H.Chen,L.Wang/AppliedMathematicsandComputation239(2014)320–325321
time-varyingdelayfunctionisrequiredtobelessthanoneshouldbeimposed.AlthoughChenhasdiscussedtheexponentialstabilityinmeansquaremomentforneutralstochasticlinearsystemwithtime-varyingdelaybyusingtheLMIapproachin[4,5],theobtainedresultsarenotsuitableforthefasttime-varyingdelay,whichmeansthatthederivativevalueofthetime-varyingdelayfunctionislargerthanoneandthetime-varyingdelayfunctionisonlymeasurable.Somedeterministicneutralsystemswithtime-varyingdelayhaveextensivelybeeninvestigatedbyusingtheLMIandeventheobtainedresultscanbeappliedforthefasttime-varyingdelaycase,buttheproposedmethodscannotbeeasilyusedtoanalyzethestabilityanalysisforneutralstochasticsystemswithtime-varyingdelayssincetheexistenceoftheneutralitemandthepresenceofstochasticperturbationcanmaketheproblemberathercomplicated,see[13]andthereferencestherein.Thus,howtoobtainsomesimplesuf cientcriteriaensuringtheexponentialstabilityinmeansquaremomentforneutralstochasticlinearsystemwiththefasttime-varyingdelaystillremainsanimportantandchallengingproblem.
Inthispaper,thisproblemwillbesolved.Byestablishingacoupleddelay-integralinequalityandusingthestochasticanalytictechnique,theexponentialstabilitycriteriaforneutralstochasticlinearsystemwithtime-varyingdelayaregivenintermsofthealgebraicinequality.Whatismore,therestrictiveconditiononthetime-varyingdelayfunctioncanberemoved,whichmeansthatafasttime-varyingdelayisallowed.Finally,anillustrativenumericalexampleisprovidedtoshowtheeffectivenessofthederivedresult.
Notations:Inthispaper,RnandRmÂnarethen-dimensionalEuclideanspaceandthesetofmÂn-dimensionalrealmatrix,respectively.jÁjdenotestheEuclideannormandkÁkpresentsthenÂn-dimensionalrealmatrixnorm.ðX;I;PÞisacompletedprobabilityspace,whereXisthesamplespace,Iisar-algebraofsubsetsofX,andPistheprobabilitymeasure.
nn
DenotebyL2I0ð½Àh;0 ;RÞðh>0ÞisthefamilyofallI0-measurableCð½Àh;0 ;RÞ-valuedrandomvariablesn¼fnðhÞ:
Àh6h60gsuchthatsuph2½Àh;0 EfjnðhÞj2g<þ1,whereEfÁgstandsforthemathematicalexpectationoperator.Rþdenotesthesetofallrealnon-negativenumbersandðRþÞ ðRþÞÂðRþÞÂÁÁÁÂðRþÞ.
| {z }
n
n
2.Problemstatementandmainresult
Inthispaper,weconsidertheneutralstochasticlinearsystem:
&
d½xðtÞÀDxðtÀhðtÞÞ ¼½ÀCxðtÞþAxðtÀhðtÞÞ dtþHxðtÞdBðtÞ;tP0;xðhÞ¼wðhÞ;
T
h2½Àh;0 ;
T
ð1Þ
nnTTTT
wherexðtÞ¼½xTandxðtÀhðtÞÞ¼½xTarethestate1ðtÞ;x2ðtÞ;...;xnðtÞ 2R1ðtÀh1ðtÞÞ;x2ðtÀh2ðtÞÞ;...;xnðtÀhnðtÞÞ 2R
vectorandthedelayedstatevector,respectively.BðtÞis1-dimensionalBrownianmotionde nedonthecomplete2nTT
probabilityspaceðX;I;PÞ.wðÁÞ¼½wT1ðÁÞ;w2ðÁÞ;...;wnðÁÞ 2LI0ð½Àh;0 ;RÞisthegiveninitialcondition.Thematrices
T
C¼diagfc1;c2;...;cng>0,A¼ðaijÞnÂnandH¼ðhijÞnÂn.ThematrixD¼ðdijÞnÂndenotestheneutralitemwithitsnormkDk<1.Thetime-varyingdelayshjðtÞðj¼1;2;...;nÞaresomeboundedfunctionsde nedontheinterval½0;þ1Þsatisfyingthefollowingassumptions:06hjðtÞ6hjandh¼maxfh1;h2;...;hng.Thesystem(1)canbewrittenasfollows
"#8nnnXXX><d½xðtÞÀdijxjðtÀhjðtÞÞ ¼ÀcixiðtÞþaijxjðtÀhjðtÞÞdtþhijxjðtÞdBðtÞ;tP0;i
j¼1j¼1j¼1>:
xiðhÞ¼wiðhÞ;h2½Àh;0 :
ð2Þ
Toobtaintheexponentialstabilityinmeansquaremomentforsystem(1),weneedthefollowingLemma:
Lemma1.Forci>0ði¼1;2;...;nÞ,thereexistan-dimensionalpositivevector:k¼½k1;k2;...;kn >0,twomatrices:
W¼ðwijÞnÂnandV¼ðvijÞnÂn,wherewij>0;vij>0;ð16i6j6nÞ,andonen-dimensionalvectorfunctionyðÁÞ¼½yT1ðÁÞ;Pn
vPnTþnj¼1ijT
<1,thefollowinginequalities:yTj¼1wijþ2ðÁÞ;...;ynðÁÞ :½Àh;þ1Þ!ðRÞ.If8nnXXR><kieÀcitþwijsupyjðtþhÞþvij0teÀciðtÀsÞsupyjðsþhÞds;tP0;
yiðtÞ6h2½Às;0 h2½Às;0 j¼1j¼1
>:Àct
t2½Àh;0 kiei;
i
ð3Þ
holdfori¼1;2;...;n.Thenwehave:yiðtÞ6MeÀltðtPÀhÞ;l¼minfl1;l2;...;lng,whereliisapositiverootofthealgebra
Pn &'
vPnkðcÀlÞj¼1ij
elih¼1;ði¼1;2;...;nÞ,respectively,andM¼maxi¼1;2;...;niiih;ki>0.equation:j¼1wijþi
i
e
j¼1ij
v
Proof.LettingFðmiÞ¼
Pn
j¼1wij
þ
Pn
ii
j¼1ij
v
emihÀ1,wehaveFð0ÞFðciÀÞ<0holds.Thatis,thereexistsapositiveconstant
li2ð0;ciÞ,suchthatFðliÞ¼0.
Foranye>0andletting