李志鴻，何秀麗

(河海大學(xué) 理學(xué)院，江蘇 南京 210098 )

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基于馬氏切換Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)輸入到狀態(tài)穩(wěn)定的新判據(jù)

李志鴻，何秀麗

(河海大學(xué) 理學(xué)院，江蘇 南京 210098 )

摘要：針對(duì)具有馬氏切換的Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性問題，應(yīng)用Lyapunov函數(shù)理論和Halanay不等式技巧，提出了神經(jīng)網(wǎng)絡(luò)輸入到狀態(tài)穩(wěn)定性的新判據(jù)，并通過數(shù)值例子來驗(yàn)證所得結(jié)論的正確性和有效性.

關(guān)鍵詞：Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)； 輸入到狀態(tài)穩(wěn)定； 馬氏切換；Halanay微分不等式

1983年，Cohen和Grossberg提出了一種神經(jīng)網(wǎng)絡(luò)模型[1]，被稱為Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)，由于其廣泛應(yīng)用在不同領(lǐng)域，如模式識(shí)別、記憶與信號(hào)處理、圖象處理與計(jì)算技術(shù)等領(lǐng)域，因而得到了高度的重視.但是在實(shí)際應(yīng)用中輸入、切換是不可避免的，這些因素往往導(dǎo)致整個(gè)系統(tǒng)的震蕩，甚至不穩(wěn)定. 因此有關(guān)輸入[2-4]、切換[5-6]Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)的研究已逐漸引起學(xué)者的關(guān)注. 最常見的切換就是馬氏切換[7，10]，研究帶有馬氏切換的神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性問題具有極其重要的意義. 本文推廣了Halandy不等式[8]，構(gòu)造了更一般的Lyapunov函數(shù)，得出的主要結(jié)果改進(jìn)和推廣了已有文獻(xiàn)的一些結(jié)論[9-10]，不但推導(dǎo)出模型指數(shù)輸入到狀態(tài)穩(wěn)定的新判別準(zhǔn)則，而且給出解的指數(shù)收斂速度的估計(jì).

1模型和預(yù)備知識(shí)

考慮Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)模型

(1)

其中，k=1,2,…,n,n 表示網(wǎng)絡(luò)中神經(jīng)元的個(gè)數(shù)，xk(t)表示第k個(gè)神經(jīng)元在t時(shí)刻的狀態(tài)，hk表示放大函數(shù)，dk表示適當(dāng)?shù)男袨楹瘮?shù)，A(i)=(akl(i))n×n和B(i)=(bkl(i))n×n分別表示連接權(quán)矩陣和時(shí)滯連接權(quán)矩陣，fl為神經(jīng)元的激勵(lì)函數(shù)，uk(i,t)表示網(wǎng)絡(luò)的一些外部輸入，τ(t)表示時(shí)滯，且滿足條件0≤τ(t)≤τ，τ′(t)≤ρ<1，τ和ρ都是正常數(shù).

LV(x(t),t,i)=Vt(x(t),t,i)+Vx(x(t),t,i)×{H(x(t))[-D(i,x(t))+

要求神經(jīng)網(wǎng)絡(luò)模型(1)滿足以下假設(shè)

定義1[9]若存在一個(gè)KL—函數(shù)β：R+×R+→R+和一個(gè)K—函數(shù)γ(·)使得x(t；ξ,u(t))滿足

|x(t；ξ,u(t))|≤β(t,‖ξ‖τ)+γ(‖u‖∞)

則稱系統(tǒng)(1)是輸入到狀態(tài)穩(wěn)定的.

定義2[9]若存在一個(gè)常數(shù)λ>0和2個(gè)K—函數(shù)β,γ使得x(t;ξ,u(t))滿足

|x(t;ξ,u(t))|≤β(‖ξ‖τ)exp(-λt)+γ(‖u‖∞)

則稱系統(tǒng)(1)是指數(shù)輸入到狀態(tài)穩(wěn)定的.

定義3[11]設(shè)g是R→R的連續(xù)函數(shù)，g的Dini-導(dǎo)數(shù)可表示為

2穩(wěn)定性分析

D+V(r(t),x(t))≤-α1V(r(t),x(t))+α2V(r(t-τ),x(t-τ))+γ(|u|[0,t]),

(2)

那么

(3)

如果上式不成立，則必然存在t*使得

由定義3，有

x(t*))≤-α1EV(r(t*),x(t*))+α2EV(r(t*-τ),x(t*-τ))+(α1-α2)γt*.

(7)

由式(5)有,EV(r(t*),x(t*))-γt*=V(r(0),x(0))exp(-λ*t*). 又由式(4)有，W(r(t*-τ),x(t*-τ))≤H(r(t*-τ),x(t*-τ))=V(r(0),x(0))exp(-λ*(t*-τ))=V(r(0),x(0))exp(λ*τ)exp(-λ*t*),即EV(r(t*-τ),x(t*-τ))-γt*-τ≤V(r(0),x(0))exp(λ*τ)exp(-λ*t*). 因此，把以上2式分別代入式(7)得

D+W(r(t*),x(t*))≤-α1[EV(r(t*),x(t*))-γt*]+α2[EV(r(t*-τ),x(t*-τ))-γt*]≤-

α1V(r(0),x(0))exp(-λ*t*)+α2V(r(0),x(0))exp(λ*τ)exp(-λ*t*)(-α1+α2exp(λ*τ)V(r(0),x(0))exp(-λ*t*))<-

λ*V(r(0),x(0))exp(-λ*t*))=D+H(t*).

與式(6)矛盾, 故式(3)成立.

證畢.

定理1在假設(shè)1～4之下，如果存在正定矩陣S(l)和正對(duì)角矩陣P(l)=diag{p1(l),p2(l),…,pn(l)},Q(l)=diag{q1(l),q2(l),…,qn(l)}, 使得

并且c1δ2>c2ητ2，那么系統(tǒng)(1)是輸入到狀態(tài)穩(wěn)定的,其收斂率至少是λ/2，其中

證明考慮如下Lyapunov函數(shù)

(8)

由式(8)可得

c1|x|2≤V(x(t),l)≤c2|x|2,

(9)

根據(jù)假設(shè)1～4以及LV的定義，有

LV(x(t),l)=2xT(t)S(l)[H(x(t))(-D(l,x(t))+A(l)f(x(t))+B(l)f(x(t-τ(t)))+u(l,t))]+

2xT(t)S(l)[H(x(t))(-D(l,x(t))+A(l)f(x(t))+B(l)f(x(t))+u(l,t))]+

2fT(x(t))Q(l)[H(x(t))(-D(l,x(t))+A(l)f(x(t))+B(l)f(x(t))+u(l,t))]+

2xT(t)S(l)H(x(t))B(l)(f(x(t-τ(t))-f(x(t)))+2fT(x(t))Q(l)H(x(t))B(l)×

(xT(t),fT(x(t)))W(l)(x(t),f(x(t)))+δ|x(t)|2+2δ|f(x(t))|2+

由式(1)可得

利用廣義Ito公式把上式代入式(10)可得

根據(jù)引理1，得

.

證畢.

3數(shù)值例子

考慮帶有時(shí)滯的二階Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)系統(tǒng)

h1(x1(t))=0.9+0.1sinx1(t),h2(x2(t))=0.7+0.3cosx2(t),

取τ(t)=0.8≤τ=1，可以計(jì)算出

c1=2,c2=2.22,δ=4.15,α1=1.87,α2=0.32.

顯然,α1>α2, 滿足引理1的所有條件.模擬表明，圖1是輸入到狀態(tài)穩(wěn)定的；當(dāng)沒有輸入時(shí)，圖2也是穩(wěn)定的，因此系統(tǒng)(1)是輸入到狀態(tài)穩(wěn)定的.

參考文獻(xiàn)：

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[3] Zhu Q X, Cao J D, Rakkiyappan R. Exponential input-to-state stability of stochastic cohen-grossberg neural networks with mixed delays [J].Springer Nonlinear Dyn,2015,79(2):1 085-1 098.

[4] Zhu S, Shen Y. Two algebraic criteria for input-to-state stability of recurrent neural networks with time-varying delays [J]. Neural Comput Applic,2013,22(9):1 163-1 169.

[5] Choon K A. Passive learning and input-to-state stability of switched Hopfield neural networks with time-delay [J]. Information Sciences,2010,180:4 582-4 594.

[6] Xu Y, Luo W W, Zhong K, et al. Mean square input-to-state stability of a general class of stochastic recurrent neural networks with markovian switching [J].Neural Comput Applic,2014,25(2):1 657-1 663.

[7] Huang H, Ho D W C, Qu Y Z. Robust stability of stochastic delayed additive neural networks with markovian switching [J]. Neural Netw, 2007, 20(5):799-809.

[8] Halanay A. Differential equations: stability, oscillation, timelags [M].New York: Academic,1966.

[9] Zhou W S, Teng L Y, Xu D Y. Mean-square exponentially input-to-state stability of stochastic cohen-grossberg neural networks with time-varying delays [J].Neurocomputing,2015,153(4):54-61.

[10] Shen Y, Wang J. Almost sure exponential stability of recurrent neural networks with markovian switching [J]. IEEE Transactions on neural networks,2009,20(5):840-855.

[11] Liao X X, Luo Q, Zeng Z G, et al. Global exponential stability in Lagrange sense for recurrent neural networks with time delays [J].Nonlinear Analysis: Real World Applications,2008,9(3):1 535-1 557.

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Input to State Stability Based on Cohen-Grossberg Neural Network with Markovian Switching

Li Zhihong, He Xiuli

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

Abstract：Aimed at the stability of Cohen Grossberg neural networks with Markovian switching, in the report, Lyapunov-Krasovskii functional method and Halanay differential inequalities technique were used, and the new criteria on the input to the state stability of neutral network was proposed. Lastly, the numerical examples were used to illustrate the correctness and efficiency of the results.

Keywords：input to state stability; Cohen-Grossberg neural networks; Halanay differential inequalities; Markovian switching

收稿日期：2015-11-04

基金項(xiàng)目：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)

作者簡(jiǎn)介：李志鴻(1992-), 女, 江蘇泰州人，河海大學(xué)2014級(jí)碩士研究生, 研究方向：神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性分析，E-mail:1353285605@qq.com. 通信作者： 何秀麗(1979-), 女, 湖北黃岡人，博士研究生, 講師, 研究方向：隨機(jī)動(dòng)力系統(tǒng)，E-mail:hexiu00@163.com

文章編號(hào)：1004-1729(2016)01-0012-07

中圖分類號(hào)：O175

文獻(xiàn)標(biāo)志碼：ADOl：10.15886/j.cnki.hdxbzkb.2016.0003