2011 Seventh International Conference on Computational Intelligence and Security

中立型

定理一是为下面证明做准备

Stabilityanalysisofneutral-typefuzzyneuralnetworkswithdistributeddelays

ShujunLong

CollegeofMathematicsandInformationScience

LeshanNormalUniversityLeshan614004,P.R.ChinaEmail:longer207@yahoo.com.cn

LipingJia

CollegeofMathematicsandInformationScience

LeshanNormalUniversityLeshan614004,P.R.ChinaEmail:lpjia@foxmail.com

Abstract—Inthispaper,aclassofneutral-typefuzzyneuralnetworkswithdistributeddelayisconsidered.Byusingthepropertiesof matrix,anewdifferential-integro-differenceinequalityisestablished.Basedontheinequalityandbyusingthepropertiesoffuzzyoperator,weobtainthesuf cientconditionensuringtheexponentialstabilityoftheequilibriumpointoftheconsideredneuralnetworks.

Keywords-Exponentialstability;Fuzzyneuralnetworks;Neutral-type;Differential-integro-differenceinequality.

I.INTRODUCTION

Recently,theanalysisofstabilityofcellularneuralnet-works(CNN)withdelayshasreceivedmuchattention,andlotsofresultsonstabilityhavebeenreportedsinceCNNwasintroducedinthe1980s[1]–[3].However,besidesdelayeffects,inmathematicalmodelingofrealworldproblems,wewillencountersomeotherinconveniences,forexample,thecomplexityandtheuncertaintyorvagueness.Fuzzytheoryisconsideredasamoresuitablemethodforthesakeoftakingvaguenessintoconsideration.BasedonthetraditionalCNN,Yangetal.introducedfuzzycellularneuralnetwork(FCNN)in1996[4],[5],whichcombinedfuzzylogicwiththestructureoftraditionalCNNandmaintainedlocalconnectednessamongcells.Inrecentyears,variousinterestingresultsonthestabilityofFCNNhavebeenreported[6]–[8].Tothebestofourknowledge,noresultonneutral-typeFCNNwithdistributeddelayhasbeenreportedintheliteratures.Motivatedbytheabovediscussion,themainpurposeofthispaperistostudytheexponentialstabilityofneutral-typeFCNNwithdistributeddelay.Byestablishinganewdifferential-integro-differenceinequalityandusingthepropertiesoffuzzyoperator,weobtainthesuf cientconditionensuringtheexponentialstabilityoftheequilibriumpointofconsideredneuralnetworks.

II.MODELDESCRIPTIONANDPRELIMINARIESThroughoutthispaper,weusethefollowingnotations.Letbethespaceof -dimensionalnonnegativerealcolumnvectors, bethespaceof -dimensionalrealcolumn

Δ

vectors, ={1,2,..., },and × denotesthesetof × realmatrices.Usually denotesan × unitmatrix.For , ∈ × ,thenotation ≥ ( > )meansthateachpairofcorrespondingelementsofAandBsatis es

+

theinequality“≥(>)”.Especially, ∈ × iscalledanonnegativematrixif ≥0,and iscalledapositivevectorif >0. denotesthe -throwvectorofmatrix . ° =( ) × istheHadamardproductorSchurproductofthematrices =( ) × and =( ) × .

Δ

= [( ∞,0], ]denotesthefamilyofallcontinu-ous -valuedboundedfunctions.

[ , ]={ : → isboundedandcontinuousforallbutatmosta nitenumberofpoints ∈ andatthesepoints ∈ , ( +)and ( )exist, ( +)= ( )},where isaninterval, ( +)and ( )denotetheright-handandleft-handlimitsofthefunction ( ),respectively.

Δ

Especially,let = [( ∞,0], ]. ={ ( ): → ∣ +=[0,∞), ( )iscontinu-∫+∞

ousandsatis es0 ∣ ( )∣ <∞,where >0isaconstant}.

For ∈ , ∈ ,wede ne[ ( )]∞=sup ∞< ≤0{ ( + )}, ∈ ,[ ]+=(∣ 1∣,...,∣ ∣) ,[ ( )]∞=([ 1( )]∞,...,[ ( )]∞) ,[ ( )]+=∞

++

[[ ( )]]∞,and ( )denotestheupper-right-handderivativeof ( )attime .

For ∈ or ∈ ,weintroducethefollowingnorm

∥ ∥∞=max{max∣ ( )∣}.

1≤ ≤ ∞< ≤0

Inthispaper,weconsiderthefollowingstateequation ∑ ( ( ) ( ( )))′= ( ) =1 ∑∑ + ( ( ))+ + + =1 =1 =1

∫ + ∞ ( ) ( ( )) + =1 =1 ∫ + ∞ ( ) ( ( )) , ≥0, =1

( )= ( ), ∞< ≤0,

(1)

where ∈ , , , denotethestate,inputandbiasofthe -thneuron,respectively. ( )denotesthesignal

propagationfunctionofthe -thunit. >0representstheratewithwhich -thneuronwillresetitspotentialtotherestingstateinisolationwhendisconnectedfromthenetworkandexternalinputs. , areelementsof

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