Feedback Control of Traveling Wave Solutions of th

arXiv:nlin/0308021v1[nlin.PS]18Aug2003FeedbackControlofTravelingWaveSolutionsoftheComplexGinzburgLandauEquationK.A.Montgomery†andM.Silber††DepartmentofEngineeringSciencesandAppliedMathematics,NorthwesternUniversity,Evanston,IL60208USAAbstract.Throughalinearstabilityanalysis,weinvestigatetheeffectivenessofanoninvasivefeedbackcontrolschemeaimedatstabilizingtravelingwavesolutionsReiKx+iωtoftheone-dimensionalcomplexGinzburgLandauequation(CGLE)intheBenjamin-Feirunstableregime.Thefeedbackcontrolisageneralizationofthetime-delaymethodofPyragas[1],whichwasproposedbyLu,YuandHarrison[2]inthesettingofnonlinearoptics.Itinvolvesbothspatialshifts,bythewavelengthofthetargetedtravelingwave,andatimedelaythatcoincideswiththetemporalperiodofthetravelingwave.Wederiveasinglenecessaryandsufficientstabilitycriterionwhichdetermineswhetheratravelingwaveisstabletoallperturbationwavenumbers.Thiscriterionhasthebenefitthatitdeterminesanoptimalvalueforthetime-delayfeedbackparameter.ForvariouscoefficientsintheCGLEweusethisalgebraicstabilitycriteriontonumericallydeterminestableregionsinthe(K,ρ)–parameterplane,whereρisthefeedbackparameterassociatedwiththespatialtranslation.Wefindthatthecombinationofthetwofeedbacksgreatlyenlargestheparameterregimewherestabilizationispossible,andthatthestabilityregionstaketheformofstabilitytonguesinthe(K,ρ)–plane.WediscusspossibleresonancemechanismsthatcouldaccountforthespacingwithKofthestabilitytongues.Submittedto:Nonlinearity11.IntroductionAcurrentchallengetoourunderstandingofpattern-formingsystemsliesinourabilitytocontrolthespatio-temporalchaosthatmanyofthesesystemsnaturallyexhibit.Themathematicalexistenceofaplethoraofsimplespatialorspatio-temporalpatternsformanynonequilibrium,spatiallyextendedsystemsiswell-establishedonthebasisofequivariantbifurcationtheory[3].However,thesesimplepatternsoftenprovetobeunstableinagivensystem,whichevolvesinsteadtoastateofspatio-temporalchaos.Perhapsthesimplestandbeststudiedmanifestationofspatio-temporalchaosisthatassociatedwiththeone-dimensionalcomplexGinzburgLandauequation(CGLE)[4],auniversalamplitudeequationthatdescribesspatially-extendedsystemsinthevicinityofaHopfbifurcation.Intheso-calledBenjamin-Feirunstableregime,thesimplesolutions(e.g.travelingplanewavesandspatially-homogeneousoscillations)areallunstabletolong-waveperturbations.ThefocusofthispaperisalinearstabilityanalysisofBenjamin-FeirunstabletravelingwavesolutionsoftheCGLEinthepresenceofafeedbackcontrolschemethatwasoriginallyproposedbyLu,YuandHarrison[2]inthesettingofnonlinearoptics.Thefeedbackapproachisnoninvasive,meaningthatthefeedbacksignaldecaystozerooncethetargetedtravelingwavestateofthesystemisrealized.Feedbackcontrolmethodsaimedatstabilizingtheunstableperiodicorbitsassociatedwithlow-dimensionalchaoticattractorshavebeenextensivelyinvestigatedformorethantenyearsnow,andhaveprovedespeciallyeffectivefornonlinearopticalsystems.ThesimpleapproachtakenbyOtt,GrebogiandYorke[5]reliesonapplyingsmallperturbationstoasystemparameterthathelpmaintainthesysteminaneighborhoodofthedesiredunstableperiodicorbit.However,thisapproach,whichrequiresanactivemonitoringofthestateofthesystemsothatthefeedbackisappropriatelyadjusted,canproveimpracticalinsystemsthatevolvetoorapidly.Autoadjustingfeedbackcontrolmethods,inwhichthefeedbackisbaseduponcurrentandpaststatesofasystem,haveprovenusefulforrapidlyevolvingsystemsbecausetheyadjustautomaticallytorapidchangesofthesystemandrequirenoactivemonitoring.OneautoadjustingfeedbacktechniquethathasattractedconsiderableattentionwasintroducedbyPyragas[1].Inthisapproachthefeedbackisproportionaltothedifferencebetweenthecurrentandpaststatesofasystem,i.e.thefeedbackisF=γ(x(t)−x(t−Δt)),whereΔtistheperiodofthetargetedunstableperiodicorbit.Themethodpossessesacoupleofpropertieswhichmakeitattractiveexperimentally.First,ifastatewiththedesiredperiodicityisstabilizedthefeedbacktermvanishesandcontrolisachievedinareasonablynoninvasivemanner.Second,thistypeoffeedbackcontrolmaybestraightforwardtoimplementinthelaboratorywhenfeedbackloopsarepractical.Themethodhasbeenimplementedsuccessfullyinavarietyofexperimentalsystemsincludingelectronic[6,7],laser[8],plasma[9,10,11],andchemicalsystems[12,13,14].AnumberofmodificationsofthemethodofPyragashavealsobeeninvestigated.Forinstance,Labateetal.[15]addedafiltertotheirtimedelayedfeedbackschemeforaCO2laserandsuccessfullystabilizedperiodicbehavior;thefilterrejectedoneofthecharacteristicfrequenciesassociatedwithaquasiperiodicroutetochaosinthissystem.InyetanotherdirectionSocolaretal.[16]proposedamethodof“extendedtimedelayautosynchronization”whichincorporatesinformationaboutthestateofthesystematmanyearliertimest−nΔt(prescribedbypositiveintegersn).Thisextendedtimedelayfeedbackcansuccessfullystabilizetravelingwavesolutionsoftheone-dimensionalCGLE[17],althoughitfailsintwo-dimensions[18].Forspatiallyextendedpatternformingsystems,modificationsofthetime-delayautosynchronizationschemeofPyragashavebeenproposedwhichtakeintoaccountnotonlythetemporalperiodicity,butalsothespatialperiodicityofthetargetedpattern.