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JOptimTheoryAppl(2016)169:801–824 DOI10.1007/s10957-016-0929-7 Convergence Rate for a Gauss Collocation Method Applied to Unconstrained Optimal Control WilliamW.Hager1 · HongyanHou2 · AnilV.Rao3 Received:1June2015/Accepted:19March2016/Publishedonline:30March2016 ©SpringerScience+BusinessMediaNewYork2016 Abstract Alocalconvergencerateisestablishedforanorthogonalcollocationmethod based on Gauss quadrature applied to an unconstrained optimal control problem. If thecontinuousproblemhasasmoothsolutionandtheHamiltoniansatisfiesastrong convexitycondition,thenthediscreteproblempossessesalocalminimizerinaneigh- borhoodofthecontinuoussolution,andasthenumberofcollocationpointsincreases, the discrete solution converges to the continuous solution at the collocation points, exponentiallyfastinthesup-norm.Numericalexamplesillustratingtheconvergence theoryareprovided. Keywords Gauss collocation method · Convergence rate · Optimal control · Orthogonalcollocation MathematicsSubjectClassification 49M25·49M37·65K05·90C30 B WilliamW.Hager hager@ufl.edu HongyanHou [email protected] AnilV.Rao anilvrao@ufl.edu 1 DepartmentofMathematics,UniversityofFlorida,P.O.Box118105,Gainesville, FL32611-8105,USA 2 DepartmentofChemicalEngineering,CarnegieMellonUniversity,5000ForbesAvenue, Pittsburgh,PA15213,USA 3 DepartmentofMechanicalandAerospaceEngineering,UniversityofFlorida, P.O.Box116250,Gainesville,FL32611-6250,USA 123 802 JOptimTheoryAppl(2016)169:801–824 1 Introduction Theconvergencerateisdeterminedfortheorthogonalcollocationmethoddevelopedin [1,2]wherethecollocationpointsaretheGaussquadratureabscissas,orequivalently, the roots of a Legendre polynomial. Other sets of collocation points that have been studied in the literature include the Lobatto quadrature points [3,4], the Chebyshev quadrature points [5,6], the Radau quadrature points [7–10], and extrema of Jacobi polynomials[11].Weshowthatifthecontinuousproblemhasasmoothsolutionand the Hamiltonian satisfies a strong convexity assumption, then the discrete problem has a local minimizer that converges exponentially fast at the collocation points to asolutionofthecontinuouscontrolproblem.Inotherwork[12,13],Kangconsiders controlsystemsinfeedbacklinearizablenormalformandshowsthatwhentheLobatto discretizedcontrolproblemisaugmentedwithboundsonthestatesandcontrolandon certainLegendrepolynomialexpansioncoefficients,thentheobjectivesinthediscrete problemconvergetotheoptimalobjectiveofthecontinuousproblematanexponential rate. Kang’s analysis does not involve coercivity assumptions, but instead employs boundsinthediscreteproblem.Also,in[14]aconsistencyresultisestablishedfora schemebasedonLobattocollocation. Thesecollocationschemesbasedontherootsoforthogonalpolynomialsortheir derivativesallemployapproximationstothestatebasedonglobalpolynomials.Pre- viously,convergencerateshavebeenobtainedwhentheapproximatingspaceconsists ofpiecewisepolynomialsasin[15–21].Inthisearlierwork,convergenceisachieved bylettingthemeshspacingtendtozero,whilekeepingfixedthedegreeoftheapprox- imatingpolynomialsoneachmeshinterval.Incontrast,forschemesbasedonglobal polynomials, convergence is achieved by letting the degree of the approximating polynomials and the number of collocation points tend to infinity. This leads to an exponentialconvergencerateforproblemswithasmoothsolution. 2 Preliminaries Aconvergencerateisestablishedforanorthogonalcollocationmethodappliedtoan unconstrainedcontrolproblemoftheform ⎫ minimize C(x(1)) ⎬ subjectto x˙(t)=f(x(t),u(t)), t ∈[−1,1], (1) ⎭ x(−1)=x , (x,u)∈C1(Rn)×C0(Rm), 0 d where the state x(t) ∈ Rn, x˙ := x, the control u(t) ∈ Rm, f : Rn × Rm → dt Rn, C : Rn → R, and x is the initial condition, which we assume is given; Ck 0 denotesthespaceofk timescontinuouslydifferentiablefunctions.Althoughthereis nointegral(Lagrange)costtermappearingin(1),itcouldbeincludedintheobjective by augmenting the state with an (n + 1)st component xn+1 with initial condition xn+1(−1) = 0andwithdynamicsequaltotheintegrandoftheLagrangeterm.Itis assumedthatf andC areatleastcontinuous.LetP denotethespaceofpolynomials N 123 JOptimTheoryAppl(2016)169:801–824 803 ofdegreeatmost N definedontheinterval[−1,+1],andletPn denotethen-fold N CartesianproductP ×···×P .Weanalyzeadiscreteapproximationto(1)ofthe N N form minimize C(x(1)) subjectto x˙(τ )=f(x(τ ),u ), 1≤i ≤ N, (2) i i i x(−1)=x , x∈Pn. 0 N Thecollocationpointsτ ,1≤i ≤ N,arewheretheequationshouldbesatisfied,and i u isthecontrolapproximationattimeτ .ThedimensionofP isN+1,whilethere i i N are N +1equationsin(2)correspondingtothecollocateddynamicsat N pointsand theinitialcondition.Inthispaper,wecollocateattheGaussquadraturepoints,which aresymmetricaboutt =0andsatisfy −1<τ <τ <···<τ <+1. 1 2 N Inaddition,theanalysisutilizestwononcollocatedpointsτ0 =−1andτN+1 =+1. Tostateourconvergenceresultsinapreciseway,weneedtointroduceafunction space setting. Let Ck(Rn) denote the space of k times continuously differentiable functions x : [−1,+1] → Rn. For the space of continuous functions, we use the sup-norm(cid:5)·(cid:5)∞givenby (cid:5)x(cid:5)∞ =sup{|x(t)|:t ∈[−1,+1]}, (3) where|·|istheEuclideannorm.Fory∈Rn,letBρ(y)denotetheballwithcentery andradiusρ: Bρ(y)={x∈Rn :|x−y|≤ρ}. Thefollowingregularityassumptionisassumedtoholdthroughoutthepaper. Smoothness.Theproblem(1)hasalocalminimizer(x∗,u∗)inC1(Rn)×C0(Rm), andthefirsttwoderivativesof f and C arecontinuousontheclosureofanopenset (cid:4) and on Bρ(x∗(1)) respectively, where (cid:4) ⊂ Rm+n has the property that for some ρ >0, Bρ(x∗(t),u∗(t))⊂(cid:4)forallt ∈[−1,+1]. Letλ∗denotethesolutionofthelinearcostateequation λ˙∗(t)=−∇ H(x∗(t),u∗(t),λ∗(t)), λ∗(1)=∇C(x∗(1)), (4) x where H is the Hamiltonian defined by H(x,u,λ) = λTf(x,u). Here ∇C denotes the gradient of C. By the first-order optimality conditions (Pontryagin’s minimum principle),wehave ∇ H(x∗(t),u∗(t),λ∗(t))=0 (5) u forallt ∈[−1,+1]. 123 804 JOptimTheoryAppl(2016)169:801–824 Sincethediscretecollocationproblem(2)isfinitedimensional,thefirst-orderopti- mality conditions (Karush–Kuhn–Tucker conditions) imply that when a constraint qualification holds [22],the gradient of theLagrangian vanishes. By theanalysis in [2],thegradientoftheLagrangianvanishesifandonlyifthereexistsλ ∈ Pn such N that λ˙(τ )=−∇ H(x(τ ),u ,λ(τ )), 1≤i ≤ N, (6) i x i i i λ(1)=∇C(x(1)), (7) 0 =∇ H(x(τ ),u ,λ(τ )), 1≤i ≤ N. (8) u i i i Theassumptionsthatplayakeyroleintheconvergenceanalysisarethefollowing: (A1) Thematrix∇2C(x∗(1))ispositivesemidefiniteandforsomeα >0,thesmallest eigenvalueoftheHessianmatrix ∇2 H(x∗(t),u∗(t),λ∗(t)) (x,u) isgreaterthanα,uniformlyfort ∈[−1,+1]. (A2) TheJacobianofthedynamicssatisfies (cid:5)∇xf(x∗(t),u∗(t))(cid:5)∞ ≤β <1/2 and (cid:5)∇xf(x∗(t),u∗(t))T(cid:5)∞ ≤β forallt ∈[−1,+1]where(cid:5)·(cid:5)∞isthematrixsup-norm(largestabsoluterow sum),andtheJacobian∇ f isannbynmatrixwhoseithrowis(∇ f )T. x x i Thecoercivityassumption(A1)ensuresthatthesolutionofthediscreteproblemisa stationarypoint.Thecondition(A2)arisesinthetheoreticalconvergenceanalysissince thesolutionoftheproblem(1)isapproximatedbypolynomialsdefinedontheentire timeinterval[−1,+1].Incontrast,whenthesolutionisapproximatedbypiecewise polynomialsasin[15–21],thisconditionisnotneeded.Tomotivatewhythiscondition arises,supposethatthesystemdynamicsfhastheformf(x,u)=Ax+g(u)whereAis annbynmatrix.Sincethedynamicsarelinearinthestate,itfollowsthatforanygiven u with g(u) absolutely integrable, we can invert the continuous dynamics to obtain thestatexasafunctionofthecontrolu.Thedynamicsinthediscreteapproximation (2)arealsoalinearfunctionofthediscretestateevaluatedatthecollocationpoints; however,theinvertibilityofthematrixinthediscretedynamicsisnotobvious.Ifthe systemmatrixsatisfies(cid:5)A(cid:5)∞ <1/2,thenwewillshowthatthematrixforthediscrete dynamicsisinvertiblewithitsnormuniformlyboundedasafunctionofN,thedegree ofthepolynomials.Consequently,itispossibletosolveforthestateasafunctionof thecontrolinthediscreteproblem(2).When(cid:5)A(cid:5)∞ >1/2,thematrixforthediscrete dynamicscouldbesingularforsomechoiceof N.Ingeneral,(A2)couldbereplaced byanyconditionthatensuresthesolvabilityofthelinearizeddynamicsforthestate in terms of the control, and the stability of this solution under perturbations in the dynamics. Whenthedynamicsofthecontrolproblemarenonlinear,theconvergenceanalysis leadsustostudythelinearizeddynamicsaround(x∗,u∗),and(A2)impliesthatthe 123 JOptimTheoryAppl(2016)169:801–824 805 linearizeddynamicsinthediscreteproblem(2)areinvertiblewithrespecttothestate. In contrast, if the global polynomials are replaced by piecewise polynomials on a uniformmeshofwidthh,then(A2)isreplacedby (cid:5)∇xf(x∗(t),u∗(t))(cid:5)∞ <1/h and (cid:5)∇xf(x∗(t),u∗(t))T(cid:5)∞ <1/h, whichissatisfiedwhenh issufficientlysmall.Inotherwords,whenh issufficiently small, it is possible to solve for the discrete states in terms of the discrete controls, independentofthedegreeofthepolynomialsoneachmeshinterval.Theseschemes where both the mesh width h and the degree of the polynomials p on each mesh intervalarefreeparametershavebeenreferredtoashp-collocationmethods[9,10]. Inadditiontothetwoassumptions,theanalysisemploystwopropertiesoftheGauss collocationscheme.Letω ,1 ≤ j ≤ N,denotetheGaussquadratureweights,and j for1≤i ≤ N and0≤ j ≤ N,define (cid:5)N τ −τ D = L˙ (τ ), where L (τ):= k . (9) ij j i j τ −τ k=0 j k k(cid:10)=j D is a differentiation matrix in the sense that (Dp) = p˙(τ ), 1 ≤ i ≤ N, where i i p ∈ P isthepolynomialthatsatisfies p(τ ) = p for0 ≤ j ≤ N.Thesubmatrix N j j D1:N consistingofthetailing N columnsofDhasthefollowingproperties: (P1) D1:N isinvertibleand(cid:5)D−1:1N(cid:5)∞ ≤2. (P2) If W is the diagonal matrix containing the Gauss quadrature weights ω on thediagonal,√thentherowsofthematrix[W1/2D1:N]−1 haveEuclideannorm boundedby 2. ThefactthatD1:N isinvertibleisestablishedin[2,Prop.1].Theboundsonthenorms in (P1) and (P2), however, are more subtle. We refer to (P1) and (P2) as properties rather than assumptions since the matrices are readily evaluated, and we can check numericallythat(P1)and(P2)arealwayssatisfied.Infact,numericallywefindthat (cid:5)D−1:1N(cid:5)∞ =1+τN wherethelastGaussquadratureabscissaτN approaches+1asN tendsto∞.Ontheotherhand,wedonotyethaveageneralprooffortheproperties (P1)and(P2).AprizeforobtainingaproofisexplainedonWilliamHager’sWebsite (GoogleWilliamHager10,000yen).Incontrasttotheseproperties,conditions(A1) and(A2)areassumptionsthatareonlysatisfiedbycertaincontrolproblems. IfxN ∈Pn isasolutionof(2)associatedwiththediscretecontrolsu ,1≤i ≤ N, N i andifλN ∈Pn satisfies(6)–(8),thenwedefine N XN =[xN(−1), xN(τ ), ...,xN(τ ), xN(+1) ], 1 N X∗ =[x∗(−1), x∗(τ ), ...,x∗(τ ), x∗(+1) ], 1 N UN =[ u , ...,u ], 1 N U∗ =[ u∗(τ ), ...,u∗(τ ) ], 1 N (cid:4)N =[λN(−1),λN(τ ),...,λN(τ ),λN(+1)], 1 N (cid:4)∗ =[λ∗(−1), λ∗(τ ), ...,λ∗(τ ), λ∗(+1) ]. 1 N 123 806 JOptimTheoryAppl(2016)169:801–824 The following convergence result relative to the vector ∞-norm (largest absolute element)isestablished: Theorem2.1 If (x∗,u∗) is a local minimizer for the continuous problem (1) with (x∗,λ∗) ∈ Cη for some η ≥ 4 and both (A1)–(A2) and (P1)–(P2) hold, then for N sufficiently large, the discrete problem (2) has a stationary point (XN,UN) and an associateddiscretecostate(cid:4)N,andthereexistsaconstantcindependentof N andη suchthat (cid:6) (cid:7) max (cid:5)XN −X∗(cid:5)∞,(cid:5)UN −U∗(cid:5)∞,(cid:5)(cid:4)N −(cid:4)∗(cid:5)∞ (cid:8) (cid:9) (cid:8) (cid:9) ≤ c p−3 (cid:5)x∗(p)(cid:5)∞+(cid:5)λ∗(p)(cid:5)∞ , p =min{η,N +1}. (10) N Moreover, if ∇2C(x∗(1)) is positive definite, then (XN,UN) is a local minimizer of (2).Herethesuperscript(p)denotesthe(p)thderivative. Althoughthediscreteproblemonlypossessesdiscretecontrolsatthecollocation points−1<τ <+1,1≤i ≤ N,anestimateforthediscretecontrolatt =−1and i t =+1couldbeobtainedfromtheminimumprinciple(5)sincewehaveestimatesfor thediscretestateandcostateattheendpoints.Alternatively,polynomialinterpolation couldbeusedtoobtainestimatesforthecontrolattheendpointsoftheinterval. The paper is organized as follows. In Sect. 3, the discrete optimization problem (2) is reformulated as a nonlinear system of equations obtained from the first-order optimality conditions, and a general approach to convergence analysis is presented. Section4obtainsanestimateforhowcloselythesolutiontothecontinuousproblem satisfiesthefirst-orderoptimalityconditionsforthediscreteproblem.Section5proves thatthelinearizationofthediscretecontrolproblemaroundasolutionofthecontinuous problemisinvertible.Section6establishesanL2stabilitypropertyforthelinearization, ∞ whileSect.7strengthensthenormto L .Thisstabilitypropertyisthebasisforthe proofofTheorem2.1.Numericalexamplesillustratingtheexponentialconvergence resultaregiveninSect.8. Notation.Themeaningofthenorm(cid:5)·(cid:5)∞isbasedoncontext.Ifx∈C0(Rn),then (cid:5)x(cid:5)∞ denotesthemaximumof|x(t)|overt ∈ [−1,+1],where|·|istheEuclidean norm. If A ∈ Rm×n, then (cid:5)A(cid:5)∞ is the largest absolute row sum (the matrix norm induced by the vector sup-norm). For a vector v ∈ Rm, (cid:5)v(cid:5)∞ is the maximum of |v | over 1 ≤ i ≤ m. We let |A| denote the matrix norm induced by the Euclidean i vectornorm.ThedimensionoftheidentitymatrixIisoftenclearfromcontext;when necessary,thedimensionofIisspecifiedbyasubscript.Forexample,I isthen by n n identitymatrix.∇C denotesthegradient,acolumnvector,while∇2C denotesthe Hessianmatrix.Throughoutthepaper,cdenotesagenericconstantwhichhasdifferent valuesindifferentequations.ThevalueofthisconstantisalwaysindependentofN.1 denotesavectorwhoseentriesareallequaltoone,while0isavectorwhoseentriesare allequaltozero,andtheirdimensionshouldbeclearfromcontext.Finally,X¯ ∈RnN isthevectorobtainedbyverticallystackingX ∈Rn,1≤i ≤ N. i 123 JOptimTheoryAppl(2016)169:801–824 807 3 AbstractSetting Asshownin[2],thediscreteproblem(2)canbereformulatedasthenonlinearpro- grammingproblem minimize C(cid:10)(XN+1) subjectto Nj=0DijXj =(cid:10)f(Xi,Ui), 1≤i ≤ N, X0 =x0, (11) XN+1 =X0+ Nj=1ωjf(Xj,Uj). As indicated before Theorem 2.1, X corresponds to xN(τ ). Also, Garg et al. [2] i i showthattheequationsobtainedbysettingthegradientoftheLagrangiantozeroare equivalenttothesystemofequations N(cid:11)+1 D†(cid:4) =−∇ H(X ,U ,(cid:4) ), 1≤i ≤ N, (12) ij j x i i i j=1 (cid:4)N+1 =∇C(XN+1), (13) 0 =∇ H(X ,U ,(cid:4) ), 1≤i ≤ N, (14) u i i i where (cid:12) (cid:13) ω D† =− j D , 1≤i ≤ N, 1≤ j ≤ N, (15) ij ω ji i (cid:11)N D† =− D†, 1≤i ≤ N. (16) i,N+1 ij j=1 Here (cid:4) corresponds to λN(τ ). The relationship between the discrete costate (cid:4) , i i i theKKTmultipliersλ ,1 ≤ i ≤ N,associatedwiththediscretedynamics,andthe i multiplierλN+1associatedwiththeequationforXN+1 is ωi(cid:4)i =λi +ωiλN+1 when 1≤i ≤ N, and (cid:4)N+1 =λN+1. (17) Thefirst-orderoptimalityconditionsforthenonlinearprogram(11)consistofthe Eqs.(12)–(14)andtheconstraintsin(11).ThissystemcanbewrittenasT(X,U,(cid:4))= 0where (T ,T ,...,T )(X,U,(cid:4))∈RnN ×Rn ×RnN ×Rn ×RmN. 1 2 5 ThefivecomponentsofT aredefinedasfollows: ⎛ ⎞ (cid:11)N T (X,U,(cid:4))=⎝ D X ⎠−f(X ,U ), 1≤i ≤ N, 1i ij j i i j=0 (cid:11)N T2(X,U,(cid:4))=XN+1−X0− ωjf(Xj,Uj), j=1 123 808 JOptimTheoryAppl(2016)169:801–824 ⎛ ⎞ N(cid:11)+1 T (X,U,(cid:4))=⎝ D†(cid:4) ⎠+∇ H(X ,U ,(cid:4) ), 1≤i ≤ N, 3i ij j x i i i j=1 T4(X,U,(cid:4))=(cid:4)N+1−∇xC(XN+1), T (X,U,(cid:4))=∇ H(X ,U ,(cid:4) ), 1≤i ≤ N. 5i u i i i Note that in formulating T, we treat X as a constant whose value is the given 0 startingconditionx .Alternatively,wecouldtreatX asanunknownandthenexpand 0 0 T tohaveasixthcomponentX −x .WiththisexpansionofT,weneedtointroduce 0 0 anadditionalmultiplier(cid:4) fortheconstraintX −x ;itturnsoutthat(cid:4) approximates 0 0 0 0 λ∗(−1).Toachieveaslightsimplificationintheanalysis,weemployafive-component T andtreatX asaconstant,notanunknown. 0 The proof of Theorem 2.1 reduces to a study of solutions to T(X,U,(cid:4)) = 0 in a neighborhood of (X∗,U∗,(cid:4)∗). Our analysis is based on Dontchev et al. [17, Proposition3.1], which we simplify below to take into account the structure of our T.OtherresultslikethisarecontainedinTheorem3.1of[16],inProposition5.1of Hager[19],andinTheorem2.1ofHager[23]. Proposition3.1 LetX beaBanachspaceandY bealinearnormedspacewiththe norms in both spaces denoted (cid:5)·(cid:5). Let T: X (cid:13)−→ Y with T continuously Fréchet differentiableinB (θ∗)forsomeθ∗ ∈X andr >0.Supposethat r (cid:5)∇T(θ)−∇T(θ∗)(cid:5)≤εforallθ ∈B (θ∗) r (cid:18) (cid:19) (cid:20)(cid:18) where∇T(θ∗)isinvertible,anddefineμ:=(cid:5)∇T(θ∗)−1(cid:5).Ifεμ<1and(cid:18)T θ∗ (cid:18)≤ (1−με)r/μ,thenthereexistsauniqueθ ∈ B (θ∗)suchthatT(θ) = 0.Moreover, r wehavetheestimate μ (cid:18) (cid:19) (cid:20)(cid:18) (cid:5)θ −θ∗(cid:5)≤ (cid:18)T θ∗ (cid:18)≤r. (18) 1−με WeapplyProposition3.1withθ∗ = (X∗,U∗,(cid:4)∗)andθ(cid:18)=(cid:19)(X(cid:20)N(cid:18),UN,(cid:4)N).The key steps in the analysis are the estimation of the residual (cid:18)T θ∗ (cid:18), the proof that ∇T(θ∗)isinvertible,andthederivationofaboundfor(cid:5)∇T(θ∗)−1(cid:5)thatisindependent of N.Inourcontext,weusethe∞-normforbothX andY.Inparticular, (cid:5)θ(cid:5)=(cid:5)(X,U,(cid:4))(cid:5)∞ =max{(cid:5)X(cid:5)∞,(cid:5)U(cid:5)∞,(cid:5)(cid:4)(cid:5)∞}. (19) Forthisnorm,theleftsideof(10)andtheleftsideof(18)arethesame.Thenormon Y entersintotheestimationofboththeresidual(cid:5)T(θ∗)(cid:5)in(18)andtheparameter μ := (cid:5)∇T(θ∗)−1(cid:5). In our context, we think of an element of Y as a vector with componentsy ,1 ≤ i ≤ 3N +2,wherey ∈ Rn for1 ≤ i ≤ 2N +2andy ∈ Rm i i i for i > 2N +2. For example, T (X,U,(cid:4)) ∈ RnN corresponds to the components 1 y ∈Rn,1≤i ≤ N. i 4 AnalysisoftheResidual Wenowestablishaboundfortheresidual. 123 JOptimTheoryAppl(2016)169:801–824 809 Lemma4.1 If(x∗,λ∗)∈Cηand p =min{η,N+1}≥4,thenthereexitsaconstant cindependentof N andηsuchthat (cid:8) (cid:9) (cid:8) (cid:9) (cid:5)T(X∗,U∗,(cid:4)∗)(cid:5)∞ ≤ c p−3 (cid:5)x∗(p)(cid:5)∞+(cid:5)λ∗(p)(cid:5)∞ . (20) N Proof BythedefinitionofT,T (X∗,U∗,(cid:4)∗)=0sincex∗andλ∗satisfytheboundary 4 conditionin(4).Likewise,T (X∗,U∗,(cid:4)∗) = 0since(5)holdsforallt ∈ [−1,+1]; 5 inparticular,(5)holdsatthecollocationpoints. NowletusconsiderT .ByGargetal.[2,Eq.(7)], 1 (cid:11)N D X∗ =x˙I(τ ), 1≤i ≤ N, ij j i j=0 wherexI ∈ Pn isthe(interpolating)polynomialthatpassesthroughx∗(τ )for0 ≤ N i i ≤ N. Since x∗ satisfies the dynamics of (1), it follows that f(X∗,U∗) = x˙∗(τ ). i i i Hence,wehave T (X∗,U∗,(cid:4)∗)=x˙I(τ )−x˙∗(τ ). (21) 1i i i ByHageretal.[24,Prop.2.1],wehave (cid:8) (cid:9) (cid:5)x˙I −x˙∗(cid:5)∞ ≤ 1+2N2 inf (cid:5)x˙ −q˙(cid:5)∞+N2(1+LN) inf (cid:5)x−q(cid:5)∞, (22) q∈Pn q∈Pn N N where LN istheLebesgu√econstantofthepointsetτi,0≤i ≤ N.In[24,Thm.4.1], weshowthat L = O( N).ItfollowsfromElschner[25,Prop.3.1]thatforsome N constantc,independentof N,wehave (cid:12) (cid:13) η q∈inPfn (cid:5)x−q(cid:5)∞ ≤ N c+1 (cid:5)x∗(η)(cid:5)∞ (23) N and (cid:8) (cid:9) inf (cid:5)x˙ −q˙(cid:5)∞ ≤ c η−1(cid:5)x∗(η)(cid:5)∞ (24) q∈Pn N N whenever N +1≥η.Inthecasethat N +1<η,thesmoothnessrequirementηcan berelaxedto N +1toobtainsimilarbounds.Sincethefirstbound(23)isdominated by(24),itfollowsthatthefirsttermin(22)dominatesthesecondterm,andwehave (cid:8) (cid:9) (cid:5)x˙I −x˙∗(cid:5)∞ ≤ c p−3(cid:5)x∗(p)(cid:5)∞ (25) N where p = min{η,N +1}. The bound (25) is valid in both cases N +1 ≥ η and N +1<η.By(21)and(25),T (X∗,U∗,(cid:4)∗)complieswiththebound(20). 1 123 810 JOptimTheoryAppl(2016)169:801–824 Next,letusconsider (cid:11)N T (X∗,U∗,(cid:4)∗)=x∗(1)−x∗(−1)− ω f(x∗(τ ),u∗(τ )). (26) 2 j j j j=1 Bythefundamentaltheoremofcalculusandthefactthat N-pointGaussquadrature isexactforpolynomialsofdegreeupto2N −1,wehave (cid:21) 1 (cid:11)N 0=xI(1)−xI(−1)− x˙I(t)dt =xI(1)−x∗(−1)− ω x˙I(τ ) (27) j j −1 j=1 sincexI(−1)=x∗(−1).Subtract(27)from(26)toobtain (cid:11)N (cid:8) (cid:9) T (X∗,U∗,(cid:4)∗)=x∗(1)−xI(1)+ ω x˙I(τ )−x˙∗(τ ) . (28) 2 j j j j=1 Sinceω >0andtheirsumis2, j (cid:22) (cid:22) (cid:22) (cid:22) (cid:22)(cid:11)I (cid:8) (cid:9)(cid:22) (cid:22)(cid:22) ωj x˙I(τj)−x˙∗(τj) (cid:22)(cid:22)≤2(cid:5)x˙I −x˙∗(cid:5)∞. (29) (cid:22) (cid:22) j=1 Likewise, (cid:22)(cid:22)(cid:21) +1(cid:8) (cid:9) (cid:22)(cid:22) |x∗(1)−xI(1)|=(cid:22)(cid:22) x˙∗(t)−x˙I(t) dt(cid:22)(cid:22)≤2(cid:5)x˙∗−x˙I(cid:5)∞. (30) −1 Wecombine(28)–(30)and(25)toseethatT (X∗,U∗,(cid:4)∗)complieswiththebound 2 (20). Finally,letusconsiderT .ByGargetal.[2,Thm.1], 3 N(cid:11)+1 D†(cid:4)∗ =λ˙I(τ ), 1≤i ≤ N, ij j i j=1 whereλI ∈Pn isthe(interpolating)polynomialthatpassesthrough(cid:4)∗ =λ(τ )for N j j 1 ≤ j ≤ N +1.Sinceλ∗ satisfies(4),itfollowsthatλ˙∗(τ ) =−∇ H(X∗,U∗,(cid:4)∗). i x i i i Hence,wehave T (X∗,U∗,(cid:4)∗)=λ˙I(τ )−λ˙∗(τ ). 3i i i InexactlythesamewaythatT in(21)washandled,weconcludethatT (X∗,U∗,(cid:4)∗) 1 3 complieswiththebound(20).Thiscompletestheproof. (cid:14)(cid:15) 123

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Keywords Gauss collocation method · Convergence rate · Optimal control · [email protected]. Anil V. Rao .. A prize for obtaining a proof is explained on William Hager's Web site. (Google William Hager 10,000 yen).
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