William W. Hager, Hongyan Hou, and Anil V. Rao

24 Pages·2016·0.62 MB·English

by  William W. Hager

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
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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
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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].
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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
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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
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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
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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
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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.
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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
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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)
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