王繼禹，賈秀玲，佘連兵

(1.河南理工大學(xué)萬(wàn)方科技學(xué)院,公共基礎(chǔ)部，河南鄭州　451400；2.六盤水師范學(xué)院數(shù)學(xué)系，貴州六盤水　553004)

?

含有變時(shí)滯的非自治BAM神經(jīng)網(wǎng)絡(luò)周期解的全局指數(shù)穩(wěn)定性

王繼禹1，賈秀玲1，佘連兵2

(1.河南理工大學(xué)萬(wàn)方科技學(xué)院,公共基礎(chǔ)部，河南鄭州451400；2.六盤水師范學(xué)院數(shù)學(xué)系，貴州六盤水553004)

利用M矩陣?yán)碚撘约癏alanay不等式技巧，給出了一類含有變時(shí)滯的非自治BAM神經(jīng)網(wǎng)絡(luò)周期解的全局指數(shù)穩(wěn)定的充分條件，這些條件去掉了對(duì)激活函數(shù)的有界性、單調(diào)性和可微性的要求.

BAM神經(jīng)網(wǎng)絡(luò)；周期解；M矩陣；全局指數(shù)穩(wěn)定性

0　引言

雙向聯(lián)想記憶神經(jīng)網(wǎng)絡(luò)(BAM)模型自從1988年被Kosko[1]提出以來，已在模式識(shí)別、智能處理、優(yōu)化計(jì)算以及復(fù)雜控制等領(lǐng)域得到廣泛應(yīng)用.雙向聯(lián)想記憶神經(jīng)網(wǎng)絡(luò)已經(jīng)引起了眾多研究者的關(guān)注[2-7],但在現(xiàn)實(shí)應(yīng)用中，人們發(fā)現(xiàn)時(shí)滯往往不可避免地出現(xiàn)在神經(jīng)網(wǎng)絡(luò)中，從而引起許多振蕩和不穩(wěn)定現(xiàn)象.文獻(xiàn)[7]對(duì)時(shí)滯的雜交雙向聯(lián)想記憶神經(jīng)網(wǎng)絡(luò)(BAM)周期解的穩(wěn)定性進(jìn)行了研究，給出了一些新的結(jié)果.受文獻(xiàn)[2-7]的啟發(fā)，本文研究如下神經(jīng)網(wǎng)絡(luò)模型：

(1)

其中i=1,2,…,n;j=1,2,…,m.這里xi(t),yj(t)分別表示第i和第j個(gè)神經(jīng)元在t時(shí)刻神經(jīng)細(xì)胞的狀態(tài)；ai(t)>0,bj(t)>0,cij(t),dji(t),pij(t),qji(t)表示t時(shí)刻的聯(lián)接權(quán)值；fj,gi為激活函數(shù)；τij(t),σji(t)是以ω為周期的信號(hào)傳輸時(shí)滯；Ii(t),Jj(t)分別是t時(shí)刻的外部輸入.

系統(tǒng)(1)的初始條件為

這里φi(t),ψj(t)是定義在[-σ,0]和[-τ,0]上的實(shí)值連續(xù)函數(shù).

1　預(yù)備知識(shí)

本文假設(shè)激活函數(shù)、信號(hào)傳輸時(shí)滯函數(shù)和聯(lián)接權(quán)值函數(shù)滿足如下條件：

(H1)存在正常數(shù)Li,Mj，使對(duì)任意的x,y∈R.

(H2)0≤τij(t)≤τ,0≤σji(t)≤σ.

為方便起見，記

定義1若存在常數(shù)κ≥1和ε>0滿足

則稱系統(tǒng)(1)是指數(shù)穩(wěn)定的，其中

這里r>1,r∈R.記

則(x*(t),y*(t))T是系統(tǒng)(1)的一個(gè)ω周期解.

定義2設(shè)實(shí)矩陣S=(sij)n×n的非主對(duì)角線元非正，且逆矩陣為非負(fù)矩陣，即當(dāng)i≠j時(shí)，sij≤0且S-1≥0，則稱S為M矩陣.

引理1[7]若a>0,b>0,p>0,q>0,p+q=1,則apbq≤pa+qb.

引理2[7]設(shè)Z(t)滿足不等式:

2　周期解的全局指數(shù)穩(wěn)定性

定理1若(H1)～(H3)成立，且存在實(shí)數(shù)αij,α*ij,βij,β*ij,ηji,η*ji,δji,δ*ji和正整數(shù)r>1,使得

為M矩陣，則模型(1)的周期解全局指數(shù)穩(wěn)定.在矩陣Σ中,

證明模型(1)可改寫為：

若令

其中r為正整數(shù),則由(H1)～(H3)和引理1可知

同理可得

令

設(shè)

則

由條件可知

也即

同理

所以

于是由定義1可知，模型(1)的周期解全局指數(shù)穩(wěn)定.】

3　例子

考慮下列帶有時(shí)變時(shí)滯的BAM神經(jīng)網(wǎng)絡(luò)周期解的全局指數(shù)穩(wěn)定性:

(2)

是M矩陣，因此模型(2)的周期解全局指數(shù)穩(wěn)定.

[1]KOSKO B.Bi-dierctional associative memories[J].IEEETransanctionSystemManCybernet,1988,18(1)：49.

[2]CAO J D,MENG D F.Exponential stability of delayed bidirectional associative memory neural networks[J].AppliedMathematicsandComputation,2003,135:102.

[3]CAO J D.Global asymptotic stability of delayed bi-directional associative memory neural networks[J].AppliedMathematicsandComputation,2003,142:333.

[4]CAO J D,LIANG J J.Global asytmptotic stability of bi-directional associative memory neural networks with distributed delays[J].AppliedMathematicsandComputation,2004,152:415.

[5]LI Y T,YANG C B.Global exponential stability analysis on impulisive BAM neural networks with distributed delays[J].JournalofMathematicalAnalysisandApplications,2006,324:1125.

[6]龐麗艷.時(shí)標(biāo)上帶有多變時(shí)滯的BAM神經(jīng)網(wǎng)絡(luò)的反周期解的存在性和指數(shù)穩(wěn)定性[J].西北師范大學(xué)學(xué)報(bào)(自然科學(xué)版)，2014,50(5):15.

[7]LI Y T,WANG J Y.An analysis on global exponential stability and the existence of periodic solutions for non-autonomous hybrid BAM neural networks with distributed delays and impulses[J].ComputersandMathematicswithApplications,2008,56:2256.

(責(zé)任編輯馬宇鴻)

An analysis on global exponential stability of periodic solutions for non-autonomous BAM neural networks with time-varying delays

WANG Ji-yu1,JIA Xiu-ling1,SHE Lian-bing2

(1.Department of Public Basic Education,Wanfang College of Science and Technology,Henan Polytechnic University, Zhengzhou 451400,Henna,China; 2.Department of Mathematics，Liupanshui Normal University,Liupanshui 553004,Guizhou,China)

Applying theM-matrix theory,Halanay inequality technique and some analysis techniques,some sufficient conditions are obtained for the global exponential stability of periodic solutions for non-autonomous BAM neural networks with time-varying delays.The results remove the assumptions of the boundedness,monotonicity or differentiability of the activation functions,and in some cases,the stability criteria can be easily checked.

BAM neural network;periodic solution;M-matrix;global exponential stability

10.16783/j.cnki.nwnuz.2016.05.003

2015-09-04;修改稿收到日期:2015-12-03

國(guó)家自然科學(xué)基金資助項(xiàng)目(11361074);河南省教育廳重點(diǎn)科研項(xiàng)目(15A110027)；河南省基礎(chǔ)與前沿技術(shù)項(xiàng)目(142300410384)

王繼禹(1981—)，男，河南南陽(yáng)人，講師，碩士.主要研究方向?yàn)榉汉⒎址匠潭ㄐ岳碚?

E-mail:jywang1981@163.com

O 175.1

A

1001-988Ⅹ(2016)05-0010-04