云南民族大学数学与计算机科学学院，云南 昆明

收稿日期：2021年7月15日；录用日期：2021年8月18日；发布日期：2021年8月25日

摘要

本文主要研究一类新的二重积分不等式并用于时滞系统的稳定性分析。首先，通过引入零等式得到新的二重积分不等式。然后，利用增广向量构造李雅普诺夫泛函(LKF)，再通过不等式对泛函导数中的积分项进行处理，得到保守性更低的稳定性条件。接着用更加宽松的二次函数负决定引理得到新的稳定条件。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

时变时滞，增广的李雅普诺夫泛函，自由矩阵不等式，时滞分割

A Novel Double Integral Inequalities Applied to Time-Delay Systems

Yongjia Ye, Lianglin Xiong, Haiyang Zhang

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

Received: Jul. 15^th, 2021; accepted: Aug. 18^th, 2021; published: Aug. 25^th, 2021

ABSTRACT

This article is concerned with a novel double integral inequalities applied to time-delay systems. Firstly, two zero-value equations have been introduced to estimate the upper bound of double integral inequality. Secondly, augmented vectors are used to construct Lyapunov function, and the inequality is utilized to estimate the derivative of functional, and then relax quadratic function negative-determination lemma is used to obtain stability criterion. Finally, examples are given to show the effectiveness of the obtained result.

Keywords:Time-Varying Delay, Augmented Lyapunov Functional, Free-Base-Matrix Inequality, Delay Partition

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

时滞是现实世界中不可避免的一种自然现象。我们在很多领域都会遇到时滞，例如信号处理，模式识别，组合优化等等 [1] [2] [3]。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]，因此研究时滞系统的稳定性是非常有意义的。

到目前为止，研究具有低保守性稳定性标准的时滞系统依旧是个热门话题。学者们针对时滞系统的稳定性研究主要采用李雅普诺夫函数方法。通过利用李雅普诺夫稳定性理论 [5]，构造恰当的LKF以及尽可能紧密地估计泛函的导数来得到低保守性的稳定性标准。对于构造LKF方法，目前比较常见的是构造带有增广的LKF [6]，以及利用时滞分割方法构造泛函 [7] 和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下界。为了处理这些积分，学者们研究了许多估计技术，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。虽然这些不等式有效地降低系统的保守性，但是还有改进的空间，文献 [14] 中通过引入增广的LKF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，在目前的工作中利用零等式构造具有增广的二重积分不等式，目前尚未被考虑。事实上，高阶积分项对降低时滞系统的保守性扮演着十分重要的作用，例如构造带有三重积分的LKF，在对这个泛函求导后会产生二重积分项，估计这个二重积分项会将一重积分状态信息以及系统的状态信息引入并使它们与二重积分状态信息相互交叉，因此建立一个新的带有自由矩阵的具有增广的二重积分不等式对降低时滞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相关项能有效克服系统的保守性，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LKF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LKF，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 $n\times m$ 的矩阵值； $\text{sym}\left\{X\right\}=X+X^{\text{T}}$ ； $\text{diag}\{\cdots \}$ 表示对角矩阵； $\text{col}\{\cdots \}$ 表示列向量； $X^{\text{T}}$ 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线性系统

$\{\begin{array}{l}\stackrel{\dot{}}{x}\left(t\right)=Ax\left(t\right)+Bx\left(t-h\left(t\right)\right),\\ x\left(t\right)=\phi \left(t\right),t\in \left[-h,0\right]\end{array}$ (1)

其中 $x\left(t\right)\in R^n$ 是系统的状态向量； $A\in R^{n\times n}$ 和 $B\in R^{n\times n}$ 是已知的实数矩阵；初始条件 $\phi \left(t\right)$ 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)

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

$T_1=f\left(h_i\right)<0\left(i=1,2\right)$，$T_2=-{\beta }^2{h}_{12}^2{a}_2+f\left(h_1\right)<0$，$T_3=-{\left(1-\beta \right)}^2{h}_{12}^2{a}_2+f\left(h_2\right)<0$，(3)

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] $w\left(t\right)$ 是一个可微函数： $\left[a,b\right]\to R^n$，若存在对称正定矩阵 $H\in R^{n\times n}$，对于任意矩阵L，M，则下列的不等式成立：

${\displaystyle {\int }_a^b{w}^{\text{T}}\left(s\right)Hw\left(s\right)\text{d}s}\ge -\text{sym}\left\{{\sigma }_0^{\text{T}}L{\sigma }_1+{\sigma }_0^{\text{T}}M{\sigma }_2\right\}-\left(b-a\right){\sigma }_0^{\text{T}}\left(\frac{3L{H}^{-1}{L}^{\text{T}}+M{H}^{-1}{M}^{\text{T}}}{3}\right){\sigma }_0$ (4)

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

引理3 [9] 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

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

${\displaystyle {\int }_a^b{\displaystyle {\int }_{\beta }^b{x}^{\text{T}}\left(s\right)Rx\left(s\right)\text{d}s\text{d}\beta }}\ge \frac{2}{{\left(b-a\right)}^2}{\left({\displaystyle {\int }_a^b{\displaystyle {\int }_{\beta }^{b}x\left(s\right)\text{d}s\text{d}\beta }}\right)}^{\text{T}}R\left({\displaystyle {\int }_a^b{\displaystyle {\int }_{\beta }^{b}x\left(s\right)\text{d}s\text{d}\beta }}\right)$，

其中 ${\Omega }_1={\displaystyle {\int }_a^{b}x\left(s\right)\text{d}s}-\frac{2}{b-a}{\displaystyle {\int }_a^b{\displaystyle {\int }_{\beta }^{b}x\left(s\right)\text{d}s\text{d}\beta }}$。

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

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

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

$-{\displaystyle {\int }_a^b{\displaystyle {\int }_{\theta }^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}\le {\xi }^{\text{T}}\Omega \xi$ (5)

其中

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

$w_2=\frac{1}{c-a}{\displaystyle {\int }_a^{c}x\left(s\right)\text{d}s}$，$v_1=\frac{1}{{\left(b-c\right)}^2}{\displaystyle {\int }_c^b{\displaystyle {\int }_{\theta }^{b}x\left(s\right)\text{d}s\text{d}\theta }}$，$v_2=\frac{1}{{\left(c-a\right)}^2}{\displaystyle {\int }_a^c{\displaystyle {\int }_{\theta }^{c}x\left(s\right)\text{d}s\text{d}\theta }}$，

$\begin{array}{c}\Omega =\left(c-a\right)e_2^{\text{T}}{X}_2{e}_2+\left(b-c\right)e_1^{\text{T}}{X}_1{e}_1-\left(b-c\right)e_4^{\text{T}}{X}_1{e}_4-\left(c-a\right)e_5^{\text{T}}{X}_2{e}_5-\left(c-a\right)\left(b-a\right)e_4^{\text{T}}{Y}_1{e}_4\\ -\text{sym}\{{\left(b-c\right)}^2{e}_6^{\text{T}}{Y}_1{e}_6+2\left(b-a\right)e_6^{\text{T}}\left(Y_1+X_1\right)\left(e_1-e_4\right)+{\left(e_1-e_4\right)}^{\text{T}}{Y}_3\left(e_1-e_4\right)+{\left(c-a\right)}^2{e}_7^{\text{T}}{Y}_1{e}_7\\ +2\left(c-a\right)e_7^{\text{T}}\left(Y_2+X_2\right)\left(e_2-e_5\right)+{\left(e_2-e_5\right)}^{\text{T}}{Y}_3\left(e_2-e_5\right)+\left(c-a\right)e_4^{\text{T}}{Y}_2\left(e_1-e_2\right)\}\\ -\frac{c-a}{b-a}{\left(e_1-e_2\right)}^{\text{T}}{Y}_3\left(e_1-e_2\right),\end{array}$

$e_i=\left[\begin{array}{ccc}{0}_{n,\left(i-1\right)n}& I_n& 0_{n,\left(7-i\right)n}\end{array}\right]\left(i=1,2,\cdots ,7\right)$。

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

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

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

将上面两个零等式加入 $-{\displaystyle {\int }_a^b{\displaystyle {\int }_{\theta }^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}$。

可得

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

其中

$-{\displaystyle {\int }_a^b{\displaystyle {\int }_{\theta }^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}=-{\displaystyle {\int }_a^c{\displaystyle {\int }_{\theta }^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}-{\displaystyle {\int }_c^b{\displaystyle {\int }_{\theta }^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}$

$-{\displaystyle {\int }_a^c{\displaystyle {\int }_{\theta }^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}=-{\displaystyle {\int }_a^c{\displaystyle {\int }_{\theta }^{\text{c}}{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}-{\displaystyle {\int }_a^c{\displaystyle {\int }_c^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}$

$-{\displaystyle {\int }_c^b{\displaystyle {\int }_{\theta }^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}=-{\displaystyle {\int }_c^b{\displaystyle {\int }_{\theta }^c{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}-{\displaystyle {\int }_c^b{\displaystyle {\int }_c^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s\text{d}\theta }}$

合并同类项后可以得到

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

利用Jensen不等式可得

$-{\displaystyle {\int }_c^b{x}^{\text{T}}\left(s\right)X_{1}x\left(s\right)\text{d}s}\le -\left(b-c\right)w_1^{\text{T}}{X}_1{w}_1$

$-{\displaystyle {\int }_a^c{x}^{\text{T}}\left(s\right)X_{2}x\left(s\right)\text{d}s}\le -\left(c-a\right)w_2^{\text{T}}{X}_2{w}_2$

由引理3，可以得到

$-{\displaystyle {\int }_c^b{\displaystyle {\int }_{\theta }^b{W}^{\text{T}}\left(s\right){\Omega }_{1}W\left(s\right)\text{d}s\text{d}\theta }}\le -2{\left[\begin{array}{c}\left(b-c\right)v_1\\ x\left(b\right)-w_1\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}{Y}_1& Y_2+X_1\\ *& Y_3\end{array}\right]\left[\begin{array}{c}\left(b-c\right)v_1\\ x\left(b\right)-w_1\end{array}\right]$

$-{\displaystyle {\int }_a^c{\displaystyle {\int }_{\theta }^c{W}^{\text{T}}\left(s\right){\Omega }_{2}W\left(s\right)\text{d}s\text{d}\theta }}\le -2{\left[\begin{array}{c}\left(c-a\right)v_2\\ x\left(c\right)-w_2\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}{Y}_1& Y_2+X_2\\ *& Y_3\end{array}\right]\left[\begin{array}{c}\left(c-a\right)v_2\\ x\left(c\right)-w_2\end{array}\right]$

$\begin{array}{c}-\left(c-a\right){\displaystyle {\int }_c^b{W}^{\text{T}}\left(s\right){\Omega }_{0}W\left(s\right)\text{d}s}\le -\left(c-a\right)\{\frac{1}{b-c}{\left[\begin{array}{c}\left(b-c\right)w_1\\ x\left(b\right)-x\left(c\right)\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}{Y}_1& Y_2\\ *& Y_3\end{array}\right]\left[\begin{array}{c}\left(b-c\right)w_1\\ x\left(b\right)-x\left(c\right)\end{array}\right]\\ +\frac{3}{b-c}{\left[\begin{array}{c}\left(b-c\right)\left(w_1-2v_1\right)\\ 2w_1-x\left(b\right)-x\left(c\right)\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}{Y}_1& Y_2\\ *& Y_3\end{array}\right]\left[\begin{array}{c}\left(b-c\right)\left(w_1-2v_1\right)\\ 2w_1-x\left(b\right)-x\left(c\right)\end{array}\right]\}\end{array}$

整理可得

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

同时加上和减去 ${\left(x\left(c\right)-x\left(a\right)\right)}^{\text{T}}{Y}_3\left(x\left(c\right)-x\left(a\right)\right)+3{\left(2w_2-x\left(c\right)-x\left(a\right)\right)}^{\text{T}}{Y}_3\left(2w_2-x\left(c\right)-x\left(a\right)\right)$，运用互凸组合可以得到

$\begin{array}{c}\Phi \le -2{\left(b-c\right)}^2{v}_1^{\text{T}}{Y}_1{v}_1-4\left(b-c\right)v_1^{\text{T}}\left(Y_1+X_2\right)\left(x\left(b\right)-w_1\right)-2{\left(x\left(b\right)-w_1\right)}^{\text{T}}{Y}_3\left(x\left(b\right)-w_1\right)\\ -2{\left(c-a\right)}^2{v}_2^{\text{T}}{Y}_1{v}_2-4\left(c-a\right)v_2^{\text{T}}\left(Y_2+X_2\right)\left(x\left(c\right)-w_2\right)-2{\left(x\left(c\right)-w_2\right)}^{\text{T}}{Y}_3\left(x\left(c\right)-w_2\right)\\ -\left(c-a\right)\left(b-c\right)w_1^{\text{T}}{Y}_1{w}_1-2\left(c-a\right)w_1^{\text{T}}{Y}_2\left(x\left(b\right)-x\left(c\right)\right)-\frac{c-a}{b-a}{\left(x\left(b\right)-x\left(c\right)\right)}^{\text{T}}{Y}_3\left(x\left(b\right)-x\left(c\right)\right)\\ -3\left(c-a\right)\left(b-c\right){\left(w_1-2v_1\right)}^{\text{T}}{Y}_1\left(w_1-2v_1\right)-6\left(c-a\right){\left(w_1-2v_1\right)}^{\text{T}}{Y}_2\left(2w_1-x\left(b\right)-x\left(c\right)\right)\\ -\frac{3\left(c-a\right)}{b-c}{\left(2w_1-x\left(b\right)-x\left(c\right)\right)}^{\text{T}}{Y}_3\left(2w_1-x\left(b\right)-x\left(c\right)\right)+\left(b-c\right)x^{\text{T}}\left(b\right)X_{1}x\left(b\right)\\ +\left(c-a\right)x^{\text{T}}\left(c\right)X_{2}x\left(c\right)-\left(b-c\right)w_1^{\text{T}}{X}_1{w}_1-\left(c-a\right)w_2^{\text{T}}{X}_2{w}_2\end{array}$

因此，结论成立，证毕。

目前已有的文献中，处理带增广的二重积分不等式都是直接用Jenson不等式处理，带有很强的保守性。在本篇文章中，结合前人研究以及所学知识，通过引入两个零等式来估计积分的上限，引入自由矩阵让这个积分拥有更多的自由度。

为了便于后面的研究，需要对带有分母的积分项进行处理，因此在这个不等式中，引入 ${\left(x\left(c\right)-x\left(a\right)\right)}^{\text{T}}{Y}_3\left(x\left(c\right)-x\left(a\right)\right)+3{\left(2w_2-x\left(c\right)-x\left(a\right)\right)}^{\text{T}}{Y}_3\left(2w_2-x\left(c\right)-x\left(a\right)\right)$ 让它与后面这一项组合 $-\left(c-a\right)\left\{\frac{1}{b-a}{\left(x\left(b\right)-x\left(c\right)\right)}^{\text{T}}{Y}_3\left(x\left(b\right)-x\left(c\right)\right)+\frac{3}{b-c}{\left(2w_1-x\left(b\right)-x\left(c\right)\right)}^{\text{T}}{Y}_3\left(2w_1-x\left(b\right)-x\left(c\right)\right)\right\}$，然后巧妙地 运用互凸组合进行化简和放缩。通过这一系列的处理后，我们可以得到一个拥有多个松弛矩阵和大量状态向量交叉信息的具有增广的二重积分不等式。利用这个不等式对二重积分进行放缩，在降低保守性具有非常良好的效果。

3. 理论结果及证明

在这个部分当中，基于上面提到的二重积分不等式引理，我们通过恰当地构造李雅普诺夫泛函得到关于系统(1)的新的稳定标准。

为了简化向量和矩阵表示给出以下术语：

${\xi }_1\left(t\right)=\text{col}\left\{x\left(t\right),{\displaystyle {\int }_{t-h}^{t}x\left(s\right)\text{d}s}\right\}$，${\xi }_2\left(t\right)=\text{col}\left\{x\left(t\right),\stackrel{\dot{}}{x}\left(t\right)\right\}$，${\xi }_3\left(t\right)=\text{col}\left\{x\left(s\right),{\displaystyle {\int }_s^{t}x\left(u\right)\text{d}u}\right\}$，

${\xi }_4\left(t\right)=\text{col}\left\{x\left(t\right)-x\left(t-h\left(t\right)\right),x\left(t\right)+x\left(t-h\left(t\right)\right)-2w_1,x\left(t\right)-x\left(t-h\left(t\right)\right)+6w_1-12v_1\right\}$，

${\xi }_5\left(t\right)=\text{col}\left\{x\left(t-h\left(t\right)\right)-x\left(t-h\right),x\left(t-h\left(t\right)\right)+x\left(t-h\right)-2w_2,x\left(t-h\left(t\right)\right)-x\left(t-h\right)+6w_2-12v_2\right\}$，

$r_1=\frac{1}{h\left(t\right)}{\displaystyle {\int }_{t-h\left(t\right)}^{t}x\left(s\right)\text{d}s}$，$r_2=\frac{1}{t-h\left(t\right)}{\displaystyle {\int }_{t-h}^{t-h\left(t\right)}x\left(s\right)\text{d}s}$，

$t_1=\frac{1}{h{\left(t\right)}^2}{\displaystyle {\int }_{t-h\left(t\right)}^t{\displaystyle {\int }_{\theta }^{t}x\left(s\right)\text{d}s\text{d}\theta }}$，$t_2=\frac{1}{{\left(t-h\left(t\right)\right)}^2}{\displaystyle {\int }_{t-h}^{t-h\left(t\right)}{\displaystyle {\int }_{\theta }^{t-h\left(t\right)}x\left(s\right)\text{d}s\text{d}\theta }}$

定理1 对于给定的实数h和u，若存在对称正定矩阵 $U\in R^{2n\times 2n}$，$Q_i\in R^{2n\times 2n}\left(i=1,2,3,4\right)$，$R\in R^{n\times n}$，$P\in R^{2n\times 2n}$，$Z_i\in R^{n\times n}\left(i=1,2,3\right)$，$Z=\left[\begin{array}{cc}{Z}_1& Z_2\\ *& Z_3\end{array}\right]\in R^{2n\times 2n}$，$M_i\in R^{n\times n}\left(i=1,2\right)$，任意矩阵 $S\in R^{3n\times 3n}$，$L_i\in R^{9n\times 2n}\left(i=1,2\right)$，$M_j\in R^{9n\times 2n}\left(j=3,4\right)$，使得下列线性矩阵不等式成立：

${\Gamma }_1=\left[\begin{array}{cc}\bar{R}& S\\ S& \bar{R}\end{array}\right]>0$ (5)

${\Gamma }_2=\left[\begin{array}{cc}{Z}_1& Z_2+M_i\\ *& Z_3\end{array}\right]>0\left(i=1,2\right)$ (6)

${\Gamma }_3\left(h\left(t\right)=0\right)={\left({\delta }^{\perp }\right)}^{\text{T}}\left[\begin{array}{ccccc}\Xi \left(h\left(t\right)\right)& \sqrt{h\left(t\right)}{L}_1& \sqrt{h\left(t\right)}{M}_3& \sqrt{h-h\left(t\right)}{L}_2& \sqrt{h-h\left(t\right)}{M}_4\\ *& -P& 0& 0& 0\\ *& *& -\frac{1}{3}P& 0& 0\\ *& *& *& -P& 0\\ *& *& *& *& -\frac{1}{3}P\end{array}\right]\left({\delta }^{\perp }\right)<0$ (7)

${\Gamma }_4\left(h\left(t\right)=h\right)={\left({\delta }^{\perp }\right)}^{\text{T}}\left[\begin{array}{ccccc}\Xi \left(h\left(t\right)\right)& \sqrt{h\left(t\right)}{L}_1& \sqrt{h\left(t\right)}{M}_3& \sqrt{h-h\left(t\right)}{L}_2& \sqrt{h-h\left(t\right)}{M}_4\\ *& -P& 0& 0& 0\\ *& *& -\frac{1}{3}P& 0& 0\\ *& *& *& -P& 0\\ *& *& *& *& -\frac{1}{3}P\end{array}\right]\left({\delta }^{\perp }\right)<0$ (8)

${\Gamma }_5={\left({\delta }^{\perp }\right)}^{\text{T}}\left(-{\beta }^2{h}_2^2{J}_2+{\Gamma }_3\right)\left({\delta }^{\perp }\right)<0$ (9)

${\Gamma }_6={\left({\delta }^{\perp }\right)}^{\text{T}}\left(-{\left(1-\beta \right)}^2{h}_2^2{J}_2+{\Gamma }_4\right)\left({\delta }^{\perp }\right)<0$ (10)

则时变时滞系统(1)是渐近稳定的，其中

$\Xi \left(h\left(t\right)\right)={\displaystyle \underset{n=1}{\overset{6}{\sum }}{\Pi }_n}$，$\xi \left(t\right)=\text{col}\left\{x\left(t\right),x\left(t-h\left(t\right)\right),x\left(t-h\right),w_1,w_2,v_1,v_2,\stackrel{\dot{}}{x}\left(t-h\left(t\right)\right),\stackrel{\dot{}}{x}\left(t-h\right)\right\}$，

$\bar{R}=\text{diag}\left\{R,3R,5R\right\}$，$e_i=\left[\begin{array}{ccc}{0}_{n,\left(i-1\right)n}& I_n& 0_{n,\left(9-i\right)n}\end{array}\right]\left(i=1,2,\cdots ,9\right)$，$f_0=A{e}_1+B{e}_2$，$G_4=\text{col}\left\{e_2,e_8\right\}$，

$G_1=\text{col}\left\{e_1,h\left(t\right)e_4+\left(h-h\left(t\right)\right)e_5\right\}$，$G_2=\text{col}\left\{f_0,e_1-e_3\right\}$，$G_3=\text{col}\left\{e_1,f_0\right\}$，$G_{19}=\text{col}\left\{e_0,e_4\right\}$，

$G_5=\text{col}\left\{e_3,e_9\right\}$，$G_6=\text{col}\left\{e_1,e_0\right\}$，$G_7=\text{col}\left\{e_2,h^2\left(t\right)e_4\right\}$，$G_8=\text{col}\left\{e_3,h\left(t\right)e_4+\left(h-h\left(t\right)\right)e_5\right\}$，

$G_9=\text{col}\left\{e_0,e_1\right\}$，$G_{10}=\text{col}\left\{h\left(t\right)e_4,h\left(t\right)e_6\right\}$，$G_{18}=\text{col}\left\{-\left(h-h\left(t\right)\right)e_5+2\left(h-h\left(t\right)\right)e_7,e_2+e_3-2e_5\right\}$，

$G_{11}=\text{col}\left\{h\left(t\right)e_4+\left(h-h\left(t\right)\right)e_5,h^2\left(t\right)e_6+{\left(h-h\left(t\right)\right)}^2{e}_7+h\left(t\right)\left(h-h\left(t\right)\right)e_4\right\}$，$G_{20}=\text{col}\left\{e_0,e_4-e_5\right\}$，

$G_{12}=\text{col}\left\{e_1-e_2,e_1+e_2-2e_4,e_1-e_2+6e_4-12e_6\right\}$，$G_{17}=\text{col}\left\{\left(h-h\left(t\right)\right)e_5,e_2-e_3\right\}$，$e_0=0_{n\times 9n}$，

$G_{13}=\text{col}\left\{e_2-e_3,e_2+e_3-2e_5,e_2-e_3+6e_5-12e_7\right\}$，$G_{16}=\text{col}\left\{-h\left(t\right)e_4+2h\left(t\right)e_6,e_1+e_2-2e_4\right\}$，

$G_{14}=\text{col}\left\{e_1,e_2,e_3,e_4,e_5,e_6,e_7,e_8,e_9\right\}$，$G_{15}=\text{col}\left\{h\left(t\right)e_4,e_1-e_2\right\}$，$G_{21}=\text{col}\left\{e_0,e_6+e_7-e_4\right\}$，

${\Pi }_1=\text{sym}\left\{G_{1}U{G}_2\right\}$，${\Pi }_2=G_3^{\text{T}}{Q}_1{G}_3-\left(1-u\right)G_4^{\text{T}}{Q}_1{G}_4+G_3^{\text{T}}{Q}_2{G}_3-G_5^{\text{T}}{Q}_2{G}_5$，${\Pi }_6=\frac{h^2}{2}{G}_3^{\text{T}}Z{G}_3+{\Upsilon }_1$，

${\Pi }_3=G_6^{\text{T}}{Q}_3{G}_6-\left(1-u\right)G_7^{\text{T}}{Q}_3{G}_7+G_6^{\text{T}}{Q}_4{G}_6-G_8^{\text{T}}{Q}_4{G}_8+\text{sym}\left\{G_{10}^{\text{T}}{Q}_3{G}_9+G_{11}^{\text{T}}{Q}_4{G}_9\right\}$，

${\Pi }_4=h^2{f}_{0}R{f}_0-{\left[\begin{array}{c}{G}_{12}\\ G_{13}\end{array}\right]}^{\text{T}}{\Gamma }_1\left[\begin{array}{c}{G}_{12}\\ G_{13}\end{array}\right]$，${\Pi }_5=h{G}_3^{\text{T}}P{G}_3+\text{sym}\left\{{\eta }_0^{\text{T}}{L}_1{\eta }_1+{\eta }_0^{\text{T}}{M}_3{\eta }_2+{\eta }_0^{\text{T}}{L}_2{\eta }_3+{\eta }_0^{\text{T}}{M}_4{\eta }_4\right\}$，

$\begin{array}{c}{\Upsilon }_1=\left(h-h\left(t\right)\right)e_2^{\text{T}}{M}_2{e}_2+h\left(t\right)e_1^{\text{T}}{M}_1{e}_1-h\left(t\right)e_4^{\text{T}}{M}_1{e}_4-\left(h-h\left(t\right)\right)e_5^{\text{T}}{M}_2{e}_5-h\left(t\right)\left(h-h\left(t\right)\right)e_4^{\text{T}}{Z}_1{e}_4\\ -\text{sym}\{h{\left(t\right)}^2{e}_6^{\text{T}}{Z}_1{e}_6+2h\left(t\right)e_6^{\text{T}}\left(Z_1+M_1\right)\left(e_1-e_4\right)+{\left(e_1-e_4\right)}^{\text{T}}{Z}_3\left(e_1-e_4\right)+{\left(h-h\left(t\right)\right)}^2{e}_7^{\text{T}}{Z}_1{e}_7\\ +2\left(h-h\left(t\right)\right)e_7^{\text{T}}\left(Z_2+M_2\right)\left(e_2-e_5\right)+{\left(e_2-e_5\right)}^{\text{T}}{Z}_3\left(e_2-e_5\right)+\left(c-a\right)e_4^{\text{T}}{Z}_2\left(e_1-e_2\right)\}\\ -\frac{h-h\left(t\right)}{h}{\left(e_1-e_2\right)}^{\text{T}}{Z}_3\left(e_1-e_2\right),\end{array}$

${\Upsilon }_2=\text{sym}\left\{h\left(t\right)G_{14}^{\text{T}}\left(\frac{3L_1{P}^{-1}{L}_1+M_3{P}^{-1}{M}_3}{3}\right)G_{14}+\left(h-h\left(t\right)\right)G_{14}^{\text{T}}\left(\frac{3L_2{P}^{-1}{L}_2+M_4{P}^{-1}{M}_4}{3}\right)G_{14}\right\}$ (11)

证明：构造如下李雅普诺夫泛函

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

$V_1\left(t\right)={\xi }_1^{\text{T}}\left(t\right)U{\xi }_1(\; t\; )$

$V_2\left(t\right)={\displaystyle {\int }_{t-h\left(t\right)}^t{\xi }_2^{\text{T}}\left(s\right)Q_1}{\xi }_2\left(s\right)\text{d}s+{\displaystyle {\int }_{t-h}^t{\xi }_2^{\text{T}}\left(s\right)Q_2}{\xi }_2\left(s\right)\text{d}s$

$V_3={\displaystyle {\int }_{t-h\left(t\right)}^t{\xi }_3^{\text{T}}\left(t,s\right)Q_3}{\xi }_3\left(t,s\right)\text{d}s+{\displaystyle {\int }_{t-h}^t{\xi }_3^{\text{T}}\left(t,s\right)Q_4}{\xi }_3\left(t,s\right)\text{d}s$，$V_4=h{\displaystyle {\int }_{t-h}^t{\displaystyle {\int }_{\theta }^t\stackrel{\dot{}}{x}\left(s\right)R\stackrel{\dot{}}{x}\left(s\right)\text{d}s\text{d}\theta }}$，