﻿ 混合时滞随机Hopfield神经网络的均方渐近稳定性

# 混合时滞随机Hopfield神经网络的均方渐近稳定性Mean Square Asymptotic Stability of Stochastic Hopfield Neural Networks with Mixed Delays

Abstract: This paper considers a stochastic Hopfield neural network model with mixed delays, the mixed delays of the model are composed of constant fixed delay and continuously distributed delay. Li and Ding (2017) introduced this model and discussed its properties. In this paper, we will continue to study this model. Therefore, the main purpose of this paper is to obtain the criteria for the mean-square asymptotic stability of stochastic Hopfield neural networks with mixed delays through research and analysis. In addition, the methods we used are Lyapunov function method, Itô’s formula method and inequality method. First of all, we construct a suitable Lyapunov function. Then we apply Itô’s formula to the Lyapunov function. By calculation, we obtain the condition for judging the mean-square asymptotic stability of stochastic Hopfield neural networks with mixed delays. Lastly, we give an example to verify the results we obtained.

1. 引言

2. 模型，符号和假设

$\left\{\begin{array}{l}\text{d}{x}_{i}\left(t\right)=\left[-{c}_{i}{x}_{i}\left(t\right)+{\sum }_{j=1}^{n}{a}_{ij}{f}_{j}\left({x}_{j}\left(t\right)\right)+{\sum }_{j=1}^{n}{b}_{ij}{g}_{j}\left({x}_{j}\left(t-{\tau }_{j}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\sum }_{j=1}^{n}{d}_{ij}{\int }_{-\infty }^{t}{k}_{ij}\left(t-s\right){h}_{j}\left({x}_{j}\left(s\right)\right)\text{d}s\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\sum }_{j=1}^{n}{\sigma }_{ij}\left({x}_{j}\left(t\right),{x}_{j}\left(t-{\tau }_{j}\right),{\int }_{-\infty }^{t}{k}_{ij}\left(t-s\right){\phi }_{j}\left({x}_{j}\left(s\right)\right)\text{d}s\right)\text{d}{\omega }_{j}\left(t\right),\\ {x}_{i}\left(u\right)={\xi }_{i}\left(u\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\infty (1)

$\left\{\begin{array}{l}\text{d}x\left(t\right)=\left[-Cx\left(t\right)+Af\left(x\left(t\right)\right)+Bg\left(x\left(t-\tau \right)\right)+D{\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\sigma \left(x\left(t\right),x\left(t-\tau \right),{\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)\text{d}\omega \left(t\right),\\ x\left(u\right)=\xi \left(u\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\infty (2)

${C}^{2,1}\left[{R}^{n}×{R}^{+}\to {R}^{+}\right]$ 定义了一族 ${R}^{n}×{R}^{+}$ 上对x二次可微，对t一次可微的非负函数 $V\left(x,t\right)$。定义

$\begin{array}{c}LV\left(x\left(t\right),t\right)={V}_{t}\left(x,t\right)+{V}_{x}\left(x,t\right)\left[-Cx\left(t\right)+Af\left(x\left(t\right)\right)+Bg\left(x\left(t-\tau \right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+D{\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}\text{trace}\left[{\sigma }^{\text{T}}\left(x\left(t\right),x\left(t-\tau \right),{\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×{V}_{xx}\left(x,t\right)\sigma \left(x\left(t\right),x\left(t-\tau \right),{\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)\right],\end{array}$ (3)

2.1. 假设

a) 函数 ${f}_{j}\left(x\right)$${g}_{j}\left(x\right)$${h}_{j}\left(x\right)$${\phi }_{j}\left(x\right)$${\sigma }_{ij}\left(x,y,z\right)$ 满足：

${f}_{j}\left(0\right)={g}_{j}\left(0\right)={\phi }_{j}\left(0\right)=0$，且 ${\sigma }_{ij}\left(0,0,0\right)=0$$i,j=1,\cdots ,n$

b) 函数 ${f}_{j}\left(x\right)$${g}_{j}\left(x\right)$${h}_{j}\left(x\right)$${\phi }_{j}\left(x\right)$ 是利普希茨连续的，即，存在非负常数 ${L}_{j}$${M}_{j}$${N}_{j}$${U}_{j}$ 使得

$|{f}_{j}\left(x\right)-{f}_{j}\left(y\right)|\le {L}_{j}|x-y|,$ (4)

$|{g}_{j}\left(x\right)-{g}_{j}\left(y\right)|\le {M}_{j}|x-y|,$ (5)

$|{h}_{j}\left(x\right)-{h}_{j}\left(y\right)|\le {N}_{j}|x-y|,$ (6)

$|{\phi }_{j}\left(x\right)-{\phi }_{j}\left(y\right)|\le {U}_{j}|x-y|.$ (7)

c) 下面的不等式是成立的：

(8)

2.2. 定义

3. 主要的结果

$\begin{array}{l}-2PC+PA{Q}^{-1}{A}^{\text{T}}P+LQL+PB{Q}^{-1}{B}^{\text{T}}P+{P}_{2}\\ +PD{Q}^{-1}{D}^{\text{T}}P+MQM+NQN+{P}_{1}+URU<0,\end{array}$

$\begin{array}{l}LV\left(x\left(t\right),t\right)\\ =2{x}^{\text{T}}\left(t\right)P\left[-Cx\left(t\right)+Af\left(x\left(t\right)\right)+Bg\left(x\left(t-\tau \right)\right)+D{\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{x}^{\text{T}}\left(t\right)\left({P}_{2}+MQM\right)x\left(t\right)-{x}^{\text{T}}\left(t-\tau \right)\left({P}_{2}+MQM\right)x\left(t-\tau \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{q}_{j}{\int }_{0}^{\infty }{k}_{ij}\left(\eta \right){h}_{j}^{2}\left({x}_{j}\left(t\right)\right)\text{d}\eta -\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{q}_{j}{\int }_{0}^{\infty }{k}_{ij}\left(\eta \right){h}_{j}^{2}\left({x}_{j}\left(t-\eta \right)\right)\text{d}\eta \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{r}_{j}{\int }_{0}^{\infty }{k}_{ij}\left(\xi \right){\phi }_{j}^{2}\left({x}_{j}\left(t\right)\right)\text{d}\xi -\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{r}_{j}{\int }_{0}^{\infty }{k}_{ij}\left(\xi \right){\phi }_{j}^{2}\left({x}_{j}\left(t-\xi \right)\right)\text{d}\xi \end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+\text{trace}\left[{\sigma }^{\text{T}}\left(x\left(t\right),x\left(t-\tau \right),{\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }×P\sigma \left(x\left(t\right),x\left(t-\tau \right),{\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)\right]\\ =-2{x}^{\text{T}}\left(t\right)PCx\left(t\right)+2{x}^{\text{T}}\left(t\right)PAf\left(x\left(t\right)\right)+2{x}^{\text{T}}\left(t\right)PBg\left(x\left(t-\tau \right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+2{x}^{\text{T}}PD{\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s+{h}^{\text{T}}\left(x\left(t\right)\right)Qh\left(x\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{x}^{\text{T}}\left(t\right)\left({P}_{2}+MQM\right)x\left(t\right)-{x}^{\text{T}}\left(t-\tau \right)\left({P}_{2}+MQM\right)x\left(t-\tau \right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+2{x}^{\text{T}}PD{\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s+{x}^{\text{T}}\left(t\right)\left({P}_{2}+MQM\right)x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-{x}^{\text{T}}\left(t-\tau \right)\left({P}_{2}+MQM\right)x\left(t-\tau \right)+{h}^{\text{T}}\left(x\left(t\right)\right)Qh\left(x\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{q}_{j}{\int }_{0}^{\infty }{k}_{ij}\left(\eta \right){h}_{j}^{2}\left({x}_{j}\left(t-\eta \right)\right)\text{d}\eta +{\phi }^{\text{T}}\left(x\left(t\right)\right)R\phi \left(x\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-\underset{i=1}{\overset{n}{\sum }}\underset{j=1}{\overset{n}{\sum }}{r}_{j}{\int }_{0}^{\infty }{k}_{ij}\left(\xi \right){\phi }_{j}^{2}\left({x}_{j}\left(t-\xi \right)\right)\text{d}\xi +{x}^{\text{T}}\left(t\right){P}_{1}x\left(t\right)+{x}^{\text{T}}\left(t-\tau \right){P}_{2}x\left(t-\tau \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{\left({\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)}^{\text{T}}R\left({\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)\end{array}$

$\begin{array}{l}=-2{x}^{\text{T}}\left(t\right)PCx\left(t\right)+2{x}^{\text{T}}\left(t\right)PAf\left(x\left(t\right)\right)+2{x}^{\text{T}}\left(t\right)PBg\left(x\left(t-\tau \right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+2{x}^{\text{T}}PD{\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s+{x}^{\text{T}}\left(t\right)\left({P}_{2}+MQM\right)x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-{x}^{\text{T}}\left(t-\tau \right)\left({P}_{2}+MQM\right)x\left(t-\tau \right)+{x}^{\text{T}}\left(t\right)NQNx\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-{\left({\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s\right)}^{\text{T}}Q\left({\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }-{\left({\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)}^{\text{T}}R\left({\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{x}^{\text{T}}\left(t\right)URUx\left(t\right)+{x}^{\text{T}}\left(t\right){P}_{1}x\left(t\right)+{x}^{\text{T}}\left(t-\tau \right){P}_{2}x\left(t-\tau \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }+{\left({\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)}^{T}R\left({\int }_{-\infty }^{t}K\left(t-s\right)\phi \left(x\left(s\right)\right)\text{d}s\right)\end{array}$ (9)

$\begin{array}{c}2{x}^{\text{T}}\left(t\right)PAf\left(x\left(t\right)\right)\le {x}^{\text{T}}\left(t\right)PA{Q}^{-1}{A}^{\text{T}}{P}^{\text{T}}x\left(t\right)+{f}^{\text{T}}\left(x\left(t\right)\right)Qf\left(x\left(t\right)\right)\\ \le {x}^{\text{T}}\left(t\right)\left(PA{Q}^{-1}{A}^{\text{T}}P+LQL\right)x\left(t\right),\end{array}$ (10)

$\begin{array}{c}2{x}^{\text{T}}\left(t\right)PBg\left(x\left(t-\tau \right)\right)\le {x}^{\text{T}}\left(t\right)PB{Q}^{-1}{B}^{\text{T}}{P}^{\text{T}}x\left(t\right)+{g}^{\text{T}}\left(x\left(t-\tau \right)\right)Qg\left(x\left(t-\tau \right)\right)\\ \le {x}^{\text{T}}\left(t\right)\left(PB{Q}^{-1}{B}^{\text{T}}Px\left(t\right)+x\left(t-\tau \right)MQM\right)x\left(t-\tau \right),\end{array}$ (11)

$\begin{array}{l}2{x}^{\text{T}}\left(t\right)PD{\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s\\ \le {x}^{\text{T}}\left(t\right)PD{Q}^{-1}{D}^{\text{T}}Px\left(t\right)+{\left({\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s\right)}^{\text{T}}P\left({\int }_{-\infty }^{t}K\left(t-s\right)h\left(x\left(s\right)\right)\text{d}s\right).\end{array}$ (12)

$\begin{array}{c}LV\left(x\left(t\right),t\right)\le {x}^{\text{T}}\left(t\right)\left[-2PC+PA{Q}^{-1}{A}^{\text{T}}P+LQL+PB{Q}^{-1}{B}^{\text{T}}P+{P}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+PD{Q}^{-1}DP+MQM+NQN+{P}_{1}+URU\right]x\left(t\right)\end{array}$

4. 例子

$\begin{array}{l}\text{d}{x}_{1}\left(t\right)=\left[-{c}_{1}{x}_{1}\left(t\right)+{a}_{11}{f}_{1}\left({x}_{1}\left(t\right)\right)+{a}_{12}{f}_{2}\left({x}_{2}\left(t\right)\right)+{b}_{11}{g}_{1}\left({x}_{1}\left(t-{\tau }_{1}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{b}_{12}{g}_{2}\left({x}_{2}\left(t-{\tau }_{2}\right)\right)+{d}_{11}{\int }_{-\infty }^{t}{k}_{11}\left(t-s\right){h}_{1}\left({x}_{1}\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{d}_{12}{\int }_{-\infty }^{t}{k}_{12}\left(t-s\right){h}_{2}\left({x}_{2}\left(s\right)\right)\text{d}s\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\sigma }_{11}\left({x}_{1}\left(t\right),{x}_{1}\left(t-{\tau }_{1}\right),{\int }_{-\infty }^{t}{k}_{11}\left(t-s\right){\phi }_{1}\left({x}_{1}\left(s\right)\right)\text{d}s\right)\text{d}{\omega }_{1}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\sigma }_{12}\left({x}_{2}\left(t\right),{x}_{2}\left(t-{\tau }_{2}\right),{\int }_{-\infty }^{t}{k}_{12}\left(t-s\right){\phi }_{2}\left({x}_{2}\left(s\right)\right)\text{d}s\right)\text{d}{\omega }_{2}\left(t\right),\end{array}$

$\begin{array}{l}\text{d}{x}_{2}\left(t\right)=\left[-{c}_{2}{x}_{2}\left(t\right)+{a}_{21}{f}_{1}\left({x}_{1}\left(t\right)\right)+{a}_{22}{f}_{2}\left({x}_{2}\left(t\right)\right)+{b}_{21}{g}_{1}\left({x}_{1}\left(t-{\tau }_{1}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{b}_{22}{g}_{2}\left({x}_{2}\left(t-{\tau }_{2}\right)\right)+{d}_{21}{\int }_{-\infty }^{t}{k}_{21}\left(t-s\right){h}_{1}\left({x}_{1}\left(s\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{d}_{22}{\int }_{-\infty }^{t}{k}_{22}\left(t-s\right){h}_{2}\left({x}_{2}\left(s\right)\right)\text{d}s\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\sigma }_{21}\left({x}_{1}\left(t\right),{x}_{1}\left(t-{\tau }_{1}\right),{\int }_{-\infty }^{t}{k}_{21}\left(t-s\right){\phi }_{1}\left({x}_{1}\left(s\right)\right)\text{d}s\right)\text{d}{\omega }_{1}\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\sigma }_{22}\left({x}_{2}\left(t\right),{x}_{2}\left(t-{\tau }_{2}\right),{\int }_{-\infty }^{t}{k}_{22}\left(t-s\right){\phi }_{2}\left({x}_{2}\left(s\right)\right)\text{d}s\right)\text{d}{\omega }_{2}\left(t\right),\end{array}$ (13)

$C=\left(\begin{array}{cc}{c}_{11}& 0\\ 0& {c}_{22}\end{array}\right)=\left(\begin{array}{cc}6& 0\\ 0& 8\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}A=\left(\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right)=\left(\begin{array}{cc}0.2& 0.1\\ 0.3& 0.4\end{array}\right),$

$B=\left(\begin{array}{cc}{b}_{11}& {b}_{12}\\ {b}_{21}& {b}_{22}\end{array}\right)=\left(\begin{array}{cc}0.4& 0.1\\ 0.2& 0.2\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}D=\left(\begin{array}{cc}{d}_{11}& {d}_{12}\\ {d}_{21}& {d}_{22}\end{array}\right)=\left(\begin{array}{cc}0.1& 0.2\\ 0.4& 0.3\end{array}\right).$

$\begin{array}{l}{\sigma }_{12}\left({x}_{2}\left(t\right),{x}_{2}\left(t-{\tau }_{2}\right),{\int }_{-\infty }^{t}{k}_{12}\left(t-s\right){\phi }_{2}\left({x}_{2}\left(s\right)\right)\text{d}s\right)\\ =\frac{{x}_{2}\left(t\right)}{4}+\frac{{x}_{2}\left(t-{\tau }_{2}\right)}{2}+\frac{1}{2}{\int }_{-\infty }^{t}{k}_{12}\left(t-s\right){\phi }_{2}\left({x}_{2}\left(s\right)\right)\text{d}s,\end{array}$

$\begin{array}{l}{\sigma }_{21}\left({x}_{1}\left(t\right),{x}_{1}\left(t-{\tau }_{1}\right),{\int }_{-\infty }^{t}{k}_{21}\left(t-s\right){\phi }_{1}\left({x}_{1}\left(s\right)\right)\text{d}s\right)\\ =\frac{{x}_{1}\left(t\right)}{2}+\frac{{x}_{1}\left(t-{\tau }_{1}\right)}{4}+\frac{1}{2}{\int }_{-\infty }^{t}{k}_{21}\left(t-s\right){\phi }_{1}\left({x}_{1}\left(s\right)\right)\text{d}s,\end{array}$

$\begin{array}{l}{\sigma }_{22}\left({x}_{2}\left(t\right),{x}_{2}\left(t-{\tau }_{2}\right),{\int }_{-\infty }^{t}{k}_{22}\left(t-s\right){\phi }_{2}\left({x}_{2}\left(s\right)\right)\text{d}s\right)\\ =\frac{{x}_{1}\left(t\right)}{2}+\frac{{x}_{1}\left(t-{\tau }_{1}\right)}{2}+\frac{1}{4}{\int }_{-\infty }^{t}{k}_{22}\left(t-s\right){\phi }_{2}\left({x}_{2}\left(s\right)\right)\text{d}s.\end{array}$

${f}_{j}\left(x\right)={g}_{j}\left(x\right)={h}_{j}\left(x\right)={\phi }_{j}\left(x\right)=\mathrm{tanh}\left(x\right)$${k}_{ij}\left(t\right)={\text{e}}^{-t}$$i,j=1,2$${\tau }_{1}={\tau }_{2}=1$，然后我们有 ${L}_{1}={L}_{2}=1$${M}_{1}={M}_{2}=1$${N}_{1}={N}_{2}=1$${U}_{1}={U}_{2}=1$

$\begin{array}{l}-2PC+PA{Q}^{-1}{A}^{\text{T}}P+LQL+PB{Q}^{-1}{B}^{\text{T}}P+{P}_{2}\\ +PD{Q}^{-1}{D}^{\text{T}}P+MQM+NQN+{P}_{1}+URU=\left(\begin{array}{cc}-7.43& 0.3\\ 0.3& -11.02\end{array}\right)<0,\end{array}$

5. 结论

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https://doi.org/10.1073/pnas.79.8.2554

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