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Convergence Analysis of Orthogonal Collocation Methods for Unconstrained Optimal Control

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Title:
Convergence Analysis of Orthogonal Collocation Methods for Unconstrained Optimal Control
Creator:
Hou, Hongyan
Publisher:
University of Florida
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Hager, William Ward
Committee Members:
Chen, Yunmei
Zhang, Lei
Pilyugin, Sergei S
Rao, Anil
Graduation Date:
8/10/2013

Subjects

Subjects / Keywords:
Approximation ( jstor )
Control theory ( jstor )
Degrees of polynomials ( jstor )
Gaussian quadratures ( jstor )
Interpolation ( jstor )
Matrices ( jstor )
Optimal control ( jstor )
Polynomials ( jstor )
Quadratic programming ( jstor )
Textual collocation ( jstor )
convergence
gaussian-collocation-mehtod
lebesgue-constant

Notes

General Note:
This dissertation develops convergence theories for three numerical methods: the  Gauss collocation method, the Radau collocation method and the hp-collocation method applied to unconstrained optimal control problems. Under assumptions of coercivity and smoothness, all three collocation methods have local minimizers and corresponding Lagrange multipliers that converge in the sup-norm to a local minimizer and costate of the continuous- time optimal control problem. Both the Gauss and Radau collocation methods have error estimate of the form O(1/N^(l-5/2)) where l is the number of continuous derivatives in the solution and N is the degree of the polynomials in the Gauss or Radau collocation scheme. The error estimate for the hp-collocation method has the form of O(h^l/N^(l-5/2)) where h is the length of each subinterval, N is the degree of polynomial in each interval and l is the number of continuous derivatives in the solution. The convergence analysis for the three collocation methods is based on the application of an abstract implicit function theorem in nonlinear spaces and the Lipschitz stability of quadratic programming problems. The analysis requires an estimate of Lebesgue constants associated with three sets of points: (1) the Radau quadrature points, (2) the Radau quadrature points augmented by the point +1, (3) the Gauss quadrature points augmented by the point -1. The estimation for points set (2) is O(log N), and for points sets (1) and (3) are O(N^1/2) where N is the number of Gauss or Radau quadrature points. These results are extension of Szego’s analysis of the Lebesgue constants for interpolation schemes based on the roots of Jacobi polynomials.

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Copyright Hou, Hongyan. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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8/31/2015

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CONVERGENCEANALYSISOFORTHOGONALCOLLOCATIONMETHODSFORUNCONSTRAINEDOPTIMALCONTROLByHONGYANHOUADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013HongyanHou 2

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Toallwhonurturedmyintellectualcuriosity,academicinterests,andsenseofscholarshipthroughoutmylifetime,makingthismilestonepossible 3

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ACKNOWLEDGMENTS IthankmanypeopleattheUniversityofFloridaforhelpingmetocompletemyPh.D.study.Foremost,IsincerelythankmyadvisorDr.WilliamHagerforhiswiseandpatientguidanceduringmystudy.Withouthisgeneroushelpandenthusiasticencouragement,thisresearchcouldnothavebeenaccomplished.IwouldalsoliketoexpressmygreatappreciationtoDr.AnilRaoandhisstudentsintheMechanicalandAerospaceEngineeringDepartment.Ifitisnotfortheirgreataccomplishmentinsolvingoptimalcontrolproblems,myresearchwouldnotbeinitiatedandprogressed.ManythankstoDr.LeiZhangforhisvaluableandconstructivesuggestionsaboutmystudyandacademiccareer.IalsogreatlythankDr.SergeiPilyugin,Dr.SergeiShavanovandDr.YunmeiChenforbeingonmyPh.D.committeeandverysupportivetomyresearch.Igreatlythankmyparentsforeverythingtheyhavegiventomeandtheirforeverencouragementandsupportinmylifeandstudy.IwouldliketogivemydeepestthankstoMingXue.Thetwoyearsapartwerealonganddifcultjourneyforus,butnomatterwhenandnomatterwhere,heisalwayssupportiveandhelpful.Hisencouragementgivesmethecouragetogetthroughallthedifcultiesandkeepupwithhardwork.Ithankeveryonewhohavegenerouslyofferedtheirhelpinmylifeandstudy. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2LEBESGUECONSTANTSARISINGINORTHOGONALCOLLOCATIONMETHODS ...................................... 16 2.1BackgroundofOrthogonalCollocationMethods ............... 16 2.2AnalysisoftheResidual ............................ 19 2.3ResultsandExtensionsofSzego'sBook .................. 21 2.4LebesgueConstantfortheGaussQuadraturePoints ............ 23 2.5LebesgueConstantfortheRadauQuadraturePoints ............ 30 2.6LebesgueConstantfortheRadauQuadraturePointsPlus-1 ....... 45 2.7TightnessofEstimates ............................. 60 2.8ConcludingRemarks .............................. 61 3CONVERGENCEOFAGAUSSCOLLOCATIONMETHODFORUNCONSTRAINEDOPTIMALCONTROL ..................... 63 3.1BackgroundofGaussCollocationMethod .................. 63 3.2TheControlProblemandGaussCollocationMethod ............ 63 3.3ConvergenceResult .............................. 70 3.4AbstractSetting ................................. 70 3.5ApproximationPreliminaries .......................... 72 3.6AnalysisofResidualandStationarity ..................... 73 3.7Invertibility .................................... 79 3.8ProofofTheorem 3.1 ............................. 95 3.9ConcludingRemarks .............................. 95 4CONVERGENCEFOFARADAUCOLLOCATIONMETHODFORUNCONSTRAINEDOPTIMALCONTROL ..................... 96 4.1BackgroundofRadauCollocationMethod .................. 96 4.2TheControlProblemandRadauCollocationMethod ............ 96 4.3ConvergenceResult .............................. 105 4.4AbstractSetting ................................. 106 4.5ApproximationPreliminaries .......................... 107 5

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4.6AnalysisofResidualandStationarity ..................... 109 4.7Invertibility .................................... 116 4.8ProofofTheorem 4.1 ............................. 132 4.9ConcludingRemarks .............................. 132 5CONVERGENCEANALYSISFORAHPCOLLOCATIONMETHODFORUNCONSTRAINEDOPTIMALCONTROL ..................... 133 5.1BackgroundofhpCollocationMethod .................... 133 5.2TheControlProblemandhpCollocationMethod .............. 134 5.3ConvergenceResult .............................. 142 5.4AbstractSetting ................................. 143 5.5ApproximationPreliminaries .......................... 145 5.6AnalysisofResidualandStationarity ..................... 147 5.7Invertibility .................................... 156 5.8ProofofTheorem 5.1 ............................. 182 5.9ConcludingRemarks .............................. 183 6CONCLUSIONS ................................... 184 APPENDIX ACOMPUTATIONOFTHEMINIMUM-PSLFILTER ................. 186 BSYNTHESISOFWFORST-CDMA ........................ 187 REFERENCES ....................................... 189 BIOGRAPHICALSKETCH ................................ 193 6

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LISTOFTABLES Table page 2-1Theanalysisof( 2 )isorganizedinto8cases. ................ 47 A-1InnitynormofD)]TJ /F6 7.97 Tf 6.58 0 Td[(11:Nwithrespecttonumberofinterpolationnodes ....... 186 A-2InnitynormofDy)]TJ /F6 7.97 Tf 12.7 0 Td[(11:Nwithrespecttonumberofinterpolationnodes ....... 186 A-3MaximalofEuclidiannormofarowofW1 2D1:N)]TJ /F6 7.97 Tf 6.58 0 Td[(1withrespecttonumberofinterpolationnodes .................................. 186 A-4MaximalofEuclidiannormofarowofW1 2Dy1:N)]TJ /F6 7.97 Tf 6.58 0 Td[(1withrespecttonumberofinterpolationnodes .................................. 186 B-1InnitynormofD)]TJ /F6 7.97 Tf 6.58 0 Td[(12:N+1withrespecttonumberofinterpolationnodes ...... 187 B-2InnitynormOf(Dz))]TJ /F6 7.97 Tf 6.58 0 Td[(1withrespecttonumberofinterpolationnodes ...... 187 B-3Maximalsquarevalueofrowvectornormwithrespecttonumberofnodes .. 187 B-4Maximalsquarevalueofrowvectornormwithrespecttonumberofnodes .. 188 7

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LISTOFFIGURES Figure page 2-1LeastsquaresapproximationtotheLebesgueconstantforthepointset(P1)(Gaussquadraturepointsaugmentedwith)]TJ /F10 11.955 Tf 9.3 0 Td[(1)usingcurvesoftheformap N+b ............................................ 61 2-2LeastsquaresapproximationtotheLebesgueconstantforthepointset(P2)(Radauquadraturepoints)usingcurvesoftheformap N+b .......... 61 2-3LeastsquaresapproximationtotheLebesgueconstantforthepointset(P3)(Radauquadraturepointsaugmentedwith)]TJ /F10 11.955 Tf 9.3 0 Td[(1)usingcurvesoftheformalogN+b ............................................ 62 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCONVERGENCEANALYSISOFORTHOGONALCOLLOCATIONMETHODSFORUNCONSTRAINEDOPTIMALCONTROLByHongyanHouAugust2013Chair:WilliamW.HagerMajor:Mathematics Thisdissertationdevelopsconvergencetheoriesforthreenumericalmethods:theGausscollocationmethod,theRadaucollocationmethodandthehp-collocationmethodappliedtounconstrainedoptimalcontrolproblems.Underassumptionsofcoercivityandsmoothness,allthreecollocationmethodshavelocalminimizersandcorrespondingLagrangemultipliersthatconvergeinthesup-normtoalocalminimizerandcostateofthecontinuous-timeoptimalcontrolproblem.BoththeGaussandRadaucollocationmethodshaveerrorestimateoftheformO1 Nl)]TJ /F9 5.978 Tf 6.95 2.35 Td[(5 2wherelisthenumberofcontinuousderivativesinthesolutionandNisthedegreeofthepolynomialsintheGaussorRadaucollocationscheme.Theerrorestimateforthehp-collocationmethodhastheformofOhl Nl)]TJ /F9 5.978 Tf 6.96 2.34 Td[(5 2wherehisthelengthofeachsubinterval,Nisthedegreeofpolynomialineachintervalandlisthenumberofcontinuousderivativesinthesolution. TheconvergenceanalysisforthethreecollocationmethodsisbasedontheapplicationofanabstractimplicitfunctiontheoreminnonlinearspacesandtheLipschitzstabilityofquadraticprogrammingproblems.TheanalysisrequiresanestimateofLebesgueconstantsassociatedwiththreesetsofpoints:(1)theRadauquadraturepoints,(2)theRadauquadraturepointsaugmentedbythepoint+1,(3)theGaussquadraturepointsaugmentedbythepoint)]TJ /F10 11.955 Tf 9.3 0 Td[(1.Theestimationforpointsset(2)isO(logN),andforpointssets(1)and(3)areON1 2whereNisthenumberofGauss 9

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orRadauquadraturepoints.TheseresultsareextensionofSzego'sanalysisoftheLebesgueconstantsforinterpolationschemesbasedontherootsofJacobipolynomials. 10

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CHAPTER1INTRODUCTION ThePontryagin'sminimumprinciple[ 1 40 ]andcalculusofvariations[ 7 ]givetherstordernecessaryconditionsforthesolutionofoptimalcontrolproblems.However,itisdifculttosolvethenecessaryconditionstogetananalyticalsolution.Numericaltechniquesprovideapproximationstotheanalyticalsolution.Therearegenerallytwonumericalapproaches.Oneistoapproximatetherst-ordernecessaryconditionsoftheoptimalcontrolproblems.Thisapproachiscalledindirectmethod.Theotherapproachistoapproximatetheoptimalcontrolproblemsdirectly.Thisiscalledthedirectmethod.Overthelasttwodecades,thedirectcollocationmethodshavebeenmorepopularinthenumericalsolutionsforoptimalcontrolproblems.Thedirectcollocationmethodswererstdevelopedashmethodswherethetimeintervalisdividedintosubintervalsandthestateisapproximatedusingthesamexed-degreepolynomialineachsubinterval.Theconvergenceofhmethodscanbeachievedbyreningthemeshinterval[ 6 13 14 ].Morerecently,theGaussianquadratureorthogonalcollocationmethods[ 4 5 16 20 22 23 34 37 39 42 47 51 ],aredevelopedasaclassofpmethodsusingasingleintervalandaglobalapproximationpolynomials.Theconvergencerateofpmethodsareachievedbyincreasingthedegreeoftheglobalpolynomial[ 8 24 46 ].InaGaussianquadratureorthogonalcollocationmethods,theoptimalcontrolproblemistranscribedtoanonlinearprogrammingproblem(NLP)byparameterizingthestateandcontrolusingglobalpolynomialsandcollocatingthedifferential-algebraicequationsusingnodesobtainedfromaGaussianquadrature.TheGaussandRadaucollocationmethods[ 4 5 35 37 ]aretwowelldevelopedGaussianquadraturecollocationmethodswhereeitherGaussquadraturepointsorRadauquadraturepointsareusedascollocationpoints.Notethattheuseofglobalpolynomialsismostappropriateforproblemswithsmoothsolutions.Forproblemswherethesolutionsarenonsmoothornotwellapproximatedbyglobalpolynomialsofareasonablylowdegree,itispreferabletouseapiecewise 11

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polynomialapproach(see[ 9 ]and[ 10 ])wherethetimeinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1]ispartitionedintosubintervalsandadifferentpolynomialisusedovereachsubinterval.Continuityispreservedbymatchingthevaluesofthestateattheinterfacebetweensubintervals.Thisapproachiscalledthehpcollocationmethod[ 11 12 43 ]. NumericalexperienceshowsthattheconvergenceratesoftheGauss,theRadauandthehpcollocationmethodhavebeenveryfast[ 4 5 25 26 ],however,therewasnomathematicaltheorytosubstantiatetherapidexponentialconvergenceobservedinpractice.ThisdissertationdevelopsaconvergencetheoryfortheGauss,theRadauandthehpcollocationmethodsappliedtounconstrainedcontrolproblems. Intheconvergenceanalysisofthethreecollocationmethodsforoptimalcontrolproblems,theLebesgueconstantfortheassociatedinterpolationschemeplaysakeyroleintheerrorestimate.Inchapter1,theLebesgueconstantsareestimatedin3importantspecialcases: (P1)f1,...,NgaretheGaussquadraturepointsandtheLebesgueconstantisestimatedrelativetof0,1,...,Ng,where0=)]TJ /F10 11.955 Tf 9.3 0 Td[(1. (P2)TheLebesgueconstantisestimatedrelativetoRadauquadraturepointsf1,...,Ng,where1>)]TJ /F10 11.955 Tf 9.3 0 Td[(1andN=+1. (P3)TheLebesgueconstantisestimatedrelativetof0,...,Ng,wheref1,...,NgaretheRadauquadraturepoints,where0=)]TJ /F10 11.955 Tf 9.3 0 Td[(1andN=+1. TheGaussquadraturepointsareareobtainedfromtherootsofaLegendrepolynomialPNandtheRadauquadraturepointsaretherootsofPN+PN+1.ByusingthedenitionofJacobipolynomialintroducedinSzego0sbook[ 45 ],GaussquadraturepointsandRadauquadraturepointsexcludingtheendpointaretherootsofJacobipolynomialP(,)N.Gausspointscorrespondingto==0andRadaupointsexcludingendpointcorrespondingto=1and=0.Thesetwosetsofpointsaredenedonthedomain[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1],butdiffersignicantlyinthattheGaussquadraturepointsincludeneitheroftheendpoints,andtheRadauquadraturepointsincludeoneoftheendpoints.Inaddition, 12

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theRadauquadraturepointsareasymmetricrelativetotheoriginandarenotuniqueinthattheycanbedenedusingeithertheinitialpointortheterminalpoint[ 25 ].Thesetofpointsusingtheterminalpoint=+1usuallycalledippedRadauquadraturepoints[ 26 ]incontrasttotheRadaupointsusingtheinitialpoint=)]TJ /F10 11.955 Tf 9.3 0 Td[(1. AccordingtoSzego'sresults,theLebesgueconstantfortheGaussquadraturepointsisON1 2.TheanalysisinthischaptershowsthattheLebesgueconstantfortheGaussquadraturepointsaugmentedwith)]TJ /F10 11.955 Tf 9.29 0 Td[(1isalsoON1 2(pointsset(P1)above).SzegoshowsthattheLebesgueconstantfortheJacobipolynomialP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1isON3 2.OuranalysisshowsthattheLebesgueconstantfortherootsofP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1augmentedby+1(pointsset(P2)above)isON1 2.Finally,itisshowninthischapterthattheLebesgueconstantfortherootsofP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1augmentedby+1and)]TJ /F10 11.955 Tf 9.3 0 Td[(1isO(logN)(pointsset(P3)above).NumericalresultsareprovidedtoshowthattheestimateforthethreeLebesgueconstantsaretight. TheGausscollocationmethodisaclassofdirectcollocationmethodwheretheoptimalcontrolproblemistranscribedtoanonlinearprogrammingproblem(NLP)byparameterizingthestateandcontrolusingglobalpolynomialsandcollocatingthedifferential-algebraicequationsusingnodesobtainedfromaGaussquadraturescheme[ 4 5 ].InChapter3,theconvergencetheoryoftheGausscollocationmethodisprovided.Underassumptionsofcoercivityandsmoothness,theerrorestimateofthestate,thecontrolandthecostatehastheformO1 Nk)]TJ /F9 5.978 Tf 6.95 2.34 Td[(5 2wherekisthenumberofcontinuousderivativesinthesolutionandNisthedegreeofthepolynomialsinthecollocationscheme.Herethenormisthediscretesup-norminRN.TheproofisbasedonanestimateofLebesgueconstantfortheGausscollocationpointsgiveinChapter2,andtheabstracterrorestimatefordiscreteapproximationsgivenin[ 14 ]. TheRadaucollocationmethodisalsoawelldevelopeddirectcollocationmethodforsolvingoptimalcontrolproblems.InaRadaucollocationmethod,theoptimalcontrolproblemistranscribedtoanonlinearprogrammingproblem(NLP)byparameterizing 13

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thestateandcontrolusingglobalpolynomialsandcollocatingthedifferential-algebraicequationsusingnodesobtainedfromaRadauquadraturescheme[ 25 26 ].InChapter4,theconvergencetheoryofRadaucollocationmethodisprovided.Underassumptionsofcoercivityandsmoothness,theerrorestimateofthestate,thecontrolandthecostatehastheformO1 Nk)]TJ /F9 5.978 Tf 6.95 2.34 Td[(5 2wherekisthenumberofcontinuousderivativesinthesolutionandNisthedegreeofthepolynomialsinthecollocationscheme.TheproofisbasedonanestimateofLebesgueconstantfortheRadaucollocationpointsgiveinChapter1,andtheabstracterrorestimatefordiscreteapproximationsgivenin[ 14 ]. Forproblemswherethesolutionsareinnitelysmoothandwellbehaved,theGaussandRadaucollocationmethodhavesimplestructuresandconvergeexponentiallyfast.Theconvergenceisachievedbyincreasingthenumberofcollocationpoints.Thereareseverallimitationsofusingaglobalpolynomial.First,anaccurateapproximationmayrequireanunreasonablyhighdegreeglobalpolynomial.Second,thedensityofthediscreteNLPgrowsquicklyandtheproblembecomescomputationallyintractable.Incontrasttopmethods,thecommonlyusedcollocationmethodsforoptimalcontrolproblemisthehmethod,whereaxedlow-degreepolynomialisusedasstateapproximationandtheproblemisdividedintomanyintervals.Convergenceofthehmethodsareachievedbyincreasingthenumberofmeshintervals[ 6 13 14 ].Whilehmethodsarecomputationallymoretractablethanpmethods,theymayrequirealargenumberofmeshintervalsinordertoachieveanacceptableaccuracybecauseexponentialconvergenceislost.Inordertomakethecomputationmoretractableandalsohaveafastconvergencerate,anhpcollocationmethodwasdevelopedin[ 2 3 12 27 29 ].Thehpcollocationmethodisacombinationofthepmethodandhmethod.Morespecically,thetimeinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1]ispartitionedintosubintervalsandadifferentpolynomialisusedovereachsubinterval.Continuityispreservedbymatchingthevaluesofthestateattheinterfacebetweensubintervals.ThehpcollocationmethodstudiedinthisdissertationusesaRadaucollocationmethodoneachsubinterval.Under 14

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acoercivityandsmoothnessassumptions,theerrorinthediscretesolutiontendstozeroeitherthedegreeofthepolynomialoneachsubintervalisincreasedorbythenumberofmeshintervalsisincreased.Theerrorestimateofthestate,thecontrolandthecostatehastheformOhl Nl)]TJ /F9 5.978 Tf 6.95 2.35 Td[(5 2wherehisthelengthofeachmeshinterval,Nisthedegreeofthepolynomialsineachsubintervalandlisthenumberofcontinuousderivativesinthesolution.TheanalysisreliesontheconvergencetheoryofRadaucollocationmethodinChapter3andmathematicalinduction. Theoutlineofthisdissertationisasfollows: First,inChapter2,Lebesgueconstantsassociatedwiththreesetsofpointsareestimated.ThesethreesetsofpointsareRadauquadraturepoints,Radauquadraturepointsaugmentedwith+1,andGausspointsaugmentedwith)]TJ /F10 11.955 Tf 9.3 0 Td[(1.TheJacobipolynomialsandtherelatedpropertiesareintroduced. InChapter3,GausscollocationmethodisintroducedandtheconvergencetheoremofGausscollocationmethodappliedtounconstrainedoptimalcontrolproblemispresentedandproved. Chapter4beginswithabriefintroductionofRadaucollocationmethod.TheinvertibilityofadifferentialmatrixrelatedwithRadauquadraturepointsisproved.AconvergencetheoremofRadaucollocationmethodappliedtounconstrainedoptimalcontrolproblemispresentedandproved. Finally,Chapter5introducesthehpcollocationmethodanditsconvergencetheorem.ThehpcollocationmethodstudiedinthischapterbasedonaRadaucollocationmethodoneachsubinterval.Theproofoftheconvergencetheoremisprovided. 15

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CHAPTER2LEBESGUECONSTANTSARISINGINORTHOGONALCOLLOCATIONMETHODS 2.1BackgroundofOrthogonalCollocationMethods IntheconvergenceanalysisofGaussianquadraturecollocationmethodsforoptimalcontrolproblems,theLebesgueconstantfortheassociatedinterpolationschemeplaysakeyroleintheerrorestimate.InthisChapter,wegiveboundsontheLebesgueconstantsthatariseinGaussandRadaucollocationschemesandshowhowtheyariseincollocationschemesingeneral. Asanillustration,considerthefollowingscalarrstorderdifferentialequation _x()=f(x()),g(x()]TJ /F10 11.955 Tf 9.3 0 Td[(1),x(+1))=0 (2) wheref:R!Randg:R2!R.Forarst-orderdifferentialequation,thereistypicallyaone-parameterfamilyofsolutions.Thisonedegreeoffreedomwillbespeciedthroughtheconditiong(x()]TJ /F10 11.955 Tf 9.3 0 Td[(1),x(+1))=0.Theproblem( 2 )isintroducedinordertomotivatehowtheLebesgueconstantentersintotheerrorestimationforacollocationscheme.InourtargetapplicationtoaGaussorRadauquadratureschemediscretizationofanoptimalcontrolproblem,wenotonlysatisfyadifferentialequation,butalsooptimizeanobjectivefunction. Letusconsideracollocationschemefor( 2 ),wherethedifferentialequationiscollocatedatNpointsf1,2,...,Ng()]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1].Inaddition,weintroduceanotherpoint0=)]TJ /F10 11.955 Tf 9.29 0 Td[(1.LetLi,i=0,...,N,betheLagrangeinterpolationpolynomialsdenedby Li()=NYj=0j6=i)]TJ /F14 11.955 Tf 11.95 0 Td[(j i)]TJ /F14 11.955 Tf 11.95 0 Td[(j.(2) IfyisapolynomialofdegreeatmostN,thenwecanwrite y()=NXi=0yiLi(),(2) 16

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whereyi=y(i).Differentiatingtheseries( 2 )andevaluatingatthecollocationpointsk,k=1,2,,N,weget _y(k)=NXi=0yi_Li(k)=NXi=0Dkiyi=(Dy)k,(2) whereDki=_Li(k)andy2RN+1isthevectorwithcomponenty0,...,yN.TherectangularN(N+1)matrixDformedbythecoefcientsDkiisthedifferentiationmatrix.Ifwerequirethatysatises( 2 )atthecollocationpoints1,...,N,thenweobtain Dy=f(y),(2) wheref(y)2RNisgivenbyfi(y)=f(yi). Equation( 2 )yieldsNequationsinN+1unknowns.An(N+1)stequationisobtainedfromtheboundaryconditiong(y()]TJ /F10 11.955 Tf 9.3 0 Td[(1),y(+1))=0. IfN=+1,thisreducesto g(y0,yN)=0, sincey0=y()]TJ /F10 11.955 Tf 9.3 0 Td[(1)andyN=y(+1). IfN<+1,thenbythefundamentaltheoremofcalculusandbytheexactnessofGaussquadratureforpolynomialsofdegree(N)]TJ /F10 11.955 Tf 11.95 0 Td[(1),wehave y(+1)=y()]TJ /F10 11.955 Tf 9.3 0 Td[(1)+Z1)]TJ /F6 7.97 Tf 6.58 0 Td[(1_y()d=y()]TJ /F10 11.955 Tf 9.3 0 Td[(1)+NXi=1!i_y(i), (2) where!iistheGaussquadratureweights.LetDidenotetheithrowofmatrixD.By( 2 )and( 2 ),( 2 )canbeexpressedas y(+1)=y0+NXi=1!iDiy=y0+NXi=1!if(yi). 17

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Hence,thediscretizedversionof( 2 )isDy=f(y)andg(y0,yN)=0,ifN=+1g(y0,yN+1)=0,yN+1=y0+NXi=1!if(yi),ifN<+1 Thelocalexistenceofasolutiontothediscretizedproblemanditsconvergencetoasolutionofcontinuousproblem( 2 )dependsontheLebesgueconstantfortheLagrangepolynomialassociatedwithf0,...,Ng.InthisChapter,weshowtherelationbetweentheerrorandtheLebesgueconstantandweestimatetheLebesgueconstantin3importantspecialcases: (P1)f1,...,NgaretheGaussquadraturepointsandtheLebesgueconstantisestimatedrelativetof0,1,...,Ng. (P2)TheLebesgueconstantisestimatedrelativetoRadauquadraturepointsf1,...,Ng,where1>)]TJ /F10 11.955 Tf 9.3 0 Td[(1andN=+1. (P3)TheLebesgueconstantisestimatedrelativetof0,...,Ng,wheref1,...,NgaretheRadauquadraturepoints. AccordingtoSzego'sresults,theLebesgueconstantfortheGaussquadraturepointsisON1 2.WeshowthattheLebesgueconstantfortheGaussquadraturepointsaugmentedwith)]TJ /F10 11.955 Tf 9.3 0 Td[(1isalsoON1 2(pointsset(P1)above).SzegoshowsthattheLebesgueconstantfortheJacobipolynomialP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1isON3 2.WeshowthattheLebesgueconstantfortherootsofP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1augmentedby+1(pointsset(P2)above)isON1 2.Finally,weshowthattheLebesgueconstantfortherootsofP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1augmentedby+1and)]TJ /F10 11.955 Tf 9.3 0 Td[(1isO(logN)(pointsset(P3)above).WegivenumericalresultstoshowthatourestimatefortheLebesgueconstanttight. Notation.Ck[a,b]isthecollectionofktimescontinuouslydifferentialfunctionsontheinterval[a,b],y(k)denotesthek-thderivativeofy,andPNdenotesthesetofpolynomialsofdegreeatmostN.kkdenotesthesup-norm.Thesymbolisdened 18

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asfollows:ifpn,qnarecomplex,qn6=0,andthesequencejpnj jqnjhasanitepositivelimitasn!1,wewritepnqn. 2.2AnalysisoftheResidual Akeystepintheconvergenceanalysisofadiscreteapproximationto( 2 )istheestimationoftheresidual.Thatis,howcloselydoestheinterpolantofanexactsolutionof( 2 )satisfytheequation( 2 ),(forexample,see[ 33 ]).Letx()beasolutionof( 2 ),andletxbeavectorwithcomponentsx(0),...,x(N).Substitutexinto( 2 )toobtaintheresidualr=Dy)]TJ /F4 11.955 Tf 12.07 0 Td[(f(y).By( 2 ),thei-thcomponentoftheresidualcanbeexpressedasri=Dix)]TJ /F4 11.955 Tf 11.95 0 Td[(fi(x)=Dix)]TJ /F10 11.955 Tf 13.53 0 Td[(_x(i)=_xN(i))]TJ /F10 11.955 Tf 13.53 0 Td[(_x(i),i=1,2,...,N, wherexNisthepolynomialthatinterpolatesxatx.Thisresidualisnowestimated. Let`NdenotetheLebesgueconstantdenedby `N=max2[)]TJ /F6 7.97 Tf 6.58 0 Td[(1,1]NXi=0jLi()j,(2) whereLi()isdenedin( 2 ).Givenanarbitraryfunctiony2C[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1],letyI2PNbetheinterpolatingpolynomialwhichsatisesyI(i)=y(i),for0iN.Weestablishthefollowingtheorems: Theorem2.1. Foranyy2C1[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1],wehave max1iN_y(i))]TJ /F10 11.955 Tf 13.64 0 Td[(_yI(i)_y)]TJ /F10 11.955 Tf 13.64 0 Td[(_yI)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+2N2`Ninfz2PNk_y)]TJ /F10 11.955 Tf 13.49 0 Td[(_zk.(2) Proof. Givenz2PN,wehave_z2PN)]TJ /F6 7.97 Tf 6.59 0 Td[(1and _y)]TJ /F10 11.955 Tf 13.64 0 Td[(_yIk_y)]TJ /F10 11.955 Tf 13.49 0 Td[(_zk+_z)]TJ /F10 11.955 Tf 13.64 0 Td[(_yI.(2) Letp2PNbedenedby p()=yI()]TJ /F10 11.955 Tf 9.29 0 Td[(1)+Z)]TJ /F6 7.97 Tf 6.58 0 Td[(1_z(s)ds=y()]TJ /F10 11.955 Tf 9.3 0 Td[(1)+Z)]TJ /F6 7.97 Tf 6.58 0 Td[(1_z(s)ds.(2) 19

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Hence,_p=_zandbyMarkov'sinequality[ 41 ],wehave _z)]TJ /F10 11.955 Tf 13.64 0 Td[(_yI=_p)]TJ /F10 11.955 Tf 13.64 0 Td[(_yIN2p)]TJ /F3 11.955 Tf 11.96 0 Td[(yI=N2NXi=0(p(i))]TJ /F3 11.955 Tf 11.96 0 Td[(y(i))Li()N2`Nmax0iNjp(i))]TJ /F3 11.955 Tf 11.96 0 Td[(y(i)j. (2) Bythefundamentaltheoremofcalculusand( 2 ), jp(i))]TJ /F3 11.955 Tf 11.96 0 Td[(y(i)j=Zi)]TJ /F6 7.97 Tf 6.59 0 Td[(1(_p(s))]TJ /F10 11.955 Tf 13.64 0 Td[(_y(s))dsZi)]TJ /F6 7.97 Tf 6.59 0 Td[(1j_z(s))]TJ /F10 11.955 Tf 13.63 0 Td[(_y(s)jds2k_z)]TJ /F10 11.955 Tf 13.64 0 Td[(_yk. Substitutethisinto( 2 )gives _z)]TJ /F10 11.955 Tf 13.64 0 Td[(_yI2N2`Nk_z)]TJ /F10 11.955 Tf 13.64 0 Td[(_yk.(2) Sincez2PNisarbitrary,( 2 )and( 2 )yields( 2 ). Anestimatefortherightsideof( 2 )canbeobtainedfromaresultwhichoriginatesfromJackson[ 38 ]andwhichissummarizedbelow: Theorem2.2. Lety()2Ck[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1]andNk,then infp2PNky)]TJ /F3 11.955 Tf 11.95 0 Td[(pk2ck Nky(k),ck=6k+1ek(1+k))]TJ /F6 7.97 Tf 6.59 0 Td[(1.(2) Proof. Itisshownin[ 44 ,Thm.1.5]that infp2PNky)]TJ /F3 11.955 Tf 11.96 0 Td[(pkck Nk!k1 N)]TJ /F3 11.955 Tf 11.96 0 Td[(k,(2) where!kisthemodulusofcontinuityofy(k).Bythedenitionofmodulusofcontinuity,wehave!k1 N)]TJ /F3 11.955 Tf 11.96 0 Td[(k=supy(k)(1))]TJ /F3 11.955 Tf 11.96 0 Td[(y(k)(2):1,22[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1],j1)]TJ /F14 11.955 Tf 11.96 0 Td[(2j1 N)]TJ /F3 11.955 Tf 11.95 0 Td[(k. Sincey(k)(1))]TJ /F3 11.955 Tf 11.95 0 Td[(y(k)(2)2y(k) ( 2 )followsfrom( 2 ). 20

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Bytheabovetwotheorems,weseethattheanalysisoftheresidualrireducestotheestimationofLebesgueconstant`N. Remark1.TheremainderoftheanalysisreliesheavilyonresultsestablishedbySzegoin[ 45 ].Hence,tofacilitatetheuseofSzego'sresult,wewillreindexthezerosaccordingtohisconversion.Inparticular,thequadraturepointsarenowindecreasingorder1>2>>N2()]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1]andweintroduceanotherpointN+1=)]TJ /F10 11.955 Tf 9.3 0 Td[(1.Likewise,theLagrangepolynomialsarenowL1,L2,...,LN+1andtheLebesgueconstantin( 2 )becomes `N=max2[)]TJ /F6 7.97 Tf 6.59 0 Td[(1,+1]N+1Xi=1jLi()j.(2) 2.3ResultsandExtensionsofSzego'sBook GaborSzegohasgivenanestimateofLebesgueconstantassociatedwithaquadratureschemebasedontherootsoftheJacobipolynomialP(,)N().Thesearepolynomialsontheinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1]thatareorthogonalrelativetotheweightfunction(1)]TJ /F14 11.955 Tf 12.23 0 Td[()(1+),where>)]TJ /F10 11.955 Tf 9.29 0 Td[(1and>)]TJ /F10 11.955 Tf 9.3 0 Td[(1(see[ 45 ]).OneofSzego'sresultsthatweexploitisthefollowing(see[ 45 ,p.338]): Theorem2.3. Let=max(,),theLebesgueconstantfortheinterpolationpolynomi-alsbasedontherootsofJacobipolynomialP(,)N()isO(N+1 2)if>)]TJ /F6 7.97 Tf 10.5 4.7 Td[(1 2andO(logN)if)]TJ /F6 7.97 Tf 23.12 4.71 Td[(1 2. Remark2.Inthespecialcase==0,P(0,0)N()isamultipleofLegendrepolynomialand=0.Hence,theLebesgueconstantforinterpolationattheGaussquadraturepointsisON1 2.Theorem 2.3 ismainlydevelopedfromthefollowingpropositionsforJacobipolynomials: Proposition2.1. Letandbearbitraryandreal,andc1axedpositiveconstant.AsNtendsto1,wehave P(,)N(cos)=8>><>>:)]TJ /F15 7.97 Tf 6.58 0 Td[()]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2,ifc1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2,O(N),if0c1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1.(2) 21

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Proof. See[ 45 ,(7.32.5)]. Proposition2.2. P(,)N()=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)NP(,)N()]TJ /F14 11.955 Tf 9.3 0 Td[(),2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1].(2) Proof. See[ 45 ,(4.1.3)]. Proposition2.3. Let>)]TJ /F10 11.955 Tf 9.3 0 Td[(1,>)]TJ /F10 11.955 Tf 9.3 0 Td[(1,andlet0<1<2<><>>:()]TJ /F14 11.955 Tf 11.95 0 Td[())]TJ /F15 7.97 Tf 6.59 0 Td[()]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2,if 2)]TJ /F3 11.955 Tf 11.95 0 Td[(c1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1,O)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(N,if)]TJ /F3 11.955 Tf 11.96 0 Td[(c1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1.(2) Proof. For2h 2,i,wehave)]TJ /F14 11.955 Tf 11.95 0 Td[(2h0, 2i,thenby( 2 ), P(,)N(cos)=P(,)N()]TJ /F10 11.955 Tf 11.29 0 Td[(cos()]TJ /F14 11.955 Tf 11.95 0 Td[())=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)NP(,)N(cos()]TJ /F14 11.955 Tf 11.96 0 Td[()). ByProposition 2.1 ,wegettheresultin( 2 )for2h 2,i. Corollary2. Let>)]TJ /F10 11.955 Tf 9.29 0 Td[(1,>)]TJ /F10 11.955 Tf 9.3 0 Td[(1,andlet0<1<2<
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Hence, P(,)0N(cosi)=P(,)0N()]TJ /F10 11.955 Tf 11.3 0 Td[(cos()]TJ /F14 11.955 Tf 11.96 0 Td[(i))=P(,)0N(cos()]TJ /F14 11.955 Tf 11.95 0 Td[(i)).(2) SinceP(,)N(cos()]TJ /F14 11.955 Tf 12.53 0 Td[(i))=P(,)N()]TJ /F10 11.955 Tf 11.29 0 Td[(cosi)=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)N+1P(,)N(cosi)=0,weseethat)]TJ /F14 11.955 Tf 12.69 0 Td[(i< 2isthezeroofP(,)N(cos).ByProposition 2.3 ,P(,)0N(cos()]TJ /F14 11.955 Tf 11.95 0 Td[(i))N1 2()]TJ /F14 11.955 Tf 11.96 0 Td[(i))]TJ /F15 7.97 Tf 6.59 0 Td[()]TJ /F9 5.978 Tf 7.78 3.25 Td[(3 2,andby( 2 ),theproofiscomplete. 2.4LebesgueConstantfortheGaussQuadraturePoints InthissectionweestimatetheLebesgueconstantassociatedwithinterpolationat1>2>>N>N+1,where1,...,NaretheGaussquadraturepointson()]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1)andN+1=)]TJ /F10 11.955 Tf 9.3 0 Td[(1.Itfollowsfromtheremark2thattheLebesgueconstantassociatedwiththeGaussquadraturepointsisON1 2.NotethattheLebesgueconstantthatwearegoingtoestimateasdenedin( 2 )isnotthesameastheLebesgueconstantfortheGaussquadraturepointssincetheLagrangepolynomialalsoinvolvesthepoint)]TJ /F10 11.955 Tf 9.3 0 Td[(1aswellastheGaussquadraturepoints.SinceP(0,0)N()isfrequentlyusedinthissection,wedropthesuperscripts,andletPN()standsforP(0,0)N()inthissection. Theorem2.4. LetLi,i=1,...,N+1,betheLagrangeinterpolationpolynomialsasdenedin( 2 )andsupposetheinterpolationpointsaretheGaussquadraturepointsf1,...,NgandN+1=)]TJ /F10 11.955 Tf 9.3 0 Td[(1.ThentheLebesgueconstant`Nhasthefollowingbound:`N:=max2[)]TJ /F6 7.97 Tf 6.59 0 Td[(1,1]N+1Xi=1jLi()j=ON1 2. Proof. By( 2 )and( 2 )with==0,wehave PN(cos)=8>>>>>>>>>>>><>>>>>>>>>>>>:)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2,ifc1N)]TJ /F6 7.97 Tf 6.58 0 Td[(1 2,O(1),if0c1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1,()]TJ /F14 11.955 Tf 11.95 0 Td[())]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2ON)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2,if 2)]TJ /F3 11.955 Tf 11.96 0 Td[(c1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1,O(1),if)]TJ /F3 11.955 Tf 11.95 0 Td[(c1N)]TJ /F6 7.97 Tf 6.58 0 Td[(1.9>>>>>>>>>>>>=>>>>>>>>>>>>;(2) 23

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Denel()=()]TJ /F14 11.955 Tf 11.96 0 Td[(1)()]TJ /F14 11.955 Tf 11.96 0 Td[(2)...()]TJ /F14 11.955 Tf 11.95 0 Td[(N),andL()=(+1)l(). EvaluatingthederivativeofL()ati(i=1,...,N+1),wehaveL0(i)=l(i)+(i+1)l0(i)=8>><>>:(i+1)l0(i),i=1,2,...,N,l()]TJ /F10 11.955 Tf 9.3 0 Td[(1),i=N+1. Hence,theLagrangepolynomialLi()denedin( 2 )canbeexpressedas Li()=L() L0(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)=8>>><>>>:(+1)l() (i+1)l0(i)()]TJ /F14 11.955 Tf 11.95 0 Td[(i),i=1,2,...,N,l() l()]TJ /F10 11.955 Tf 9.3 0 Td[(1),i=N+1.(2) Since1,...,NarerootsofPN(),wehave l()=PN() KN, whereKNistheleadingcoefcientofPN().Insertingthisin( 2 )yields Li()=8>>><>>>:(+1)PN() (i+1)P0N(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i),i=1,2,...,N,PN() PN()]TJ /F10 11.955 Tf 9.29 0 Td[(1),i=N+1.(2) WeestimatejLN+1()jrst.In[ 45 ,(4.1.4)],itisshownthatPN()]TJ /F10 11.955 Tf 9.29 0 Td[(1)=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)N. SincePN()isatmostO(1),itfollowsthatjLN+1()j=PN() PN()]TJ /F10 11.955 Tf 9.3 0 Td[(1)=jPN()j=O(1). SincejLN+1()j=O(1),theproofofTheorem 2.4 isreducedtoshowmax2[)]TJ /F6 7.97 Tf 6.59 0 Td[(1,1]NXi=1jLi()j=ON1 2.Thesummationtermisfurtherpartitionedintothetermscorrespondingto 24

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positiveandnegativequadraturepoints: NXi=1jLi()j=Xi0jLi()j+X)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2andji)]TJ /F14 11.955 Tf 11.95 0 Td[(j 2 Case1a.ji)]TJ /F14 11.955 Tf 11.96 0 Td[(j> 2. 25

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Sincej1+j<2andPN()=ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2,weget X)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2jLi()j=X)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2(+1)PN() (i+1)P0N(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)4 X)]TJ /F6 7.97 Tf 6.59 0 Td[(18 2()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2.(2) By[ 45 ,(6.21.5)], 2i)]TJ /F10 11.955 Tf 11.96 0 Td[(1 2N+1i2i 2N+1,i=1,2,...,N. Itfollowsthat 2N+1)]TJ /F10 11.955 Tf 11.96 0 Td[(2i 2N+1)]TJ /F14 11.955 Tf 11.95 0 Td[(i2N+2)]TJ /F10 11.955 Tf 11.96 0 Td[(2i 2N+1. Wemakechangeofvariablesj=N+1)]TJ /F3 11.955 Tf 12.16 0 Td[(i.Since1iN,wehave1jN.Theaboveinequalitybecomes 2j)]TJ /F10 11.955 Tf 11.95 0 Td[(1 2N+1)]TJ /F14 11.955 Tf 11.95 0 Td[(i2j 2N+1. Wecombineitwiththeinequality2j)]TJ /F10 11.955 Tf 11.95 0 Td[(1 2N+1>j 3N>j Ntoobtain )]TJ /F14 11.955 Tf 11.95 0 Td[(i>j N.(2) 26

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By( 2 )and( 2 ),wehave j1+ij>8 2()]TJ /F14 11.955 Tf 11.96 0 Td[(i)28 2j N2.(2) NowweestimatejP0N(i)j.BythesymmetryoftheGaussquadraturepoints,wehave)]TJ /F14 11.955 Tf 9.3 0 Td[(i=N+1)]TJ /F7 7.97 Tf 6.58 0 Td[(i=j>0. By( 2 )and( 2 ),for==0,wehave jP0N(i)j=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)N+1P0N()]TJ /F14 11.955 Tf 9.3 0 Td[(i)=jP0N(j)jj)]TJ /F9 5.978 Tf 7.78 3.25 Td[(3 2N2.(2) Wecombine( 2 )and( 2 )toobtain X)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2jLi()j=O(1). (2) Case1b.ji)]TJ /F14 11.955 Tf 11.96 0 Td[(j 2. 27

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Sincejj<1)]TJ /F14 11.955 Tf 11.74 0 Td[(andji)]TJ /F14 11.955 Tf 11.73 0 Td[(j 2,wehavej1+j<2andj1+ij> 2.N+1=)]TJ /F10 11.955 Tf 9.3 0 Td[(1isnotincludedinthiscase.Itfollowsthat Xi<0ji)]TJ /F15 7.97 Tf 6.59 0 Td[(j 2jLi()j=Xi<0ji)]TJ /F15 7.97 Tf 6.58 0 Td[(j 2(+1)PN() (i+1)P0N(i)()]TJ /F14 11.955 Tf 11.95 0 Td[(i)4 Xji)]TJ /F15 7.97 Tf 6.58 0 Td[(j 2jPN()j j)]TJ /F14 11.955 Tf 11.95 0 Td[(ijjP0N(i)j. (2) In[ 45 ,p.336],itfollowsthatXji)]TJ /F15 7.97 Tf 6.58 0 Td[(j 2jPN()j j)]TJ /F14 11.955 Tf 11.95 0 Td[(ijjP0N(i)j=O(logN). Hence,( 2 )yields Xi<0ji)]TJ /F15 7.97 Tf 6.58 0 Td[(j 2jLi()j=O(logN).(2)( 2 )and( 2 )combinetogiveXi<0jLi()j=O(logN)whenjj1)]TJ /F14 11.955 Tf 11.96 0 Td[((Case1). Case2.1)]TJ /F14 11.955 Tf 11.96 0 Td[(1. Sincewearefocusingoni<0and1)]TJ /F14 11.955 Tf 11.96 0 Td[(1,wehaveji)]TJ /F14 11.955 Tf 11.95 0 Td[(j>1 2>,and X)]TJ /F6 7.97 Tf 6.58 0 Td[(1
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Thetermsin( 2 )forwhichji)]TJ /F14 11.955 Tf 12.24 0 Td[(j>haveboundedbyON1 2by( 2 )and( 2 ).Hence,wefocusonthefollowingtermsinthesumwhereiiscloseto: X)]TJ /F6 7.97 Tf 6.58 0 Td[(1,wehave+1 i+1<1,and Xi+1>0,then X)]TJ /F6 7.97 Tf 6.59 0 Td[(1i+1jLi()jX)]TJ /F6 7.97 Tf 6.59 0 Td[(1i+12()]TJ /F14 11.955 Tf 11.95 0 Td[(i) (i+1)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)PN() P0N(i)X)]TJ /F6 7.97 Tf 6.59 0 Td[(1
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If)]TJ /F14 11.955 Tf 11.95 0 Td[(ii+1,then X)]TJ /F6 7.97 Tf 6.59 0 Td[(12>>N>)]TJ /F10 11.955 Tf 9.3 0 Td[(1.TheRadauquadraturepointswith1=+1aretherootsofpolynomialR()=PN()]TJ /F14 11.955 Tf 9.3 0 Td[()+PN)]TJ /F6 7.97 Tf 6.58 0 Td[(1()]TJ /F14 11.955 Tf 9.29 0 Td[(),wherePN()denotestheLegendrepolynomialofdegreeN.SincePN()]TJ /F10 11.955 Tf 9.3 0 Td[(1)=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)N,itfollowsthat+1isonerootofR().By( 2 )and[ 45 ,(4.5.4)],wehave P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1P(0,1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()]TJ /F14 11.955 Tf 9.3 0 Td[()=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1PN()]TJ /F14 11.955 Tf 9.3 0 Td[()+PN)]TJ /F6 7.97 Tf 6.58 0 Td[(1()]TJ /F14 11.955 Tf 9.3 0 Td[() 1)]TJ /F14 11.955 Tf 11.95 0 Td[(=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1R() 1)]TJ /F14 11.955 Tf 11.95 0 Td[(. (2) 30

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Hence,theremainingRadauquadraturepointsf2,...,NgaretherootsoftheJacobipolynomialP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1().InthissectionweestimatetheLebesgueconstantassociatewiththeRadauquadraturepoints. Theorem2.5. LetLi,i=1,...,N,betheLagrangeinterpolationpolynomialsasdenedin( 2 )andsupposetheinterpolationpointsaretheRadauquadraturepointsf1,...,Ng.ThentheLebesgueconstant`Nhasthefollowingestimate: `N=max2[)]TJ /F6 7.97 Tf 6.58 0 Td[(1,1]NXi=1jLi()j=ON1 2.(2) Proof. By( 2 )and( 2 ),with=1and=0,wehave P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(cos)=8>>>>>>>>>>>><>>>>>>>>>>>>:)]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2,ifc1N)]TJ /F6 7.97 Tf 6.58 0 Td[(1 2,O(N),if0c1N)]TJ /F6 7.97 Tf 6.59 0 Td[(1,()]TJ /F14 11.955 Tf 11.95 0 Td[())]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2,if 2)]TJ /F3 11.955 Tf 11.95 0 Td[(c1N)]TJ /F6 7.97 Tf 6.58 0 Td[(1,O(1),if)]TJ /F3 11.955 Tf 11.95 0 Td[(c1N)]TJ /F6 7.97 Tf 6.58 0 Td[(1.9>>>>>>>>>>>>=>>>>>>>>>>>>;(2) Dene l()=()]TJ /F14 11.955 Tf 11.96 0 Td[(2)()]TJ /F14 11.955 Tf 11.96 0 Td[(3)...()]TJ /F14 11.955 Tf 11.95 0 Td[(N),andL()=()]TJ /F10 11.955 Tf 11.96 0 Td[(1)l(). EvaluatingthederivativeofL()ati(i=1,...,N),wehaveL0(i)=l(i)+(i)]TJ /F10 11.955 Tf 11.95 0 Td[(1)l0(i)=8>><>>:(i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)l0(i),i=2,3,...,N,l(1),i=1. Hence,theLagrangepolynomialLi()denedin( 2 )canbeexpressedas Li()=L() L0(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)=8>>><>>>:()]TJ /F10 11.955 Tf 11.96 0 Td[(1)l() (i)]TJ /F10 11.955 Tf 11.95 0 Td[(1)l0(i)()]TJ /F14 11.955 Tf 11.95 0 Td[(i),i=2,3,...,N,l() l(1),i=1.(2) 31

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Since2,...,NarerootsofP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(),wehave l()=P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() KN)]TJ /F6 7.97 Tf 6.59 0 Td[(1, whereKN)]TJ /F6 7.97 Tf 6.59 0 Td[(1istheleadingcoefcientofP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1().Insertingthisin( 2 )yields Li()=8>>>><>>>>:()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i),i=2,3,...,N,P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1),i=1.(2) WerstestimatejL1()j.P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()isatmostO(N)by( 2 )andin[ 45 ,(4.1.1)],itisshownthatP(1,0)N)]TJ /F6 7.97 Tf 6.59 -.01 Td[(1(1)=0B@N)]TJ /F10 11.955 Tf 11.95 0 Td[(1+1N)]TJ /F10 11.955 Tf 11.96 0 Td[(11CA=N. ItfollowsthatjL1()j=P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1)=O(1). SincejL1()j=O(1),theproofofTheorem 2.5 isreducedtoshow max2[)]TJ /F6 7.97 Tf 6.59 0 Td[(1,1]NXi=2jLi()j=ON1 2.Thetermsinthesumarepartitionedinto3setsdependingonthelocationof2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1].Letbeaxedsmallnumberwith0<<1 2. Case1a.jj<1)]TJ /F14 11.955 Tf 11.95 0 Td[(andji)]TJ /F14 11.955 Tf 11.96 0 Td[(j 2. Inthiscase,wehave 2.(2) 32

PAGE 33

Therefore, Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij 2jLi()j=Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij 2()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2(2)]TJ /F14 11.955 Tf 11.95 0 Td[() Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij 2P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() ()]TJ /F14 11.955 Tf 11.95 0 Td[(i)P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)=2(2)]TJ /F14 11.955 Tf 11.95 0 Td[() Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij 2jJi()j. (2) whereJi(),2iN,aretheLagrangeinterpolatingpolynomialsassociatewiththezerosf2,...,NgoftheJacobipolynomialP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1().Notethatthei=1termdoesnotappearinthesum( 2 )since1=1andj)]TJ /F10 11.955 Tf 12.46 0 Td[(1j>by( 2 ).In[ 45 ,(14.4.6)]andintheparagraphthatfollows(14.4.6),itisobservedthatthesumoftheLagrangepolynomialsin( 2 )associatedwiththezerosoftheJacobipolynomialsisO(logN).Hence,wehavePj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij 2jJi()j=O(logN). Case1b.jj<1)]TJ /F14 11.955 Tf 11.95 0 Td[(,ji)]TJ /F14 11.955 Tf 11.95 0 Td[(j> 2,and)]TJ /F10 11.955 Tf 9.3 0 Td[(1
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forNsufcientlarge.Inthiscase,( 2 )againholdsduetotheconstraintsoni.Therefore, Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij> 2)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2)]TJ /F6 7.97 Tf 6.59 0 Td[(1 2,( 2 )isby( 2 ),( 2 )isby( 2 )andtherstestimatein( 2 ),and( 2 )isduetothefactthatthereareatmostNtermsinthesumandthetermsareuniformlybounded. Case1c.jj<1)]TJ /F14 11.955 Tf 11.95 0 Td[(,ji)]TJ /F14 11.955 Tf 11.95 0 Td[(j> 2and1)]TJ /F14 11.955 Tf 13.47 8.08 Td[( 2i<1. Inthiscase,wehavej)]TJ /F10 11.955 Tf 11.96 0 Td[(1j<2)]TJ /F14 11.955 Tf 11.96 0 Td[(.Hence, Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij> 21)]TJ /F19 5.978 Tf 7.78 3.26 Td[( 2i<1jLi()j=Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij> 21)]TJ /F19 5.978 Tf 7.78 3.26 Td[( 2i<1()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2(2)]TJ /F14 11.955 Tf 11.96 0 Td[() X1)]TJ /F19 5.978 Tf 7.79 3.26 Td[( 2i<1P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i) (2) Since<1 2,wehavei3 4.Hence0
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Withthislowerboundfor1)]TJ /F14 11.955 Tf 11.96 0 Td[(i,( 2 )yields Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij> 21)]TJ /F19 5.978 Tf 7.78 3.26 Td[( 2i<1jLi()jON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2X1)]TJ /F19 5.978 Tf 7.79 3.26 Td[( 2i<11 2iN1 2)]TJ /F9 5.978 Tf 7.78 3.25 Td[(5 2i (2) =O)]TJ /F3 11.955 Tf 5.47 -9.69 Td[(N)]TJ /F6 7.97 Tf 6.58 0 Td[(1X1)]TJ /F19 5.978 Tf 7.78 3.26 Td[( 2i<11 2i=O(1),. (2) where( 2 )followsfrom( 2 )and( 2 ),and( 2 )isduetothefactthatthereareatmostNtermsinthesumandi 3.WecombineCase1a,Case1bandCase1ctogetforjj<1)]TJ /F14 11.955 Tf 11.95 0 Td[(,PNi=2jLi()j=O(logN). Case2a.2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+]andji)]TJ /F14 11.955 Tf 11.95 0 Td[(j. Inthiscase,wehaveji)]TJ /F10 11.955 Tf 11.96 0 Td[(1j>2)]TJ /F10 11.955 Tf 11.96 0 Td[(2andj)]TJ /F10 11.955 Tf 11.96 0 Td[(1j<2.Itfollowsthat Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjLi()j=Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2 2)]TJ /F10 11.955 Tf 11.96 0 Td[(2Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ijP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i) (2) In[ 45 ,p.338]isshowsthat NXi=2P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.95 0 Td[(i)=ON1 2, for)]TJ /F10 11.955 Tf 9.29 0 Td[(1)]TJ /F10 11.955 Tf 21.92 0 Td[(1+.Hence,( 2 )yieldsXj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjLi()j=ON1 2. Case2b.2[)]TJ /F10 11.955 Tf 9.29 0 Td[(1,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+],ji)]TJ /F14 11.955 Tf 11.96 0 Td[(j>andi0. Inthiscase,j)]TJ /F10 11.955 Tf 11.96 0 Td[(1j2andji)]TJ /F10 11.955 Tf 11.95 0 Td[(1j>1.Hence, Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij>i0jLi()j=Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>i0()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2 Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i). (2) 35

PAGE 36

SinceP(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)=N1 2()]TJ /F14 11.955 Tf 11.95 0 Td[(i))]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2by( 2 )andP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()=O(1)by( 2 ),wehave Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij>i0jLi()j=ON)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>()]TJ /F14 11.955 Tf 11.96 0 Td[(i)3 2=O(N1 2). Case2c.2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+],ji)]TJ /F14 11.955 Tf 11.95 0 Td[(j>and0000000i)]TJ /F6 7.97 Tf 6.59 0 Td[(1>0i)]TJ /F6 7.97 Tf 6.58 0 Td[(1>i>00i>)]TJ /F10 11.955 Tf 9.3 0 Td[(1 fori=2...,N.Denecos0i=0iandcos00i=00i,by[ 45 ,(6.21.5)],wehave 2i)]TJ /F10 11.955 Tf 11.96 0 Td[(3 2N)]TJ /F10 11.955 Tf 11.95 0 Td[(10i)]TJ /F6 7.97 Tf 6.58 0 Td[(12i)]TJ /F10 11.955 Tf 11.96 0 Td[(2 2N)]TJ /F10 11.955 Tf 11.95 0 Td[(1and2i)]TJ /F10 11.955 Tf 11.96 0 Td[(1 2N+100i2i 2N+1.(2) 36

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Recallthatcosi=i.Sincecos(t),t2[0,],ismonotonedecreasing,itfollowsthat0i)]TJ /F6 7.97 Tf 6.59 0 Td[(12i)]TJ /F10 11.955 Tf 11.96 0 Td[(3 2N>2i)]TJ /F10 11.955 Tf 11.96 0 Td[(3 N>i 2Nand2i 2N+1<2i 2N=i N, fori=2,...,N,wehave i 2N
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Withthisbound,wehave jLi()j=()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2 2(i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() )]TJ /F14 11.955 Tf 11.95 0 Td[(i=O)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(21 2iN1 2)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2iP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() )]TJ /F14 11.955 Tf 11.95 0 Td[(i (2) =O)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(2p i N1 2P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() )]TJ /F14 11.955 Tf 11.96 0 Td[(i=O)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(2p i NP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() )]TJ /F14 11.955 Tf 11.95 0 Td[(i. (2) where( 2 )isby( 2 )and( 2 ),( 2 )isby( 2 ).Wegivetwodifferentboundsforlastfactorin( 2 ).WhenweformsummationofjLi()j,weneedtousedifferentboundsfordifferentvaluesofi. Ourrstboundisbasedonthemeanvaluetheorem: P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() )]TJ /F14 11.955 Tf 11.96 0 Td[(iP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1())]TJ /F3 11.955 Tf 11.95 0 Td[(P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i) )]TJ /F14 11.955 Tf 11.95 0 Td[(i=N+1 2P(2,1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(2(cos),(2) whereisbetweenandi.Thelastequalityin( 2 )usestheformulagivenin[ 45 ,(4.21.7)]forthederivativeofP(,)N()intermsofP(+1,+1)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(). Oursecondboundisby( 2 ), P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() )]TJ /F14 11.955 Tf 11.95 0 Td[(i.52P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(cos) 2)]TJ /F14 11.955 Tf 11.96 0 Td[(2i.(2) WiththesepreliminaryestimatesforjLi()j,wenowobtainboundsforPj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjLi()jwhen2[1)]TJ /F14 11.955 Tf 12.16 0 Td[(,1].Werstconsiderthecasethatisverycloseto1inthesensethat2h0,c Niforanygivenc>0,wherecos=.Sincei20, 2i,itfollowsthatin( 2 ),2h0, 2i,andhence,P(2,1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(2(cos)=O)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(N2by( 2 ).Wecombinethiswith( 2 )and( 2 )toobtain jLi()j=O)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(2N2p i=O(1)p i,(2) 38

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Thebound( 2 )isvalidforanyi(i=2,...,N).AnotherboundforjLi()jisobtainedfrom( 2 )and( 2 ): jLi()j=O)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(2p i NP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(cos) 2)]TJ /F14 11.955 Tf 11.96 0 Td[(2i.(2) Ifi4c,thenby( 2 ),wehave i>i 2N2c N>2. Hence, 2i)]TJ /F14 11.955 Tf 11.95 0 Td[(2=2i)]TJ /F14 11.955 Tf 11.95 0 Td[(23 42i>3 16i N2. Combinethiswith( 2 )toobtain jLi()j=O)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(2Ni)]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2jP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(cos)j=O(1)i)]TJ /F9 5.978 Tf 7.79 3.26 Td[(3 2,(2) sincejP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(cos)j=O(N)by( 2 )and=O)]TJ /F6 7.97 Tf 7.75 -4.98 Td[(1 N. WepartitionthesumoftheLagrangepolynomialsasfollows: Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(iji6=1jLi()j=Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij1
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Thepreviousanalysisfor2h0,c Niappliestoanychoiceofc>0,butfor2c N, 3i,wechoosec=5maxfc1,c2g,wherec1isdenedin( 2 )andc2isdenedin( 2 ).ThesumofLagrangepolynomialsissplitintothefollowingtwosums: Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(iji6=1jLi()j=XjN)]TJ /F7 7.97 Tf 6.58 0 Td[(ij2c2j)]TJ /F15 7.97 Tf 6.59 0 Td[(iji6=1jLi()j+XjN)]TJ /F7 7.97 Tf 6.58 0 Td[(ij>2c2j)]TJ /F15 7.97 Tf 6.58 0 Td[(iji6=1jLi()j.(2) First,wefocusonboundingtherstsumontherightsideof( 2 ).Thenumberoftermsintherstsumisnitedependingonlyonc2,notonN.By( 2 )and( 2 ),wehave jLi()j=O)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(2p iP(2,1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(2(cos).(2) SincejN)]TJ /F3 11.955 Tf 13.12 0 Td[(ij2c2,wehavei2[N)]TJ /F10 11.955 Tf 13.12 0 Td[(2c2,N+2c2].By( 2 ),wehavei2)]TJ /F10 11.955 Tf 13.15 8.09 Td[(3c2 N,+3c2 N.Sinceisbetweenandi,wehave )]TJ /F10 11.955 Tf 13.15 8.09 Td[(3c2 N.(2) Fromthedenitionc=5maxfc1,c2g,itfollowsthatc2c 5andby( 2 ),)]TJ /F10 11.955 Tf 13.23 8.09 Td[(3c2 N)]TJ /F10 11.955 Tf 14.75 8.09 Td[(3c 5N.Since>c Nandc5c1,wehave2c 5N2c1 N>c1 N.By( 2 )with=2and=1,wehave P(2,1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(2(cos)=)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2ON)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2.(2) WhenjN)]TJ /F3 11.955 Tf 11.28 0 Td[(ij2c2,wehaveiN)]TJ /F10 11.955 Tf 11.28 0 Td[(2c2.Since>c N,itfollowsthatic)]TJ /F10 11.955 Tf 11.27 0 Td[(2c23c2orc2i 3sincec5c2.Intherstsumof( 2 ),jN)]TJ /F3 11.955 Tf 12.18 0 Td[(ij2c2.Sincec2i 3,wehave i 3N<<5i 3N.(2) By( 2 ),( 2 )andthefactthatisbetweenandi,wehave i 2N5i 3N.(2) 40

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Also,by( 2 ), N i<5 3.(2) Wecombine( 2 )( 2 )toobtain jLi()j=2p i)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2ON)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2=2p ii 2N)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2=O(1)N i2=O(1). Therstrelationisby( 2 )and( 2 ),thesecondis( 2 )andthethirdis( 2 ).Therefore,therstsumin( 2 )isO(1). Nowweconsiderthesecondsumin( 2 ).SinceP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(cos)=)]TJ /F9 5.978 Tf 7.79 3.26 Td[(3 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2by( 2 ),itfollowsfrom( 2 )and( 2 )that jLi()j=O)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(2p i NP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(cos) 2)]TJ /F14 11.955 Tf 11.96 0 Td[(2i=p p iON)]TJ /F9 5.978 Tf 7.78 3.25 Td[(3 21 2)]TJ /F14 11.955 Tf 11.95 0 Td[(2i.(2) Werstgiveanupperboundfor1 2)]TJ /F14 11.955 Tf 11.96 0 Td[(2iasfollows: 2)]TJ /F14 11.955 Tf 11.96 0 Td[(2i 2)]TJ /F8 11.955 Tf 11.95 16.86 Td[(i N2=(N+Ni)(N)]TJ /F3 11.955 Tf 11.96 0 Td[(Ni) (N+i)(N)]TJ /F3 11.955 Tf 11.96 0 Td[(i)=1+Ni)]TJ /F3 11.955 Tf 11.96 0 Td[(i N+i1+i)]TJ /F3 11.955 Tf 11.96 0 Td[(Ni N)]TJ /F3 11.955 Tf 11.96 0 Td[(i (2) Since>c Nandc=5maxfc1,c2g,wehave >c N5c2 N.(2) Itfollowsfrom( 2 )and( 2 )that Ni)]TJ /F3 11.955 Tf 11.96 0 Td[(i N+i
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Hence, 1+Ni)]TJ /F3 11.955 Tf 11.96 0 Td[(i N+i>1)]TJ /F10 11.955 Tf 13.16 8.08 Td[(1 5=4 5.(2) SincejN)]TJ /F3 11.955 Tf 11.96 0 Td[(ij>2c2intherstsumof( 2 )and( 2 )holds,wehave i)]TJ /F3 11.955 Tf 11.96 0 Td[(Ni N)]TJ /F3 11.955 Tf 11.95 0 Td[(i1)]TJ /F10 11.955 Tf 13.16 8.09 Td[(1 2=1 2.(2) By( 2 )and( 2 ),wegetalowerboundfor( 2 ),2)]TJ /F14 11.955 Tf 11.96 0 Td[(2i 2)]TJ /F8 11.955 Tf 11.96 16.86 Td[(i N2>2 5, whichyields 1 2)]TJ /F14 11.955 Tf 11.96 0 Td[(2i<5 21 2)]TJ /F8 11.955 Tf 11.96 16.85 Td[(i N2. Itfollowsfrom( 2 )that jLi()j=p p iON)]TJ /F9 5.978 Tf 7.79 3.26 Td[(3 21 2)]TJ /F8 11.955 Tf 11.95 16.85 Td[(i N2=p iOp N1 (N)2)]TJ /F10 11.955 Tf 11.96 0 Td[((i)2(2) 42

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IfiN 2,then(N)2)]TJ /F10 11.955 Tf 11.96 0 Td[((i)23 4(N)2.Itfollowsfrom( 2 )that Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjN)]TJ /F7 7.97 Tf 6.59 0 Td[(ij>2c2iN 2jLi()j=O(N))]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2XiN 2p i=O(N))]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2ZN 2+10p xdx=O(N))]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2N 2+13 2=O(1). (2) Ifi>N 2,wehave i)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 22c2i>N 2jLi()j=Op NXj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjN)]TJ /F7 7.97 Tf 6.59 0 Td[(ij>2c2i>N 2p i (N+i)(N)]TJ /F3 11.955 Tf 11.96 0 Td[(i)=Op NXj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjN)]TJ /F7 7.97 Tf 6.59 0 Td[(ij>2c2i>N 2p i (i)jN)]TJ /F3 11.955 Tf 11.96 0 Td[(ij=p NN 2)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2XjN)]TJ /F7 7.97 Tf 6.58 0 Td[(ij>2c21 jN)]TJ /F3 11.955 Tf 11.96 0 Td[(ij (2) =O(1)XjN)]TJ /F7 7.97 Tf 6.59 0 Td[(ij>2c21 jN)]TJ /F3 11.955 Tf 11.96 0 Td[(ij, (2) 43

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where( 2 )followsfrom( 2 )and( 2 )isestimatedasfollows: XjN)]TJ /F7 7.97 Tf 6.59 0 Td[(ij>2c21 jN)]TJ /F3 11.955 Tf 11.95 0 Td[(ij=XN)]TJ /F7 7.97 Tf 6.59 0 Td[(i>2c21 N)]TJ /F3 11.955 Tf 11.96 0 Td[(i+XN)]TJ /F7 7.97 Tf 6.59 0 Td[(i<)]TJ /F6 7.97 Tf 6.59 0 Td[(2c21 i)]TJ /F3 11.955 Tf 11.96 0 Td[(N=XiN+2c2 1 i)]TJ /F3 11.955 Tf 11.95 0 Td[(N< 1 N)]TJ /F14 11.955 Tf 11.95 0 Td[(+ZN)]TJ /F9 5.978 Tf 5.76 0 Td[(2c2 11 N)]TJ /F3 11.955 Tf 11.96 0 Td[(xdx!+ 1 2c2+ZNN+2c2 1 x)]TJ /F3 11.955 Tf 11.96 0 Td[(Ndx!=O(logN). (2) Wecombine( 2 )and( 2 )toobtainthesecondsumin( 2 )isO(logN).Therefore,wehavePj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjLi()j=O(logN)for2[1)]TJ /F14 11.955 Tf 11.95 0 Td[(,1]. Case3b.2[1)]TJ /F14 11.955 Tf 11.95 0 Td[(,1]andji)]TJ /F14 11.955 Tf 11.95 0 Td[(j>. Inthiscase,wehave1)]TJ /F14 11.955 Tf 11.96 0 Td[(i<2.Withthisinequality,wehave Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>jLi()j=Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)<1 2X()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)=1 2Xi0()]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)+1 2Xi<0()]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)=2 22Xi0P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() N1 2)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2i+2 22Xi<0P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() N1 2()]TJ /F14 11.955 Tf 11.95 0 Td[(i))]TJ /F9 5.978 Tf 7.79 3.26 Td[(3 2, (2) where( 2 )isby( 2 ),( 2 )and( 2 ). Theestimateof( 2 )isdeterminedbythelocationofonh0, 3i.If0c1 N,P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()=O(N)by( 2 ).Itfollowsfrom( 2 )that Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij>jLi()j=O)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(2N1 2Xi05 2i+O)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(2N1 2Xi<0()]TJ /F14 11.955 Tf 11.95 0 Td[(i)3 2=O(N)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2)+O(N)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2)=O(N)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2), (2) where( 2 )isduetotheboundc1 Nandthefactthateachterminthesumisboundedby5 2andthereareatmostNtermsineachsum. 44

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Ifc1 N< 3,P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()=)]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2O(N)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2)by( 2 ).Itfollowsfrom( 2 )that Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij>jLi()j=O)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(2)]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2N)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xi05 2i+O)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(2)]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xi<0()]TJ /F14 11.955 Tf 11.96 0 Td[(i)3 2=O(1 2)+O(1 2)=O(1). Hence,wehavePj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>jLi()j=O(1)for2[1)]TJ /F14 11.955 Tf 12.1 0 Td[(,1].WecombinethiswiththeresultofCase3atoobtainPNi=2jLi()j=O(logN)for2[1)]TJ /F14 11.955 Tf 11.96 0 Td[(,1]. TogetherCase1aCase3bcompletetheproof. 2.6LebesgueConstantfortheRadauQuadraturePointsPlus-1 Inthissection,weaddanextrapointN+1=)]TJ /F10 11.955 Tf 9.3 0 Td[(1totheRadauquadraturepoints1>2>>N,where1=1.WeestimatetheLebesgueconstantwithrespecttotheinterpolationpointsetf1,...,N+1g. Theorem2.6. LetLi,i=1,...,N+1,betheLagrangeinterpolationpolynomialsasdenedin( 2 )andsupposetheinterpolationpointsaref1,...,N+1gwhereN+1=)]TJ /F10 11.955 Tf 9.3 0 Td[(1andf1,...,NgaretheRadauquadraturepoints(1=1).ThentheLebesgueconstant`Nhasthefollowingestimate: `N=max2[)]TJ /F6 7.97 Tf 6.59 0 Td[(1,1]N+1Xi=1jLi()j=O(logN).(2) Proof. Dene l()=()]TJ /F14 11.955 Tf 11.95 0 Td[(2)()]TJ /F14 11.955 Tf 11.95 0 Td[(2)...()]TJ /F14 11.955 Tf 11.96 0 Td[(N),andL()=()]TJ /F10 11.955 Tf 11.95 0 Td[(1)(+1)l(). EvaluatingthederivativeofL()ati(i=1,...,N+1),wehaveL0(i)=2il(i)+(2i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)l0(i)=8>>>>>>><>>>>>>>:(2i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)l0(i),i=2,3,...,N,2l(1),i=1,)]TJ /F10 11.955 Tf 9.29 0 Td[(2l()]TJ /F10 11.955 Tf 9.3 0 Td[(1),i=N+1. 45

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Hence,theLagrangepolynomialLi()denedin( 2 )canbeexpressedas Li()=L() L0(i)()]TJ /F14 11.955 Tf 11.95 0 Td[(i)=8>>>>>>>><>>>>>>>>:(2)]TJ /F10 11.955 Tf 11.95 0 Td[(1)l() (2i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)l0(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i),i=2,3,...,N,(+1)l() 2l(1),i=1,()]TJ /F10 11.955 Tf 11.96 0 Td[(1)l() )]TJ /F10 11.955 Tf 9.3 0 Td[(2l()]TJ /F10 11.955 Tf 9.3 0 Td[(1),i=N+1.(2) Since2,...,NarerootsofP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(),wehave l()=P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() KN)]TJ /F6 7.97 Tf 6.59 0 Td[(1, whereKN)]TJ /F6 7.97 Tf 6.59 0 Td[(1istheleadingcoefcientofP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1().Insertingthisin( 2 )yields Li()=8>>>>>>>>>><>>>>>>>>>>:(2)]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (2i)]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i),i=2,3,...,N,(+1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() 2P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1),i=1,()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() )]TJ /F10 11.955 Tf 9.29 0 Td[(2P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()]TJ /F10 11.955 Tf 9.3 0 Td[(1),i=N+1.(2) WerstestimatejL1()j.P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()isatmostO(N)by( 2 )andin[ 45 ,(4.1.1)],itisshownthatP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1)=0B@N)]TJ /F10 11.955 Tf 11.95 0 Td[(1+1N)]TJ /F10 11.955 Tf 11.96 0 Td[(11CA=N. ItfollowsthatjL1()j=(+1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() 2P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1)=(+1)O(N) 2N=O(1). ThenweestimatejLN+1()j.In[ 45 ,(4.1.4)],itisshownthatP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()]TJ /F10 11.955 Tf 9.3 0 Td[(1)=()]TJ /F10 11.955 Tf 9.3 0 Td[(1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(10B@N)]TJ /F10 11.955 Tf 11.96 0 Td[(1N)]TJ /F10 11.955 Tf 11.96 0 Td[(11CA=()]TJ /F10 11.955 Tf 9.29 0 Td[(1)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1. 46

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Table2-1. Theanalysisof( 2 )isorganizedinto8cases. ConstraintsonConstraintsoni )]TJ /F10 11.955 Tf 9.3 0 Td[(1+<<1)]TJ /F14 11.955 Tf 11.96 0 Td[(ji)]TJ /F14 11.955 Tf 11.96 0 Td[(j 2ji)]TJ /F14 11.955 Tf 11.96 0 Td[(j> 2 jij<1)]TJ /F15 7.97 Tf 13.34 4.71 Td[( 21)]TJ /F15 7.97 Tf 13.34 4.71 Td[( 2i<1)]TJ /F10 11.955 Tf 9.3 0 Td[(1Case2aCase2b1)]TJ /F14 11.955 Tf 11.96 0 Td[(1ji)]TJ /F14 11.955 Tf 11.96 0 Td[(jji)]TJ /F14 11.955 Tf 11.95 0 Td[(j>Case3aCase3b Wecombinetheaboveequationwith( 2 )toobtain jLN+1()j=()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() )]TJ /F10 11.955 Tf 9.3 0 Td[(2P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1()]TJ /F10 11.955 Tf 9.3 0 Td[(1)2P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() 4.(2) If2h 2,i,P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()isatmostO(1)by( 2 ).Itfollowsfrom( 2 )thatjLN+1()j=O(1).If2h0,c1 Ni,P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1()=O(N)by( 2 ).SinceisO1 N,wehavejLN+1()j=O1 Nby( 2 ).If2hc1 N, 2i,P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1()=)]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2by( 2 ).Itfollowsfrom( 2 )thatjLN+1()j=1 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2=ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2.Hence,alltogether,wehavejLN+1()j=O(1). SincewehaveshownthatjL1()j=O(1)andjLN+1()j=O(1),inordertoproveTheorem 2.6 ,weonlyneedtoshowthat max2[)]TJ /F6 7.97 Tf 6.58 0 Td[(1,1]NXi=2jLi()j=O(logN).(2) Letbeanyxednumbersatisfying0<<1 2.Thetermsinthesumarepartitionedinto3setsdependingonthelocationof2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1]andthevalueofiasindicatedinTable 2-1 .Forexample,inCase1a,weconsiderany2[)]TJ /F10 11.955 Tf 9.29 0 Td[(1+,1)]TJ /F14 11.955 Tf 12.03 0 Td[(]andwefocusonthesummationoveribetween2andNforwhichji)]TJ /F14 11.955 Tf 11.96 0 Td[(j 2. Case1a.jj<1)]TJ /F14 11.955 Tf 11.95 0 Td[(andji)]TJ /F14 11.955 Tf 11.96 0 Td[(j 2. Inthiscase,wehave j2)]TJ /F10 11.955 Tf 11.95 0 Td[(1j<1,andj2i)]TJ /F10 11.955 Tf 11.96 0 Td[(1j>2 4.(2) 47

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Since1=1andN+1=)]TJ /F10 11.955 Tf 9.3 0 Td[(1donotsatisfy( 2 ),theyarenotincludedinthiscase.Therefore, Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij 2jLi()j=Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij 2(2)]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (2i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.95 0 Td[(i)4 2Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij 2P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() ()]TJ /F14 11.955 Tf 11.96 0 Td[(i)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i). (2) Theestimationfor( 2 )isthesameasSection5,Case1a.WehavePj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij 2jLi()j=O(logN). Case1b.jj<1)]TJ /F14 11.955 Tf 11.95 0 Td[(,ji)]TJ /F14 11.955 Tf 11.95 0 Td[(j> 2,andjij<1)]TJ /F14 11.955 Tf 13.47 8.08 Td[( 2. Inthiscase,( 2 )againholdsand1andN+1arenotincluded.Sincejj<1)]TJ /F14 11.955 Tf 10.91 0 Td[(,thenby( 2 ),P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1()=ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2forNsufcientlarge.Therefore, Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij> 2jij<1)]TJ /F19 5.978 Tf 7.78 3.26 Td[( 2jLi()j=Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij> 2jij<1)]TJ /F19 5.978 Tf 7.78 3.26 Td[( 2(2)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (2i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)8 3Xjij<1)]TJ /F19 5.978 Tf 7.79 3.26 Td[( 2P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)=ON)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2Xjij<1)]TJ /F19 5.978 Tf 7.78 3.26 Td[( 21 P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)=O(1). By( 2 )( 2 ). Case1c.jj<1)]TJ /F14 11.955 Tf 11.95 0 Td[(,ji)]TJ /F14 11.955 Tf 11.95 0 Td[(j> 2,and1)]TJ /F14 11.955 Tf 13.48 8.09 Td[( 2i<1. Inthiscase,wehavej2)]TJ /F10 11.955 Tf 12.04 0 Td[(1j<1andji+1j2)]TJ /F14 11.955 Tf 13.57 8.08 Td[( 2.Since1)]TJ /F14 11.955 Tf 13.57 8.08 Td[( 2i<1,1=1andN+1=)]TJ /F10 11.955 Tf 9.3 0 Td[(1arenotincludedinthiscase.Hence, Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij> 21)]TJ /F19 5.978 Tf 7.79 3.26 Td[( 2i<1jLi()j=Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij> 21)]TJ /F19 5.978 Tf 7.78 3.26 Td[( 2i<1(2)]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i+1)(i)]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2 (2)]TJ /F15 7.97 Tf 13.34 4.71 Td[( 2)X1)]TJ /F19 5.978 Tf 7.78 3.26 Td[( 2i<1P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)=O(1). 48

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ByCase1cinSection5. Case1d.jj<1)]TJ /F14 11.955 Tf 11.95 0 Td[(,ji)]TJ /F14 11.955 Tf 11.95 0 Td[(j> 2,and)]TJ /F10 11.955 Tf 9.3 0 Td[(1j 3N>j Nand2j 2N)]TJ /F10 11.955 Tf 11.95 0 Td[(1<2j N,wehave j N<)]TJ /F14 11.955 Tf 11.96 0 Td[(i<2j Nforj=1,...,N)]TJ /F10 11.955 Tf 11.96 0 Td[(1.(2) Sincejj<1)]TJ /F14 11.955 Tf 11.96 0 Td[(,thenby( 2 ), P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()=ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2(2) 49

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forNsufcientlarge.Itfollowsfrom( 2 )that Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij 2)]TJ /F6 7.97 Tf 6.59 0 Td[(1
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By( 2 )andfor2iN,wehave jLi()j=()]TJ /F10 11.955 Tf 11.96 0 Td[(1)(+1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)(i+1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)1 1)]TJ /F14 11.955 Tf 11.96 0 Td[((+1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i+1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.95 0 Td[(i) (2) =O)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[()21 ()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2N1 2()]TJ /F14 11.955 Tf 11.95 0 Td[(i))]TJ /F9 5.978 Tf 7.79 3.26 Td[(3 2P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() )]TJ /F14 11.955 Tf 11.95 0 Td[(i (2) =O)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[()2 N1 2()]TJ /F14 11.955 Tf 11.96 0 Td[(i)1 2P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() )]TJ /F14 11.955 Tf 11.96 0 Td[(i=O)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[()2 p jP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() )]TJ /F14 11.955 Tf 11.95 0 Td[(i. (2) where( 2 )yields( 2 ),( 2 )isdeducedfrom( 2 ),( 2 )and( 2 ),and( 2 )isby( 2 ).Recallthatj=N+1)]TJ /F3 11.955 Tf 11.95 0 Td[(i,andj=N)]TJ /F10 11.955 Tf 11.96 0 Td[(1,...,1fori=2,...,N. Wemakeuseoftwodifferentboundsforthelastfactorin( 2 ),andwhenwesumjLi()j,weusedifferentboundsfordifferenti.Oneboundisalreadygivenin( 2 ).Oursecondbound,basedon( 2 ),is P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() )]TJ /F14 11.955 Tf 11.95 0 Td[(i.52P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(cos) ()]TJ /F14 11.955 Tf 11.96 0 Td[()2)]TJ /F10 11.955 Tf 11.95 0 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2.(2) WiththesepreliminaryestimatesforjLi()j,wenowobtainboundsforPj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjLi()jwhen2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+].Werstconsiderthecasethat=cosisverycloseto)]TJ /F10 11.955 Tf 9.3 0 Td[(1inthesensethat2h)]TJ /F3 11.955 Tf 14.53 8.09 Td[(c N,i,where c=5maxfc1,c2g,(2) wherec1isdenedin( 2 )andc2isdenedin( 2 ).Sincei2 2,,22 3,andisbetweenandiby( 2 ),wehave2h 2,i.By( 2 ),P(2,1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(2(cos)=O(N).Wecombinethiswith( 2 )and( 2 )toobtain jLi()j=O)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[()2 p jN+1 2O(N)=O(1) p j.(2) 51

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Thebound( 2 )isvalidforalli=2,...,N.AnotherboundforjLi()jisobtainedfrom( 2 )and( 2 ): jLi()j=O)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[()2 p jP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(cos) ()]TJ /F14 11.955 Tf 11.95 0 Td[()2)]TJ /F10 11.955 Tf 11.96 0 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[(i)2.(2) Ifj2c,thenby( 2 ),wehave )]TJ /F14 11.955 Tf 11.95 0 Td[(i>j N2c N>2()]TJ /F14 11.955 Tf 11.96 0 Td[(). Hence, ()]TJ /F14 11.955 Tf 11.95 0 Td[(i)2)]TJ /F10 11.955 Tf 11.96 0 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[()2=()]TJ /F14 11.955 Tf 11.95 0 Td[(i)2)]TJ /F10 11.955 Tf 11.96 0 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[()23 4()]TJ /F14 11.955 Tf 11.95 0 Td[(i)2>3 4j N2. Combinethiswith( 2 )toobtain jLi()j=O)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[()2N2j)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2jP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(cos)j=O(1)j)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2,(2) sincejP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(cos)j=O(1)by( 2 )and)]TJ /F14 11.955 Tf 11.76 0 Td[(=O1 N.Theboundin( 2 )isvalidforallj2cwhiletheboundin( 2 )isvalidforallj. WepartitionthesumoftheLagrangepolynomialsasfollows: Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(iji6=1,N+1jLi()j=Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij1j<2cjLi()j+Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij2cjN)]TJ /F6 7.97 Tf 6.59 0 Td[(1jLi()jO(1)X1j<2c1 p j+O(1)X2cjN)]TJ /F6 7.97 Tf 6.58 0 Td[(1j)]TJ /F9 5.978 Tf 7.78 3.25 Td[(5 2, (2) by( 2 )and( 2 ).TherstsumisO(1)since1j<2c.Thesecondsumisboundedasfollows: X2cjN)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)]TJ /F9 5.978 Tf 7.78 3.25 Td[(5 2N)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xj=1j)]TJ /F9 5.978 Tf 7.78 3.25 Td[(5 21+ZN)]TJ /F6 7.97 Tf 6.59 0 Td[(11x)]TJ /F9 5.978 Tf 7.79 3.25 Td[(5 2dx=O(1). Itfollowsfrom( 2 )thatXj)]TJ /F15 7.97 Tf 6.58 0 Td[(iji6=1,N+1jLi()j=O(1)forc N. 52

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RecallthatinCase2a,weconsider2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+].Sofar,wehaveconsider=cosnear)]TJ /F10 11.955 Tf 9.3 0 Td[(1inthesensethat2h)]TJ /F3 11.955 Tf 14.53 8.08 Td[(c N,i.Nowweconsiderawayfrom)]TJ /F10 11.955 Tf 9.3 0 Td[(1inthesensethat22 3,)]TJ /F3 11.955 Tf 14.53 8.09 Td[(c N.Since<1 2,thiswillcoverall2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+].ThesumofLagrangepolynomialsissplitintothefollowingtwosums: Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(iji6=1,N+1jLi()j=XjN)]TJ /F7 7.97 Tf 6.58 0 Td[(ij2c2j)]TJ /F15 7.97 Tf 6.58 0 Td[(iji6=1,N+1jLi()j+XjN)]TJ /F7 7.97 Tf 6.58 0 Td[(ij>2c2j)]TJ /F15 7.97 Tf 6.58 0 Td[(iji6=1,N+1jLi()j,(2) wherec2appearsinthedenition( 2 )ofc.First,wefocusonboundingtherstsumontherightsideof( 2 ).Thenumberoftermsintherstsumisnitedependingonlyonc2,notonN.By( 2 )and( 2 ),wehave jLi()j=O)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[()2 p jN+1 2P(2,1)N)]TJ /F6 7.97 Tf 6.59 0 Td[(2(cos).(2) SincejN)]TJ /F3 11.955 Tf 11.96 0 Td[(ij2c2,wehave i2[N)]TJ /F10 11.955 Tf 11.96 0 Td[(2c2,N+2c2].(2) By( 2 ),wehavei2i N)]TJ /F3 11.955 Tf 13.15 8.09 Td[(c2 N,i N+c2 N2)]TJ /F10 11.955 Tf 13.15 8.09 Td[(3c2 N,+3c2 Nby( 2 ).Sinceisbetweenandi,wehave+3c2 N.Since22 3,)]TJ /F3 11.955 Tf 14.53 8.09 Td[(c N,itfollowsthat<)]TJ /F3 11.955 Tf 13.57 8.09 Td[(c Nand <)]TJ /F3 11.955 Tf 14.53 8.08 Td[(c N+3c2 N.(2) Fromthedenitionc=5maxfc1,c2g,itfollowsthatc2c 5andby( 2 ),<)]TJ /F10 11.955 Tf 14.66 8.09 Td[(2c 5N.Sincec5c1,wehave<)]TJ /F10 11.955 Tf 13.15 8.09 Td[(2c1 N<)]TJ /F3 11.955 Tf 13.15 8.09 Td[(c1 N.By( 2 )with=2and=1,wehave P(2,1)N)]TJ /F6 7.97 Tf 6.58 0 Td[(2(cos)=()]TJ /F14 11.955 Tf 11.95 0 Td[())]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2.(2) WhenjN)]TJ /F3 11.955 Tf 12.94 0 Td[(ij2c2,wehaveiN+2c2.Since<)]TJ /F3 11.955 Tf 15.51 8.09 Td[(c N,itfollowsthati
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Intherstsumof( 2 ),jN)]TJ /F3 11.955 Tf 11.62 0 Td[(ij2c2yieldsi N)]TJ /F10 11.955 Tf 12.81 8.09 Td[(2c2 Ni N+2c2 N.Bythebound( 2 )forc2,itfollowsthat(5i)]TJ /F10 11.955 Tf 11.96 0 Td[(2N) 3N(2N+i) 3N.Hence,wehave N)]TJ /F3 11.955 Tf 11.96 0 Td[(i 3N<)]TJ /F14 11.955 Tf 11.95 0 Td[(<5N)]TJ /F10 11.955 Tf 11.95 0 Td[(5i 3N. Bythedenition,j=N+1)]TJ /F3 11.955 Tf 11.96 0 Td[(i,wehave (j)]TJ /F10 11.955 Tf 11.96 0 Td[(1) 3N<)]TJ /F14 11.955 Tf 11.96 0 Td[(<5(j)]TJ /F10 11.955 Tf 11.96 0 Td[(1) 3N.(2) By( 2 ),( 2 ),andthefactthatisbetweenandi,wehave j)]TJ /F10 11.955 Tf 11.96 0 Td[(1 N<)]TJ /F14 11.955 Tf 11.96 0 Td[(<2j N.(2) Also,by( 2 ), N()]TJ /F14 11.955 Tf 11.96 0 Td[() j)]TJ /F10 11.955 Tf 11.96 0 Td[(1<5 3.(2) By( 2 ),weobtain jLi()j=()]TJ /F14 11.955 Tf 11.96 0 Td[()2 p jN+1 2()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F9 5.978 Tf 7.79 3.26 Td[(3 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2 (2) =()]TJ /F14 11.955 Tf 11.96 0 Td[()2 p jN+1 2j)]TJ /F10 11.955 Tf 11.95 0 Td[(1 N)]TJ /F9 5.978 Tf 7.79 3.26 Td[(3 2ON)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2 (2) =O(1)N()]TJ /F14 11.955 Tf 11.96 0 Td[() j)]TJ /F10 11.955 Tf 11.96 0 Td[(12=O(1), (2) The( 2 )comesfrom( 2 ),( 2 )comesfrom( 2 )and( 2 )is( 2 ).Therefore,therstsumin( 2 )isO(1). Nowweconsiderthesecondsumin( 2 ).SinceP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(cos)=()]TJ /F14 11.955 Tf -401.67 -23.91 Td[())]TJ /F9 5.978 Tf 7.78 3.25 Td[(1 2ON)]TJ /F9 5.978 Tf 7.78 3.25 Td[(1 2by( 2 ),itfollowsfrom( 2 )and( 2 )that jLi()j=O)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[()2 p jP(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(cos) ()]TJ /F14 11.955 Tf 11.96 0 Td[()2)]TJ /F10 11.955 Tf 11.95 0 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[(i)2=()]TJ /F14 11.955 Tf 11.95 0 Td[()3 2ON)]TJ /F9 5.978 Tf 7.78 3.25 Td[(1 2 p jj()]TJ /F14 11.955 Tf 11.95 0 Td[()2)]TJ /F10 11.955 Tf 11.96 0 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2j.(2) 54

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Toobtainanupperboundfor1 ()]TJ /F14 11.955 Tf 11.96 0 Td[()2)]TJ /F10 11.955 Tf 11.95 0 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2,werstexploittheidentity( 2 )withi=j)]TJ /F10 11.955 Tf 11.95 0 Td[(1toobtain ()]TJ /F14 11.955 Tf 11.95 0 Td[()2)]TJ /F10 11.955 Tf 11.96 0 Td[(()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2 ()]TJ /F14 11.955 Tf 11.95 0 Td[()2)]TJ /F8 11.955 Tf 11.96 13.27 Td[((j)]TJ /F6 7.97 Tf 6.58 0 Td[(1) N2=(N()]TJ /F14 11.955 Tf 11.96 0 Td[()+N()]TJ /F14 11.955 Tf 11.96 0 Td[(i))(N()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F3 11.955 Tf 11.95 0 Td[(N()]TJ /F14 11.955 Tf 11.96 0 Td[(i)) (N()]TJ /F14 11.955 Tf 11.96 0 Td[()+(j)]TJ /F10 11.955 Tf 11.95 0 Td[(1))(N()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F10 11.955 Tf 11.95 0 Td[((j)]TJ /F10 11.955 Tf 11.96 0 Td[(1))=1+N()]TJ /F14 11.955 Tf 11.95 0 Td[(i))]TJ /F10 11.955 Tf 11.96 0 Td[((j)]TJ /F10 11.955 Tf 11.95 0 Td[(1) N()]TJ /F14 11.955 Tf 11.96 0 Td[()+(j)]TJ /F10 11.955 Tf 11.96 0 Td[(1)1+(j)]TJ /F10 11.955 Tf 11.95 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(N()]TJ /F14 11.955 Tf 11.96 0 Td[(i) N()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F10 11.955 Tf 11.95 0 Td[((j)]TJ /F10 11.955 Tf 11.96 0 Td[(1) (2) Since)]TJ /F14 11.955 Tf 11.95 0 Td[(>c Nandc=5maxfc1,c2g,wehave )]TJ /F14 11.955 Tf 11.96 0 Td[(>c N5c2 N.(2) Bythedenitionj=N+1)]TJ /F3 11.955 Tf 11.96 0 Td[(i,wefollowsfrom( 2 )that jN()]TJ /F14 11.955 Tf 11.95 0 Td[(i))]TJ /F10 11.955 Tf 11.96 0 Td[((j)]TJ /F10 11.955 Tf 11.96 0 Td[(1)jc2.(2)( 2 )and( 2 )togetheryield N()]TJ /F14 11.955 Tf 11.96 0 Td[(i))]TJ /F10 11.955 Tf 11.95 0 Td[((j)]TJ /F10 11.955 Tf 11.96 0 Td[(1) N()]TJ /F14 11.955 Tf 11.95 0 Td[()+(j)]TJ /F10 11.955 Tf 11.95 0 Td[(1)1)]TJ /F10 11.955 Tf 13.15 8.08 Td[(1 5=4 5.(2) Inthesecondsumof( 2 ),wehavejN)]TJ /F3 11.955 Tf 10.56 0 Td[(ij>2c2.Inserti=N+1)]TJ /F3 11.955 Tf 10.56 0 Td[(j,thisbecomesjN()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F10 11.955 Tf 11.96 0 Td[((j)]TJ /F10 11.955 Tf 11.95 0 Td[(1)j>2c2.Wecombinethiswith( 2 )toobtain (j)]TJ /F10 11.955 Tf 11.96 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(N()]TJ /F14 11.955 Tf 11.95 0 Td[(i) N()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F10 11.955 Tf 11.95 0 Td[((j)]TJ /F10 11.955 Tf 11.95 0 Td[(1)1)]TJ /F10 11.955 Tf 13.15 8.09 Td[(1 2=1 2.(2) By( 2 )and( 2 ),wegetalowerboundfor( 2 ),()]TJ /F14 11.955 Tf 11.96 0 Td[()2)]TJ /F10 11.955 Tf 11.95 0 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[(i)2 ()]TJ /F14 11.955 Tf 11.96 0 Td[()2)]TJ /F8 11.955 Tf 11.95 13.27 Td[((j)]TJ /F6 7.97 Tf 6.59 0 Td[(1) N2>2 5, 55

PAGE 56

whichyields 1 ()]TJ /F14 11.955 Tf 11.95 0 Td[()2)]TJ /F10 11.955 Tf 11.96 0 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[(i)2<5 21 ()]TJ /F14 11.955 Tf 11.96 0 Td[()2)]TJ /F8 11.955 Tf 11.95 13.27 Td[((j)]TJ /F6 7.97 Tf 6.59 0 Td[(1) N2. Itfollowsfrom( 2 )that jLi()j=()]TJ /F14 11.955 Tf 11.95 0 Td[()3 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2 p j()]TJ /F14 11.955 Tf 11.95 0 Td[()2)]TJ /F8 11.955 Tf 11.96 13.27 Td[((j)]TJ /F6 7.97 Tf 6.58 0 Td[(1) N2.(2) If(j)]TJ /F10 11.955 Tf 11.96 0 Td[(1) N)]TJ /F14 11.955 Tf 11.96 0 Td[( 2,then()]TJ /F14 11.955 Tf 11.95 0 Td[()2)]TJ /F8 11.955 Tf 11.96 13.27 Td[((j)]TJ /F6 7.97 Tf 6.59 0 Td[(1) N23 4()]TJ /F14 11.955 Tf 11.96 0 Td[()2.Itfollowsfrom( 2 )that Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjN)]TJ /F7 7.97 Tf 6.59 0 Td[(ij>2c2(j)]TJ /F9 5.978 Tf 5.76 0 Td[(1) N)]TJ /F19 5.978 Tf 5.76 0 Td[( 2jLi()j=X(j)]TJ /F9 5.978 Tf 5.75 0 Td[(1) N)]TJ /F19 5.978 Tf 5.76 0 Td[( 2()]TJ /F14 11.955 Tf 11.95 0 Td[()3 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2 p j()]TJ /F14 11.955 Tf 11.96 0 Td[()2=O 1 p N()]TJ /F14 11.955 Tf 11.95 0 Td[()!X(j)]TJ /F9 5.978 Tf 5.75 0 Td[(1) N)]TJ /F19 5.978 Tf 5.75 0 Td[( 21 p j=O 1 p N()]TJ /F14 11.955 Tf 11.95 0 Td[()! 1+ZN()]TJ /F19 5.978 Tf 5.75 0 Td[() 2+11x)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2dx!=O 1 p N()]TJ /F14 11.955 Tf 11.95 0 Td[()! 2r N()]TJ /F14 11.955 Tf 11.96 0 Td[() 2+1)]TJ /F10 11.955 Tf 11.95 0 Td[(1!=O(1). (2) If(j)]TJ /F10 11.955 Tf 11.96 0 Td[(1) N>)]TJ /F14 11.955 Tf 11.96 0 Td[( 2,wehave (j)]TJ /F10 11.955 Tf 11.96 0 Td[(1))]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2
PAGE 57

Bythedenitionofj,jN)]TJ /F3 11.955 Tf 11.99 0 Td[(ij>2c2isequivalenttojN()]TJ /F14 11.955 Tf 12 0 Td[())]TJ /F10 11.955 Tf 12 0 Td[((j)]TJ /F10 11.955 Tf 11.99 0 Td[(1)j>2c2.Hence,( 2 )yields Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjN)]TJ /F7 7.97 Tf 6.59 0 Td[(ij>2c2(j)]TJ /F9 5.978 Tf 5.76 0 Td[(1) N>)]TJ /F19 5.978 Tf 5.76 0 Td[( 2jLi()j=XjN()]TJ /F15 7.97 Tf 6.59 0 Td[())]TJ /F6 7.97 Tf 6.59 0 Td[((j)]TJ /F6 7.97 Tf 6.59 0 Td[(1)j>2c2(j)]TJ /F9 5.978 Tf 5.76 0 Td[(1) N>)]TJ /F19 5.978 Tf 5.76 0 Td[( 2()]TJ /F14 11.955 Tf 11.96 0 Td[()3 2ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2j)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2 ()]TJ /F14 11.955 Tf 11.95 0 Td[()+(j)]TJ /F6 7.97 Tf 6.59 0 Td[(1) N()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F8 11.955 Tf 11.95 13.27 Td[((j)]TJ /F6 7.97 Tf 6.59 0 Td[(1) N=XjN()]TJ /F15 7.97 Tf 6.59 0 Td[())]TJ /F6 7.97 Tf 6.59 0 Td[((j)]TJ /F6 7.97 Tf 6.59 0 Td[(1)j>2c2(j)]TJ /F9 5.978 Tf 5.76 0 Td[(1) N>)]TJ /F19 5.978 Tf 5.76 0 Td[( 2()]TJ /F14 11.955 Tf 11.96 0 Td[()3 2ON1 2j)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2 ()]TJ /F14 11.955 Tf 11.96 0 Td[()jN()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F10 11.955 Tf 11.96 0 Td[((j)]TJ /F10 11.955 Tf 11.95 0 Td[(1)j=XjN()]TJ /F15 7.97 Tf 6.59 0 Td[())]TJ /F6 7.97 Tf 6.59 0 Td[((j)]TJ /F6 7.97 Tf 6.59 0 Td[(1)j>2c2(j)]TJ /F9 5.978 Tf 5.76 0 Td[(1) N>)]TJ /F19 5.978 Tf 5.76 0 Td[( 2O(N()]TJ /F14 11.955 Tf 11.95 0 Td[())1 2N()]TJ /F15 7.97 Tf 6.59 0 Td[() 2)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2 jN()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F10 11.955 Tf 11.95 0 Td[((j)]TJ /F10 11.955 Tf 11.96 0 Td[(1)j (2) =O(1)XjN()]TJ /F15 7.97 Tf 6.58 0 Td[())]TJ /F6 7.97 Tf 6.58 0 Td[((j)]TJ /F6 7.97 Tf 6.58 0 Td[(1)j>2c21 jN()]TJ /F14 11.955 Tf 11.95 0 Td[())]TJ /F10 11.955 Tf 11.96 0 Td[((j)]TJ /F10 11.955 Tf 11.96 0 Td[(1)j. (2) Where( 2 )followsfrom( 2 )and( 2 )isestimatedasfollows: XjN()]TJ /F15 7.97 Tf 6.59 0 Td[())]TJ /F6 7.97 Tf 6.59 0 Td[((j)]TJ /F6 7.97 Tf 6.59 0 Td[(1)j>2c21 jN()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F10 11.955 Tf 11.95 0 Td[((j)]TJ /F10 11.955 Tf 11.95 0 Td[(1)j=XN()]TJ /F15 7.97 Tf 6.58 0 Td[())]TJ /F6 7.97 Tf 6.58 0 Td[((j)]TJ /F6 7.97 Tf 6.58 0 Td[(1)>2c21 N()]TJ /F14 11.955 Tf 11.95 0 Td[())]TJ /F10 11.955 Tf 11.95 0 Td[((j)]TJ /F10 11.955 Tf 11.96 0 Td[(1)+XN()]TJ /F15 7.97 Tf 6.59 0 Td[())]TJ /F6 7.97 Tf 6.59 0 Td[((j)]TJ /F6 7.97 Tf 6.59 0 Td[(1)<)]TJ /F6 7.97 Tf 6.58 0 Td[(2c21 (j)]TJ /F10 11.955 Tf 11.96 0 Td[(1))]TJ /F3 11.955 Tf 11.96 0 Td[(N()]TJ /F14 11.955 Tf 11.95 0 Td[()=Xj)]TJ /F6 7.97 Tf 6.59 0 Td[(1N()]TJ /F19 5.978 Tf 5.75 0 Td[()+2c2 1 (j)]TJ /F10 11.955 Tf 11.95 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(N()]TJ /F14 11.955 Tf 11.96 0 Td[()= 1 2c2+ZN()]TJ /F19 5.978 Tf 5.76 0 Td[())]TJ /F9 5.978 Tf 5.75 0 Td[(2c2 01 N()]TJ /F14 11.955 Tf 11.96 0 Td[())]TJ /F3 11.955 Tf 11.96 0 Td[(xdx!+ 1 2c2+ZN)]TJ /F6 7.97 Tf 6.59 0 Td[(2N()]TJ /F19 5.978 Tf 5.76 0 Td[()+2c2 1 x)]TJ /F3 11.955 Tf 11.96 0 Td[(N()]TJ /F14 11.955 Tf 11.95 0 Td[()dx!=O(logN). (2) Herethe1 2c2termsabovecorrespondtothersttermsinthesumsinthepreviousequation,analogoustotherelationPNi=1i)]TJ /F6 7.97 Tf 6.58 0 Td[(11+RN1x)]TJ /F6 7.97 Tf 6.59 0 Td[(1dx. Wecombine( 2 )and( 2 )toobtainthesecondsumin( 2 )isO(logN).Therefore,wehavePj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjLi()j=O(logN)for[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+]. Case2b.2[)]TJ /F10 11.955 Tf 9.29 0 Td[(1,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+]andji)]TJ /F14 11.955 Tf 11.95 0 Td[(j>. 57

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Inthiscase,wehave 2)]TJ /F14 11.955 Tf 11.95 0 Td[(j)]TJ /F10 11.955 Tf 11.96 0 Td[(1j2,andjLi()j=Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij>()]TJ /F10 11.955 Tf 11.96 0 Td[(1)(+1)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.95 0 Td[(1)(i+1)P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2 2Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>(+1)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i) (2) =O)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[()20BB@Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij>i0P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)+Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>i>0P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)1CCA (2) =O)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[()20BB@Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij>i0P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() N1 2()]TJ /F14 11.955 Tf 11.96 0 Td[(i))]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2+Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>i>0P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() 2iN1 2)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2i1CCA, (2) where( 2 )isby( 2 ),and( 2 )isby( 2 ).Therstsummationin( 2 )isobtainedfrom( 2 ).Sinceji)]TJ /F10 11.955 Tf 12.33 0 Td[(1j1wheni0,andthesecondsummationin( 2 )isobtainedfrom( 2 )and( 2 ). Wegivetwodifferentboundsfor( 2 )dependingthevalueof.If2 3)]TJ /F3 11.955 Tf 13.15 8.09 Td[(c1 N,thenP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()=()]TJ /F14 11.955 Tf 11.95 0 Td[())]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2O(N)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2)by( 2 ),andby( 2 ),wehave Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>jLi()j=O()]TJ /F14 11.955 Tf 11.96 0 Td[()3 2N)]TJ /F6 7.97 Tf 6.58 0 Td[(1 Xi0()]TJ /F14 11.955 Tf 11.96 0 Td[(i)3 2+Xi>01 2i!=O(1). SinceeachterminthesumsareboundedandthesumsareO(N). If)]TJ /F3 11.955 Tf 13.15 8.09 Td[(c1 N<,thenP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()=O(1)by( 2 ).Itfollowsfrom( 2 )that Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij>jLi()j=O)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(()]TJ /F14 11.955 Tf 11.95 0 Td[()2N)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2 Xi0()]TJ /F14 11.955 Tf 11.95 0 Td[(i)3 2+Xi>01 2i!=ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2. 58

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Since)]TJ /F14 11.955 Tf 11.95 0 Td[(=O1 NandthesumsareagainO(N). WecombineCase2aand2btoobtainfor2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,)]TJ /F10 11.955 Tf 9.3 0 Td[(1+],PNi=2jLi()j=O(logN). Case3a.2[1)]TJ /F14 11.955 Tf 11.96 0 Td[(,1]andji)]TJ /F14 11.955 Tf 11.96 0 Td[(j. Inthiscase,wehave2)]TJ /F14 11.955 Tf 12.71 0 Td[(+12and2)]TJ /F10 11.955 Tf 12.72 0 Td[(2i+1<2.Withtheseinequalities,wehave Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ijjLi()j=Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij(+1)()]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i+1)(i)]TJ /F10 11.955 Tf 11.95 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.95 0 Td[(i)=1 1)]TJ /F14 11.955 Tf 11.95 0 Td[(Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)=O(logN), wherethelastrelationisbyCase3aofSection5. Case3b.2[1)]TJ /F14 11.955 Tf 11.95 0 Td[(,1]andji)]TJ /F14 11.955 Tf 11.95 0 Td[(j>. Inthiscase,wehave 2)]TJ /F14 11.955 Tf 11.95 0 Td[(+12andjLi()j=Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>(+1)()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i+1)(i)]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i)()]TJ /F14 11.955 Tf 11.96 0 Td[(i)2 2Xj)]TJ /F15 7.97 Tf 6.59 0 Td[(ij>()]TJ /F10 11.955 Tf 11.96 0 Td[(1)P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i+1)P(1,0)0N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i) (2) =O)]TJ /F14 11.955 Tf 5.47 -9.68 Td[(2 Xi0P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() (i+1)P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)+Xi<0P(1,0)N)]TJ /F6 7.97 Tf 6.58 0 Td[(1() (i+1)P(1,0)0N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)! (2) =O)]TJ /F14 11.955 Tf 5.47 -9.69 Td[(2 Xi0P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() N1 2)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2i+Xi<0P(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1() ()]TJ /F14 11.955 Tf 11.95 0 Td[(i)2N1 2()]TJ /F14 11.955 Tf 11.96 0 Td[(i))]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2! (2) where( 2 )isfrom( 2 )and( 2 )isby( 2 ).Therstsummationin( 2 )isobtainedfrom( 2 ).Sinceji+1j1fori0.Thesecondsummationin( 2 )isobtainedfrom( 2 )and( 2 ). 59

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Again,ourboundfor( 2 )dependsonthelocationof.If0c1 N,thenP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()=O(N)by( 2 ).Itfollowsfrom( 2 )that Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>jLi()j=2ON1 2 Xi05 2i+Xi<0()]TJ /F14 11.955 Tf 11.95 0 Td[(i))]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2!. (2) TherstsummationintheaboveequationisO(N)sinceeachterminthesumisboundedby5 2andthereareatmostNterms.Thesecondsummationin( 2 )isboundedasfollows Xi<0()]TJ /F14 11.955 Tf 11.95 0 Td[(i))]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2jLi()j=2ON1 2(O(N)+O(N))=ON)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2. Ifc1 N< 3,thenP(1,0)N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()=)]TJ /F9 5.978 Tf 7.78 3.26 Td[(3 2ON)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2by( 2 ).Itfollowsfrom( 2 )that Xj)]TJ /F15 7.97 Tf 6.58 0 Td[(ij>jLi()j=1 2O)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1 Xi05 2i+Xi<0()]TJ /F14 11.955 Tf 11.95 0 Td[(i))]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2!=O(1), by( 2 ). WecombineCase3abtoobtainPNi=2jLi()j=O(logN),for2[1)]TJ /F14 11.955 Tf 12.47 0 Td[(,1].AlltogetherCase1abcompletetheproof. 2.7TightnessofEstimates InthisChapter,wehaveestablishedthefollowingupperboundsfortheLebesgueconstantassociatedwiththreepointsetsintroducedinSection 2.1 :(P1)ON1 2(Theorem 2.4 ),(P2)ON1 2(Theorem 2.5 ),(P3)O(logN)(Theorem 2.6 ).Wenowobservethattheseestimatesarealltight.Foreachofthepointsets,wenumericallyevaluatetheLebesgueconstantforNbetween2and30.Forpointssets(P1)and(P2),wettheresulting29pointsbyacurveoftheformap N+b,whereaandbarechosen 60

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Figure2-1. LeastsquaresapproximationtotheLebesgueconstantforthepointset(P1)(Gaussquadraturepointsaugmentedwith)]TJ /F10 11.955 Tf 9.3 0 Td[(1)usingcurvesoftheformap N+b Figure2-2. LeastsquaresapproximationtotheLebesgueconstantforthepointset(P2)(Radauquadraturepoints)usingcurvesoftheformap N+b togivealeastsquaresttothe29points.InFigure 2-1 and 2-2 ,weshowthebestttingcurveforpointset(P1)and(P2)respectively.InFigure 2-3 ,weshowthebestttopointset(P3)usingcurveoftheformalogN+b.Basedontheclosetbetweenthedatapointsandthecurves,ourestimatesfortheLebesgueconstantsallappeartobetight. 2.8ConcludingRemarks WeobtaintheLebesgueconstantforthreesetsofinterpolationpoints(P1),(P2)and(P3)thatariseinGaussianquadraturecollocationdiscretizationofcontrolproblem.Inparticular,theLebesgueconstantforthepointset(P1)(Gaussquadraturepoints 61

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Figure2-3. LeastsquaresapproximationtotheLebesgueconstantforthepointset(P3)(Radauquadraturepointsaugmentedwith)]TJ /F10 11.955 Tf 9.3 0 Td[(1)usingcurvesoftheformalogN+b augmentedbythepoint)]TJ /F10 11.955 Tf 9.29 0 Td[(1)isON1 2asshowninTheorem 2.4 .TheLebesgueconstantforthepointset(P2)(Radauquadraturepoints)isON1 2asshowninTheorem 2.5 .TheLebesgueconstantforthepointset(P3)(Radauquadraturepointsaugmentedwith)]TJ /F10 11.955 Tf 9.3 0 Td[(1)isO(logN)asshowninTheorem 2.6 .BasedonnumericaltsshowninSection 2.7 ,theseestimatesallappeartobetight. 62

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CHAPTER3CONVERGENCEOFAGAUSSCOLLOCATIONMETHODFORUNCONSTRAINEDOPTIMALCONTROL 3.1BackgroundofGaussCollocationMethod Inrecentyears,Gaussianquadratureorthogonalcollocationmethodshavebecomepopularapproachforsolvingoptimalcontrolproblems.TheGaussianquadraturecollocationmethodsareaclassofdirectcollocationwheretheoptimalcontrolproblemistranscribedtoanonlinearprogrammingproblem(NLP)byparameterizingthestateandcontrolusingglobalpolynomialsandcollocatingthedifferential-algebraicequationsusingnodesobtainedfromaquadraturescheme[ 4 5 16 20 22 23 34 37 39 42 47 51 ].TheGausscollocationmethodusetheGaussquadraturepointsascollocationpoints. InthisChapter,weanalyzetheconvergenceoftheGausscollocationmethod.Underassumptionsofcoercivityandsmoothness,weestablishaerrorestimateoftheformO1 Nk)]TJ /F9 5.978 Tf 6.95 2.34 Td[(5 2wherekisthenumberofcontinuousderivativesinthesolutionandNisthedegreeofthepolynomialsinthecollocationscheme.Herethenormisthediscretesup-norminRN.TheproofisbasedonanestimateofLebesgueconstantfortheGausscollocationpointsgiveinChapter2,andtheabstracterrorestimatefordiscreteapproximationsgivenin[ 14 ]. 3.2TheControlProblemandGaussCollocationMethod Weconsiderthefollowingunconstrainedoptimalcontrolproblem: minimizeC(x(1))subjectto_x(t)=f(x(t),u(t)),t2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1]x()]TJ /F10 11.955 Tf 9.29 0 Td[(1)=x0,(3) wherethestatex(t)2Rn,_xd dtx,thecontrolu(t)2Rm,f:RnRm!Rn,C:Rn!Randx0istheinitialcondition,whichweassumeisgiven. 63

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Notation.ThroughouttheChapter,Ckisthecollectionofreal-valuedktimescontinuouslydifferentialfunctionsontheinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1].y(k)denotesthek-thderivativeofyfory2Ck,and_ydenotethederivativeofy.LetjjdenotestheabsolutevalueofscalarsortheEuclideannormofvectors.Letkk1denotestheuniformnormovertheinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1],thatis,kyk1=max)]TJ /F6 7.97 Tf 6.58 0 Td[(1t1jy(t)j,foranycontinuousfunctionyon[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1].PNdenotesthesetofpolynomialsofdegreeatmostN.Weassumeallvectorfunctionsoftimearerowvectors:x(t)=(x1(t),x2(t),...,xn(t)),u(t)=(u1(t),u2(t),...,um(t)).Givenvectorsaandb2Rn,thenotationha,bidenotestheinnerproductofaandbinRn.B(x)istheclosedballcenteredatxwithradius.ThetransposeofamatrixAisAT.Iff:Rn!Rm,thenrfisthembynJacobianmatrixwhosei-throwisrfi.Inparticular,thegradientofascalarvaluedfunctionisarowvector.LetL1denotethespaceofessentiallyboundedfunctionsandletW1,1denotethespaceofLipschitzcontinuousfunctionsdenedon[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1].Further,cisageneralconstantthathasdifferentvalueindifferentrelations. WenowpresenttheassumptionsthatareemployedintheanalysisoftheGausscollocationschemeof( 3 ). (A1)Smoothness.Problem( 3 )hasalocalminimizer(x,u)whichliesinCk+1(Rn)L1(Rm),forsomek3.ThereexistsanopensetRnRmand>0suchthatB(x(t),u(t))foreveryt2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1].ThersttwoderivativesoffareLipschitzcontinuousin,andthersttwoderivativesofCareLipschitzcontinuousinB(x(1)).Furthermore,thereexitsassociatedcostate2W1,1(Rn)forwhichthefollowingequations(Pontryagin'sminimumprinciple)aresatisedforx=x,u=uand=: (1)=rC(x(1)), (3) _(t)=rxH(x(t),u(t),(t)), (3) 0=ruH(x(t),u(t),(t)). (3) 64

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whereHistheHamiltoniandenedbyH(x(t),u(t),(t))=h(t),f(x(t),u(t))i.Finally,2Ck+1(Rn). (A2)Coercivity.Forsome>0,thesmallesteigenvaluesofthematricesbelowaregreaterthan: V=rxxC(x(1))and0B@Q(t)S(t)ST(t)R(t)1CAforallt2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1],(3) where Q(t)=rxxH(x(t),u(t),(t)),S(t)=ruxH(x(t),u(t),(t)),R(t)=ruuH(x(t),u(t),(t)). (A3).kA(t)k11 4forallt2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1],whereA(t)=rxf(x(t),u(t)). WenowintroducetheGausscollocationmethod[ 4 5 26 ].ConsiderNGaussquadraturepoints(t1,t2,,tN)ontheinterval()]TJ /F10 11.955 Tf 9.3 0 Td[(1,1)andtwoadditionalpointst0=)]TJ /F10 11.955 Tf 9.3 0 Td[(1andtN+1=1.TheLagrangeinterpolatingpolynomialsrelativetot0,t1,,tNaregivenby Li(t)=NYj=0j6=it)]TJ /F3 11.955 Tf 11.96 0 Td[(tj ti)]TJ /F3 11.955 Tf 11.95 0 Td[(tj,i=0,,N.(3) Thej-thcomponentofthestatex(t)isapproximatedbythefollowingN-thdegreepolynomial: xNj(t)=NXi=0xijLi(t)(3) wherexijisanapproximationtoxj(ti).Differentiatingtheseries( 3 )andevaluatingatthecollocationpointtk,k=1,2,,N,wehave _xNj(tk)=NXi=0xij_Li(tk)=NXi=0Dkixij,whereDki=_Li(tk).(3) LetDbetheNbyN+1matrixwhose(k,i)elementisDki.ThematrixDisreferredtoasthedifferentiationmatrix.LetDjdenotethej-thcolumnofD,0jN,andletDj:k 65

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bethesubmatrixofDformedbycolumnsofjthroughk.Thefollowingpropositionisestablishedin[ 26 ]: Proposition3.1. ThematrixD1:Nisinvertible. Inouranalysisofconvergence,weneedtoboundbothD)]TJ /F6 7.97 Tf 6.58 0 Td[(11:NandW1 2D1:N)]TJ /F6 7.97 Tf 6.58 0 Td[(1,whereWisthediagonalmatrixwhosediagonalelements!i,1iN,aretheGaussquadratureweights.Numerically,wehaveevaluatedD1:NandD)]TJ /F6 7.97 Tf 6.59 0 Td[(11:N,andfoundthat D)]TJ /F6 7.97 Tf 6.59 0 Td[(11:N12,(3) and W1 2D1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1i2p 2,1iN,(3) whereW1 2D1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1idenotestheithrowofthematrixW1 2D1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Althoughwedonothaveaproofofthesetworesults,wenumericallycomputedD1:NandD)]TJ /F6 7.97 Tf 6.59 0 Td[(11:NforNupto200andfoundthatkD)]TJ /F6 7.97 Tf 6.59 0 Td[(11:Nk1andW1 2D1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1i2aremonotonicallyincreasingfunctionsofNthatapproachthelimitsgivenabove. LetXibearowvectorformedbyxij,1jn.LetUibearowvectorwhosecomponentsareanapproximationtothecontrolu(ti),1iN.Ourdiscreteapproximationtothesystemdynamics_x(t)=f(x(t),u(t))isobtainedbyevaluatingthesystemdynamicsateachcollocationpointandreplacing_x(ti)=(_x1(ti),...,_xn(ti))byitsdiscreteapproximation)]TJ /F10 11.955 Tf 7.06 -9.68 Td[(_xN1(tk),...,_xNn(tk).By( 3 )andthenotationofXiandUi,wehave NXj=0DijXj=f(Xi,Ui),1iN.(3) SincetheGaussquadratureisexactforpolynomialsuptodegreeof2N)]TJ /F10 11.955 Tf 12.59 0 Td[(1[ 21 ],wehave xNj(1)=xNj()]TJ /F10 11.955 Tf 9.3 0 Td[(1)+Z1)]TJ /F6 7.97 Tf 6.59 0 Td[(1_xNj(t)dt=xNj()]TJ /F10 11.955 Tf 9.29 0 Td[(1)+NXi=1!i_xNj(ti),1jn, (3) 66

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where!i,1iN,aretheGaussquadratureweights.LetXN+1denotethevectorformedbyxNj(1),1jn.By( 3 )( 3 ),thestateatthenaltimecanbeapproximatedby XN+1=X0+NXi=1!if(Xi,Ui).(3) By( 3 )and( 3 ),thecontinuous-timenonlinearoptimalcontrolproblemcanbeapproximatedbythefollowingnite-dimensionalnonlinearprogrammingproblem minimizeC(XN+1)subjecttoPNj=0DijXj=f(Xi,Ui),1iN,XN+1=X0+PNi=1!if(Xi,Ui),X0=x0.9>>>>>>>=>>>>>>>;(3) Wenowdeveloptherst-orderoptimalityconditionfor( 3 ).LetbeanNbynmatrixofLagrangemultiplierswhosei-throwiisassociatedwithsystemdynamicPNj=0DijXj=f(Xi,Ui),andletN+1bearowvectorofLagrangemultipliersassociatedwiththeequationforXN+1.TheLagrangianassociatedwith( 3 )is L(,X,U)=C(XN+1)+NXi=1*i, NXj=0DijXj)]TJ /F4 11.955 Tf 11.96 0 Td[(f(Xi,Ui)!++*N+1, XN+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X0)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!if(Xi,Ui)!+, (3) andtherst-orderoptimalityconditionsare: NXj=1Djij=(i+N+1!i)rXf(Xi,Ui),1iN, (3) N+1=rXC(XN+1), (3) rUh(i+N+1!i),f(Xi,Ui)i=0,1iN. (3) 67

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Inordertorelatethecontinuousrstordercondition( 3 )( 3 )tothediscretecondition( 3 )( 3 ),weintroducetransformedadjointvariables: i=1 !ii+N+1when1iN,andN+1=N+1.(3) Bysubstituting( 3 )into( 3 )and( 3 ),wehave, N+1=rXC(XN+1),andrUH(Xi,Ui,i)=0,1iN.(3) LetDybetheNbyN+1matrixdenedby Dyij=)]TJ /F14 11.955 Tf 10.49 8.09 Td[(!j !iDji,1iN,1jN, (3) Dyi,N+1=)]TJ /F7 7.97 Tf 16.63 14.94 Td[(NXj=1Dyij,1iN. (3) By( 3 )and( 3 ),( 3 )becomes N+1Xj=1Dyijj=rXH(Xi,Ui,i),1iN.(3)( 3 )and( 3 )togetherconsistsoftherstorderoptimalityconditionsofthenonlinearprogrammingproblem( 3 ).Theyhavethesamestructureasthenecessaryconditionsforthecontinuouscontrolproblem. LetDy1:NdenotetherstNcolumnsofDy.ThefollowingpropositionsofDyandDy1:Nareimportantfortheconvergenceproofinthenextsection.Fortheproofofthefollowingpropositions,pleasesee[ 26 ]. Proposition3.2. ThematrixDyisadifferentiationmatrixforspaceofpolynomialsofdegreeN.Moreprecisely,ifqisapolynomialofdegreeatmostNandq2RN+1isavectorwithi-thcomponentqi=q(ti),1iN+1,then (Dyq)i=_q(ti),1iN. Proposition3.3. ThematrixDy1:Nisinvertible. 68

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Let1denoteavectorwhosecomponentsareallequalto1.WehavethefollowingpropositionofDy. Proposition3.4. DyN+1=)]TJ /F4 11.955 Tf 9.3 0 Td[(Dy1:N1;equivalently,)]TJ /F4 11.955 Tf 9.3 0 Td[(Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(11:NDyN+1=1. Proof. BythedenitionofDyin( 3 )and( 3 ),wehaveDyN+1=)]TJ /F4 11.955 Tf 9.3 0 Td[(Dy1:N1.SinceDy1:NisinvertiblebyProposition 3.3 ,wehave)]TJ /F4 11.955 Tf 9.3 0 Td[(Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(11:NDyN+1=1. Inouranalysisofconvergence,weneedtoboundbothDy)]TJ /F6 7.97 Tf 12.7 0 Td[(11:NandW1 2Dy1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Numerically,wehaveevaluatedDy1:NandDy)]TJ /F6 7.97 Tf 12.7 0 Td[(11:N,andfoundthat Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(11:N12,(3) and W1 2Dy1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1i2p 2,1iN,(3) whereW1 2Dy1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1idenotestheithrowofthematrixW1 2Dy1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Althoughwedonothaveaproofofthesetworesults,wenumericallycomputedDy1:NandDy)]TJ /F6 7.97 Tf 12.71 0 Td[(11:NforNupto200andfoundthatkDy)]TJ /F6 7.97 Tf 12.7 0 Td[(11:Nk1andW1 2Dy1:N)]TJ /F6 7.97 Tf 6.58 0 Td[(1i2aremonotonicallyincreasingfunctionsofNthatapproachthelimitsgivenabove. Inordertoanalyzethediscreteproblem( 3 ),weneedtointroducediscreteanaloguesoftheL1norms.Inparticular,forasequencez=z1,z2,...,zNwhosei-thelementisavectorzi2Rn,wedenetheL1normasthefollowing: kzk1=sup1iNjzij, wherejjistheEuclideannormforvectors.WedeneX=(X0,X1,...,XN+1),U=(U1,...,UN)and=(1,...,N+1).BythedenitionofL1norm,wehave kXk1=sup0iN+1jXij,kUk1=sup1iNjUijandkk1=sup1iN+1jij. 69

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3.3ConvergenceResult Let(x(t),u(t))bethelocalsolutionofthecontinuousoptimalproblem( 3 )and(t)betheassociatedcostate.WedenevectorsequencesX,UandbyXi=x(ti),0iN+1,Ui=u(ti),1iNandi=(ti),1iN+1.Weshowthatundersuitableassumptions,thenonlinearprogrammingproblem( 3 )hasanextremepoint)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XN,UNandtheassociatedLagrangemultiplierNwhichconvergesto(X,U)andexponentiallyfastintheL1norm. Theorem3.1. Suppose( 3 )hasalocalminimizerxandu,andthereexistssuchthatthePontryagin'sminimumprincipleholds.Iftheassumptions(A1)(A3)hold,then( 3 )hasanextremepoint)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XN,UNandassociatedLagrangemultiplierNsatisfying kXN)]TJ /F4 11.955 Tf 11.95 0 Td[(Xk1+kUN)]TJ /F4 11.955 Tf 11.96 0 Td[(Uk1+kN)]TJ /F20 11.955 Tf 11.95 0 Td[(k1c Nk)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2,(3) whereNisthenumberofGausspointsandcisaconstantindependentofN. 3.4AbstractSetting TheproofofTheorem 3.1 isbasedon[ 14 ,Proposition3.1].Forourconvergenceproof,werestatethepropositioninthefollowingform: Proposition3.5. LetXbeaBanachspaceandYbealinearnormedspacewiththenormsinbothspacesdenotedkk.LetT:X7)165(!YwithTcontinuouslyFrechetdifferentiableinBr()forsome2Xandr>0.Supposethatthefollowingconditionsholdforsomescalars"and, (P1)krT())-222(rT()k"forall2Br(). (P2)rT())]TJ /F6 7.97 Tf 6.58 0 Td[(1issingle-valuedandLipschitzcontinuouswithLipschitzconstant. If"<1andkT()k(1)]TJ /F14 11.955 Tf 11.97 0 Td[(")r=,thenthereexistsaunique2Br()suchthatT()=0.Moreover,wehavetheestimate k)]TJ /F14 11.955 Tf 11.96 0 Td[(k 1)]TJ /F14 11.955 Tf 11.95 0 Td[("kT()k.(3) 70

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WeapplyProposition 3.5 totherstorderconditions( 3 )and( 3 ).WeshowthatwhenNissufcientlarge,theassumptionsofthetheoremaresatisedwithconstantsindependentofN.First,weneedtospecifythenormedspaceX,YandthemapT.ThespaceXconsistsof3-tupleswhosecomponentsarevectorsequences=(X,U,).ThenormisthediscreteL1normgivenby kk1=k(X,U,)k1=maxfkXk1,kUk1,kk1g.(3) ThemappingTisselectedinthefollowingway: T(X,U,)=0BBBBBBBBBB@PNj=0DijXj)]TJ /F4 11.955 Tf 11.96 0 Td[(f(Xi,Ui),1iN,XN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X0)]TJ /F8 11.955 Tf 11.95 8.97 Td[(PNi=1!if(Xi,Ui),PN+1j=1Dyijj+rXH(Xi,Ui,i),1iN,rUH(Xi,Ui,i),1iN,N+1)-222(rXC(XN+1)1CCCCCCCCCCA.(3) LetYdenotethespaceassociatedwiththevecomponentsofT.Yisaspaceof5-tuplesofnitesequencesinL1RnL1L1Rn.Let=(X,U,).Then rT()=0BBBBBBBBBB@PNj=0DijXj)]TJ /F4 11.955 Tf 11.95 0 Td[(XiATi)]TJ /F4 11.955 Tf 11.95 0 Td[(UiBTi,1iN,XN+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X0)]TJ /F8 11.955 Tf 11.95 8.97 Td[(PNi=1!i(XiATi+UiBTi)PN+1j=1Dyijj+iATi+XiQTi+UiSTi,1iN,iBi+XiSTi+UiRTi,1iNN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(XN+1VT1CCCCCCCCCCA.(3) where Ai=rxf(x(ti),u(ti)),Bi=ruf(x(ti),u(ti)),Qi=rxxH(x(ti),u(ti),(ti)),Si=ruxH(x(ti),u(ti),(ti)),Ri=ruuH(x(ti),u(ti),(ti)),V=r2xC(x(1)).9>>>>=>>>>;(3) 71

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3.5ApproximationPreliminaries ToproveTheorem 3.1 ,weneedtoshowtheassumptionsofProposition 3.5 holdforNsufcientlarge.TheestimateofTheorem 3.1 isaconsequenceof( 3 ).InverifyingtheassumptionsofProposition 3.5 ,weutilizeseveralapproximationpropertiesofLagrangeinterpolationpolynomials.Inthissection,wepulltogetherthesepropertiesthatareexploitedthroughouttheanalysis. Lemma1. If`NistheLebesgueconstantdenedby `N=maxt2[)]TJ /F6 7.97 Tf 6.59 0 Td[(1,1]NXj=0jLj(t)j,(3) whereLj(t)istheLagrangeinterpolationpolynomialdenedin( 3 ),then`N=ON1 2. Proof. SeeTheorem 2.4 Lemma2. Foranyy2C1[)]TJ /F10 11.955 Tf 9.29 0 Td[(1,1],letyN2PNbetheinterpolatingpolynomialdenedasyN(ti)=PNi=0y(ti)Li(t),whereLi(t)isdenedin( 3 ).Then, _y)]TJ /F10 11.955 Tf 13.64 0 Td[(_yN1(1+2N2`N)infq2PN)]TJ /F9 5.978 Tf 5.76 0 Td[(1k_y)]TJ /F3 11.955 Tf 11.96 0 Td[(qk1, where`NistheLebesgueconstantdenedin( 3 ). Proof. SeeTheorem 2.1 Lemma3. (Jackson'sTheorem).Lety(t)2Cjandn>j0,then infp2Pnky)]TJ /F3 11.955 Tf 11.95 0 Td[(pk12cjky(j)k1 nj,(3) wherecj=6j+1ej(1+j))]TJ /F6 7.97 Tf 6.58 0 Td[(1. Proof. SeeTheorem 2.2 72

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3.6AnalysisofResidualandStationarity Inthissection,webegintoshowthatProposition 3.5 isapplicabletotheGausscollocationmethodbyverifyingassumptions(P1)andestimatingkT()k.FirstweestimatekT()k. Lemma4. Ifsmoothnessholds,thenthereexitsaconstantcindependentofN,suchthat kT()kc Nk)]TJ /F9 5.978 Tf 7.79 3.25 Td[(5 2,(3) wherekisdenedinassumption(A1). Proof. ThelasttwocomponentsofT()are0.Sincebytherstorderoptimalityconditionsforthecontinuous-timecontrolproblem( 3 )and( 3 ): (1))-222(rxC(x(1))=0,ruH(x(ti),u(ti),(ti))=0, NowweanalyzetherstcomponentsofT().LetxN(t)denotethevectorwhosejthcomponentis:xNj(t)=PNi=0xijLi(t),wherexij=xj(ti).BythedenitionofD,wehaveDi0x(t0)+Di1x(t1)++DiNx(tN)=_xN(ti).Sincexsatisesthedynamicf(x(ti),u(ti))=_x(ti),Hence,wehave Di0x(t0)+Di1x(t1)++DiNx(tN))]TJ /F4 11.955 Tf 11.95 0 Td[(f(x(ti),u(ti))=_xN(ti))]TJ /F10 11.955 Tf 13.2 0 Td[(_x(ti),(3) Sincex(t)2Ck+1[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1],forsomek3,wehave_x(t)2Ck[)]TJ /F10 11.955 Tf 9.29 0 Td[(1,1].ByLemmas 1 3 ,( 3 )canbeboundedasfollows: _xNj)]TJ /F10 11.955 Tf 13.54 0 Td[(_xj11+N2ON1 2infq2PN)]TJ /F9 5.978 Tf 5.76 0 Td[(1k_xj)]TJ /F3 11.955 Tf 11.95 0 Td[(qk11+N2ON1 2c(xj)(k+1)1 Nk=c(xj)(k+1)1O1 Nk)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2+1 Nkc Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. (3) 73

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Sinceweareusingthesup-norm,( 3 )yields _xN(ti))]TJ /F10 11.955 Tf 13.2 0 Td[(_x(ti)=vuut nXj=1_xNj(ti))]TJ /F10 11.955 Tf 13.53 0 Td[(_xj(ti)2vuut nXj=1c Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 22c Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2 (3) Since( 3 )holdsforeachi,i=1,...,N,wededucethatthesup-normoftherstcomponentofT()isboundedbyc Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. NowweestimatethesecondcomponentofT(): x(1))]TJ /F4 11.955 Tf 11.95 0 Td[(x()]TJ /F10 11.955 Tf 9.29 0 Td[(1))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!if(x(ti),u(ti))(3) SinceN-pointGaussquadratureisexactforpolynomialofdegreeupto2N)]TJ /F10 11.955 Tf 11.95 0 Td[(1,wehave xN(1))]TJ /F4 11.955 Tf 11.96 0 Td[(xN()]TJ /F10 11.955 Tf 9.3 0 Td[(1))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1!i_xN(ti)=xN(1))]TJ /F4 11.955 Tf 11.95 0 Td[(xN()]TJ /F10 11.955 Tf 9.3 0 Td[(1))]TJ /F8 11.955 Tf 11.95 16.27 Td[(Z1)]TJ /F6 7.97 Tf 6.59 0 Td[(1_xN(t)dt=0.(3) Wesubtract( 3 )from( 3 )toobtain x(1))]TJ /F4 11.955 Tf 11.95 0 Td[(xN(1)+NXi=1!i)]TJ /F10 11.955 Tf 6.73 -9.68 Td[(_xN(ti))]TJ /F10 11.955 Tf 13.2 0 Td[(_x(ti)(3) By( 3 ),_xN(ti))]TJ /F10 11.955 Tf 13.21 0 Td[(_x(ti)isboundedbyc Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2.Wenowestimatex(1))]TJ /F4 11.955 Tf 11.96 0 Td[(xN(1).Letq2PN,thenq(t)=PNi=0q(ti)Li(t),whereLi(t)isdenedin( 3 ).Specially,fort=1,wehaveq(1)=PNi=0q(ti)Li(1).Similarly,forthej-thcomponentofxN(1),wehave 74

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xNj(1)=PNi=0xNj(ti)Li(1)=PNi=0xj(ti)Li(1).Hence, xj(1))]TJ /F3 11.955 Tf 11.95 0 Td[(xNj(1)=xj(1))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=0xj(ti)Li(1)=xj(1))]TJ /F3 11.955 Tf 11.95 0 Td[(q(1)+NXi=0q(ti)Li(1))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=0xj(ti)Li(1)=xj(1))]TJ /F3 11.955 Tf 11.95 0 Td[(q(1))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=0(xj(ti))]TJ /F3 11.955 Tf 11.96 0 Td[(q(ti))Li(1)xj(1))]TJ /F3 11.955 Tf 11.95 0 Td[(q(1)+kxj)]TJ /F3 11.955 Tf 11.96 0 Td[(qk1NXi=0jLi(1)jkxj)]TJ /F3 11.955 Tf 11.96 0 Td[(qk1(1+`N) (3) SinceqisarbitrarilychoseninPN,( 3 )yields jxj(1))]TJ /F3 11.955 Tf 11.95 0 Td[(xNj(1)jinfq2PNkxj)]TJ /F3 11.955 Tf 11.96 0 Td[(qk1(1+`N).(3) ByLemma 2 ,wehave infq2PNkxj)]TJ /F3 11.955 Tf 11.95 0 Td[(qk1ck(xj)(k+1)k1 Nk+1.(3) Wecombine( 3 )and( 3 )toobtain xj(1))]TJ /F3 11.955 Tf 11.95 0 Td[(xNj(1)ck(xj)(k+1)k1 Nk+1(1+`N)c Nk+1 2.(3) Hence,by( 3 )wehave x(1))]TJ /F4 11.955 Tf 11.96 0 Td[(xN(1)=vuut nXj=1xj(1))]TJ /F3 11.955 Tf 11.95 0 Td[(xNj(1)2c Nk+1 2.(3) 75

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Then,by( 3 )and( 3 ),( 3 )isboundedby x(1))]TJ /F4 11.955 Tf 11.96 0 Td[(xN(1)+NXi=1!i)]TJ /F10 11.955 Tf 6.72 -9.68 Td[(_xN(ti))]TJ /F10 11.955 Tf 13.2 0 Td[(_x(ti)x(1))]TJ /F4 11.955 Tf 11.96 0 Td[(xN(1)+NXi=1!i_xN(ti))]TJ /F10 11.955 Tf 13.2 0 Td[(_x(ti)c Nk+1 2+c Nk)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2NXi=1!ic Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. NowweestimatethethirdcomponentofT().LetNj(t)denotetheN-thdegreepolynomialsatisfyingNj(ti)=j(ti),for1iN,andletN(t)bethevectorwhosejthcomponentisNj(t).Thenthei-thentryofthethirdcomponentofT()equals Dyi1(t1)+Dyi2(t2)++DyiN+1(1)+rx(ti)f(x(ti),u(ti))T=_N(ti))]TJ /F10 11.955 Tf 14.22 2.65 Td[(_(ti), Bytheassumption,(t)2Ck+1[)]TJ /F10 11.955 Tf 9.29 0 Td[(1,1],forsomek3.ByasimilarargumentastheestimateoftherstcomponentofT(),wegettheL1normofthecostateresidualisboundedbyc Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. CombiningtheestimateofthevecomponentsofT(),wecompletetheproofof( 3 ). Next,weestablishthecondition(P2)oftheProposition 3.5 Lemma5. Ifsmoothnessholds,theforeach">0,thereexistsr>0suchthatforall2Br(),then krT())-222(rT()k,(3) wherekkisthematrixnorminducedbytheL1normonXandY,andrisindependentofN. Proof. Throughouttheproof,weneedtoconsiderthedifferencebetweenderivativesoffandHevaluatedat(X,U,)andthesamederivativesevaluatedatapoint(X,U,). 76

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Forfurtherreferences,letAandBbeblockdiagonalmatriceswithi-thblocksgivenby Ai=Ai)-222(rXf(Xi,Ui)andBi=Bi)-222(rUf(Xi,Ui),(3) whereAi=rxf(Xi,Ui)andBi=ruf(Xi,Ui).Similarly,letQ,SandRbeblockdiagonalmatriceswiththeithblocksgivenby Qi=rXXH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.96 0 Td[(Qi,Si=rXUH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.95 0 Td[(Si,Ri=rXXH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.96 0 Td[(Ri,9>>>>=>>>>;(3) whereQi=rxxH(Xi,Ui,i),Si=ruxH(Xi,Ui,i),andRi=ruuH(Xi,Ui,i). LetrT()idenotetheiblockofrT()appearingin( 3 ).Withthisnotation,(rT())-222(rT())1=A B 0, Bythedenitionofmatrixnorm,wehave k(rT())-222(rT())1k=max1iNmaxkyk1kvk1kAiy+Bivk2max1iNmaxkyk1kAiyk2+maxkvk1kBivk2=max1iN(kAik2+kBik2).max1iN0B@krXf(Xi,Ui))-222(rxf(Xi,Ui)k2+krUf(Xi,Ui))-222(ruf(Xi,Ui)k21CA, (3) where( 3 )isby( 3 ).Inordertomake( 3 )smallerthan",wechoosearadiusr,suchthatk)]TJ /F20 11.955 Tf 13.6 0 Td[(k1
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ThesecondcomponentofrT())-352(rT()isarowvectorwhoserstNcomponentsare !i(rXf(Xi,Ui))]TJ /F4 11.955 Tf 11.95 0 Td[(Ai),1iN, andwhosenextNcomponentsare !i(rUf(Xi,Ui))]TJ /F4 11.955 Tf 11.96 0 Td[(Bi),1iN. Dene !=0BBBB@!1In...!NIn1CCCCA. Thenwecanwrite(rT())-222(rT())2intermsof!and(rT())-222(rT())1: (rT())-222(rT())2=!(rT())-222(rT())1. Takethematrixnormonbothsidesoflastequation,wehave k(rT())-222(rT())2k=k!(rT())-222(rT())1kk!kk(rT())-222(rT())1k. (3) Since!i2foreachi,( 3 )becomes k(rT())-221(rT())2k2k(rT())-221(rT())1k. Sincek(rT())-333(rT())1k<"wheniscloseto,wecanguaranteethatk(rT())-222(rT())2k<"wheniscloseto. ThethirdcomponentofrT())-222(rT()isamatrixofthefollowingform: rT())-221(rT()3=Q S A, 78

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whereAisdenedin( 3 ),andQandSaredenedin( 3 ).Bythesimilarargumentasfor(rT())-222(rT())1,weobtain k(rT())-222(rT())3kmax1iN0B@krXf(Xi,Ui))]TJ /F4 11.955 Tf 11.96 0 Td[(Aik2+krXXH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.95 0 Td[(Qik2+krXUH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.95 0 Td[(Sik2.1CA. Bychoosingrsufcientlysmall,wehavek(rT())-222(rT())3k<"fork)]TJ /F20 11.955 Tf 11.95 0 Td[(k1
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(p,s,q,r,z)2Y,weshowthatthereexistsaunique2Xsuchthat rT()+=0,(3) andweobtainaboundforkk1intermsofkk1,wherekk1isdenedin( 3 )and kk1=maxfkpk1,jsj,kqk1,krk1,jzjg.(3) Werstrelatethesolutionof( 3 )tothesolutionofaquadraticprogrammingprobleminwhichtheparameterappearsintheconstraintsandinthecostfunction.WeshowthatthequadraticprogrammingproblemhasauniquesolutiondependingLipschitzcontinuouslyontheparameter. By( 3 ),( 3 )hasthefollowingform: 0BBBBBBBBBB@PNj=0DijXj)]TJ /F4 11.955 Tf 11.95 0 Td[(XiATi)]TJ /F4 11.955 Tf 11.95 0 Td[(UiBTi+pi=0,1iN,XN+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X0)]TJ /F8 11.955 Tf 11.96 8.96 Td[(PNi=1!i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XiATi+UiBTi+s=0,PN+1j=1Dyijj+iATi+XiQTi+UiSTi+qi=0,1iN,iBi+XiSTi+UiRTi+ri=0,1iN,N+1)]TJ /F4 11.955 Tf 11.95 0 Td[(XN+1VT+z=0.1CCCCCCCCCCA (3) Wedenethefollowingcolumnvector: X=0BBBBBBB@XT1XT2...XTN1CCCCCCCA,U=0BBBBBBB@UT1UT2...UTN,1CCCCCCCAq=0BBBBBBB@!1qT1!2qT2...!NqTN1CCCCCCCA,r=0BBBBBBB@!1rT1!2rT2...!NrTN1CCCCCCCA. Letusconsiderthefollowingquadraticprogrammingproblem: minimizeB(X,U)+qTX+rTU)]TJ /F4 11.955 Tf 11.96 0 Td[(zXTN+1subjecttoPNj=0DijXj=XiATi+UiBTi)]TJ /F4 11.955 Tf 11.95 0 Td[(pi,1iN,XN+1=X0+PNi=1!i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XiATi+UiBTi)]TJ /F4 11.955 Tf 11.95 0 Td[(s,X0=x0,9>>>>>>>=>>>>>>>;(3) 80

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where B(X,U)=1 2 XN+1VXTN+1+NXi=1!i)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XiQiXTi+2XiSiUTi+UiRiUTi!.(3) Inthefollowing,werstshowthatBisstronglyconvex,hence( 3 )hasauniquesolution.MoreovertheKKTconditionsassociatewith( 3 )canbeexpressedintheformof( 3 ).SinceBisconvex,itfollowsthatanysolutionof( 3 )alsoyieldsasolutionof( 3 )(aswellasthemultipliersassociatedwiththeconstraints). Lemma6. Ifthecoercivityholds,thenthequadraticformBdenedin( 3 )isstronglyconvex. Proof. BythecoercivityassumptionandthedenitionofmatricesQi,SiandRiin( 3 ),wehave XN+1VXTN+1jXN+1j2, and XiQiXTi+2XiSiUTi+UiRiUTi)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jXij2+jUij2,1iN. Hence, B(X,U)jXN+1j2+NXi=1!i)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jXij2+jUij2(3) Ifwedenethe!-normsofXandUasfollows: kXk2!=NXi=1!ijXij2+jXN+1j2andkUk2!=NXi=1!ijUij2.(3) Itfollowsfrom( 3 )thatB(X,U))]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kXk2!+kUk2!.Hence,Bisstronglyconvex. Lemma7. Thequadraticprogrammingproblem( 3 )and( 3 )haveidenticaluniquesolution. 81

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Proof. TheLagrangianassociatewith( 3 )is L(,X,U)=B(X,U)+qTX+rTU)]TJ /F4 11.955 Tf 11.95 0 Td[(zXTN+1)]TJ /F7 7.97 Tf 16.63 14.94 Td[(NXi=1i NXj=0DijXj)]TJ /F4 11.955 Tf 11.96 0 Td[(XiATi)]TJ /F4 11.955 Tf 11.96 0 Td[(UiBTi+pi!T)]TJ /F4 11.955 Tf 9.29 0 Td[(N+1 XN+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X0)]TJ /F7 7.97 Tf 17.29 14.95 Td[(NXi=1!i(XiATi+UiBTi)+s!T. WenowformulatetheKKToptimalityconditionsfor( 3 ).TheseconditionscorrespondtosettingthederivativeoftheLagrangianwithrespecttoX,Uandtozero. ThepartialderivativeoftheLagrangianwithrespecttoXiis: 0=rXiL(,X,U)=!i)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XiQTi+UiSTi)]TJ /F7 7.97 Tf 17.29 14.95 Td[(NXj=1jDji+iATi+!iN+1ATi+!iqi=!i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XiQTi+UiSTi)]TJ /F7 7.97 Tf 17.29 14.94 Td[(NXj=1j !j!j !iDji!i+!ii !i+N+1ATi+!iqi. AftersubstitutingforDjiintermsofDyijusing( 3 )andforiintermsofiusing( 3 ),weobtain )]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XiQTi+UiSTi+iATi+NXj=1(i)]TJ /F20 11.955 Tf 11.96 0 Td[(N+1)Dyij+qi=0,(3) since!i6=0.By( 3 ),wehave NXj=1(i)]TJ /F20 11.955 Tf 11.96 0 Td[(N+1)Dyij=NXj=1Dyiji+DyiN+1N+1=N+1Xj=1Dyiji. (3) Wesubstitute( 3 )into( 3 )toobtain )]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XiQTi+UiSTi+iATi+N+1Xj=1Dyiji+qi=0, whichisthethirdequationin( 3 ). 82

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ThepartialderivativeoftheLagrangianwithrespecttoXN+1is: 0=rXN+1L(,X,U)=XN+1VT)]TJ /F4 11.955 Tf 11.96 0 Td[(N+1)]TJ /F4 11.955 Tf 11.96 0 Td[(z=XN+1VT)]TJ /F20 11.955 Tf 11.96 0 Td[(N+1)]TJ /F4 11.955 Tf 11.95 0 Td[(z, whichyieldsthelastequationin( 3 ). ThepartialderivativeoftheLagrangianwithrespecttoUiis 0=rUiL(,X,U)=!i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XiSTi+UiRTi+iBTi+!iN+1BTi+!iri=!i)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XiSTi+UiRTi+!ii !i+N+1BTi+!iri=!i)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XiSTi+UiRTi+iBTi+ri, by( 3 ).Since!i6=0,wedivide!itoobtain XiSTi+UiRTi+iBTi+ri=0, whichisthefourthequationin( 3 ).ThepartialderivativeoftheLagrangianwithrespecttoiyieldstherstandsecondequationsof( 3 ). Hence,theKKTconditionsassociatewith( 3 )canbeexpressedintheformof( 3 ).SinceBisstronglyconvexbyLemma 6 ,( 3 )hasauniquesolution.Also,itfollowsthatanysolutionof( 3 )alsoyieldsasolutionof( 3 ).Hence,thequadraticprogrammingproblem( 3 )and( 3 )haveidenticaluniquesolution. Inthefollowing,weshowkrT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kisboundedbyaconstantwhichdoesnotdependonN.Ourproofisbasedon[ 30 ,Lemma1],whichisstatedbelow: Lemma8. Letbeasymmetric,continuousbilinearformdenedonanonempty,closedconvexsubsetKofaHilbertspaceV,andlethiVdenotetheHilbertspaceinnerproduct.Ifthereexist>0suchthat (w)]TJ /F3 11.955 Tf 11.96 0 Td[(v,w)]TJ /F3 11.955 Tf 11.96 0 Td[(v)hw)]TJ /F3 11.955 Tf 11.96 0 Td[(v,w)]TJ /F3 11.955 Tf 11.96 0 Td[(viVforallw,v2K, 83

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thenforany2V,thequadraticprogram minimize1 2(v,v))-222(h,viVsubjecttov2K(3) hasauniquesolutionw.Thissolutionistheuniquew2Kthatsatisesthevariationalinequality(w,v)]TJ /F3 11.955 Tf 12.3 0 Td[(w)h,v)]TJ /F3 11.955 Tf 12.29 0 Td[(wiVforallv2K.Ifwidenotesthesolutionof( 3 )correspondingto=i,fori=1andi=2,thenwehave kw1)]TJ /F3 11.955 Tf 11.95 0 Td[(w2kk1)]TJ /F14 11.955 Tf 11.96 0 Td[(2k, wherekkisthenorminducedbytheHilbertspaceinnerproduct. Proof. See[ 30 ,Lemma1]. TheapplicationofLemma 8 tothequadraticprogrammingproblem( 3 )proceedsasfollows.Werstshowthatiftheassumption(A3)holds,thenforallU2L1,thereexistsauniqueXfeasiblein( 3 ).Inotherwords,wecanwriteXintermsofU.Then,byLemma 8 ,weobtainaboundforkUk!intermsofkk!.TherelationbetweenXandUaregivenbythefollowingtwolemmas.First,wemakethefollowingnotations:D=D1:NIn,A=0BBBBBBB@A1A2...AN1CCCCCCCA,B=0BBBBBBB@B1B2...BN1CCCCCCCA,Q=0BBBBBBB@Q1Q2...QN1CCCCCCCA,S=0BBBBBBB@S1S2...SN1CCCCCCCA, 84

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X0=0BBBBBBB@D10XT0D20XT0...DN0XT01CCCCCCCA,p=0BBBBBBB@pT1pT2...pTN1CCCCCCCA, wheredenotestheKroneckerproduct. Lemma9. ThematrixI)]TJ /F10 11.955 Tf 13.82 2.66 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1Aisinvertibleand)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.56 2.66 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(112,assuming(A3)and( 3 )hold. Proof. ByProposition 3.1 and( 3 ),D1:NisinvertibleandkD)]TJ /F6 7.97 Tf 6.59 0 Td[(11:Nk1<2.Hence,D=D1:NInisalsoinvertible.D)]TJ /F6 7.97 Tf 6.59 0 Td[(1=D)]TJ /F6 7.97 Tf 6.58 0 Td[(11:NInandD)]TJ /F6 7.97 Tf 6.59 0 Td[(11=D)]TJ /F6 7.97 Tf 6.58 0 Td[(11:N1<2.Byassumption(A3),kAk11 4.Hence,kD)]TJ /F6 7.97 Tf 6.59 0 Td[(1Ak1kD)]TJ /F6 7.97 Tf 6.59 0 Td[(1k1kAk11 2.By[ 32 ,p.351],I)]TJ /F10 11.955 Tf 13.56 2.66 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1Aisinvertibleand(I)]TJ /F10 11.955 Tf 13.56 2.66 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A))]TJ /F6 7.97 Tf 6.58 0 Td[(112. Lemma10. If(A3)holds,thenforallU2L1,thereexistsauniqueXfeasiblein( 3 ).LetXbefeasiblein( 3 )andletX0bethestate,associatedwiththecontrolU0andperturbationsp0ands0,thenwehave X)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X01c)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!+kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!,(3) and XN+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X0N+1c)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!+kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!+js)]TJ /F4 11.955 Tf 11.95 0 Td[(s0j,(3) wherecisaconstantindependentofN. Proof. By( 3 ), (D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)X=BU)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X0)]TJ /F10 11.955 Tf 12.17 0 Td[(p.(3) ByLemma 9 ,D)]TJ /F10 11.955 Tf 13.21 2.65 Td[(A=D)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.56 2.65 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1Aisinvertible.Itfollowsfrom( 3 )that X=)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.65 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 6.73 -7.03 Td[(BU)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X0)]TJ /F10 11.955 Tf 12.17 0 Td[(p.(3) 85

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Let!=(!1In,!2In,...,!NIn).TheequationforXN+1in( 3 )canbewrittenas XTN+1=XT0+!AX+!BU)]TJ /F4 11.955 Tf 11.95 0 Td[(sT.(3) Substituting( 3 )into( 3 ),wehave XTN+1=!A)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.65 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1+!BU)]TJ /F10 11.955 Tf 13.33 0 Td[(!A)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.65 Td[(A)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F10 11.955 Tf 6.72 -7.03 Td[(X0+p+XT0)]TJ /F4 11.955 Tf 11.95 0 Td[(sT.(3) By( 3 )and( 3 ),weobtainthatforallU2L1,thereexistsauniqueXfeasiblein( 3 ).Now,weshowthat( 3 )and( 3 )hold.By( 3 ),wehave: X)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X0=)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.58 0 Td[(1B)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U0)]TJ /F8 11.955 Tf 11.96 9.69 Td[()]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.56 2.65 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A)]TJ /F6 7.97 Tf 6.58 0 Td[(1D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U0)]TJ /F8 11.955 Tf 11.95 9.68 Td[()]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.57 2.65 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(1D)]TJ /F6 7.97 Tf 6.58 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0). Wetaketheinnitynormofbothsidesofthelastequationtoobtain X)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X01)]TJ /F4 11.955 Tf 5.47 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(11D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 11.955 Tf 6.9 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U01+)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(11D)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)12)]TJ 5.48 .48 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U01+D)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)1, (3) where( 3 )isbyLemma 9 .Inthefollowing,weestimateD)]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U01andD)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)1separately.LetW=WIn,andWisthediagonalmatrixwiththeGaussquadratureweights!i,1iN,onitsdiagonal.Observethat D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0=hD)]TJ /F6 7.97 Tf 6.58 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2ihW1 2B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0i=Mv,(3) whereM=D)]TJ /F6 7.97 Tf 6.59 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2andv=W1 2B)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U0.BytheSchwartzinequality, kMvk1=maxijMivjkvk2maxikMik2,(3) whereMiistheithrowofM.SinceW1 2commuteswithB,wehave kvk2=W1 2B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U02=BW1 2)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U02ckU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!,(3) 86

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wherec=maxkBik2,1iN,andW1 2)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U02=kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!.ObservethatM=D)]TJ /F6 7.97 Tf 6.58 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2=W1 2D1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1In.Moreover,by( 3 ),eachrowofthematrixW1 2D1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1hasEuclidiannormsmallerthanp 2.Hence,M=W1 2D1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1InsatiseskMik2p 2.By( 3 ), kMvk1p 2kvk2.(3) Wecombine( 3 ),( 3 )and( 3 )toobtain D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U01ckU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!.(3) Bythesameanalysis,wehave D)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)=D)]TJ /F6 7.97 Tf 6.58 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2W1 2(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0).(3) and D)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)1p 2kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!.(3) Wecombine( 3 ),( 3 )and( 3 )tocompletetheproofof( 3 ). By( 3 ),thechangeXN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X0N+1inthestatesatises XTN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X0N+1T=!A)]TJ /F10 11.955 Tf 6.73 -7.03 Td[(X)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X0+!B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U0)]TJ /F8 11.955 Tf 11.95 9.68 Td[()]TJ /F4 11.955 Tf 5.48 -9.68 Td[(sT)]TJ /F4 11.955 Tf 11.95 0 Td[(s0. WetaketheEuclidiannormofbothsidestoobtain XTN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X0TN+1!A)]TJ /F10 11.955 Tf 6.72 -7.02 Td[(X)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X0+!B)]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0+sT)]TJ /F4 11.955 Tf 11.95 0 Td[(s0.(3) First,weobservethat !A)]TJ /F10 11.955 Tf 6.72 -7.03 Td[(X)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X0=j!1A1(X1)]TJ /F4 11.955 Tf 11.96 0 Td[(X01)++!NAN(XN)]TJ /F4 11.955 Tf 11.95 0 Td[(X0N)jcmaxijXi)]TJ /F4 11.955 Tf 11.96 0 Td[(X0ijj!1++!Nj=2cmaxijXi)]TJ /F4 11.955 Tf 11.95 0 Td[(X0ij, (3) 87

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wherec=maxikAik2,andj!1++!Nj=2.SincemaxijXi)]TJ /F4 11.955 Tf 11.95 0 Td[(X0ijnkX)]TJ /F10 11.955 Tf 12.62 2.66 Td[(X0k1,wherenisthedimensionofXi,itfollowsthat !A)]TJ /F10 11.955 Tf 6.73 -7.03 Td[(X)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X0ckX)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X0k1. (3) Next, !B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.65 Td[(U0=p !BW1 2)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U0p 2maxikBik2kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!,=ckU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!, (3) wherec=p 2maxikBik2.Wecombine( 3 ),( 3 )and( 3 )tocompletetheproofof( 3 ). AftersubstitutingforXandXN+1using( 3 )and( 3 )respectively,theobjectivefunctionin( 3 )becomeaquadraticinUandthelineartermintheobjectivefunctionisLU,where L=1 2X0)]TJ /F4 11.955 Tf 11.95 0 Td[(s)]TJ /F8 11.955 Tf 11.95 13.27 Td[(!A)]TJ /F10 11.955 Tf 7.09 -7.02 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 6.73 -7.02 Td[(X0+pTV)]TJ /F4 11.955 Tf 11.95 0 Td[(z!A)]TJ /F10 11.955 Tf 7.1 -7.02 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1+!B)]TJ /F8 11.955 Tf 19.26 13.27 Td[()]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F10 11.955 Tf 6.72 -7.03 Td[(X0+pTW1 2Q)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1B+S+qT)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.65 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1B+rT. (3) ByLemma 9 and( 3 ),theinnitynormofthematrix )]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.58 0 Td[(1=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(1D)]TJ /F6 7.97 Tf 6.58 0 Td[(1isboundedby4,andbythesmoothnessassumption,thematricesA,B,QandVarealluniformlybounded.LetUdenotetheoptimalsolutionof( 3 ),andletU0denotetheoptimalsolutionassociatedwiththeperturbation0=(p0,s0,q0,r0,z0).ByLemma 8 ,wehave kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!c(kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k!+kr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k!+js)]TJ /F4 11.955 Tf 11.95 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j),(3) 88

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wherecisaconstantindependentofNandkk!isdenedin( 3 ).Observethat kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!= NXi=1!ijpi)]TJ /F4 11.955 Tf 11.96 0 Td[(p0ij2!1 2max1iNjpi)]TJ /F4 11.955 Tf 11.96 0 Td[(p0ij NXi=1!i!1 2=p 2kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k1.(3) Similarly, kq)]TJ /F10 11.955 Tf 12.17 0 Td[(q0k!p 2kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k1andkr)]TJ /F10 11.955 Tf 11.14 0 Td[(r0k!p 2kr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k1.(3) Hence,( 3 )yields kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!c(kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k1+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k1+kr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k1+js)]TJ /F4 11.955 Tf 11.95 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j).(3) ThegoalofthissectionistoshowkrT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kisboundedbyaconstantwhichisindependentofN.However,( 3 )isnotsufcientforustoreachthisgoalsincethecontrolUliesinL1and( 3 )onlygivesaboundofthe!-normofU.Inthefollowing,weneedtoanalyzeofkU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k1.First,letusdene Dy=Dy1:NIn,DyN+1=DyN+1In,=0BBBB@T1...TN1CCCCA,(3) Now,weintroducethefollowinglemmas: Lemma11. If(A3)and( 3 )hold,thenthematrixDyandI+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1AareinvertiblewithDy)]TJ /F6 7.97 Tf 12.7 0 Td[(112and)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(112. Proof. ByProposition 3.3 and( 3 ),Dy1:NisinvertibleandkDy)]TJ /F6 7.97 Tf 12.7 0 Td[(11:Nk1<2.Hence,Dy=Dy1:NInisalsoinvertible.Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1=Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(11:NInandDy)]TJ /F6 7.97 Tf 12.7 0 Td[(11=Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(11:N1<2.Byassumption(A3),kAk11 4.Hence, kDy)]TJ /F6 7.97 Tf 12.7 0 Td[(1Ak1kDy)]TJ /F6 7.97 Tf 12.71 0 Td[(1k1kAk11 2.By[ 32 ,p.351],I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1Aisinvertibleand(I+Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(1A))]TJ /F6 7.97 Tf 6.58 0 Td[(112. Lemma12. If(A3)holds,thenforall(X,U)feasiblein( 3 ),thereexistsauniqueadjointvariablesuchthattheKKTconditionsof( 3 )hold.Letbetheadjoint 89

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variableassociatewith(X,U)andtheperturbationandlet0betheadjointvariableassociatewith(X0,U0)andtheperturbation0,thenwehave )]TJ /F10 11.955 Tf 12.83 2.66 Td[(01c)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!+js)]TJ /F4 11.955 Tf 11.95 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j,(3) and N+1)]TJ /F20 11.955 Tf 11.96 0 Td[(0N+1c)]TJ /F2 11.955 Tf 5.47 -9.69 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j,(3) wherecisaconstantindependentofN. Proof. ByLemma 7 ,theKKTconditionsofthequadraticprogrammingproblem( 3 )areidenticalto( 3 ).Bythelastequationof( 3 ), N+1=XN+1VT)]TJ /F4 11.955 Tf 11.95 0 Td[(z.(3) Withthenotationofthematricesin( 3 ),thethirdequationin( 3 )canbewrittenas )]TJ /F10 11.955 Tf 7.09 -7.03 Td[(Dy+A=)]TJ /F10 11.955 Tf 10.91 2.65 Td[(QX)]TJ /F10 11.955 Tf 12.47 2.65 Td[(SU)]TJ /F10 11.955 Tf 13.57 2.65 Td[(DyN+1TN+1)]TJ /F10 11.955 Tf 15.12 2.65 Td[(W)]TJ /F6 7.97 Tf 6.58 0 Td[(1q.(3) ByLemma 11 ,Dy+A=Dy)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1Aisinvertible.Itfollowsfrom( 3 )that =)]TJ /F10 11.955 Tf 19.26 0 Td[((Dy+A))]TJ /F6 7.97 Tf 6.59 0 Td[(1QX)]TJ /F10 11.955 Tf 11.95 0 Td[((Dy+A))]TJ /F6 7.97 Tf 6.58 0 Td[(1SU)]TJ /F10 11.955 Tf 11.95 0 Td[((Dy+A))]TJ /F6 7.97 Tf 6.58 0 Td[(1DyN+1TN+1)]TJ /F10 11.955 Tf 19.26 0 Td[((Dy+A))]TJ /F6 7.97 Tf 6.59 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1q. (3) By( 3 )and( 3 ),weobtainthatforall(X,U)feasiblein( 3 ),thereexistsauniqueadjointvariablesuchthattheKKTconditionsof( 3 )hold. Nowweshowthat( 3 )and( 3 )hold. By( 3 ),wehave N+1)]TJ /F20 11.955 Tf 11.96 0 Td[(0N+1=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XN+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X0N+1VT)]TJ /F10 11.955 Tf 11.96 -.17 Td[((z)]TJ /F4 11.955 Tf 11.95 0 Td[(z0). WetaketheEuclidiannormofbothsidesofthelastequationtoobtain N+1)]TJ /F20 11.955 Tf 11.96 0 Td[(0N+1cXN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X0N+1+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j, 90

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wherec=kVk2.Wecombinethiswith( 3 )toobtain N+1)]TJ /F20 11.955 Tf 11.95 0 Td[(0N+1c)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!+kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!+js)]TJ /F4 11.955 Tf 11.95 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j, whichis( 3 ). By( 3 ),wehave )]TJ /F10 11.955 Tf 12.82 2.65 Td[(0=)]TJ /F8 11.955 Tf 19.26 9.68 Td[()]TJ /F10 11.955 Tf 7.09 -7.03 Td[(Dy+A)]TJ /F6 7.97 Tf 6.58 0 Td[(1Q)]TJ /F10 11.955 Tf 6.72 -7.03 Td[(X)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X0)]TJ /F10 11.955 Tf 11.96 0 Td[((Dy+A))]TJ /F6 7.97 Tf 6.59 0 Td[(1S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U0)]TJ /F8 11.955 Tf 19.26 9.69 Td[()]TJ /F10 11.955 Tf 7.09 -7.03 Td[(Dy+A)]TJ /F6 7.97 Tf 6.58 0 Td[(1DyN+1)]TJ /F20 11.955 Tf 5.48 -9.69 Td[(TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[(0N+1T)]TJ /F8 11.955 Tf 19.26 9.69 Td[()]TJ /F10 11.955 Tf 7.09 -7.03 Td[(Dy+A)]TJ /F6 7.97 Tf 6.58 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0). (3) Since)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(Dy+A)]TJ /F6 7.97 Tf 6.59 0 Td[(1=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1A)]TJ /F6 7.97 Tf 6.58 0 Td[(1Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1,wesubstitute)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(Dy+A)]TJ /F6 7.97 Tf 6.58 0 Td[(1in( 3 )by)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(1Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1.Moreover,observethatDy)]TJ /F6 7.97 Tf 12.7 0 Td[(1DyN+1=Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(11:NDyN+1In=)]TJ /F4 11.955 Tf 9.3 0 Td[(1InbyProposition 3.4 .Wedenote1In=1.Itfollowsfrom( 3 )that )]TJ /F10 11.955 Tf 12.83 2.66 Td[(0=)]TJ /F10 11.955 Tf 19.26 0 Td[((I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1A))]TJ /F6 7.97 Tf 6.59 0 Td[(1Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1Q)]TJ /F10 11.955 Tf 6.73 -7.03 Td[(X)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X0)]TJ /F10 11.955 Tf 11.96 0 Td[((I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1A))]TJ /F6 7.97 Tf 6.59 0 Td[(1Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0+(I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1A))]TJ /F6 7.97 Tf 6.59 0 Td[(11)]TJ /F20 11.955 Tf 5.48 -9.68 Td[(TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[(0TN+1)]TJ /F10 11.955 Tf 19.26 0 Td[((I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1A))]TJ /F6 7.97 Tf 6.59 0 Td[(1Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.58 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0). Wetaketheinnitynormofbothsidesofthelastequationtoobtain )]TJ /F10 11.955 Tf 12.83 2.66 Td[(01(I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1A))]TJ /F6 7.97 Tf 6.59 0 Td[(1Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(1Q1X)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X01+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I+Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(11Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(1S)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.65 Td[(U01+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I+Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(111TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[(0TN+1+)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I+Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(11Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)1c)]TJ 5.48 .48 Td[(X)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X01+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U01+TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[(0TN+1+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)1, (3) wherethelastinequalityisbecause)]TJ /F4 11.955 Tf 5.47 -9.68 Td[(I+Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1A)]TJ /F6 7.97 Tf 6.58 0 Td[(112byLemma 11 91

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Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(11=Dy)]TJ /F6 7.97 Tf 12.71 0 Td[(11<2by( 3 ),andQ1isuniformlybounded.Inthefollowing,weestimateDy)]TJ /F6 7.97 Tf 12.7 0 Td[(1S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U01andDy)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)1.Observethat Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0=hDy)]TJ /F6 7.97 Tf 12.71 0 Td[(1W)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2ihW1 2S)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U0i. SinceDy)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2=W1 2Dy1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1In,andeachrowofthematrixW1 2Dy1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1hasEuclidiannormsmallerthanp 2by( 3 ),wededucethateachrowofthematrixDy)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2hasEuclidiannormsmallerthanp 2.SinceW1 2commuteswithS,wehave W1 2S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U02=SW1 2)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U02cU)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U0! wherec=maxkSik2,1iNandW1 2)]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U02=U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0!.Hence,bySchwartzinequality,wehave Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1S)]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U01cU)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0!.(3) Bythesameanalysis,wehave Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)=Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2W1 2W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0). and Dy)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)1p 2kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k!.(3) Wesubstitute( 3 )and( 3 )into( 3 )toobtain )]TJ /F10 11.955 Tf 12.82 2.65 Td[(01c)]TJ 5.48 .48 Td[(X)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X01+kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!+TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[(0TN+1+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k!.(3) By( 3 )and( 3 ),( 3 )reducesto )]TJ /F10 11.955 Tf 12.82 2.66 Td[(01c)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j, whichis( 3 ). Nowweintroducethefollowinglemma: 92

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Lemma13. LetUbefeasiblein( 3 ),andletU0bethecontrolassociatedwiththeperturbations0=(p0,s0,q0,r0,z0),thenwehave kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k1c(kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k1+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k1+kr)]TJ /F4 11.955 Tf 11.96 0 Td[(r0k1+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j),(3) wherecisaconstantindependentofN. Proof. Bythecoercivityassumption(A2),wehave: UiRiUTiUiUTi,for1iN. ThefourthequationofrT()+=0in( 3 )is iBi+XiSTi+UiRTi+ri=0,1iN.(3) Thisisthenecessaryoptimalityconditionforthequadraticprogrammingproblem minUi2Rm1 2UiRiUTi+)]TJ /F20 11.955 Tf 5.48 -9.68 Td[(iBi+XiSTi+riUTi. ByLemma 8 ,wehave jUi)]TJ /F4 11.955 Tf 11.96 0 Td[(U0ij(ji)]TJ /F20 11.955 Tf 11.95 0 Td[(0ij+jXi)]TJ /F4 11.955 Tf 11.96 0 Td[(X0ij+jri)]TJ /F4 11.955 Tf 11.95 0 Td[(r0ij),(3) wherejjdenotesEuclidiannorm.Inthefollowing,wegiveboundsofeachtermintherightsideof( 3 )intermsofk)]TJ /F20 11.955 Tf 11.96 0 Td[(0k1. BythedenitionofL1-norm,wehavejri)]TJ /F4 11.955 Tf 11.96 0 Td[(r0ijkr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k1.ObservethatjXi)]TJ /F4 11.955 Tf 11.95 0 Td[(X0ij=vuut nXj=1(xij)]TJ /F3 11.955 Tf 11.96 0 Td[(x0ij)2max1jnjxij)]TJ /F3 11.955 Tf 11.96 0 Td[(x0ijjp nkX)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X0k1p n. Wecombinethiswith( 3 )toobtain jXi)]TJ /F4 11.955 Tf 11.95 0 Td[(X0ijc)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!.(3) 93

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Inthesamefashionwehaveji)]TJ /F20 11.955 Tf 11.95 0 Td[(0ij=vuut nXj=1(ij)]TJ /F14 11.955 Tf 11.96 0 Td[(0ij)2max1jnjij)]TJ /F14 11.955 Tf 11.96 0 Td[(0ijjp nk)]TJ /F10 11.955 Tf 12.83 2.66 Td[(0k1p n. By( 3 ),wehave ji)]TJ /F20 11.955 Tf 11.96 0 Td[(0ij)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j.(3) Wecombine( 3 ),( 3 )and( 3 )toobtainthatforeachi,1iN,wehave jUi)]TJ /F4 11.955 Tf 11.96 0 Td[(U0ij)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j. SincekU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k1=max1iNjUi)]TJ /F4 11.955 Tf 11.96 0 Td[(U0ij,theaboveinequalityyields kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k1c)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j. Wecombinethiswith( 3 )( 3 )tocompletetheproof. Nowwegivethemainresultofthissection: Lemma14. Iftheassumption(A1)(A3)hold,thenkrT())]TJ /F6 7.97 Tf 6.58 0 Td[(1kisboundedbyaconstantindependentofN. Proof. Wecombine( 3 )and( 3 )with( 3 )( 3 )toobtain kX)]TJ /F4 11.955 Tf 11.95 0 Td[(X0k1c(kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k1+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k1+kr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k1+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j).(3) Wecombine( 3 )and( 3 )with( 3 )( 3 )toobtain k)]TJ /F20 11.955 Tf 11.95 0 Td[(0k1c(kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k1+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k1+kr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k1+js)]TJ /F4 11.955 Tf 11.95 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j).(3) By( 3 ),( 3 ),( 3 )andthedenitionofkk1in( 3 )andkk1in( 3 ),wehave k)]TJ /F20 11.955 Tf 11.95 0 Td[(0k1ck)]TJ /F20 11.955 Tf 11.95 0 Td[(0k1, whichshowsthatkrT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kc,isaconstantindependentofN. 94

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3.8ProofofTheorem 3.1 WenowcollectresultstoproveTheorem 3.1 usingProposition 3.5 andthecorrespondencewiththecontrolproblemdescribedinsection 3.4 .ReferringtoLemma 14 ,let=krT())]TJ /F6 7.97 Tf 6.59 0 Td[(1k,andchoosesmallenoughsuchthat<1.ChoosersmallenoughsuchthatbyLemma 5 ,krT())-221(rT()k,forall2Br().Finally,byLemma 4 ,forNsufcientlarge,kT()k(1)]TJ /F14 11.955 Tf 12.46 0 Td[()=.SincealltheconditionsofProposition 3.5 aresatised,weobtaintheconclusionof( 3 );WecombinethiswithLemma 4 toobtaintheestimateof( 3 )ofTheorem 3.1 3.9ConcludingRemarks InthisChapter,weprovetheconvergenceofaGausscollocationmethodappliedtoanunconstrainedcontrolproblem.Weshowthatundertheassumptionofcoercivityandsmoothness,thediscretenonlinearprogrammingproblemhasanextremepointandassociatedtransformedadjointvariablewhichconvergetothesolutionoftheoptimalcontrolproblemattherateO1 Nk)]TJ /F9 5.978 Tf 6.95 2.34 Td[(5 2,wherekisthenumberofcontinuousderivativesinthesolutionandNisthenumberoftheGaussquadraturepoints. 95

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CHAPTER4CONVERGENCEFOFARADAUCOLLOCATIONMETHODFORUNCONSTRAINEDOPTIMALCONTROL 4.1BackgroundofRadauCollocationMethod Inrecentyears,theRadaucollocationmethodhasbeenpopularlyusedasanumericalmethodforsolvingoptimalcontrolproblems.Itisadirectcollocationwheretheoptimalcontrolproblemistranscribedtoanonlinearprogrammingproblem(NLP)byparameterizingthestateandcontrolusingglobalpolynomialsandcollocatingthedifferential-algebraicequationsusingnodesobtainedfromaRadauquadraturescheme[ 25 26 ]. InthisChapter,weanalyzeaconvergenceoftheRadaucollocationmethod.Underassumptionsofcoercivityandsmoothness,weestablishanerrorestimateoftheformO1 Nk)]TJ /F9 5.978 Tf 6.96 2.35 Td[(5 2wherekisthenumberofcontinuousderivativesinthesolutionandNisthedegreeofthepolynomialsinthecollocationscheme.Herethenormisthediscretesup-norminRN.TheproofisbasedonanestimateofLebesgueconstantfortheRadaucollocationpointsgiveinChapter2,andtheabstracterrorestimatefordiscreteapproximationsgivenin[ 14 ]. 4.2TheControlProblemandRadauCollocationMethod Weconsiderthefollowingunconstrainedoptimalcontrolproblem: minimizeC(x(1))subjectto_x(t)=f(x(t),u(t)),t2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1]x()]TJ /F10 11.955 Tf 9.29 0 Td[(1)=x0,(4) wherethestatex(t)2Rn,_xd dtx,thecontrolu(t)2Rm,f:RnRm!Rn,C:Rn!Randx0istheinitialcondition,whichweassumeisgiven. Notation.ThroughouttheChapter,Ckisthecollectionofreal-valuedktimescontinuouslydifferentialfunctionsontheinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1].y(k)denotesthek-thderivativeofyfory2Ck,and_ydenotethederivativeofy.Letjjdenotesthe 96

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absolutevalueofscalarsortheEuclideannormofvectors.Letkk1denotestheuniformnormovertheinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1],thatis,kyk1=max)]TJ /F6 7.97 Tf 6.58 0 Td[(1t1jy(t)j,foranycontinuousfunctionyon[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1].PNdenotesthesetofpolynomialsofdegreeatmostN.Weassumeallvectorfunctionsoftimearerowvectors:x(t)=(x1(t),x2(t),...,xn(t)),u(t)=(u1(t),u2(t),...,um(t)).Givenvectorsaandb2Rn,thenotationha,bidenotestheinnerproductofaandbinRn.B(x)istheclosedballcenteredatxwithradius.ThetransposeofamatrixAisAT.Iff:Rn!Rm,thenrfisthembynJacobianmatrixwhosei-throwisrfi.Inparticular,thegradientofascalarvaluedfunctionisarowvector.LetL1denotethespaceofessentiallyboundedfunctionsandletW1,1denotethespaceofLipschitzcontinuousfunctionsdenedon[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1].Further,cisageneralconstantthathasdifferentvalueindifferentrelations. WenowpresenttheassumptionsthatareemployedintheanalysisoftheRadaucollocationschemeof( 4 ). (A1)Smoothness.Problem( 4 )hasalocalminimizer(x,u)whichliesinCk+1(Rn)L1(Rm),forsomek3.ThereexistsanopensetRnRmand>0suchthatB(x(t),u(t))foreveryt2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1].ThersttwoderivativesoffareLipschitzcontinuousin,andthersttwoderivativesofCareLipschitzcontinuousinB(x(1)).Furthermore,thereexitsassociatedcostate2Ck+1(Rn)and2Rnforwhichthefollowingequations(Pontryagin'sminimumprinciple)aresatisedforx=x,u=u,=,and=: ()]TJ /F10 11.955 Tf 9.3 0 Td[(1)=, (4) (1)=rC(x(1)), (4) _(t)=rxH(x(t),u(t),(t)), (4) 0=ruH(x(t),u(t),(t)). (4) whereHistheHamiltoniandenedbyH(x,u,)=h,f(x,u)i. 97

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(A2)Coercivity.Forsome>0,thesmallesteigenvaluesofthematricesbelowaregreaterthan: V=rxxC(x(1))and0B@Q(t)S(t)ST(t)R(t)1CAforallt2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1],(4) where Q(t)=rxxH(x(t),u(t),(t)),S(t)=ruxH(x(t),u(t),(t)),R(t)=ruuH(x(t),u(t),(t)). (A3).kA(t)k11 8forallt2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1],whereA(t)=rxf(x(t),u(t)). WenowintroducetheRadaucollocationmethod[ 25 26 ].ConsiderNRadauquadraturepoints(t1,t2,,tN)ontheinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1],witht1=)]TJ /F10 11.955 Tf 9.29 0 Td[(1andtN<+1.WeintroduceanadditionalnoncollocatedpointtN+1=1whichisusedtodescribetheapproximationtothestatevariable.TheLagrangeinterpolatingpolynomialsrelativetot1,t2,,tN+1aregivenby Li(t)=N+1Yj=1j6=it)]TJ /F3 11.955 Tf 11.95 0 Td[(tj ti)]TJ /F3 11.955 Tf 11.96 0 Td[(tj,i=1,N+1.(4) Thej-thcomponentofthestatexj(t)isapproximatedbythefollowingN-thdegreepolynomial: xNj(t)=N+1Xi=1xijLi(t).(4) wherexijisanapproximationtoxj(ti).Differentiatingtheseries( 4 )andevaluatingatthecollocationpointtk,k=1,2,,N,wehave _xNj(tk)=N+1Xi=1xij_Li(tk)=N+1Xi=1Dkixij,whereDki=_Li(tk).(4) LetDbetheNbyN+1matrixwhose(k,i)elementisDki.LetDjdenotethej-thcolumnofD,1jN+1,andletDj:kbethesubmatrixofDformedbycolumnsofjthroughk. 98

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Let1denoteavectorwhosecomponentsareallequalto1.Thefollowingpropositionisestablishedin[ 25 ]: Proposition4.1. ThematrixD1:NandD2:N+1obtainedbydeletingeithertherstorthelastcolumnofDareinvertible.Moreover,D1=0. Inouranalysisofconvergence,weneedtoboundbothD)]TJ /F6 7.97 Tf 6.58 0 Td[(12:N+1andW1 2D2:N+1)]TJ /F6 7.97 Tf 6.58 0 Td[(1,whereWisthediagonalmatrixwhosediagonalelements!i,1iN,aretheRadauquadratureweights.Numerically,wehaveevaluatedD2:N+1andD)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1,andfoundthat D)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+11=2,(4) and max1iNW1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1i2=p 2,(4) whereW1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1idenotestheithrowofthematrixW1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Althoughwedonothaveaproofofthesetworesults,wenumericallycomputedD2:N+1andD)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1forNupto200andfoundthatkD)]TJ /F6 7.97 Tf 6.58 0 Td[(12:N+1k1andW1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1i2areboundbytheconstantsgivenabove. LetXi(1iN+1)bearowvectorwithcomponentsxij,1jn.LetUi(1iN)bearowvectorwhosecomponentsareanapproximationtothecontrolu(ti),1iN.Ourdiscreteapproximationtothesystemdynamics_x(t)=f(x(t),u(t))isobtainedbyevaluatingthesystemdynamicsateachcollocationpointandreplacing_x(ti)=(_x1(ti),...,_xn(ti))byitsdiscreteapproximation(_xN1(tk),...,_xNn(tk)).By( 4 )andthenotationofXiandUi,wehave N+1Xj=1DijXj=f(Xi,Ui),1iN.(4) 99

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Thecontinuous-timenonlinearoptimalcontrolproblemcanbeapproximatedbythefollowingnite-dimensionalnonlinearprogrammingproblem minimizeC(XN+1)subjecttoPN+1j=1DijXj=f(Xi,Ui),1iN,X1=x0.9>>>>=>>>>;(4) Wenowdeveloptherst-orderoptimalityconditionfor( 4 ).LetbeanNbynmatrixofLagrangemultiplierswhosei-throwiisassociatedwithsystemdynamicPN+1j=1DijXj=f(Xi,Ui),andletbearowvectorofLagrangemultipliersassociatedwiththeinitialstatecondition.TheLagrangianassociatedwith( 4 )is L(,X,U)=C(XN+1)+NXi=1*i,f(Xi,Ui))]TJ /F7 7.97 Tf 11.96 14.94 Td[(N+1Xj=1DijXj++h,x0)]TJ /F4 11.955 Tf 11.96 0 Td[(X1i,(4) andtherst-orderoptimalityconditionsare: NXj=1Djij=irXf(Xi,Ui),2iN, (4) NXj=1Dj1j=1rXf(X1,U1))]TJ /F20 11.955 Tf 11.96 0 Td[(, (4) rC(XN+1)=DTN+1, (4) irUf(Xi,Ui)=0,1iN. (4) Inordertorelatethecontinuousrst-orderconditions( 4 )( 4 )tothediscretecondition( 4 )( 4 ),weintroducetransformedadjointvariables: i=1 !ii,for1iN,andN+1=DTN+1.(4) LetDybeanNbyNmatrixdenedby Dy11=)]TJ /F3 11.955 Tf 9.29 0 Td[(D11)]TJ /F10 11.955 Tf 16.13 8.09 Td[(1 !1andDyij=)]TJ /F14 11.955 Tf 10.49 8.09 Td[(!j !iDjiotherwise.(4) 100

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Bysubstituting( 4 )and( 4 )into( 4 )( 4 ),wehave rC(XN+1)=N+1, (4) NXj=1Dyijj=rXH(Xi,Ui,i)2iN, (4) NXj=1Dy1jj=rXH(X1,U1,1)+1 !1()]TJ /F20 11.955 Tf 11.96 0 Td[(1), (4) rUH(Xi,Ui,i)=0,1iN. (4) ByProposition 4.1 ,D1=0,whichimplies DN+1=)]TJ /F7 7.97 Tf 16.63 14.95 Td[(NXj=1Dj.(4) BythedenitionofN+1in( 4 )and( 4 ),wehave N+1=NXi=1iDi,N+1=)]TJ /F7 7.97 Tf 16.64 14.94 Td[(NXi=1NXj=1iDij=1 !1+NXi=1NXj=1iDyji!j !i=1 !1+NXi=1NXj=1jDyij!i !j (4) =1+NXi=1NXj=1!ijDyij (4) =)]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1!irXH(Xi,Ui,i), (4) where( 4 )isby( 4 ),( 4 )followsfrom( 4 )and( 4 )isby( 4 )and( 4 ).( 4 )-( 4 )and( 4 )consistsoftherstorderoptimalityconditionsofthenonlinearprogrammingproblem( 4 ).Theyhavethesamestructureasthenecessaryconditionsforthecontinuouscontrolproblem. Proposition4.2. ThematrixDyisthedifferentiationmatrixforspaceofpolynomialsofdegreeN)]TJ /F10 11.955 Tf 11.14 0 Td[(1evaluatedati,1iN.Intheotherwords,ifpisapolynomialofdegreeatmostN)]TJ /F10 11.955 Tf 11.96 0 Td[(1andifp2RNisavectorwithi-thcomponentpi=p(i),then (Dyp)i=_p(i),1iN. 101

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Proof. See[ 25 ]. Let Dz=Dy+0BBBBBBB@1 !10...01CCCCCCCA, then,wehavethefollowingproperty: Proposition4.3. Dzisinvertibleanditsinversehasthefollowingform: 0BBBBBBB@!1)]TJ /F14 11.955 Tf 9.29 0 Td[(!1~L2(1)...)]TJ /F14 11.955 Tf 9.3 0 Td[(!1~LN(1)!1R2)]TJ /F6 7.97 Tf 6.59 0 Td[(1~L2()d)]TJ /F14 11.955 Tf 11.96 0 Td[(!1~L2(1)...R2)]TJ /F6 7.97 Tf 6.59 0 Td[(1~LN()d)]TJ /F14 11.955 Tf 11.96 0 Td[(!1~LN(1)...!1RN)]TJ /F6 7.97 Tf 6.59 0 Td[(1~L2()d)]TJ /F14 11.955 Tf 11.96 0 Td[(!1~L2(1)...RN)]TJ /F6 7.97 Tf 6.59 0 Td[(1~LN()d)]TJ /F14 11.955 Tf 11.96 0 Td[(!1~LN(1)1CCCCCCCA,(4) where~LiistheLagrangeinterpolationpolynomialrelativetothequadraturepoints2,...,N,whichisgivenas ~Li()=NYj=2j6=i)]TJ /F14 11.955 Tf 11.96 0 Td[(j i)]TJ /F14 11.955 Tf 11.96 0 Td[(j,i=2,...,N. Proof. Supposethatthereexistssomep2RNwhichsatisesDzp=0.Letp2PN)]TJ /F6 7.97 Tf 6.58 0 Td[(1,betheuniquepolynomialwhichsatisesp(i)=pi,1iN,wherepiistheithcomponentofp.ByPropositionof 4.2 ,wehave 0=Dzp=Dyp+0BBBBBBB@1 !1p10...01CCCCCCCA=0BBBBBBB@_p()]TJ /F10 11.955 Tf 9.3 0 Td[(1)+1 !1p()]TJ /F10 11.955 Tf 9.29 0 Td[(1)_p(2)..._p(N)1CCCCCCCA.(4) 102

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Observefrom( 4 )that_phasN)]TJ /F10 11.955 Tf 12.65 0 Td[(1zerosf2,...,Ng.Since_p2PN)]TJ /F6 7.97 Tf 6.59 0 Td[(2,weobtain_p=0,whichmeansthatpisconstant.By( 4 ),wealsohave_p()]TJ /F10 11.955 Tf 9.3 0 Td[(1)+1 !1p()]TJ /F10 11.955 Tf 9.3 0 Td[(1)=0,whichyieldsp()]TJ /F10 11.955 Tf 9.3 0 Td[(1)=0.Sincepisconstant,wehavep=0,whichyieldsDzisinvertible. Denote(Dz))]TJ /F6 7.97 Tf 6.59 0 Td[(1=0BBBB@11...1N...N1...NN1CCCCA.Inthefollowing,wecalculateij.BythedenitionofDz,wehave Dz1=Dy1+0BBBBBBB@1 !10...01CCCCCCCA=0BBBBBBB@1 !10...01CCCCCCCA. Wetaketheinverseof(Dz)ofbothsidesoflastequationtoobtain (Dz))]TJ /F6 7.97 Tf 6.59 0 Td[(10BBBBBBB@1 !10...01CCCCCCCA=1,(4) whichyields11=21==N1=!1. Letq2PN)]TJ /F6 7.97 Tf 6.58 0 Td[(1.Wedeneqi=q(i),and_qi=_q(i),1iN.Then Dzq=_q+0BBBBBBB@1 !10...01CCCCCCCAq1.(4) TakingtheinverseofDzofbothsidesof( 4 )andbyusingof( 4 ),weobtain q)]TJ /F4 11.955 Tf 11.95 0 Td[(1q1=(Dz))]TJ /F6 7.97 Tf 6.59 0 Td[(1_q.(4) 103

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Bythedenitionof~Lj,_q2PN)]TJ /F6 7.97 Tf 6.59 0 Td[(2canbewrittenas _q()=NXj=2_qj~Lj().(4) Weintegrate( 4 )from)]TJ /F10 11.955 Tf 9.29 0 Td[(1toitoobtain: qi)]TJ /F4 11.955 Tf 11.96 0 Td[(q1=q(i))]TJ /F3 11.955 Tf 11.96 0 Td[(q()]TJ /F10 11.955 Tf 9.3 0 Td[(1)=NXj=2_qjAij,(4) whereAij=Ri)]TJ /F6 7.97 Tf 6.59 0 Td[(1~Lj()dandi=2,...,N. In( 4 ),iftaking_q2=1and_qi=0fori=3,...,N,weobtain_q1=_q(1)=~L2(1).By( 4 ),wehave~L2(1)11+12=0.Since11=!1,wehave12=)]TJ /F14 11.955 Tf 9.3 0 Td[(!1~L2(1).Similarly,weobtainthat(13,...,1N)=()]TJ /F14 11.955 Tf 9.3 0 Td[(!1L3(1),)]TJ /F14 11.955 Tf 28.56 0 Td[(!1LN(1)).By( 4 )and( 4 ),wehave: 0BBBBBBB@A22...A2NA32...A3N...AN2...ANN1CCCCCCCA0BBBBBBB@_q2_q3..._qN1CCCCCCCA=0BBBBBBB@!122...2N!132...3N...!1N2...NN1CCCCCCCA0BBBBBBB@_q1_q2..._qN1CCCCCCCA.(4) Taking_q1=L2(1),_q2=1and_q3==_qN=0in( 4 ),wehave 0BBBB@A22...AN21CCCCA=0BBBB@!1...!11CCCCAL2(1)+0BBBB@22...N21CCCCA, whichyields 0BBBB@22...N21CCCCA=0BBBB@A22)]TJ /F14 11.955 Tf 11.96 0 Td[(!1L2(1)...AN2)]TJ /F14 11.955 Tf 11.95 0 Td[(!1L2(1)1CCCCA. 104

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Similarly,wehave 0BBBB@2j...Nj1CCCCA=0BBBB@A2j)]TJ /F14 11.955 Tf 11.96 0 Td[(!1Lj(1)...ANj)]TJ /F14 11.955 Tf 11.95 0 Td[(!1Lj(1)1CCCCA,3iN. BythedenitionofAij,weprovethatformof(Dz))]TJ /F6 7.97 Tf 6.58 0 Td[(1is( 4 ). Inouranalysisofconvergence,weneedtoboundbothDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1andW1 2Dz)]TJ /F6 7.97 Tf 6.58 0 Td[(1.Numerically,wehaveevaluatedDzandDz)]TJ /F6 7.97 Tf 12.71 0 Td[(1,andfoundthat Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(112,(4) and max1iNW1 2Dz)]TJ /F6 7.97 Tf 6.58 0 Td[(1i2p 2,(4) whereW1 2Dz)]TJ /F6 7.97 Tf 6.59 0 Td[(1idenotestheithrowofthematrixW1 2Dz)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Althoughwedonothaveaproofofthesetworesults,wenumericallycomputedDzandDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1forNupto200andfoundthatkDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1k1andW1 2Dz)]TJ /F6 7.97 Tf 6.59 0 Td[(1i2aremonotonicallyincreasingfunctionsofNthatapproachthelimitsgivenabove. Inordertoanalyzethediscreteproblem( 4 ),weneedtointroducediscreteanaloguesoftheL1norms.Inparticular,forasequencez=z1,z2,...,zNwhosei-thelementisavectorzi2Rn,wedenetheL1normasthefollowing: kzk1=sup1iNjzij, wherejjistheEuclideannorm. 4.3ConvergenceResult Let(x(t),u(t))bethelocalsolutionofthecontinuousoptimalproblem( 4 )and(t)betheassociatedcostate.WedenevectorsequencesX,UandbyXi=x(ti),1iN+1,Ui=u(ti),1iNandi=(ti),1iN+1.Weshowthatundersuitableassumptions,thenonlinearprogrammingproblem( 4 )has 105

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anextremepoint)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XN,UNandtheassociatedLagrangemultiplierNwhichconvergesto(X,U)andexponentiallyfastintheL1norm. Theorem4.1. Suppose( 4 )hasalocalminimizerxandu,andthereexistssuchthatthePontryagin'sminimumprincipleholds.Iftheassumptions(A1)(A3)hold,then( 4 )hasanextremepoint)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XN,UNandassociatedLagrangemultiplierNsatisfying kXN)]TJ /F4 11.955 Tf 11.95 0 Td[(Xk1+kUN)]TJ /F4 11.955 Tf 11.96 0 Td[(Uk1+kN)]TJ /F20 11.955 Tf 11.95 0 Td[(k1c Nk)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2,(4) whereNisthenumberofGausspointsandcisaconstantindependentofN. 4.4AbstractSetting TheproofofTheorem 4.1 isbasedon[ 14 ,Proposition3.1].Forourconvergenceproof,werestatethepropositioninthefollowingform: Proposition4.4. LetXbeaBanachspaceandYbealinearnormedspacewiththenormsinbothspacesdenotedkk.LetT:X7)165(!YwithTcontinuouslyFrechetdifferentiableinBr()forsome2Xandr>0.Supposethat (P1)rT()isinvertible. (P2)Forsomescalar",krT())-222(rT()k"forall2Br(). If"<1,where=krT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kandkT()k(1)]TJ /F14 11.955 Tf 12.33 0 Td[(")r=,thenthereexistsaunique2Br()suchthatT()=0.Moreover,wehavetheestimate k)]TJ /F14 11.955 Tf 11.96 0 Td[(k 1)]TJ /F14 11.955 Tf 11.95 0 Td[("kT()k.(4) WeapplyProposition 4.4 toproveourTheorem 4.1 .First,weneedtospecifythenormedspaceX,YandthemapT.ThespaceXconsistsof3-tupleswhosecomponentsarevectorsequences=(X,U,),whereX=(X1,...,XN+1),U=(U1,...,UN)and=(1,...,N+1).Let=(X,U,).ThenormisthediscreteL1normgivenby kk1=k(X,U,)k1=maxfkXk1,kUk1,kk1g.(4) 106

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ThemappingTisselectedinthefollowingway: T(X,U,)=0BBBBBBBBBBBBBB@PN+1j=1DijXj)]TJ /F4 11.955 Tf 11.95 0 Td[(f(Xi,Ui),1iN,X1)]TJ /F4 11.955 Tf 11.96 0 Td[(x0,PNj=1Dy1jj+rXH(X1,U1,1))]TJ /F10 11.955 Tf 16.12 8.09 Td[(1 !1()]TJ /F20 11.955 Tf 11.96 0 Td[(1),PNj=1Dyijj+rXH(Xi,Ui,i),2iN,rUH(Xi,Ui,i),1iN,rXC(XN+1))]TJ /F20 11.955 Tf 11.95 0 Td[(1+PNi=1!irXH(Xi,Ui,i)1CCCCCCCCCCCCCCA.(4) LetYdenotethespaceassociatedwiththesixcomponentsofT.YisaspacewhoseelementsaresequencesandeachelementofthesequenceliesinL1RnRnL1L1Rn. rT()=0BBBBBBBBBBBBBBBBBBBBBBB@N+1Xj=1DijXj)]TJ /F4 11.955 Tf 11.96 0 Td[(XiATi)]TJ /F4 11.955 Tf 11.96 0 Td[(UiBTi,1iN,X1,NXj=1Dy1jj+1AT1+X1QT1+U1ST1)]TJ /F10 11.955 Tf 16.13 8.09 Td[(1 !1()]TJ /F20 11.955 Tf 11.95 0 Td[(1),NXj=1Dyijj+iATi+XiQTi+UiSTi,2iN,iBTi+XiSTi+UiRTi,1iN,XN+1VT)]TJ /F20 11.955 Tf 11.96 0 Td[(+NXi=1!i)]TJ /F20 11.955 Tf 5.48 -9.69 Td[(iATi+XiQTi+UiSTi1CCCCCCCCCCCCCCCCCCCCCCCA.(4) where Ai=rxf(Xi,Ui),Bi=ruf(Xi,Ui),Qi=rxxH(Xi,Ui,i),Si=ruxH(Xi,Ui,i),Ri=ruuH(Xi,Ui,i),V=r2xC)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XN+1.9>>>>=>>>>;(4) 4.5ApproximationPreliminaries ToproveTheorem 4.1 ,weneedtoshowtheassumptionsofProposition 4.4 holdforNsufcientlarge.TheestimateofTheorem 4.1 isaconsequenceof( 4 ).In 107

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verifyingtheassumptionsofProposition 4.4 ,weutilizeseveralapproximationpropertiesofLagrangeinterpolationpolynomials.Inthissection,wepulltogetherthesepropertiesthatareexploitedthroughouttheanalysis. Lemma15. If`NistheLebesgueconstantdenedby `N=maxt2[)]TJ /F6 7.97 Tf 6.59 0 Td[(1,1]N+1Xj=1jLj(t)j,(4) whereLj(t)istheLagrangeinterpolationpolynomialdenedin( 4 ),then`N=O(logN). Proof. SeeTheorem 2.6 Lemma16. Let`N)]TJ /F6 7.97 Tf 6.59 0 Td[(1betheLebesgueconstantdenedby `N)]TJ /F6 7.97 Tf 6.59 0 Td[(1=maxt2[)]TJ /F6 7.97 Tf 6.59 0 Td[(1,1]NXj=1Lj(t),(4) whereLi,i=1,...,N,aretheLagrangepolynomialsdenedas Li(t)=NYj=1j6=it)]TJ /F3 11.955 Tf 11.95 0 Td[(tj ti)]TJ /F3 11.955 Tf 11.95 0 Td[(tj,(4) whereti,i=1,...,N,aretheRadauquadraturepoints.Then,`N)]TJ /F6 7.97 Tf 6.59 0 Td[(1=ON1 2. Proof. SeeTheorem 2.5 Lemma17. Foranyy2C1[)]TJ /F10 11.955 Tf 9.29 0 Td[(1,1],letyN2PNbetheinterpolatingpolynomialdenedasyN(ti)=PN+1i=1y(ti)Li(t),whereLi(t)isdenedin( 4 ).Then, _y)]TJ /F10 11.955 Tf 13.63 0 Td[(_yN1)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+2N2`Ninfq2PN)]TJ /F9 5.978 Tf 5.76 0 Td[(1k_y)]TJ /F3 11.955 Tf 11.96 0 Td[(qk1, where`NistheLebesgueconstantdenedin( 4 ). Proof. SeeTheorem 2.1 108

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Lemma18. (Jackson'sTheorem).Lety(t)2Cjandn>j0,then infp2Pnky)]TJ /F3 11.955 Tf 11.95 0 Td[(pk12cjky(j)k1 nj,(4) wherecj=6j+1ej(1+j))]TJ /F6 7.97 Tf 6.58 0 Td[(1. Proof. SeeTheorem 2.2 4.6AnalysisofResidualandStationarity Inthissection,webegintoshowthatProposition 4 isapplicabletotheRadaucollocationmethodbyestimatingkT()k. Lemma19. Ifsmoothnessholds,thenthereexitsaconstantcindependentofN,suchthat kT()kc Nk)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2,(4) wherekisdenedinassumption(A1). Proof. Bytherst-orderoptimalityconditionsforthecontinuous-timecontrolproblem( 4 ),wehave x()]TJ /F10 11.955 Tf 9.29 0 Td[(1))]TJ /F4 11.955 Tf 11.95 0 Td[(x0=0,andruH(x(ti),u(ti),(ti))=0. Hence,thesecondandfthcomponentsofT()are0.NowweanalyzetherstcomponentsofT().Theithelementoftherstcomponentis T()1i=N+1Xj=1DijXj)]TJ /F4 11.955 Tf 11.95 0 Td[(f(Xi,Ui).(4) LetxN(t)denotethevectorwhosejthcomponentisxNj(t)=PN+1i=1xijLi(t),wherexij=xj(ti).BythedenitionofD,wehavePN+1j=1DijXj=_xN(ti).Sincexsatisesthedynamicsf(x(ti),u(ti))=_x(ti),( 4 )reducesto T()1i=_xN(ti))]TJ /F10 11.955 Tf 13.2 0 Td[(_x(ti). (4) 109

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Sincex(t)2Ck+1[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1],forsomek3,wehave_x(t)2Ck[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1].ByLemma 15 ,Lemma 17 andLemma 18 ,( 4 )canbeboundedasfollows: _xNj(ti))]TJ /F10 11.955 Tf 13.2 0 Td[(_xj(ti)_xNj)]TJ /F10 11.955 Tf 13.53 0 Td[(_xj1)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+2N2O(logN)infq2PN)]TJ /F9 5.978 Tf 5.76 0 Td[(1_xj)]TJ /F3 11.955 Tf 11.95 0 Td[(q)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+2N2O(logN)c(xj)(k+1) Nkc(xj)(k+1)logN Nk)]TJ /F6 7.97 Tf 6.59 0 Td[(2clogN Nk)]TJ /F6 7.97 Tf 6.58 0 Td[(2. Hence,kT()1k1clogN Nk)]TJ /F6 7.97 Tf 6.59 0 Td[(2. NowweestimatethefourthcomponentofT().Theithelementofthefourthcomponentis T()4i=NXj=1Dyijj+rXH(Xi,Ui,i).(4) LetN)]TJ /F6 7.97 Tf 6.59 0 Td[(1(t)denotethevectorwhosejthcomponentisN)]TJ /F6 7.97 Tf 6.59 0 Td[(1j(t)=PNi=1ijLi(t),whereij=j(ti)andLi(t)isdenedin( 4 ).BytheProposition 4.2 ,wehavePNj=1Dyijj=_N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(ti).SinceisatisesrxH(Xi,Ui,i)=)]TJ /F10 11.955 Tf 11.56 2.65 Td[(_(ti),( 4 )reducesto T()4i=_N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(ti))]TJ /F10 11.955 Tf 14.22 2.66 Td[(_(ti), (4) for2iN.Since(t)2Ck+1[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1],forsomek3,wehave_(t)2Ck[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1].ByLemmas 16 ,Lemma 17 andLemma 18 ,( 4 )canbeboundedasfollows: _N)]TJ /F6 7.97 Tf 6.58 0 Td[(1j(ti))]TJ /F10 11.955 Tf 14.22 2.66 Td[(_j(ti)_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)]TJ /F10 11.955 Tf 13.62 2.66 Td[(_j11+2(N)]TJ /F10 11.955 Tf 11.95 0 Td[(1)2ON1 2infq2PN)]TJ /F9 5.978 Tf 5.75 0 Td[(2_j)]TJ /F3 11.955 Tf 11.95 0 Td[(q11+2(N)]TJ /F10 11.955 Tf 11.95 0 Td[(1)2ON1 2c(j)(k+1)1 Nkc Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. Hence,kT()4k1c Nk)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2. 110

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NowestimatethethirdcomponentofT(): NXj=1Dy1jj+rxH(X1,U1,1))]TJ /F10 11.955 Tf 16.12 8.09 Td[(1 !1()]TJ /F20 11.955 Tf 11.96 0 Td[(1),(4) By( 4 ),wehave)]TJ /F20 11.955 Tf 12.28 0 Td[(1=0.Hence,theestimateof( 4 )reducestotheestimateofPNj=1Dy1jj+rxH(X1,U1,1).ByasimilarargumentastheestimateofthefourthcomponentofT(),wegettheL1normofthethirdcomponentisboundedbyc Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. NowweestimatethelastcomponentofT(): rXC)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XN+1)]TJ /F20 11.955 Tf 11.95 0 Td[(+NXi=1!irXH(Xi,Ui,i).(4) By( 4 ),( 4 )and( 4 ),( 4 )becomes (1))]TJ /F20 11.955 Tf 11.95 0 Td[(()]TJ /F10 11.955 Tf 9.29 0 Td[(1))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!i_(ti).(4) SinceN-pointRadauquadratureisexactforpolynomialofdegreeupto2N)]TJ /F10 11.955 Tf 12.13 0 Td[(2[ 31 ],wehave N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1))]TJ /F20 11.955 Tf 11.95 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()]TJ /F10 11.955 Tf 9.3 0 Td[(1))]TJ /F7 7.97 Tf 17.29 14.94 Td[(NXi=1!i_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(ti)=N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1))]TJ /F20 11.955 Tf 11.95 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()]TJ /F10 11.955 Tf 9.3 0 Td[(1))]TJ /F8 11.955 Tf 11.95 16.27 Td[(Z1)]TJ /F6 7.97 Tf 6.59 0 Td[(1_N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(t)dt=0. (4) Wesubtract( 4 )from( 4 )toobtain (1))]TJ /F20 11.955 Tf 11.95 0 Td[(N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1)+NXi=1!i)]TJ /F10 11.955 Tf 7.75 -7.03 Td[(_N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(ti))]TJ /F10 11.955 Tf 14.22 2.66 Td[(_(ti).(4) BytheestimationforthefourthcomponentofT(), _N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(ti))]TJ /F10 11.955 Tf 14.22 2.65 Td[(_(ti)c Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2.(4) Wenowestimate(1))]TJ /F20 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1).Letq2PN)]TJ /F6 7.97 Tf 6.59 0 Td[(1,thenq(t)=PNi=1q(ti)Li(t),whereLi(t)isdenedin( 4 ).Inparticular,fort=1,wehaveq(1)=PNi=1q(ti)Li(1).Similarly,forthej-thcomponentofN)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1),wehaveNj(1)=PNi=1Nj(ti)Li(1)= 111

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PNi=1j(ti)Li(1).Hence, j(1))]TJ /F14 11.955 Tf 11.95 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1j(1)=j(1))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1j(ti)Li(1)=j(1))]TJ /F3 11.955 Tf 11.96 0 Td[(q(1)+NXi=1q(ti)Li(1))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1j(ti)Li(1)=j(1))]TJ /F3 11.955 Tf 11.96 0 Td[(q(1))]TJ /F7 7.97 Tf 17.29 14.94 Td[(NXi=1)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(j(ti))]TJ /F3 11.955 Tf 11.96 0 Td[(q(ti)Li(1)j(1))]TJ /F3 11.955 Tf 11.96 0 Td[(q(1)+j)]TJ /F3 11.955 Tf 11.95 0 Td[(q1NXi=1Li(1)j)]TJ /F3 11.955 Tf 11.95 0 Td[(q1(1+`N)]TJ /F6 7.97 Tf 6.59 0 Td[(1). (4) SinceqisarbitraryinPN)]TJ /F6 7.97 Tf 6.59 0 Td[(1,( 4 )yields j(1))]TJ /F14 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1j(1)infq2PN)]TJ /F9 5.978 Tf 5.75 0 Td[(1j)]TJ /F3 11.955 Tf 11.95 0 Td[(q1(1+`N)]TJ /F6 7.97 Tf 6.59 0 Td[(1).(4) ByLemma 18 ,wehave infq2PN)]TJ /F9 5.978 Tf 5.75 0 Td[(1j)]TJ /F3 11.955 Tf 11.95 0 Td[(q1c(j)(k+1)1 Nk+1,(4) Wecombine( 4 ),( 4 )andLemma 16 toobtain j(1))]TJ /F14 11.955 Tf 11.95 0 Td[(N)]TJ /F6 7.97 Tf 6.58 0 Td[(1j(1)c(j)(k+1)1 Nk+1(1+`N)]TJ /F6 7.97 Tf 6.58 0 Td[(1)c Nk+1 2.(4) Hence,by( 4 )wehave (1))]TJ /F20 11.955 Tf 11.95 0 Td[(N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1)c Nk+1 2.(4) 112

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Then,by( 4 )and( 4 ),( 4 )isboundedasfollows: (1))]TJ /F20 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1)+NXi=1!i)]TJ /F10 11.955 Tf 7.75 -7.02 Td[(_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(ti))]TJ /F10 11.955 Tf 14.22 2.66 Td[(_(ti)(1))]TJ /F20 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1)+NXi=1!i_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(ti))]TJ /F10 11.955 Tf 14.22 2.66 Td[(_(ti)c Nk+1 2+c Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2NXi=1!ic Nk)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. TheseestimatesforthesixcomponentsofT()yieldskT()k1c Nk)]TJ /F9 5.978 Tf 7.78 3.25 Td[(5 2. Next,weestablishthecondition(P2)oftheProposition 5.4 Lemma20. Ifsmoothnessholds,theforeach">0,thereexistsr>0suchthatforall2Br(),then krT())-222(rT()k,(4) wherekkisthematrixnorminducedbytheL1normonXandY,andrisindependentofN. Proof. Throughouttheproof,weneedtoconsiderthedifferencebetweenderivativesoffandHevaluatedat(X,U,,)andthesamederivativesevaluatedatapoint(X,U,,).Forfurtherreferences,letAandBbeblockdiagonalmatriceswithi-thblocksgivenby Ai=Ai)-222(rXf(Xi,Ui)andBi=Bi)-222(rUf(Xi,Ui),(4) whereAi=rxf(Xi,Ui)andBi=ruf(Xi,Ui).Similarly,letQ,SandRbeblockdiagonalmatriceswiththeithblocksgivenby Qi=rXXH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.96 0 Td[(Qi,Si=rXUH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.95 0 Td[(Si,Ri=rXXH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.96 0 Td[(Ri,9>>>>=>>>>;(4) whereQi=rxxH(Xi,Ui,i),Si=ruxH(Xi,Ui,i),andRi=ruuH(Xi,Ui,i). 113

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LetrT()idenotetheiblockofrT()appearingin( 4 ).Withthisnotation,(rT())-222(rT())1=A B 0 0, Bythedenitionofmatrixnorm,wehave k(rT())-222(rT())1k=max1iNmaxkyk1kvk1kAiy+Bivk2max1iNmaxkyk1kAiyk2+maxkvk1kBivk2=max1iN)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kAik2+kBik2max1iN0B@krXf(Xi,Ui))-221(rxf(Xi,Ui)k2+krUf(Xi,Ui))-221(ruf(Xi,Ui)k21CA, (4) where( 4 )isby( 4 ).Inordertomake( 4 )smallerthan",wechoosearadiusr,suchthatk)]TJ /F20 11.955 Tf 13.33 0 Td[(k1
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ThefourthcomponentofrT())-221(rT()isamatrixofthefollowingform: (rT())-222(rT())4=Q S A 0, whereAisdenedin( 4 ),andQandSaredenedin( 4 ).Bythesimilarargumentasfor(rT())-222(rT())1,weobtain k(rT())-222(rT())4kmax2iN)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(krXf(Xi,Ui))-222(rxf(Xi,Ui)k2+krXXH(Xi,Ui,i))-222(rxxH(Xi,Ui,i)k2+krXUH(Xi,Ui,i))-221(ruxH(Xi,Ui,i)k2. Bychoosingrsufcientlysmall,wehavek(rT())-222(rT())3k<"fork)]TJ /F20 11.955 Tf 11.95 0 Td[(k1
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!i(rXf(Xi,Ui))]TJ /F4 11.955 Tf 11.96 0 Td[(Ai),1iN.Bythedenitionofmatrixnorm,wehave k(rT())-222(rT())6k=maxkyik1kvik1krik1NXi=1!i(rXXH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.96 0 Td[(Qi)yi+(rXXC(XN+1))]TJ /F4 11.955 Tf 11.95 0 Td[(V)yN+1+NXi=1!i(rXUH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.95 0 Td[(Si)vi+NXi=1!i(rXf(Xi,Ui))]TJ /F4 11.955 Tf 11.95 0 Td[(Ai)ri2maxkyik1NXi=1!i(rXXH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.96 0 Td[(Qi)yi2+maxkyN+1k1k(rXXC(XN+1))]TJ /F4 11.955 Tf 11.96 0 Td[(V)yN+1k2+maxkvik1NXi=1!i(rXUH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.96 0 Td[(Si)vi2+maxkrik1NXi=1!i(rXf(Xi,Ui))]TJ /F4 11.955 Tf 11.96 0 Td[(Ai)ri22max1iNkrXXH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.95 0 Td[(Qik2+krXXC(XN+1))]TJ /F4 11.955 Tf 11.96 0 Td[(Vk2+2max1iNkrXUH(Xi,Ui,i))]TJ /F4 11.955 Tf 11.96 0 Td[(Sik2+2max1iNkrXf(Xi,Ui))]TJ /F4 11.955 Tf 11.96 0 Td[(Aik2. (4) ThelastinequalityisbecausePNi=1!i=2.Bytheassumption,thesecondderivativeofCiscontinuous,hence,whenk)]TJ /F20 11.955 Tf 13.49 0 Td[(k1
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Werstrelatethesolutionof( 4 )tothesolutionofaquadraticprogrammingprobleminwhichtheparameterappearsintheconstraintsandinthecostfunction.WeshowthatthequadraticprogrammingproblemhasauniquesolutiondependingLipschitzcontinuouslyontheparameter. By( 4 ),( 4 )hasthefollowingform: 0BBBBBBBBBBBBBBBBBBBBBBB@N+1Xj=1DijXj)]TJ /F4 11.955 Tf 11.96 0 Td[(XiATi)]TJ /F4 11.955 Tf 11.96 0 Td[(UiBTi+pi=0,1iN,X1+y=0,NXj=1Dy1jj+1AT1+X1QT1+U1ST1)]TJ /F10 11.955 Tf 16.12 8.09 Td[(1 !1()]TJ /F20 11.955 Tf 11.96 0 Td[(1)+s=0,NXj=1Dyijj+iATi+XiQTi+UiSTi+qi=0,2iN,iBTi+XiSTi+UiRTi+ri=0,1iN,XN+1VT)]TJ /F20 11.955 Tf 11.95 0 Td[(+NXi=1!i(iATi+XiQTi+UiSTi)+z=01CCCCCCCCCCCCCCCCCCCCCCCA.(4) Wedenethefollowingcolumnvector: X1:N=0BBBBBBB@XT1XT2...XTN1CCCCCCCA,U=0BBBBBBB@UT1UT2...UTN1CCCCCCCA,q=0BBBBBBB@!1sT!2qT2...!NqTN1CCCCCCCA,r=0BBBBBBB@!1rT1!2rT2...!NrTN1CCCCCCCA. Letusconsiderthefollowingquadraticprogrammingproblem: minimizeB(X,U)+qTX1:N+z)]TJ /F14 11.955 Tf 11.95 0 Td[(!1s)]TJ /F8 11.955 Tf 11.95 8.97 Td[(PNi=2!iqiXTN+1+rTUsubjecttoPN+1j=1DijXj)]TJ /F4 11.955 Tf 11.96 0 Td[(XiATi)]TJ /F4 11.955 Tf 11.95 0 Td[(UiBTi+pi=0,1iN,X1+y=0,9>>>>>>>=>>>>>>>;(4) 117

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where B(X,U)=1 2 XN+1VXTN+1+NXi=1!i)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XiQiXTi+2XiSiUTi+UiRiUTi!.(4) Inthefollowing,werstshowthatBisstronglyconvex,hence( 4 )hasauniquesolution.MoreovertheKKTconditionsassociatewith( 4 )canbeexpressedintheformof( 4 ).SinceBisconvex,itfollowsthatanysolutionof( 4 )alsoyieldsasolutionof( 4 )(aswellasthemultipliersassociatedwiththeconstraints). Lemma21. Ifthecoercivityholds,thenthequadraticformBdenedin( 4 )isstronglyconvex. Proof. BythecoercivityassumptionandthedenitionofmatricesQi,SiandRiin( 4 ),wehave XN+1VXTN+1jXN+1j2, and XiQiXTi+2XiSiUTi+UiRiUTi)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jXij2+jUij2,1iN. Hence, B(X,U) 2jXN+1j2+ 2NXi=1!i)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jXij2+jUij2(4) Ifwedenethe!-normsofXandUasfollows: kXk2!=NXi=1!ijXij2+jXN+1j2andkUk2!=NXi=1!ijUij2.(4) Itfollowsfrom( 4 )thatB(X,U) 2)]TJ /F2 11.955 Tf 5.47 -9.69 Td[(kXk2!+kUk2!.Hence,Bisstronglyconvex. Lemma22. Thequadraticprogrammingproblem( 4 )and( 4 )haveidenticaluniquesolution. 118

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Proof. TheLagrangianassociatewith( 4 )is L(,X,U)=B(X,U)+qTX1:N+ z)]TJ /F14 11.955 Tf 11.95 0 Td[(!1s)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=2!iqi!XTN+1+rTU)]TJ /F7 7.97 Tf 16.63 14.95 Td[(NXi=1i N+1Xj=1DijXj)]TJ /F4 11.955 Tf 11.96 0 Td[(XiATi)]TJ /F4 11.955 Tf 11.95 0 Td[(UiBTi+pi!T)]TJ /F20 11.955 Tf 11.95 0 Td[((X1+y)T. WenowformulatetheKKToptimalityconditionsfor( 4 ).TheseconditionscorrespondtosettingthederivativeoftheLagrangianwithrespecttoX,U,andtozero. ThepartialderivativeoftheLagrangianwithrespecttoX1is: 0=rX1L=!1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(X1QT1+U1ST1+!1s)]TJ /F7 7.97 Tf 17.29 14.94 Td[(NXi=1iDi1+1AT1)]TJ /F20 11.955 Tf 11.96 0 Td[(=!1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(X1QT1+U1ST1)]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1i !i!i !1Di1!1+!11 !1AT1)]TJ /F20 11.955 Tf 11.95 0 Td[(+!1s AftersubstitutingforDi1intermsofDy1iusing( 4 ),for1intermsof1using( 4 )anddividing!1bothsidesofthelastequation,weobtain )]TJ /F4 11.955 Tf 5.48 -9.68 Td[(X1QT1+U1ST1+1AT1+NXi=1Dy1ii+1 !1(1)]TJ /F20 11.955 Tf 11.95 0 Td[()+s=0,(4) whichisthethirdequationin( 4 ). ThepartialderivativeoftheLagrangianwithrespecttoXi,2iN,is: 0=rXiL=!i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XiQTi+UiSTi)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXj=1jDji+iATi+!iqi=!i)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XiQTi+UiSTi)]TJ /F7 7.97 Tf 17.29 14.95 Td[(NXj=1j !j!j !iDji!i+!ii !iATi+!iqi AftersubstitutingforDjiintermsofDyijusing( 4 ),foriintermsofiusing( 4 )anddividing!ibothsidesofthelastequation,weobtain XiQTi+UiSTi+iATi+NXj=1Dyijj+qi=0,(4) whichisthefourthequationin( 4 ). 119

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ThepartialderivativeoftheLagrangianwithrespecttoXN+1is: 0=rXN+1L=XN+1VT)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1iDiN+1+ z)]TJ /F14 11.955 Tf 11.95 0 Td[(!1s)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=2!iqi!. WesubstituteDiN+1intheaboveequationby( 4 )toobtain 0=XN+1VT+NXi=1NXj=1iDij+ z)]TJ /F14 11.955 Tf 11.95 0 Td[(!1s)]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=2!iqi!=XN+1VT+NXi=1NXj=1i !i!i !jDij!j+ z)]TJ /F14 11.955 Tf 11.95 0 Td[(!1s)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=2!iqi!=XN+1VT)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXj=1!jNXi=1Dyjii)]TJ /F20 11.955 Tf 11.96 0 Td[(1+ z)]TJ /F14 11.955 Tf 11.95 0 Td[(!1s)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=2!iqi!, (4) BysubstitutePNi=1Dyjiiin( 4 )by( 4 )and( 4 ),weobtain 0=XN+1VT)]TJ /F20 11.955 Tf 11.96 0 Td[(+NXj=1!j)]TJ /F20 11.955 Tf 5.48 -9.69 Td[(jATj+XjQTj+UjSTj+z, whichyieldsthelastequationin( 4 ). ThepartialderivativeoftheLagrangianwithrespecttoUiis: 0=rUiL=!i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XiSTi+UiRTi+iBTi+!iri=!i)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(XiSTi+UiRTi+!ii !iBTi+!iri=!i)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XiSTi+UiRTi+iBTi+!iri, (4) by( 4 ).Since!i6=0,wedivide!itoobtain XiSTi+UiRTi+iBTi+ri=0, whichisthefthequationin( 4 ). ThepartialderivativeoftheLagrangianwithrespecttoi,1iN,andyieldtherstandsecondequationsof( 4 ). Hence,theKKTconditionsassociatewith( 4 )canbeexpressedintheformof( 4 ).SinceBisstronglyconvexbyLemma 21 ,( 4 )hasauniquesolution.Also,it 120

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followsthatanysolutionof( 4 )alsoyieldsasolutionof( 4 ).Hence,thequadraticprogrammingproblem( 4 )and( 4 )haveidenticaluniquesolution. Inthefollowing,weshowkrT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kisboundedbyaconstantwhichdoesnotdependonN.Ourproofisbasedon[ 30 ,Lemma1],whichisstatedbelow: Lemma23. Letbeasymmetric,continuousbilinearformdenedonanonempty,closedconvexsubsetKofaHilbertspaceV,andlethiVdenotetheHilbertspaceinnerproduct.Ifthereexist>0suchthat (w)]TJ /F3 11.955 Tf 11.96 0 Td[(v,w)]TJ /F3 11.955 Tf 11.96 0 Td[(v)hw)]TJ /F3 11.955 Tf 11.96 0 Td[(v,w)]TJ /F3 11.955 Tf 11.96 0 Td[(viVforallw,v2K, thenforany2V,thequadraticprogram minimize1 2(v,v))-222(h,viVsubjecttov2K(4) hasauniquesolutionw.Thissolutionistheuniquew2Kthatsatisesthevariationalinequality(w,v)]TJ /F3 11.955 Tf 12.3 0 Td[(w)h,v)]TJ /F3 11.955 Tf 12.29 0 Td[(wiVforallv2K.Ifwidenotesthesolutionof( 4 )correspondingto=i,fori=1andi=2,thenwehave kw1)]TJ /F3 11.955 Tf 11.95 0 Td[(w2kk1)]TJ /F14 11.955 Tf 11.96 0 Td[(2k, wherekkisthenorminducedbytheHilbertspaceinnerproduct. Proof. See[ 30 ,Lemma1]. TheapplicationofLemma 23 tothequadraticprogrammingproblem( 4 )proceedsasfollows.Werstshowthatiftheassumption(A3)holds,thenforallU2L1,thereexistsauniqueXfeasiblein( 4 ).Inotherwords,wecanwriteXintermsofU.Then,byLemma 23 ,weobtainaboundforkUk!intermsofkk!.Therelationbetween 121

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XandUaregivenbythefollowingtwolemmas.First,wemakethefollowingnotations:X2:N+1=0BBBBBBB@XT2XT3...XTN+11CCCCCCCA,A=0BBBBBBB@0A20......AN01CCCCCCCA,A1=0BBBBBBB@A10...01CCCCCCCA,B=0BBBB@B1...BN1CCCCA,Q=0BBBB@Q1...QN1CCCCA,p=0BBBB@pT1...pTN1CCCCA,S=0BBBB@S1...SN1CCCCAIn=0BBBBBBB@0In0......In01CCCCCCCA,I1=0BBBBBBB@In0...01CCCCCCCA,IN=(0,...,0,In),D=D2:N+1In,D1=D1In,W=WIn. wheredenotestheKroneckerproduct. Lemma24. ThematrixI)]TJ /F10 11.955 Tf 13.61 2.65 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1Aisinvertibleand)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.57 2.65 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(114 3,assuming(A3)and( 4 )hold. Proof. ByProposition 4.1 and( 4 ),D2:N+1isinvertibleandkD)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1k1=2.Hence,D=D2:N+1Inisalsoinvertible.D)]TJ /F6 7.97 Tf 6.59 0 Td[(1=D)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1InandD)]TJ /F6 7.97 Tf 6.59 0 Td[(11=D)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+11=2.Byassumption(A3),kAk11 8.Hence, kD)]TJ /F6 7.97 Tf 6.59 0 Td[(1Ak1kD)]TJ /F6 7.97 Tf 6.58 0 Td[(1k1kAk11 4.(4) By[ 32 ,p.351],I)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1Aisinvertileand(I)]TJ /F10 11.955 Tf 13.56 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A))]TJ /F6 7.97 Tf 6.59 0 Td[(114 3. Lemma25. If(A3)holds,thenforallU2L1,thereexistsauniqueXfeasiblein( 4 ).LetXbefeasiblein( 4 )andletX0bethestate,associatedwiththecontrolU0and 122

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perturbationsp0andy0,thenwehave X2:N+1)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X02:N+11c)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!+kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!+jy)]TJ /F4 11.955 Tf 11.95 0 Td[(y0j,(4) wherecisaconstantindependentofN. Proof. By( 4 ), (D)]TJ /F10 11.955 Tf 13.2 2.65 Td[(A)X2:N+1=BU+)]TJ /F10 11.955 Tf 6.72 -7.03 Td[(A1)]TJ /F10 11.955 Tf 13.57 2.65 Td[(D1XT1)]TJ /F10 11.955 Tf 12.17 0 Td[(p.(4) ByLemma 24 ,D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A=D)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1Aisinvertible.Itfollowsfrom( 4 )that X2:N+1=)]TJ /F10 11.955 Tf 7.09 -7.02 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 6.73 -7.02 Td[(BU+)]TJ /F10 11.955 Tf 6.73 -7.02 Td[(A1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D1XT1)]TJ /F10 11.955 Tf 12.17 0 Td[(p.(4) Hence,forallU2L1,thereexistsauniqueXfeasiblein( 4 ).Now,weshowthat( 4 )holds.By( 4 ),wehave: X2:N+1)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X02:N+1=)]TJ /F10 11.955 Tf 7.09 -7.02 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.58 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0+)]TJ /F10 11.955 Tf 7.09 -7.02 Td[(D)]TJ /F10 11.955 Tf 13.21 2.66 Td[(A)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F10 11.955 Tf 6.72 -7.02 Td[(A1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D1)]TJ /F4 11.955 Tf 12.95 -9.68 Td[(XT1)]TJ /F4 11.955 Tf 11.95 0 Td[(X10T)]TJ /F8 11.955 Tf 11.96 9.68 Td[()]TJ /F10 11.955 Tf 7.09 -7.02 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A)]TJ /F6 7.97 Tf 6.58 0 Td[(1D)]TJ /F6 7.97 Tf 6.58 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U0+)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1D1)]TJ /F4 11.955 Tf 12.95 -9.69 Td[(XT1)]TJ /F4 11.955 Tf 11.96 0 Td[(X10T)]TJ /F8 11.955 Tf 11.29 9.68 Td[()]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.57 2.65 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(1D)]TJ /F6 7.97 Tf 6.58 0 Td[(1(p)]TJ /F10 11.955 Tf 12.18 0 Td[(p0). Wetaketheinnitynormofbothsidesofthelastequationtoobtain X2:N+1)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X02:N+11)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(11D)]TJ /F6 7.97 Tf 6.58 0 Td[(1B)]TJ /F10 11.955 Tf 6.9 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U01+)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(I)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(11D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1D11XT1)]TJ /F4 11.955 Tf 11.95 0 Td[(X10T+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.57 2.65 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A)]TJ /F6 7.97 Tf 6.59 0 Td[(11D)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)14 3)]TJ 5.48 .48 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U01+D)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)1+D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1D11XT1)]TJ /F4 11.955 Tf 11.95 0 Td[(X10T, (4) 123

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where( 4 )isbyLemma 24 .Inthefollowing,weestimateD)]TJ /F6 7.97 Tf 6.58 0 Td[(1B)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U01,D)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)1andD)]TJ /F6 7.97 Tf 6.59 0 Td[(1A1)]TJ /F10 11.955 Tf 13.56 2.65 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1D11XT1)]TJ /F4 11.955 Tf 11.96 0 Td[(X01separately.Observethat D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0=hD)]TJ /F6 7.97 Tf 6.58 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2ihW1 2B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0i=Mv,(4) whereM=D)]TJ /F6 7.97 Tf 6.59 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2andv=W1 2B)]TJ /F10 11.955 Tf 6.9 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U0.BytheSchwartzinequality, kMvk1=maxijMivjkvk2maxikMik2,(4) whereMiistheithrowofM.SinceW1 2commuteswithB,wehave kvk2=W1 2B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U02=BW1 2)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U02ckU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!,(4) wherec=maxkBik2,1iN,andW1 2)]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U02=kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!.ObservethatM=D)]TJ /F6 7.97 Tf 6.59 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2=W1 2D2:N+1)]TJ /F6 7.97 Tf 6.58 0 Td[(1In.Moreover,by( 4 ),eachrowofthematrixW1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1hasEuclidiannormsmallerthanp 2.Hence,M=W1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1InsatiseskMik2p 2.By( 4 ), kMvk1p 2kvk2.(4) Wecombine( 4 ),( 4 )and( 4 )toobtain D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U01ckU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!.(4) Bythesameanalysis,wehave D)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)=D)]TJ /F6 7.97 Tf 6.58 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.25 Td[(1 2W1 2(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0).(4) and D)]TJ /F6 7.97 Tf 6.59 0 Td[(1(p)]TJ /F10 11.955 Tf 12.17 0 Td[(p0)1p 2kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!.(4) 124

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ByProposition 4.1 ,D1=0,weobtainD2:N+11=)]TJ /F4 11.955 Tf 9.3 0 Td[(D1.SinceD2:N+1isinvertible,wehaveD)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1D1=)]TJ /F4 11.955 Tf 9.3 0 Td[(1.Hence,D)]TJ /F6 7.97 Tf 6.58 0 Td[(1D)]TJ /F6 7.97 Tf 6.59 0 Td[(11=D)]TJ /F6 7.97 Tf 6.58 0 Td[(12:N+1D1In1=1.By( 4 ),wehave D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A1)]TJ /F10 11.955 Tf 13.56 2.66 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1D11XT1)]TJ /F4 11.955 Tf 11.96 0 Td[(X10TcXT1)]TJ /F4 11.955 Tf 11.96 0 Td[(X10T.(4) Wecombine( 4 ),( 4 )and( 4 )tocompletetheproof. BythedenitionofX1:NandX2:N+1,wehave X1:N=I1XT1+InX2:N+1.(4) AfterreplacingX1:Nintheobjectivefunctionin( 4 )using( 4 )and( 4 ),weobtainaquadraticinUandthelineartermintheobjectivefunctionisLU,where L= 1 2IN)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.58 0 Td[(1\000A1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D1XT1)]TJ /F10 11.955 Tf 12.17 0 Td[(pV+z)]TJ /F14 11.955 Tf 11.95 0 Td[(!1s)]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=2!iqi!IN)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.58 0 Td[(1B+I1XT1+In)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1\000A1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D1XT1)]TJ /F10 11.955 Tf 12.17 0 Td[(p1 2WQIn)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1B+WS+qTIn)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.66 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1B+rT. ByLemma 24 ,theinnitynormofthematrix)]TJ /F10 11.955 Tf 7.09 -7.03 Td[(D)]TJ /F10 11.955 Tf 13.2 2.65 Td[(A)]TJ /F6 7.97 Tf 6.59 0 Td[(1=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(I)]TJ /F10 11.955 Tf 13.56 2.65 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A)]TJ /F6 7.97 Tf 6.58 0 Td[(1D)]TJ /F6 7.97 Tf 6.59 0 Td[(1isbounded,andbythesmoothnessassumption,thematricesA,A1,B,Q,SandVarealluniformlybounded.LetUdenotetheoptimalsolutionof( 4 ),andletU0denotetheoptimalsolutionassociatedwiththeperturbation0=(p0,y0,s0,q0,r0,z0).ByLemma 23 ,wehave kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!c(kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k!+kr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k!+jy)]TJ /F4 11.955 Tf 11.95 0 Td[(y0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j), (4) wherecisaconstantindependentofNandkk!isdenedin( 4 ).Observethat kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!= NXi=1!ijpi)]TJ /F4 11.955 Tf 11.95 0 Td[(p0ij2!1 2max1iNjpi)]TJ /F4 11.955 Tf 11.96 0 Td[(p0ij NXi=1!i!1 2=p 2kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k1. (4) 125

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Similarly, kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k!p 2kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k1andkr)]TJ /F4 11.955 Tf 11.96 0 Td[(r0k!p 2kr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k1.(4) Hence,( 4 )yields kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!c(kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k1+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k1+kr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k1+jy)]TJ /F4 11.955 Tf 11.95 0 Td[(y0j+js)]TJ /F4 11.955 Tf 11.95 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j). (4) ThegoalofthissectionistoshowkrT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kisboundedbyaconstantwhichisindependentofN.However,( 4 )isnotsufcientforustoreachthisgoalsincethecontrolUliesinL1and( 4 )onlygivesaboundofthe!-normofU.Inthefollowing,weneedtoanalyzeofkU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k1.First,letusdene Dz=DzIn,~A=0BBBB@A1...AN1CCCCA,~In=0BBBB@In...In1CCCCA,=0BBBB@T1...TN1CCCCA,!=(!1In,!2In,...,!NIn). Now,weintroducethefollowinglemmas: Lemma26. If(A3)and( 4 )hold,thenthematrixDzandI+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 11.65 2.65 Td[(~In!~AareinvertiblewithDz)]TJ /F6 7.97 Tf 12.7 0 Td[(112andI+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.8 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.58 0 Td[(112. Proof. ByProposition 4.3 and( 4 ),DzisinvertibleandkDz)]TJ /F6 7.97 Tf 12.71 0 Td[(1k1<2.Hence,Dz=DzInisalsoinvertible.Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1=Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1InandDz)]TJ /F6 7.97 Tf 12.7 0 Td[(11=Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(11<2.Byassumption(A3),kAk11 8.Hence,kDz)]TJ /F6 7.97 Tf 12.71 0 Td[(1Ak1kDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1k1kAk11 4and~In!~A1max1iNkAik1PNi=1!i1 4.By[ 32 ,p.351],I+Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1~A)]TJ /F10 11.955 Tf 10.86 2.65 Td[(~In!~AisinvertibleandI+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.58 0 Td[(112. Lemma27. If(A3)holds,thenforall(X,U)feasiblein( 4 ),thereexistsauniqueadjointvariablesuchthattheKKTconditionsof( 4 )hold.Letbetheadjointvariableassociatewith(X,U)andtheperturbationandlet0betheadjointvariable 126

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associatewith(X0,U0)andtheperturbation0,thenwehave )]TJ /F10 11.955 Tf 12.83 2.66 Td[(01c)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j,(4) wherecisaconstantindependentofN. Proof. ByLemma 22 ,theKKTconditionsofthequadraticprogrammingproblem( 4 )areidenticalto( 4 ).Bythelastequationof( 4 ), =XN+1VT+NXi=1!i)]TJ /F20 11.955 Tf 5.48 -9.69 Td[(iATi+XiQTi+UiSTi+z. Bytheaboveequation,thethirdandfourthcomponentsof( 4 )canbewrittenas Dz+~A)]TJ /F10 11.955 Tf 16.13 8.08 Td[(1 !1I1!~A=)]TJ /F10 11.955 Tf 10.91 2.65 Td[(Q+1 !1I1!QX1:N+)]TJ /F10 11.955 Tf 9.81 2.65 Td[(S+1 !1I1!SU+1 !1I1)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(VXTN+1+zT)]TJ /F10 11.955 Tf 15.12 2.65 Td[(W)]TJ /F6 7.97 Tf 6.59 0 Td[(1q. (4) SinceDz)]TJ /F6 7.97 Tf 12.7 0 Td[(11 !1~I1=~Inby( 4 ),wehaveDz+~A)]TJ /F10 11.955 Tf 16.4 8.09 Td[(1 !1I1!~A=DzI+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.65 Td[(~In!~A,whichisinvertiblebyLemma( 26 ).Itfollowsfrom( 4 )that =I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.65 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 10.91 2.65 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1Q+~In!QX1:N+I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S+~In!SU+I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(1~In)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(VXTN+1+zT)]TJ /F8 11.955 Tf 11.29 13.27 Td[(I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.65 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(1Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.58 0 Td[(1q. (4) By( 4 ),weobtainthatforall(X,U)feasiblein( 4 ),thereexistsauniqueadjointvariablesuchthattheKKTconditionsof( 4 )hold. 127

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Nowweshow( 4 )hold.By( 4 ),wehave )]TJ /F10 11.955 Tf 12.83 2.66 Td[(0=I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1Q+~In!Q)]TJ /F10 11.955 Tf 6.73 -7.03 Td[(X1:N)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X01:N+I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.65 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 10.91 2.65 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S+~In!S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U0+I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(1~InV)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XTN+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X0TN+1+I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(1~In)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(zT)]TJ /F4 11.955 Tf 11.95 0 Td[(z0T)]TJ /F8 11.955 Tf 11.29 13.27 Td[(I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.65 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(1Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.58 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0). Wetaketheinnitynormofbothsidesofthelastequation.SinceI+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.65 Td[(~In!~A)]TJ /F6 7.97 Tf 6.58 0 Td[(112byLemma 26 ,Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(11=Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(11<2by( 4 ),andQ1andjVj1areuniformlybounded,weobtain )]TJ /F10 11.955 Tf 12.83 2.66 Td[(01I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(11)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1Q+~In!Q1X1:N)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X01:N1+I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.65 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(11)]TJ /F10 11.955 Tf 10.91 2.65 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S+~In!S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U01+I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(11~InV1XTN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X0TN+1+I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(11~In1zT)]TJ /F4 11.955 Tf 11.96 0 Td[(z0T+I+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A)]TJ /F10 11.955 Tf 10.79 2.66 Td[(~In!~A)]TJ /F6 7.97 Tf 6.59 0 Td[(11Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.58 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)1cX2:N+1)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X02:N+11+jy)]TJ /F4 11.955 Tf 11.95 0 Td[(y0j+)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S+~In!S)]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U01+zT)]TJ /F4 11.955 Tf 11.95 0 Td[(z0T+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)1, (4) wherein( 4 ),weusethefollowinginequalityX1:N)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X01:N1+XTN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X0TN+1c)]TJ 5.48 .47 Td[(X2:N+1)]TJ /F10 11.955 Tf 13.21 2.66 Td[(X02:N+11+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0j.Inthefollowing,weestimate )]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S+~In!S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U01andDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)1.Observethat Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0=hDz)]TJ /F6 7.97 Tf 12.71 0 Td[(1W)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2ihW1 2S)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U0i.(4) 128

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SinceDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2=W1 2Dz)]TJ /F6 7.97 Tf 6.59 0 Td[(1In,andeachrowofthematrixW1 2Dz)]TJ /F6 7.97 Tf 6.58 0 Td[(1hasEuclidiannormsmallerthanp 2by( 4 ),wededucethateachrowofthematrixDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2hasEuclidiannormsmallerthanp 2.SinceW1 2commuteswithS,wehave W1 2S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.65 Td[(U02=SW1 2)]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.65 Td[(U02cU)]TJ /F10 11.955 Tf 13.39 2.65 Td[(U0! wherec=maxkSik2,1iNandW1 2)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U02=U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0!.Hence,bySchwartzinequality,wehave Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U01cU)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0!.(4) Observethat~In!S)]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.39 2.66 Td[(U0=~In!1 2SW1 2)]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0andeachrowofthematrix~In!1 2hasEuclidiannormsmallerthanp 2.Byasimilarestimateof( 4 ),Wehave k~In!S)]TJ /F10 11.955 Tf 6.9 -7.02 Td[(U)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0k1cU)]TJ /F10 11.955 Tf 13.38 2.66 Td[(U0!.(4) Bythesameanalysis,wehave Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)=Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2W1 2W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0). and Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F6 7.97 Tf 6.59 0 Td[(1(q)]TJ /F10 11.955 Tf 12.17 0 Td[(q0)1p 2kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k!.(4) Wesubstitute( 4 )( 4 )into( 4 )toobtain )]TJ /F10 11.955 Tf 12.82 2.65 Td[(01c)]TJ 5.48 .48 Td[(X2:N+1)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X02:N+11+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0j+kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k!. (4) By( 4 ),( 4 )reducesto )]TJ /F10 11.955 Tf 12.82 2.66 Td[(01c)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j, whichis( 4 ). 129

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Nowweintroducethefollowinglemma: Lemma28. LetUbefeasiblein( 4 ),andletU0bethecontrolassociatedwiththeperturbations0=(p0,y0,s0,q0,r0,z0),thenwehave kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k1c(kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k1+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k1+kr)]TJ /F4 11.955 Tf 11.96 0 Td[(r0k1+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0j+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j), (4) wherecisaconstantindependentofN. Proof. Bythecoercivityassumption(A2),wehave: UiRiUTiUiUTi,1iN. ThefthequationofrT()+=0in( 4 )is iBi+XiSTi+UiRTi+ri=0,1iN.(4) Thisisthenecessaryoptimalityconditionforthequadraticprogrammingproblem minUi2Rm1 2UiRiUTi+)]TJ /F20 11.955 Tf 5.48 -9.68 Td[(iBi+XiSTi+riUTi. ByLemma 23 ,wehave jUi)]TJ /F4 11.955 Tf 11.96 0 Td[(U0ij(ji)]TJ /F20 11.955 Tf 11.95 0 Td[(0ij+jXi)]TJ /F4 11.955 Tf 11.96 0 Td[(X0ij+jri)]TJ /F4 11.955 Tf 11.95 0 Td[(r0ij),(4) wherejjdenotesEuclidiannorm.Inthefollowing,wegiveboundsofeachtermintherightsideof( 4 )intermsofk)]TJ /F20 11.955 Tf 11.95 0 Td[(0k1. BythedenitionofL1-norm,wehavejri)]TJ /F4 11.955 Tf 11.96 0 Td[(r0ijkr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k1.Observethat jXi)]TJ /F4 11.955 Tf 11.95 0 Td[(X0ij=vuut nXj=1(xij)]TJ /F3 11.955 Tf 11.95 0 Td[(x0ij)2max1jnjxij)]TJ /F3 11.955 Tf 11.95 0 Td[(x0ijjp n)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kX2:N+1)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X02:N+1k1+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0jp n. (4) 130

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Wecombinethiswith( 4 )toobtain jXi)]TJ /F4 11.955 Tf 11.95 0 Td[(X0ijc)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0j.(4) Inthesamefashionwehaveji)]TJ /F20 11.955 Tf 11.95 0 Td[(0ij=vuut nXj=1(ij)]TJ /F14 11.955 Tf 11.96 0 Td[(0ij)2max1jnjij)]TJ /F14 11.955 Tf 11.96 0 Td[(0ijjp nk)]TJ /F10 11.955 Tf 12.83 2.66 Td[(0k1p n. By( 4 ),wehave ji)]TJ /F20 11.955 Tf 11.96 0 Td[(0ij)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j.(4) Wecombine( 4 ),( 4 )and( 4 )toobtainthatforeachi,1iN,wehave jUi)]TJ /F4 11.955 Tf 11.96 0 Td[(U0ij)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.95 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k!+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j+kr)]TJ /F4 11.955 Tf 11.96 0 Td[(r0k1. SincekU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k1=max1iNjUi)]TJ /F4 11.955 Tf 11.96 0 Td[(U0ij,theaboveinequalityyields kU)]TJ /F4 11.955 Tf 11.96 0 Td[(U0k1c)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(kU)]TJ /F4 11.955 Tf 11.95 0 Td[(U0k!+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k!+kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k!+jy)]TJ /F4 11.955 Tf 11.95 0 Td[(y0j+jz)]TJ /F4 11.955 Tf 11.96 0 Td[(z0j+kr)]TJ /F4 11.955 Tf 11.95 0 Td[(r0k1. Wecombinethiswith( 4 )( 4 )tocompletetheproof. Nowwegivethemainresultofthissection: Lemma29. Iftheassumption(A1)(A3)hold,thenkrT())]TJ /F6 7.97 Tf 6.58 0 Td[(1kisboundedbyaconstantindependentofN. Proof. Wecombine( 4 )and( 4 )( 4 )toobtain kX)]TJ /F4 11.955 Tf 11.96 0 Td[(X0k1c(kp)]TJ /F4 11.955 Tf 11.96 0 Td[(p0k1+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k1+kr)]TJ /F4 11.955 Tf 11.96 0 Td[(r0k1+jy)]TJ /F4 11.955 Tf 11.95 0 Td[(y0j+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j). (4) Wecombine( 4 )and( 4 )( 4 )toobtain k)]TJ /F20 11.955 Tf 11.96 0 Td[(0k1c(kp)]TJ /F4 11.955 Tf 11.95 0 Td[(p0k1+kq)]TJ /F4 11.955 Tf 11.96 0 Td[(q0k1+kr)]TJ /F4 11.955 Tf 11.96 0 Td[(r0k1+jy)]TJ /F4 11.955 Tf 11.96 0 Td[(y0j+js)]TJ /F4 11.955 Tf 11.96 0 Td[(s0j+jz)]TJ /F4 11.955 Tf 11.95 0 Td[(z0j). (4) 131

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By( 4 ),( 4 ),( 4 )andthedenitionofkk1in( 4 )andkk1in( 4 ),wehave k)]TJ /F20 11.955 Tf 11.95 0 Td[(0k1ck)]TJ /F20 11.955 Tf 11.95 0 Td[(0k1, whichshowsthatkrT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kc,isaconstantindependentofN. 4.8ProofofTheorem 4.1 WenowcollectresultstoproveTheorem 4.1 usingProposition 4.4 andthecorrespondencewiththecontrolproblemdescribedinsection 4.4 .ReferringtoLemma 29 ,let=krT())]TJ /F6 7.97 Tf 6.59 0 Td[(1k,andchoosesmallenoughsuchthat<1.ChoosersmallenoughsuchthatbyLemma 20 ,krT())-222(rT()k,forall2Br().Finally,byLemma 19 ,forNsufcientlarge,kT()k(1)]TJ /F14 11.955 Tf 12.31 0 Td[()=.SincealltheconditionsofProposition 4.4 aresatised,weobtaintheconclusionof( 4 );WecombinethiswithLemma 19 toobtaintheestimateof( 4 )ofTheorem 4.1 4.9ConcludingRemarks InthisChapter,weprovetheconvergenceofaRadaucollocationmethodappliedtoanunconstrainedcontrolproblem.Weshowthatundertheassumptionofcoercivityandsmoothness,thediscretenonlinearprogrammingproblemhasanextremepointandassociatedtransformedadjointvariablewhichconvergetothesolutionoftheoptimalcontrolproblemattherateO1 Nk)]TJ /F9 5.978 Tf 6.95 2.35 Td[(5 2,wherekisthenumberofcontinuousderivativesinthesolutionandNisthenumberoftheRadauquadraturepoints. 132

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CHAPTER5CONVERGENCEANALYSISFORAHPCOLLOCATIONMETHODFORUNCONSTRAINEDOPTIMALCONTROL 5.1BackgroundofhpCollocationMethod Inrecentyears,directGaussianquadratureorthogonalcollocationmethodshavebecomepopularapproachforsolvingoptimalcontrolproblems[ 4 5 16 20 22 23 34 37 39 42 47 51 ].InaGaussianquadraturecollocationmethod,theoptimalcontrolproblemsistranscribedtoanonlinearprogrammingproblem(NLP)byparameterizingthestateandcontrolusingglobalpolynomialsandcollocatingthedifferential-algebraicequationsusingnodesobtainedfromaquadraturescheme.Forproblemswherethesolutionsareinnitelysmoothandwellbehaved,aGaussianquadraturecollocationmethodhasasimplestructureandconvergesatexponentialrate[ 4 5 25 26 ].Themostwell-developedGaussianquadraturecollocationmethodsaretheGausscollocationmethod[ 4 5 ],theRadaucollocationmethod[ 25 26 39 ]andtheLobattocollocationmethod[ 16 ]. TheGaussianquadraturecollocationmethodsaretypicallyemployedaspmethodswhereasinglemeshintervalisused,andtheconvergenceisachievedbyincreasingthedegreeoftheglobalpolynomial[ 8 24 46 ].Thereareseverallimitationsofusingaglobalpolynomial.First,anaccurateapproximationmayrequireanunreasonablyhighdegreeglobalpolynomial.Second,thedensityofthediscreteNLPgrowsquicklyandtheproblembecomescomputationallyintractable.Incontrasttopmethods,thecommonlyusedcollocationmethodsforoptimalcontrolproblemisthehmethod,whereaxedlow-degreepolynomialisusedasstateapproximationandtheproblemisdividedintomanyintervals.Convergenceofthehmethodsareachievedbyincreasingthenumberofmeshintervals[ 6 13 14 ].Whilehmethodsarecomputationallymoretractablethanpmethods,theymayrequirealargenumberofmeshintervalsinordertoachieveanacceptableaccuracybecauseexponentialconvergenceislost.Inordertomakethecomputationmoretractableandalsohasafastconvergencerate,anhp-collocation 133

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methodisdeveloped[ 2 3 12 27 29 ].Thehp-collocationmethodisacombinationofthepmethodandhmethod.Morespecically,thetimeinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1]ispartitionedintosubintervalsandadifferentpolynomialisusedovereachsubinterval.Continuityispreservedbymatchingthevaluesofthestateattheinterfacebetweensubintervals. TheobjectiveofthisChapteristoprovidetheconvergenceanalysisforanhp-collocationmethodbasedonRadauquadrature.Ourresultshowsthat,underacoercivityandsmoothnessassumptions,theaccuracyofthediscreteNLPisgreatlyimprovedeitherbyimprovingthedegreeoftheapproximationpolynomialoneachsubintervalorbyincreasingthenumberofmeshintervals.Ouranalysisreliesonthepreviousestablishedconvergenceresultforap-methodbasedonRadauquadrature. 5.2TheControlProblemandhpCollocationMethod Weconsiderthefollowingunconstrainedoptimalcontrolproblem:minimizeC(x(1))subjectto_x(t)=f(x(t),u(t)),t2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1] (5)x()]TJ /F10 11.955 Tf 9.3 0 Td[(1)=x0, wherethestatex(t)2Rn,_xd dtx,thecontrolu(t)2Rm,f:RnRm!Rn,C:Rn!Randx0istheinitialcondition,whichweassumeisgiven. Notation.ThroughouttheChapter,Clisthecollectionofreal-valuedltimescontinuouslydifferentialfunctionsontheinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1].y(k)denotesthek-thderivativeofyfory2Ck,and_ydenotethederivativeofy.LetjjdenotestheabsolutevalueofscalarsortheEuclideannormofvectors.Letkk1denotestheuniformnormovertheinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1],thatis,kyk1=max)]TJ /F6 7.97 Tf 6.58 0 Td[(1t1jy(t)j,foranycontinuousfunctionyon[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1].PNdenotesthesetofpolynomialsofdegreeatmostN.Weassumeallvectorfunctionsoftimearerowvectors:x(t)=(x1(t),x2(t),...,xn(t)),u(t)=(u1(t),u2(t),...,um(t)).Givenvectorsaandb2Rn,thenotationha,bidenotestheinnerproductofaandbinRn.B(x)istheclosedballcenteredatxwithradius. 134

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ThetransposeofamatrixAisAT.Iff:Rn!Rm,thenrfisthembynJacobianmatrixwhosei-throwisrfi.Inparticular,thegradientofascalarvaluedfunctionisarowvector.LetL1denotethespaceofessentiallyboundedfunctionsandletW1,1denotethespaceofLipschitzcontinuousfunctionsdenedon[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1].Further,cisageneralconstantthathasdifferentvalueindifferentrelations. Thisfollowinghpcollocationschemeisintroducedin[ 11 12 ].Supposethetimeinterval[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1]isequallydividedintoameshconsideringKmeshintervals[tk)]TJ /F6 7.97 Tf 6.59 0 Td[(1,tk],k=1,...,K,where(t0,...,tK)arethemeshpoints;themeshpointshavethepropertythat)]TJ /F10 11.955 Tf 9.3 0 Td[(1=t0
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Finally,itisrequiredthatthestatebecontinuousattheinterfaceofmeshintervals;thatis,x)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t)]TJ /F7 7.97 Tf -.92 -8.28 Td[(k=x)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t+k,k=1,...,K)]TJ /F10 11.955 Tf 11.95 0 Td[(1. Wenowpresenttheassumptionsthatareemployedintheanalysisofthehpcollocationschemeof( 5 ). (A1)Smoothness.Problem( 5 )hasalocalminimizer(x,u)whichliesinCl+1(Rn)L1(Rm),forsomel3.ThereexistsanopensetRnRmand>0suchthatB(x(t),u(t))foreveryt2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1].ThersttwoderivativesoffareLipschitzcontinuousin,andthersttwoderivativesofCareLipschitzcontinuousinB(x(1)).Furthermore,thereexitsassociatedcostate2Cl+1(Rn)and2Rnforwhichthefollowingequations(Pontryagin'sminimumprinciple)aresatisedforx=x,u=u,=,and=: ()]TJ /F10 11.955 Tf 9.3 0 Td[(1)=, (5) (1)=rC(x(1)), (5) _(t)=rxH(x(t),u(t),(t)), (5) 0=ruH(x(t),u(t),(t)). (5) whereHistheHamiltoniandenedbyH(x,u,)=h,f(x,u)i. (A2)Coercivity.Forsome>0,thesmallesteigenvaluesofthematricesbelowaregreaterthan: V=rxxC(x(1))and0B@Q(t)S(t)ST(t)R(t)1CAforallt2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,1],(5) where Q(t)=rxxH(x(t),u(t),(t)),S(t)=ruxH(x(t),u(t),(t)),R(t)=ruuH(x(t),u(t),(t)). 136

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(A3).kA(t)k11 4hforallt2[)]TJ /F10 11.955 Tf 9.3 0 Td[(1,+1],whereA(t)=rxf(x(t),u(t)). ThehpcollocationmethodinthisChapterisdevelopedbydiscretizingtheoptimalcontrolproblemusingthepreviousdevelopedRadaucollocationMethod[ 34 ]. Ineachmeshintervalk2[1,...,K],wechooseNtheRadauquadraturepoints(1,...,N)ascollocationpoints,where1=)]TJ /F10 11.955 Tf 9.29 0 Td[(1.weintroduceN+1=+1asanon-collocationpoint.TheLagrangeinterpolatingpolynomialsrelativeto1,1,,N+1aregivenby Li()=N+1Yj=1j6=i)]TJ /F14 11.955 Tf 11.95 0 Td[(j i)]TJ /F14 11.955 Tf 11.95 0 Td[(j,i=1,N+1.(5) Thestateofthecontinuousoptimalcontrolproblemisapproximatedwithintheeachmeshintervalas x(k)()X(k)()=N+1Xi=1X(k)iLi(),(5)X(k)i,i=1,...,N+1isanapproximationofx(k)(i).Differentiating( 5 )andevaluatingatthecollocationpointj,j=1,2,,N,wehave _X(k)(j)=N+1Xi=1X(k)i_Li(j)=N+1Xi=1X(k)iDji,whereDji=_Li(j).(5) LetDbetheNbyN+1matrixwhose(j,i)elementisDji.LetDjdenotethej-thcolumnofD,1jN+1,andletDj:kbethesubmatrixofDformedbycolumnsofjthroughk.Let1denoteavectorwhosecomponentsareallequalto1.Thefollowingpropositionisestablishedin[ 25 ]: Proposition5.1. ThematrixD1:NandD2:N+1obtainedbydeletingeithertherstorthelastcolumnofDareinvertible.Moreover,D1=0. Inouranalysisofconvergence,weneedtoboundbothD)]TJ /F6 7.97 Tf 6.58 0 Td[(12:N+1andW1 2D2:N+1)]TJ /F6 7.97 Tf 6.58 0 Td[(1,whereWisthediagonalmatrixwhosediagonalelements!i,1iN,aretheRadauquadratureweights.Numerically,wehaveevaluatedD2:N+1andD)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1,andfoundthat D)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+11=2,(5) 137

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and max1iNW1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1i2=p 2,(5) whereW1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1idenotestheithrowofthematrixW1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Althoughwedonothaveaproofofthesetworesults,wenumericallycomputedD2:N+1andD)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1forNupto200andfoundthatkD)]TJ /F6 7.97 Tf 6.58 0 Td[(12:N+1k1andW1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1i2areboundbytheconstantsgivenabove. LetU(k)i(1iN)bearowvectorwhosecomponentsareanapproximationtothecontrolu(k)(i),1iN.Ourdiscreteapproximationtothesystemdynamics( 5 )isobtainedbyevaluatingthesystemdynamicsateachcollocationpointandreplacing_x(k)(i)byitsdiscreteapproximation_X(k)i.By( 5 )andthenotationofX(k)iandU(k)i,wehave N+1Xj=1DijX(k)j=h 2fX(k)i,U(k)i,1iN.(5) ContinuityacrossthemeshpointsismaintainedbytheconditionX(k)N+1=X(k+1)1. Thecontinuous-timenonlinearoptimalcontrolproblemcanbeapproximatedbythefollowingnite-dimensionalnonlinearprogrammingproblem minimizeCX(K)N+1subjecttoPN+1j=1DijX(k)j=h 2fX(k)i,U(k)i,1iN,1kK,X(1)1=x0,X(k)N+1=X(k+1)1,1kK)]TJ /F10 11.955 Tf 11.96 0 Td[(1.9>>>>>>>=>>>>>>>;(5) Wenowdeveloptherst-orderoptimalityconditionfor( 5 ).Let(k),k=1,...,K,beanNbynmatrixofLagrangemultiplierswhosei-throw(k)iisassociatedwithsystemdynamicPN+1j=1DijX(k)j=h 2fX(k)i,U(k)i.Let(0)bearowvectorofLagrangemultipliersassociatedwiththestateattheinitialtime.TheLagrangianassociatedwith 138

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( 5 )isL(,X,U)=C(X(K)N+1)+KXk=1NXi=1*(k)i, h 2fX(k)i,U(k)i)]TJ /F7 7.97 Tf 11.95 14.94 Td[(N+1Xj=1DijX(k)j!++D(0),x0)]TJ /F4 11.955 Tf 11.96 0 Td[(X(1)1E, whereX(k)N+1=X(k+1)1,for1kK)]TJ /F10 11.955 Tf 12.23 0 Td[(1.Therstorderoptimalityconditionsof( 5 )are rCX(K)N+1=DTN+1(K),rX(K)N+1L=0 (5) NXi=1Di1(1)i=h 2rXHX(1)1,U(1)1,(1)1)]TJ /F20 11.955 Tf 11.95 0 Td[((0),rX(1)1L=0 (5) NXi=1Di1(k)i+NXi=1DiN+1(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)i=h 2rXHX(k)1,U(k)1,(k)1, (5) 2kK,rX(k)1L=0NXi=1Dij(k)i=h 2rXHX(k)j,U(k)j,(k)j, (5) 2jN,1kK,rX(k)jL=0rUHX(k)i,U(k)i,(k)i=0,1iN,1kK,rU(k)iL=0. (5) Inordertorelatethecontinuousrst-orderconditions( 5 )( 5 )tothediscretecondition( 5 )( 5 ),weintroducetransformedadjointvariables: (k)i=1 !i(k)i,1iN,1kK, (5) (k)N+1=DTN+1(k),1kK. (5) LetDybeaNbyNmatrix,denedasfollows: Dy11=)]TJ /F3 11.955 Tf 9.29 0 Td[(D11)]TJ /F10 11.955 Tf 16.13 8.09 Td[(1 !1andDyij=)]TJ /F14 11.955 Tf 10.49 8.09 Td[(!j !iDjiotherwise.(5) 139

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Bysubstituting( 5 ),( 5 )and( 5 )into( 5 )-( 5 ),wehave rC(X(K)N+1)=(K)N+1, (5) NXj=1Dy1j(k)j=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(h 2rXHX(k)1,U(k)1,(k)1+1 !1(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)N+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)1, (5) 1kKNXj=1Dyij(k)j=)]TJ /F3 11.955 Tf 10.49 8.09 Td[(h 2rXHX(k)i,U(k)i,(k)i,2iN,1kK, (5) rUHX(k)i,U(k)i,(k)i=0,1iN,1kK, (5) where(0)N+1=(0).ByProposition 5.1 ,D1=0,whichimplies DN+1=)]TJ /F7 7.97 Tf 16.63 14.95 Td[(NXj=1Dj.(5) Bythedenitionof(k)N+1in( 5 )and( 5 ),wehave (k)N+1=NXi=1(k)iDi,N+1=)]TJ /F7 7.97 Tf 16.64 14.95 Td[(NXi=1NXj=1(k)iDij=(k)1 !1+NXi=1NXj=1(k)iDyji!j !i=(k)1 !1+NXi=1NXj=1(k)jDyij!i !j (5) =(k)1+NXi=1NXj=1!i(k)jDyij (5) =(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1)]TJ /F3 11.955 Tf 13.15 8.08 Td[(h 2NXi=1!irXHX(k)i,U(k)i,(k)i, (5) where( 5 )isby( 5 ),( 5 )followsfrom( 5 )and( 5 )isby( 5 )and( 5 ).( 5 )-( 5 )and( 5 )consistsoftherstorderoptimalityconditionsofthenonlinearprogrammingproblem( 5 ).Theyhavethesamestructureasthenecessaryconditionsforthecontinuouscontrolproblem. Proposition5.2. ThematrixDyisthedifferentiationmatrixforspaceofpolynomialsofdegreeN)]TJ /F10 11.955 Tf 11.14 0 Td[(1evaluatedati,1iN.Intheotherwords,ifpisapolynomialofdegree 140

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atmostN)]TJ /F10 11.955 Tf 11.96 0 Td[(1andifp2RNisavectorwithi-thcomponentpi=p(i),then (Dyp)i=_p(i),1iN. Proof. See[ 25 ]. Let Dz=Dy+0BBBBBBB@1 !10...01CCCCCCCA, then,wehavethefollowingproperty: Proposition5.3. Dzisinvertibleanditsinversehasthefollowingform: 0BBBBBBB@!1)]TJ /F14 11.955 Tf 9.29 0 Td[(!1~L2(1)...)]TJ /F14 11.955 Tf 9.3 0 Td[(!1~LN(1)!1R2)]TJ /F6 7.97 Tf 6.59 0 Td[(1~L2()d)]TJ /F14 11.955 Tf 11.96 0 Td[(!1~L2(1)...R2)]TJ /F6 7.97 Tf 6.59 0 Td[(1~LN()d)]TJ /F14 11.955 Tf 11.96 0 Td[(!1~LN(1)...!1RN)]TJ /F6 7.97 Tf 6.59 0 Td[(1~L2()d)]TJ /F14 11.955 Tf 11.96 0 Td[(!1~L2(1)...RN)]TJ /F6 7.97 Tf 6.59 0 Td[(1~LN()d)]TJ /F14 11.955 Tf 11.96 0 Td[(!1~LN(1)1CCCCCCCA,(5) where~LiistheLagrangeinterpolationpolynomialrelativetothequadraturepoints2,...,N,whichisgivenas ~Li()=NYj=2j6=i)]TJ /F14 11.955 Tf 11.96 0 Td[(j i)]TJ /F14 11.955 Tf 11.96 0 Td[(j,i=2,...,N. Proof. SeeProposition 4.3 Inouranalysisofconvergence,weneedtoboundbothDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1andW1 2Dz)]TJ /F6 7.97 Tf 6.58 0 Td[(1.Numerically,wehaveevaluatedDzandDz)]TJ /F6 7.97 Tf 12.71 0 Td[(1,andfoundthat Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(112,(5) 141

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and max1iNW1 2Dz)]TJ /F6 7.97 Tf 6.58 0 Td[(1i2p 2,(5) whereW1 2Dz)]TJ /F6 7.97 Tf 6.59 0 Td[(1idenotestheithrowofthematrixW1 2Dz)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Althoughwedonothaveaproofofthesetworesults,wenumericallycomputedDzandDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1forNupto200andfoundthatkDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1k1andW1 2Dz)]TJ /F6 7.97 Tf 6.59 0 Td[(1i2aremonotonicallyincreasingfunctionsofNthatapproachthelimitsgivenabove. Inordertoanalyzethediscreteproblem( 5 ),weneedtointroducediscreteanaloguesoftheL1norms.Inparticular,forasequencez=z1,z2,...,zNwhosei-thelementisavectorzi2Rn,wedenetheL1normasthefollowing: kzk1=sup1iNjzij, wherejjistheEuclideannorm. 5.3ConvergenceResult Let(x(t),u(t))bethelocalsolutionofthecontinuousoptimalproblem( 5 )and(t)betheassociatedcostate.WedenevectorsequencesX(k),U(k)and(k)byX(k)i=x(k)(i),1iN+1,U(k)i=u(k)(i),1iNand(k)i=(k)(i),1iN+1.Weshowthatundersuitableassumptions,thenonlinearprogrammingproblem( 5 )hasanextremepoint)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XN(k),UN(k)andtheassociatedLagrangemultiplierN(k)whichconvergesto)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(X(k),U(k)and(k)exponentiallyfastintheL1norm. Theorem5.1. Suppose( 5 )hasalocalminimizerxandu,andthereexistssuchthatthePontryagin'sminimumprincipleholds.Iftheassumptions(A1)(A3)hold,then( 5 )hasanextremepoint)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(XN(k),UN(k)andassociatedLagrangemultiplierN(k)satisfying max1kK)]TJ 5.48 .48 Td[(XN(k))]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)1+UN(k))]TJ /F4 11.955 Tf 11.95 0 Td[(U(k)1+N(k))]TJ /F20 11.955 Tf 11.95 0 Td[((k)1chl Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. (5) 142

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whereKisthenumberofmashintervals,histhelengthofeachmashinterval,NisthenumberofRadaupointsoneachmashintervallisdenedin(A1)andcisaconstantindependentofNandK 5.4AbstractSetting TheproofofTheorem 5.1 isbasedon[ 14 ,Proposition3.1].Forourconvergenceproof,werestatethepropositioninthefollowingform: Proposition5.4. LetXbeaBanachspaceandYbealinearnormedspacewiththenormsinbothspacesdenotedkk.LetT:X7)165(!YwithTcontinuouslyFrechetdifferentiableinBr()forsome2Xandr>0.Supposethat (P1)rT()isinvertible. (P2)Forsomescalar",krT())-222(rT()k"forall2Br(). If"<1,where=krT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kandkT()k(1)]TJ /F14 11.955 Tf 12.33 0 Td[(")r=,thenthereexistsaunique2Br()suchthatT()=0.Moreover,wehavetheestimate k)]TJ /F14 11.955 Tf 11.96 0 Td[(k 1)]TJ /F14 11.955 Tf 11.95 0 Td[("kT()k.(5) WeapplyProposition 5.4 toproveourTheorem 5.1 .First,weneedtospecifythenormedspaceX,YandthemapT.ThespaceXconsistsof3-tupleswhosecomponentsarevectorsequences=(X,U,),whereX=X(k)1,...,X(k)N+1,U=U(k)1,...,U(k)N,=(k)1,...,(k)N+1,and1kK.Let=(X,U,).ThenormisthediscreteL1normgivenby kk1=k(X,U,)k1=maxfkXk1,kUk1,kk1g.(5) 143

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ThemappingTisselectedinthefollowingway: T(X,U,)=0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@N+1Xj=1DijX(k)j)]TJ /F3 11.955 Tf 13.16 8.09 Td[(h 2fX(k)i,U(k)i,(1iN,1kk)X(1)1)]TJ /F4 11.955 Tf 11.95 0 Td[(x0,X(k)N+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k+1)1,(1kK)]TJ /F10 11.955 Tf 11.95 0 Td[(1)rCX(K)N+1)]TJ /F20 11.955 Tf 11.96 0 Td[((K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1+h 2NXi=1!irXHX(K)i,U(K)i,(K)i,NXj=1Dy1j(k)j+h 2rXHX(k)1,U(k)1,(k)1)]TJ /F10 11.955 Tf 16.13 8.09 Td[(1 !1(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)N+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)1,(1kK)NXj=1Dyij(k)j+h 2rXHX(k)i,U(k)i,(k)i,(2iN,1kK),rUHX(k)i,U(k)i,(k)i,(1iN,1kK),(k)N+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1+h 2NXi=1!irXHX(k)i,U(k)i,(k)i,(1kK).1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA(5) LetYdenotethespaceassociatedwiththeeightcomponentsofT.YisaspacewhoseelementsaresequencesandeachelementofthesequenceliesinL1RnL1RnL1L1L1L1. 144

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rT()=0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@N+1Xj=1DijX(k)j)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2X(k)iA(k)Ti)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2U(k)iB(k)Ti,(1iN,1kK)X(1)1,X(k)N+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k+1)1,(1kK)]TJ /F10 11.955 Tf 11.96 0 Td[(1)X(K)N+1VT)]TJ /F20 11.955 Tf 11.96 0 Td[((K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1+h 2NXi=1!i(K)iA(K)Ti+X(K)iQ(K)Ti+U(K)iS(K)Ti,NXj=1Dy1j(k)j+h 2(k)1A(k)T1+h 2X(k)1Q(k)T1+h 2U(k)1S(k)T1)]TJ /F10 11.955 Tf 13.47 8.09 Td[(1 !1(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)1,(1kK)NXj=1Dyij(k)j+h 2(k)iA(k)Ti+h 2X(k)iQ(k)Ti+h 2U(k)iS(k)Ti,(2iN,1kK)(k)iB(k)Ti+X(k)iS(k)Ti+U(k)iR(k)Ti,(1iN,1kK)(k)N+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1+h 2NXi=1!i(k)iA(k)Ti+X(k)iQ(k)Ti+U(k)iS(k)Ti,(1kK)1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA(5) where A(k)i=rxfX(k)i,U(k)i,B(k)i=rufX(k)i,U(k)i,Q(k)i=rxxHX(k)i,U(k)i,(k)i,S(k)i=ruxHX(k)i,U(k)i,(k)i,R(k)i=ruuHX(k)i,U(k)i,(k)i,V=r2xCX(K)N+1.9>>>>=>>>>;(5) 5.5ApproximationPreliminaries ToproveTheorem 5.1 ,weneedtoshowtheassumptionsofProposition 5.4 holdforNsufcientlarge.TheestimateofTheorem 5.1 isaconsequenceof( 5 ).In 145

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verifyingtheassumptionsofProposition 5.4 ,weutilizeseveralapproximationpropertiesofLagrangeinterpolationpolynomials.Inthissection,wepulltogetherthesepropertiesthatareexploitedthroughouttheanalysis. Lemma30. If`NistheLebesgueconstantdenedby `N=max2[)]TJ /F6 7.97 Tf 6.58 0 Td[(1,1]N+1Xj=1jLj()j,(5) whereLj(t)istheLagrangeinterpolationpolynomialdenedin( 5 ),then`N=O(logN). Proof. SeeTheorem 2.6 Lemma31. Let`N)]TJ /F6 7.97 Tf 6.59 0 Td[(1betheLebesgueconstantdenedby `N)]TJ /F6 7.97 Tf 6.58 0 Td[(1=max2[)]TJ /F6 7.97 Tf 6.59 0 Td[(1,1]NXj=1Lj(),(5) whereLi,i=1,...,N,aretheLagrangepolynomialsdenedas Li()=NYj=1j6=i)]TJ /F14 11.955 Tf 11.95 0 Td[(j i)]TJ /F14 11.955 Tf 11.95 0 Td[(j,(5) wherei,i=1,...,N,aretheRadauquadraturepoints.Then,`N)]TJ /F6 7.97 Tf 6.59 0 Td[(1=ON1 2. Proof. SeeTheorem 2.5 Lemma32. Foranyy2C1[)]TJ /F10 11.955 Tf 9.29 0 Td[(1,1],letyN2PNbetheinterpolatingpolynomialdenedasyN(ti)=PN+1i=1y(ti)Li(t),whereLi(t)isdenedin( 5 ).Then, _y)]TJ /F10 11.955 Tf 13.63 0 Td[(_yN1)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+2N2`Ninfq2PN)]TJ /F9 5.978 Tf 5.76 0 Td[(1k_y)]TJ /F3 11.955 Tf 11.96 0 Td[(qk1, where`NistheLebesgueconstantdenedin( 5 ). Proof. SeeTheorem 2.1 146

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Lemma33. (Jackson'sTheorem).Lety(t)2Cjandn>j0,then infp2Pnky)]TJ /F3 11.955 Tf 11.95 0 Td[(pk12cj njdjy dtj1,(5) wherecj=6j+1ej(1+j))]TJ /F6 7.97 Tf 6.58 0 Td[(1. Proof. SeeTheorem 2.2 5.6AnalysisofResidualandStationarity Inthissection,webegintoshowthatProposition 5 isapplicabletotheRadaucollocationmethodbyestimatingkT()k. Lemma34. Ifsmoothnessholds,thenthereexitsaconstantcindependentofNandK,suchthat kT()kchl+1 Nl)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2,(5) wherelisdenedinassumption(A1),andhisthelengthofmeshinterval. Proof. Bytherst-orderoptimalityconditionsforthecontinuous-timecontrolproblem( 5 )andthecontinuousrequirementofattheinterfaceofthemeshinterval,wehave x()]TJ /F10 11.955 Tf 9.3 0 Td[(1))]TJ /F4 11.955 Tf 11.95 0 Td[(x0=0,ruH)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(x(k)(i),u(k)(i),(k)(i)=0,x(k)(N+1)=x(k+1)(1). Hence,thesecond,thirdandseventhcomponentsofT()are0. NowweanalyzetherstcomponentsofT().Sincetheanalysisforeachk,1kKarethesame.Forsimplicity,wefocusonthekthmeshintervalandomittheupperindex(k)inthefollowingproof.Theithelementoftherstcomponentis T()1i=N+1Xj=1DijXj)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2f(Xi,Ui).(5) LetxN()denotethevectorwhosejthcomponentisxNj()=PN+1i=1xijLi(),wherexij=xj(i).BythedenitionofD,wehavePN+1j=1DijXj=_xN(i).Sincexsatisesthe 147

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dynamicsf(Xi,Ui)=_x(ti),( 5 )reducesto T()1i=_xN(i))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2_x(ti). (5) By( 5 ),wehaveh 2_x(ti)=_x(i), hence,( 5 )becomesT()1i=_xN(i))]TJ /F10 11.955 Tf 13.21 0 Td[(_x(i). ByLemma 30 ,Lemma 32 andLemma 33 ,( 5 )canbeboundedasfollows: _xNj(i))]TJ /F10 11.955 Tf 13.2 0 Td[(_xj(i)_xNj)]TJ /F10 11.955 Tf 13.54 0 Td[(_xj1)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+2N2O(logN)infq2PN)]TJ /F9 5.978 Tf 5.75 0 Td[(1_xj)]TJ /F3 11.955 Tf 11.95 0 Td[(q1)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+2N2O(logN)c Nldl+1xj dl+11clogN Nl)]TJ /F6 7.97 Tf 6.58 0 Td[(2dl+1xj dl+11 (5) By( 5 ),wehavedl+1xj dl+1()=h 2l+1dl+1xj dtl+1(t). Hence,( 5 )becomes _xNj(i))]TJ /F10 11.955 Tf 13.2 0 Td[(_xj(i)clogN Nl)]TJ /F6 7.97 Tf 6.59 0 Td[(2h 2l+1dl+1xj dtl+1(t)1chl+1logN Nl)]TJ /F6 7.97 Tf 6.58 0 Td[(2. Sincetheaboveinequalityistrueforeachiandk,wehavekT()1k1chl+1logN Nl)]TJ /F6 7.97 Tf 6.59 0 Td[(2. NowweestimatethesixthcomponentofT().Theithelementofthesixcomponentis T()6i=NXj=1Dyijj+h 2rXH(Xi,Ui,i).(5) LetN)]TJ /F6 7.97 Tf 6.59 0 Td[(1()denotethevectorwhosejthcomponentisN)]TJ /F6 7.97 Tf 6.58 0 Td[(1j()=PNi=1ijLi(),whereij=j(i)andLi()isdenedin( 5 ).BytheProposition 5.2 ,wehavePNj=1Dyijj=_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i).SinceisatisesrxH(Xi,Ui,i)=)]TJ /F10 11.955 Tf 11.56 2.66 Td[(_(ti)by( 5 ),( 5 ) 148

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reducesto T()6i=_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2_(ti)=_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i))]TJ /F10 11.955 Tf 14.22 2.66 Td[(_(i), (5) wherewechangevariablein( 5 )by( 5 ).ByLemmas 31 ,Lemma 32 andLemma 33 ,( 5 )canbeboundedasfollows: _N)]TJ /F6 7.97 Tf 6.58 0 Td[(1j(ti))]TJ /F10 11.955 Tf 14.22 2.65 Td[(_j(ti)_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1j)]TJ /F10 11.955 Tf 13.62 2.65 Td[(_j11+2(N)]TJ /F10 11.955 Tf 11.95 0 Td[(1)2ON1 2infq2PN)]TJ /F9 5.978 Tf 5.75 0 Td[(2_j)]TJ /F3 11.955 Tf 11.95 0 Td[(q11+2(N)]TJ /F10 11.955 Tf 11.95 0 Td[(1)2ON1 2c Nldl+1j dl+11c Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2dl+1j dl+11. (5) By( 5 ),wehavedl+1j dl+1()=h 2l+1dl+1j dtl+1(t). Hence,( 5 )becomes _N)]TJ /F6 7.97 Tf 6.59 0 Td[(1j(ti))]TJ /F10 11.955 Tf 14.22 2.66 Td[(_j(ti)c Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2h 2l+1dl+1j dtl+1(t)chl+1 Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. Sincetheaboveinequalityistrueforeachiandk,wehavekT()6k1chl+1 Nl)]TJ /F9 5.978 Tf 7.79 3.26 Td[(5 2. NowestimatethefthcomponentofT().Since(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)(N+1)=(k)(1),thefthcomponentbecomes NXj=1Dy1jj+rxH(X1,U1,1)(5) ByasimilarargumentastheestimateofthesixthcomponentofT(),wegettheL1normofthefthcomponentisboundedbychl+1 Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. NowestimatethefourthcomponentofT().Sincewerequire(t)iscontinuousattheinterfaceofmashinterval,wehave(K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(N+1)=(K)(1).By( 5 )and 149

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( 5 ),wehave T()4=rCX(K)N+1)]TJ /F20 11.955 Tf 11.96 0 Td[((K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1+h 2NXi=1!irXHX(K)i,U(K)i,(K)i=(K)(N+1))]TJ /F20 11.955 Tf 11.96 0 Td[((K)(1))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2NXi=1!i_(K)(ti)=(K)(1))]TJ /F20 11.955 Tf 11.96 0 Td[((K)()]TJ /F10 11.955 Tf 9.3 0 Td[(1))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1!i_(K)(i) (5) where( 5 )isbychangingvariable( 5 ). SinceN-pointRadauquadratureisexactforpolynomialofdegreeupto2N)]TJ /F10 11.955 Tf 12.12 0 Td[(2,wehave N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1))]TJ /F20 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()]TJ /F10 11.955 Tf 9.3 0 Td[(1))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!i_N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(i)=N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1))]TJ /F20 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()]TJ /F10 11.955 Tf 9.3 0 Td[(1))]TJ /F8 11.955 Tf 11.96 16.27 Td[(Z1)]TJ /F6 7.97 Tf 6.59 0 Td[(1_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1()dt=0. (5) Wesubtract( 5 )from( 5 )toobtain T()4=(K)(1))]TJ /F20 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1)+NXi=1!i)]TJ /F10 11.955 Tf 7.75 -7.03 Td[(_N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i))]TJ /F10 11.955 Tf 14.22 2.65 Td[(_(K)(i).(5) BytheestimationforthesixthcomponentofT(), _N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(i))]TJ /F10 11.955 Tf 14.22 2.65 Td[(_(K)(i)chl+1 Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2.(5) Wenowestimate(K)(1))]TJ /F20 11.955 Tf 11.95 0 Td[(N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1).Letq2PN)]TJ /F6 7.97 Tf 6.59 0 Td[(1,thenq()=PNi=1q(i)Li(),whereLi()isdenedin( 5 ).Inparticular,for=1,wehaveq(1)=PNi=1q(i)Li(1).Similarly,forthej-thcomponentofN)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1),wehaveNj(1)=PNi=1Nj(i)Li(1)= 150

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PNi=1(K)j(i)Li(1).Hence, (K)j(1))]TJ /F14 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1j(1)=(K)j(1))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1(K)j(i)Li(1)=(K)j(1))]TJ /F3 11.955 Tf 11.96 0 Td[(q(1)+NXi=1q(ti)Li(1))]TJ /F7 7.97 Tf 17.29 14.95 Td[(NXi=1(K)j(ti)Li(1)=(K)j(1))]TJ /F3 11.955 Tf 11.96 0 Td[(q(1))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1(K)j(ti))]TJ /F3 11.955 Tf 11.95 0 Td[(q(ti)Li(1)(K)j(1))]TJ /F3 11.955 Tf 11.96 0 Td[(q(1)+(K)j)]TJ /F3 11.955 Tf 11.96 0 Td[(q1NXi=1Li(1)(K)j)]TJ /F3 11.955 Tf 11.96 0 Td[(q1(1+`N)]TJ /F6 7.97 Tf 6.59 0 Td[(1). (5) SinceqisarbitraryinPN)]TJ /F6 7.97 Tf 6.59 0 Td[(1,( 5 )yields (K)j(1))]TJ /F14 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1j(1)infq2PN)]TJ /F9 5.978 Tf 5.75 0 Td[(1(K)j)]TJ /F3 11.955 Tf 11.96 0 Td[(q1(1+`N)]TJ /F6 7.97 Tf 6.58 0 Td[(1).(5) ByLemma 33 andchangingofvariable,wehave infq2PN)]TJ /F9 5.978 Tf 5.75 0 Td[(1j)]TJ /F3 11.955 Tf 11.95 0 Td[(q1chl+1 Nl+1dl+1(k)j dtl+11chl+1 Nl+1.(5) Wecombine( 5 ),( 5 )andLemma 31 toobtain (K)j(1))]TJ /F14 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1j(1)chl+1 Nl+1(1+`N)]TJ /F6 7.97 Tf 6.58 0 Td[(1)chl+1 Nk+1 2.(5) Hence,by( 5 )wehave (K)(1))]TJ /F20 11.955 Tf 11.96 0 Td[(N)]TJ /F6 7.97 Tf 6.59 0 Td[(1(1)chl+1 Nk+1 2.(5) 151

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Then,by( 5 )and( 5 ),( 5 )isboundedasfollows: jT()4j=(K)(1))]TJ /F20 11.955 Tf 11.95 0 Td[(N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1)+NXi=1!i)]TJ /F10 11.955 Tf 7.75 -7.02 Td[(_N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(ti))]TJ /F10 11.955 Tf 14.22 2.66 Td[(_(K)(ti)(K)(1))]TJ /F20 11.955 Tf 11.95 0 Td[(N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(1)+NXi=1!i_N)]TJ /F6 7.97 Tf 6.58 0 Td[(1(ti))]TJ /F10 11.955 Tf 14.23 2.66 Td[(_(K)(ti)chl+1 Nl+1 2+chl+1 Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2NXi=1!ichl+1 Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. NowweestimatethelastcomponentofT().Bythecontinuityrequirementof(t)attheinterfaceofmashintervalandby( 5 )and( 5 ),wehave (k)N+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)N+1+h 2NXi=1!irXHX(k)i,U(k)i,(k)i=(k)(N+1))]TJ /F20 11.955 Tf 11.95 0 Td[((k)(1))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2NXi=1!i_(k)(ti)=(k)(1))]TJ /F20 11.955 Tf 11.95 0 Td[((k)()]TJ /F10 11.955 Tf 9.3 0 Td[(1))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1!i_(k)(i). (5) Similartotheproofofthefourthcomponent,weobtainthat( 5 )issmallerthanchl+1 Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. WecombinetheestimatesfortheeighthcomponentsofT()toobtainthatkT()k1chl+1 Nl)]TJ /F9 5.978 Tf 7.78 3.26 Td[(5 2. Next,weestablishthecondition(P2)oftheProposition 5.4 Lemma35. Ifsmoothnessholds,theforeach">0,thereexistsr>0suchthatforall2Br(),then krT())-222(rT()k,(5) wherekkisthematrixnorminducedbytheL1normonXandY,andrisindependentofN. Proof. Throughouttheproof,weneedtoconsiderthedifferencebetweenderivativesoffandHevaluatedat(X,U,)andthesamederivativesevaluatedatapoint(X,U,).Forfurtherreferences,letA(k)andB(k),1kK,beblockdiagonalmatriceswith 152

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i-thblocksgivenby A(k)i=h 2A(k)i)-221(rXfX(k)i,U(k)i,B(k)i=h 2B(k)i)-221(rUfX(k)i,U(k)i,(5) whereA(k)i=rxfX(k)i,U(k)iandB(k)i=rufX(k)i,U(k)i.Similarly,letQ(k),S(k)andR(k)beblockdiagonalmatriceswiththeithblocksgivenby Q(k)i=h 2rXXHX(k)i,U(k)i,(k)i)]TJ /F4 11.955 Tf 11.96 0 Td[(Q(k)i,S(k)i=h 2rXUHX(k)i,U(k)i,(k)i)]TJ /F4 11.955 Tf 11.96 0 Td[(S(k)i,R(k)i=h 2rXXHX(k)i,U(k)i,(k)i)]TJ /F4 11.955 Tf 11.95 0 Td[(R(k)i,9>>>>>=>>>>>;(5) whereQ(k)i=rxxHX(k)i,U(k)i,(k)i,S(k)i=ruxHX(k)i,U(k)i,(k)i,andR(k)i=ruuHX(k)i,U(k)i,(k)i. LetrT()idenotetheiblockofrT()appearingin( 5 ).WiththisnotationrT())-222(rT()1=A B 0, whereAandBareblockdiagonalmatriceswhosek-thblockareA(k)andB(k),1kK.Bythedenitionofmatrixnorm,wehave k(rT())-222(rT())1k=max1iN1kKmaxkyk1kvk1A(k)iy+B(k)iv2max1iN1kKA(k)i2+B(k)i2max1iN1kKh 2rxfX(k)i,U(k)i)-222(rXfX(k)i,U(k)i2+rufX(k)i,U(k)i)-222(rUfX(k)i,U(k)i2, (5) where( 5 )isby( 5 ).Inordertomake( 5 )smallerthan",wechoosearadiusr,suchthatk)]TJ /F20 11.955 Tf 11.95 0 Td[(k1
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wehaveX(k)i)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)i
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Bychoosingrsufcientlysmall,wehavek(rT())-222(rT())6k<"fork)]TJ /F20 11.955 Tf 11.95 0 Td[(k1
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k(rT())-221(rT())7kmax1kKmaxkyk1kvk1krk1NXi=1!iQ(k)iy+S(k)iv+A(k)ir2maxi,kQ(k)i2+S(k)i2+A(k)i2 Bychoosingk)]TJ /F20 11.955 Tf 11.95 0 Td[(k1
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By( 5 ),( 5 )hasthefollowingform: 0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@N+1Xj=1DijX(k)j)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2X(k)iA(k)Ti)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2U(k)iB(k)Ti+h 2p(k)i=0,(1iN,1kK)X(1)1+h 2y=0,X(k)N+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k+1)1+h 2s(k)=0,(1kK)]TJ /F10 11.955 Tf 11.95 0 Td[(1)X(K)N+1VT)]TJ /F20 11.955 Tf 11.96 0 Td[((K)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1+h 2NXi=1!i(K)iA(K)Ti+X(K)iQ(K)Ti+U(K)iS(K)Ti+h 2z(K)=0,NXi=1Dy1i(k)i+h 2(k)1A(k)T1+h 2X(k)1Q(k)T1+h 2U(k)1S(k)T1)]TJ /F10 11.955 Tf 13.47 8.08 Td[(1 !1(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)1+h 2q(k)1=0,(1kK)NXj=1Dyij(k)j+h 2(k)iA(k)Ti+h 2X(k)iQ(k)Ti+h 2U(k)iS(k)Ti+h 2q(k)i=0,(2iN,1kK)(k)iB(k)Ti+X(k)iS(k)Ti+U(k)iR(k)Ti+h 2r(k)i=0,(1iN,1kK)(k)N+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1+h 2NXi=1!i(k)iA(k)Ti+X(k)iQ(k)Ti+U(k)iS(k)Ti+h 2z(k)=0,(1kK)1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA.(5) Wedenethefollowingcolumnvector: X(k)1:N=0BBBBBBB@X(k)T1X(k)T2...X(k)TN1CCCCCCCA,U(k)=0BBBBBBB@U(k)T1U(k)T2...U(k)TN1CCCCCCCA,q(k)=0BBBBBBB@q(k)T1q(k)T2...q(k)TN1CCCCCCCA,r(k)=0BBBBBBB@r(k)T1r(k)T2...r(k)TN1CCCCCCCA. 157

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Letusconsiderthefollowingquadraticprogrammingproblem: minimizeB(X,U)+h 2KXk=1q(k)TWX(k)1:N+h 22KXk=1r(k)TWU(k)+h 2KXk=1 z(k))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1!iq(k)i!X(k)TN+1,subjecttoPN+1j=1DijX(k)j)]TJ /F3 11.955 Tf 13.16 8.09 Td[(h 2X(k)iA(k)Ti)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2U(k)iB(k)Ti+h 2p(k)i=0,(1iN,1kK)X(1)1+h 2y=0,X(k)N+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k+1)1+h 2s(k)=0(1kK)]TJ /F10 11.955 Tf 11.96 0 Td[(1),9>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>;(5) where B(X,U)=h 4KXk=1NXi=1!iX(k)iQ(k)iX(k)Ti+2X(k)iS(k)iU(k)Ti+U(k)iR(k)iU(k)Ti+1 2X(K)N+1VX(K)TN+1. (5) Inthefollowing,werstshowthatBisstronglyconvex,hence( 5 )hasauniquesolution.MoreovertheKKTconditionsassociatewith( 5 )canbeexpressedintheformof( 5 ).SinceBisconvex,itfollowsthatanysolutionof( 5 )alsoyieldsasolutionof( 5 )(aswellasthemultipliersassociatedwiththeconstraints). Lemma36. Ifthecoercivityholds,thenthequadraticformBdenedin( 5 )isstronglyconvex. Proof. BythecoercivityassumptionandthedenitionofmatricesQ(k)i,S(k)i,R(k)iandVin( 5 ),wehave X(K)N+1VX(K)TN+1X(K)N+12, and X(k)iQ(k)iX(k)Ti+2X(k)iS(k)iU(k)Ti+U(k)iR(k)iU(k)TiX(k)i2+U(k)i2, 158

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where1iNand1kK.Hence, B(X,U) 2 X(K)N+12+h 2KXk=1NXi=1!iX(k)i2+U(k)i2!(5) Ifwedenethe!-normsofXandUasfollows: kXk2!=h 2KXk=1NXi=1!iX(k)i2+X(K)N+12andkUk2!=h 2KXk=1NXi=1!iU(k)i2.(5) Itfollowsfrom( 5 )thatB(X,U) 2kXk2!+kUk2!.Hence,Bisstronglyconvex. Lemma37. Thequadraticprogrammingproblem( 5 )and( 5 )haveidenticaluniquesolution. Proof. TheLagrangianassociatewith( 5 )is L(,X,U)=B(X,U)+h 2KXk=1Wq(k)TX(k)1:N+h 22KXk=1Wr(k)TU(k)+h 2KXk=1 z(k))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!iq(k)i!X(k)TN+1)]TJ /F7 7.97 Tf 16.47 14.94 Td[(KXk=1NXi=1(k)i N+1Xj=1DijX(k)j)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2X(k)iA(k)Ti)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2U(k)iB(k)Ti+h 2p(k)i!T+K)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xk=1(k)N+1X(k)N+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k+1)1+h 2s(k)T)]TJ /F20 11.955 Tf 11.96 0 Td[((0)N+1X(1)1+h 2yT. WenowformulatetheKKToptimalityconditionsfor( 5 ).TheseconditionscorrespondtosettingthederivativeoftheLagrangianwithrespecttoX,U,andtozero. 159

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ThepartialderivativeoftheLagrangianwithrespecttoX(k)1(1kK)is: 0=rX(k)1L=h 2!1X(k)1Q(k)T1+U(k)1S(k)T1+q(k)1)]TJ /F7 7.97 Tf 17.29 14.95 Td[(NXi=1(k)iDi1+h 2(k)1A(k)T1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)N+1=h 2!1X(k)1Q(k)T1+U(k)1S(k)T1+q(k)1)]TJ /F7 7.97 Tf 17.29 14.95 Td[(NXi=1 (k)i !i!i !1Di1!!1+h 2!1(k)1 !1A(k)T1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)N+1. AftersubstitutingforDi1intermsofDy1iusing( 5 ),for1intermsof1using( 5 )anddividing!1bothsidesofthelastequation,weobtain NXi=1Dy1i(k)i+h 2(k)1A(k)T1+h 2X(k)1Q(k)T1+h 2U(k)1S(k)T1)]TJ /F10 11.955 Tf 13.46 8.08 Td[(1 !1(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)1+h 2q(k)1=0, (5) where1kK,whichistheforthequationin( 5 ). ThepartialderivativeoftheLagrangianwithrespecttoX(k)i,2iN,1kKis: 0=rX(k)iL=h 2!iX(k)iQ(k)Ti+U(k)iS(k)Ti+q(k)i)]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXj=1(k)jDji+h 2(k)iA(k)Ti=h 2!iX(k)iQ(k)Ti+U(k)iS(k)Ti+q(k)i)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXj=1 (k)j !j!j !iDji!!i+h 2!i(k)i !iA(k)Ti=!i NXj=1Dyij(k)j+h 2(k)iA(k)Ti+h 2X(k)iQ(k)Ti+h 2U(k)iS(k)Ti+h 2q(k)i! (5) where( 5 )isby( 5 )and( 5 ).Wedivide!ibothsidesof( 5 )toobtainthefthequationin( 5 ). ThepartialderivativeoftheLagrangianwithrespecttoX(K)N+1is: 0=rX(K)N+1L=X(K)N+1VT)]TJ /F7 7.97 Tf 17.29 14.94 Td[(NXi=1(K)iDiN+1+h 2 z(K))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!iq(K)i!. 160

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WesubstituteDiN+1intheaboveequationby( 5 )toobtain 0=X(K)N+1VT+NXi=1NXj=1(K)iDij+h 2 z(K))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!iq(K)i!=X(K)N+1VT+NXj=1NXi=1 i(K) !i!i !jDij!!j+h 2 z(K))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1!iq(K)i!=X(K)N+1VT)]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXj=1!jNXi=1Dyji(K)i)]TJ /F20 11.955 Tf 11.96 0 Td[((K)1+h 2 z(K))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!iq(K)i! (5) BysubstitutePNi=1DyjiKiin( 5 )by( 5 )and( 5 ),weobtainthethirdequationin( 5 ). ThepartialderivativeoftheLagrangianwithrespecttoX(k)N+1(1kK)]TJ /F10 11.955 Tf 11.95 0 Td[(1)is: 0=rX(k)N+1L=(k)N+1)]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1(k)iDiN+1+h 2 z(k))]TJ /F7 7.97 Tf 17.3 14.95 Td[(NXi=1!iq(k)i!. (5) ByasimiliaranalysisasthepartialderivativeoftheLagrangianwithrespecttoX(K)N+1,wecanverifythatthe( 5 )isequaltothelastequationof( 5 ). ThepartialderivativeoftheLagrangianwithrespecttoU(k)i,1iN,1kKis: 0=rU(k)iL=h 2!iX(k)iS(k)Ti+U(k)iR(k)Ti+h 2(k)iB(k)Ti+h 22!ir(k)i=h 2!iX(k)iS(k)Ti+U(k)iR(k)Ti+(k)iB(k)Ti+h 2r(k)i. (5) Sinceh 2!i6=0,( 5 )yieldsthesixthequationin( 5 ). ThepartialderivativeoftheLagrangianwithrespectto(k)i,1iN,1kKand(0)N+1,wecangettherstandsecondcomponentof( 5 ). Hence,theKKTconditionsassociatewith( 5 )canbeexpressedintheformof( 5 ).SinceBisstronglyconvexbyLemma 36 ,( 5 )hasauniquesolution.Also,itfollowsthatanysolutionof( 5 )alsoyieldsasolutionof( 5 ).Hence,thequadraticprogrammingproblem( 5 )and( 5 )haveidenticaluniquesolution. 161

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Inthefollowing,weusemathematicalinductiontoshowthatkrT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kisboundedbyaconstantwhichdoesnotdependonN.Ourproofisbasedon[ 30 ,Lemma1],whichisstatedbelow: Lemma38. Letbeasymmetric,continuousbilinearformdenedonanonempty,closedconvexsubsetKofaHilbertspaceV,andlethiVdenotetheHilbertspaceinnerproduct.Ifthereexist>0suchthat (w)]TJ /F3 11.955 Tf 11.96 0 Td[(v,w)]TJ /F3 11.955 Tf 11.96 0 Td[(v)hw)]TJ /F3 11.955 Tf 11.96 0 Td[(v,w)]TJ /F3 11.955 Tf 11.96 0 Td[(viVforallw,v2K, thenforany2V,thequadraticprogram minimize1 2(v,v))-222(h,viVsubjecttov2K(5) hasauniquesolutionw.Thissolutionistheuniquew2Kthatsatisesthevariationalinequality(w,v)]TJ /F3 11.955 Tf 12.3 0 Td[(w)h,v)]TJ /F3 11.955 Tf 12.29 0 Td[(wiVforallv2K.Ifwidenotesthesolutionof( 5 )correspondingto=i,fori=1andi=2,thenwehave kw1)]TJ /F3 11.955 Tf 11.95 0 Td[(w2kk1)]TJ /F14 11.955 Tf 11.96 0 Td[(2k, wherekkisthenorminducedbytheHilbertspaceinnerproduct. Proof. See[ 30 ,Lemma1]. Inordertocompletetheproofbymathematicalinduction,werstfocusonthekthmeshinterval.TheinitialstateX(k)1isknownwhenweanalyzethekthintervalby 162

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X(k)1=X(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1.By( 5 ),X(k),U(k)and(k)satisfythefollowingequations: 0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@N+1Xj=1DijX(k)j)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2X(k)iA(k)Ti)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2U(k)iB(k)Ti+h 2p(k)i=0,(1iN)X(k)1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2s(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=0,X(k)N+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k+1)1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2s(k)=0,NXi=1Dy1i(k)i+h 2(k)1A(k)T1+h 2X(k)1Q(k)T1+h 2U(k)1S(k)T1)]TJ /F10 11.955 Tf 13.47 8.09 Td[(1 !1(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)1+h 2q(k)1=0,NXj=1Dyij(k)j+h 2(k)iA(k)Ti+h 2X(k)iQ(k)Ti+h 2U(k)iS(k)Ti+h 2q(k)i=0,(2iN)(k)iB(k)Ti+X(k)iS(k)Ti+U(k)iR(k)Ti+h 2r(k)i=0,(1iN)(k)N+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1+h 2NXi=1!i(k)iA(k)Ti+X(k)iQ(k)Ti+U(k)iS(k)Ti+h 2z(k)=0.1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA(5) Wedenethe!-normsofX(k)andU(k)asfollows: X(k)2!=NXi=1!iX(k)i2andU(k)2!=NXi=1!iU(k)i2.(5) LetB(k)(X,U)=1 2NXi=1!iX(k)iQ(k)iX(k)Ti+2X(k)iS(k)iU(k)Ti+U(k)iR(k)iU(k)Ti. Ifthecoercivityassumptionholds,thenB(k)(X,U) 2X(k)2!+U(k)2!.Hence,B(k)(X,U)isconvex.ByasimilaranalysistotheproofofLemma 37 ,weobtainthat 163

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( 5 )hasidenticaluniquesolutiontothefollowingquadraticprogrammingproblem minimizeh 2B(k)(X,U)+h 2q(k)TWX(k)1:N+h 22r(k)TWU(k)+h 2 z(k))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!iq(k)i!X(k)TN+1,subjecttoPN+1j=1DijX(k)j)]TJ /F3 11.955 Tf 13.16 8.09 Td[(h 2X(k)iA(k)Ti)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2U(k)iB(k)Ti+h 2p(k)i=0,(1iN)X(k)1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)N+1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2s(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)=0,X(k)N+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k+1)1+h 2s(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)=0.9>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>;(5) TheapplicationofLemma 38 tothequadraticprogrammingproblem( 5 )proceedsasfollows.Werstshowthatiftheassumption(A3)holds,thenforallU(k)2L1,thereexistsauniqueX(k)feasiblein( 5 ).Inotherwords,wecanwriteX(k)intermsofU(k).Then,byLemma 38 ,weobtainaboundforkU(k)k!intermsofk(k)k!,where(k)denotestheperturbationofonthekthinterval.TherelationbetweenX(k)andU(k)aregivenbythefollowingtwolemmas.First,wemakethefollowingnotations:X(k)2:N+1=0BBBB@X(k)T2...X(k)TN+11CCCCA,A(k)=0BBBBBBB@0A(k)20......A(k)N01CCCCCCCA,A(k)1=0BBBBBBB@A(k)10...01CCCCCCCA,B(k)=0BBBB@B(k)1...B(k)N1CCCCA,Q(k)=0BBBB@Q(k)1...Q(k)N1CCCCA,p(k)=0BBBB@p(k)1...p(k)N1CCCCA,S(k)=0BBBB@S(k)1...S(k)N1CCCCAIn=0BBBBBBB@0In0......In01CCCCCCCA,I1=0BBBBBBB@In0...01CCCCCCCA, 164

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IN=(0,...,0,In),D=D2:N+1In,D1=D1In,W=WIn. wheredenotestheKroneckerproduct. Lemma39. If(A3)and( 5 )hold,thenthematrixI)]TJ /F3 11.955 Tf 14.09 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k)isinvertibleandI)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(111+hc,wherecisaconstantindependentofN. Proof. ByProposition 5.1 and( 5 ),D2:N+1isinvertibleandkD)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1k1=2.Hence,D=D2:N+1Inisalsoinvertible.D)]TJ /F6 7.97 Tf 6.59 0 Td[(1=D)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1InandD)]TJ /F6 7.97 Tf 6.59 0 Td[(11=D)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+11=2.Byassumption(A3),kA(k)k11 2h.Hence, kD)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k)k1kD)]TJ /F6 7.97 Tf 6.58 0 Td[(1k1kA(k)k11 h.(5) By[ 32 ,p.351],I)]TJ /F3 11.955 Tf 13.2 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k)isinvertibleandI)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k))]TJ /F6 7.97 Tf 6.58 0 Td[(111+hc,forsomeconstantcindependentofN. Lemma40. If(A3)holds,thenforallU(k)2L1,thereexistsauniqueX(k)feasiblein( 5 ).LetX(k)befeasiblein( 5 )andletX(k)0bethestate,associatedwiththecontrolU(k)0andperturbationsp(k)0,thenwehave X(k)2:N+1)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X(k)02:N+11hc)]TJ 5.48 .48 Td[(U(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)0!+p(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)0!+(1+hc)X(k)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)0T11, (5) wherecisaconstantindependentofN. Proof. By( 5 ), D)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2A(k)X(k)2:N+1=h 2B(k)U(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2p(k)+h 2A(k)1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D1X(k)T1.(5) ByLemma 39 ,D)]TJ /F3 11.955 Tf 13.15 8.08 Td[(h 2A(k)=DI)]TJ /F3 11.955 Tf 13.15 8.08 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k)isinvertible.Itfollowsfrom( 5 )that X(k)2:N+1=D)]TJ /F3 11.955 Tf 13.15 8.08 Td[(h 2A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1h 2B(k)U(k))]TJ /F3 11.955 Tf 13.15 8.08 Td[(h 2p(k)+h 2A(k)1)]TJ /F10 11.955 Tf 13.57 2.65 Td[(D1X(k)T1. (5) 165

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Hence,forallU(k)2L1,thereexistsauniqueX(k)feasiblein( 5 ).Now,weshowthat( 5 )holds.By( 5 ),wehave: X(k)2:N+1)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X(k)02:N+1=h 2D)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1B(k))]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.65 Td[(U(k)0)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 5.7 -9.68 Td[(p(k))]TJ /F10 11.955 Tf 12.17 0 Td[(p(k)0+D)]TJ /F3 11.955 Tf 13.16 8.09 Td[(h 2A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1h 2A(k)1)]TJ /F10 11.955 Tf 13.56 2.66 Td[(D1X(k)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0T1=h 2I)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B(k))]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U(k))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(k)0)]TJ /F3 11.955 Tf 10.49 8.09 Td[(h 2I)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1D)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 5.7 -9.68 Td[(p(k))]TJ /F10 11.955 Tf 12.17 0 Td[(p(k)0+I)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k)1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1D1X(k)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0T1. Wetaketheinnitynormofbothsidesofthelastequationtoobtain X(k)2:N+1)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X(k)02:N+11h 2I)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(11D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)01+h 2I)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(11D)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 5.7 -9.69 Td[(p(k))]TJ /F10 11.955 Tf 12.17 0 Td[(p(k)01+I)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(11h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k)1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1D11X(k)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0T11hc)]TJ 5.47 .48 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(k)01+D)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 5.7 -9.69 Td[(p(k))]TJ /F10 11.955 Tf 12.17 0 Td[(p(k)01+(1+hc)h 2D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A(k)1)]TJ /F10 11.955 Tf 13.57 2.65 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1D11X(k)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)0T11, (5) where( 5 )isbyLemma 39 .Inthefollowing,weestimateD)]TJ /F6 7.97 Tf 6.58 0 Td[(1B(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)01,D)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 5.7 -9.68 Td[(p(k))]TJ /F10 11.955 Tf 12.17 0 Td[(p(k)01andh 2D)]TJ /F6 7.97 Tf 6.58 0 Td[(1A(k)1)]TJ /F10 11.955 Tf 13.57 2.65 Td[(D)]TJ /F6 7.97 Tf 6.59 0 Td[(1D11X(k)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0T11separately.Observethat D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(k)0=hD)]TJ /F6 7.97 Tf 6.59 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2ihW1 2B(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0i=Mv, (5) 166

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whereM=D)]TJ /F6 7.97 Tf 6.59 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2andv=W1 2B(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0.BytheSchwartzinequality, kMvk1=maxijMivjkvk2maxikMik2,(5) whereMiistheithrowofM.SinceW1 2commuteswithB,wehave kvk2=W1 2B(k))]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)02=B(k)W1 2)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(k)02cU(k))]TJ /F4 11.955 Tf 11.95 0 Td[(U(k)0!, (5) wherec=maxkB(k)ik2,1iN,andW1 2)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(k)02=U(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)0!.ObservethatM=D)]TJ /F6 7.97 Tf 6.58 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.25 Td[(1 2=W1 2D2:N+1)]TJ /F6 7.97 Tf 6.58 0 Td[(1In.Moreover,by( 5 ),eachrowofthematrixW1 2D2:N+1)]TJ /F6 7.97 Tf 6.58 0 Td[(1hasEuclidiannormsmallerthanp 2.Hence,M=W1 2D2:N+1)]TJ /F6 7.97 Tf 6.59 0 Td[(1InsatiseskMik2p 2.By( 5 ), kMvk1p 2kvk2.(5) Wecombine( 5 ),( 5 )and( 5 )toobtain D)]TJ /F6 7.97 Tf 6.59 0 Td[(1B(k))]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)01cU(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)0!.(5) Bythesameanalysis,wehave D)]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F10 11.955 Tf 5.69 -9.69 Td[(p(k))]TJ /F10 11.955 Tf 12.17 0 Td[(p(k)0=D)]TJ /F6 7.97 Tf 6.58 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2W1 2)]TJ /F10 11.955 Tf 5.7 -9.69 Td[(p(k))]TJ /F10 11.955 Tf 12.17 0 Td[(p(k)0.(5) and D)]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 5.7 -9.69 Td[(p(k))]TJ /F10 11.955 Tf 12.17 0 Td[(p(k)01p 2p(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)0!.(5) ByProposition 5.1 ,D1=0,weobtainD2:N+11=)]TJ /F4 11.955 Tf 9.3 0 Td[(D1.SinceD2:N+1isinvertible,wehaveD)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1D1=)]TJ /F4 11.955 Tf 9.3 0 Td[(1.Hence,D)]TJ /F6 7.97 Tf 6.58 0 Td[(1D)]TJ /F6 7.97 Tf 6.59 0 Td[(111=D)]TJ /F6 7.97 Tf 6.58 0 Td[(12:N+1D1In1=1.Hence, h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A1)]TJ /F10 11.955 Tf 13.57 2.65 Td[(D)]TJ /F6 7.97 Tf 6.58 0 Td[(1D11X(k)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0T11(1+hc)X(k)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)0T11. (5) 167

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Wecombine( 5 ),( 5 )and( 5 )tocompletetheproof. BythedenitionofX(k)1:NandX(k)2:N+1,wehave X(k)1:N=I1X(k)T1+InX(k)2:N+1.(5) AfterreplacingX(k)1:Nintheobjectivefunctionin( 5 )using( 5 )and( 5 ),weobtainaquadraticinU(k)andthelineartermintheobjectivefunctionisLU(k),where L=h 4 I1X(k)T1+InD)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2A(k))]TJ /F6 7.97 Tf 6.58 0 Td[(1h 2A(k)1)]TJ /F10 11.955 Tf 13.57 2.66 Td[(D1X(k)T1)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2p(k)!W h 2Q(k)InD)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1B(k)+2S(k)!+h2 4 q(k)TWIn+ z(k))]TJ /F7 7.97 Tf 17.3 14.94 Td[(NXi=1!iq(k)i!IN!D)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1B(k)+h2 4r(k)TW. ByLemma 39 ,theinnitynormofthematrix D)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1=I)]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2D)]TJ /F6 7.97 Tf 6.59 0 Td[(1A(k))]TJ /F6 7.97 Tf 6.58 0 Td[(1D)]TJ /F6 7.97 Tf 6.58 0 Td[(1isbounded,andbythesmoothnessassumption,thematricesA(k),A(k)1,B(k),Q(k),andS(k)arealluniformlybounded.LetU(k)denotetheoptimalsolutionof( 5 ),andletU(k)0denotetheoptimalsolutionassociatedwiththeperturbation(k)0=)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(p(k)0,y(k)0,q(k)0,r(k)0,z(k)0.ByLemma 38 ,wehave U(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)0!cX(k)1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)011+hp(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)0!+hq(k))]TJ /F4 11.955 Tf 11.96 0 Td[(q(k)0!+hr(k))]TJ /F4 11.955 Tf 11.95 0 Td[(r(k)0!+hz(k))]TJ /F4 11.955 Tf 11.95 0 Td[(z(k)0, (5) wherecisaconstantindependentofNandh,andkk!isdenedin( 5 ).Observethat p(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)0!= NXi=1!ip(k)i)]TJ /F4 11.955 Tf 11.95 0 Td[(p(k)0i2!1 2max1iNp(k)i)]TJ /F4 11.955 Tf 11.95 0 Td[(p(k)0i NXi=1!i!1 2=p 2p(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)01. 168

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Similarly, q(k))]TJ /F4 11.955 Tf 11.95 0 Td[(q(k)0!p 2q(k))]TJ /F4 11.955 Tf 11.95 0 Td[(q(k)01,r(k))]TJ /F4 11.955 Tf 11.95 0 Td[(r(k)0!p 2r(k))]TJ /F4 11.955 Tf 11.96 0 Td[(r(k)01.(5) Hence,( 5 )yields U(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)0!cX(k)1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)011+hp(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)01+hq(k))]TJ /F4 11.955 Tf 11.96 0 Td[(q(k)01+hr(k))]TJ /F4 11.955 Tf 11.95 0 Td[(r(k)01+hz(k))]TJ /F4 11.955 Tf 11.95 0 Td[(z(k)0, (5) ThegoalofthissectionistoshowkrT())]TJ /F6 7.97 Tf 6.59 0 Td[(1kisboundedbyaconstantwhichisindependentofNandh.However,( 5 )isnotsufcientforustoreachthisgoalsincethecontrolUliesinL1and( 5 )onlygivesaboundofthe!-normofU(k).Inthefollowing,weneedtoanalyzeofU(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)01.First,weintroducethefollowinglemma: Lemma41. LetX(k)befeasiblein( 5 )andletX(k)0bethestate,associatedwiththecontrolU(k)0andperturbations(k)0,thenwehave X(k)1:N+1)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X(k)01:N+11cmax1jkp(j))]TJ /F4 11.955 Tf 11.95 0 Td[(p(j)01k K+cmax1jkq(j))]TJ /F4 11.955 Tf 11.95 0 Td[(q(j)01k K+cmax1jkr(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01k K+cmax1jkz(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0k K+cmax1jks(j))]TJ /F4 11.955 Tf 11.95 0 Td[(s(j)01k K+cX(1)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)0T11, (5) wherecisaconstantindependentofNandK,andX(k)1:N+1isacolumnvectorbystackingX(k)Ti,1iN+1. 169

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Proof. Inordertoshow( 5 )hold,werstshowthefollowinginequalityholds: X(k)Ti)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)0Ti1kXj=1)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(k)...)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(j+1)hc(j))]TJ 5.48 .48 Td[(p(j))]TJ /F4 11.955 Tf 11.95 0 Td[(p(j)01+q(j))]TJ /F4 11.955 Tf 11.96 0 Td[(q(j)01+r(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01+z(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0+kXj=2)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(j)h 2s(j))]TJ /F4 11.955 Tf 11.95 0 Td[(s(j)01+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(1)X(1)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)0T11. (5) wherec(j),1jk,isaconstantindependentofhandN.Weusethesuperindex(j)todenotethatthetheconstantc(j)isrelatedtothej-thinterval. Inthefollowing,weprove( 5 )bymathematicalinduction.ByLemma 40 ,wehave X(k)2:N+1)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X(k)02:N+11hc(k))]TJ 5.48 .48 Td[(U(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)0!+p(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)0!+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)X(k)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)0T11. Hence,foreachi,1iN+1,wehave X(k)Ti)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0Ti1hc(k))]TJ 5.48 .48 Td[(U(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)0!+p(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)0!+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)X(k)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0T11. WereplaceU(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)0!intheaboveinequalityby( 5 )toobtain X(k)Ti)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0Ti1hc(k))]TJ 5.48 .48 Td[(p(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)01+q(k))]TJ /F4 11.955 Tf 11.96 0 Td[(q(k)01+r(k))]TJ /F4 11.955 Tf 11.96 0 Td[(r(k)01+z(k))]TJ /F4 11.955 Tf 11.95 0 Td[(z(k)0+)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(k)X(k)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0T11. (5) Bymakingk=1in( 5 ),weknowthat( 5 )istruefork=1.Wesuppose( 5 )istruefork)]TJ /F10 11.955 Tf 12.28 0 Td[(1.SinceX(k)1=X(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)N+1+h 2s(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)andX(k)01=X(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0N+1+h 2s(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0, 170

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( 5 )canbefurtherwrittenas X(k)Ti)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)0Ti1hc(k))]TJ 5.48 .48 Td[(p(k))]TJ /F4 11.955 Tf 11.95 0 Td[(p(k)01+q(k))]TJ /F4 11.955 Tf 11.96 0 Td[(q(k)01+r(k))]TJ /F4 11.955 Tf 11.95 0 Td[(r(k)01+z(k))]TJ /F4 11.955 Tf 11.95 0 Td[(z(k)0+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)X(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)TN+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0TN+11+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)h 2s(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1))]TJ /F4 11.955 Tf 11.96 0 Td[(s(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)01. (5) Sincewesupposethat( 5 )isholdfork)]TJ /F10 11.955 Tf 11.95 0 Td[(1,wehave X(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)TN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0TN+11k)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xj=1)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)...)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(j+1)hc(j))]TJ 5.48 .48 Td[(p(j))]TJ /F4 11.955 Tf 11.95 0 Td[(p(j)01+q(j))]TJ /F4 11.955 Tf 11.96 0 Td[(q(j)01+r(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01+z(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0+k)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xj=2)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(j)h 2s(j))]TJ /F4 11.955 Tf 11.95 0 Td[(s(j)01+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(1)X(1)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)0T11. (5) ByreplacingX(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)TN+1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)0TN+11in( 5 )by( 5 )weshowthat( 5 )isholdfork.Hence,weprovethat( 5 )istrueforall1kK. Nowweshowthat( 5 )isholdby( 5 ).Sinceh=2 K,wehave )]TJ /F10 11.955 Tf 5.47 -9.68 Td[(1+hc(K)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(1)1+cm KKecm=c,(5) wherecm=2max1jKc(j).By( 5 ),thecoefcientsin( 5 )areboundedasfollows: )]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(1))]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(K)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(1)c.(5) kXj=1)]TJ /F10 11.955 Tf 5.47 -9.69 Td[(1+hc(k)...)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(j+1)hc(j)p(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01cmax1jkp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01kXj=1h=cmax1jkp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01k K. (5) 171

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Thecoefcientsforq(j))]TJ /F4 11.955 Tf 11.95 0 Td[(q(j)01,r(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01,s(j))]TJ /F4 11.955 Tf 11.96 0 Td[(s(j)01,z(j))]TJ /F4 11.955 Tf 11.95 0 Td[(z(j)0in( 5 )canbeanalyzedinthesamewayasin( 5 ).Wereplacethecoefcientsin( 5 )by( 5 )and( 5 )tocompletetheproofof( 5 ). Letusdene Dz=DzIn,~A(k)=0BBBB@A(k)1...A(k)N1CCCCA,~In=0BBBB@In...In1CCCCA,(k)=0BBBB@(k)01...(k)0N1CCCCA,!=(!1In,!2In,...,!NIn). Now,weintroducethefollowinglemmas: Lemma42. If(A3)and( 5 )hold,thenthematrixDzandI+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.47 8.09 Td[(h 2~In!~A(k)areinvertiblewithDz)]TJ /F6 7.97 Tf 12.71 0 Td[(112andI+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.08 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(11(1+hc),wherecisaconstantwhichindependentofhandN. Proof. ByProposition 5.3 and( 5 ),DzisinvertibleandkDz)]TJ /F6 7.97 Tf 12.71 0 Td[(1k1<2.Hence,Dz=DzInisalsoinvertible.Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1=Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1InandDz)]TJ /F6 7.97 Tf 12.7 0 Td[(11=Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(11<2.Byassumption(A3),kA(k)k11 4h.Hence,h 2Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1A(k)1h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(11A(k)11 4andh 2~In!~A(k)1h 2max1iNkA(k)ik1NXi=1!i1 4.By[ 32 ,p.351],I+h 2Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.2 8.08 Td[(h 2~In!~A(k)isinvertibleandI+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(11(1+hc). Lemma43. If(A3)holds,thenforall)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(X(k),U(k)feasiblein( 5 ),thereexistsauniqueadjointvariable(k)suchthattheKKTconditionsof( 5 )hold.Let(k)betheadjointvariableassociatewith)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(X(k),U(k)andtheperturbation(k)andlet(k)0betheadjoint 172

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variableassociatewith)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(X(k)0,U(k)0andtheperturbation(k)0,thenwehave (k))]TJ /F10 11.955 Tf 12.83 2.66 Td[((k)01cmax1jkp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01k K+cmax1jkq(j))]TJ /F4 11.955 Tf 11.95 0 Td[(q(j)01k K+cmax1jkr(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01k K+cmax1jks(j))]TJ /F4 11.955 Tf 11.96 0 Td[(s(j)01k K+cmax1jkz(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0k K+cX(1)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)0T11+(1+hc)(k)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)0TN+11. (5) wherecisaconstantindependentofN. Proof. ByLemma 37 ,theKKTconditionsofthequadraticprogrammingproblem( 5 )areidenticalto( 5 ).Bythelastequationof( 5 ), (k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)TN+1=(k)TN+1+h 2!~A(k)(k)+h 2!Q(k)X(k)1:N+h 2!S(k)U(k)+h 2z(k)T.(5) Bytheaboveequation,thethirdandfourthcomponentsof( 5 )canbewrittenas Dz+h 2~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 21 !1I1!~A(k)(k)=h 2)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Q(k)+1 !1I1!Q(k)X(k)1:N+h 2)]TJ /F10 11.955 Tf 9.82 2.66 Td[(S(k)+1 !1I1!S(k)U(k)+1 !1I1(k)TN+1+h 2zT)]TJ /F3 11.955 Tf 13.15 8.08 Td[(h 2q(k). (5) SinceDz)]TJ /F6 7.97 Tf 12.7 0 Td[(11 !1~I1=~Inby( 4 ),wehaveDz+h 2~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 21 !1I1!~A(k)=DzI+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k), 173

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whichisinvertiblebyLemma( 42 ).Itfollowsfrom( 5 )that (k)=h 2I+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1Q(k)+~In!Q(k)X(k)1:N+h 2I+h 2Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S(k)+~In!S(k)U(k)+I+h 2Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.08 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1~In(k)TN+1+h 2z(k)T)]TJ /F3 11.955 Tf 10.49 8.09 Td[(h 2I+h 2Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1q(k). (5) By( 5 ),weobtainthatforall)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(X(k),U(k)feasiblein( 5 ),thereexistsauniqueadjointvariable(k)suchthattheKKTconditionsof( 5 )hold. Nowweshow( 5 )hold.By( 5 ),wehave (k))]TJ /F10 11.955 Tf 12.83 2.66 Td[((k)0=h 2I+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1Q(k)+~In!Q(k)X(k)1:N)]TJ /F10 11.955 Tf 13.21 2.66 Td[(X(k)01:N+h 2I+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.58 0 Td[(1)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1S(k)+~In!S(k))]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0+I+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.08 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(1~In(k)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)0TN+1+h 2I+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.58 0 Td[(1~In)]TJ /F4 11.955 Tf 5.47 -9.69 Td[(z(k)T)]TJ /F4 11.955 Tf 11.95 0 Td[(z(k)0T)]TJ /F3 11.955 Tf 10.5 8.09 Td[(h 2I+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.58 0 Td[(1Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1)]TJ /F10 11.955 Tf 5.7 -9.68 Td[(q(k))]TJ /F10 11.955 Tf 12.17 0 Td[(q(k)0. Wetaketheinnitynormofbothsidesoflastequation.Since I+h 2Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1~A(k))]TJ /F3 11.955 Tf 13.15 8.09 Td[(h 2~In!~A(k))]TJ /F6 7.97 Tf 6.59 0 Td[(11(1+hc)byLemma 42 ,Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(11=Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(11<2by( 5 ),andQ(k)1andS(k)1areuniformlybounded,weobtain (k))]TJ /F10 11.955 Tf 12.83 2.66 Td[((k)01hcX(k)1:N)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X(k)01:N1+)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S(k)+~In!S(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(k)01+Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1)]TJ /F10 11.955 Tf 5.69 -9.68 Td[(q(k))]TJ /F10 11.955 Tf 12.17 0 Td[(q(k)01+z(k)T)]TJ /F4 11.955 Tf 11.95 0 Td[(z(k)0T+(1+hc)(k)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)0TN+11. (5) 174

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Inthefollowing,weestimate)]TJ /F10 11.955 Tf 10.91 2.66 Td[(Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S(k)+~In!S(k))]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)01and Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1)]TJ /F10 11.955 Tf 5.69 -9.69 Td[(q(k))]TJ /F10 11.955 Tf 12.18 0 Td[(q(k)01.Observethat Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S(k))]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U(k))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(k)0=hDz)]TJ /F6 7.97 Tf 12.71 0 Td[(1W)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2ihW1 2S(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0i. (5) SinceDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2=W1 2Dz)]TJ /F6 7.97 Tf 6.59 0 Td[(1In,andeachrowofthematrixW1 2Dz)]TJ /F6 7.97 Tf 6.58 0 Td[(1hasEuclidiannormsmallerthanp 2by( 5 ),wededucethateachrowofthematrixDz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F9 5.978 Tf 7.78 3.26 Td[(1 2hasEuclidiannormsmallerthanp 2.SinceW1 2commuteswithS(k),wehave W1 2S(k))]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)02=S(k)W1 2)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)02cU(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0! wherec=maxkS(k)ik2,1iNandW1 2)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)02=U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0!.Hence,bySchwartzinequality,wehave Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1S(k))]TJ /F10 11.955 Tf 6.9 -7.02 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)01cU(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0!.(5) Observethat~In!S(k))]TJ /F10 11.955 Tf 6.9 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.65 Td[(U(k)0=~In!1 2S(k)W1 2)]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.39 2.65 Td[(U(k)0andeachrowofthematrix~In!1 2hasEuclidiannormsmallerthanp 2.Byasimilarestimateof( 5 ),Wehave k~In!S(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0k1cU(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0!.(5) Wecombine( 5 )and( 5 )toobtainthat Dz)]TJ /F6 7.97 Tf 12.71 0 Td[(1S(k)+~In!S(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.65 Td[(U(k)01cU(k))]TJ /F10 11.955 Tf 13.38 2.65 Td[(U(k)0!.(5) Bythesameanalysis,wehave Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1)]TJ /F10 11.955 Tf 5.69 -9.69 Td[(q(k))]TJ /F10 11.955 Tf 12.17 0 Td[(q(k)0=Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1W)]TJ /F9 5.978 Tf 7.79 3.26 Td[(1 2W1 2)]TJ /F10 11.955 Tf 5.69 -9.69 Td[(q(k))]TJ /F10 11.955 Tf 12.17 0 Td[(q(k)0. and Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(1)]TJ /F10 11.955 Tf 5.7 -9.68 Td[(q(k))]TJ /F10 11.955 Tf 12.17 0 Td[(q(k)01p 2q(k))]TJ /F10 11.955 Tf 12.17 0 Td[(q(k)0!.(5) 175

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Wesubstitute( 5 )and( 5 )into( 5 )toobtain (k))]TJ /F10 11.955 Tf 12.83 2.66 Td[((k)01hcX(k)1:N)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X(k)01:N1+U(k))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(k)0!+q(k))]TJ /F10 11.955 Tf 12.18 0 Td[(q(k)0!+z(k)T)]TJ /F4 11.955 Tf 11.96 0 Td[(z(k)0T+(1+hc)(k)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)0TN+11. (5) By( 5 )and( 5 ),( 5 )reduceto (k))]TJ /F10 11.955 Tf 12.83 2.66 Td[((k)01hcX(k)1:N)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X(k)01:N1+p(k))]TJ /F10 11.955 Tf 12.17 0 Td[(p(k)01+q(k))]TJ /F10 11.955 Tf 12.17 0 Td[(q(k)01+r(k))]TJ /F10 11.955 Tf 11.13 0 Td[(r(k)01+z(k)T)]TJ /F4 11.955 Tf 11.96 0 Td[(z(k)0T+(1+hc)(k)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)0TN+11. (5) WereplaceX(k)1:N)]TJ /F10 11.955 Tf 13.21 2.66 Td[(X(k)01:N1intheaboveinequalityby( 5 )tocompletetheproofof( 5 ). Nowweintroducethefollowinglemma: Lemma44. LetU(k)befeasiblein( 5 ),andletU(k)0bethecontrolassociatedwiththeperturbations(k)0,thenwehave U(k))]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)01cmax1jkp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01k K+cmax1jkq(j))]TJ /F4 11.955 Tf 11.95 0 Td[(q(j)01k K+cmax1jkr(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01k K+cmax1jks(j))]TJ /F4 11.955 Tf 11.96 0 Td[(s(j)01k K+cmax1jkz(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0k K+cX(1)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)0T11+(1+hc)(k)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)0TN+11. (5) wherecisaconstantindependentofN. Proof. Bythecoercivityassumption(A2),wehave U(k)iR(k)iU(k)TiU(k)iU(k)Ti,for1iN. Considerthefthcomponentof( 5 ) (k)iB(k)Ti+X(k)iS(k)Ti+U(k)iR(k)Ti+h 2r(k)i=0.(5) 176

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( 5 )isthenecessaryoptimalityconditionforthequadraticprogrammingproblem minU(k)i2Rm1 2U(k)iR(k)iU(k)Ti+((k)iB(k)i+X(k)iS(k)Ti+h 2r(k)i)U(k)Ti. ByLemma 38 ,wehave U(k)i)]TJ /F4 11.955 Tf 11.95 0 Td[(U(k)0i(k)i)]TJ /F20 11.955 Tf 11.95 0 Td[((k)0i+X(k)i)]TJ /F4 11.955 Tf 11.96 0 Td[(X(k)0i+h 2r(k)i)]TJ /F4 11.955 Tf 11.96 0 Td[(r(k)0i,(5) where1iN.By( 5 )and( 5 ),( 5 )yield U(k)i)]TJ /F4 11.955 Tf 11.96 0 Td[(U(k)0icmax1jkp(j))]TJ /F4 11.955 Tf 11.95 0 Td[(p(j)01k K+cmax1jkq(j))]TJ /F4 11.955 Tf 11.95 0 Td[(q(j)01k K+cmax1jkr(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01k K+cmax1jkz(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0k K+cX(1)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(1)0T11+(1+hc)(k)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)0TN+11. (5) Since( 5 )istrueforeachi,1iN.Bythedenitionofinnitynorm,weprove( 5 )hold. Lemma45. Let(k)betheadjointvariableassociatewith)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(X(k),U(k)andthepertur-bation(k)andlet(k)0betheadjointvariableassociatewith)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(X(k)0,U(k)0andtheperturbation(k)0,thenwehave max1jk(j))]TJ /F10 11.955 Tf 12.83 2.65 Td[((j)01c(k)max1jkp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01k K+c(k)max1jkq(j))]TJ /F4 11.955 Tf 11.96 0 Td[(q(j)01k K+c(k)max1jkr(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01k K+c(k)max1jks(j))]TJ /F4 11.955 Tf 11.96 0 Td[(s(j)01k K+c(k)max1jkz(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0k K+cX(1)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(1)0T11+)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(1)...)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(k)(k)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)0TN+11. (5) wherec(k)isaconstantindependentofNandK. 177

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Proof. Weprove( 5 )bymathematicalinduction.By( 5 ),( 5 )istruefork=1.Wesuppose( 5 )istruefork)]TJ /F10 11.955 Tf 11.96 0 Td[(1.By( 5 ),wehave (k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0TN+1=(k)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)0TN+1+h 2!~A(k))]TJ /F10 11.955 Tf 6.36 -7.03 Td[((k))]TJ /F10 11.955 Tf 12.83 2.66 Td[((k)0+h 2!Q(k)X(k)1:N)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X(k)01:N+h 2!S(k))]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0+h 2)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(z(k)T)]TJ /F4 11.955 Tf 11.95 0 Td[(z(k)T0. Wetaketheinnitynormofbothsidesofthelastequationtoobtain (k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0TN+11hc(k))]TJ /F10 11.955 Tf 12.83 2.65 Td[((k)01+X(k)1:N)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X(k)01:N1+!Q(k))]TJ /F10 11.955 Tf 6.9 -7.02 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)01+z(k))]TJ /F4 11.955 Tf 11.96 0 Td[(z(k)0+(k)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)0TN+11. (5) Since!Q(k))]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0=!1 2Q(k)W1 2)]TJ /F10 11.955 Tf 6.91 -7.02 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0,bySchwartzinequality,wehave !Q(k))]TJ /F10 11.955 Tf 6.91 -7.03 Td[(U(k))]TJ /F10 11.955 Tf 13.38 2.65 Td[(U(k)01cU(k))]TJ /F10 11.955 Tf 13.39 2.65 Td[(U(k)0!. Hence,( 5 )yields (k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0TN+11hc(k))]TJ /F10 11.955 Tf 12.83 2.66 Td[((k)01+X(k)1:N)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X(k)01:N1+U(k))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(k)0!+z(k))]TJ /F4 11.955 Tf 11.96 0 Td[(z(k)0+(k)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)0TN+11. (5) Wereplace(k))]TJ /F10 11.955 Tf 12.83 2.65 Td[((k)01andU(k))]TJ /F10 11.955 Tf 13.38 2.65 Td[(U(k)0!in( 5 )by( 5 )and( 5 )toobtain (k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0TN+11hcX(k)1:N)]TJ /F10 11.955 Tf 13.2 2.66 Td[(X(k)01:N1+p(k))]TJ /F4 11.955 Tf 11.95 0 Td[(p(k)01+q(k))]TJ /F4 11.955 Tf 11.96 0 Td[(q(k)01+r(k))]TJ /F4 11.955 Tf 11.96 0 Td[(r(k)01+z(k))]TJ /F4 11.955 Tf 11.96 0 Td[(z(k)0+(1+hc)(k)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)0TN+11. (5) 178

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Sincewesupposethat( 5 )istruefork)]TJ /F10 11.955 Tf 11.96 0 Td[(1,wehave max1jk)]TJ /F6 7.97 Tf 6.58 0 Td[(1(j))]TJ /F10 11.955 Tf 12.82 2.66 Td[((j)01c(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1p(j))]TJ /F4 11.955 Tf 11.95 0 Td[(p(j)01k)]TJ /F10 11.955 Tf 11.96 0 Td[(1 K+c(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1q(j))]TJ /F4 11.955 Tf 11.96 0 Td[(q(j)01k)]TJ /F10 11.955 Tf 11.96 0 Td[(1 K+c(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1r(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01k)]TJ /F10 11.955 Tf 11.95 0 Td[(1 K+c(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1s(j))]TJ /F4 11.955 Tf 11.96 0 Td[(s(j)01k)]TJ /F10 11.955 Tf 11.95 0 Td[(1 K+c(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1z(j))]TJ /F4 11.955 Tf 11.95 0 Td[(z(j)0k)]TJ /F10 11.955 Tf 11.95 0 Td[(1 K+cX(1)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(1)0T11+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(1)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)0TN+11. (5) By( 5 ),thelasttermin( 5 )canbewrittenas )]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(1)...)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)0TN+11hc(k)X(k)1:N)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X(k)01:N1+p(k))]TJ /F4 11.955 Tf 11.96 0 Td[(p(k)01+q(k))]TJ /F4 11.955 Tf 11.95 0 Td[(q(k)01+r(k))]TJ /F4 11.955 Tf 11.95 0 Td[(r(k)01+z(k))]TJ /F4 11.955 Tf 11.96 0 Td[(z(k)0+)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(1)...)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(k)(k)TN+1)]TJ /F20 11.955 Tf 11.96 0 Td[((k)0TN+11, (5) wherein( 5 ),weusethefactthat)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(1)...)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)isboundedbysomeconstantc.Wecombine( 5 ),( 5 )and( 5 )toobtain max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1(j))]TJ /F10 11.955 Tf 12.83 2.66 Td[((j)01c(k)max1jkp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01k K+c(k)max1jkq(j))]TJ /F4 11.955 Tf 11.96 0 Td[(q(j)01k K+c(k)max1jkr(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01k K+c(k)max1jks(j))]TJ /F4 11.955 Tf 11.96 0 Td[(s(j)01k K+c(k)max1jkz(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0k K+cX(1)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(1)0T11+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(1)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(k)(k)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)0TN+11. (5) Wecombine( 5 )with( 5 )toobtainthat( 5 )istruefork.Hence( 5 )holdsforeachk,1kK. 179

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Lemma46. LetU(k)befeasiblein( 5 ),andletU(k)0bethecontrolassociatedwiththeperturbations(k)0,thenwehave max1jkU(j))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(j)01c(k)max1jkp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01k K+c(k)max1jkq(j))]TJ /F4 11.955 Tf 11.96 0 Td[(q(j)01k K+c(k)max1jkr(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01k K+c(k)max1jks(j))]TJ /F4 11.955 Tf 11.96 0 Td[(s(j)01k K+c(k)max1jkz(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0k K+cX(1)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(1)0T11+)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(1)...)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(k)(k)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)0TN+11. (5) wherec(k)isaconstantindependentofNandK. Proof. Weprove( 5 )bymathematicalinduction.By( 5 ),( 5 )istruefork=1.Wesuppose( 5 )istruefork)]TJ /F10 11.955 Tf 11.96 0 Td[(1.Hence, max1jk)]TJ /F6 7.97 Tf 6.58 0 Td[(1U(j))]TJ /F10 11.955 Tf 13.38 2.66 Td[(U(j)01c(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1p(j))]TJ /F4 11.955 Tf 11.95 0 Td[(p(j)01k)]TJ /F10 11.955 Tf 11.96 0 Td[(1 K+c(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1q(j))]TJ /F4 11.955 Tf 11.96 0 Td[(q(j)01k)]TJ /F10 11.955 Tf 11.96 0 Td[(1 K+c(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1r(j))]TJ /F4 11.955 Tf 11.96 0 Td[(r(j)01k)]TJ /F10 11.955 Tf 11.95 0 Td[(1 K+c(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1s(j))]TJ /F4 11.955 Tf 11.96 0 Td[(s(j)01k)]TJ /F10 11.955 Tf 11.95 0 Td[(1 K+c(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)max1jk)]TJ /F6 7.97 Tf 6.59 0 Td[(1z(j))]TJ /F4 11.955 Tf 11.95 0 Td[(z(j)0k)]TJ /F10 11.955 Tf 11.95 0 Td[(1 K+cX(1)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(1)0T11+)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(1)...)]TJ /F10 11.955 Tf 5.48 -9.69 Td[(1+hc(k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)(k)]TJ /F6 7.97 Tf 6.59 0 Td[(1)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((k)]TJ /F6 7.97 Tf 6.58 0 Td[(1)0TN+11. (5) SimilartotheproofforLemma 45 ,wereplacethelasttermin( 5 )by( 5 )toshowthat( 5 )istruefork.Hence( 5 )holdsforeachk,1kK. 180

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Since( 5 )holdsforeachk,1kK,specially,forK,wehave max1jKU(j))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(j)01c(K)max1jKp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01+c(K)max1jKq(j))]TJ /F4 11.955 Tf 11.96 0 Td[(q(j)01+c(K)max1jKr(j))]TJ /F4 11.955 Tf 11.95 0 Td[(r(j)01+c(K)max1jKs(j))]TJ /F4 11.955 Tf 11.96 0 Td[(s(j)01+c(K)max1jKz(j))]TJ /F4 11.955 Tf 11.96 0 Td[(z(j)0+cX(1)T1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(1)0T11+)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(1)...)]TJ /F10 11.955 Tf 5.48 -9.68 Td[(1+hc(K)(K)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((K)0TN+11. (5) Bythethirdandlastequationin( 5 ),wehave (K)N+1=X(K)N+1VT. Hence, (K)TN+1)]TJ /F20 11.955 Tf 11.95 0 Td[((K)0TN+11cX(K)TN+1)]TJ /F4 11.955 Tf 11.96 0 Td[(X(K)0TN+11 Wesubstitutetheaboveinequalityin( 5 )andcombinetheresultwith( 5 )toobtainthefollowinglemma: Lemma47. LetU(k)befeasiblein( 5 ),andletU(k)0bethecontrolassociatedwiththeperturbations(k)0,thenwehave max1jKU(j))]TJ /F10 11.955 Tf 13.39 2.66 Td[(U(j)01cmax1jKp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01+max1jKq(j))]TJ /F4 11.955 Tf 11.95 0 Td[(q(j)01+max1jKr(j))]TJ /F4 11.955 Tf 11.95 0 Td[(r(j)01+max1jKs(j))]TJ /F4 11.955 Tf 11.95 0 Td[(s(j)01+max1jKz(j))]TJ /F4 11.955 Tf 11.95 0 Td[(z(j)0+X(1)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)0T11. (5) wherecisaconstantindependentofNandK. Similar,byLemma 45 ,wehavethefollowinglemma: Lemma48. Let(k)betheadjointvariableassociatewith)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(X(k),U(k)andthepertur-bation(k)andlet(k)0betheadjointvariableassociatewith)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(X(k)0,U(k)0andthe 181

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perturbation(k)0,thenwehave max1jK(j))]TJ /F10 11.955 Tf 12.83 2.66 Td[((j)01cmax1jKp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01+max1jKq(j))]TJ /F4 11.955 Tf 11.95 0 Td[(q(j)01+max1jKr(j))]TJ /F4 11.955 Tf 11.95 0 Td[(r(j)01+max1jKs(j))]TJ /F4 11.955 Tf 11.95 0 Td[(s(j)01+max1jKz(j))]TJ /F4 11.955 Tf 11.95 0 Td[(z(j)0+X(1)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)0T11. (5) wherecisaconstantindependentofNandK. BasedonLemma 41 ,wefurtherhavethefollowinglemmaholds Lemma49. LetX(k)befeasiblein( 5 )andletX(k)0bethestate,associatedwiththecontrolU(k)0andperturbationsp(k)0,thenwehave max1jKX(j)1:N+1)]TJ /F10 11.955 Tf 13.2 2.65 Td[(X(j)01:N+11cmax1jKp(j))]TJ /F4 11.955 Tf 11.96 0 Td[(p(j)01+max1jKq(j))]TJ /F4 11.955 Tf 11.95 0 Td[(q(j)01+max1jKr(j))]TJ /F4 11.955 Tf 11.95 0 Td[(r(j)01+max1jKs(j))]TJ /F4 11.955 Tf 11.95 0 Td[(s(j)01+max1jKz(j))]TJ /F4 11.955 Tf 11.95 0 Td[(z(j)0+X(1)T1)]TJ /F4 11.955 Tf 11.95 0 Td[(X(1)0T11. (5) wherecisaconstantindependentofNandK. WecombineLemma 47 49 toobtainthemainresultofthissection: Lemma50. Iftheassumption(A1)(A3)hold,thenkrT())]TJ /F6 7.97 Tf 6.58 0 Td[(1kisboundedbyaconstantindependentofNandK. 5.8ProofofTheorem 5.1 WenowcollectresultstoproveTheorem 5.1 usingProposition 5.4 andthecorrespondencewiththecontrolproblemdescribedinsection 5.4 .ReferringtoLemma 50 ,let=krT())]TJ /F6 7.97 Tf 6.59 0 Td[(1k,andchoosesmallenoughsuchthat<1.ChoosersmallenoughsuchthatbyLemma 35 ,krT())-222(rT()k,forall2Br().Finally,byLemma 34 ,forNsufcientlarge,kT()k(1)]TJ /F14 11.955 Tf 12.31 0 Td[()=.SincealltheconditionsofProposition 5.4 aresatised,weobtaintheconclusionof( 5 );WecombinethiswithLemma 34 toobtaintheestimateof( 5 )ofTheorem 5.1 182

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5.9ConcludingRemarks InthisChapter,weprovetheconvergenceofahp-collocationmethodappliedtoanunconstrainedcontrolproblem.Weshowthatundertheassumptionofcoercivityandsmoothness,thediscretenonlinearprogrammingproblemhasanextremepointandassociatedtransformedadjointvariablewhichconvergetothesolutionoftheoptimalcontrolproblemattherateOhl Nl)]TJ /F9 5.978 Tf 6.95 2.35 Td[(5 2,whereldenotesthenumberofcontinuousderivativesofthestate,NisthenumberoftheRadauquadraturepointsoneachintervalandhisthelengthofeachinterval. 183

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CHAPTER6CONCLUSIONS Thisdissertationtheoreticallyanalyzestheconvergenceratesofthreecollocationmethods:theGausscollocationmethod,theRadaucollocationmethodandthehpcollocationmethodappliedtounconstrainedoptimalcontrolproblem.TheproofoftheconvergenceratesofGaussandRadaucollocationmethodsarebasedontheapplicationofanabstractimplicitfunctiontheoreminthenonlinearspaces,anestimationofLebesgueconstantsassociatedwithGaussquadraturepointsandRadauquadraturepointsandtheLipschitzstabilityinquadraticprogrammingproblems.TheproofoftheconvergencerateofhpcollocationmethodisbasedontheconvergenceanalysisfortheRadaucollocationmethodsoneachsubintervalandthemathematicalinductions.Theanalysisemployssomeresultsthathavebeenestablishedbynumericalcomputation.Forexample,bynumericalcomputationwecanshowthatthesup-normtheGaussdifferentialmatrixandtheRadaudifferentialmatrixareboundedby2,independentofN.Ideally,onewouldlikeatheoremwhichprovesthisupperboundof2.Butfromapracticalperspective,Nwillneverbeverylarge;andinthiscase,wecansimplycomputethesup-normofthesematrices;thenormsarealwayslessthan2andtendtowards2asNbecomeslarge. TheLebesgueconstantsforthreesetsofinterpolationpointsthatariseinGaussianquadraturecollocationdiscretizationofcontrolproblemareestimatedinChapter2.Inparticular,theLebesgueconstantforGaussquadraturepointsaugmentedbythepoint)]TJ /F10 11.955 Tf 9.3 0 Td[(1isON1 2asshowninTheorem 2.4 .TheLebesgueconstantfortheRadauquadraturepointsisON1 2asshowninTheorem 2.5 .TheLebesgueconstantfortheRadauquadraturepointsaugmentedwith)]TJ /F10 11.955 Tf 9.3 0 Td[(1isO(logN)asshowninTheorem 2.6 .BasedonnumericaltsshowninSection 2.7 ,theseestimatesallappeartobetight. InChapter3andChapter4,weprovedtheconvergenceoftheGaussandRadaucollocationmethodsappliedtoanunconstrainedcontrolproblemrespectively.We 184

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showthatundertheassumptionofcoercivityandsmoothness,thediscretenonlinearprogrammingproblemobtainedbytheGaussandRadaucollocationmethodsbothhaveextremepointsandassociatedtransformedadjointvariableswhichconvergetothesolutionoftheoptimalcontrolproblemattherateofO1 Nk)]TJ /F9 5.978 Tf 6.95 2.35 Td[(5 2,wherekisthenumberofcontinuousderivativesinthesolutionandNisthenumberoftheGaussorRadauquadraturepoints. InChapter5,weprovideaconvergencetheoryofthehpcollocationmethodappliedtotheunconstrainedcontrolproblem.Thetheoryshowsthatundertheassumptionofcoercivityandsmoothness,thediscretenonlinearprogrammingproblemhasanextremepointandassociatedtransformedadjointvariablewhichconvergetothesolutionoftheoptimalcontrolproblemattherateOhl Nl)]TJ /F9 5.978 Tf 6.95 2.35 Td[(5 2,whereldenotesthenumberofcontinuousderivativesofthestate,NisthenumberoftheRadauquadraturepointsoneachintervalandhisthelengthofmeshinterval. 185

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APPENDIXACOMPUTATIONOFTHEMINIMUM-PSLFILTER Inthispart,welistthenumericalresultsusedinourChapter3.Theseresultshavenotbeenprovedtheoretically.However,numerically,wecanshowtheyaretrue. MaximumofEuclidiannormofarowinthematrixW1 2D1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1issmallerthanp 21.4142.ThisresultischeckedforNupto200andtheresultforevery25pointsisshowntable A-3 MaximumofEuclidiannormofarowinthematrixW1 2Dy1:N)]TJ /F6 7.97 Tf 6.59 0 Td[(1issmallerthanp 21.4142.ThisresultischeckedforNupto200andtheresultforevery25pointsisshownintable A-4 TableA-1. InnitynormofD)]TJ /F6 7.97 Tf 6.58 0 Td[(11:Nwithrespecttonumberofinterpolationnodes nodes255075100125150175200 norm1.99551.99881.99941.99971.99981.99981.99991.9999 TableA-2. InnitynormofDy)]TJ /F6 7.97 Tf 12.7 0 Td[(11:Nwithrespecttonumberofinterpolationnodes nodes255075100125150175200 norm1.99551.99881.99941.99971.99981.99981.99991.9999 TableA-3. MaximalofEuclidiannormofarowofW1 2D1:N)]TJ /F6 7.97 Tf 6.58 0 Td[(1withrespecttonumberofinterpolationnodes nodes255075100125150175200 norm1.41221.41371.41391.41401.41411.41411.41411.4141 TableA-4. MaximalofEuclidiannormofarowofW1 2Dy1:N)]TJ /F6 7.97 Tf 6.58 0 Td[(1withrespecttonumberofinterpolationnodes nodes255075100125150175200 norm1.41221.41371.41391.41401.41411.41411.41411.4141 186

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APPENDIXBSYNTHESISOFWFORST-CDMA Inthispart,welistthenumericalresultsusedinChapter4.Theseresultshavenotbeenprovedtheoretically.However,numerically,wecanshowtheyaretrue. (1).kD)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+1k1=2. (2).k(Dz))]TJ /F6 7.97 Tf 6.58 0 Td[(1k12. (3).TheEuclidiannormofeveryrowinmatrix D)]TJ /F6 7.97 Tf 6.59 0 Td[(12:N+10BBBB@1 p !1...1 p !N1CCCCA isboundedbyp 2,whichisshownin B-3 (4).TheEuclidiannormofeveryrowinmatrix Dz)]TJ /F6 7.97 Tf 12.7 0 Td[(10BBBB@1 p !1...1 p !N1CCCCA isboundedbyp 2,whichisshownin B-4 TableB-1. InnitynormofD)]TJ /F6 7.97 Tf 6.58 0 Td[(12:N+1withrespecttonumberofinterpolationnodes numberofnodes510152025303540 norm2.00002.00002.00002.00002.00002.00002.00002.0000 TableB-2. InnitynormOf(Dz))]TJ /F6 7.97 Tf 6.58 0 Td[(1withrespecttonumberofinterpolationnodes numberofnodes510152025303540 norm1.82281.96441.98531.99201.99501.99661.99751.9981 TableB-3. Maximalsquarevalueofrowvectornormwithrespecttonumberofnodes numberofnodes510152025303540 norm2.00002.00002.00002.00002.00002.00002.00002.0000 187

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TableB-4. Maximalsquarevalueofrowvectornormwithrespecttonumberofnodes numberofnodes510152025303540 norm1.75371.95311.98091.98971.99351.99561.99681.9976 188

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[41] Markov,V.A.UberPolynome,dieineinemgegebenenIntervallemoglichstwenigvonNullabweichen.Math.Ann.77(1916):185. [42] Rao,AnilV.,Benson,DavidA.,Darby,Christopher,Patterson,MichaelA.,Francolin,Camila,Sanders,Ilyssa,andHuntington,G.T.Algorithm902:GPOPS,AMATLABSoftwareforSolvingMultiple-PhaseOptimalControlProblemsUsingtheGaussPseudospectralMethod.ACMTransactionsonMathematicalSoftware37(2010).2:1. [43] Rao,AnilV.,Patterson,Michael.A.,andHager,WilliamW.Aph-CollocationSchemeforOptimalControl.Automatica,Submitted,2013. [44] Rivlin,T.J.AnIntroductionToTheApproximationOfFunctions.NewYork:DoverPublications,1969. [45] Szego,G.OrthogonalPolynomials.Providence,RI:AmericanMathematicalSociety,1939. [46] Trefethen,LloydN.andDavidBau,III.NumericalLinearAlgebra.SIAM,1997. [47] Vlassenbroeck,J.AChebyshevPolynomialMethodforOptimalControlwithStateConstraints.Automatica24(1988).4:499. [48] Vlassenbroeck,J.andDoreen,R.Van.AChebyshevTechniqueforSolvingNonlinearOptimalControlProblems.IEEETransactionsonAutomaticControl33(1988).4:333. [49] Williams,Paul.ApplicationofPseudospectralmethodsforRecedingHorizonControl.JournalofGuidance,Control,andDynamics27(2004).2:310. [50] .JacobiPseudospectralMethodforSolvingOptimalControlProblems.JournalofGuidance,Control,andDynamics27(2004).2:293. [51] .Hermite-Legendre-Gauss-LobattoDirectTranscriptionMethodsinTrajectoryOptimization.JournalofGuidance,Control,andDynamics32(2009).4:1392. 192

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BIOGRAPHICALSKETCH HongyanHoureceivedherPh.D.degreesinmathematicsfromUniversityofFlorida,Gainesville,FL,in2013andtheM.S.andB.S.degreesinmathematicsfromNankaiUniversity,Tianjin,China,in2008and2005respectively.Herresearchinterestsincludesnumericaloptimizationandoptimalcontrol. 193