孫小淇， 王林山

(1.中國(guó)海洋大學(xué)信息科學(xué)與工程學(xué)院，山東 青島 266100； 2.中國(guó)海洋大學(xué)數(shù)學(xué)科學(xué)學(xué)院，山東 青島 266100)

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S-分布時(shí)滯的隨機(jī)Hopfield神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性?

孫小淇1， 王林山2??

(1.中國(guó)海洋大學(xué)信息科學(xué)與工程學(xué)院，山東 青島 266100； 2.中國(guó)海洋大學(xué)數(shù)學(xué)科學(xué)學(xué)院，山東 青島 266100)

研究一類具有S-分布時(shí)滯的隨機(jī)Hopfield神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性問(wèn)題。通過(guò)構(gòu)造隨機(jī)Lyapunov泛函與隨機(jī)分析技巧相結(jié)合的方法得到了實(shí)用有效的判別準(zhǔn)則.具有S-分布時(shí)滯的Hopfield神經(jīng)網(wǎng)絡(luò)解決了具有離散時(shí)滯的Hopfield神經(jīng)網(wǎng)絡(luò)和具有連續(xù)分布時(shí)滯的Hopfield神經(jīng)網(wǎng)絡(luò)不能相互包含的問(wèn)題。且本文在已有文獻(xiàn)的系統(tǒng)模型中加入了隨機(jī)干擾項(xiàng)，證明了該隨機(jī)Hopfield神經(jīng)網(wǎng)絡(luò)全局解的存在唯一性及其全局均方魯棒指數(shù)穩(wěn)定性，使其具有更廣泛的實(shí)際應(yīng)用價(jià)值，推廣了相關(guān)文獻(xiàn)中的結(jié)果。

神經(jīng)網(wǎng)絡(luò)； S-分布時(shí)滯； 全局均方魯棒指數(shù)穩(wěn)定性

引用格式：孫小淇，王林山. S-分布時(shí)滯的隨機(jī)Hopfield神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性[J].中國(guó)海洋大學(xué)學(xué)報(bào)(自然科學(xué)版), 2016, 46(10):139-142.

SUN Xiao-Qi， WANG Lin-Shan. Stability of stochastic Hopfield neural network with S-type distributed delays [J].Periodical of Ocean University of China, 2016, 46(10):139-142.

1982年美國(guó)生物物理學(xué)家J. Hopfield提出了具有聯(lián)想記憶功能，能量定律和動(dòng)力方程等特點(diǎn)并且可以在集成電路上實(shí)現(xiàn)的Hopfield神經(jīng)網(wǎng)絡(luò)模型[1]，這些特點(diǎn)奠定了這種網(wǎng)絡(luò)的輝煌前景。此后，眾多學(xué)者對(duì)Hopfield神經(jīng)網(wǎng)絡(luò)進(jìn)行了深入的研究，研究成果增長(zhǎng)迅速[2]。特別是關(guān)于網(wǎng)絡(luò)的穩(wěn)定性研究引起了人們的關(guān)注[3-7]。文獻(xiàn)[8-10]運(yùn)用Lyapunov函數(shù)與Razumikhin條件相結(jié)的方法研究了隨機(jī)時(shí)滯Hopfield神經(jīng)網(wǎng)絡(luò)的指數(shù)穩(wěn)定性，給出了依賴于時(shí)滯的穩(wěn)定性判據(jù)。具有離散時(shí)滯和分布時(shí)滯的神經(jīng)網(wǎng)絡(luò)是相互獨(dú)立的，而具有S-分布時(shí)滯的神經(jīng)網(wǎng)絡(luò)卻蘊(yùn)含了二者。文獻(xiàn)[11-13]研究了具有S-分布時(shí)滯的Hopfield神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性，隨后關(guān)于這種網(wǎng)絡(luò)的穩(wěn)定性的研究文獻(xiàn)大量涌現(xiàn)。但是據(jù)作者所知，關(guān)于S-分布時(shí)滯的隨機(jī)Hopfield神經(jīng)網(wǎng)絡(luò)穩(wěn)定性研究相對(duì)較少，其原因是隨機(jī)擾動(dòng)的引入，給研究這類網(wǎng)絡(luò)帶來(lái)了較大的困難。本文運(yùn)用隨機(jī)Lyapunov泛函和隨機(jī)分析技巧相結(jié)合的方法，研究了S-分布時(shí)滯隨機(jī)Hopfield神經(jīng)網(wǎng)絡(luò)的全局均方魯棒指數(shù)穩(wěn)定性，給出了易于驗(yàn)證的穩(wěn)定性判據(jù)。推廣了相關(guān)文獻(xiàn)中的結(jié)果。

1　主要結(jié)果

考慮如下S-分布時(shí)滯隨機(jī)Hopfield神經(jīng)網(wǎng)絡(luò)

(1)

(2)

(3)

初始條件φ=(φ1(t),φ2(t),…,φn)T:[-r,0]→Rn是F0可測(cè)的，且右連續(xù)。

引理1[14]考慮如下隨機(jī)泛函微分方程

(4)

若滿足下列條件：

(5)

(6)

dV(t,φ)=(Vt(t,φ(0))+Vx(t,φ(0))f(t,φ)+

LV(t,φ)dt+Vx(t,φ(0))g(t,φ)dWt。

(7)

(8)

其中

LV(t,φ)=Vt(t,φ(0))+Vx(t,φ(0))f(t,φ)+

(9)

定義1如果存在正常數(shù)P，β，使得系統(tǒng)(1)存在滿足條件(2)的解，且這個(gè)解滿足：

(10)

則稱系統(tǒng)(1)是全局均方魯棒指數(shù)穩(wěn)定的。

定理1假設(shè)下列條件成立：

(A1)設(shè) fj(0)=σij(0,0)=0且存在常數(shù)lj>0，cij>0,dij>0,i,j=1,2，…，n，使得對(duì)任意ν,μ,x,y∈R，有

(11)

(12)

(A2)下列不等式成立

(13)

則滿足條件(A1)～(A2)的系統(tǒng)(1)存在唯一的全局解，且系統(tǒng)(1)是均方魯棒指數(shù)穩(wěn)定的。

證明

Ⅰ系統(tǒng)(1)存在唯一全局解

令

(14)

(15)

(16)

從而

(17)

同理由(A1)得

(18)

令

(19)

則 (5)式成立。從而由引理 1 知，則系統(tǒng)(1)存在唯一連續(xù)的全局解x(t),t≥0。

Ⅱ 系統(tǒng)(1)均方魯棒指數(shù)穩(wěn)定

定義

(20)

由(A2)可知

(21)

由H(u)在(0,+∞)上連續(xù)，且當(dāng)u→+∞時(shí)，H(u)→-∞。故存在u*∈[0,+∞)，滿足

(22)

定義Lyapunov泛函

(23)

由(2), (7), (23)和 (A1)得

θ)dηj(θ))dwj(t)≤

(24)

由(8)和 (22)可知

θ)dηj(θ))dwj(s)≤

(25)

上式兩端取數(shù)學(xué)期望得

即

注1 如果擴(kuò)散系數(shù)σij=0,i,j=1,2,…,n，則系統(tǒng)(1)轉(zhuǎn)化為文獻(xiàn)[2]中第三章研究的系統(tǒng)，因此文獻(xiàn)[2] 第三章研究問(wèn)題是本文的特例。

實(shí)例

kj=1。j=1,2。則顯然滿足定理中條件(A1)，且可取

l1=l2=d11=d22=c11=c22=1,

則有

滿足定理中條件(A2)，因此該系統(tǒng)是均方魯棒均方指數(shù)穩(wěn)定的。

[1]Hopfield J J. Neurons with graded response have collective computational properties like those of two-state neurons [J]. Proceedings of the national academy of sciences, 1984, 81(10): 3088-3092.

[2]王林山. 時(shí)滯遞歸神經(jīng)網(wǎng)絡(luò)(Delayed recurrent neural network)[M]. 北京： 科學(xué)出版社， 2008.

Wang L.Delayed Recurrent Neural Network[M]. Beijing： Science Press, 2008.

[3]Forti M, Tesi A. New conditions for global stability of neural networks with application to linear and quadratic programming problems [J]. Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, 1995, 42(7): 354-366.

[4]Wan L, Sun J. Mean square exponential stability of stochastic delayed Hopfield neural networks [J]. Physics Letters A, 2005, 343(4): 306-318.

[5]Wang Z, Shu H, Fang J, et al. Robust stability for stochastic Hopfield neural networks with time delays [J]. Nonlinear Analysis: Real World Applications, 2006, 7(5): 1119-1128.

[6]Duan S, Hu W, Li C, et al. Exponential Stability of Discrete-Time Delayed Hopfield Neural Networks with Stochastic Perturbations and Impulses [J]. Results in Mathematics, 2012, 62(1-2): 73-87.

[7]Pradeep C, Vinodkumar A, Rakkiyappan R. Delay-dependent exponential stability results for uncertain stochastic Hopfield neural networks with interval time-varying delays [J]. Arabian Journal of Mathematics, 2012, 1(2): 227-239.

[8]沈軼, 廖曉昕. Hopfield 型時(shí)滯神經(jīng)網(wǎng)絡(luò)的指數(shù)穩(wěn)定性[J]. 數(shù)學(xué)物理學(xué)報(bào): A 輯, 1999, 19(2): 211-218.

Shen Y, Liao X.Exponential Stability of Hopfield Neural Networks[J].Acta Mathematica Scientia, 1999, 19(2)： 211-218.

[9]沈軼, 廖曉昕. 非線性隨機(jī)時(shí)滯系統(tǒng)族的魯棒穩(wěn)定性[J]. 自動(dòng)化學(xué)報(bào), 1999, 25(4): 537-542.

Shen Y,Liao X.Robust stability of a family of nonlinear stochastic delay systems[J].Acta Automatica Sinica, 1999, 25(4): 537-542.

[10]Wallis G. Stability criteria for unsupervised temporal association networks [J]. IEEE Transactions on Neural Networks, 2005, 16(2): 301.

[11]Wang L, Xu D. Global asymptotic stability of bidirectional associative memory neural networks with S-type distributed delays [J]. International Journal of Systems Science, 2002, 33(11): 869-877.

[12]Wang Y, Lu C, Ji G, et al. Global exponential stability of high-order Hopfield-type neural networks with S-type distributed time delays [J]. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(8): 3319-3325.

[13]Zhang R J， Wang L S． Global exponential robust stability of interval cellular neural networks with S-type distributed delays[J]．Mathematical and Computer Modelling， 2009， 50： 380-385.

[14]Mao X R． Stochastic Differential Equation and Application (second edition)[M]． Chichester： Horwood Publishing， 2007.

AMS Subject Classifications：00A69； 03B30； 03C05

責(zé)任編輯陳呈超

Stability of Stochastic Hopfield Neural Network with S-Type Distributed Delays

SUN Xiao-Qi1， WANG Lin-Shan2

(1.College of Information Science and Engineering, Ocean University of China, Qingdao 266100， China; 2.School of Mathematical sciences, Ocean University of China, Qingdao 266100， China)

This paper is studied the stochastic Hopfield neural network with S-type distributed delays and investigated stability problems of this neural network. Some sufficient conditions on global robust exponential stability in mean square are established in this paper. The means are mainly constructing the suitable Lyapunov functional and applying the stochastic analysis techniques. Because the systems with discrete time delays and the systems with continuously distributed delays do not contain each other. However, S-distributed delays are introducted in stochastic neural network with time delays. It effectively solves the problem that discrete and distributed delays issues not included in the mutual. More even, the existence and uniqueness of solutions and the global robust exponential stability in mean square of the system are proved, which are promoted the results of the relevant literature. An example was given to show the correctnessof the conclusions.

neural networks； S-type distributed delays； global robust exponential stability in mean square

國(guó)家自然科學(xué)基金項(xiàng)目(11171374)； 山東省自然科學(xué)基金重點(diǎn)項(xiàng)目(ZR2011AZ001)資助

2014-10-12；

2015-06-12

孫小淇(1986-)，女，博士生。E-mail:sunxiaoqi@live.com.

??通訊作者： E-mail：Wangls@ouc.edu.com

TP183

A

1672-5174(2016)10-139-04

10.16441/j.cnki.hdxb.20140231

Supported by National Natural Science Foundation of China(11171374)；Shandong Municipal Natural Science Foundation(ZR2011AZ001)