云南民族大学数学与计算机科学学院，云南 昆明

收稿日期：2020年12月14日；录用日期：2021年1月15日；发布日期：2021年1月22日

摘要

本文主要研究一类一重积分不等式的改进并将其用于分析时滞神经网络稳定性的问题。首先，借助交互凸组合不等式和多个零等式，改进一类一重积分不等式。其次，根据研究的时变时滞系统的属性，构造一类新的含有更多时滞信息的增广型Lyapunov泛函。然后，利用改进的积分不等式及其它分析技巧，获得保守性较低的稳定性判据。最后，通过一个数值例子验证所得结果的有效性和优越性。

关键词

交互凸组合不等式，增广的Lyapunov泛函，零等式，改进的一重积分不等式

The Application of Improved Single Integral Inequality in Neural Networks with Time-Delayed

Yingying Liu, Lianglin Xiong^*, Mei Zou, Li Cai, Zhen Wang

School of Mathematics and Computer Science, Yunnan Minzu University, Kunming Yunnan

Received: Dec. 14^th, 2020; accepted: Jan. 15^th, 2021; published: Jan. 22^nd, 2021

ABSTRACT

This paper is concerned with an improved single integral inequality with application in the time-delayed neural network. First of all, an improved single integral inequality is proposed by virtue of the reciprocally convex combination inequality and zero-qualities. Secondly, relay on the characteristics of the system more information about time-delay is considered in constructing the augmented Lyapunov functional. Then, a less conservative stability condition is obtained through utilizing the improved integral inequality and other analysis techniques. In the end, to verify the effectiveness and superiority of the derived results, a numerical example is provided.

Keywords:Reciprocally Convex Combination Inequality, Augmented Lyapunov Functional, Zero-Qualities, An Improved Single Integral Inequality

Copyright © 2021 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

时滞神经网络的稳定性问题是神经网络处理联想记忆、信号处理和一些优化问题等多个实际问题的基础 [1] [2] [3]。一方面，在应用神经网络之前，应确保稳定性；另一方面，在神经网络的实现中，由于神经元的有限切换速率就不可避免的出现时滞，很多相关研究表明时滞会导致不良的动态行为，这些行为会使得系统的性能下降甚至不稳定 [4]，因此研究时滞神经网络的稳定性是非常有意义的。

到目前为止，学者们针对时滞神经网络系统的稳定性研究主要采用Lyapunov泛函方法。通过利用Lyapunov稳定性理论 [5]，构造恰当的Lyapunov泛函以及尽可能精细地估计泛函的导数来得到低保守性的稳定性准则。对于构造Lyapunov泛函方法，目前比较常见的有构造带有增广的Lyapunov泛函 [6]，利用时滞分割方法构造泛函 [7] 以及构造带有多重积分项的泛函 [8]。对于估计泛函的导数，主要是估计求导后产生的积分项，力求获得求导后更严格的上界。为了处理这些积分，学者们主要通过不等式放缩的方法，例如Jensen不等式 [9]，基于Wirtinger的积分不等式 [10]，基于自由矩阵的积分不等式 [11]，改进的自由矩阵积分不等式 [12] 以及互凸不等式 [13] 等等。这些不等式有效地降低了时滞神经网络的时滞依赖稳定性条件，不过仍然有改进的空间。

文献 [14] 中通过引入增广的Lyapunov泛函并应用于Wirtinger不等式中，得到了保守性较低的时滞相关稳定性标准，文献 [15] 利用零等式改进具有增广的一重积分不等式，与直接使用Jensen不等式放缩相比，运用零等式可以巧妙加入自由矩阵，从而提供更大的自由度进而得到更紧的时滞上界。然而，文献 [15] 中不等式的放缩只带有一重积分信息，具有一定的保守性。众所周知，高阶积分项在降低时滞系统的保守性中扮演着十分重要的作用，例如在估计一重积分不等式中引入二重积分项，这时二重积分状态信息会与一重状态信息以及系统的状态信息交叉，从而得到更大的时滞上界，因此建立一个新的带有自由矩阵和二重积分项的具有增广的一重积分不等式对降低时滞神经网络系统的保守性具有深远的意义。此外，值得一提的是在增广向量中引入s相关项能有效克服条件的保守性，因为它将一重积分放入增广向量中并把这个增广向量放入带有一重积分的Lyapunov泛函中，在对这个泛函求导后能够获得更多状态信息。因此，本文拟通过新的积分不等式处理技巧结合更符合系统的Lyapunov泛函，获得保守性更低的稳定性条件。

为了简化表述，我们有必要做如下的符号说明： $R^n$ 表示n维欧式空间； $R^{n\times m}$ 代表 $n\times m$ 的矩阵； $\text{sym}\left\{X\right\}=X+X^{\text{T}}$ ； $\text{diag}\{\cdots \}$ 表示对角矩阵； $\text{col}\{\cdots \}$ 列向量 $X^{\text{T}}$，和 $X^{-1}$ 分别矩阵的转置和逆矩阵； $*$ 表示矩阵的对称项。

2. 预备知识

2.1. 系统描述

考虑一般的带有时变时滞的神经网络系统

$\{\begin{array}{l}\stackrel{\dot{}}{x}\left(t\right)=-Ax\left(t\right)+W_{1}f\left(W_{0}x\left(t\right)\right)+W_{2}f\left(W_{0}x\left(t-h\left(t\right)\right)\right),\\ x\left(\theta \right)=\varphi \left(\theta \right),\theta \in \left[-h,0\right].\end{array}$ (1)

其中 $x\left(t\right)\in R^n$ 是系统的状态向量 $A\in R^{n\times n}$，$W_0\in R^{n\times n}$，$W_1\in R^{n\times n}$，和 $W_2\in R^{n\times n}$ 是已知的实数矩阵，初始条件 $\phi \left(\theta \right)$ 是连续的微分函数，h和u是已知的常数，且满足如下条件：

$0<h\left(t\right)<h$,$\stackrel{\dot{}}{h}\left(t\right)<u$. (2)

$f\left(W_{0}x\left(t\right)\right)=\text{col}\left\{f\left(W_{01}x\left(t\right)\right),f\left(W_{02}x\left(t\right)\right),\cdots ,f\left(W_{0n}x\left(t\right)\right)\right\}$ 是神经元的激活函数 $W_{0i}$ 是 $W_0$ 的第i行，假设 $f\left(0\right)=0$ 且满足Lipschitz条件

$l_i^{-}\le \frac{f_i\left(a\right)-f_i\left(b\right)}{a-b}\le l_i^{+}$, (3)

其中， $l_i^{-}$ 和 $l_i^{+}$ 是一些常数，且 $L^{-}=\text{diag}\left\{l_1^{-},l_2^{-},\cdots ,l_n^{-}\right\}$，$L^{+}=\text{diag}\left\{l_1^{+},l_2^{+},\cdots ,l_n^{+}\right\}$。

2.2. 引理

在分析系统(1)的时滞依赖稳定性判据之前，本节首先引入如下一些重要的引理：

引理1. [16] 给定一个二次函数 $f\left(y\right)=a_2{y}^2+a_{1}y+a_0$，当 $\beta \in \left[0,1\right]$ 时，如果有

$T_i=f\left(h_i\right)<0\left(i=1,2\right)$,$T_3=-{\beta }^2{h}_{12}^2{a}_2+f\left(h_1\right)<0$,$T_4=-{\left(1-\beta \right)}^2{h}_{12}^2{a}_2+f\left(h_2\right)<0$,

那么 $f\left(y\right)<0\left(h_1\le y\le h_2\right)$ 成立，其中 $h_{12}=h_2-h_1$。

引理2. [12] 设 $w\left(t\right)$ 是一个可微函数： $\left[a,b\right]\to R^n$，那么存在对称正定矩阵 $H\in R^{n\times n}$，对于任意具有恰当维度的矩阵L，M，有下列的不等式成立：

${\displaystyle {\int }_a^b{\stackrel{\dot{}}{w}}^{\text{T}}\left(s\right)H\stackrel{\dot{}}{w}\left(s\right)\text{d}s}\ge -\text{sym}\left\{{\sigma }_0^{\text{T}}L{\sigma }_1+{\sigma }_0^{\text{T}}M{\sigma }_2\right\}-\left(b-a\right){\sigma }_0^{\text{T}}\left(\frac{3L{H}^{-1}{L}^{\text{T}}+M{H}^{-1}{M}^{\text{T}}}{3}\right){\sigma }_0$

其中 ${\sigma }_1=x\left(b\right)-x\left(a\right)$，${\sigma }_2=x\left(b\right)+x\left(a\right)-\frac{2}{b-a}{\displaystyle {\int }_a^{b}x\left(s\right)\text{d}s}$，${\sigma }_0$ 是任意向量。

引理3. [17] 若对称正定矩阵 $R>0$，$x\left(t\right)$ 是一个可微函数： $\left[a,b\right]\to R^n$，则下列不等式成立

${\displaystyle {\int }_a^b{x}^{\text{T}}\left(s\right)Rx\left(s\right)\text{d}s}\ge \frac{1}{b-a}{\left({\displaystyle {\int }_a^{b}x\left(s\right)\text{d}s}\right)}^{\text{T}}R\left({\displaystyle {\int }_a^{b}x\left(s\right)\text{d}s}\right)+\frac{3}{b-a}{\Omega }_1^{\text{T}}R{\Omega }_1$,

${\displaystyle {\int }_a^b{\stackrel{\dot{}}{x}}^{\text{T}}\left(s\right)R\stackrel{\dot{}}{x}\left(s\right)\text{d}s}\ge \frac{1}{b-a}{\Omega }_2^{\text{T}}R{\Omega }_2+\frac{3}{b-a}{\Omega }_3^{\text{T}}R{\Omega }_3$,

其中

${\Omega }_1={\displaystyle {\int }_a^{b}x\left(s\right)\text{d}s}-\frac{2}{b-a}{\displaystyle {\int }_a^b{\displaystyle {\int }_{\beta }^{b}x\left(s\right)\text{d}s\text{d}\beta }}$,${\Omega }_2=x\left(b\right)-x\left(a\right)$,${\Omega }_3=x\left(b\right)+x\left(a\right)-\frac{2}{b-a}{\displaystyle {\int }_a^{b}x\left(s\right)\text{d}s}$.

为了能得到保守性更低的稳定性条件，本文首先引入一个改进的一重积分不等式如下：

引理4. 若半正定对称矩阵 $Y_i\in R^{n\times n}\left(i=1,3\right)$，$X_i\in R^{n\times n}\left(i=1,2\right)$，任意矩阵 $Y_2\in R^{n\times n}$，$X_3\in R^{n\times n}$，$X_4\in R^{2n\times 2n}$，$x\left(t\right)$ 是一个可微函数： $\left[a,b\right]\to R^n$，满足：

${\Omega }_0=\left[\begin{array}{cc}{Y}_1& Y_2\\ *& Y_3\end{array}\right]\ge 0$,${\Omega }_1=\left[\begin{array}{cc}{Y}_1& Y_2+X_1\\ *& Y_3\end{array}\right]\ge 0$,${\Omega }_2=\left[\begin{array}{cc}{Y}_1& Y_2+X_2\\ *& Y_3\end{array}\right]\ge 0$,

${\Omega }_3=\left[\begin{array}{cc}{Y}_3& X_3\\ *& Y_3\end{array}\right]\ge 0$,${\Omega }_4=\left[\begin{array}{cc}{\Omega }_1& X_4\\ *& {\Omega }_2\end{array}\right]\ge 0$,

那么，对任意向量 $c\in \left[a,b\right]$，有下面的不等式成立

$\Phi =-{\displaystyle {\int }_a^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s}\le {\xi }^{\text{T}}\Omega \xi$. (4)

其中

$W\left(s\right)=\text{col}\left\{x\left(s\right),\stackrel{\dot{}}{x}\left(s\right)\right\}$,$\xi =\text{col}\left\{x\left(b\right),x\left(c\right),x\left(a\right),w_1,w_2,v_1,v_2\right\}$,$w_1=\frac{1}{b-c}{\displaystyle {\int }_c^{b}x\left(s\right)\text{d}s}$,

$w_2=\frac{1}{c-a}{\displaystyle {\int }_a^{c}x\left(s\right)\text{d}s}$,$v_1=\frac{1}{b-c}{\displaystyle {\int }_c^b{\displaystyle {\int }_{\theta }^{b}x\left(s\right)\text{d}s\text{d}\theta }}$,$v_2=\frac{1}{c-a}{\displaystyle {\int }_a^c{\displaystyle {\int }_{\theta }^{c}x\left(s\right)\text{d}s\text{d}\theta }}$,

$e_i=\left[\begin{array}{ccc}{0}_{n,\left(i-1\right)n}& I_n& 0_{n,\left(7-i\right)n}\end{array}\right]\left(i=1,2,\cdots ,7\right)$,${\zeta }_1=\text{col}\left\{e_1-e_2,e_2-e_3\right\}$,

${\zeta }_2=\text{col}\left\{\left(b-c\right)e_4-2e_6,2e_4-e_1-e_2,\left(c-a\right)e_5-2e_7,2e_5-e_2-e_3\right\}$,

$\begin{array}{c}\Omega =\text{sym}\left\{-e_4^{\text{T}}\left(Y_2+X_1\right)\left(e_1-e_2\right)-e_5^{\text{T}}\left(Y_2+X_2\right)\left(e_2-e_3\right)\right\}-\left(b-c\right)e_4^{\text{T}}{Y}_1{e}_4-\left(c-a\right)e_5^{\text{T}}{Y}_1{e}_5\\ -e_2^{\text{T}}\left(X_1-X_2\right)e_2+e_1^{\text{T}}{X}_1{e}_1-e_3^{\text{T}}{X}_2{e}_3-\frac{1}{b-a}{\zeta }_1^{\text{T}}{\Omega }_3{\zeta }_1-\frac{1}{b-a}{\zeta }_2^{\text{T}}{\Omega }_4{\zeta }_2\end{array}$.

证明：对于半正定对称矩阵 $X_i\in R^{n\times n}\left(i=1,2\right)$，有下面的零等式成立

$0=-2{\displaystyle {\int }_c^b{x}^{\text{T}}\left(s\right)X_1\stackrel{\dot{}}{x}\left(s\right)\text{d}s}+x^{\text{T}}\left(b\right)X_{1}x\left(b\right)-x^{\text{T}}\left(c\right)X_{1}x\left(c\right)$,

$0=-2{\displaystyle {\int }_a^c{x}^{\text{T}}\left(s\right)X_2\stackrel{\dot{}}{x}\left(s\right)\text{d}s}+x^{\text{T}}\left(c\right)X_{2}x\left(c\right)-x^{\text{T}}\left(a\right)X_{2}x\left(a\right)$.

将上面两个零等式加入二重积分，合并同类项后可以得到

$\begin{array}{c}\Phi =-{\displaystyle {\int }_a^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s}-2{\displaystyle {\int }_c^b{x}^{\text{T}}\left(s\right)X_1\stackrel{\dot{}}{x}\left(s\right)\text{d}s}-2{\displaystyle {\int }_a^c{x}^{\text{T}}\left(s\right)X_2\stackrel{\dot{}}{x}\left(s\right)\text{d}s}\\ -x^{\text{T}}\left(c\right)\left(X_1-X_2\right)x\left(c\right)+x^{\text{T}}\left(b\right)X_{1}x\left(b\right)-x^{\text{T}}\left(a\right)X_{2}x\left(a\right)\\ =-{\displaystyle {\int }_c^b{W}^{\text{T}}\left(s\right){\Omega }_{1}W\left(s\right)\text{d}s}-{\displaystyle {\int }_a^c{W}^{\text{T}}\left(s\right){\Omega }_{2}W\left(s\right)\text{d}s}-x^{\text{T}}\left(c\right)\left(X_1-X_2\right)x\left(c\right)\\ +x^{\text{T}}\left(b\right)X_{1}x\left(b\right)-x^{\text{T}}\left(a\right)X_{2}x(\; a\; )\end{array}$

利用引理3中的不等式放缩，展开计算如下

$\begin{array}{c}\Phi \le -\frac{1}{b-c}{\left({\displaystyle {\int }_c^{b}W\left(s\right)\text{d}s}\right)}^{\text{T}}{\Omega }_1\left({\displaystyle {\int }_c^{b}W\left(s\right)\text{d}s}\right)-\frac{1}{c-a}{\left({\displaystyle {\int }_a^{c}W\left(s\right)\text{d}s}\right)}^{\text{T}}{\Omega }_2\left({\displaystyle {\int }_a^{c}W\left(s\right)\text{d}s}\right)\\ -\frac{3}{b-c}{\left({\displaystyle {\int }_c^{b}W\left(s\right)\text{d}s}-\frac{2}{b-c}{\displaystyle {\int }_c^b{\displaystyle {\int }_{\beta }^{b}W\left(s\right)\text{d}s\text{d}\beta }}\right)}^{\text{T}}{\Omega }_1\left({\displaystyle {\int }_c^{b}W\left(s\right)\text{d}s}-\frac{2}{b-c}{\displaystyle {\int }_c^b{\displaystyle {\int }_{\beta }^{b}W\left(s\right)\text{d}s\text{d}\beta }}\right)\\ -\frac{3}{c-a}{\left({\displaystyle {\int }_a^{c}W\left(s\right)\text{d}s}-\frac{2}{c-a}{\displaystyle {\int }_a^c{\displaystyle {\int }_{\beta }^{c}W\left(s\right)\text{d}s\text{d}\beta }}\right)}^{\text{T}}{\Omega }_2\left({\displaystyle {\int }_a^{c}W\left(s\right)\text{d}s}-\frac{2}{c-a}{\displaystyle {\int }_a^c{\displaystyle {\int }_{\beta }^{c}W\left(s\right)\text{d}s\text{d}\beta }}\right)\\ -x^{\text{T}}\left(c\right)\left(X_1-X_2\right)x\left(c\right)+x^{\text{T}}\left(b\right)X_{1}x\left(b\right)-x^{\text{T}}\left(a\right)X_{2}x(\; a\; )\end{array}$

$\begin{array}{l}=-\left(b-c\right)w_1^{\text{T}}{Y}_1{w}_1-2w_1^{\text{T}}\left(Y_2+X_1\right)\left(x\left(b\right)-x\left(c\right)\right)-\frac{1}{b-c}{\left(x\left(b\right)-x\left(c\right)\right)}^{\text{T}}{Y}_3\left(x\left(b\right)-x\left(c\right)\right)\\ -\left(c-a\right)w_2^{\text{T}}{Y}_1{w}_2-2w_2^{\text{T}}\left(Y_2+X_2\right)\left(x\left(c\right)-x\left(a\right)\right)-\frac{1}{c-a}{\left(x\left(c\right)-x\left(a\right)\right)}^{\text{T}}{Y}_3\left(x\left(c\right)-x\left(a\right)\right)\\ -\frac{3}{b-c}{\left[\begin{array}{c}\left(b-c\right)w_1-2v_1\\ 2w_1-x\left(b\right)-x\left(c\right)\end{array}\right]}^T\left[\begin{array}{cc}{Y}_1& Y_2+X_1\\ *& Y_3\end{array}\right]\left[\begin{array}{c}\left(b-c\right)w_1-2v_1\\ 2w_1-x\left(b\right)-x\left(c\right)\end{array}\right]\\ -\frac{3}{c-a}{\left[\begin{array}{c}\left(c-a\right)w_2-2v_2\\ 2w_2-x\left(c\right)-x\left(a\right)\end{array}\right]}^T\left[\begin{array}{cc}{Y}_1& Y_2+X_2\\ *& Y_3\end{array}\right]\left[\begin{array}{c}\left(c-a\right)w_2-2v_2\\ 2w_2-x\left(c\right)-x\left(a\right)\end{array}\right]\\ -x^{\text{T}}\left(c\right)\left(X_1-X_2\right)x\left(c\right)+x^{\text{T}}\left(b\right)X_{1}x\left(b\right)-x^{\text{T}}\left(a\right)X_{2}x(\; a\; )\end{array}$

利用交互凸组合不等式可得

$\begin{array}{c}\Phi \le -\left(b-c\right)w_1^{\text{T}}{Y}_1{w}_1-2w_1^{\text{T}}\left(Y_2+X_1\right)\left(x\left(b\right)-x\left(c\right)\right)-\left(c-a\right)w_2^{\text{T}}{Y}_1{w}_2\\ -2w_2^{\text{T}}\left(Y_2+X_2\right)\left(x\left(c\right)-x\left(a\right)\right)-x^{\text{T}}\left(c\right)\left(X_1-X_2\right)x\left(c\right)+x^{\text{T}}\left(b\right)X_{1}x\left(b\right)\\ -x^{\text{T}}\left(a\right)X_{2}x\left(a\right)-\frac{1}{b-a}{\left[\begin{array}{c}x\left(b\right)-x\left(c\right)\\ x\left(c\right)-x\left(a\right)\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}{Y}_3& X_3\\ *& Y_3\end{array}\right]\left[\begin{array}{c}x\left(b\right)-x\left(c\right)\\ x\left(c\right)-x\left(a\right)\end{array}\right]\\ -\frac{3}{b-a}{\left[\begin{array}{c}\left(b-c\right)w_1-2v_1\\ 2w_1-x\left(b\right)-x\left(c\right)\\ \left(c-a\right)w_2-2v_2\\ 2w_2-x\left(c\right)-x\left(a\right)\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}{\Omega }_1& X_4\\ *& {\Omega }_2\end{array}\right]\left[\begin{array}{c}\left(b-c\right)w_1-2v_1\\ 2w_1-x\left(b\right)-x\left(c\right)\\ \left(c-a\right)w_2-2v_2\\ 2w_2-x\left(c\right)-x\left(a\right)\end{array}\right]\\ ={\xi }^{\text{T}}\Omega \xi \end{array}$

结论成立，证毕。

3. 理论结果及证明

在本节当中，基于上面提到的一重积分不等式引理以及广义的自由矩阵积分不等式引理，我们通过恰当地构造Lyapunov泛函分析得到关于系统(1)的新的时滞依赖条件。

为了简化向量和矩阵表示给出以下术语：

${\varpi }_1\left(t\right)=\frac{1}{h\left(t\right)}{\displaystyle {\int }_{t-h\left(t\right)}^{t}x\left(s\right)\text{d}s}$,${\varpi }_2\left(t\right)=\frac{1}{t-h\left(t\right)}{\displaystyle {\int }_{t-h}^{t-h\left(t\right)}x\left(s\right)\text{d}s}$,${\vartheta }_1\left(t\right)=\frac{1}{h\left(t\right)}{\displaystyle {\int }_{t-h\left(t\right)}^t{\displaystyle {\int }_{\theta }^{t}x\left(s\right)\text{d}s\text{d}\theta }}$,

${\vartheta }_2\left(t\right)=\frac{1}{t-h\left(t\right)}{\displaystyle {\int }_{t-h}^{t-h\left(t\right)}{\displaystyle {\int }_{\theta }^{t-h\left(t\right)}x\left(s\right)\text{d}s\text{d}\theta }}$,${\eta }_2\left(t\right)=\text{col}\left\{x\left(t\right),\stackrel{\dot{}}{x}\left(t\right),f\left(W_{0}x\left(t\right)\right)\right\}$,

${\eta }_1\left(t,s\right)=\text{col}\left\{x\left(s\right),\stackrel{\dot{}}{x}\left(s\right),f\left(W_{0}x\left(s\right)\right),x\left(t\right),x\left(t-h\right),{\displaystyle {\int }_s^{t}x\left(\theta \right)\text{d}\theta }\right\}$,${\eta }_3\left(t\right)=\text{col}\left\{x\left(t\right),\stackrel{\dot{}}{x}\left(t\right)\right\}$,

${\xi }_1\left(t\right)=\text{col}\left\{x\left(t\right),x\left(t-h\right),{\varpi }_1\left(t\right),{\vartheta }_1\left(t\right),{\displaystyle {\int }_{t-h}^{t}x\left(s\right)\text{d}s},{\displaystyle {\int }_{t-h\left(t\right)}^{t}f\left(W_{0}x\left(s\right)\right)\text{d}s},{\displaystyle {\int }_{t-h}^{t}f\left(W_{0}x\left(s\right)\right)\text{d}s}\right\}$,

$\begin{array}{l}\xi \left(t\right)=\text{col}\{x\left(t\right),x\left(t-h\left(t\right)\right),x\left(t-h\right),{\varpi }_1\left(t\right),{\varpi }_2\left(t\right),{\vartheta }_1\left(t\right),{\vartheta }_2\left(t\right),\stackrel{\dot{}}{x}\left(t-h\left(t\right)\right),\stackrel{\dot{}}{x}\left(t-h\right),{}_{{}_{}{}^{}}{}^{}\\ f\left(W_{0}x\left(t\right)\right),f\left(W_{0}x\left(t-h\left(t\right)\right)\right),{\displaystyle {\int }_{t-h\left(t\right)}^{t}f\left(W_{0}x\left(s\right)\right)\text{d}s},{\displaystyle {\int }_{t-h}^{t-h\left(t\right)}f\left(W_{0}x\left(s\right)\right)\text{d}s}\}\end{array}$,

$g_{i1}\left(s\right)=f_i\left(s\right)-l_i^{-}s$,$g_{i2}\left(s\right)=l_i^{+}s-f_i\left(s\right)$.

定理1 对于给定的实数h和u，若存在对称正定矩阵 $P\in R^{7n\times 7n}$，$Q_1\in R^{5n\times 5n}$，$Q_2\in R^{5n\times 5n}$，$R\in R^{3n\times 3n}$，$Z=\left[\begin{array}{cc}{Z}_1& Z_2\\ *& Z_3\end{array}\right]\in R^{2n\times 2n}$，$N_i\in R^{n\times n}\left(i=1,2\right)$，半正定对角矩阵 $T_{12}\in R^{n\times n}$，$T_{13}\in R^{n\times n}$，$T_{23}\in R^{n\times n}$，$V_k\in R^{n\times n}\left(k=1,2,3,4\right)$，$D_i\in R^{n\times n}$ 任意矩阵 $M_k\in R^{n\times n}$，$S_1\in R^{3n\times 3n}$，$S_2\in R^{n\times n}$，$S_3\in R^{2n\times 2n}$ 得下列线性矩阵不等式成立

${\Gamma }_m=\left[\begin{array}{cc}{Z}_1& Z_2+N_i\\ *& Z_3\end{array}\right]>0\left(m=1,2\right)$,${\Gamma }_3=\left[\begin{array}{cc}{Z}_3& S_2\\ *& Z_3\end{array}\right]>0$，${\Gamma }_4=\left[\begin{array}{cc}{\Gamma }_1& S_3\\ *& {\Gamma }_2\end{array}\right]>0$,(5)

${\Gamma }_5=\left[\begin{array}{cc}R& S_1\\ *& R\end{array}\right]>0$, (6)

${\Gamma }_6\left(h\left(t\right)=0\right)={\left({\delta }^{\perp }\right)}^{\text{T}}\left[\begin{array}{ccc}\Xi & \sqrt{h}{M}_3& \sqrt{h}{M}_4\\ *& -H& 0\\ *& *& -3H\end{array}\right]\left({\delta }^{\perp }\right)<0$, (7)

${\Gamma }_7\left(h\left(t\right)=h\right)={\left({\delta }^{\perp }\right)}^{\text{T}}\left[\begin{array}{ccc}\Xi & \sqrt{h}{M}_1& \sqrt{h}{M}_2\\ *& -H& 0\\ *& *& -3H\end{array}\right]\left({\delta }^{\perp }\right)<0$, (8)

${\Gamma }_8={\left({\delta }^{\perp }\right)}^{\text{T}}\left(-{\gamma }^2{h}^2{J}_2+{\Gamma }_6\right)\left({\delta }^{\perp }\right)<0$,${\Gamma }_9={\left({\delta }^{\perp }\right)}^{\text{T}}\left(-{\left(1-\gamma \right)}^2{h}^2{J}_2+{\Gamma }_7\right)\left({\delta }^{\perp }\right)<0$,(9)

则时滞神经网络系统(1)是渐近稳定的。

其中

$\Xi \left(h\left(t\right)\right)={\displaystyle \underset{n=1}{\overset{8}{\sum }}{\Pi }_n}$,${\Pi }_1=\text{sym}\left\{G_1^{\text{T}}P{G}_2\right\}$,${\Pi }_3=h^2{G}_9^{\text{T}}R{G}_9-G_{10}^{\text{T}}{\Gamma }_5{G}_{10}$,

${\Pi }_2=G_3^{\text{T}}{Q}_1{G}_3-\left(1-u\right)G_4^{\text{T}}{Q}_1{G}_4+G_3^{\text{T}}{Q}_2{G}_3-G_5^{\text{T}}{Q}_2{G}_5+\text{sym}\left\{G_7^{\text{T}}{Q}_1{G}_6+G_8^{\text{T}}{Q}_2{G}_6\right\}$,

$\begin{array}{c}{\Pi }_4=h{G}_{11}^{\text{T}}Z{G}_{11}-\frac{1}{h}{G}_{12}^{\text{T}}{\Gamma }_3{G}_{12}-\frac{1}{h}{G}_{13}^{\text{T}}{\Gamma }_4{G}_{13}+\text{sym}\left\{-e_4^{\text{T}}\left(Y_2+X_1\right)\left(e_1-e_2\right)-e_5^{\text{T}}\left(Y_2+X_2\right)\left(e_2-e_3\right)\right\}\\ -h\left(t\right)e_4^{\text{T}}{Y}_1{e}_4-\left(h-h\left(t\right)\right)e_5^{\text{T}}{Y}_1{e}_5-e_2^{\text{T}}\left(X_1-X_2\right)e_2+e_1^{\text{T}}{X}_1{e}_1-e_3^{\text{T}}{X}_2{e}_3\end{array}$,

${\Pi }_5=h{f}_{0}H{f}_0+\text{sym}\left\{G_{14}^{\text{T}}{M}_1{G}_{15}+G_{14}^{\text{T}}{M}_2{G}_{16}+G_{14}^{\text{T}}{M}_3{G}_{17}+G_{14}^{\text{T}}{M}_4{G}_{18}\right\}$,

$\begin{array}{l}{\Pi }_6=\text{sym}\left\{\left[{\left(e_{10}-L_2{W}_0{e}_1\right)}^{\text{T}}{V}_1+{\left(L_1{W}_0{e}_1-e_{10}\right)}^{\text{T}}{V}_2\right]W_0{f}_0\right\}\\ +\text{sym}\left\{\left[{\left(e_{12}-L_2{W}_0{e}_3\right)}^{\text{T}}{V}_3+{\left(L_1{W}_0{e}_3-e_{12}\right)}^{\text{T}}{V}_4\right]W_0{e}_9\right\}\end{array}$,

$\begin{array}{c}{\Pi }_7=\text{sym}\left\{{\left[e_{10}-e_{11}-L_2{W}_0\left(e_1-e_2\right)\right]}^{\text{T}}{T}_{12}\left[L_1{W}_0\left(e_1-e_2\right)-\left(e_{10}-e_{11}\right)\right]\right\}\\ +\text{sym}\left\{{\left[e_{10}-e_{12}-L_2{W}_0\left(e_1-e_3\right)\right]}^{\text{T}}{T}_{13}\left[L_1{W}_0\left(e_1-e_3\right)-\left(e_{10}-e_{12}\right)\right]\right\}\\ +\text{sym}\left\{{\left[e_{11}-e_{12}-L_2{W}_0\left(e_2-e_3\right)\right]}^{\text{T}}{T}_{23}\left[L_1{W}_0\left(e_2-e_3\right)-\left(e_{11}-e_{12}\right)\right]\right\}\\ +\text{sym}\left\{{\left(e_{10}-L_2{W}_0{e}_1\right)}^{\text{T}}{T}_1\left(L_1{W}_0{e}_1-e_{10}\right)\right\}+\text{sym}\left\{{\left(e_{11}-L_2{W}_0{e}_2\right)}^{\text{T}}{T}_2\left(L_1{W}_0{e}_2-e_{11}\right)\right\}\\ +\text{sym}\left\{{\left(e_{12}-L_2{W}_0{e}_3\right)}^{\text{T}}{T}_3\left(L_1{W}_0{e}_3-e_{12}\right)\right\}\end{array}$,

${\Pi }_8=\text{sym}\left\{{\left(e_{13}-L_2{e}_4\right)}^{\text{T}}{D}_1\left(L_1{e}_4-e_{13}\right)\right\}+\text{sym}\left\{{\left(e_{14}-L_2{e}_5\right)}^{\text{T}}{D}_2\left(L_1{e}_5-e_{14}\right)\right\}$,

$\Upsilon =h\left(t\right)G_{14}^{\text{T}}\left(\frac{3M_1{H}^{-1}{M}_1^{\text{T}}+M_2{H}^{-1}{M}_2^{\text{T}}}{3}\right)G_{14}+\left(h-h\left(t\right)\right)G_{14}^{\text{T}}\left(\frac{3M_3{H}^{-1}{M}_3^{\text{T}}+M_4{H}^{-1}{M}_4^{\text{T}}}{3}\right)G_{14}$,

$e_i=\left[\begin{array}{ccc}{0}_{n,\left(i-1\right)n}& I_n& 0_{n,\left(9-i\right)n}\end{array}\right]\left(i=1,2,\cdots ,9\right)$,

$G_1=\text{col}\left\{e_1,e_3,h\left(t\right)e_4,h\left(t\right)e_6,h\left(t\right)e_4+\left(h-h\left(t\right)\right)e_5,e_{13},e_{13}+e_{14}\right\}$,$G_4=\text{col}\left\{e_2,e_8,e_{11},e_1,e_3,h\left(t\right)e_4\right\}$,

$G_2=\text{col}\left\{f_0,e_9,e_1-\left(1-u\right)e_2,h\left(t\right)\left(e_1-\left(1-u\right)e_4\right),e_1-e_3,e_{10}-\left(1-u\right)e_{11},e_{10}-e_{12}\right\}$,

$G_3=\text{col}\left\{e_1,f_0,e_{10},e_1,e_3,e_0\right\}$,$G_5=\text{col}\left\{e_3,e_9,e_{12},e_1,e_3,h\left(t\right)e_4+\left(h-h\left(t\right)\right)e_5\right\}$,

$G_6=\text{col}\left\{e_0,e_0,e_0,f_0,e_9,e_1\right\}$,$G_7=\text{col}\left\{h\left(t\right)e_4,e_1-e_2,e_{13},h\left(t\right)e_1,h\left(t\right)e_3,h\left(t\right)e_6\right\}$,

$G_8=\text{col}\left\{h\left(t\right)e_4+\left(h-h\left(t\right)\right)e_5,e_1-e_3,e_{13}+e_{14},h{e}_1,h{e}_3,h\left(t\right)e_6+\left(h-h\left(t\right)\right)e_7+h\left(t\right)\left(h-h\left(t\right)\right)e_4\right\}$,

$G_9=\text{col}\left\{e_1,f_0,e_{10}\right\}$,$G_{10}=\text{col}\left\{h\left(t\right)e_4,e_1-e_2,e_{13},\left(h-h\left(t\right)\right)e_5,e_2-e_3,e_{14}\right\}$,$G_{11}=\text{col}\left\{e_1,f_0\right\}$,

$G_{12}=\text{col}\left\{e_1-e_2,e_2-e_3\right\}$,$G_{13}=\text{col}\left\{h\left(t\right)e_4-2e_6,2e_4-e_1-e_2,\left(h-h\left(t\right)\right)e_5-2e_7,2e_5-e_2-e_3\right\}$,

$G_{14}=\text{col}\left\{e_1,e_2,e_3,e_4,e_5,e_6,e_7,e_8,e_9,e_{10},e_{11},e_{12},e_{13},e_{14}\right\}$,$G_{15}=e_1-e_2$,$G_{16}=e_1+e_2-2e_4$,

$G_{17}=e_2-e_3$,$G_{18}=e_2+e_3-2e_5$,$G_{19}=\text{col}\left\{e_0,e_0,e_4,e_6,e_4-e_5,e_0,e_0\right\}$,

$G_{20}=\text{col}\left\{e_0,e_0,e_0,e_1-\left(1-u\right)e_4,e_0,e_0,e_0\right\}$,$G_{21}=\text{col}\left\{e_0,e_0,e_0,e_0,e_0,e_4\right\}$,

$G_{22}=\text{col}\left\{e_0,e_0,e_0,e_0,e_0,e_4-e_5\right\}$,$G_{23}=\text{col}\left\{e_4,e_0,e_0,-e_5,e_0,e_0\right\}$,$G_{24}=\text{col}\left\{e_4,e_0,-e_5,e_0\right\}$.

证明:构造如下Lyapunov泛函

$V\left(t\right)={\displaystyle \underset{d=1}{\overset{6}{\sum }}{V}_d}$,

$V_1\left(t\right)={\xi }_1^{\text{T}}\left(t\right)P{\xi }_1\left(t\right)$,$V_2\left(t\right)={\displaystyle {\int }_{t-h\left(t\right)}^t{\eta }_1^{\text{T}}\left(t,s\right)Q_1}{\eta }_1\left(t,s\right)\text{d}s+{\displaystyle {\int }_{t-h}^t{\eta }_1^{\text{T}}\left(t,s\right)Q_2}{\eta }_1\left(t,s\right)\text{d}s$,

$V_3\left(t\right)=h{\displaystyle {\int }_{t-h}^t{\displaystyle {\int }_{\theta }^t{\eta }_2^{\text{T}}\left(s\right)R{\eta }_2\left(s\right)\text{d}s\text{d}\theta }}$,$V_4\left(t\right)={\displaystyle {\int }_{t-h}^t{\displaystyle {\int }_{\theta }^t{\eta }_3^{\text{T}}\left(s\right)Z{\eta }_3\left(s\right)\text{d}s\text{d}\theta }}$,

$V_5\left(t\right)={\displaystyle {\int }_{t-h}^t{\displaystyle {\int }_{\theta }^t{\stackrel{\dot{}}{x}}^{\text{T}}\left(s\right)H\stackrel{\dot{}}{x}\left(s\right)\text{d}s\text{d}\theta }}$,

$V_6\left(t\right)=2{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle {\int }_0^{W_{0i}x\left(t\right)}\left[V_{1i}{g}_{i1}\left(s\right)+V_{2i}{g}_{i2}\left(s\right)\right]\text{d}s}}+2{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle {\int }_0^{W_{0i}x\left(t-h\right)}\left[V_{3i}{g}_{i1}\left(s\right)+V_{4i}{g}_{i2}\left(s\right)\right]\text{d}s}}$.

对 $V\left(t\right)$ 求导，可得

${\stackrel{\dot{}}{V}}_1\left(t\right)=2{\xi }_1^{\text{T}}\left(t\right)P{\stackrel{\dot{}}{\xi }}_1\left(t\right)$,${\stackrel{\dot{}}{V}}_3\left(t\right)=h^2{\eta }_2^{\text{T}}\left(t\right)R{\eta }_2\left(t\right)-{\displaystyle {\int }_{t-h}^t{\eta }_2^{\text{T}}\left(s\right)R{\eta }_2\left(s\right)\text{d}s}$,