Vol. 07  No. 12 ( 2018 ), Article ID: 28302 , 9 pages
10.12677/AAM.2018.712194

A New Criterion for Stability of Hopfield Neural Network Based on Gronwall Integral Inequality

Xingshou Huang, Ricai Luo, Wusheng Wang

School of Mathematics and Statistics, Hechi University, Yizhou Guangxi

Received: Dec. 2nd, 2018; accepted: Dec. 22nd, 2018; published: Dec. 29th, 2018

ABSTRACT

When people study Hopfield time-delay neural network, Lyapunov function is usually used to analyze the stability of the system. But, in this paper, we study the stability of Hopfield neural network by using Gronwall integral inequalities, and obtain the new criterion of global exponential stability of Hopfield neural network and its delay system. Finally, we demonstrate the validity of the results by a numerical example.

Keywords:Gronwall Integral Inequalities, Hopfield Neural Networks, Time Delays, Exponential Stability

1. 引言

Gronwall不等式在微分方程定性理论研究中也发挥极其重要的作用，并且得到不断的推广和广泛应用 [23] - [38] ，但是用于研究神经网络系统的稳定性比较少见。

$\stackrel{˙}{x}\left(t\right)=-Ax\left(t\right)+Bg\left(x\left(t\right)\right)+Cg\left(x\left(t-\tau \right)\right)+I$ (1)

$‖M‖=\sqrt{{\lambda }_{\mathrm{max}}\left({A}^{\text{T}}A\right)}$ ，其中， ${\lambda }_{\mathrm{max}}\left({A}^{\text{T}}A\right)$ 表示 ${A}^{\text{T}}A$ 的最大特征值， $‖x‖=\sqrt{{x}_{1}^{2}+{x}_{2}^{2}+\cdots +{x}_{n}^{2}}$

$\stackrel{˙}{y}\left(t\right)=-Ay\left(t\right)+Bf\left(y\left(t\right)\right)+Cf\left(y\left(t-\tau \right)\right)$ , (2)

$u\left(t\right)\le K+{\int }_{\alpha }^{t}u\left(s\right)g\left(s\right)\text{d}s,\text{\hspace{0.17em}}\alpha \le t\le \beta$

$u\left(t\right)\le K\cdot \mathrm{exp}\left[{\int }_{\alpha }^{t}g\left(s\right)\text{d}s\right],\text{\hspace{0.17em}}\alpha \le t\le \beta$ .

2. 稳定性分析

2.1. 无时滞的情形分析

$\stackrel{˙}{y}\left(t\right)=-Ay\left(t\right)+Bf\left(y\left(t\right)\right)$ (3)

$\stackrel{˙}{y}\left(t\right)=-Ay\left(t\right)$ (4)

$\mathrm{exp}\left[-At\right]=\left[\begin{array}{cccc}{\text{e}}^{-{a}_{1}t}& & & 0\\ & {\text{e}}^{-{a}_{2}t}& & \\ & & \ddots & \\ 0& & & {\text{e}}^{-{a}_{n}t}\end{array}\right]$ ,

I) $M=\sqrt{n}\cdot \mathrm{max}\left\{|{\eta }_{1}|,|{\eta }_{2}|,\cdots ,|{\eta }_{n}|\right\}$ ，显然，M > 0。

II) $\sigma =\mathrm{min}\left\{{a}_{1},{a}_{2},\cdots ,{a}_{n}\right\}$

$y\left(t\right)=Y\left(t\right)+{\int }_{0}^{t}{e}^{-A\left(t-s\right)}Bf\left(y\left(s\right)\right)\text{d}s$ ，实际上，这里 $Y\left(t\right)={e}^{-At}\eta$ ，两边取范数

$\begin{array}{c}‖y\left(t\right)‖=‖Y\left(t\right)‖+{\int }_{0}^{t}‖{e}^{-A\left(t-s\right)}Bf\left(y\left(s\right)\right)‖\text{d}s\\ \le M{\text{e}}^{-\sigma t}+{\int }_{0}^{t}‖{e}^{-A\left(t-s\right)}‖‖B‖‖f\left(y\left(s\right)\right)‖\text{d}s\\ \le M{\text{e}}^{-\sigma t}+L‖B‖{\int }_{0}^{t}{\text{e}}^{-\sigma \left(t-s\right)}‖y\left(s\right)‖\text{d}s\end{array}$

${\text{e}}^{\sigma t}‖y\left(t\right)‖\le M+L‖B‖{\int }_{0}^{t}{\text{e}}^{\sigma s}‖y\left(s\right)‖\text{d}s$

${\text{e}}^{\sigma t}‖y\left(t\right)‖\le M{\text{e}}^{L‖B‖{\int }_{0}^{t}\text{d}s}=M{\text{e}}^{L‖B‖t}$

2.2. 具有时滞的情况分析

$D\left(y\right)=\left[\begin{array}{cccc}{f}_{1}\left({y}_{1}\right)/{y}_{1}& & & 0\\ & {f}_{2}\left({y}_{2}\right)/{y}_{2}& & \\ & & \ddots & \\ 0& & & {f}_{n}\left({y}_{n}\right)/{y}_{n}\end{array}\right]$ (5)

$\stackrel{˙}{y}\left(t\right)=\left(-A+BD\left(y\right)\right)y\left(t\right)$ (6)

$‖\phi \left(t\right)‖\le ‖\eta ‖{\text{e}}^{-{\sigma }_{1}t}$ (7)

$\stackrel{˙}{y}\left(t\right)=\left(-A+BD\left(y\right)\right)y\left(t\right)+Cf\left(y\left(t-\tau \right)\right)$ (8)

$t=0$ 时，对应初值为 $\eta =\left({\eta }_{1},{\eta }_{2},\cdots ,{\eta }_{n}\right)$ ，方程(8)的解可以表示为

$y\left(t\right)=\phi \left(t\right)+{\int }_{0}^{t}{e}^{\left(L‖B‖E-A\right)\left(t-s\right)}Cf\left(y\left(s-\tau \right)\right)\text{d}s$

$\begin{array}{c}‖y\left(t\right)‖\le ‖\phi \left(t\right)‖+{\int }_{0}^{t}‖{e}^{\left(L‖B‖E-A\right)\left(t-s\right)}Cf\left(y\left(s-\tau \right)\right)‖\text{d}s\\ \le ‖\phi \left(t\right)‖+{\int }_{0}^{t}‖{e}^{\left(L‖B‖E-A\right)\left(t-s\right)}‖‖C‖‖f\left(y\left(s-\tau \right)\right)‖\text{d}s\\ \le ‖\eta ‖{\text{e}}^{-{\sigma }_{1}t}+{\int }_{-\tau }^{t-\tau }{\text{e}}^{\left(L‖B‖-\sigma \right)\left(t-\tau -s\right)}‖C‖L‖y\left(s\right)‖\text{d}s\\ =‖\eta ‖{\text{e}}^{-{\sigma }_{1}t}+{\text{e}}^{{\sigma }_{1}t}‖C‖L{\int }_{-\tau }^{t-\tau }{\text{e}}^{-{\sigma }_{1}\left(t-s\right)}‖y\left(s\right)‖\text{d}s\end{array}$

$\begin{array}{c}{\text{e}}^{{\sigma }_{1}t}‖y\left(t\right)‖=‖\eta ‖+{\text{e}}^{{\sigma }_{1}t}‖C‖L{\int }_{-\tau }^{t-\tau }{\text{e}}^{{\sigma }_{1}t}{\text{e}}^{-{\sigma }_{1}\left(t-s\right)}‖y\left(s\right)‖\text{d}s\\ =‖\eta ‖+{\text{e}}^{{\sigma }_{1}t}‖C‖L{\int }_{-\tau }^{t-\tau }{\text{e}}^{{\sigma }_{1}s}‖y\left(s\right)‖\text{d}s\end{array}$

${\text{e}}^{{\sigma }_{1}t}‖y\left(t\right)‖\le ‖\eta ‖{\text{e}}^{{\text{e}}^{{\sigma }_{1}\tau }‖C‖L{\int }_{-\tau }^{t-\tau }\text{d}s}=‖\eta ‖{\text{e}}^{{\text{e}}^{{\sigma }_{1}\tau }‖C‖Lt}$

3. 数值仿真

$\left(\begin{array}{c}\stackrel{˙}{x}\left(t\right)\\ \stackrel{˙}{y}\left(t\right)\end{array}\right)=-A\left(\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right)+B\left(\begin{array}{c}{f}_{1}\left(x\left(t\right)\right)\\ {f}_{2}\left(y\left(t\right)\right)\end{array}\right)$ (9)

Figure 1. The state rail diagram of system (9) ( $L‖B‖-\sigma <0$ )

$\left(\begin{array}{c}\stackrel{˙}{x}\left(t\right)\\ \stackrel{˙}{y}\left(t\right)\end{array}\right)=-A\left(\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right)+B\left(\begin{array}{c}{f}_{1}\left(x\left(t\right)\right)\\ {f}_{2}\left(y\left(t\right)\right)\end{array}\right)+C\left(\begin{array}{c}{f}_{1}\left(x\left(t-\tau \right)\right)\\ {f}_{2}\left(y\left(t-\tau \right)\right)\end{array}\right)$ (10)

Figure 2. The state rail diagram of system (9) ( $L‖B‖-\sigma >0$ )

Figure 3. The state rail diagram of system (10) ( ${\text{e}}^{{\sigma }_{1}\tau }‖C‖L-{\sigma }_{1}<0$ )

Figure 4. The state rail diagram of system (10) ( ${\text{e}}^{{\sigma }_{1}\tau }‖C‖L-{\sigma }_{1}>0$ )

A New Criterion for Stability of Hopfield Neural Network Based on Gronwall Integral Inequality[J]. 应用数学进展, 2018, 07(12): 1658-1666. https://doi.org/10.12677/AAM.2018.712194

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