ODE-波动方程耦合系统半离散降阶有限差分格式的一致指数稳定性

Uniformly Exponential Stability for the Semi-Discrete Order Reduction Finite Difference Schemes of ODE-Wave Equation Coupled System

摘要: 本文研究了由线性常微分方程和一维波动方程耦合系统的一致指数稳定性，对其构建了一个半离散有限差分格式，并进一步研究了离散格式的一致指数稳定性。首先采用Backstepping控制法将初始耦合系统转化为一个等价的目标系统，同时推导出使原耦合系统稳定的控制律。接着为目标系统构造适当的Lyapunov函数，采用Lyapunov稳定性分析法证明目标系统的一致指数稳定性。为了更方便地研究目标系统对应的空间半离散有限差分格式，先通过变量代换将目标系统简化为与之等价的低阶耦合系统，并针对该低阶系统构造空间半离散有限差分格式，设计相应的Lyapunov函数，证明了半离散系统的一致指数稳定性，从而表明所提出的基于空间半离散的有限差分格式能够有效保持原连续耦合系统的指数稳定特性。

Abstract: This paper analyzes the uniform exponential stability of a coupled system consisting of linear ordinary differential equations and a one-dimensional wave equation. A semi-discrete finite difference scheme is constructed for this system, and the uniform exponential stability of the discrete scheme is further studied. Firstly, the Backstepping control method is employed to transform the original coupled system into an equivalent target system, while deriving the control law that stabilizes the original coupled system. Subsequently, an appropriate Lyapunov function is constructed for the target system, and the Lyapunov stability analysis method is used to prove the uniform exponential stability of the target system. In order to more conveniently study the spatially semi-discrete finite difference scheme corresponding to the target system, the target system is first simplified into an equivalent lower-order coupled system through variable substitution. A spatially semi-discrete finite difference scheme is then constructed for this lower-order system, and a corresponding Lyapunov function is designed to prove the uniform exponential stability of the semi-discrete system. This demonstrates that the proposed spatially semi-discrete finite difference scheme effectively preserves the exponential stability characteristics of the original continuous coupled system.

1. 引言

在控制系统中常常会出现耦合系统，其中常微分方程组与偏微分方程组的耦合是一种基本的耦合方式。近几十年来关于耦合系统的控制器设计问题越来越受到学者们的关注，目前常用的一种方式是利用Backstepping控制法将原始系统转化为相应的目标系统，且通过设计出合适的控制律来得到一个稳定的耦合控制系统。Susto G A [1]等人为一系列常微分方程与Neumann边界条件的热方程或者波动方程的耦合系统设计出了可应用的控制率。文献研究了ODE-反应–扩散耦合系统的边界反馈控制问题[2]。文献在[3]中对ODE-KdV在低阶处耦合的闭环系统的稳定性进行了研究，随后Yang Y H等人为ODE-KdV在二阶导数处耦合的系统设计了稳定控制律[4]。还有大量学者更进一步研究了带扰动[5]或者带任意时滞[6]的ODE-PDE耦合系统的控制问题。但对于耦合系统的离散问题，相关研究成果目前仍较少。

关于复杂边界值问题的数值求解已有了丰硕的研究成果。文献中用混合有限元法研究了波动方程的可控性问题[7]，这种基于传统有限元法的改造使得构造更加简单，文献中用有限体积法证明了边界带导数的梁方程的指数稳定性[8]。在所有的数值逼近方法中，有限差分法因为其原理简单、使用广泛，成为了当下使用最普遍的一种数值离散方式。在大多数情况下，只对控制系统的空间变量进行离散而保持时间变量连续，通过这种方法，就可以将无限维ODE系统和无限维PDE系统转化成一个ODE方程组问题。在进行半离散的过程中，一个很关键的问题是离散系统是否能够保留连续系统的一致指数稳定性等重要性质。众所周知，当考虑空间步长的时候，由于在离散过程中可能会产生在连续层级不会出现的高频振荡，所以经典有限差分格式不一定可以保持原系统的指数衰减性[9]。为了修正这个问题，学者们相继研究出了滤波法[10]、双重网格法[11]等方法。Sorin Micu通过引入数值阻尼项得到了一维边界耗散波动方程具备一致指数稳定特性的半离散差分格式[12]，Bouslous H等人将这种方法进一步推广到二维边界耗散波动方程[13]，并得到相应的空间半离散有限差分格。Liu等人对一维边界控制波动方程提出一种基于降阶法的有限差分格式[14]，由于变量代换是可逆的，所以降阶后的系统与原系统是完全等价的，而前者的阶数更低，这种方法的优势是一方面误差只产生于离散过程中用低阶差分算子代替微分算子的过程，这与一般的半离散有限差分法是完全一致的，因此并未产生其他多余的误差，另一方面相比数值粘性法，这种方法无需引入新的粘性项，从而简化了证明过程。而后这种思想又相继被运用到薛定谔方程[15]、带边界阻尼的波动方程[16]以及带尖端质量的波动方程中[17]。

2. 记号及引理

2.1. 记号

首先将本文范数定义如下，设$X\left(t\right)={\left(x^1\left(t\right),x^2\left(t\right),\cdots ,x^n\left(t\right)\right)}^{\text{T}}$ 是欧几里得空间上的n维向量值函数，其欧几里得范数为

$\left|X\left(t\right)\right|={\left(X^{\text{T}}\left(t\right)X\left(t\right)\right)}^{\frac{1}{2}},$

定义在$L^2\left(0,1\right)$ 的函数$u\left(x,t\right)$ 的$L^2$ 范数为

$\Vert u\left(x,t\right)\Vert ={\left({\displaystyle {\int }_0^1{u}^2\left(x,t\right)\text{d}x}\right)}^{\frac{1}{2}},$

令$\mathscr{H}$ $={ℝ}^n\times H^1\left(0,1\right)\times L^2\left(0,1\right)$ 为系统的能量状态空间，其连续形式和离散形式的范数分别定义为

$\begin{array}{c}{\Vert \left(X\left(t\right),w_x\left(x,t\right),w_t\left(x,t\right)\right)\Vert }_{\mathscr{H}}={\left({\left|X\left(t\right)\right|}^2+{\Vert w_x\left(x,t\right)\Vert }^2+{\Vert w_t\left(x,t\right)\Vert }^2\right)}^{\frac{1}{2}}\\ ={\left({\left|X\left(t\right)\right|}^2+{\displaystyle {\int }_0^1{w}_x^2\left(x,t\right)\text{d}x}+{\displaystyle {\int }_0^1{w}_t^2\left(x,t\right)\text{d}x}\right)}^{\frac{1}{2}},\end{array}$

以及

${\Vert \left(X\left(t\right),u\left(x,t\right),v\left(x,t\right)\right)\Vert }_h={\left({\left|X\left(t\right)\right|}^2+h{\displaystyle \sum _{j=0}^N{u}_j^2\left(t\right)+h{\displaystyle \sum _{j=0}^N{v}_j^2\left(t\right)}}\right)}^{\frac{1}{2}}.$

为了叙述方便，引入如下记号。若用${\left\{U_j\right\}}_j$ 表示在${\left\{x_j\right\}}_j$ 处满足$U_j\left(t\right)=u\left(x_j,t\right)$ 的网格函数，那么中心算子、一阶差分算子和二阶差分算子可分别表示为

$U_{j+\frac{1}{2}}\left(t\right)=\frac{U_{j+1}\left(t\right)+U_j\left(t\right)}{2},$

${\delta }_x{U}_{j+\frac{1}{2}}\left(t\right)=\frac{U_{j+1}\left(t\right)-U_j\left(t\right)}{h},$

${\delta }_x^2{U}_j\left(t\right)=\frac{{\delta }_x{U}_{j+\frac{1}{2}}\left(t\right)-{\delta }_x{U}_{j-\frac{1}{2}}\left(t\right)}{h}=\frac{U_{j+1}-2U_j+U_{j-1}}{h^2}.$

2.2. 引理

引理2.1 [14] (Gronwall不等式)令$y\left(t\right)$ 是$y\left(t\right)$ 上的非负且绝对连续的函数，并且满足如下不等式

$y^{\prime }\left(t\right)\le g\left(t\right)y\left(t\right),\forall t\in \left(0,\infty \right),$

且$g\left(t\right)$ 是$\left[0,\infty \right]$ 上的非负可积函数，那么有

$y\left(t\right)\le y\left(0\right){\text{e}}^{{\displaystyle {\int }_0^{t}g\left(s\right)\text{d}s}},\forall t\in \left(0,\infty \right).$

引理2.2 [14]对于定义在${\left\{x_j\right\}}_j$ 处的网格函数${\left\{W_j\right\}}_j$ 、${\left\{U_j\right\}}_j$ 和${\left\{V_j\right\}}_j$ 下面两个求和式都成立

$h{\displaystyle \sum _{j=0}^N{\delta }_x}{U}_{j+\frac{1}{2}}{V}_{j+\frac{1}{2}}+h{\displaystyle \sum _{j=0}^N{\delta }_x}{V}_{j+\frac{1}{2}}{U}_{j+\frac{1}{2}}=U_{N+1}{V}_{N+1}-U_0{V}_0,$

$\begin{array}{l}h{\displaystyle \sum _{j=0}^N{\delta }_x}{U}_{j+\frac{1}{2}}{V}_{j+\frac{1}{2}}{W}_{j+\frac{1}{2}}+h{\displaystyle \sum _{j=0}^N{U}_{j+\frac{1}{2}}}{\delta }_x{V}_{j+\frac{1}{2}}{W}_{j+\frac{1}{2}}+h{\displaystyle \sum _{j=0}^N{U}_{j+\frac{1}{2}}}{V}_{j+\frac{1}{2}}{\delta }_x{W}_{j+\frac{1}{2}}\\ =U_{N+1}{V}_{N+1}{W}_{N+1}-U_0{V}_0{W}_0-\frac{1}{4}{\displaystyle \sum _{j=0}^N\left(U_{j+1}-U_j\right)}\left(V_{j+1}-V_j\right)\left(W_{j+1}-W_j\right).\end{array}$

3. 连续系统的指数稳定性

本章考虑如下ODE-Wave方程左侧边界耦合控制系统

$\{\begin{array}{l}\dot{X}\left(t\right)=AX\left(t\right)+B{w}_x\left(0,t\right),t>0,\\ w_{tt}\left(x,t\right)=w_{xx}\left(x,t\right),0<x<1,t>0,\\ w\left(0,t\right)=0,t>0,\\ w_x\left(1,t\right)=U\left(t\right),t>0,\\ w\left(x,0\right)=w^0\left(x\right),w_t\left(x,0\right)=w^1\left(x\right),0<x<1,\end{array}$ (1)

其中$X\left(t\right)\in {ℝ}^n$ 和$u\in ℝ$ 分别是常微分方程子系统和波动方程子系统的状态变量，$U\left(t\right)$ 是控制输入，$A\in {ℝ}^{n\times n}$ ，$B\in {ℝ}^{n\times 1}$ 使得$\left(A,B\right)$ 是可控的。耦合发生在两个系统的左侧接点。

文献通过变量代换

$\{\begin{array}{l}X\left(t\right)=X\left(t\right),\\ \phi \left(x,t\right)=w\left(x,t\right)-{\displaystyle {\int }_0^{x}k}\left(x,y\right)w\left(y,t\right)\text{d}y-{\displaystyle {\int }_0^{x}l}\left(x,y\right)w_t\left(y,t\right)\text{d}y-\gamma \left(x\right)X\left(t\right),\end{array}$ (2)

将系统(1)转换为如下目标系统

$\{\begin{array}{l}\dot{X}\left(t\right)=\left(A+BK\right)X\left(t\right)+B{\phi }_x\left(0,t\right),t>0,\\ {\phi }_{tt}\left(x,t\right)={\phi }_{xx}\left(x,t\right),0<x<1,t>0,\\ \phi \left(0,t\right)=0,t>0,\\ {\phi }_x\left(1,t\right)=-k{\phi }_t\left(t\right),t>0,\\ \phi \left(x,0\right)={\phi }^0\left(x\right),{\phi }_t\left(x,0\right)={\phi }^1\left(x\right),0<x<1.\end{array}$ (3)

其中核函数

$k\left(x,y\right)=KM\left(x-y\right)AB,l\left(x,y\right)=KM\left(x-y\right)B,$ (4)

和

$\gamma \left(x\right)=K\left[\begin{array}{cc}0& I\end{array}\right]{\text{e}}^{\left[\begin{array}{cc}0& A^2\\ I& 0\end{array}\right]x}\left[\begin{array}{c}A\\ I\end{array}\right],$ (5)

同时利用反步控制理论给出在控制

$U(t)=c\left[KB\left(1,t\right)-w_t\left(1,t\right)\right]+\rho \left(1\right)X\left(t\right)+{\displaystyle {\int }_0^D\rho }\left(1-x\right)\left[ABw\left(x,t\right)+B{w}_t\left(x,t\right)\right]\text{d}x,$ (6)

下，闭环耦合系统(1)在$\Vert \left(X\left(t\right),u\left(x,t\right),v\left(x,t\right)\right)\Vert$ 范数下是一致指数稳定的。文献进一步指出变换(2)的逆变换为

$\{\begin{array}{l}X\left(t\right)=X\left(t\right),\\ w\left(x,t\right)=\phi \left(x,t\right)-{\displaystyle {\int }_0^x\phi }\left(x-y\right)w\left(y\right)\text{d}y-{\displaystyle {\int }_0^{x}n}\left(x-y\right)w_t\left(y\right)\text{d}y-\psi \left(x\right)X\left(t\right),\end{array}$ (7)

其中

$\psi \left(x\right)=K\left[\begin{array}{cc}0& I\end{array}\right]{\text{e}}^{\left[\begin{array}{cc}0& {\left(A+BK\right)}^2\\ I& 0\end{array}\right]x}\left[\begin{array}{c}I\\ 0\end{array}\right],$ (8)

以及核函数为

$n\left(s\right)=\psi \left(s\right)B,\varphi \left(s\right)=\psi \left(s\right)AB,$ (9)

通过变换(2)和(7)的具体形式可知二者都是有界变换，所以下面只要证明目标系统的指数稳定性就能说明在控制(6)下，原耦合系统(1)也是一致指数稳定的。

首先为闭环耦合系统(3)定义如下Lyapunov函数

$\begin{array}{c}V\left(t\right)=X^{\text{T}}\left(t\right)PX\left(t\right)+\frac{a}{2}{\displaystyle {\int }_0^1\left[{\phi }_x^2\left(x,t\right)+{\phi }_t^2\left(x,t\right)\right]\text{d}x}\\ +a\delta {\displaystyle {\int }_0^1\left(x+1\right){\phi }_x\left(x,t\right){\phi }_t\left(x,t\right)\text{d}x},\end{array}$ (10)

其中$\delta$ 是任意小的参数，a的具体形式在后面会给出，$P=P^{\text{T}}>0$ 是李雅普诺夫方程 $P\left(A+BK\right)+{\left(A+BK\right)}^{\text{T}}P=-I$ 的正定矩阵解，I是n阶矩阵单位。

引理3.1由(10)定义的$V\left(t\right)$ 满足

$c_1{\Vert \left(X\left(t\right),{\phi }_x\left(x,t\right),{\phi }_t\left(x,t\right)\right)\Vert }_{\mathscr{H}}^2\le V\left(t\right)\le c_2{\Vert \left(X\left(t\right),{\phi }_x\left(x,t\right),{\phi }_t\left(x,t\right)\right)\Vert }_{\mathscr{H}}^2,$

其中对于任意的$0<\delta <\frac{1}{2}$ 有$c_1=\min \left\{{\lambda }_{\min }\left(P\right),a\left(\frac{1}{2}-\delta \right)\right\}$ ，$c_2=\max \left\{{\lambda }_{\max }\left(P\right),a\left(\frac{1}{2}+\delta \right)\right\}$ 。

由杨氏不等式和事实${\lambda }_{\min }\left(P\right){\left|X\left(t\right)\right|}^2\le X^{\text{T}}\left(t\right)PX\left(t\right)\le {\lambda }_{\max }\left(P\right){\left|X\left(t\right)\right|}^2$ 易证。

引理3.2由(10)定义的$V\left(t\right)$ 满足$\dot{V}\left(t\right)\le -\eta V\left(t\right)$ ，其中$\eta =\frac{\min \left\{1,a\delta \right\}}{2c_2}$ 。

证明：$V\left(t\right)$ 关于t求导得到

$\begin{array}{c}\dot{V}\left(t\right)=-{\left|X\left(t\right)\right|}^2+2X^{\text{T}}\left(t\right)PB{\phi }_x\left(0,t\right)+a{\displaystyle {\int }_0^1\left[{\phi }_x\left(x,t\right){\phi }_{xt}\left(x,t\right)+{\phi }_t\left(x,t\right){\phi }_{xx}\left(x,t\right)\right]\text{d}x}\\ +a\delta {\displaystyle {\int }_0^1\left(x+1\right){\phi }_{xt}\left(x,t\right){\phi }_t\left(x,t\right)\text{d}x}+a\delta {\displaystyle {\int }_0^1\left(x+1\right){\phi }_x\left(x,t\right){\phi }_{xx}\left(x,t\right)\text{d}x},\end{array}$(11)

先对第二项进行估计，根据柯西不等式和杨氏不等式可以得到

$2X\left(t\right)PB{\phi }_x\left(0,t\right)\le \frac{1}{2}{\left|X\left(t\right)\right|}^2+2{\left|PB\right|}^2{\phi }_x^2\left(0,t\right),$ (12)

第三部分由分部积分容易得到

$a{\displaystyle {\int }_0^1\left[{\phi }_x\left(x,t\right){\phi }_{xt}\left(x,t\right)+{\phi }_t\left(x,t\right){\phi }_{xx}\left(x,t\right)\right]\text{d}x}=-ak{\phi }_t^2\left(1,t\right),$ (13)

对第四项和第五项同时使用分部积分得到

$\begin{array}{l}a\delta {\displaystyle {\int }_0^1\left(x+1\right){\phi }_{xt}\left(x,t\right){\phi }_t\left(x,t\right)\text{d}x}+a\delta {\displaystyle {\int }_0^1\left(x+1\right){\phi }_x\left(x,t\right){\phi }_{xx}\left(x,t\right)\text{d}x}\\ =a\delta \left(1+k^2\right){\phi }_t^2\left(x,t\right)-\frac{a\delta }{2}{\phi }_x^2\left(0,t\right)-\frac{a\delta }{2}{\displaystyle {\int }_0^1{\phi }_t^2\left(x,t\right)\text{d}x}-\frac{a\delta }{2}{\displaystyle {\int }_0^1{\phi }_x^2\left(x,t\right)\text{d}x},\end{array}$ (14)

由式(12)、式(13)和式(14)可得

$\begin{array}{c}\dot{V}\left(t\right)\le -\frac{1}{2}{\left|X\left(t\right)\right|}^2+2{\left|PB\right|}^2{\phi }_x^2\left(0,t\right)-ak{\phi }_t^2\left(1,t\right)+a\delta \left(1+k^2\right){\phi }_t^2\left(x,t\right)\\ -\frac{a\delta }{2}{\phi }_x^2\left(0,t\right)-\frac{a\delta }{2}{\displaystyle {\int }_0^1{\phi }_t^2\left(x,t\right)\text{d}x}-\frac{a\delta }{2}{\displaystyle {\int }_0^1{\phi }_x^2\left(x,t\right)\text{d}x}\\ =-\frac{1}{2}{\left|X\left(t\right)\right|}^2-\frac{a\delta }{2}{\displaystyle {\int }_0^1{\phi }_t^2\left(x,t\right)\text{d}x}-\frac{a\delta }{2}{\displaystyle {\int }_0^1{\phi }_x^2\left(x,t\right)\text{d}x}\\ -a\left[k-\delta \left(1+k^2\right)\right]{\phi }_t^2\left(1,t\right)-\left(\frac{a\delta }{2}-2{\left|PB\right|}^2\right){\phi }_x^2\left(0,t\right)\\ \le -\frac{\min \left\{1,a\delta \right\}}{2}{\Vert \left(X\left(t\right),{\phi }_x\left(t\right),{\phi }_t\left(t\right)\right)\Vert }_{\mathscr{H}}^2\\ \le -\eta V\left(t\right),\end{array}$(15)

其中$0<\delta <\frac{1}{2}$ ，$a>\max \left\{2,8{\left|PB\right|}^2\right\}$ ，$\eta =\frac{\min \left\{1,a\delta \right\}}{2c_2}$ ，$c_2$ 在引理2.1中已给出。

定理3.1闭环耦合系统(3)在$\Vert \left(X\left(t\right),w_x\left(x,t\right),w_t\left(x,t\right)\right)\Vert$ 范数下是一致指数稳定的。

证明：由引理(2.1)和引理(3.2)容易推出目标系统的Lyapunov函数$V\left(t\right)$ 满足

$V\left(t\right)\le V\left(0\right){\text{e}}^{-\eta t},$ (16)

进一步结合引理(2.1)可知

${\Vert X\left(t\right),{\phi }_x\left(x,t\right),{\phi }_t\left(x,t\right)\Vert }_{\mathscr{H}}^2\le \frac{c_2}{c_1}{\text{e}}^{-\eta t}{\Vert X\left(t\right),{\phi }_x\left(x,0\right),{\phi }_t\left(x,0\right)\Vert }_{\mathscr{H}}^2$ (17)

结合变换(2)和(7)的有界性，可知系统(1)在范数${\Vert X\left(t\right),w_x\left(x,t\right),w_t\left(x,t\right)\Vert }_{\mathscr{H}}^2$ 下同样具有一致指数稳定特性。

4. 离散系统的一致指数稳定性

本节通过引入中间变量将目标耦合系统转化为等价的低阶耦合系统，由于这种变换是可逆的，所以我们只需建立等价系统的半离散化有限差分格式，然后利用Lyapunov函数得到其一致指数稳定性就可说明原耦合系统也具备相同的性质。

4.1. 空间半离散格式的建立

首先引入如下两个新的变量

$u\left(x,t\right)={\phi }_x\left(x,t\right),v\left(x,t\right)={\phi }_t\left(x,t\right),$

则目标系统(3)就可变形为

$\{\begin{array}{l}\dot{X}\left(t\right)=\left(A+BK\right)X\left(t\right)+Bu\left(0,t\right),t>0,\\ v_t\left(x,t\right)=u_x\left(x,t\right),0<x<1,t>0,\\ u_t\left(x,t\right)=v_x\left(x,t\right),0<x<1,t>0,\\ v\left(0,t\right)=0,t>0,\\ u\left(1,t\right)+kv\left(1,t\right)=0,t>0,\\ u\left(x,0\right)=u^0\left(x\right),v\left(x,0\right)=v^0\left(x\right),0<x<1,\end{array}$ (18)

相较于原始系统，系统(18)关于时间变量和空间变量的导数从二阶降到了一阶，更重要的是耦合系统(18)中波动方程子系统的右侧边界不再含有关于空间变量的导数，于是就将常微分方程与波动方程在高阶处的耦合转换为在低阶处耦合从而简化后续的证明过程。

为了得到系统(18)的空间半离散化有限差分格式，首先取定正整数$N\in {\mathbb{N}}^{*}$ ，对区间$\left[0,1\right]$ 进行等距划分，

$0=x_0<x_1<\cdots <x_N<x_{N+1},$

其中$h=\frac{1}{N+1}$ 是空间步长。令$I_j=\left[x_j,x_{j+1}\right]\left(j=0,\cdots ,N\right)$ 有${\displaystyle {\cup }_{j=0}^N{I}_j}=\left[0,1\right]$ ，$I_j$ 的中点用$x_{j+\frac{1}{2}}=\left(j+\frac{1}{2}\right)h$ 表示。容易看出系统在$x_{j+\frac{1}{2}}$ 处也成立，因此可以得到

$\{\begin{array}{l}\dot{X}\left(t\right)=\left(A+BK\right)X\left(t\right)+Bu\left(x_0,t\right),t>0,\\ v_t\left(x_{j+\frac{1}{2}},t\right)-u_x\left(x_{j+\frac{1}{2}},t\right)=0,j=0,\cdots ,N,t>0,\\ u_t\left(x_{j+\frac{1}{2}},t\right)-v_x\left(x_{j+\frac{1}{2}},t\right)=0,j=0,\cdots ,N,t>0,\\ v\left(x_0,t\right)=0,t>0,\\ u\left(x_{N+1},t\right)+kv\left(x_{N+1},t\right)=0,t>0,\\ u\left(x_{j+\frac{1}{2}},0\right)=u^0\left(x_{j+\frac{1}{2}}\right),v\left(x_{j+\frac{1}{2}},0\right)=v^0\left(x_{j+\frac{1}{2}}\right),j=0,\cdots ,N,\end{array}$ (19)

用${\left\{W_j\right\}}_j$ 、${\left\{U_j\right\}}_j$ 、${\left\{V_j\right\}}_j$ 表示在${\left\{x_j\right\}}_j$ 处满足下列条件的网格函数

$W_j\left(t\right)=w\left(x_j,t\right),U_j\left(t\right)=u\left(x_j,t\right),V_j\left(t\right)=v\left(x_j,t\right),$

用$u^{\prime }$ 和$v^{\prime }$ 表示对时间的导数，并用差分算子${\delta }_x$ 代替掉微分算子$\partial x$ ，那么系统(19)就可再次变形为

$\{\begin{array}{l}\dot{X}\left(t\right)=\left(A+BK\right)X\left(t\right)+B{U}_0\left(t\right),t>0,\\ {V^{\prime }}_{j+\frac{1}{2}}\left(t\right)-{\delta }_x{U}_{j+\frac{1}{2}}\left(t\right)={\rm O}\left(h^2\right),j=0,\cdots ,N,t>0,\\ {U^{\prime }}_{j+\frac{1}{2}}\left(t\right)-{\delta }_x{V}_{j+\frac{1}{2}}\left(t\right)={\rm O}\left(h^2\right),j=0,\cdots ,N,t>0,\\ V_0\left(t\right)=0,t>0,\\ U_{N+1}\left(t\right)+k{V}_{N+1}\left(t\right)=0,t>0,\\ U_{j+\frac{1}{2}}\left(0\right)=U_{j+\frac{1}{2}}^0,V_{j+\frac{1}{2}}\left(0\right)=V_{j+\frac{1}{2}}^0,j=0,\cdots ,N,\end{array}$ (20)

舍弃掉高阶无穷小项，同时用$u_j\left(t\right)$ 、$v_j\left(t\right)$ 分别代替$U_j\left(t\right)$ 、$V_j\left(t\right)$ 后得到降阶系统(20)的空间半离散有限差分格式

$\{\begin{array}{l}\dot{X}\left(t\right)=\left(A+BK\right)X\left(t\right)+B{u}_0\left(t\right),t>0,\\ {v^{\prime }}_{j+\frac{1}{2}}\left(t\right)-{\delta }_x{u}_{j+\frac{1}{2}}\left(t\right)=0,j=0,\cdots ,N,t>0,\\ {u^{\prime }}_{j+\frac{1}{2}}\left(t\right)-{\delta }_x{v}_{j+\frac{1}{2}}\left(t\right)=0,j=0,\cdots ,N,t>0,\\ v_0\left(x_0,t\right)=0,t>0,\\ u_{N+1}\left(t\right)+k{v}_{N+1}\left(t\right)=0,t>0,\\ u_{j+\frac{1}{2}}\left(0\right)=u_{j+\frac{1}{2}}^0,v_{j+\frac{1}{2}}\left(0\right)=v_{j+\frac{1}{2}}^0,j=0,\cdots ,N,\end{array}$ (21)

可以看出，在降阶过程中使用的变量代换是可逆的，因此闭环系统(3)与(18)是等价的，由此可以得知系统(3)的半离散有限差分格式与离散系统(21)是等价的，因此只需对离散系统(21)的一致指数衰减性进行说明。

4.2. 差分格式的一致指数稳定性

首先为离散系统(21)构造相应离散形式的Lyapunov函数

$V_h\left(t\right)=X^{\text{T}}\left(t\right)PX\left(t\right)+\frac{ah}{2}{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}^2}\left(t\right)+\frac{ah}{2}{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}^2}\left(t\right)+a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right)u_{j+\frac{1}{2}}}\left(t\right)v_{j+\frac{1}{2}}\left(t\right),$ (22)

其中$\delta$ 可以任意小，a是需要被确定的参数。Lyapunov函数(22)在范数${\Vert \left(X\left(t\right),u\left(x,t\right),v\left(x,t\right)\right)\Vert }_h$ 下其实是目标系统的Lyapunov函数(10)的自然离散，这使得离散系统与连续系统在一致指数稳定性上的证明过程是类似的。

引理4.1由(22)定义的$V_h\left(t\right)$ 满足

$c_1{\Vert X\left(t\right),u\left(x,t\right),v\left(x,t\right)\Vert }_h^2\le V_h\left(t\right)\le c_2{\Vert X\left(t\right),u\left(x,t\right),v\left(x,t\right)\Vert }_h^2,$

其中对于任意的$0<\delta <\frac{1}{2}$ 有$c_1=\min \left\{{\lambda }_{\min }\left(P\right),a\left(\frac{1}{2}-\delta \right)\right\}$ ，$c_2=\min \left\{{\lambda }_{\max }\left(P\right),a\left(\frac{1}{2}+\delta \right)\right\}$ 。

证明：由柯西不等式和杨氏不等式得到

$\begin{array}{c}\left|a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right)u_{j+\frac{1}{2}}\left(t\right)v_{j+\frac{1}{2}}\left(t\right)}\right|\le 2a\delta h{\displaystyle \sum _{j=0}^N\left|u_{j+\frac{1}{2}}\left(t\right)v_{j+\frac{1}{2}}\left(t\right)\right|}\\ \le a\delta \left[h{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}^2}\left(t\right)+h{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}^2}\left(t\right)\right]\end{array}$ (23)

结合事实${\lambda }_{\min }\left(P\right){\left|X\left(t\right)\right|}^2\le X^{\text{T}}\left(t\right)PX\left(t\right)\le {\lambda }_{\max }\left(P\right){\left|X\left(t\right)\right|}^2$ 可得不等式

${\lambda }_{\min }\left(P\right){\left|X\left(t\right)\right|}^2+a\left(\frac{1}{2}-\delta \right)h\left[{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}^2}\left(t\right)+{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}^2}\left(t\right)\right]\le V_h\left(t\right),$ (24)

$V_h\left(t\right)\le {\lambda }_{\max }\left(P\right){\left|X\left(t\right)\right|}^2+a\left(\frac{1}{2}+\delta \right)h\left[{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}^2}\left(t\right)+{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}^2}\left(t\right)\right]$ (25)

令$c_1=\min \left\{{\lambda }_{\min }\left(P\right),a\left(\frac{1}{2}-\delta \right)\right\}$ ，$c_2=\min \left\{{\lambda }_{\max }\left(P\right),a\left(\frac{1}{2}+\delta \right)\right\}$ ，证毕。

引理4.2由定义的$V_h\left(t\right)$ 满足${\dot{V}}_h\left(t\right)\le -\eta V_h\left(t\right)$ ，其中$\eta =\frac{\min \left\{1,a\delta \right\}}{2c_2}$ 。

证明：$V_h\left(t\right)$ 关于t求导可得

$\begin{array}{c}\dot{V}\left(t\right)=-{\left|X\left(t\right)\right|}^2+2X^{\text{T}}\left(t\right)PB{u}_0\left(t\right)+ah{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}\left(t\right){\delta }_x{v}_{j+\frac{1}{2}}\left(t\right)}+ah{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}\left(t\right){\delta }_x{u}_{j+\frac{1}{2}}\left(t\right)}\\ +a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\delta }_x{v}_{j+\frac{1}{2}}\left(t\right)v_{j+\frac{1}{2}}\left(t\right)}+a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\delta }_x{u}_{j+\frac{1}{2}}}\left(t\right)u_{j+\frac{1}{2}}\left(t\right),\end{array}$(26)

先来估计第二项，由柯西不等式和杨氏不等式得到

$2X^{\text{T}}\left(t\right)PB{u}_0\left(t\right)\le \frac{1}{2}{\left|X\left(t\right)\right|}^2+2{\left|PB\right|}^2{u}_0^2\left(t\right),$ (27)

第三项和第四项利用引理(1.2)第一式可得

$ah{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}\left(t\right){\delta }_x}{v}_{j+\frac{1}{2}}\left(t\right)+ah{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}\left(t\right){\delta }_x}{u}_{j+\frac{1}{2}}\left(t\right)=-ak{v}_{N+1}^2\left(t\right),$ (28)

再来估计第五项

$\begin{array}{l}a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right)v_{j+\frac{1}{2}}\left(t\right){\delta }_x{v}_{j+\frac{1}{2}}\left(t\right)}\\ =-a\delta h{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}^2\left(t\right)}-a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\delta }_x{v}_{j+\frac{1}{2}}\left(t\right)v_{j+\frac{1}{2}}\left(t\right)}+2a\delta v_{N+1}^2\left(t\right)\\ -\frac{a\delta }{4}{\displaystyle \sum _{j=0}^N\left(x_{j+1}-x_j\right){\left(v_{j+1}-v_j\right)}^2}\\ \le -a\delta h{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}^2\left(t\right)}-a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\delta }_x{v}_{j+\frac{1}{2}}\left(t\right)v_{j+\frac{1}{2}}\left(t\right)}+2a\delta v_{N+1}^2\left(t\right),\end{array}$ (29)

其中用到了引理(2)的第二式，对第六项进行完全相同的处理得到

$\begin{array}{l}a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right)u_{j+\frac{1}{2}}\left(t\right){\delta }_x{u}_{j+\frac{1}{2}}\left(t\right)}\\ \le -a\delta h{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}^2\left(t\right)}-a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\delta }_x{u}_{j+\frac{1}{2}}\left(t\right)u_{j+\frac{1}{2}}\left(t\right)}+2a\delta u_{N+1}^2\left(t\right)-a\delta u_0^2\left(t\right),\end{array}$ (30)

联立式(29)和式(30)可得

$\begin{array}{l}2a\delta \left[h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right)u_{j+\frac{1}{2}}\left(t\right){\delta }_x{u}_{j+\frac{1}{2}}\left(t\right)}+h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right)v_{j+\frac{1}{2}}\left(t\right){\delta }_x{v}_{j+\frac{1}{2}}\left(t\right)}\right]\\ \le -a\delta \left[h{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}^2\left(t\right)}+h{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}^2\left(t\right)}\right]+2a\delta \left(1+k^2\right)v_{N+1}^2\left(t\right)-a\delta u_0^2\left(t\right),\end{array}$ (31)

将式(27)、式(28)和式(31)结合起来得到

$\begin{array}{c}{\dot{V}}_h\left(t\right)\le -\frac{1}{2}{\left|X\left(t\right)\right|}^2+2{\left|PB\right|}^2{u}_0^2\left(t\right)-\frac{a\delta }{2}\left[h{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}^2\left(t\right)}+h{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}^2\left(t\right)}\right]\\ -ak{v}_{N+1}^2\left(t\right)+2a\delta \left(1+k^2\right)v_{N+1}^2\left(t\right)-\frac{a\delta }{2}{u}_0^2\left(t\right)\\ =-\frac{1}{2}\left[{\left|X\left(t\right)\right|}^2+a\delta h{\displaystyle \sum _{j=0}^N{v}_{j+\frac{1}{2}}^2\left(t\right)}+a\delta h{\displaystyle \sum _{j=0}^N{u}_{j+\frac{1}{2}}^2\left(t\right)}\right]\\ -a\left[k-2\delta \left(1+k^2\right)\right]v_{N+1}^2\left(t\right)-\left[\frac{a\delta }{2}-2{\left|PB\right|}^2\right]u_0^2\left(t\right)\\ \le -\frac{\min \left\{1,a\delta \right\}}{2}{\Vert \left(X\left(t\right),u\left(t\right),v\left(t\right)\right)\Vert }_h^2\\ \le -\eta V_h\left(t\right),\end{array}$ (32)

其中$a>\max \left\{2,8{\left|PB\right|}^2\right\}$ ，$\eta =\frac{1}{2c_1}$ ，$\eta =\frac{\min \left\{1,a\delta \right\}}{2c_2}$ 。

定理4.2存在正常数$M=\frac{c_2}{c_1}$ 和$\eta =\frac{\min \left\{1,a\delta \right\}}{2c_2}$ 使得离散系统在${\Vert \left(X\left(t\right),u\left(x,t\right),v\left(x,t\right)\right)\Vert }_h$ 范数下是一致指数稳定的，换句话说，

$V_h\left(t\right)\le M{V}_h\left(0\right){\text{e}}^{-\eta t},$

其中$c_1$ ，$c_2$ 在引理(4.1)中已给出。

根据引理(2.1)、引理(4.1)和引理(4.2)很容易得到上述不等式。

下面进一步说明离散格式的收敛性。首先定义误差函数

${\xi }_i\left(t\right)=U_i\left(t\right)-u_i\left(t\right)$ ，${\eta }_i\left(t\right)=V_i\left(t\right)-v_i\left(t\right)$

将(20)与(21)相减后得到如下误差系统

$\{\begin{array}{l}\dot{X}\left(t\right)=\left(A+BK\right)X\left(t\right)+B{\xi }_0\left(t\right),t>0,\\ {{\eta }^{\prime }}_{j+\frac{1}{2}}\left(t\right)-{\delta }_x{\xi }_{j+\frac{1}{2}}\left(t\right)=s_{j+\frac{1}{2}}\left(t\right),j=0,\cdots ,N,t>0,\\ {{\xi }^{\prime }}_{j+\frac{1}{2}}\left(t\right)-{\delta }_x{\eta }_{j+\frac{1}{2}}\left(t\right)=r_{j+\frac{1}{2}}\left(t\right),j=0,\cdots ,N,t>0,\\ {\eta }_0\left(t\right)=0,t>0,\\ {\xi }_{N+1}\left(t\right)+k{\eta }_{N+1}\left(t\right)=0,t>0,\\ {\xi }_{j+\frac{1}{2}}\left(0\right)={\xi }_{j+\frac{1}{2}}^0,{\eta }_{j+\frac{1}{2}}\left(0\right)={\eta }_{j+\frac{1}{2}}^0,j=0,\cdots ,N,\end{array}$ (33)

根据Taylor级数可知其中

$r_{j+\frac{1}{2}}\left(t\right)=\left(-\frac{1}{8}{u^{\prime }}_{xx}\left({\zeta }_1,t\right)+\frac{1}{24}{v}_{xxx}\left({\zeta }_2,t\right)\right)h^2,$

$s_{j+\frac{1}{2}}\left(t\right)=\left(-\frac{1}{8}{v^{\prime }}_{xx}\left({\zeta }_1,t\right)+\frac{1}{24}{u}_{xxx}\left({\zeta }_2,t\right)\right)h^2,$

其中${\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4\in \left(x_j,x_{j+\frac{1}{2}}\right)$ 。

为误差系统定义如下离散形式的Lyapunov函数

$G_h\left(t\right)=X^{\text{T}}\left(t\right)PX\left(t\right)+\frac{ah}{2}{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}^2}\left(t\right)+\frac{ah}{2}{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}^2}\left(t\right)+a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\xi }_{j+\frac{1}{2}}}\left(t\right){\eta }_{j+\frac{1}{2}}\left(t\right),$ (34)

采用与引理(4.1)类似的过程可以得到

$c_1{\Vert X\left(t\right),\xi \left(x,t\right),\eta \left(x,t\right)\Vert }_h^2\le G_h\left(t\right)\le c_2{\Vert X\left(t\right),\xi \left(x,t\right),\eta \left(x,t\right)\Vert }_h^2,$ (35)

其中$c_1=\min \left\{{\lambda }_{\min }\left(P\right),a\left(\frac{1}{2}-\delta \right)\right\}$ ，$c_2=\min \left\{{\lambda }_{\max }\left(P\right),a\left(\frac{1}{2}+\delta \right)\right\}$ 。

定理4.3 设$u,v\in C^1\left(\left[0,\infty \right);C^3\left[0,1\right]\right)$ 是降阶后的等价系统(18)的解，${\left\{u_j\left(t\right)\right\}}_j,{\left\{v_j\left(t\right)\right\}}_j$ 对应的差分格式(21)的解，${\left\{{\xi }_j\left(t\right)\right\}}_j,{\left\{{\eta }_j\left(t\right)\right\}}_j$ 是误差系统(33)的解，那么有下列误差不等式成立

${\Vert X\left(t\right),\xi \left(x,t\right),\eta \left(x,t\right)\Vert }_h^2\le \frac{c_2}{c_1}{\text{e}}^{-\gamma t}{\Vert X\left(0\right),\xi \left(x,0\right),\eta \left(x,0\right)\Vert }_h^2+\frac{a\epsilon }{2c_1}\left(1+2\delta \right)C^2{h}^4,$

其中$\gamma =\frac{\min \left\{1,a\delta \right\}}{2c_2}-\frac{a\epsilon }{2c_1}\left(1+2\delta \right)$ ，$0<\epsilon <\frac{\min \left\{1,a\delta \right\}{c}_1}{a{c}_2\left(1+2\delta \right)}$ 从而保证$\gamma >0$ ，$c_1$ 和$c_2$ 在(35)中已给出。

证明：由$r_{j+\frac{1}{2}}\left(t\right)$ ，$s_{j+\frac{1}{2}}\left(t\right)$ 的具体形式可知存在不依赖于$h$ 的$C$ 满足

$\left|r_{j+\frac{1}{2}}\left(t\right)\right|\le C{h}^2,\left|s_{j+\frac{1}{2}}\left(t\right)\right|\le C{h}^2,j=0,1,\cdots ,N,t>0,$ (36)

用${\widehat{r}}_{j+\frac{1}{2}}\left(t\right)$ 和${\widehat{s}}_{j+\frac{1}{2}}\left(t\right)$ 分别代替${\left|r_{j+\frac{1}{2}}\left(t\right)\right|}_{0\le j\le N}$ 和${\left|s_{j+\frac{1}{2}}\left(t\right)\right|}_{0\le j\le N}$ ，令$G_h\left(t\right)$ 对$t$ 求导可得

$\begin{array}{c}{\dot{G}}_h\left(t\right)=-{\left|X\left(t\right)\right|}^2+2X^{\text{T}}\left(t\right)PB{\xi }_0\left(t\right)+ah{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}\left(t\right){\delta }_x{\eta }_{j+\frac{1}{2}}\left(t\right)}\\ +ah{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}\left(t\right){\delta }_x{\xi }_{j+\frac{1}{2}}\left(t\right)}+ah{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}\left(t\right){\widehat{r}}_{j+\frac{1}{2}}\left(t\right)}+ah{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}\left(t\right){\widehat{s}}_{j+\frac{1}{2}}\left(t\right)}\end{array}$

$\begin{array}{c}+a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\delta }_x{\eta }_{j+\frac{1}{2}}\left(t\right){\eta }_{j+\frac{1}{2}}\left(t\right)}+a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\delta }_x{\xi }_{j+\frac{1}{2}}\left(t\right){\xi }_{j+\frac{1}{2}}\left(t\right)}\\ +a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\widehat{r}}_{j+\frac{1}{2}}\left(t\right){\eta }_{j+\frac{1}{2}}\left(t\right)}+a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\xi }_{j+\frac{1}{2}}}\left(t\right){\widehat{s}}_{j+\frac{1}{2}}\left(t\right),\end{array}$(37)

对第二项进行估计

$2X^{\text{T}}\left(t\right)PB{\xi }_0\left(t\right)\le \frac{1}{2}{\left|X\left(t\right)\right|}^2+2{\left|PB\right|}^2{\xi }_0^2\left(t\right),$ (38)

第三项和第四项利用引理(2.2)的第一式处理得到

$ah{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}\left(t\right){\delta }_x{\eta }_{j+\frac{1}{2}}\left(t\right)}+ah{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}\left(t\right){\delta }_x{\xi }_{j+\frac{1}{2}}\left(t\right)}=-ak{\eta }_{N+1}^2\left(t\right),$ (39)

类似于式子(29)、式(30)和式(31)的过程可得

$\begin{array}{l}2a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\delta }_x{\eta }_{j+\frac{1}{2}}\left(t\right){\eta }_{j+\frac{1}{2}}\left(t\right)}+a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\delta }_x{\xi }_{j+\frac{1}{2}}\left(t\right){\xi }_{j+\frac{1}{2}}\left(t\right)}\\ \le -a\delta \left[h{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}^2\left(t\right)}+h{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}^2\left(t\right)}\right]+2a\delta \left(1+k^2\right){\eta }_{N+1}^2\left(t\right)-a\delta {\xi }_0^2\left(t\right),\end{array}$ (40)

再对剩余的其他项进行估计

$\begin{array}{l}ah{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}\left(t\right){\widehat{r}}_{j+\frac{1}{2}}\left(t\right)}+ah{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}\left(t\right){\widehat{s}}_{j+\frac{1}{2}}\left(t\right)}\\ +a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\widehat{r}}_{j+\frac{1}{2}}\left(t\right){\eta }_{j+\frac{1}{2}}\left(t\right)}+a\delta h{\displaystyle \sum _{j=0}^N\left(x_{j+\frac{1}{2}}+1\right){\xi }_{j+\frac{1}{2}}\left(t\right){\widehat{s}}_{j+\frac{1}{2}}\left(t\right)}\\ \le \frac{a\epsilon }{2}h{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}^2\left(t\right)+\frac{a}{2\epsilon }h{\displaystyle \sum _{j=0}^N{\widehat{r}}_{j+\frac{1}{2}}^2\left(t\right)}}+\frac{a\epsilon }{2}h{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}^2\left(t\right)+\frac{a}{2\epsilon }h{\displaystyle \sum _{j=0}^N{\widehat{s}}_{j+\frac{1}{2}}^2\left(t\right)}}\\ +a\delta \epsilon h{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}^2\left(t\right)+\frac{a\delta }{\epsilon }h{\displaystyle \sum _{j=0}^N{\widehat{r}}_{j+\frac{1}{2}}^2\left(t\right)}}+a\delta \epsilon h{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}^2\left(t\right)+\frac{a\delta }{\epsilon }h{\displaystyle \sum _{j=0}^N{\widehat{s}}_{j+\frac{1}{2}}^2\left(t\right)}}\\ \le \frac{a\epsilon }{2}\left(1+2\delta \right){\Vert X\left(t\right),\xi \left(x,t\right),\eta \left(x,t\right)\Vert }_h^2+\frac{a\epsilon }{2}\left(1+2\delta \right)\left[{\Vert {\widehat{r}}_{j+\frac{1}{2}}\Vert }^2+{\Vert {\widehat{s}}_{j+\frac{1}{2}}\Vert }^2\right]\\ \le \frac{a\epsilon }{2c_1}\left(1+2\delta \right)G_h\left(t\right)+\frac{a\epsilon }{2}\left(1+2\delta \right)\left[{\Vert {\widehat{r}}_{j+\frac{1}{2}}\Vert }^2+{\Vert {\widehat{s}}_{j+\frac{1}{2}}\Vert }^2\right],\end{array}$ (41)

其中用到了杨氏不等式$ab\le \frac{a^2\epsilon }{2}+\frac{b^2}{2\epsilon }$ ，以及$0<\epsilon <\frac{\min \left\{1,a\delta \right\}{c}_1}{a{c}_2\left(1+2\delta \right)}$ ，结合式(38)、式(39)、式(40)和式(41)，有

$\begin{array}{c}{\dot{G}}_h\left(t\right)\le \frac{1}{2}{\left|X\left(t\right)\right|}^2+2{\left|PB\right|}^2{\xi }_0^2\left(t\right)-ak{\eta }_{N+1}^2\left(t\right)-\frac{a\delta }{2}\left[h{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}^2\left(t\right)+h{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}^2\left(t\right)}}\right]\\ +2a\delta \left(1+k^2\right){\eta }_{N+1}^2\left(t\right)-\frac{a\delta }{2}{\xi }_0^2\left(t\right)+\frac{a\epsilon }{2c_1}\left(1+2\delta \right)G_h\left(t\right)+\frac{a\epsilon }{2}\left(1+2\delta \right)\left[{\Vert {\widehat{r}}_{j+\frac{1}{2}}\Vert }^2+{\Vert {\widehat{s}}_{j+\frac{1}{2}}\Vert }^2\right],\end{array}$ (42)

进一步利用不等式$c_1{\Vert X\left(t\right),\xi \left(x,t\right),\eta \left(x,t\right)\Vert }_h^2\le G_h\left(t\right)\le c_2{\Vert X\left(t\right),\xi \left(x,t\right),\eta \left(x,t\right)\Vert }_h^2$ 对(42)进行放缩可以得到

$G_h\left(t\right)\le -\frac{1}{2}\left[{\left|X\left(t\right)\right|}^2+a\delta h{\displaystyle \sum _{j=0}^N{\xi }_{j+\frac{1}{2}}^2\left(t\right)+a\delta h{\displaystyle \sum _{j=0}^N{\eta }_{j+\frac{1}{2}}^2\left(t\right)}}\right]-a\left[k-2\delta \left(1+k^2\right)\right]{\eta }_{N+1}^2\left(t\right)$

$\begin{array}{c}-\left[\frac{a\delta }{2}-2{\left|PB\right|}^2\right]{\xi }_0^2\left(t\right)+\frac{a\epsilon }{2c_1}\left(1+2\delta \right)G_h\left(t\right)+\frac{a\epsilon }{2}\left(1+2\delta \right)\left[{\Vert {\widehat{r}}_{j+\frac{1}{2}}\Vert }^2+{\Vert {\widehat{s}}_{j+\frac{1}{2}}\Vert }^2\right]\\ \le -\frac{\min \left\{1,a\delta \right\}}{2c_2}{G}_h\left(t\right)+\frac{a\epsilon }{2c_1}\left(1+2\delta \right)G_h\left(t\right)+\frac{a\epsilon }{2}\left(1+2\delta \right)\left[{\Vert {\widehat{r}}_{j+\frac{1}{2}}\Vert }^2+{\Vert {\widehat{s}}_{j+\frac{1}{2}}\Vert }^2\right]\\ \le \left[-\frac{\min \left\{1,a\delta \right\}}{2c_2}+\frac{a\epsilon }{2c_1}\left(1+2\delta \right)\right]G_h\left(t\right)+\frac{a\epsilon }{2}\left(1+2\delta \right)\left[{\Vert {\widehat{r}}_{j+\frac{1}{2}}\Vert }^2+{\Vert {\widehat{s}}_{j+\frac{1}{2}}\Vert }^2\right]\\ =-\gamma G_h\left(t\right)+\frac{a\epsilon }{2}\left(1+2\delta \right)\left[{\Vert {\widehat{r}}_{j+\frac{1}{2}}\Vert }^2+{\Vert {\widehat{s}}_{j+\frac{1}{2}}\Vert }^2\right],\end{array}$ (43)

其中$\gamma =\frac{\min \left\{1,a\delta \right\}}{2c_2}-\frac{a\epsilon }{2c_1}\left(1+2\delta \right)$ ，由$\epsilon$ 的限制条件可知$\gamma >0$ ，根据引理(2.1)，由(43)可得

$G_h\left(t\right)\le {\text{e}}^{-\gamma t}{G}_h\left(0\right)+\frac{a\epsilon }{2}\left(1+2\delta \right){\displaystyle {\int }_0^t{\text{e}}^{-\gamma \left(t-s\right)}}\left[{\Vert \widehat{r}\left(\tau \right)\Vert }^2+{\Vert \widehat{s}\left(\tau \right)\Vert }^2\right]\text{d}\tau ,$(44)

再利用(35)可进一步得，对$\forall t>0$ 有

${\Vert X\left(t\right),\xi \left(x,t\right),\eta \left(x,t\right)\Vert }_h^2\le \frac{c_2}{c_1}{\text{e}}^{-\gamma t}{\Vert X\left(0\right),\xi \left(x,0\right),\eta \left(x,0\right)\Vert }_h^2+\frac{a\epsilon }{2c_1}\left(1+2\delta \right)C^2{h}^4,$ (45)

定理证明完毕。

由定理4.3可以看出，若初始条件是确定的，那么差分格式(21)的解二阶收敛到降阶系统(18)的精确解。

5. 结论

本文研究了ODE-Wave方程在左侧以Numman形式连接的耦合系统，为其构造了一个空间半离散有限差分格式，并证明了连续系统和离散系统的一致指数稳定性。利用Backstepping控制法将原始系统转换为等价的目标系统并设计出控制$U\left(t\right)$ ，通过Lyapunov函数法证明目标系统的一致指数稳定性，再结合变换及其逆变换的有界性说明在控制$U\left(t\right)$ 下的闭环控制系统也是指数稳定的。在研究离散系统的稳定性时，先用变量代换将目标系统转化成了等价的低阶闭环系统。对于等价的闭环系统使用有限差分法进行空间上的半离散，并为其构造相应的Lyapunov函数证明该格式是一致指数稳定的。最后说明了该空间半离散有限差分格式是二阶收敛的，通过引入误差函数建立等价降阶系统与相应的半离散格式的误差系统，并为其构造相应的Lyapunov函数，而后利用杨氏不等式等基本放缩方法得到误差不等式，明确表明在初始条件确定的前提下，半离散空间有限差分格式的解二阶收敛到降阶系统。