趙 雁

(樂山職業(yè)技術(shù)學(xué)院 電子信息工程系, 四川 樂山 614000)

自從1983年M. Cohen等[1]提出Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)以來，由于它在信號和圖像處理，聯(lián)想記憶，組合優(yōu)化等中的廣泛應(yīng)用，因此受到了廣泛的關(guān)注和研究[2-13].由于脈沖，隨機(jī)干擾和反應(yīng)擴(kuò)散，時滯Cohen-Grossberg脈沖隨機(jī)反應(yīng)擴(kuò)散神經(jīng)網(wǎng)絡(luò)的平衡點(diǎn)往往不存在，這時,往往研究其吸引集的存在性[14-16].事實(shí)上，在實(shí)際應(yīng)用當(dāng)中,這種穩(wěn)定就能夠滿足人們的需求[17].利用Ito公式，時滯微分不等式和M-矩陣性質(zhì),獲得了變時滯Cohen-Grossberg脈沖隨機(jī)反應(yīng)擴(kuò)散神經(jīng)網(wǎng)絡(luò)的吸引集存在的充分條件.

1 預(yù)備知識

為了方便，引入一些符號和定義.

Rn是實(shí)n維列向量空間.N{1,2,…,n}.R+[0,+∞).Rm×n記作m×n實(shí)矩陣集.用T記作n×n的單位矩陣.對A,B∈Rm×n或者A,B∈Rn,A≥B(A>B)表示A和B中的每一個對應(yīng)元素都滿足不等式“≥(>)”.特別地,如果矩陣A≥0和向量z>0,則他們分別被稱為非負(fù)矩陣和正向量.E為數(shù)學(xué)期望.

令C[X,Y]表示從集合X到集合Y的連續(xù)映射集.并記CC[[-τ,0],Rn].

PC[X,Y]={φ(t):X→Y|φ(t)除可數(shù)點(diǎn)外函數(shù)連續(xù),并且在這些可數(shù)點(diǎn)當(dāng)中，函數(shù)φ(t)的左右極限φ-(t),φ+(t)存在，滿足φ+(t)=φ(t)}.并記PCPC[[-τ,0],Rn].

對φ(t)∈C或φ(t)∈PC,定義

[φ(t)]τ=([φ1(t)]τ,…,[φn(t)]τ)T,

討論如下變時滯Cohen-Grossberg脈沖隨機(jī)反應(yīng)擴(kuò)散神經(jīng)網(wǎng)絡(luò)：

記u=(u1,…,un)T和L2(X)為標(biāo)量值勒貝格可測函數(shù)集.記

為X的L2模.進(jìn)一步，定義模‖u‖為

其中

引理1.1[18]如果ai≥0,bi≥0,i∈N,p>0,q>0,并且1/p+1/q=1,那么

引理1.2設(shè)J是非負(fù)向量，P=(pij)n×n，其中pij≥0(i≠j),Q=(qij)n×n≥0，D=-(P+Q)是非奇異M-矩陣.在初值條件u(t0+s)∈C,s∈[-τ,0]下，設(shè)u(t)=(u1(t),…,un(t))∈C[[t0,∞),Rn]滿足下面不等式條件:

D+u(t)≤Pu(t)+Q[u(t)]τ+J,t≥t0. (2)

如果初值條件滿足

u(t)≤kze-λ(t-t0)-(P+Q)-1J,

t∈[t0-τ,t0],

(3)

其中，k≥0,z=(z1,z2,…,zn)T>0，正數(shù)λ由下面不等式?jīng)Q定

[λI+P+Qeλτ]z<0,

(4)

那么

u(t)≤kze-λ(t-t0)-(P+Q)-1J,t≥t0. (5)

為了獲得所需結(jié)果，需要下列條件：

(A1) 對?j∈N和x∈Rn，都有|fj(x)|≤αj|x|和|gj(x)|≤βj|x|,

(A2) 對?i∈N和s1,s2∈R(s1≠s2)，(ci(s1)-ci(s2))/(s1-s2)≥γi>0,

(A4) 存在非負(fù)常數(shù)νij和μij，使得對?u,v∈Rn有

trace[(σi(u,v))(σi(u,v))T]≤

(A5) 存在正數(shù)λ和一向量z滿足

其中

k=1,2,…;

(6)

(A7)

k=1,2,…,

(7)

其中，δk≥1和vk≥1滿足

k=1,2,…,

(8)

其中

(9)

2 主要結(jié)果

由邊界條件和格林公式得

由條件(A1)～(A5)和不等式|ab|≤a2/2+b2/2得

那么可得

(11)

對充分小的△t>0可得

(12)

由((11))和(12)有

(13)

從上式可得

即

(14)

(15)

t∈[-τ,0],i∈N，

(16)

那么由(14)～(16)式和引理1.2可得

0≤t

(17)

假設(shè)對所有的m=1,2,…,k,下列不等式成立

v0v1…vm-1ρi,t∈[tm-1,tm),i∈N,

(18)

其中δ0=v0=1.由(A6)～(A7)和(18)式和引理1.1可得

i∈N.

(19)

由(18)～(19)式和δk,vk≥1可得

v0v1…vk-1vkρi,t∈[tk-τ,tk],i∈N. (20)

另一方面，由(14)式和vk≥1可得

(21)

由(15),(20)～(21)式和引理1.2可得

v0v1…vk-1vkρi,t∈[tk,tk+1),i∈N.

由遞推法可得

i∈N,t∈[tk,tk+1),k=0,1,2….

所以，定理2.1證明完畢.

3 例子

例3.1考慮下面模型

條件(A5)的參數(shù)如下:

取z=(1,1)和λ=0.1可得

設(shè)α1k=α2k=e0.2k/3,β1k+β2k=2e0.2k/3和tk-tk-1=8k,那么

顯然，定理2.1的所有條件成立.故

是(22)式的吸引集.

[1] Cohen M, Grossberg S. Absolute stability of global pattern formulation and parallel memory storage by competitive neural net networks[J]. IEEE Trans Syst Man Cybernet,1983,13:815-826.

[2] 龍述君. 具有分布時滯的脈沖Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)的指數(shù)穩(wěn)定性[J]. 四川師范大學(xué)學(xué)報(bào):自然科學(xué)版,2009,32(1):68-71.

[3] 龍述君,張永新,向麗. 具有混合時滯的隨機(jī)細(xì)胞神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性分析[J]. 四川師范大學(xué)學(xué)報(bào):自然科學(xué)版,2012,35(6):796-801.

[4] Cao J, Liang J. Boundedness and stability for Cohen-Grossberg neural network with time-varying delays[J]. J Math Anal Appl,2004,296:665-685.

[5] Lu K, Xu D, Yang Z. Global attraction and stability for Cohen-Grossberg neural networks with delays[J]. Neural Networks,2006,19:1538-1549.

[6] Song Q, Cao J. Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays[J]. J Comput Appl Math,2006,197:188-203.

[7] Yuan K, Cao J. An analysis of global asymptotic stability of delayed Cohen-Grossberg neural networks via nonsmooth analysis[J]. IEEE Trans Circ Syst,2005,52I(9):1854-1861.

[8] Wan L, Zhou Q. Exponential stability of stochastic reaction-diffusion Cohen-Grossberg neural networks with delays[J]. Appl Math Comput,2008,206:818-824.

[9] Wang D S, Huang L H. Periodicity and global exponential stability of generalized Cohen-Grossberg neural networks with discontinuous activations and mixed delays[J]. Neural Networks,2014,51:80-95.

[10] Mathiyalagan K, Park J H, Sakthivel R, et al. Delay fractioning approach to robust exponential stability of fuzzy Cohen-Grossberg neural networks[J]. Appl Math Comput,2014,230:451-463.

[11] Ke Y Q, Miao C F. Stability analysis of inertial Cohen-Grossberg-type neural networks with time delays[J]. Neurocomputing,2013,117:196-205.

[12] Wan L, Zhou Q H. Asymptotic behaviors of stochastic Cohen-Grossberg neural networks with mixed time-delays[J]. Appl Math Comput,2013,225:541-549.

[13] Wang J L, Wu H N, Guo L. Stability analysis of reaction-diffusion Cohen-Grossberg neural networks under impulsive control[J]. Neurocomputing,2013,106:21-30.

[14] Zhao H Y. Invariant set and attractor of nonautonomous functional differential systems[J]. J Math Anal Appl,2003,282:437-443.

[15] Xu L G, Xu D Y.p-attracting andp-invariant sets for a class of impulsive stochastic functional differential equations[J]. Comput Math Appl,2009,57:54-61.

[16] Huang Y M, Zhu W, Xu D Y. Invariant and attracting set of fuzzy cellular neural networks with variable delays[J]. Appl Math Lett,2009:478-483.

[17] 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 Anal:RWA,2008,9:1535-1557.

[18] Berberian S K. Fundamentals of Real Analysis[M]. New York:Springer-Verlag,1999.