Existence and global exponential stability of periodic solution

AppliedMathematicsandComputationjournalhomepage:www.elsevier.com/locate/amcExistenceandglobalexponentialstabilityofperiodicsolutionforimpulsiveCohen–Grossberg-typeBAMneuralnetworkswithcontinuouslydistributeddelays

XiaodiLi

SchoolofMathematicalSciences,XiamenUniversity,Xiamen361005,PRChinaarticleinfoabstract

ThispaperisconcernedwiththeexistenceandglobalexponentialstabilityofperiodicsolutionforaclassofimpulsiveCohen–Grossberg-typeBAMneuralnetworkswithcontin-uouslydistributeddelays.SomesufficientconditionsensuringtheexistenceandglobalexponentialstabilityofperiodicsolutionarederivedbyconstructingasuitableLyapunovfunctionandanewdifferentialinequality.TheproposedmethodcanalsobeappliedtostudytheimpulsiveCohen–Grossberg-typeBAMneuralnetworkswithfinitedistributeddelays.Theresultsinthispaperextendandimprovetheearlierpublications.Finally,twoexampleswithnumericalsimulationsaregiventodemonstratetheobtainedresults.Ó2009ElsevierInc.Allrightsreserved.Keywords:Cohen–Grossberg-typeBAMneuralnetworksImpulseExistenceGlobalexponentialstabilityPeriodicsolutionContinuouslydistributeddelays1.IntroductionInrecentyears,considerableattentionhasbeenpaidtoinvestigatethedynamicsofartificialneuralnetworksbecauseoftheirimportantapplicationsindifferentfieldssuchasimageprocessing,automaticcontrol,engineering,patternrecognition,andoptimizationproblems.Manyimportantandinterestingresultshavebeenobtainedin[1–14,17,18,38–41]andrefer-encestherein.TheCohen–Grossberg-typeBAMneuralnetworksmodel(i.e.,theBAMmodelthatpossessesCohen–Grossbergdynamics),initiallyproposedbyCohenandGrossberg[4],havetheirpromisingpotentialforthetasksofparallelcomputa-tion,associativememoryandhavegreatabilitytosolvedifficultoptimizationproblems.Insuchapplications,itisofprimeimportancetoensurethattheequilibriumpointofdesignedneuralnetworksarestable[7].Nowtherehavebeenmanyre-sultsonthestabilityandconvergenceofequilibriumpointofCohen–Grossberg-typeBAMneuralnetworkswithdelays,see[19–23].Forexample,byconstructingsomesuitableLyapunovfunctionals,FengandPlamondon[19]investigatedtheasymptoticstabilityofaclassofCohen–Grossberg-typeBAMneuralnetworkswithconstantdelays.In[20],CaoandSongfurtherinvestigatedtheglobalexponentialstabilityforCohen–Grossberg-typeBAMneuralnetworkswithtime-varyingde-laysbyusingLyapunovfunction,M-matrixtheoryandinequalitytechnique.Inaddition,theresearchofneuralnetworkswithdelaysinvolvesnotonlythedynamicanalysisofequilibriumpointbutalsothatofperiodicoscillatorysolution.Inpractice,thedynamicbehaviorofperiodicoscillatorysolutionisveryimportantinlearningtheory[11,12],whichismotivatedbythefactthatlearningusuallyrequiresrepetition.Moreover,itiswellknownthatanequilibriumpointcanbeviewedasaspecialperiodicsolutionofneuralnetworkswitharbitraryperiod.Inthissense,theanalysisofperiodicsolutionsofneuralnetworkswithdelaystobemoregeneralthanthatofequilibriumpoint.SeveralimportantresultsforperiodicsolutionsofneuralnetworkswithdelayshavebeenobtainedinRefs.[24–37].Asiswellknown,impulsiveeffectswidelyexistinmanydynamicalsystemsinvolvingsuchareasaspopulationdynamics,automaticcontrol,drugadministrationandsoon.Forexample,inimplementationofelectronicnetworksinwhichstateisE-mailaddress:sodymath@163.com0096-3003/$-seefrontmatterÓ2009ElsevierInc.Allrightsreserved.doi:10.1016/j.amc.2009.05.005X.Li/AppliedMathematicsandComputation215(2009)292–307293subjecttoinstantaneousperturbationsandexperiencesabruptchangeatcertainmoments,whichmaybecausedbyswitch-ingphenomenon,frequencychangeorothersuddennoise,thatis,doesexhibitimpulsiveeffects,see[13–17,21–27].Inpar-ticular,periodicimpulsiveeffectsisalsolikelytoexistinsomedynamicalsystems.Forexample,intheprocessofpartialdischargeonlineMonitoringofhigh-voltagetransformer,periodicimpulsiveinterferencewilloccursunavoidablyduetothefrequentswitchingofSiliconControlledRectifiertriggerandtheinfluenceofunknownfactorsingroundnetwork[15,16]andanumberofmodelsinengineeringcanbeformulatedasnon-autonomoussystemswithperiodicimpulsiveef-fects[24,26,27].Eveninbiologicalneuralnetworks,periodicimpulsiveeffectsmaybeunavoidable.Forexample,whenastimulifromthebodyortheexternalfactorsisreceivedbyreceptorsinaperiodicenvironment,theelectricalimpulseswillbeconveyedtotheneuralnetandperiodicimpulsiveeffectsarisenaturallyinthenet[10].Someinterestingresultsontheexistenceandstabilityofperiodicsolutionsofartificialneuralnetworkswithperiodicimpulsiveeffectshavebeengained,see[24,25,31]andreferencestherein.However,tothebestofourknowledge,fewauthorshaveconsideredtheproblemsofexistenceandglobalexponentialstabilityofperiodicsolutionofCohen–Grossberg-typeBAMneuralnetworkswithcon-tinuouslydistributeddelaysandperiodicimpulsiveeffects.Motivatedbytheabovediscussion,inthispaper,wearetoinvestigatetheexistenceandglobalexponentialstabilityofperiodicsolutionofCohen–Grossberg-typeBAMneuralnetworkswithcontinuouslydistributeddelaysandperiodicimpul-siveeffects.ByconstructingasuitableLyapunovfunctionandanewdifferentialinequality,somesufficientconditionscon-cerningexistenceandglobalexponentialstabilityofperiodicsolutionareobtained.Moreover,wealsoobtainsomesufficientconditionsfortheexistenceandglobalexponentialstabilityofperiodicsolutionforimpulsiveCohen–Grossberg-typeBAMneuralnetworkswithfinitedistributeddelaysbyusingtheproposedmethod.Theresultsinthispaperextendandimprovetheearlierpublications[28,30,33].Finally,twoexampleswithnumericalsimulationsaregiventodemonstratetheeffective-nessoftheobtainedresults.2.PreliminariesLetRdenotethesetofrealnumbers,Rnthen-dimensionalrealspaceequippedwiththeEuclideannormjÁj;RþthesetofpositivenumbersandZþthesetofpositiveintegralnumbers.Inthispaper,wewillinvestigatetheexistenceandglobalexponentialstabilityofperiodicsolutionforimpulsiveCohen–Grossberg-typeBAMneuralnetworkswithcontinuouslydistributeddelaysasfollows:8>dxiðtÞ>>>dt>>>>>>>>>>>>>>>>>>>>>>>>> >>n>PdyjðtÞð1Þ>>¼ÀcðyðtÞÞdðt;yðtÞÞÀwjiðtÞgiðlixiðtÀsðtÞÞÞ>jjjjdt>>i¼1>>>>!>>nR1ð2ÞP>ð2Þ>>ÀujiðtÞ0KjiðsÞgiðlixiðtÀsÞÞdsÀJjðtÞ;t–tk;j2X;>>>i¼1>>>>>n>P>ÀÀ>:DyjðtkÞ¼yjðtkÞÀyjðtÀsjiðtÀk2ZþkÞ¼ÀgjkyjðtkÞþkÞSiðxiðtkÀsÞÞ;i¼1withinitialvalues(xiðsÞ¼/1iðsÞ;yjðsÞ¼/2jðsÞ;s2ðÀ1;0󰀄;s2ðÀ1;0󰀄;i2K;j2X;whereK¼f1;2;...;ng;X¼f1;2;...;mg,theimpulsetimestksatisfy0¼t00;i2K;j2X:󰀂i;00,thereexistsq2Zþsuchthattkþx¼tkþqandcik¼ciðkþqÞ;gjk¼gjðkþqÞ;k2Zþ;i2K;j2X:HHHðH7Þhij¼maxt2½0;x󰀄jhijðtÞj;vHij¼maxt2½0;x󰀄jvijðtÞj;eij¼maxt2½0;x󰀄jeijðtÞj;wji¼maxt2½0;x󰀄jwjiðtÞj,HHHuHji¼maxt2½0;x󰀄jujiðtÞj;sji¼maxt2½0;x󰀄jsjiðtÞj;di¼mint2½0;x󰀄diðtÞ;fj¼mint2½0;x󰀄fjðtÞ:Inthispaper,weusethefollowingnormofRnþm:kzk¼nXi¼1jxijþmXj¼1jyjj;k/k¼sups2ðÀ1;0󰀄 nXi¼1j/1iðsÞjþmXj¼1!j/2jðsÞjforz¼ðx1;...;xn;y1;...;ymÞT2Rnþm;/¼ð/11;...;/1n;/21;...;/2mÞT2Cnþm.Thefollowinglemmawillbeusedforderivingourmainresults.Lemma2.1.Letp;q;randinequalitysdenotenonnegativeconstants,andfunctionf2PCðR;RþÞsatisfiesthescalarimpulsivedifferentialt–tk;tPt0;8Rr0inthecasewhenr¼þ1.Moreover,whenr¼þ1,theinterval½tÀr;t󰀄isunderstoodtobereplacedbyðÀ1;t󰀄.AssumethatRr(i)p>qþr0kðsÞds:(ii)ThereexistconstantsM>0;g>0suchthatnYk¼1ÈÉmax1;akþbkeks6MegðtnÀt0Þ;n2Zþ;wherek2ð0;g0Þsatisfiesk󰀂fðt0Þ:LettH¼infft2½t0;t1Þ;LðtÞ>󰀂LðtHÞ¼󰀂fðt0Þ;LðtÞ6󰀂fðt0Þ;t2½t0Àmaxfr;sg;tH󰀄andDþLðtHÞP0:ð4ÞSupposethatfðhtHÞ¼suptHÀs6s6tHfðsÞ;htH2½tHÀs;tH󰀄:CalculatingtheupperrightDini-derivativeDþLðtÞalongthesolutionof(2),by(3)and(4),wegetDþLðtÞjt¼tH¼DþfðtHÞekðtÀt0ÞþkfðtHÞekðtÀt0Þ !ZrHHHH6ÀpfðtÞþqfðhtHÞþrkðsÞfðtÀsÞdsekðtÀt0ÞþkfðtHÞekðtÀt0Þ !ZrHHH6ðkÀpÞfðtÞþqfðhtHÞþrkðsÞfðtÀsÞdsekðtÀt0Þ00HH<ÀqefðtÞeþrZr0ksHkðtHÀt0ÞþqekðtHÀhtHÞfðhtHÞedsHkðhtHÀt0ÞÀrfðtÞeHkðtHÀt0ÞZr0kðsÞeksdskðsÞeksfðtHÀsÞekðtHksHÀsÀt0Þ6ÀqeLðtÞþqeLðhtHÞÀrLðtÞ6ÀqeLðtÞþqefðt0ÞÀrLðtÞksHks󰀂HksZr0kðsÞedsþrksksZr0kðsÞeksLðtHÀsÞdskðsÞeksds¼0;ð5ÞZr0kðsÞedsþr󰀂fðt0ÞZr0whichcontradicts(4).SowehaveprovenLðtÞ6󰀂fðt0Þforallt2½t0;t1Þ.NowweassumethatLðtÞ6󰀂fðt0ÞNÀ1Ym¼0ÈÉ:max1;amþbmeks¼WNÀ1;t2½tNÀ1;tNÞforsomeN2Zþ;whichimpliesthatLðtÞ6WNÀ1forallt2½t0Àmaxfs;rg;tNÞ:ThenwegetÂÃkðtÀtÞÀksÀksLðtNÞ¼fðtNÞekðtNÀt0Þ6aNfðtÀÞþbfðtÀsÞeN0¼aNLðtÀNNNNÞþbNeLðtNÀsÞ6ðaNþbNeÞWNÀ1ÈÉ6max1;aNþbNeksWNÀ1¼WN:NextweshallshowthatLðtÞ6WN;t2½tN;tNþ1Þ:Supposeforthesakeofcontradictionthatthereexistssomet2½tN;tNþ1ÞsuchthatLðtÞ>WN.LettHH¼infft2½tN;tNþ1Þ;LðtÞ>WNg;thentHHPtN;LðtHHÞ¼WN;LðtÞmax󰀂i;󰀂maxi2K;j2Xfacjgi2K;j2X(nXi¼1maxhijLfjkj;j2XHð1ÞmXj¼1)gmaxwHjiLilii2Kð1Þþmax(nXi¼1maxvj2XHfð2ÞijLjkj;mXj¼1gmaxuHjiLili2Kð2Þi)Z01KðsÞds:ðH9ÞThereexistconstantsMP1;k2ð0;k0Þandg2ð0;kÞsuchthatnYk¼1maxf1;Bkg6Megtnforalln2Zþholds()nmXXmini2K;j2Xfai;cjgHð1Þgð1ÞHeksk0satisfies(6).Remark3.2.Whenvij¼0;uji¼0;eij¼0;sji¼0;cik¼0;gik¼0;i2K;j2X,thenthesystem(1)becomesthemodelana-lyzedbyLiandFan[28]andXiangandCao[30].ByCorollary3.1,wemayobservethatourresulthaswideradaptiverange.Moreover,comparedwiththesystemwithoutimpulses[28,30],thedynamicbehaviorofthesystem(1)ismorecomplexbecauseoftheeffectsofimpulsesandcontinuouslydistributeddelays.Corollary3.2.AssumethatconditionsðH1ÞÀðH8ÞholdandBk61;k2Zþ,thenthereexistsauniquex-periodicsolutionofsys-tem(1)whichisgloballyexponentiallystablewiththeapproximateexponentialconvergentratek,wherek>0satisfies(6).NextweshallconsideraclassofimpulsiveCohen–Grossberg-typeBAMneuralnetworkswithfiniteddistributeddelaysasfollows:8>dxiðtÞ>>>dt>>>>>>>>>>>>>>>>>>>>>n>PdyjðtÞð1Þ>>¼ÀcðyðtÞÞdðyðtÞÞÀwjiðtÞgiðlixiðtÀsðtÞÞÞ>jjjjdt>>i¼1>>!>>nRP>bð2Þð2Þ>>ÀujiðtÞ0KjiðsÞgiðlixiðtÀsÞÞdsÀJjðtÞ;t–tk;j2X;>>>i¼1>>>n>P>ÀÀ>:DyjðtkÞ¼yjðtkÞÀyjðtÀÞ¼ÀgðyðtÞÞþsjiðtÀk2Zþ;jjkkkkÞSiðxiðtkÀsÞÞ;i¼1ð1Þð2Þð11Þwherea;baresomepositiveconstants,thedelaykernelsKij:½0;a󰀄!Rþ;Kij:½0;b󰀄!Rþi2K;j2Xarepiecewisecontin-ð1Þð2ÞuousandsatisfyKijðsÞ6KðsÞforalls2½0;a󰀄andKjiðsÞ6KðsÞforalls2½0;b󰀄;i2K;j2X,whereKðsÞ:½0;maxfa;bg󰀄!Rþiscontinuousandintegrable.AndtheothernotationsandconditionsarethesameasthoseintheimpulsiveCohen–Gross-berg-typeBAMneuralnetworks(1).Theorem3.2.AssumethatðH1ÞÀðH4Þ;ðH6ÞandðH7Þhold,thenthereexistsauniquex-periodicsolutionofsystem(11)whichisgloballyexponentiallystableifthefollowingtwoconditionsaresatisfied:300X.Li/AppliedMathematicsandComputation215(2009)292–307ðH10Þ()nmXXmini2K;j2Xfai;cjgHfð1Þð1ÞgHHmaxhijLjkj;maxwHminfd;fg>maxjiLili󰀂i;󰀂j2Xi2Kmaxi2K;j2Xfacjgi2K;j2Xiji¼1j¼1()ZnmXXfð2Þgð2ÞþmaxmaxvHmaxuHijLjkj;jiLilii¼1j2Xj¼1i2Kmaxfa;bgKðsÞds:0ðH11ÞThereexistconstantsMP1;k>0andg2ð0;kÞsuchthatnYk¼1nmXXmini2K;j2Xfai;cjgHfð1Þgð1ÞHks;fgÀmaxfmaxhLk;maxwHminfdHijjjjiLiligeij󰀂󰀂i2K;j2Xj2Xi2Kmaxi2K;j2Xfai;cjgi¼1j¼1Znmmaxfa;bgXXfð2Þgð2ÞÀmaxfmaxvHLk;maxuHKðsÞeksds;ijjjjiLiligi¼1j2Xj¼1i2K0maxf1;Bkg6Megtnforalln2Zþholdsk>dt>>>>>>>>>>>0;t–tk;ÀÁÀ¼ÀckxðtÀk2Zþ;kÞþðsintþ0:5Þtanh0:1yðtkÀ0:3Þ;>dyðtÞ>¼Àð0:45þ0:05cosyðtÞÞ½3yðtÞÀð0:7þ0:3sintÞj0:4xðtÀsðtÞÞÀ1j>dt>>>ÃR1>>>KðsÞ0:4xðtÀsÞÀ1dsÀ4sint;t>0;t–tk;Àð0:8þ0:2costÞjj>0>>>ÀÁ:ÀDyðtkÞ¼ÀgkyðtÀk2Zþ;kÞþðcostÀ0:2Þtanh0:2xðtkÀ0:3Þ;ð12ÞtwheresðtÞ¼0:3jsin2j;ck¼gk¼À1;tk¼0:4pk;k2Zþ.Forthesimplicityofourcomputersimulations,thedelaykernelsinsystem(12)areassumedasfollows:KðsÞ¼eÀ2sfors2½0;40󰀄,andKðsÞ¼0fors>40.󰀂¼󰀂Inthecase,wehavefðuÞ¼juþ1j;gðuÞ¼juÀ1j;EðuÞ¼tanhð0:1uÞ;SðuÞ¼tanhð0:2uÞ;ac¼0:5;a¼c¼0:4;dH¼H2:5;fH¼3;h¼wH¼uH¼vH¼1;eH¼1:5;sH¼1:2;K¼eÀ2s;kð1Þ¼kð2Þ¼0:38;lð1Þ¼lð2Þ¼0:4;k0¼1:9;x¼2p;s¼0:3;Lf¼Lg¼1;LE¼0:1;LS¼0:2:Thenwecancomputethatminfa;cgminfdH;fHg¼2;󰀂;󰀂maxfacgnonoZHfð1ÞHgð1ÞHfð2ÞHgð2ÞmaxhLk;wLlþmaxvLk;uLl01KðsÞds%0:6<2:Choosek¼1>dt>>>>>>>>>dt>>>>>>:DyðtkÞ¼À2:5xðtÞþð0:1costþ0:9ÞjyðtÞþ1jþð0:4þ0:6sintÞþ3cost;t>0;t–tk;¼xðtÀk2Zþ;kÞ;¼À3yðtÞþð0:7þ0:3sintÞjxðtÞÀ1jþð0:8þ0:2costÞþ4sint;t>0;t–tk;¼yðtÀkÞ;k2Zþ;R10KðsÞjyðtÀsÞþ1jdsR10KðsÞjxðtÀsÞÀ1jdsð13Þax,y3210Non−impulsive effectsby21.5Non−impulsive effectsx(t)10.50−0.5−1−2−3−4−505101520253035t−axis40y(t)−1−1.5−2−1.5x−1−0.500.511.52cx,y3210−1−2−3−4−50510Impulsive effectsdyx(t)Impulsive effects32.521.510.50y(t)−0.5−1−1.5t−axisx1520253030−2−1.5−1−0.500.511.522.533.5Fig.1.(a)Time-seriesofthex;yofsystem(12)withoutimpulsiveeffectsfort2½À1;40󰀄.(b)Phaseportraitof2p-periodicsolutionsofsystem(12)withoutimpulsiveeffectsfort2½À40;40󰀄.(c)Time-seriesofthex;yofimpulsivesystem(12)fort2½À1;40󰀄.(d)Phaseportraitof2p-periodicsolutionsofimpulsivesystem(12)fort2½À40;40󰀄.302X.Li/AppliedMathematicsandComputation215(2009)292–307ax,y3210−1−2−3−4−50510Impulsive effectsx(t)by43210−1y(t)Impulsive effects−2t−axis40−3−2x−10123451520253035Fig.2.(a)Time-seriesofthex;yofimpulsivesystem(13)fort2½À1;40󰀄.(b)Phaseportraitof2p-periodicsolutionsofimpulsivesystem(13)fort2½À40;40󰀄.However,theresultsin[29]cannotbeapplicabletosystem(13)sinceKðsÞ¼eÀ2sandck¼gk¼À1;k2Zþ:ByourTheorem3.1,wecanobtainthatsystem(13)hasexactlyone2p-periodicsolutionwhichisgloballyexponentiallystablewiththeapproximateexponentialconvergentrate0:2:Infact,weonlyneedtochoosek¼1:2;g¼1,thentherestargumentissimilartotheproofofExample4.1.Hereweomitit.ThisfactisverifiedbythenumericalsimulationinFig.2.Fig.2aandbshowthedynamicbehaviorofthesystem(13)withsteph¼0:005andtheinitialconditions½Àe0:02sþ0:1;e0:02sþ0:1󰀄Tand½2;À2󰀄Tfors2½À40;0󰀄:ThemainnumericalschemeandalgorithmweregiveninAppendix2.Example4.2.ConsiderthefollowingimpulsiveBAMneuralnetworkmodelwithfinitedistributedelays(see[33]):8dxðtÞ>>dt>>>>>0;t–tk;t>0;t–tk;ð14ÞR10>>>xðtkÞ¼ckxðtÀ>kÞ;>>>:yðtkÞ¼gkyðtÀkÞ;k2Zþ;k2Zþ;wheretk¼f0:3þ2pk;0:6þ2pk:k¼0;1;...g:Hereweconsiderthecases:(a)ck¼gk¼2;k2Zþ;(b)ck¼gk¼À2;k2Zþ.Weshallshowthatsystem(14)hasexactlyone2p-periodicsolutionwhichisgloballyexponentiallystablewiththeapprox-imateexponentialconvergentrate0.1.󰀂¼󰀂Infact,fromsystem(14),weknowthatfðuÞ¼gðuÞ¼tanhðuÞ;ac¼a¼c¼1;dH¼fH¼1;h¼wH¼eH¼sH¼0;uH¼vH¼1;kð1Þ¼kð2Þ¼lð1Þ¼lð2Þ¼1;Lf¼Lg¼1;a¼b¼1:LetK¼0:3ð1ÀsinsÞþ0:2ð1ÀcossÞ;thenitiseasytocheckthatconditionsðH1Þ—ðH4Þ;ðH6Þ;ðH7ÞandðH10Þholdobviously.Furthermore,wechoosek¼0:5suchthatHk<1ÀZ01½0:3ð1ÀsinsÞþ0:2ð1ÀcossÞ󰀄eksds:Notethattk¼f0:3þ2pk;0:6þ2pk:k¼0;1;...g;i.e.,&t2nþ1¼0:3þ2pn;t2nþ2¼0:6þ2pn;n¼0;1;...;n¼0;1;...:Letg¼0:4;M¼2;thenfor(a)and(b),wehave2nþ1Yk¼12nþ2Yk¼1maxf1;Bkg¼22nþ1<2Â12:3453n