隨機時滯BAM神經(jīng)網(wǎng)絡的全局散逸性

馬　婧, 張啟敏

(寧夏大學數(shù)學計算機學院, 銀川 750021)

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隨機時滯BAM神經(jīng)網(wǎng)絡的全局散逸性

馬婧, 張啟敏*

(寧夏大學數(shù)學計算機學院, 銀川 750021)

摘要：把不確定性因素考慮到雙向聯(lián)想記憶神經(jīng)網(wǎng)絡(BAM)中, 得到一類帶Brown運動的隨機時滯雙向聯(lián)想記憶神經(jīng)網(wǎng)絡(BAM)模型. 在激活函數(shù)有界的條件下, 研究了隨機時滯BAM神經(jīng)網(wǎng)絡的全局散逸性. 通過Lyapunov泛函、Jensen不等式和It 公式等, 討論了隨機時滯BAM神經(jīng)網(wǎng)絡系統(tǒng)均方散逸的充分條件, 給出了該系統(tǒng)散逸的吸引集. 通過數(shù)值例子對所給出的結論進行了驗證.

關鍵詞：隨機時滯BAM神經(jīng)網(wǎng)絡； 均方散逸； 吸引集； Lyapunov泛函

考慮如下隨機時滯BAM神經(jīng)網(wǎng)絡模型:

(1)

目前主要研究了模型(1)的穩(wěn)定性問題：具有時滯的隨機BAM神經(jīng)網(wǎng)絡的指數(shù)穩(wěn)定性[1]；具有時滯和脈沖的BAM 神經(jīng)網(wǎng)絡周期解的全局穩(wěn)定性[2]；具有時變脈沖的BAM神經(jīng)網(wǎng)絡的全局漸進穩(wěn)定性[3].但是在現(xiàn)實情況中, 有的BAM神經(jīng)網(wǎng)絡的軌跡并不會接近于一個平衡點, 而是在某個時刻進入一個有限的區(qū)域并一直保持在里面, 即散逸性. 因此, 有必要研究BAM神經(jīng)網(wǎng)絡模型的散逸性.

對于確定的模型(1)(即σij(t,x,y)=0), TU等[4]研究了一類變時滯BAM神經(jīng)網(wǎng)絡的全局散逸性；WANG和ZHANG[5]討論了具有時變時滯和連續(xù)分布時滯的BAM 神經(jīng)網(wǎng)絡的全局散逸性. 然而，在神經(jīng)網(wǎng)絡的運行過程中卻存在著許多不確定因素, 就神經(jīng)網(wǎng)絡系統(tǒng)本身而言, 神經(jīng)元的反應也是隨機的, 神經(jīng)元在重復接收相同刺激時, 反應也是不同的, 小的噪聲干擾往往就能破壞整個神經(jīng)網(wǎng)絡的穩(wěn)定性, 這種小的干擾可以被布朗運動的導數(shù)所描述. 意味著研究具有噪聲干擾的BAM神經(jīng)網(wǎng)絡意義大, 更符合實際情況.本文在激活函數(shù)有界的條件下, 通過構造Lyapunov 泛函, 給出了隨機BAM 神經(jīng)網(wǎng)絡散逸的充分條件.

1預備知識

令Rn和Rn×m分別表示n維Euclidean空間和n×m階矩陣的集合.X>0表示實對稱正定矩陣. In表示n×n的單位矩陣. 對一個矩陣A,min(A)(max(A))表示矩陣A的最小(最大)特征值. 令>0,C([-,0];Rn)表示連續(xù)函數(shù)φ的全體, 且其范數(shù)定義為:‖φ‖=sup|φ(s)|, 其中|.|是Rn的Euclidean范數(shù). 令(Ω,,{t}t≥0,P)是完備的概率空間, 其域流{t}t≥0滿足通常的條件(左極限是右連續(xù)的, 并且F0包含所有的零測度集).C1,2(+×Rn;+)表示所有非負連續(xù)函數(shù)V(t,x)在+×Rn上的集合.V(t,x)對t可導, 對x二階可導.

為了方便討論, 將模型(1)表示成矩陣形式:

(2)其中:x(t)=(x1(t),x2(t),…,xn(t))T, y(t)=(y1(t),y2(t),…,yn(t))T,D=diag(d1,d2,…,dn);C=diag(c1,c2,…,cn);A=(aij)n×n,B=(bij)n×n, R=(rij)n×n, S=(sij)n×n,u=(u1,u2,…,un)T, v=(v1,v2,…,vn)T,ω(t)=(ω1(t),ω2(t),…,ωn(t))T, σ(t,x(t),x(t-))=(σij(t,x(t),x(t-)))n×n,σ(t,y(t),y(t-β))=(σij(t,y(t),y(t-β)))n×n,f(y)=(f1(y),f2(y),…,fm(y)),n(x)).

為了得到證明結果, 需要以下的假設、定義和引理.

f(y)=(f1(y),f2(y),…,fm(y))

和

假設2[2]263存在矩陣 H1≥0,H2≥0, 滿足

trace[σT(t,x(t),x(t-))σ(t,x(t),x(t-))]≤

xT(t)H1x(t)+xT(t-)H2x(t-).

引理1[5]252線性矩陣不等式

(i) S22<0, S11-S12(S22)-1(S12)T<0；

(ii) S11<0, S22-(S12)T(S11)-1S12<0.

通常的隨機系統(tǒng)dx(t)=g(t,x(t))dt+σ(t,x(t))dω(t),其中ω(t)是定義在完備概率空間(Ω,,P)上的n維布朗運動, g(.),σ(.):+×Rn→Rn.定義如下算子L:

L V(t,x)=Vt(t,x)+Vxg(t,x)+

2主要結論

(P0≤ρ1I),

(Q0≤ρ2I),

(4)

證明考慮如下Lyapunov函數(shù)

(6)

(7)

dV1=L V(t,x(t))dt+

2xT(t)P0σT(t,y(t),y(t-))dω(t),

(8)

dV2=L V(t,y(t))dt+

2yT(t)Q0σT(t,x(t),x(t-β))dω(t).

(9)

則L V1=2xT(t)P0[-Dx(t)+Af(y(t))+

Bf(y(t-))+u]+yT(t)P1y(t)-

yT(t-)P1y(t-)+fT(y(t))P2f(y(t))-

fT(y(t-))P2f(y(t-))+

trace(σT(t,y(t)y(t-))P0σ(t,y(t),y(t-))),

trace(σT(t, x(t)x(t-β))Q0σ(t, x(t), x(t-β))).

由假設1得到:

fT(y(t))f(y(t))≤yT(t)LTLy(t),

根據(jù)定理中的已知條件和引理1得到:

L V1≤-xT(t)(1-θ)(DP0+P0D)x(t)+

2xT(t)P0u-θxT(t)(DP0+P0D)x(t)+

2xT(t)P0[Af(y(t))+Bf(y(t-))]+

yT(t)P1y(t)-yT(t-)P1y(t-)+

fT(y(t))P2f(y(t))-fT(y(t-))P2f(y(t-))+

ρ1yT(t)H1y(t)+ρ1yT(t-)H2y(t-)+

γ1[yT(t)LTLy(t)-fT(y(t))f(y(t))]≤

-xT(t)(1-θ1)(DP0+P0D)x(t)+εxT(t)x(t)+

(10)

以式(10)相同的方法得到：

L V2≤-yT(t)(1-θ2)(CQ0+Q0C)y(t)+

(11)

所以

L V=L V1+L V2≤-xT(t)Φ1(x(t))+

ε-1|P0u|2-yT(t)Φ2(y(t))+μ-1|Q0v|2,

(12)

其中

ξ(t)=(x(t)f(y(t))f(y(t-))y(t)y(t-))T,

Φ1=(1-θ1(DP0+P0D)-εI),

Φ2=(1-θ2(CQ0+Q0C)-μI).

由期望的性質和Jensen不等式得到:

dEV(t)=EL V(t)dt≤[-(min(Φ1)E|x|2+

ε-1|P0u|2+μ-1|Q0v|2]dt<0,

(13)

EV(t,x(t),y(t))=EV(t0,x(t0),y(t0))+

(14)

另一方面

EV(t,x(t),y(t))≥min(P0)|Ex(t)|2+

(15)

(17)

相應的吸引集為:

證明考慮如下Lyapunov函數(shù)

(20)

(21)

類似定理1的證明即可證明推論2.

3例子

通過以下的例子對給出的結論進行驗證.考慮二階隨機BAM神經(jīng)網(wǎng)絡模型(2), 其中

取如下激活函數(shù):

f1(y1)=tanh(-0.02t),

f2(y2)=tanh(-0.02t),

f1(y1(t-))=tanh(-0.14t),

f2(y2(t-))=tanh(-0.08t),

H1=0.04I, H2=0.04I.然后, 令ε=0.3,μ=0.6,通過MATLAB中的線性矩陣不等式工具箱, 計算線性矩陣不等式(3)、(4), 得到:

4小結

引入了一類隨機時滯BAM神經(jīng)網(wǎng)絡系統(tǒng)，討論了隨機時滯BAM神經(jīng)網(wǎng)絡系統(tǒng)的均方散逸問題.該隨機模型較確定性模型更符合實際，能更好地反應神經(jīng)網(wǎng)絡的特性，在引入布朗運動的基礎上，運用Lyapunov泛函、隨機微分方程基本理論，建立了隨機時滯BAM神經(jīng)網(wǎng)絡系統(tǒng)散逸的充分條件，所得到的結論為隨機微分方程的應用提供了理論依據(jù).

圖1　t=500時的隨機BAM神經(jīng)網(wǎng)絡散逸性的數(shù)值模擬

圖2　t=300時的隨機BAM神經(jīng)網(wǎng)絡散逸性的數(shù)值模擬

參考文獻：

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[2]WANG F, SUN D. Global exponential stability and periodic solutions of BAM neural networks with time delays and impulses[J]. Neurocomputing, 2015, 155: 261-276.

[3]SAYLI M, YILMAZ E. Global robust asymptotic stability of variable-time impulsive BAM neural networks[J]. Neural Networks, 2014, 60: 67-73.

[4]TU Z W, WANG L W,ZHA Z W. Global dissipativity of a class of BAM neural networks with time-varying delays and unbounded delays[J]. Communication in Nonlinear Science & Numerical Simulation, 2013, 18: 2562-2570.

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【中文責編：莊曉瓊英文責編：肖菁】

Global Dissipativity for Stochastic BAM Neural Networks with Time Delay

MA Jing, ZHANG Qimin*

(School of Mathematics and Computer Science, Ningxia University, Yinchuan 750021, China)

Abstract：Considering the randomness, which is one of the uncertain factors in the bidirectional associative memory(BAM) neural networks system, it is obtained that a class of stochastic bidirectional associative memory(BAM) neural networks with time delay and Brownian motion. Under the condition of the bounded activation function of the equation, it discusses the global dissipativity for stochastic bidirectional associative memory (BAM) networks with time delay. By using Lyapunov functions, Jensen’s inequality and It’s formula,it provides the sufficient condition for the global dissipativity in the mean square of such stochastic bidirectional associative memory (BAM) neural networks;it also gives the attractive set of the system. Finally, the numerical example is provided to demonstrate the effectiveness of the conclusion. The conclusion is a generalization of the existing literature in the paper.

Key words：stochastic BAM neural networks; global dissipativity in mean; attractive set in mean; Lyapunov functions

中圖分類號：O193

文獻標志碼：A

文章編號:1000-5463(2016)02-0096-06

*通訊作者:張啟敏，教授，Email:zhangqimin64@sina.com.

基金項目：國家自然科學基金項目(11261043,11461053)

收稿日期：2015-06-09《華南師范大學學報(自然科學版)》網(wǎng)址：http://journal.scnu.edu.cn/n