Stability enhancement by boundary control in the K

StabilityEnhancementbyBoundaryControlintheKuramoto-SivashinskyEquationWei-JiuLiuandMiroslavKrsticDepartmentofAMESUniversityofCaliforniaatSanDiegoLaJolla,CA92093-0411fax:[e-mail]@ucsd.edu619-822-2406619-822-1374[0;1].Firstwenotethat,whiletheuncontrolledDirichletproblemisasymptoticallystablewhenan\anti-diusionpa-rameterissmall,andunstablewhenitislarge(wedeterminethecriticalvalueoftheparameter),theuncontrolledNeumannproblemisneverasymptoticallystable.WedevelopaNeumannfeedbacklawthatguaranteesL2-globalexponentialstabilityandH2-globalasymptoticstabilityforsmallvaluesoftheanti-diusionparameter.Themoreinterestingproblemofboundarystabilizationwhentheanti-diusionparameterislargeremainsopen.Ourproofofglobalexistenceanduniquenessofsolutionsoftheclosed-loopsysteminvolvesconstructionofaGreenfunctionandapplicationoftheBanachcontractionmappingprinciple.KeyWords:Kuramoto-Sivashinskyequation;boundarycontrol;stabilization;spectralanalysis.PACScodes:47.27.T,02.30.Jr,02.60.Lj.ThisworkwassupportedbygrantsfromOceofNavalResearch,AirForceOceofScienticResearch,andNationalScienceFoundation.1IntroductionInthisarticle,weaddresstheproblemofglobalboundarycontroloftheKuramoto-Sivashinskyequation(KS-equationforshort)ut+uxxxx+uxx+uux=0;0x1;t0;(1.1)wherewereferto0asthe\anti-diussionparameter.Notethatamoregeneralformut+1uxxxx+2uxx+3uux=0canalwaysbereducedto(1.1)byappropriaterescalingoft;xandu.Theequation(1.1)wasderivedindependentlybyKuramotoetal[24,25,26]asamodelforphaseturbulenceinreaction-diusionsystemsandbySivashinsky[40]asamodelforplaneamepropagation,describingthecombinedinuenceofdiusionandthermalconductionofthegasonstabilityofaplaneamefront.Sofar,ithasbeenwellunderstoodthattheKS-equationcanalsoserveasamathematicalmodelforcellularinstabilitiesinavarietyofsituations:theowofthinliquidlmsoninclinedplanes[36](inthelimitoflargesurfacetension),dendriticfrontsindilutebinaryalloys[37],andAlfvendriftwavesinplasmas[27](asanonlinearsaturationmechanismofthedissipativetrappedionmodes).Theproblemoflarge-timebehaviorofthisnonlinearfourthorderdissipativeequationhasbeenextensivelystudied.ThepioneeringworkappearstobeduetoFoiasetal[11]andNicolaenkoetal[34,35,36],whodescribedtheglobalattractorsandinertialmanifordsoftheKS-equation.Sincethen,therehasbeenanimpressiveamountofprogressonanalysisoftheKS-equation[2,3,5,6,7,9,10,12,14,15,16,17,18,21,23,28,32,39,43,45,46,47].Atthisstage,controlproblemsfortheKS-equationarelargelyunexplored.Heetal[20]havestudiednumericalaspectsofcontrollabilityandoptimalcontrol.Christodes[4]hasdevelopedlinearcontrollersbasedonaGalerkintruncationwhichachievelocalstabilization.Both[20]and[4]employdistributedcontrolandperiodicboundaryconditions.Inthispaperweareconcernedwithboundarycontrol.Westartbyshowingthat,underDirichletboundaryconditions,thetrivialsolutionu(x;t)0isunstablefor42andasymptoticallystablefor42(forthelattercasewederiveglobalexponentialdecayes-timates).ThenwemovetotheproblemwithNeumannboundarycontrol.Theuncontrolledsystemisnotasymptoticallystableevenfor42.WeintroducenonlinearboundaryfeedbackandprovethatitguaranteesL2-globalexponentialstability,H2-globalasymptoticstability,andH2-semiglobalexponentialstabilityif42.ByconstructingaGreenfunc-tionandusingtheBanachcontractionmappingprinciple,weprovethattheclosed-loopsystemhasaglobaluniqueandinnitelydierentiablesolution.Wepointoutthattheboundarystabilizationproblemfor42,i.e.,whentheuncon-trolledsystemisunstableunderbothDirichletandNeumannboundaryconditions,remainsopen.Inouropinion,thisproblemrequiresaradicallydierentapproachthantheonepresentedinthispaper.WepresentourmainresultsinSection2.Section3isdedicatedtothespectralanalysisofthelinearproblem,fromwhichwendthecriticalvalue=42.Inordertoprovethemainresults,werstestablishanewdierentialinequalityofGronwalltypeinSection4,andthenprovethemainresultsbyusingLyapunovtechniquesforthestabilitytheoremsandtheBanachcontractionmappingprincipleforthewell-posednesstheorem.22MainResultsWerstconsiderthefollowinguncontrolledequationwithDirichletboundaryconditions8:ut+uxxxx+uxx+uux=0;0x1;t0;u(0;t)=u(1;t)=0;t0;ux(0;t)=ux(1;t)=0;t0;u(x;0)=u0(x);0x1:(2.1)TheenergyE(t)ofsolutionsof(2.1)isdenedbyE(t)=Z10u(x;t)2dx;(2.2)anditshigherorderenergyV(t)isdenedbyV(t)=Z10uxx(x;t)2dx:(2.3)Thestabilityofthesystem(2.1)signicantlydependsontheanti-diusionparameter0.Roughlyspeaking,thesystemisasymptoticallystableifissmallenoughandunstableifsucientlylarge.Itisquitediculttolocatetheboundaryvaluebetweenstabilityandinstability.Thisreliesonadelicatespectralanalysisofthefollowingeigenvalueproblem1(’xxxx+’xx=’;0x1;’(0)=’(1)=’x(0)=’x(1)=0:(2.4)Foragiven2R,since@4=@x4+@2=@x2withtheaboveDirichletboundaryconditionsisaself-adjointoperator,theeigenvaluesarerealnumbers.Moreover,sincetheinver