast }),\\ D\left({\text{A}}^{\ast }\right)=\left\{{\left(\varphi ,\psi \right)}^{{\rm T}}\in H^2\left(0,1\right)\times H^1\left(0,1\right)|{\text{A}}^{\ast }\left(\varphi ,\psi \right)\in \text{H},{\varphi }^{\prime }\left(1\right)=c\psi \left(1\right)\right\}.\end{array}$ (3.7)

系统(3.7)中 $\left(\varphi ,\psi \right)\in D\left({{\rm A}}^{\ast }\right)$ 与 $\left(w,w_t\right)$ 做内积得到

$\frac{\text{d}}{\text{d}t}\langle \left(\begin{array}{l}w\\ w_t\end{array}\right),\left(\begin{array}{l}\varphi \\ \psi \end{array}\right)\rangle =\langle \left(\begin{array}{l}w\\ w_t\end{array}\right),{{\rm A}}^{\ast }\left(\begin{array}{l}\varphi \\ \psi \end{array}\right)\rangle +\langle \left(\begin{array}{l}0\\ \delta \left(x-1\right)\end{array}\right)f\left(t\right),\left(\begin{array}{l}\varphi \\ \psi \end{array}\right)\rangle ,$ (3.8)

其中 $\delta (\cdot )$ 表示狄拉克分布。因此，系统(3.5)可以写成H上的抽象发展方程

$\frac{\text{d}}{\text{d}t}{\left(w\left(\cdot ,t\right),w_t\left(\cdot ,t\right)\right)}^{{\rm T}}={\rm A}{\left(w\left(\cdot ,t\right),w_t\left(\cdot ,t\right)\right)}^{{\rm T}}+{\rm B}f\left(t\right),$ (3.9)

这里 ${\rm B}={\left(0,\delta \left(x-1\right)\right)}^{{\rm T}}$ 。众所周知，算子A在H上生成指数稳定的C₀-半群，也就是说，存在 $K>0$ ， $\mu >0$ 使得

$\Vert {\text{e}}^{{\rm A}t}\Vert \le K{\text{e}}^{-\mu t},t\ge 0.$ (3.10)

接下来我们将证明算子B对 ${\text{e}}^{{\rm A}t}$ 是允许的，只要证明 ${{\rm B}}^{\ast }$ 对 ${\text{e}}^{{{\rm A}}^{\ast }t}$ 是允许的即可。系统(3.9)的对偶系统是

$\{\begin{array}{l}{w}_{tt}^{\ast }\left(x,t\right)=w_{xx}^{\ast }\left(x,t\right),x\in \left(0,1\right),t>0,\\ w^{\ast }\left(0,t\right)=0,t>0,\\ w_x^{\ast }\left(1,t\right)=-c{w}_t^{\ast }\left(1,t\right),t>0,\\ y_0\left(t\right)=-w_t^{\ast }\left(1,t\right),t\ge 0.\end{array}$ (3.11)

因为A生成C₀-半群，所以 ${{\rm A}}^{\ast }$ 也能生成C₀-半群，即系统(3.11)存在C₀-半群解。对系统(3.11)定义能量函数如下：

$E^{\ast }\left(t\right)=\frac{1}{2}{\displaystyle {\int }_0^1\left(w_t^{\ast 2}\left(x,t\right)+w_x^{\ast 2}\left(x,t\right)\right)\text{d}x}.$

将 $E\left(t\right)$ 沿系统(3.11)的解对t求导得到

$\stackrel{\dot{}}{E}\left(t\right)=-c{\left[w_t^{\ast }\left(1,t\right)\right]}^2.$ (3.12)

对(3.12)中t从0到T积分，我们有

${\displaystyle {\int }_0^T{\left[w_t^{\ast }\left(1,t\right)\right]}^2\text{d}t}=\frac{1}{c}\left(E\left(0\right)-E\left(T\right)\right)\le \frac{1}{c}E\left(0\right).$

另一方面，直接计算可得

$\{\begin{array}{l}{{\rm A}}^{\ast -1}\left(\begin{array}{l}\varphi \\ \psi \end{array}\right)=\left(\begin{array}{c}-cx\varphi \left(1\right)+{\displaystyle {\int }_0^x{\displaystyle {\int }_s^1\psi \left(t\right)\text{d}t\text{d}s}}\\ \varphi \end{array}\right)，\\ {{\rm B}}^{\ast }{{\rm A}}^{\ast -1}\left(\begin{array}{l}\varphi \\ \psi \end{array}\right)=\left(\begin{array}{c}0\\ -\varphi \left(1\right)\end{array}\right),\forall {\left(\varphi ,\psi \right)}^{\prime }\in {\rm H}.\end{array}$ (3.13)

给出简单估计

${\left|-\phi \left(1\right)\right|}^2={\phi }^2\left(1\right)\le {\displaystyle {\int }_0^1{{\phi }^{\prime }}^2\left(x\right)\text{d}x}\le {\Vert \left(\phi ,\psi \right)\Vert }_{{\rm H}}^2.$ (3.14)

因此，从H到C， ${{\rm B}}^{\ast }{{\rm A}}^{\ast -1}$ 是有界的。这表明 ${{\rm B}}^{\ast }$ 对 ${\text{e}}^{{{\rm A}}^{\ast }t}$ 是允许的，进而B对 ${\text{e}}^{{\rm A}t}$ 是允许的。通过适定的线性无穷维系统理论( [10] )，若 $f\in L_{loc}^2\left(0,\infty \right)$ ，那么(3.9)存在唯一解 $\left(w,w_t\right)\in C\left(0,\infty ;{\rm H}\right)$ 。

(3.5)的解可以写成( [10] )

${\left(w\left(\cdot ,t\right),w_t\left(\cdot ,t\right)\right)}^{{\rm T}}={\text{e}}^{{\rm A}t}{\left(w_0,w_1\right)}^{{\rm T}}+{\displaystyle {\int }_0^t{\text{e}}^{{\rm A}\left(t-s\right)}{\rm B}f\left(s\right)\text{d}s}.$

与文献 [11] 中方程(60)的估计类似，对不依赖于初值的常数 $M_2>0$ ， ${\delta }_2>0$ ，我们有

${\Vert {\left(w\left(\cdot ,t\right),w_t\left(\cdot ,t\right)\right)}^{{\rm T}}\Vert }_{\text{H}}\le M_2{\text{e}}^{-{\delta }_{2}t}{\Vert {\left(w\left(\cdot ,0\right),w_t\left(\cdot ,0\right)\right)}^{{\rm T}}\Vert }_{\text{H}},t\ge 0.$ (3.15)

根据庞加莱不等式

$\left|w\left(x_0,t\right)\right|\le {\Vert {\left(w\left(\cdot ,t\right),w_t\left(\cdot ,t\right)\right)}^{{\rm T}}\Vert }_{{\rm H}}\le M_2{\text{e}}^{-{\delta }_{2}t}{\Vert {\left(w\left(\cdot ,0\right),w_t\left(\cdot ,0\right)\right)}^{{\rm T}}\Vert }_{{\rm H}}.$

因此，ODE部分的解

$X\left(t\right)={\text{e}}^{\left(A+BK\right)t}{X}_0+{\displaystyle {\int }_0^t{\text{e}}^{\left(A+BK\right)\left(t-s\right)}Bw\left(x_0,s\right)\text{d}s},$ (3.16)

是指数稳定的。

下面我们说明变换(3.2)是可逆的。实际上其逆变换可以写为

$\begin{array}{c}u\left(x,t\right)=w\left(x,t\right)-{\displaystyle {\int }_0^{x}n\left(x,y\right)w\left(y,t\right)\text{d}y}-{\displaystyle {\int }_0^{x}m\left(x,y\right)w_t\left(y,t\right)\text{d}y}\\ -g\left(x\right){\displaystyle {\int }_0^{x_0}h\left(y\right)w\left(y,t\right)\text{d}y}-g_1\left(x\right){\displaystyle {\int }_0^{x_0}{h}_1\left(y\right)w_t\left(y,t\right)\text{d}y}-\psi \left(x\right)X\left(t\right)，\end{array}$ (3.17)

其中函数 $m\left(x,y\right)$ ， $n\left(x,y\right)$ ， $g\left(x\right)$ ， $g_1\left(x\right)$ ， $h\left(y\right)$ ， $h_1\left(y\right)$ 和 $\psi \left(x\right)$ 满足

$\{\begin{array}{l}{n}_{xx}\left(x,y\right)-n_{yy}\left(x,y\right)=0,\\ n\left(x,x\right)=0,n\left(x,0\right)=-g\left(x\right)h\left(0\right),\\ m_{xx}\left(x,y\right)-m_{yy}\left(x,y\right)=0,\\ m\left(x,x\right)=0,m\left(x,0\right)=-g_1\left(x\right)h_1\left(0\right),\\ g^{″}\left(x\right)h\left(y\right)-g\left(x\right)h^{″}\left(y\right)=0,\\ h\left(x_0\right)=0,g\left(x\right)h^{\prime }\left(x_0\right)=\psi \left(x\right)\left(A+BK\right)B,\\ g\left(0\right)=0,n\left(x_0,y\right)+g\left(x\right)h\left(y\right)=0,\\ {g^{″}}_1\left(x\right)h_1\left(y\right)-g_1\left(x\right){h^{″}}_1\left(y\right)=0,\\ g_1\left(x_0\right)=0,g_1\left(x\right){h^{\prime }}_1\left(x_0\right)=\psi \left(x\right)B,\\ g_1\left(0\right)=0,m\left(x_0,y\right)+g_1\left(x_0\right)h_1\left(y\right)=0,\\ {\psi }^{″}\left(x\right)-\psi \left(x\right){\left(A+BK\right)}^2=0,\\ \psi \left(x_0\right)=K,\psi \left(0\right)=0,\end{array}$ (3.18)

这些函数在 [1] 中均有解。

我们证明了系统(3.3)有唯一解且解是指数稳定的。我们得到了本文的主要结果。

定理3.1:假设 $q_1<0$ ， $q_2<-a^2$ ， $k>0$ ，那么对任意的初值 ${\left(X_0,u_0,u_1,{\hat{X}}_0,{\hat{u}}_0,{\hat{u}}_1\right)}^{{\rm T}}\in {\left({\text{C}}^2\times {\rm H}\right)}^2$ ，闭环系统(2.8)都有唯一的弱解 ${\left(X\left(t\right),u\left(\cdot ,t\right),u_t\left(\cdot ,t\right),\hat{X}\left(t\right),\hat{u}\left(\cdot ,t\right),{\hat{u}}_t\left(\cdot ,t\right)\right)}^{{\rm T}}\in C\left(0,\infty ;{\left({\text{C}}^2\times {\rm H}\right)}^2\right)$ ，且解是指数稳定的：

${\Vert \left(X\left(t\right),u\left(\cdot ,t\right),u_t\left(\cdot ,t\right),\hat{X}\left(t\right),\hat{u}\left(\cdot ,t\right),{\hat{u}}_t\left(\cdot ,t\right)\right)\Vert }_{{\left(\text{C}\times {\rm H}\right)}^2}\le L{\text{e}}^{-\omega t}{\Vert \left(X_0,u_0,u_1,{\hat{X}}_0,{\hat{u}}_0,{\hat{u}}_1\right)\Vert }_{{\left(\text{C}\times {\rm H}\right)}^2},\forall t>0,$ (3.19)

其中L和 $\omega$ 是不依赖于初值的正数。