周阿麗，何秀麗，印凡成

(河海大學(xué) 理學(xué)院，江蘇 南京 210098)

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具有Markov切換的隨機(jī)Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)時(shí)滯依賴的指數(shù)同步

周阿麗，何秀麗，印凡成

(河海大學(xué) 理學(xué)院，江蘇 南京 210098)

基于隨機(jī)分析理論以及時(shí)滯依賴的反饋控制技術(shù)，首先運(yùn)用Lyapunov函數(shù)方法和Gronwall不等式，其次給出了具有Markov切換的隨機(jī)Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)時(shí)滯依賴的指數(shù)同步的充分性判據(jù)，此判據(jù)解除了傳統(tǒng)意義上時(shí)滯延遲的可微性以及其導(dǎo)數(shù)的有界性的限制，最后通過一個(gè)數(shù)值模擬實(shí)例驗(yàn)證了理論結(jié)果的正確性及有效性.

指數(shù)同步； Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)； Markov切換； 時(shí)變時(shí)滯延遲

自從Pecora[1]等最初提出耦合混沌系統(tǒng)的同步性以來，基于其在多種領(lǐng)域中的潛在應(yīng)用價(jià)值，混沌神經(jīng)網(wǎng)絡(luò)的同步性理論已被廣泛應(yīng)用在創(chuàng)造安全的交流系統(tǒng)、化學(xué)和生物系統(tǒng)以及自動(dòng)化控制等領(lǐng)域.因此，關(guān)于神經(jīng)網(wǎng)絡(luò)帶有噪聲或沒有噪聲的同步性理論也被廣泛研究[2-3].Sun等[4]利用LMI方法研究了混沌系統(tǒng)的指數(shù)同步問題,Zhou等[5]利用M矩陣方法研究了神經(jīng)網(wǎng)絡(luò)的隨機(jī)同步問題,同時(shí)自適應(yīng)性同步的問題也得到了廣泛的研究[6-7].此外，關(guān)于神經(jīng)網(wǎng)絡(luò)的指數(shù)穩(wěn)定性問題也取得了很多成果[8-10].特別地，關(guān)于Cohen-Grossberg(C-G)神經(jīng)網(wǎng)絡(luò)系統(tǒng)的指數(shù)穩(wěn)定性和同步性也取得了很大的成就[11-14].但是在現(xiàn)實(shí)應(yīng)用中，神經(jīng)網(wǎng)絡(luò)會(huì)出現(xiàn)隨機(jī)故障，導(dǎo)致鏈接權(quán)值或閥值突然被改變，此類問題可通過Markov切換來模擬,其中，Zhu等[12]運(yùn)用Halandy不等式等對(duì)C-G神經(jīng)網(wǎng)絡(luò)系統(tǒng)指數(shù)同步做了深入研究.筆者受以上研究的啟發(fā)，將Lyapunov函數(shù)方法、隨機(jī)分析技術(shù)與Grownwall不等式方法等相結(jié)合給出了具有Markov切換的變時(shí)滯隨機(jī)C-G神經(jīng)網(wǎng)絡(luò)的指數(shù)同步的充分性判據(jù),同時(shí)不再對(duì)傳統(tǒng)意義上的時(shí)滯函數(shù)的可微性以及其導(dǎo)數(shù)的有界性做要求.

本文首先介紹了具有Markov切換的C-G神經(jīng)網(wǎng)絡(luò)的驅(qū)動(dòng)和響應(yīng)系統(tǒng)以及所需的一些假設(shè)和定義，其次通過構(gòu)造Lyapunov函數(shù)并結(jié)合Grownwall不等式等技術(shù)得到了對(duì)于誤差系統(tǒng)的指數(shù)同步準(zhǔn)則，最后通過一個(gè)實(shí)例及其數(shù)值模擬驗(yàn)證了所得結(jié)論的有效性和適用性.

1 模型，符號(hào)和假設(shè)

設(shè){r(t),t≥0}為狀態(tài)空間S={i=1,2,…,N}的一個(gè)連續(xù)時(shí)間的Markov鏈，其生成元矩陣∏=(πij)n×n為

考慮下面的帶有時(shí)滯依賴的C-G神經(jīng)網(wǎng)絡(luò)模型

dx(t)={-α(x(t),r(t))[β(x(t),r(t))-C(r(t))f(x(t))-D(r(t))g(x(t-τ(t)))]+J}dt，

(1)

其中，x(t)=[x1(t),x2(t),…,xn(t)]T是n個(gè)神經(jīng)元的狀態(tài)向量,α(x(t),r(t))=diag(α1(x1(t),r(t)),…,αn(xn(t),r(t)))代表放大函數(shù)β(x(t),r(t))=[β1(x1(t),r(t)),β2(x2(t),r(t)),…,βn(xn(t),r(t))]T為行為函數(shù)，C(r(t))=(cij(r(t)))n×n和D(r(t))=(dij(r(t)))n×n分別為連接權(quán)矩陣和時(shí)滯連接權(quán)矩陣. f(x(t))=[f1(x1(t)),f2(x2(t)),…,fn(xn(t))]T和g(x(t))=[g1(x1(t)),g2(x2(t)),…,gn(xn(t))]T為神經(jīng)元的激勵(lì)函數(shù)，J=[J1,J2,…，Jn]T代表一個(gè)恒定的外部輸入變量，時(shí)滯依賴τ(t)滿足0≤τ(t)≤τ，其中τ是一個(gè)正常量.

模型(1)為驅(qū)動(dòng)系統(tǒng)，其響應(yīng)系統(tǒng)為

dy(t)= {-α(y(t),r(t))[β(y(t),r(t))-C(r(t))f(y(t))-D(r(t))g(y(t-τ(t)))]+

J+u(t,r(t))}dt+σ(t,r(t),y(t)-x(t),y(t-τ(t))-x(t-τ(t)))dw(t) ,

(2)

其中，u(t,r(t))表示控制輸入向量，w(t)是一個(gè)m維布朗運(yùn)動(dòng)，假設(shè)Markov鏈r(·)獨(dú)立于布朗運(yùn)動(dòng)w(·)，且取u(t,r(t))=K1(r(t))[f(y(t)-x(t))]+K2(r(t))[g(y(t-τ))-g(x(t-τ(t)))]，K1(r(t)),K2(r(t))為增益矩陣.

令e(t)=y(t)-x(t)為同步誤差，且A(r(t))=C(r(t))+K1(r(t))=(aij(r(t)))n×n，B(r(t))=D(r(t))+K2(r(t))=(bij(r(t)))n×n，則系統(tǒng)(1)和系統(tǒng)(2)誤差系統(tǒng)可以表示為

de(t)= {-[α(y(t),r(t))β(y(t),r(t))-α(x(t),r(t))β(x(t),r(t))]+α(y(t),r(t))A(r(t))×

[f(y(t))-f(x(t))]+α(y(t),r(t))B(r(t))[g(y(t-τ(t))-g(x(t-τ(t))]+

[α(y(t),r(t))-α(x(t),r(t))]×[C(r(t))f(x(t)+D(r(t))g(x(t-τ(t)))]}dt+

σ(t,r(t),e(t),e(t-τ(t)))dw(t).

(3)

LV(t,i,e(t))= Vt(t,i,e(t))+Ve(t,i,e(t))·F(t,i,e(t),e(t-τ(t)))+1/2trace[σT(t,i,e(t),

(4)

為了得到本文主要結(jié)果，給出以下假設(shè)：

5) f(0)≡0,g(0)≡0,σ(t,0,0)≡0.

定義1 如果存在正常數(shù)γ，λ對(duì)于t≥0有E|u(t)-v(t)|2≤γE|u(0)-v(0)|2e-λt，則稱2個(gè)耦合的神經(jīng)網(wǎng)絡(luò)系統(tǒng)(1)和(2)為指數(shù)同步的.

2 主要結(jié)論

δ1|e(t)|2≤V(t,i,e(t))≤δ2|e(t)|2

(5)以及不等數(shù)τ<τ*成立，其中τ*=(δ2/λ1)log(λ1δ1/λ2δ2)，則系統(tǒng)(1)和系統(tǒng)(2)是時(shí)滯依賴指數(shù)同步的.

(6)

由假設(shè)(1)可得

|ej(t)‖fk(yk(t))-fk(xk(t))|≤u1k|ej(t)‖ek(t)|≤u1k(|ej(t)|2+|ek(t)|2)/2,

|ej(t)‖gk(yk(t-τ(t)))- gk(xk(t-τ(t)))|≤u2k|ej(t)‖ek(t-τ(t))|≤

u2k(|ej(t)|2+|ek(t-τ(t))|2)/2，

(7)

將式(7)代入式(6)并整理可得

λ1|e(t)|2+|eτ(t)|2，

(8)

另由定理?xiàng)l件可得

(9)

對(duì)于任意t≥τ，存在整數(shù)K>2使得(K-1)τ≤t≤Kτ.

定義Lyapunove 函數(shù)V1(tm,r(tm),e,eτ)，

對(duì)于0≤m≤K-1,tm=t-mτ.

又

由式(5)和(8)條件可以得到

因此

由上述條件可得

由Gronwall不等式可得

即

因此，由定義可以得出系統(tǒng)(1)和(2)是時(shí)滯依賴指數(shù)同步的.

注：本文所研究的為時(shí)滯依賴的神經(jīng)網(wǎng)絡(luò)模型，同時(shí)不再對(duì)傳統(tǒng)意義上時(shí)滯函數(shù)的可微性以及其導(dǎo)數(shù)的有界性做要求.例如文獻(xiàn)[11]要求時(shí)滯函數(shù)可微且其導(dǎo)函數(shù)小于1，而本文定理的條件只需時(shí)滯函數(shù)有界.另外，本文所考慮的激勵(lì)函數(shù)弱化了傳統(tǒng)的Lipschitz 條件[4]及函數(shù)的單調(diào)性及連續(xù)可微性[7].

3 數(shù)值模擬

將給出具體例子來說明所得結(jié)論的有效性和可行性.

例1 考慮如下二維C-G神經(jīng)網(wǎng)絡(luò)模型

dx(t)={-α(x(t),r(t))[β(x(t),r(t))-C(r(t))f(x(t))-D(r(t))g(x(t-τ(t)))]+J}dt，dy(t)= {-α(y(t),r(t))[β(y(t),r(t))-C(r(t))f(y(t))-D(r(t))g(y(t-τ(t)))]+J+u(t)}dt+

σ(t,r(t),y(t)-x(t),y(t-τ(t))-x(t-τ(t)))dw(t),

其中,u(t,r(t))=K1(r(t))[f(y(t))-f(x(t))]+K2(r(t))[g(y(t-τ(t)))-g(x(t-τ(t)))],J=(0,0)T，x(t)=(x1(t),x2(t))T,y(t)=(y1(t),y1(t))T，w(t)是二維布朗運(yùn)動(dòng)，r(t)為Markov鏈，S={1,2}，生成元矩陣及其他參數(shù)為βj(xj(t),i)=2xj(t)(i,j=1,2)，αj(xj(t),i)=0.6+0.2cos(xj(t)),τ(t)=0.1|cos t|+1,f(xj(t))=g(xj(t))=(|xj(t)+1|-|xj(t)-1|)/2,

容易驗(yàn)證假設(shè)1)～5)成立，其他條件給出如下

圖1 a和b分別顯示出系統(tǒng)(1)和系統(tǒng)(2)的混沌現(xiàn)象，取q1=1,q2=0.5，定理中的條件均滿足.因此系統(tǒng)(1)和系統(tǒng)(2)是指數(shù)同步的，圖1 c驗(yàn)證了2個(gè)系統(tǒng)的同步性.

4 小 結(jié)

研究了具有Markov切換的隨機(jī)延遲的Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)的指數(shù)同步問題，基于隨機(jī)分析理論以及時(shí)滯依賴的反饋控制技術(shù)，首先運(yùn)用Lyapunov函數(shù)方法和Gronwall不等式，給出了具有Markov切換的變時(shí)滯隨機(jī)Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)的指數(shù)同步的充分性判據(jù)，該判據(jù)不在對(duì)時(shí)滯延遲具有可微性及導(dǎo)數(shù)有界性作要求，最后數(shù)值模擬實(shí)例驗(yàn)證了理論結(jié)果的正確性及有效性.

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Exponential Synchronization of Cohen-Grossberg Neural Networks with Time-varying Delays and Markovian Switching

Zhou Ali, He Xiuli，Yin Fancheng

(College of Science, Hohai University, Nanjing 210098, China)

Based on the stochastic analysis theory and delay-dependent feedback control technique, the Lyapunov function and Gronwall inequality were used, and the sufficient criteria of the exponential synchronization of random Cohen-Grossberg neural network time-delay dependent with Markov change was proposed, which removed the differentiability of the time varying delay and the boundness of its derivative. Lastly, a numerical example and its simulation were proposed to demonstrate the effectiveness and correctness of the theoretical results.

exponential synchronization; Markovian switching; time-varying delay; Cohen-Grossberg neural network

2016-03-29

中央高?？蒲袠I(yè)務(wù)費(fèi)青年教師科研創(chuàng)新能力培育項(xiàng)(A)(2015B19814)

周阿麗(1990-),女，安徽利辛人，河海大學(xué)2014級(jí)碩士研究生，研究方向：隨機(jī)神經(jīng)網(wǎng)絡(luò)系統(tǒng)穩(wěn)定性和同步性分析,E-mail：zhoualiahut@126.com

1004-1729(2016)03-0203-06

O 175

A DOl：10.15886/j.cnki.hdxbzkb.2016.0031