On linear differential-algebraic equations and lin

Onlineardierential-algebraicequationsandlinearizationsRoswithaMarz,BerlinAbstractOnthebackgroundofacarefulanalysisoflinearDAEs,linearizationsofnonlinearindex-2systemsareconsidered.FindingappropriatefunctionspacesandtheirtopologiesallowstoapplythestandardImplicitFunctionTheoremagain.Both,solvabilitystatementsaswellasthelocalconvergenceoftheNewton-Kantorovichmethod(quasilinearization)resultimmediately.Inparticular,thisappliesalsotofullyimplicitindex1systemswhoseleadingnullspaceisallowedtovarywithallitsarguments.Keywords.Dierentialalgebraicequations,linearization,Newton-KantorovichmethodIntroductionLinearizationplaysanimportantstandardroleintheanalysisandnumericaltreatmentofregulardierentialequations.Itisaverynicetoolforprovingsolvabilitystatements,showingasymp-toticbehaviour,describingthesensitivitywithrespecttoparametersetc.Moreover,iterativelinearizationmethodslikethestandardNewton-Kantorovichmethod,whichisalsowell-knownasquasilinearizationofBellmannandKalaba,furtherdampedandregularizedversionsofthatmethodhaveprovedtheirvalueinsolvingregularboundaryvalueproblemsforalongtime(e.g.RobertsandShipman(1972),MieleandIyer(1971),AktasandStetter(1977)).Forindex1dierentialalgebraicequationswhoseleadingnullspacedependsontimeonly,thecorrespondinglinearizationsareconsiderede.g.inMarz(1984,1986),GriepentrogandMarz(1986)andTischendorf(1994).Forindex2DAEspositiveresultsconcerningthelocalsolvabilityofinitialvalueproblemsandLyapunovstabilityvialinearizationsatconsistentvaluesresp.stationarysolutionsareobtainedinMarz(1992).Thepresentpapermainlydealswithlinearizationsofindex2DAEsalonggivenfunctionsthatarenotnecessarilysupposedtosolvetheDAE.Solvabilitystatementsforindex2DAEsaregivenunderlowsmoothnessdemands.Further,thelocalconvergenceoftheNewton-Kantorovichmethodisproved.NotethatevenfortheNewton-KantorovichprocessweareinterestedinlinearizationsalongfunctionsnotsolvingtheDAEitselfandnotnecessarilysatisfyingtherstandsecondorderconstraint.Inthiscontext,thegeometricapproachoftransferringtheDAElocallytoavectoreldonthelastorderconstraintmanifoldwillfailtobesuchausefultool,asithasbeenprovedondierentoccasions.OurmaintoolistheproposingofappropriatefunctionspacesandoperatornotionsoftheDAEproblemstoobtainFrechetderivativesthatrepresentshomeomorphismsagain.Further,standardargumentsapply.Thepaperisorganizedasfollows:x1collectssomegeneralpreliminaries.Inx2linearindex1andindex2resultsarepreparedforbeingusedbelow.Therespectivenonlinearindex1DAEsareshortlymentionedinx3.x4containsthenewpartforgeneralindex2DAEsonthebackgroundoftheexplanationsinthe1linearsection.Moreover,theindex2resultsarespeciedinx5forapplicationtofullyimplicitindex1DAEswhoseleadingnullspaceisallowedtovarywithallitsarguments.1PreliminariesGiventheDAEf(x0(t);x(t);t)=0;(1.1)wheref:IRmDJ!IRmiscontinuousandhascontinuouspartialJacobiansf0x0;f0x:IRmDJ!L(IRm),DIRmopen,Janinterval.ThenullspaceoftheleadingJacobianf0x0(y;x;t)isassumedtobeinvariantofy,x,thatiskerf0x0(y;x;t)=N(t);(y;x;t)2IRmDJ:(1.2)Moreover,letN(t)varysmoothlywitht.Thismeans,Nisspannedbyabasen1;:::;nmr2C1(J;IRm),N(t)=spanfn1(t);:::;nmr(t)g.Then,Q:=K(KTK)1KThasthepropertiesQ2C1(J;L(IRm));Q(t)2=Q(t);imQ(t)=N(t);t2J;(1.3)whereK(t):=[n1(t);:::;nmr(t)]2L(IRmr;IRm),thatis,QrepresentsaC1projectorfunc-tionontoN.Ontheotherhand,ifthereisanyprojectorfunctionQhavingtheproperties(1.3),theIVPsn0=Q0n,n(t0)=n0j,j=1;:::;mr,generateanappropriateC1base,supposedn00;:::;n0mr2IRmformabaseofN(t0)(cf.GriepentrogandMarz(1989)).Hence,theexistenceofaC1baseandaC1projectorfunction,respectively,areequivalent.Inthefollowing,wedenotebyQanyC1projectorfunctionwith(1.3),furtherP:=IQ.Assumption(1.2)simplyimpliesf(y;x;t)f(P(t)y;x;t)=1Z0f0x0(sy+(1s)P(t)y;x;t)Q(t)yds=0for(y;x;t)2IRmDJ,andfurtherf(x0(t);x(t);t)=f(P(t)x0(t);x(t);t)=f((Px)0(t)P0(t)x(t);x(t);t)forfunctionsx2C1.Thismakesclearthatthederivative(Qx)0doesnotappearin(1.1),infact.ThefunctionspaceC1N:=fx2C:Px2C1g(1.4)suggestsitselfastheverynaturaloneforthesolutionsof(1.1).WeshouldaskforC1Nsolutions,butnotforC1solutionsingeneral.Inparticular,forsemi-explicitequationsx01(t)+’(x1(t);x2(t);t)=0(x1(t);x2(t);t)=09=;(1.5)wehavesimplyP=diag(I;0),C1N:=fx2C:x12C1g.2Highersmoothnessofthesolutioncorrespondstohighersmoothnessdemandsforthegivendata,butinmostapplicationsoneisinterestedeveninlowersmoothness.Onthisbackground,(1.1)shouldbewrittenpreciselyasf((Px)0(t)P0(t)x(t);x(t);t)=0:However,forshortness,wecontinuetouse(1.1)andinterpretP(t)x0(t)asanabbreviationofP(t)((Px)0(t)P0(t)x(t))there.Next,givenaC1NfunctionxwhosetrajectoryproceedsinD.Fordierentreasonswemightbeinterestedinthelinearizationof(1.1)alongx,thatis,inthelinearequationA(t)z0(t)+B(t)z(t)=q(t);(1.6)thecontinuouscoecientsofwhicharegivenbyA(t):=f0x0(y(t);x(t);t);B(t):=f0x(y(t);x(t);t);y(t):=(Px)0(t)P0(t)x(t):Here,xisoftensupposedtobeasolution(stationaryornonstationary)oftheDAE(1.1).Withx=x+z,equation(1.1)itselfmaybedescribedapproximatelybyA(t)z0(t)+B(t)z(t)=f(y(t);x(t);t);(1.7)supposedzissmallenough(inC1N)forth