﻿ S-分布时滞静态神经网络的全局指数收敛性

# S-分布时滞静态神经网络的全局指数收敛性Global Exponential Convergence of Static Neural Networks with S-Type Distributed Delays

Abstract: This paper is concerned with the exponential convergence for a class of static neural networks with S-type distributed delays. By applying the differential inequality techniques, the sufficient conditions to ensure that all solutions of the addressed system converge exponentially to zero are established. Moreover, an example is given to show the effectiveness of the obtained results.

1. 引言

$\frac{\text{d}{x}_{i}\left(t\right)}{\text{d}t}=-{x}_{i}\left(t\right)+\underset{j=1}{\overset{n}{\sum }}{w}_{ij}{g}_{j}\left({x}_{j}\left(t\right)\right)+{I}_{i},\text{ }i=1,2,\cdots ,n.$ (1.1)

$\frac{\text{d}{y}_{i}\left(t\right)}{\text{d}t}=-{y}_{i}\left(t\right)+{g}_{i}\left(\underset{j=1}{\overset{n}{\sum }}{w}_{ij}{y}_{j}\left(t\right)+{I}_{i}\right),\text{ }i=1,2,\cdots ,n.$ (1.2)

Hopfield模型、双向联想记忆模型和CNNs模型都属于局域神经网络模型，局域神经网络模型(1.1)已被广泛研究，得到了很多深刻的理论结果。递归反向传播网络、BCOp网络、BSB网络的模型则属于静态神经网络，具有重要的应用意义，但相比之下，静态神经网络模型(1.2)的研究则较少 [5] 。

2. 主要结果

${{x}^{\prime }}_{i}\left(t\right)=-{c}_{i}\left(t\right){x}_{i}\left(t\right)+{g}_{i}\left(\underset{j=1}{\overset{n}{\sum }}{\int }_{-r}^{0}{x}_{j}\left(t+\theta \right)\text{d}{w}_{ij}\left(\theta \right)+{I}_{i}\left(t\right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n.$ (2.1)

$x\left(\sigma +\theta \right)=\varphi \left(\theta \right),\text{ }\sigma \in \text{R,}\text{\hspace{0.17em}}\theta \in \left[-r\text{,0}\right]\text{.}$ (2.2)

(H1) $\underset{t\in \text{R}}{\mathrm{inf}}{c}_{i}\left(t\right)=\underset{_}{{c}_{i}}>0,\text{\hspace{0.17em}}i\in I$

(H2) ${\int }_{-r}^{0}{x}_{j}\left(t+\theta \right)\text{d}{w}_{ij}\left(\theta \right)$ 是Lebesgue-Stieltjes可积的，且 $0\le {\int }_{-r}^{0}\text{d}{w}_{ij}\left(\theta \right)<{w}_{ij}<+\infty ,\text{\hspace{0.17em}}i,j\in I$

(H3) 存在 ${L}_{i}^{g}>0$ ，使得 $|{g}_{i}\left(u\right)|\le {L}_{i}^{g}|u|,\text{\hspace{0.17em}}u\in \text{R,}\text{\hspace{0.17em}}i\in I$

(H4) 对任意的 $i\in I$ ，存在 ${\xi }_{1},{\xi }_{2},\cdots ,{\xi }_{n}>0,\text{\hspace{0.17em}}\underset{_}{{c}_{i}}>\lambda >0$ ，满足 $\underset{i\in I}{\mathrm{max}}\left\{{\xi }_{i}{}^{-1}{L}_{i}^{g}\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{\text{e}}^{\lambda r}+1\right\}<\underset{_}{{c}_{i}}-\lambda$$|{I}_{i}\left(t\right)|\le {\text{e}}^{-\lambda t},i\in I$

(H5) $\underset{i\in I}{\mathrm{max}}\left\{\frac{1}{\underset{_}{{c}_{i}}}{\xi }_{i}^{-1}{L}_{i}^{g}\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{w}_{ij}\right\}<1$

$y\left(t\right)={\left({y}_{1}\left(t\right),{y}_{2}\left(t\right),\cdots ,{y}_{n}\left(t\right)\right)}^{\text{T}}={\left({\xi }_{1}^{-1}{x}_{1}\left(t\right),{\xi }_{2}^{-1}{x}_{2}\left(t\right),\cdots ,{\xi }_{n}^{-1}{x}_{n}\left(t\right)\right)}^{\text{T}}$

${{y}^{\prime }}_{i}\left(t\right)=-{c}_{i}\left(t\right){y}_{i}\left(t\right)+{\xi }_{i}^{-1}{g}_{i}\left(\underset{j=1}{\overset{n}{\sum }}{\int }_{-r}^{0}{x}_{j}\left(t+\theta \right)\text{d}{w}_{ij}\left(\theta \right)+{I}_{i}\left(t\right)\right),\text{\hspace{0.17em}}i\in I.$ (2.3)

$t>s>\sigma$ 时，有

${{y}^{\prime }}_{i}\left(s\right)=-{c}_{i}\left(s\right){y}_{i}\left(s\right)+{\xi }_{i}^{-1}{g}_{i}\left(\underset{j=1}{\overset{n}{\sum }}{\int }_{-r}^{0}{x}_{j}\left(s+\theta \right)\text{d}{w}_{ij}\left(\theta \right)+{I}_{i}\left(s\right)\right),\text{\hspace{0.17em}}i\in I.$ (2.4)

${y}_{i}\left(t\right)={y}_{i}\left(\sigma \right){\text{e}}^{-{\int }_{\sigma }^{t}{c}_{i}\left(u\right)\text{d}u}+{{\int }_{\sigma }^{t}\text{e}}^{-{\int }_{s}^{t}{c}_{i}\left(u\right)\text{d}u}\left[{\xi }_{i}^{-1}{g}_{i}\left(\underset{j=1}{\overset{n}{\sum }}{\int }_{-r}^{0}{x}_{j}\left(s+\theta \right)\text{d}{\omega }_{ij}\left(\theta \right)+{I}_{i}\left(s\right)\right)\right]\text{d}s,\text{\hspace{0.17em}}i\in I.$ (2.5)

$\begin{array}{c}|{y}_{i}\left(t\right)|\le {\text{e}}^{-{\int }_{\sigma }^{t}{c}_{i}\left(u\right)\text{d}u}{\xi }_{i}^{-1}|{\varphi }_{i}\left(0\right)|+|{\int }_{\sigma }^{t}{\text{e}}^{-{\int }_{s}^{t}{c}_{i}\left(u\right)\text{d}u}\left[{\xi }_{i}^{-1}{g}_{i}\left(\underset{j=1}{\overset{n}{\sum }}{\int }_{-r}^{0}{x}_{j}\left(s+\theta \right){w}_{ij}\left(\theta \right)+{I}_{i}\left(s\right)\right)\right]\text{d}s|\\ \le {\text{e}}^{-{\int }_{\sigma }^{t}\underset{_}{{c}_{i}}\text{d}u}{\xi }_{i}^{-1}|{\varphi }_{i}\left(0\right)|+{\int }_{\sigma }^{t}{\text{e}}^{-{\int }_{s}^{t}\underset{_}{{c}_{i}}\text{d}u}{\xi }_{i}^{-1}{L}_{i}^{g}\left(|\underset{j=1}{\overset{n}{\sum }}{\int }_{-r}^{0}{x}_{j}\left(s+\theta \right)\text{d}{w}_{ij}\left(\theta \right)|+|{I}_{i}\left(s\right)|\right)\text{d}s\\ \le {\text{e}}^{-\underset{_}{{c}_{i}}\left(t-\sigma \right)}{\xi }_{i}^{-1}|{\varphi }_{i}\left(0\right)|+{\xi }_{i}^{-1}{L}_{i}^{g}{\int }_{\sigma }^{t}{\text{e}}^{-\underset{_}{{c}_{i}}\left(t-s\right)}|\underset{j=1}{\overset{n}{\sum }}{\int }_{-r}^{0}{\xi }_{j}{y}_{j}\left(s+\theta \right)\text{d}{w}_{ij}\left(\theta \right)|\text{d}s+{\xi }_{i}^{-1}{L}_{i}^{g}{\int }_{\sigma }^{t}{\text{e}}^{-\underset{_}{{c}_{i}}\left(t-s\right)}{\text{e}}^{-\lambda s}\text{d}s\\ \le {\xi }_{i}^{-1}|{\varphi }_{i}\left(0\right)|+\frac{1}{\underset{_}{{c}_{i}}}{\xi }_{i}^{-1}{L}_{i}^{g}‖y‖\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{w}_{ij}+{\xi }_{i}^{-1}{L}_{i}^{g}\frac{{\text{e}}^{-\lambda \sigma }}{\lambda -\underset{_}{{c}_{i}}},\end{array}$ (2.6)

$‖y‖\le \underset{i\in I}{\mathrm{max}}\left\{{\xi }_{i}^{-1}|{\varphi }_{i}\left(0\right)|\text{+}{\xi }_{i}^{-1}{L}_{i}^{g}\frac{{\text{e}}^{-\lambda \sigma }}{\lambda -\underset{_}{{c}_{i}}}+\frac{\text{1}}{\underset{_}{{c}_{i}}}{\xi }_{i}^{-1}{L}_{i}^{g}‖y‖\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{w}_{ij}\right\},$ (2.7)

$‖y‖\le \frac{\underset{i\in I}{\mathrm{max}}\left\{{\xi }_{i}^{-1}|{\varphi }_{i}\left(0\right)|+{\xi }_{i}^{-1}{L}_{i}^{g}\frac{{\text{e}}^{-\lambda \sigma }}{\lambda -\underset{_}{{c}_{i}}}\right\}}{\underset{i\in I}{\mathrm{max}}\left\{1-\frac{\text{1}}{\underset{_}{{c}_{i}}}{\xi }_{i}^{-1}{L}_{i}^{g}\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{w}_{ij}\right\}}.$ (2.8)

y有界，因此模型(2.1)的解不会发生爆破，在 $\left[\sigma ,+\infty \right]$ 上整体存在。

$|{x}_{i}\left(t\right)|\le B{\text{e}}^{-\lambda t}.$

$|{y}_{i}\left(t\right)|=|{\xi }_{i}^{-1}{\varphi }_{i}\left(t\right)|

$‖y\left(t\right)‖ (2.9)

$|{L}_{i}^{g}{\xi }_{i}{}^{-1}{I}_{i}\left(t\right)|

$‖y\left(t\right)‖ (2.10)

$‖y\left(\eta \right)‖=|{y}_{i}\left(\eta \right)|=M\left(‖\varphi ‖+\epsilon \right){\text{e}}^{-\lambda \eta }.$ (2.11)

$|{y}_{i}\left(t\right)| (2.12)

${y}_{i}\left(t\right)={y}_{i}\left(0\right){\text{e}}^{-{\int }_{0}^{t}{c}_{i}\left(u\right)\text{d}u}+{{\int }_{0}^{t}\text{e}}^{-{\int }_{s}^{t}{c}_{i}\left(u\right)\text{d}u}{\xi }_{i}^{-1}{g}_{i}\left[\underset{j=1}{\overset{n}{\sum }}{\int }_{-r}^{0}{\xi }_{j}{y}_{j}\left(s+\theta \right)\text{d}{\omega }_{ij}\left(\theta \right)+{I}_{i}\left(s\right)\right]\text{d}s,$

$\begin{array}{c}|{y}_{i}\left(\eta \right)|=|{y}_{i}\left(0\right){\text{e}}^{-{\int }_{0}^{\eta }{c}_{i}\left(u\right)\text{d}u}+{\int }_{0}^{\eta }{\text{e}}^{-{\int }_{s}^{\eta }{c}_{i}\left(u\right)\text{d}u}{\xi }_{i}^{-1}{g}_{i}\left[\underset{j=1}{\overset{N}{\sum }}{\int }_{-r}^{0}{\xi }_{j}{y}_{j}\left(s+\theta \right)\text{d}{w}_{ij}\left(\theta \right)+{I}_{i}\left(s\right)\right]\text{d}s|\\ \le M\left(‖\varphi ‖+\epsilon \right){\text{e}}^{-\underset{_}{{c}_{i}}\eta }+{\int }_{0}^{\eta }{\text{e}}^{-\underset{_}{{c}_{i}}\left(\eta -s\right)}{\xi }_{i}^{-1}{L}_{i}^{g}\left[\underset{j=1}{\overset{n}{\sum }}|{\int }_{-r}^{0}{\xi }_{j}{y}_{j}\left(s+\theta \right)\text{d}{w}_{ij}\left(\theta \right)|+|{I}_{i}\left(s\right)|\right]\text{d}s\\ \le M\left(‖\varphi ‖+\epsilon \right){\text{e}}^{-\underset{_}{{c}_{i}}\eta }+{\int }_{0}^{\eta }{\text{e}}^{-\underset{_}{{c}_{i}}\left(\eta -s\right)}\left[{\xi }_{i}^{-1}{L}_{i}^{g}\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}M\left(‖\varphi ‖+\epsilon \right){\text{e}}^{-\lambda \left(s-r\right)}{w}_{ij}+M\left(‖\varphi ‖+\epsilon \right){\text{e}}^{-\lambda s}\right]\text{d}s\\ \le M\left(‖\varphi ‖+\epsilon \right){\text{e}}^{-\underset{_}{{c}_{i}}\eta }+M\left(‖\varphi ‖+\epsilon \right){\int }_{0}^{\eta }{\text{e}}^{-\underset{_}{{c}_{i}}\left(\eta -s\right)}\left[{\xi }_{i}^{-1}{L}_{i}^{g}\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{\text{e}}^{-\lambda \left(s-r\right)}{w}_{ij}+{\text{e}}^{-\lambda s}\right]\text{d}s\end{array}$

$\begin{array}{l}\le M\left(‖\varphi ‖+\epsilon \right)\left[{\text{e}}^{-\underset{_}{{c}_{i}}\eta }+{\text{e}}^{-\underset{_}{{c}_{i}}\eta }{\int }_{0}^{\eta }{\text{e}}^{-\left(\lambda -\underset{_}{{c}_{i}}\right)s}\left({\xi }_{i}^{-1}{L}_{i}^{g}\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{\text{e}}^{\lambda r}{w}_{ij}+1\right)\text{d}s\right]\\

$‖y\left(t\right)‖

$‖x\left(t\right)‖<\xi M\left(‖\varphi ‖+\epsilon \right){\text{e}}^{-\lambda t},$

3. 例子

$\left\{\begin{array}{l}{{x}^{\prime }}_{1}\left(t\right)=-\left(5+\mathrm{sin}t\right){x}_{1}\left(t\right)+{g}_{1}\left({\int }_{-1}^{0}{x}_{1}\left(t+\theta \right)\text{d}{w}_{11}\left(\theta \right)+{\int }_{-1}^{0}{x}_{2}\left(t+\theta \right)\text{d}{w}_{12}\left(\theta \right)+{\text{e}}^{-\frac{1}{2}t}\mathrm{sin}t\right),\\ {{x}^{\prime }}_{2}\left(t\right)=-\left(5+\mathrm{cos}t\right){x}_{2}\left(t\right)+{g}_{2}\left({\int }_{-1}^{0}{x}_{1}\left(t+\theta \right)\text{d}{w}_{21}\left(\theta \right)+{\int }_{-1}^{0}{x}_{2}\left(t+\theta \right)\text{d}{w}_{22}\left(\theta \right)+{\text{e}}^{-\frac{1}{2}t}\mathrm{cos}t\right).\end{array}$ (3.1)

${\xi }_{i}=1,i=1,2$ 。对 $\lambda =\frac{1}{2},\text{\hspace{0.17em}}i=1,2$${\xi }_{i}{}^{-1}{L}_{i}^{g}\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{\text{e}}^{\lambda r}+\text{1}=\frac{1}{6}{\text{e}}^{\frac{1}{2}}+1<\frac{7}{2}=\underset{_}{{c}_{i}}-\lambda$ 成立，即满足 $\underset{i\in I}{\mathrm{max}}\left\{{\xi }_{i}{}^{-1}{L}_{i}^{g}\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{\text{e}}^{\lambda r}+1\right\}<\underset{_}{{c}_{i}}-\lambda$ 。且容易验证 $\underset{i\in I}{\mathrm{max}}\left\{\frac{1}{\underset{_}{{c}_{i}}}{\xi }_{i}^{-1}{L}_{i}^{g}\underset{j=1}{\overset{n}{\sum }}{\xi }_{j}{w}_{ij}\right\}<1$ 成立。

[1] 马天瑾. 神经网络技术[M]. 青岛: 青岛海洋出版社, 1994.

[2] 阎平凡. 人工神经网络与模拟进化计算[M]. 北京: 清华大学出版社, 2001.

[3] 阮炯. 神经动力学模型方法与应用[M]. 北京: 科学出版社, 2001.

[4] Qiao, H., et al. (2003) A Reference Model Approach to Stability Analysis of Neural Networks. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernet-ics), 33, 925-936.
https://doi.org/10.1109/TSMCB.2002.804368

[5] 王林山. 时滞递归神经网络[M]. 北京: 科学出版社, 2007.

[6] Wang, L.S. and Xu, D.Y. (2002) Global Asymptotic Stability of Bidirectional Associative Memory Neural Networks with S-Type Distributed Delays. International Journal of Systems Science, 33, 869-877.
https://doi.org/10.1080/00207720210161777

[7] Kwon, O.M., et al. (2014) New and Improved Results on Stability of Static Neural Networks with Interval Time-Varying Delays. Applied Mathematics and Computation, 239, 346-357.
https://doi.org/10.1016/j.amc.2014.04.089

[8] Liu, B., Ma, X.L. and Jia, X.-C. (2018) Further Results on H∞ State Estimation of Static Neural Networks with Time-Varying Delay. Neurocomputing, 285, 133-140.
https://doi.org/10.1016/j.neucom.2018.01.032

[9] Manivannan, R., Samidurai, R. and Zhu, Q.X. (2017) Further Improved Results on Stability and Dissipativity Analysis of Static Impulsive Neural Networks with Interval Time-Varying Delays. Journal of the Franklin Institute, 354, 6312-6340.
https://doi.org/10.1016/j.jfranklin.2017.07.040

[10] Arbi, A., et al. (2015) Stability Analysis for Delayed High-Order Type of Hopfield Neural Networks with Impulses. Neurocomputing, 165, 312-329.
https://doi.org/10.1016/j.neucom.2015.03.021

[11] Xu, C.J. and Li, P.L. (2017) Global Exponential Convergence of Neu-tral-Type Hopfield Neural Networks with Multi-Proportional Delays and Leakage Delays. Chaos, Solitons & Fractals, 96, 139-144.
https://doi.org/10.1016/j.chaos.2017.01.012

[12] Wan, L., Zhou, Q.H. and Liu, J. (2017) Delay-Dependent Attractor Analysis of Hopfield Neural Networks with Time-Varying Delays. Chaos, Solitons & Fractals, 101, 68-72.
https://doi.org/10.1016/j.chaos.2017.05.017

Top