一類含有分布時(shí)滯的非自治細(xì)胞神經(jīng)網(wǎng)絡(luò)的全局指數(shù)穩(wěn)定性

李 賓， 龍述君

(1. 西華大學(xué) 理學(xué)院, 四川 成都 610039; 2. 樂山師范學(xué)院 數(shù)學(xué)與信息科學(xué)學(xué)院, 四川 樂山 614004)

一類含有分布時(shí)滯的非自治細(xì)胞神經(jīng)網(wǎng)絡(luò)的全局指數(shù)穩(wěn)定性

李 賓1， 龍述君2*

(1. 西華大學(xué) 理學(xué)院, 四川 成都 610039; 2. 樂山師范學(xué)院 數(shù)學(xué)與信息科學(xué)學(xué)院, 四川 樂山 614004)

研究一類具有分布時(shí)滯的非自治細(xì)胞神經(jīng)網(wǎng)絡(luò)的全局指數(shù)穩(wěn)定性，通過建立一個(gè)新的微分-積分不等式,并將其運(yùn)用到非自治細(xì)胞神經(jīng)網(wǎng)絡(luò)的穩(wěn)定性研究中,從而得到全局指數(shù)穩(wěn)定性的充分判別準(zhǔn)則.結(jié)論較以往的文獻(xiàn)結(jié)果,更具一般性,適用范圍更廣.最后,通過實(shí)例說明所獲結(jié)論的可行性和優(yōu)越性.

細(xì)胞神經(jīng)網(wǎng)絡(luò); 非自治; 全局指數(shù)穩(wěn)定; 微分-積分不等式; 分布時(shí)滯.

自從L.O.Chua等1988年在文獻(xiàn)[1-2]中首次提出細(xì)胞神經(jīng)網(wǎng)絡(luò)理論以來,其相關(guān)理論已經(jīng)成功應(yīng)用在模式識(shí)別、移動(dòng)圖像處理、運(yùn)動(dòng)目標(biāo)識(shí)別等領(lǐng)域.網(wǎng)絡(luò)系統(tǒng)的穩(wěn)定性是這些應(yīng)用必須考慮的因素，因此,吸引了很多學(xué)者對(duì)該動(dòng)力行為進(jìn)行研究并獲得許多有用的結(jié)果[3-13].在現(xiàn)實(shí)生活中,時(shí)滯現(xiàn)象是普遍存在的,時(shí)滯的存在往往會(huì)對(duì)系統(tǒng)具有一定的破壞作用,如在神經(jīng)元間信號(hào)傳輸過程中產(chǎn)生的時(shí)滯現(xiàn)象會(huì)對(duì)系統(tǒng)造成震蕩、混沌或不穩(wěn)定現(xiàn)象，因而研究時(shí)滯對(duì)動(dòng)力行為的影響是非常有必要的[14-17].

當(dāng)考慮一個(gè)長期的動(dòng)力行為時(shí),系統(tǒng)的參數(shù)常常受到環(huán)境干擾而隨時(shí)間變化,非自治微分系統(tǒng)更能準(zhǔn)確描述此類情況.對(duì)人工細(xì)胞神經(jīng)網(wǎng)絡(luò)而言,也不例外.然而對(duì)非自治神經(jīng)網(wǎng)絡(luò)的研究遠(yuǎn)比對(duì)自治神經(jīng)網(wǎng)絡(luò)的研究困難得多.人們嘗試各種方法對(duì)其進(jìn)行研究,并獲得較好結(jié)果.文獻(xiàn)[18-19]采用李亞普洛夫楔函數(shù)方法研究具有有限時(shí)滯的非自治細(xì)胞神經(jīng)網(wǎng)絡(luò)的動(dòng)力行為,獲得了系統(tǒng)穩(wěn)定性的充分判據(jù);文獻(xiàn)[8-9]通過建立新的微分不等式,運(yùn)用不等式分析技巧研究幾類非自治神經(jīng)網(wǎng)絡(luò)的動(dòng)力行為,拓寬了判定動(dòng)力行為條件的使用范圍.此外,由于具有各種軸突大小和軸突長度的大量平行路徑存在,神經(jīng)網(wǎng)絡(luò)通常具有空間延展,此類情況下,時(shí)滯往往以分布時(shí)滯呈現(xiàn)出來，因而有必要研究具有分布時(shí)滯的非自治細(xì)胞神經(jīng)網(wǎng)絡(luò)的動(dòng)力行為[20-21],但這些成果中都要求判定條件在時(shí)間變化范圍內(nèi)一致成立,這在一定程度上影響了成果的適用范圍.

基于以上分析,本文將對(duì)含有分布時(shí)滯的非自治細(xì)胞神經(jīng)網(wǎng)絡(luò)的全局指數(shù)穩(wěn)定進(jìn)行研究,得到全局指數(shù)穩(wěn)定性的充分判別準(zhǔn)則,從而推廣一些現(xiàn)有文獻(xiàn)的相關(guān)結(jié)果.

考慮如下含分布時(shí)滯的細(xì)胞神經(jīng)網(wǎng)絡(luò)模型:

i=1,2,…，n,

(1)

1 預(yù)備知識(shí)

C[X,Y]表示從拓?fù)淇臻gX到拓?fù)淇臻gY的所有連續(xù)映射全體.特別地,令CC[(-∞,0],Rn]表示所有有界連續(xù)函數(shù)φ：(-∞,0]→Rn且|.

定義1如果存在常數(shù)λgt;0,M≥1,使得對(duì)于系統(tǒng)(1)的任意2個(gè)分別滿足初值條件φ,φ∈C的解x(t,φ),y(t,φ).對(duì)任意t≥t0有

‖x(t,φ)-y(t,φ)‖≤M‖φ-φ‖e-λ(t-t0)，

則稱系統(tǒng)(1)是全局指數(shù)穩(wěn)定的.

引理1假設(shè)p(t)滿足如下含有脈沖項(xiàng)的微積分不等式

(2)

如果當(dāng)t≤t0時(shí)有

p(t)≤me-λ(t-t0)，

(3)

則

t≥t0,

(4)

其中,m為正常數(shù)，δkmax{1,|pk|+|qk|×k(s)eλsds},λ∈(0,λ0)，滿足

(5)

下面將證明

(6)

為了證明(6)式,首先證明,對(duì)任意的常數(shù)εgt;0有

(7)

假設(shè)(7)式不成立,則存在一個(gè)t*∈(t0,t1),使得

p(t*)=n(t*),D+p(t*)≥n′(t*),

(8)

p(t)≤n(t), t∈(-∞,t*].

(9)

結(jié)合(2)、(5)、(7)～(9)式,可以得到

(10)

顯然與(8)式的第二個(gè)不等式矛盾,因而(7)式成立,在(7)式中,令ε→0,得到(6)式成立.

接下來,結(jié)合(2)、(3)和(6)式,得到

(11)

則

運(yùn)用與(6)式類似的方法,得到

通過歸納得到,對(duì)任意的k∈N有

故原命題得證.

注1在引理1中,當(dāng)t≥t0時(shí),如果α(t)≡0和β(t)≡0,得到文獻(xiàn)[13]中的引理1.

2 主要結(jié)果

定理1假設(shè)如下條件成立:

(A1) 對(duì)任意i,j=1,2,…,n和l=1,2,…,m,存在kj和uijl,使得

|fj(x)-fj(y)|≤kj|x-y|,

|gijl(x)-gijl(y)|≤uijl|x-y|;

(12)

(13)

(A3) 存在常數(shù)λgt;γ≥0和h≥0,使得

(14)

其中λ滿足

(15)

則系統(tǒng)(1)是全局指數(shù)穩(wěn)定的且指數(shù)收斂率不低于λ-γ.

證明設(shè)x(t),y(t)是系統(tǒng)(1)的任意2個(gè)分別滿足初值條件φ,φ∈C的解.令

結(jié)合系統(tǒng)(1)和條件(A1)、(A2),得到

(16)

因?yàn)棣?φ∈C,則存在一個(gè)正數(shù)M≥1,使得

(17)

3 實(shí)例說明

考慮如下二維含有分布時(shí)滯的非自治細(xì)胞神經(jīng)網(wǎng)絡(luò)系統(tǒng)

(18)

明顯地,gi(s)滿足李普希茲條件且ui=1(i=1,2),得到(A2)的參數(shù)

-b1(t)≤-2.25+δ(t),

-b2(t)≤-2.5+δ(t);

|c11(t)|+|c21(t)|≤1.75,

|c12(t)|+|c22(t)|≤1.6.

下面計(jì)算

對(duì)任意tgt;t0≥0,存在正整數(shù)n≥m≥0,使得nT≤tlt;(n+1)T,mT≤t0lt;(m+1)T;令t=nT+u,t0=mT+w,其中0≤u,wlt;T.通過計(jì)算得到

特別地,當(dāng)k∈N時(shí),令

明顯地,δ(s)的周期為2π,易得

圖 1 x1(t)的狀態(tài)曲線

圖 2 x2(t)的狀態(tài)曲線

圖 3 ‖x(t)-y(t)‖的衰減曲線

注3在文獻(xiàn)[20]的條件(H2)中,令ωi=qij=rij=1(i,j=1,2);得到相應(yīng)的判別條件：

h1(t)=b1(t)-(|c11(t)|+|c21(t)|)=
1.625-cost-sint-δ(t);
h2(t)=b2(t)-(|c12(t)|+|c22(t)|)=
1.75-0.75sint-δ(t).

通過觀察圖4和5發(fā)現(xiàn),對(duì)任意的t≥t0,不存在σgt;0,使得h1(t)gt;σ,h2(t)gt;σ成立，因此,文獻(xiàn)[20]中的結(jié)論對(duì)此例是失效的.

圖 4 h1(t)對(duì)應(yīng)的圖形

圖 5 h2(t)對(duì)應(yīng)的圖形

4 結(jié)束語

本文研究一類含有分布時(shí)滯的非自治細(xì)胞神經(jīng)網(wǎng)絡(luò)的全局指數(shù)穩(wěn)定性問題,通過運(yùn)用不等式分析技巧,建立一個(gè)新的微分-積分不等式，使得神經(jīng)網(wǎng)絡(luò)的全局指數(shù)穩(wěn)定的判別準(zhǔn)則得到了進(jìn)一步放松,較之前的結(jié)果適用范圍更廣.

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2010MSC：34D23; 92B20

(編輯 鄭月蓉)

Global Exponential Stability of Non-autonomous Cellular Neutral Network Models with Distributed Delays

LI Bin1, LONG Shujun2

(1.SchoolofScience,XihuaUniversity,Chengdu610039,Sichuan;2.CollegeofMathematicsandInformationScience,LeshanNormalUnivesity,Leshan614004,Sichuan)

In this paper, we investigate the global exponential stability of non-autonomous cellular neural networks with distributed delays. We establish a new differential-integro inequality and use it in the investigation the stability of cellular neural networks to obtain a sufficient condition for the global exponential stability for the considered system. Our results improve the known results in the literature. Finally, an example is given to illustrate the effectiveness and superiority of our conclusion

cellular neural network; non-autonomous; global exponential stability; differential-integro inequality; distributed delays

O175.13

A

1001-8395(2017)06-0780-07

10.3969/j.issn.1001-8395.2017.06.012

2016-10-10

四川省教育廳創(chuàng)新團(tuán)隊(duì)項(xiàng)目(16TD0029)

*通信作者簡介：龍述君(1975—)，男，教授，主要從事運(yùn)籌學(xué)與控制論的研究，E-mail：longer207@yahoo.com.cn