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Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 895876 5 pageshttpdxdoiorg1011552013895876
Research ArticleOptimization of the Forcing Term for the Solution ofTwo-Point Boundary Value Problems
Gianni Arioli
Dipartimento di Matematica ldquoF Brioschirdquo Modellistica e Calcolo Scientifico (MOX) Politecnico di MilanoVia Bonardi 9 20133 Milano Italy
Correspondence should be addressed to Gianni Arioli gianniariolipolimiit
Received 28 August 2012 Accepted 1 January 2013
Academic Editor Alfredo Peris
Copyright copy 2013 Gianni ArioliThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We present a new numerical method for the computation of the forcing term of minimal norm such that a two-point boundaryvalue problem admits a solution The method relies on the following steps The forcing term is written as a (truncated) Chebyshevseries whose coefficients are free parameters A technique derived from automatic differentiation is used to solve the initial valueproblem so that the final value is obtained as a series of polynomials whose coefficients depend explicitly on (the coefficients of)the forcing termThen the minimization problem becomes purely algebraic and can be solved by standard methods of constrainedoptimization for example with Lagrange multipliers We provide an application of this algorithm to the planar restricted threebody problem in order to study the planning of low-thrust transfer orbits
1 Introduction
We consider a two-point boundary value problem for thenonautonomous dynamical system in R119873
1199101015840(119905) = 119891 (119910 (119905)) + 120593 (119905)
119910 (minus1) = 1199100
119910 (1) = 1199101
(1)
where 119891 isin 1198621(R119873R119873) and 120593 isin 119862([minus1 1]R119873) We intro-
duce a new numerical method for the computation of 120593 ofminimal norm among the functions representable with atruncated Chebyshev series such that (1) admits a solutionIn order to evaluate the performance of the algorithm weapply it to a well-known problem that is the study of low-thrust orbits between the Earth and theMoonWe refer to [1]and references there for a discussion of the astrodynamicalproblem
Our new approach is derived from the method intro-duced in [2 3] to consider the dependence of parameters ofthe solution of hyperbolic equations Such method heavilyrelies on a computer representation of algebras of functions
in such a way that the computer can use such functions in astraightforward and transparent way as if they were floatingpoint numbers reducing to a minimum the computationsto be done explicitly by hand The underlying ideas ofthis method stem from automatic differentiation algorithmssee [4ndash8] and recently have been used in the field ofcomputer-assisted proofs see [9 10] for some examples butits potentiality in optimization and control appears to havebeen overlooked In Section 2 we describe the approach in ageneric setting in Section 3 we describe the specific-planar-restricted three body problem that we chose as an examplein Section 4 we present the results that we obtained withthe three body problem in Section 5 we discuss the featuresof the method and compare the Taylor and the Chebyshevrepresentation
2 The Generic Problem
Consider an initial value problem in R119899
1199101015840
119888(119905) = 119891 (119905 119910 (119905)) + 120593119888 (119905)
119910119888 (minus1) = 1199100
(2)
2 Journal of Mathematics
where 119891 is an analytic function 119888 = 119888119894119895 is an 119899 times 119873 real
matrix and the forcing term 120593119888 [minus1 1] rarr R119899 depends on 119888
as follows
(120593119888 (119905))119894 =
119873minus1
sum
119895=0
119888119894119895119879119895 (119905) (3)
where 119879119895(119905) is the 119895th Chebyshev polynomial and (120593
119888(119905))119894
denotes the 119894th component of 120593119888(119905) Our first step is to
compute explicitly the dependence of 119910119888(119905) on 119888 for 119905 isin
(minus1 1]We choose a numeric method to solve the IVP in order
to focus on the issue of the parameter dependence we choosethe simplestmethod for example an explicit first-order Eulerscheme with constant time step and we consider the case 119899 =119873 = 1 We show later that it is straightforward to adapt themethod to other schemes for example a Runge-Kutta schemewith variable time step and to the case 119899119873 gt 1
Choose 119870 isin N let ℎ = 2119870 be the time step set 119905119896=
119896ℎ minus 1 and 119884119896(119888) = 119910
119888(119905119896) so that the Euler scheme reads
119884119896+1 (119888) = 119884119896 (119888) + ℎ119891 (119905119896 119884119896 (119888)) + ℎ120593119888 (119905119896) (4)
and 119884119870(119888) = 119910
119888(1) Now write
119884119896 (119888) =
+infin
sum
119895=0
119910119896119895119880119895 (119888) (5)
where 119880119895(119888)119895isinN is a basis of some space of functions119883 of our
choice In this paper we consider both the Taylor expansionfor the set of analytic functions in a disc (ie 119880
119895(119888) = 119888
119895) andthe Chebyshev expansion for the set of continuous functionsin [minus1 1] (ie 119880
119895(119888) = cos(119895 arccos 119888)) Clearly the method
can be generalized to other expansions for example Fourierseries or Legendre polynomials Now we choose an order119872 and we approximate 119884
119896(119888) with the truncated expansion
sum119872
119895=0119910119896119895119880119895(119888) Our aim is to compute the coefficients 119910
119896119895
using an automatic algorithm Since 1199100(119888) = 119910
0 then we have
11991000
= 1199100and 119910
0119895= 0 when 119895 ge 1
The core of the method consists in a generalization ofthe Taylor Models approach for a detailed description ofthe technique with different expansions see [2 3] Herewe only recall the basic ideas First we observe that it isstraightforward to implement on a computer an arithmeticof Taylor or Chebyshev polynomials of one variable Usingobject-oriented programming and operator overloading it ispossible to define a class Fun which represents a Taylor orChebyshev expansion and a set of methods which performthe basic operations and the computation of the elementaryfunctions More precisely an object Fun is represented as thelist of the coefficients It is straightforward to implement aprocedure that given a scalar 120572 and two objects Fun 119879
11198792
computes the object Fun corresponding to 120572(1198791lowast 1198792) where
lowast is either the addition or the multiplication Then givena polynomial 119901(119909) it is possible to compute the object Funcorresponding to 119901(119879
1) and since all analytic functions 119891(119909)
can be approximated with polynomials it is also possibleto compute the object Fun that better approximates 119891(119879
1)
A more algebraic-oriented way of expressing this idea isthe following let 119883 be the space of functions spanned by119880119895(119888)119895isin0119872
and let 119879 119883 rarr R119872+1 be the (invertible)map that extracts the coefficients Our approach consists inlifting the basic arithmetics of119883 toR119872+1 More precisely wecompute operators
oplus otimes R119872+1
timesR119872+1
997888rarr R119872+1
⊙ R timesR119872+1
997888rarr R119872+1
(6)
such that 119879(119891 + 119892) = 119879(119891) oplus 119879(119892) 119879(119891119892) = 119879(119891) otimes
119879(119892) and 119879(120572119892) = 120572 ⊙ 119879(119892) Once this has been done wecan make extensive use of object-oriented programming andoperator overloading in order to make this lifting completelytransparent to the user
The extension of this method to the case 119899 gt 1 isstraightforward The case 119873 gt 1 requires some additionalwork since we have to consider a multivariable Taylor orChebyshev expansion
119884119896 (119888) =
+infin
sum
1198951119895119873=0
1199101198961198951119895119873
1198801198951
(1198881) sdot sdot sdot 119880
119895119873
(119888119873) (7)
The sum of two of such expansions is done componentwiseThe multiplication is less straightforward but a very efficientalgorithm for the case of the Taylor expansion has beenintroduced in [8] and it is not too hard to extend it tothe Chebyshev (or some other) expansion We refer to ourprograms for details
With the class Fun consisting of the objects (7) andthe functions for addition and multiplication in place wecan use a standard implementation of the algorithm (4)observing that given 119884
119896 the methods of the class will take
care of the computation of 119891(119905119896 119884119896) while 120593
119888(119905119896) is explicitly
given by (3) Extending this approach to better integrationschemes say Runge-Kutta or multistep is straightforwardit suffices to implement the chosen algorithm using objectsFun and their methods By implementing and running theintegration scheme of choice with the class Fun we obtainall the coefficients 119910
1198961198951119895119873
and therefore the polynomials119884119896(119888) for all 119896 Now consider the polynomial 119884
119870(119888) solving
the boundary value problem (1) consists in computing 119888 isin
R119899119873 such that119884119870(119888) = 119910
1This amounts to solving 119899 algebraic
equations in119873 unknowns If we wish 120593119888 to be small (where
sdot is some norm of our choice) we recall that when119873 gt 1
the system 119884119870(119888) = 119910
1is underdetermined therefore we can
compute an explicit formula Φ(119888) = 120593119888 and exploit the
additional degrees of freedom to set the following constrainedminimization problem find 119888 isin R119899119873 such that
Φ (119888) = min119884119870(119888)=1199101
Φ (119888) (8)
This result can be achieved by standard constrained mini-mization algorithms for example Lagrangemultipliers Notethat this method can be easily extended to solve other kindsof conditions at 119905 = 1 let 119898 le 119899 and 119891 R119899 rarr R119898 some
Journal of Mathematics 3
polynomial functionWe can apply the same approach to find119888 isin R119899119873 such that
Φ (119888) = min119891(119884119870(119888))=0
Φ (119888) (9)
To sum up once the class Fun and some algorithm for solvinginitial value problems for ODErsquos has been implementedboundary value problems with optimization of the forcingterm are reduced to computing the constrained minumumof a polynomial the constraint being represented by apolynomial equation or system of equations We remarkthat in principle such polynomial constrainedminimizationproblems can be solved exactly (up to floating point errors)but this does not imply that the boundary condition is solvedexactly for the original problem since we are truncating theexpansion (7) and we are neglecting the truncation errorOne needs to check a posteriori for example by solving (2)with standard floating point numbers and with the forcingterm provided by (3) with 119888 = 119888 and computing the error120576 = |119910(1) minus 119910
1| or 120576 = |119891(119910(1))|
3 An Application Low-Thrust Orbits inthe Planar RTBP
In order tomodel the dynamics of a spaceship travelling froman orbit around the Earth to an orbit around the Moon weconsider the forced planar RTBP that is the system
+ 2 119910 = Ω119909+ 119891 (119905)
radic2 + 1199102
119910 minus 2 = Ω119910+ 119891 (119905)
119910
radic2 + 1199102
(10)
where
Ω(119909 119910) =1199092
2+1199102
2+
1 minus 120583
radic(119909 + 120583)2+ 1199102
+120583
radic(119909 + 120583 minus 1)2+ 1199102
+ 119862
(11)
We choose 120583 ≃ 00121506683 that is the reduced mass ofthe Earth-Moon system and 119862 = minus1600172454916536 sothat Ω(119871
1) = 0 119871
1being the Lagrangian point between the
primaries The forcing term is given by a scalar function 119891(119905)to be determined times a unit vector directed as the velocityof the spaceship In other words we assume that the thrustis always parallel to the velocity We show how to apply themethod described in the previous section to the problem ofoptimizing the travel from an orbit 167 km above the Earth toan orbit 100 km above the Moon
We choose a connecting orbit passing close to 1198711 Note
that the Jacobi function
119869 (119909 119910 119910) = 2+ 1199102minus 2Ω (119909 119910) (12)
is approximately minus60 on an orbit 167 km above the Earthand approximately minus16 on an orbit 100 km above the Moon
05
04
03
02
01
0
minus01
minus02
minus03
minus04
minus04 minus02 0 02 04 06 08 1 12
Figure 1 A complete orbit
Table 1 Result enforcing 119877 and 119881
119877 119881 120576119879
119891119879 120576
119862119891119862
04 09 103 sdot 10minus7
03012 127 sdot 10minus7
03012
04 085 232 sdot 10minus4
04215 345 sdot 10minus4
04235
045 09 NA NA 766 sdot 10minus5
02104
043 09 658 sdot 10minus10
01540 139 sdot 10minus7
01540
Furthermore in order to ldquocrossrdquo the region around theLagrangian point 119871
1 it is necessary to raise it above 0 We
build the connecting orbit as follows We start from theLyapunov orbit around 119871
1with 119869 = 01 and approximate
periodic orbits around the Moon and the Earth Then wecompute approximate connections from the Lyapunov orbitto the orbit around the Earth (backwards in time) and tothe orbit around the Moon (forward in time) Choosing theLyapunov orbit as a starting point is useful if not necessarybecause of its high instability it is much harder to aim atthe Lyapunov orbit than at a stable orbit around a primaryThese orbits are obtained by applying a constant low thrust(ie 119891(119905) = 119888 with small 119888) chosen by simple trial and error
Once these rough orbits are computed we apply thetechnique described above to compute an accurate correctionso that the trajectory satisfies some required boundaryconditions and at the same time the thrust is minimal Moreprecisely we write (10) as a system of 4 first-order equationsby setting 119907 = and 119908 = 119910 we write the modulus of theforcing term as
119891 (119905) = 119888 +
119873minus1
sum
119895=0
119888119895119879119895 (119905) (13)
we choose some initial value on the trajectory at the arbitrarytime 119905 = 0 and we solve the equation with a fourth-orderRunge-Kutta scheme up to some time 119905 = 119879 using objectsFun as scalars the coefficients 119888
119895being the independent
variables of the objects Fun Referring to Section 2 we have119899 = 4 (the dimension of the system) we choose 119873 = 4
and 119872 = 10 We rescale the time in the interval [minus1 1]and apply the method described in Section 2 to obtain(119909(119879) 119910(119879) 119907(119879) 119908(119879)) as a function of 119888
119895 expressed either
as a Taylor series or a Chebyshev series
4 Journal of Mathematics
Table 2 Result enforcing 119877 119881 and tangential trajectory
119877 119881 120576119879
120576perp
119879119891119879 120576
119862120576perp
119862119891119862
042 087 304 sdot 10minus4
848 sdot 10minus5
03627 153 sdot 10minus4
826 sdot 10minus7
03653
042 09 318 sdot 10minus6
600 sdot 10minus7
02540 402 sdot 10minus6
975 sdot 10minus6
02541
044 082 418 sdot 10minus9
337 sdot 10minus10
02044 248 sdot 10minus8
234 sdot 10minus8
02044
046 082 732 sdot 10minus8
143 sdot 10minus7
01439 377 sdot 10minus6
222 sdot 10minus6
01439
Then we choose the condition to be satisfied at119905 = 119879 We tested the following find 119888
119895 such that
(119909(119879) 119910(119879) 119907(119879) 119908(119879)) is the same as in the case 119891(119905) = 119888but 119891 is minimal find 119888
119895 such that the distance of
the spaceship to the Earth or the Moon is assignedtogether with the modulus of the final velocity that is(119909(119879) minus 120583)
2+ 119910(119879) minus 119877
2= 0 and (119907(119879))2 + 119908(119879)2 minus 1198812 = 0
and 119891 is minimal find 119888119895 such that the distance of the
spaceship to the Earth or the Moon is assigned togetherwith the modulus of the final velocity with the additionalrequirement that the final velocity is orthogonal to the lineconnecting the satellite to a primaryAll these problembelongto the class of problem described in the previous section andthe polynomial constrained minimization problems havebeen solved by the Lagrange multipliers method
4 Results
Figure 1 represents an example of an orbit computed byusing the technique Tables 1 and 2 collect some resultsobtained for a trajectory backward in time starting closeto a Lyapunov orbit around 119871
1 more precisely at the
point (119909(0) 119910(0) 119907(0) 119908(0)) = (073356888 minus0017228233
025064405 017220602) and ending close to an orbit aroundthe Earth at time 119879 = minus17 The trajectory with a constantthrust 119888 = 02 ends at 119877 = radic(119909(119879)minus120583)
2+119910(119879) = 0421852
and 119881 = radic(119907(119879))2+119908(119879)
2=0876662
For both tables we computed the thrust of minimal normnecessary to end the trajectory with the values given inthe columns 119877 and 119881 The results of Table 2 represent thesolution with the additional constraint that the final velocityis orthogonal to the segment from the spaceship to theEarth corresponding to an approximate circular orbit Wetested the accuracy of the computation as explained at theend of Section 2 that is we repeated the computation ofthe orbit with standard floating point numbers using theexplicit expression of the forcing term and we used theresult of such computation to verify that the constraints weresatisfied The values 120576
119879and 120576
119862represent the mean square
of the relative errors on 119877 and 119881 obtained with the Taylorand the Chebyshev computation respectively while 120576perp
119879and
120576perp
119862is |(119909(119879) minus 120583)119907(119879) + 119910(119879)119908(119879)| that is the error on
the orthogonality requirement The columns 119891119879 and 119891
119862
represent the norm of the forcing terms It is clear from thetables that the method can achieve very accurate results andit is also clear that the norm of the forcing term obtained withthe Taylor and the Chebyshev computation is (almost) thesame
5 Conclusions
We discuss and compare the features of the two expansionthat we tested Taylor and Chebyshev Generally speaking(see [2 3]) the main advantages of the Taylor expansionconsist in three features the algorithms are simpler andfaster they provide directly the derivatives with respectto the parameters and furthermore and since one doesnot need to know the radius of convergence a priori onecan apply them and compute a posteriori the interval ofvalidity of the computation The Taylor expansion has alsotwo main disadvantages it does not provide uniform errorsin the interval where the computation is performed andthe radius of convergence may be bounded by poles inthe complex plane The Chebyshev expansion has oppositefeatures the error is uniform in the interval and the regionof convergence is an ellipse as thin as necessary thereforethere are no problems caused by poles with nontrivialimaginary parts On the other hand the algorithm forthe multiplication is much more complicated and muchslower it does not provide any direct computation of thederivatives and finally since the Chebyshev polynomialsare defined in [minus1 1] one needs to choose a priori (via asuitable translationrescaling) the interval where the param-eter ranges finding out only a posteriori if the approxi-mation is acceptable In the application considered in [23] the Chebyshev expansion was a clear winner due tothe fact that when considering the numerical approxi-mation of shock waves and when trying to improve theresolution the problem of poles becomes the main issueHere our judgment is the opposite since the solutions ofthe odersquos that we are studying are analytic functions theissue of poles limiting the radius of convergence of Taylorseries becomes negligible while the disadvantages of theChebyshev expansion in particular the slower algorithmsand the fact that the domain has to be chosen a prioribecome very relevant Additionally we do not have evi-dence of a better performance in terms of accuracy Itis easy to find examples when either expansion performsbetter than the other one but on average the accuracy issimilar
References
[1] G Mingotti F Topputo and F Bernelli-Zazzera ldquoNumericalmethods to design low-energy low-thrust sun-perturbed trans-fers to the moonrdquo in Proceedings of the 49th Israel AnnualConference on Aerospace Sciences Tel Aviv-Haifa Israel March2009
Journal of Mathematics 5
[2] G Arioli and M Gamba ldquoAn algorithm for the studyof parameter dependence for hyperbolic systemsrdquo MOX-Report 052011 httpmoxpolimiititprogettipubblicazioniquaderni05-2011pdf
[3] G Arioli and M Gamba ldquoAutomatic computation of Cheby-shev polynomials for the study of parameter dependencefor hyperbolic systemsrdquo MOX-Report 072011 httpmoxpolimiititprogettipubblicazioniquaderni07-2011pdf
[4] K Makino and M Berz ldquoHigher order verified inclusions ofmultidimensional systems by Taylor modelsrdquo in Proceedings ofthe 3rd World Congress of Nonlinear Analysts part 5 CataniaItaly July 2000
[5] K Makino and M Berz ldquoHigher order verified inclusionsofmultidimensional systems by TaylormodelsrdquoNonlinear Anal-ysis Theory Methods and Applications vol 47 no 5 pp 3503ndash3514 2001
[6] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 6 no 3 pp 239ndash316 2003
[7] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 4 no 4 pp 379ndash456 2003
[8] M Berz and K Makino ldquoHigher order multivariate automaticdifferentiation validated computation of remainder boundsrdquoTransactions on Mathematics vol 3 no 1 pp 37ndash44 2004
[9] G Arioli and H Koch ldquoComputer-assisted methods for thestudy of stationary solutions in dissipative systems appliedto the Kuramoto-Sivashinski equationrdquo Archive for RationalMechanics and Analysis vol 197 no 3 pp 1033ndash1051 2010
[10] G Arioli and H Koch ldquoIntegration of dissipative partialdifferential equations a case studyrdquo SIAM Journal on AppliedDynamical Systems vol 9 no 3 pp 1119ndash1133 2010
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Stochastic AnalysisInternational Journal of
2 Journal of Mathematics
where 119891 is an analytic function 119888 = 119888119894119895 is an 119899 times 119873 real
matrix and the forcing term 120593119888 [minus1 1] rarr R119899 depends on 119888
as follows
(120593119888 (119905))119894 =
119873minus1
sum
119895=0
119888119894119895119879119895 (119905) (3)
where 119879119895(119905) is the 119895th Chebyshev polynomial and (120593
119888(119905))119894
denotes the 119894th component of 120593119888(119905) Our first step is to
compute explicitly the dependence of 119910119888(119905) on 119888 for 119905 isin
(minus1 1]We choose a numeric method to solve the IVP in order
to focus on the issue of the parameter dependence we choosethe simplestmethod for example an explicit first-order Eulerscheme with constant time step and we consider the case 119899 =119873 = 1 We show later that it is straightforward to adapt themethod to other schemes for example a Runge-Kutta schemewith variable time step and to the case 119899119873 gt 1
Choose 119870 isin N let ℎ = 2119870 be the time step set 119905119896=
119896ℎ minus 1 and 119884119896(119888) = 119910
119888(119905119896) so that the Euler scheme reads
119884119896+1 (119888) = 119884119896 (119888) + ℎ119891 (119905119896 119884119896 (119888)) + ℎ120593119888 (119905119896) (4)
and 119884119870(119888) = 119910
119888(1) Now write
119884119896 (119888) =
+infin
sum
119895=0
119910119896119895119880119895 (119888) (5)
where 119880119895(119888)119895isinN is a basis of some space of functions119883 of our
choice In this paper we consider both the Taylor expansionfor the set of analytic functions in a disc (ie 119880
119895(119888) = 119888
119895) andthe Chebyshev expansion for the set of continuous functionsin [minus1 1] (ie 119880
119895(119888) = cos(119895 arccos 119888)) Clearly the method
can be generalized to other expansions for example Fourierseries or Legendre polynomials Now we choose an order119872 and we approximate 119884
119896(119888) with the truncated expansion
sum119872
119895=0119910119896119895119880119895(119888) Our aim is to compute the coefficients 119910
119896119895
using an automatic algorithm Since 1199100(119888) = 119910
0 then we have
11991000
= 1199100and 119910
0119895= 0 when 119895 ge 1
The core of the method consists in a generalization ofthe Taylor Models approach for a detailed description ofthe technique with different expansions see [2 3] Herewe only recall the basic ideas First we observe that it isstraightforward to implement on a computer an arithmeticof Taylor or Chebyshev polynomials of one variable Usingobject-oriented programming and operator overloading it ispossible to define a class Fun which represents a Taylor orChebyshev expansion and a set of methods which performthe basic operations and the computation of the elementaryfunctions More precisely an object Fun is represented as thelist of the coefficients It is straightforward to implement aprocedure that given a scalar 120572 and two objects Fun 119879
11198792
computes the object Fun corresponding to 120572(1198791lowast 1198792) where
lowast is either the addition or the multiplication Then givena polynomial 119901(119909) it is possible to compute the object Funcorresponding to 119901(119879
1) and since all analytic functions 119891(119909)
can be approximated with polynomials it is also possibleto compute the object Fun that better approximates 119891(119879
1)
A more algebraic-oriented way of expressing this idea isthe following let 119883 be the space of functions spanned by119880119895(119888)119895isin0119872
and let 119879 119883 rarr R119872+1 be the (invertible)map that extracts the coefficients Our approach consists inlifting the basic arithmetics of119883 toR119872+1 More precisely wecompute operators
oplus otimes R119872+1
timesR119872+1
997888rarr R119872+1
⊙ R timesR119872+1
997888rarr R119872+1
(6)
such that 119879(119891 + 119892) = 119879(119891) oplus 119879(119892) 119879(119891119892) = 119879(119891) otimes
119879(119892) and 119879(120572119892) = 120572 ⊙ 119879(119892) Once this has been done wecan make extensive use of object-oriented programming andoperator overloading in order to make this lifting completelytransparent to the user
The extension of this method to the case 119899 gt 1 isstraightforward The case 119873 gt 1 requires some additionalwork since we have to consider a multivariable Taylor orChebyshev expansion
119884119896 (119888) =
+infin
sum
1198951119895119873=0
1199101198961198951119895119873
1198801198951
(1198881) sdot sdot sdot 119880
119895119873
(119888119873) (7)
The sum of two of such expansions is done componentwiseThe multiplication is less straightforward but a very efficientalgorithm for the case of the Taylor expansion has beenintroduced in [8] and it is not too hard to extend it tothe Chebyshev (or some other) expansion We refer to ourprograms for details
With the class Fun consisting of the objects (7) andthe functions for addition and multiplication in place wecan use a standard implementation of the algorithm (4)observing that given 119884
119896 the methods of the class will take
care of the computation of 119891(119905119896 119884119896) while 120593
119888(119905119896) is explicitly
given by (3) Extending this approach to better integrationschemes say Runge-Kutta or multistep is straightforwardit suffices to implement the chosen algorithm using objectsFun and their methods By implementing and running theintegration scheme of choice with the class Fun we obtainall the coefficients 119910
1198961198951119895119873
and therefore the polynomials119884119896(119888) for all 119896 Now consider the polynomial 119884
119870(119888) solving
the boundary value problem (1) consists in computing 119888 isin
R119899119873 such that119884119870(119888) = 119910
1This amounts to solving 119899 algebraic
equations in119873 unknowns If we wish 120593119888 to be small (where
sdot is some norm of our choice) we recall that when119873 gt 1
the system 119884119870(119888) = 119910
1is underdetermined therefore we can
compute an explicit formula Φ(119888) = 120593119888 and exploit the
additional degrees of freedom to set the following constrainedminimization problem find 119888 isin R119899119873 such that
Φ (119888) = min119884119870(119888)=1199101
Φ (119888) (8)
This result can be achieved by standard constrained mini-mization algorithms for example Lagrangemultipliers Notethat this method can be easily extended to solve other kindsof conditions at 119905 = 1 let 119898 le 119899 and 119891 R119899 rarr R119898 some
Journal of Mathematics 3
polynomial functionWe can apply the same approach to find119888 isin R119899119873 such that
Φ (119888) = min119891(119884119870(119888))=0
Φ (119888) (9)
To sum up once the class Fun and some algorithm for solvinginitial value problems for ODErsquos has been implementedboundary value problems with optimization of the forcingterm are reduced to computing the constrained minumumof a polynomial the constraint being represented by apolynomial equation or system of equations We remarkthat in principle such polynomial constrainedminimizationproblems can be solved exactly (up to floating point errors)but this does not imply that the boundary condition is solvedexactly for the original problem since we are truncating theexpansion (7) and we are neglecting the truncation errorOne needs to check a posteriori for example by solving (2)with standard floating point numbers and with the forcingterm provided by (3) with 119888 = 119888 and computing the error120576 = |119910(1) minus 119910
1| or 120576 = |119891(119910(1))|
3 An Application Low-Thrust Orbits inthe Planar RTBP
In order tomodel the dynamics of a spaceship travelling froman orbit around the Earth to an orbit around the Moon weconsider the forced planar RTBP that is the system
+ 2 119910 = Ω119909+ 119891 (119905)
radic2 + 1199102
119910 minus 2 = Ω119910+ 119891 (119905)
119910
radic2 + 1199102
(10)
where
Ω(119909 119910) =1199092
2+1199102
2+
1 minus 120583
radic(119909 + 120583)2+ 1199102
+120583
radic(119909 + 120583 minus 1)2+ 1199102
+ 119862
(11)
We choose 120583 ≃ 00121506683 that is the reduced mass ofthe Earth-Moon system and 119862 = minus1600172454916536 sothat Ω(119871
1) = 0 119871
1being the Lagrangian point between the
primaries The forcing term is given by a scalar function 119891(119905)to be determined times a unit vector directed as the velocityof the spaceship In other words we assume that the thrustis always parallel to the velocity We show how to apply themethod described in the previous section to the problem ofoptimizing the travel from an orbit 167 km above the Earth toan orbit 100 km above the Moon
We choose a connecting orbit passing close to 1198711 Note
that the Jacobi function
119869 (119909 119910 119910) = 2+ 1199102minus 2Ω (119909 119910) (12)
is approximately minus60 on an orbit 167 km above the Earthand approximately minus16 on an orbit 100 km above the Moon
05
04
03
02
01
0
minus01
minus02
minus03
minus04
minus04 minus02 0 02 04 06 08 1 12
Figure 1 A complete orbit
Table 1 Result enforcing 119877 and 119881
119877 119881 120576119879
119891119879 120576
119862119891119862
04 09 103 sdot 10minus7
03012 127 sdot 10minus7
03012
04 085 232 sdot 10minus4
04215 345 sdot 10minus4
04235
045 09 NA NA 766 sdot 10minus5
02104
043 09 658 sdot 10minus10
01540 139 sdot 10minus7
01540
Furthermore in order to ldquocrossrdquo the region around theLagrangian point 119871
1 it is necessary to raise it above 0 We
build the connecting orbit as follows We start from theLyapunov orbit around 119871
1with 119869 = 01 and approximate
periodic orbits around the Moon and the Earth Then wecompute approximate connections from the Lyapunov orbitto the orbit around the Earth (backwards in time) and tothe orbit around the Moon (forward in time) Choosing theLyapunov orbit as a starting point is useful if not necessarybecause of its high instability it is much harder to aim atthe Lyapunov orbit than at a stable orbit around a primaryThese orbits are obtained by applying a constant low thrust(ie 119891(119905) = 119888 with small 119888) chosen by simple trial and error
Once these rough orbits are computed we apply thetechnique described above to compute an accurate correctionso that the trajectory satisfies some required boundaryconditions and at the same time the thrust is minimal Moreprecisely we write (10) as a system of 4 first-order equationsby setting 119907 = and 119908 = 119910 we write the modulus of theforcing term as
119891 (119905) = 119888 +
119873minus1
sum
119895=0
119888119895119879119895 (119905) (13)
we choose some initial value on the trajectory at the arbitrarytime 119905 = 0 and we solve the equation with a fourth-orderRunge-Kutta scheme up to some time 119905 = 119879 using objectsFun as scalars the coefficients 119888
119895being the independent
variables of the objects Fun Referring to Section 2 we have119899 = 4 (the dimension of the system) we choose 119873 = 4
and 119872 = 10 We rescale the time in the interval [minus1 1]and apply the method described in Section 2 to obtain(119909(119879) 119910(119879) 119907(119879) 119908(119879)) as a function of 119888
119895 expressed either
as a Taylor series or a Chebyshev series
4 Journal of Mathematics
Table 2 Result enforcing 119877 119881 and tangential trajectory
119877 119881 120576119879
120576perp
119879119891119879 120576
119862120576perp
119862119891119862
042 087 304 sdot 10minus4
848 sdot 10minus5
03627 153 sdot 10minus4
826 sdot 10minus7
03653
042 09 318 sdot 10minus6
600 sdot 10minus7
02540 402 sdot 10minus6
975 sdot 10minus6
02541
044 082 418 sdot 10minus9
337 sdot 10minus10
02044 248 sdot 10minus8
234 sdot 10minus8
02044
046 082 732 sdot 10minus8
143 sdot 10minus7
01439 377 sdot 10minus6
222 sdot 10minus6
01439
Then we choose the condition to be satisfied at119905 = 119879 We tested the following find 119888
119895 such that
(119909(119879) 119910(119879) 119907(119879) 119908(119879)) is the same as in the case 119891(119905) = 119888but 119891 is minimal find 119888
119895 such that the distance of
the spaceship to the Earth or the Moon is assignedtogether with the modulus of the final velocity that is(119909(119879) minus 120583)
2+ 119910(119879) minus 119877
2= 0 and (119907(119879))2 + 119908(119879)2 minus 1198812 = 0
and 119891 is minimal find 119888119895 such that the distance of the
spaceship to the Earth or the Moon is assigned togetherwith the modulus of the final velocity with the additionalrequirement that the final velocity is orthogonal to the lineconnecting the satellite to a primaryAll these problembelongto the class of problem described in the previous section andthe polynomial constrained minimization problems havebeen solved by the Lagrange multipliers method
4 Results
Figure 1 represents an example of an orbit computed byusing the technique Tables 1 and 2 collect some resultsobtained for a trajectory backward in time starting closeto a Lyapunov orbit around 119871
1 more precisely at the
point (119909(0) 119910(0) 119907(0) 119908(0)) = (073356888 minus0017228233
025064405 017220602) and ending close to an orbit aroundthe Earth at time 119879 = minus17 The trajectory with a constantthrust 119888 = 02 ends at 119877 = radic(119909(119879)minus120583)
2+119910(119879) = 0421852
and 119881 = radic(119907(119879))2+119908(119879)
2=0876662
For both tables we computed the thrust of minimal normnecessary to end the trajectory with the values given inthe columns 119877 and 119881 The results of Table 2 represent thesolution with the additional constraint that the final velocityis orthogonal to the segment from the spaceship to theEarth corresponding to an approximate circular orbit Wetested the accuracy of the computation as explained at theend of Section 2 that is we repeated the computation ofthe orbit with standard floating point numbers using theexplicit expression of the forcing term and we used theresult of such computation to verify that the constraints weresatisfied The values 120576
119879and 120576
119862represent the mean square
of the relative errors on 119877 and 119881 obtained with the Taylorand the Chebyshev computation respectively while 120576perp
119879and
120576perp
119862is |(119909(119879) minus 120583)119907(119879) + 119910(119879)119908(119879)| that is the error on
the orthogonality requirement The columns 119891119879 and 119891
119862
represent the norm of the forcing terms It is clear from thetables that the method can achieve very accurate results andit is also clear that the norm of the forcing term obtained withthe Taylor and the Chebyshev computation is (almost) thesame
5 Conclusions
We discuss and compare the features of the two expansionthat we tested Taylor and Chebyshev Generally speaking(see [2 3]) the main advantages of the Taylor expansionconsist in three features the algorithms are simpler andfaster they provide directly the derivatives with respectto the parameters and furthermore and since one doesnot need to know the radius of convergence a priori onecan apply them and compute a posteriori the interval ofvalidity of the computation The Taylor expansion has alsotwo main disadvantages it does not provide uniform errorsin the interval where the computation is performed andthe radius of convergence may be bounded by poles inthe complex plane The Chebyshev expansion has oppositefeatures the error is uniform in the interval and the regionof convergence is an ellipse as thin as necessary thereforethere are no problems caused by poles with nontrivialimaginary parts On the other hand the algorithm forthe multiplication is much more complicated and muchslower it does not provide any direct computation of thederivatives and finally since the Chebyshev polynomialsare defined in [minus1 1] one needs to choose a priori (via asuitable translationrescaling) the interval where the param-eter ranges finding out only a posteriori if the approxi-mation is acceptable In the application considered in [23] the Chebyshev expansion was a clear winner due tothe fact that when considering the numerical approxi-mation of shock waves and when trying to improve theresolution the problem of poles becomes the main issueHere our judgment is the opposite since the solutions ofthe odersquos that we are studying are analytic functions theissue of poles limiting the radius of convergence of Taylorseries becomes negligible while the disadvantages of theChebyshev expansion in particular the slower algorithmsand the fact that the domain has to be chosen a prioribecome very relevant Additionally we do not have evi-dence of a better performance in terms of accuracy Itis easy to find examples when either expansion performsbetter than the other one but on average the accuracy issimilar
References
[1] G Mingotti F Topputo and F Bernelli-Zazzera ldquoNumericalmethods to design low-energy low-thrust sun-perturbed trans-fers to the moonrdquo in Proceedings of the 49th Israel AnnualConference on Aerospace Sciences Tel Aviv-Haifa Israel March2009
Journal of Mathematics 5
[2] G Arioli and M Gamba ldquoAn algorithm for the studyof parameter dependence for hyperbolic systemsrdquo MOX-Report 052011 httpmoxpolimiititprogettipubblicazioniquaderni05-2011pdf
[3] G Arioli and M Gamba ldquoAutomatic computation of Cheby-shev polynomials for the study of parameter dependencefor hyperbolic systemsrdquo MOX-Report 072011 httpmoxpolimiititprogettipubblicazioniquaderni07-2011pdf
[4] K Makino and M Berz ldquoHigher order verified inclusions ofmultidimensional systems by Taylor modelsrdquo in Proceedings ofthe 3rd World Congress of Nonlinear Analysts part 5 CataniaItaly July 2000
[5] K Makino and M Berz ldquoHigher order verified inclusionsofmultidimensional systems by TaylormodelsrdquoNonlinear Anal-ysis Theory Methods and Applications vol 47 no 5 pp 3503ndash3514 2001
[6] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 6 no 3 pp 239ndash316 2003
[7] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 4 no 4 pp 379ndash456 2003
[8] M Berz and K Makino ldquoHigher order multivariate automaticdifferentiation validated computation of remainder boundsrdquoTransactions on Mathematics vol 3 no 1 pp 37ndash44 2004
[9] G Arioli and H Koch ldquoComputer-assisted methods for thestudy of stationary solutions in dissipative systems appliedto the Kuramoto-Sivashinski equationrdquo Archive for RationalMechanics and Analysis vol 197 no 3 pp 1033ndash1051 2010
[10] G Arioli and H Koch ldquoIntegration of dissipative partialdifferential equations a case studyrdquo SIAM Journal on AppliedDynamical Systems vol 9 no 3 pp 1119ndash1133 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 3
polynomial functionWe can apply the same approach to find119888 isin R119899119873 such that
Φ (119888) = min119891(119884119870(119888))=0
Φ (119888) (9)
To sum up once the class Fun and some algorithm for solvinginitial value problems for ODErsquos has been implementedboundary value problems with optimization of the forcingterm are reduced to computing the constrained minumumof a polynomial the constraint being represented by apolynomial equation or system of equations We remarkthat in principle such polynomial constrainedminimizationproblems can be solved exactly (up to floating point errors)but this does not imply that the boundary condition is solvedexactly for the original problem since we are truncating theexpansion (7) and we are neglecting the truncation errorOne needs to check a posteriori for example by solving (2)with standard floating point numbers and with the forcingterm provided by (3) with 119888 = 119888 and computing the error120576 = |119910(1) minus 119910
1| or 120576 = |119891(119910(1))|
3 An Application Low-Thrust Orbits inthe Planar RTBP
In order tomodel the dynamics of a spaceship travelling froman orbit around the Earth to an orbit around the Moon weconsider the forced planar RTBP that is the system
+ 2 119910 = Ω119909+ 119891 (119905)
radic2 + 1199102
119910 minus 2 = Ω119910+ 119891 (119905)
119910
radic2 + 1199102
(10)
where
Ω(119909 119910) =1199092
2+1199102
2+
1 minus 120583
radic(119909 + 120583)2+ 1199102
+120583
radic(119909 + 120583 minus 1)2+ 1199102
+ 119862
(11)
We choose 120583 ≃ 00121506683 that is the reduced mass ofthe Earth-Moon system and 119862 = minus1600172454916536 sothat Ω(119871
1) = 0 119871
1being the Lagrangian point between the
primaries The forcing term is given by a scalar function 119891(119905)to be determined times a unit vector directed as the velocityof the spaceship In other words we assume that the thrustis always parallel to the velocity We show how to apply themethod described in the previous section to the problem ofoptimizing the travel from an orbit 167 km above the Earth toan orbit 100 km above the Moon
We choose a connecting orbit passing close to 1198711 Note
that the Jacobi function
119869 (119909 119910 119910) = 2+ 1199102minus 2Ω (119909 119910) (12)
is approximately minus60 on an orbit 167 km above the Earthand approximately minus16 on an orbit 100 km above the Moon
05
04
03
02
01
0
minus01
minus02
minus03
minus04
minus04 minus02 0 02 04 06 08 1 12
Figure 1 A complete orbit
Table 1 Result enforcing 119877 and 119881
119877 119881 120576119879
119891119879 120576
119862119891119862
04 09 103 sdot 10minus7
03012 127 sdot 10minus7
03012
04 085 232 sdot 10minus4
04215 345 sdot 10minus4
04235
045 09 NA NA 766 sdot 10minus5
02104
043 09 658 sdot 10minus10
01540 139 sdot 10minus7
01540
Furthermore in order to ldquocrossrdquo the region around theLagrangian point 119871
1 it is necessary to raise it above 0 We
build the connecting orbit as follows We start from theLyapunov orbit around 119871
1with 119869 = 01 and approximate
periodic orbits around the Moon and the Earth Then wecompute approximate connections from the Lyapunov orbitto the orbit around the Earth (backwards in time) and tothe orbit around the Moon (forward in time) Choosing theLyapunov orbit as a starting point is useful if not necessarybecause of its high instability it is much harder to aim atthe Lyapunov orbit than at a stable orbit around a primaryThese orbits are obtained by applying a constant low thrust(ie 119891(119905) = 119888 with small 119888) chosen by simple trial and error
Once these rough orbits are computed we apply thetechnique described above to compute an accurate correctionso that the trajectory satisfies some required boundaryconditions and at the same time the thrust is minimal Moreprecisely we write (10) as a system of 4 first-order equationsby setting 119907 = and 119908 = 119910 we write the modulus of theforcing term as
119891 (119905) = 119888 +
119873minus1
sum
119895=0
119888119895119879119895 (119905) (13)
we choose some initial value on the trajectory at the arbitrarytime 119905 = 0 and we solve the equation with a fourth-orderRunge-Kutta scheme up to some time 119905 = 119879 using objectsFun as scalars the coefficients 119888
119895being the independent
variables of the objects Fun Referring to Section 2 we have119899 = 4 (the dimension of the system) we choose 119873 = 4
and 119872 = 10 We rescale the time in the interval [minus1 1]and apply the method described in Section 2 to obtain(119909(119879) 119910(119879) 119907(119879) 119908(119879)) as a function of 119888
119895 expressed either
as a Taylor series or a Chebyshev series
4 Journal of Mathematics
Table 2 Result enforcing 119877 119881 and tangential trajectory
119877 119881 120576119879
120576perp
119879119891119879 120576
119862120576perp
119862119891119862
042 087 304 sdot 10minus4
848 sdot 10minus5
03627 153 sdot 10minus4
826 sdot 10minus7
03653
042 09 318 sdot 10minus6
600 sdot 10minus7
02540 402 sdot 10minus6
975 sdot 10minus6
02541
044 082 418 sdot 10minus9
337 sdot 10minus10
02044 248 sdot 10minus8
234 sdot 10minus8
02044
046 082 732 sdot 10minus8
143 sdot 10minus7
01439 377 sdot 10minus6
222 sdot 10minus6
01439
Then we choose the condition to be satisfied at119905 = 119879 We tested the following find 119888
119895 such that
(119909(119879) 119910(119879) 119907(119879) 119908(119879)) is the same as in the case 119891(119905) = 119888but 119891 is minimal find 119888
119895 such that the distance of
the spaceship to the Earth or the Moon is assignedtogether with the modulus of the final velocity that is(119909(119879) minus 120583)
2+ 119910(119879) minus 119877
2= 0 and (119907(119879))2 + 119908(119879)2 minus 1198812 = 0
and 119891 is minimal find 119888119895 such that the distance of the
spaceship to the Earth or the Moon is assigned togetherwith the modulus of the final velocity with the additionalrequirement that the final velocity is orthogonal to the lineconnecting the satellite to a primaryAll these problembelongto the class of problem described in the previous section andthe polynomial constrained minimization problems havebeen solved by the Lagrange multipliers method
4 Results
Figure 1 represents an example of an orbit computed byusing the technique Tables 1 and 2 collect some resultsobtained for a trajectory backward in time starting closeto a Lyapunov orbit around 119871
1 more precisely at the
point (119909(0) 119910(0) 119907(0) 119908(0)) = (073356888 minus0017228233
025064405 017220602) and ending close to an orbit aroundthe Earth at time 119879 = minus17 The trajectory with a constantthrust 119888 = 02 ends at 119877 = radic(119909(119879)minus120583)
2+119910(119879) = 0421852
and 119881 = radic(119907(119879))2+119908(119879)
2=0876662
For both tables we computed the thrust of minimal normnecessary to end the trajectory with the values given inthe columns 119877 and 119881 The results of Table 2 represent thesolution with the additional constraint that the final velocityis orthogonal to the segment from the spaceship to theEarth corresponding to an approximate circular orbit Wetested the accuracy of the computation as explained at theend of Section 2 that is we repeated the computation ofthe orbit with standard floating point numbers using theexplicit expression of the forcing term and we used theresult of such computation to verify that the constraints weresatisfied The values 120576
119879and 120576
119862represent the mean square
of the relative errors on 119877 and 119881 obtained with the Taylorand the Chebyshev computation respectively while 120576perp
119879and
120576perp
119862is |(119909(119879) minus 120583)119907(119879) + 119910(119879)119908(119879)| that is the error on
the orthogonality requirement The columns 119891119879 and 119891
119862
represent the norm of the forcing terms It is clear from thetables that the method can achieve very accurate results andit is also clear that the norm of the forcing term obtained withthe Taylor and the Chebyshev computation is (almost) thesame
5 Conclusions
We discuss and compare the features of the two expansionthat we tested Taylor and Chebyshev Generally speaking(see [2 3]) the main advantages of the Taylor expansionconsist in three features the algorithms are simpler andfaster they provide directly the derivatives with respectto the parameters and furthermore and since one doesnot need to know the radius of convergence a priori onecan apply them and compute a posteriori the interval ofvalidity of the computation The Taylor expansion has alsotwo main disadvantages it does not provide uniform errorsin the interval where the computation is performed andthe radius of convergence may be bounded by poles inthe complex plane The Chebyshev expansion has oppositefeatures the error is uniform in the interval and the regionof convergence is an ellipse as thin as necessary thereforethere are no problems caused by poles with nontrivialimaginary parts On the other hand the algorithm forthe multiplication is much more complicated and muchslower it does not provide any direct computation of thederivatives and finally since the Chebyshev polynomialsare defined in [minus1 1] one needs to choose a priori (via asuitable translationrescaling) the interval where the param-eter ranges finding out only a posteriori if the approxi-mation is acceptable In the application considered in [23] the Chebyshev expansion was a clear winner due tothe fact that when considering the numerical approxi-mation of shock waves and when trying to improve theresolution the problem of poles becomes the main issueHere our judgment is the opposite since the solutions ofthe odersquos that we are studying are analytic functions theissue of poles limiting the radius of convergence of Taylorseries becomes negligible while the disadvantages of theChebyshev expansion in particular the slower algorithmsand the fact that the domain has to be chosen a prioribecome very relevant Additionally we do not have evi-dence of a better performance in terms of accuracy Itis easy to find examples when either expansion performsbetter than the other one but on average the accuracy issimilar
References
[1] G Mingotti F Topputo and F Bernelli-Zazzera ldquoNumericalmethods to design low-energy low-thrust sun-perturbed trans-fers to the moonrdquo in Proceedings of the 49th Israel AnnualConference on Aerospace Sciences Tel Aviv-Haifa Israel March2009
Journal of Mathematics 5
[2] G Arioli and M Gamba ldquoAn algorithm for the studyof parameter dependence for hyperbolic systemsrdquo MOX-Report 052011 httpmoxpolimiititprogettipubblicazioniquaderni05-2011pdf
[3] G Arioli and M Gamba ldquoAutomatic computation of Cheby-shev polynomials for the study of parameter dependencefor hyperbolic systemsrdquo MOX-Report 072011 httpmoxpolimiititprogettipubblicazioniquaderni07-2011pdf
[4] K Makino and M Berz ldquoHigher order verified inclusions ofmultidimensional systems by Taylor modelsrdquo in Proceedings ofthe 3rd World Congress of Nonlinear Analysts part 5 CataniaItaly July 2000
[5] K Makino and M Berz ldquoHigher order verified inclusionsofmultidimensional systems by TaylormodelsrdquoNonlinear Anal-ysis Theory Methods and Applications vol 47 no 5 pp 3503ndash3514 2001
[6] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 6 no 3 pp 239ndash316 2003
[7] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 4 no 4 pp 379ndash456 2003
[8] M Berz and K Makino ldquoHigher order multivariate automaticdifferentiation validated computation of remainder boundsrdquoTransactions on Mathematics vol 3 no 1 pp 37ndash44 2004
[9] G Arioli and H Koch ldquoComputer-assisted methods for thestudy of stationary solutions in dissipative systems appliedto the Kuramoto-Sivashinski equationrdquo Archive for RationalMechanics and Analysis vol 197 no 3 pp 1033ndash1051 2010
[10] G Arioli and H Koch ldquoIntegration of dissipative partialdifferential equations a case studyrdquo SIAM Journal on AppliedDynamical Systems vol 9 no 3 pp 1119ndash1133 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Mathematics
Table 2 Result enforcing 119877 119881 and tangential trajectory
119877 119881 120576119879
120576perp
119879119891119879 120576
119862120576perp
119862119891119862
042 087 304 sdot 10minus4
848 sdot 10minus5
03627 153 sdot 10minus4
826 sdot 10minus7
03653
042 09 318 sdot 10minus6
600 sdot 10minus7
02540 402 sdot 10minus6
975 sdot 10minus6
02541
044 082 418 sdot 10minus9
337 sdot 10minus10
02044 248 sdot 10minus8
234 sdot 10minus8
02044
046 082 732 sdot 10minus8
143 sdot 10minus7
01439 377 sdot 10minus6
222 sdot 10minus6
01439
Then we choose the condition to be satisfied at119905 = 119879 We tested the following find 119888
119895 such that
(119909(119879) 119910(119879) 119907(119879) 119908(119879)) is the same as in the case 119891(119905) = 119888but 119891 is minimal find 119888
119895 such that the distance of
the spaceship to the Earth or the Moon is assignedtogether with the modulus of the final velocity that is(119909(119879) minus 120583)
2+ 119910(119879) minus 119877
2= 0 and (119907(119879))2 + 119908(119879)2 minus 1198812 = 0
and 119891 is minimal find 119888119895 such that the distance of the
spaceship to the Earth or the Moon is assigned togetherwith the modulus of the final velocity with the additionalrequirement that the final velocity is orthogonal to the lineconnecting the satellite to a primaryAll these problembelongto the class of problem described in the previous section andthe polynomial constrained minimization problems havebeen solved by the Lagrange multipliers method
4 Results
Figure 1 represents an example of an orbit computed byusing the technique Tables 1 and 2 collect some resultsobtained for a trajectory backward in time starting closeto a Lyapunov orbit around 119871
1 more precisely at the
point (119909(0) 119910(0) 119907(0) 119908(0)) = (073356888 minus0017228233
025064405 017220602) and ending close to an orbit aroundthe Earth at time 119879 = minus17 The trajectory with a constantthrust 119888 = 02 ends at 119877 = radic(119909(119879)minus120583)
2+119910(119879) = 0421852
and 119881 = radic(119907(119879))2+119908(119879)
2=0876662
For both tables we computed the thrust of minimal normnecessary to end the trajectory with the values given inthe columns 119877 and 119881 The results of Table 2 represent thesolution with the additional constraint that the final velocityis orthogonal to the segment from the spaceship to theEarth corresponding to an approximate circular orbit Wetested the accuracy of the computation as explained at theend of Section 2 that is we repeated the computation ofthe orbit with standard floating point numbers using theexplicit expression of the forcing term and we used theresult of such computation to verify that the constraints weresatisfied The values 120576
119879and 120576
119862represent the mean square
of the relative errors on 119877 and 119881 obtained with the Taylorand the Chebyshev computation respectively while 120576perp
119879and
120576perp
119862is |(119909(119879) minus 120583)119907(119879) + 119910(119879)119908(119879)| that is the error on
the orthogonality requirement The columns 119891119879 and 119891
119862
represent the norm of the forcing terms It is clear from thetables that the method can achieve very accurate results andit is also clear that the norm of the forcing term obtained withthe Taylor and the Chebyshev computation is (almost) thesame
5 Conclusions
We discuss and compare the features of the two expansionthat we tested Taylor and Chebyshev Generally speaking(see [2 3]) the main advantages of the Taylor expansionconsist in three features the algorithms are simpler andfaster they provide directly the derivatives with respectto the parameters and furthermore and since one doesnot need to know the radius of convergence a priori onecan apply them and compute a posteriori the interval ofvalidity of the computation The Taylor expansion has alsotwo main disadvantages it does not provide uniform errorsin the interval where the computation is performed andthe radius of convergence may be bounded by poles inthe complex plane The Chebyshev expansion has oppositefeatures the error is uniform in the interval and the regionof convergence is an ellipse as thin as necessary thereforethere are no problems caused by poles with nontrivialimaginary parts On the other hand the algorithm forthe multiplication is much more complicated and muchslower it does not provide any direct computation of thederivatives and finally since the Chebyshev polynomialsare defined in [minus1 1] one needs to choose a priori (via asuitable translationrescaling) the interval where the param-eter ranges finding out only a posteriori if the approxi-mation is acceptable In the application considered in [23] the Chebyshev expansion was a clear winner due tothe fact that when considering the numerical approxi-mation of shock waves and when trying to improve theresolution the problem of poles becomes the main issueHere our judgment is the opposite since the solutions ofthe odersquos that we are studying are analytic functions theissue of poles limiting the radius of convergence of Taylorseries becomes negligible while the disadvantages of theChebyshev expansion in particular the slower algorithmsand the fact that the domain has to be chosen a prioribecome very relevant Additionally we do not have evi-dence of a better performance in terms of accuracy Itis easy to find examples when either expansion performsbetter than the other one but on average the accuracy issimilar
References
[1] G Mingotti F Topputo and F Bernelli-Zazzera ldquoNumericalmethods to design low-energy low-thrust sun-perturbed trans-fers to the moonrdquo in Proceedings of the 49th Israel AnnualConference on Aerospace Sciences Tel Aviv-Haifa Israel March2009
Journal of Mathematics 5
[2] G Arioli and M Gamba ldquoAn algorithm for the studyof parameter dependence for hyperbolic systemsrdquo MOX-Report 052011 httpmoxpolimiititprogettipubblicazioniquaderni05-2011pdf
[3] G Arioli and M Gamba ldquoAutomatic computation of Cheby-shev polynomials for the study of parameter dependencefor hyperbolic systemsrdquo MOX-Report 072011 httpmoxpolimiititprogettipubblicazioniquaderni07-2011pdf
[4] K Makino and M Berz ldquoHigher order verified inclusions ofmultidimensional systems by Taylor modelsrdquo in Proceedings ofthe 3rd World Congress of Nonlinear Analysts part 5 CataniaItaly July 2000
[5] K Makino and M Berz ldquoHigher order verified inclusionsofmultidimensional systems by TaylormodelsrdquoNonlinear Anal-ysis Theory Methods and Applications vol 47 no 5 pp 3503ndash3514 2001
[6] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 6 no 3 pp 239ndash316 2003
[7] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 4 no 4 pp 379ndash456 2003
[8] M Berz and K Makino ldquoHigher order multivariate automaticdifferentiation validated computation of remainder boundsrdquoTransactions on Mathematics vol 3 no 1 pp 37ndash44 2004
[9] G Arioli and H Koch ldquoComputer-assisted methods for thestudy of stationary solutions in dissipative systems appliedto the Kuramoto-Sivashinski equationrdquo Archive for RationalMechanics and Analysis vol 197 no 3 pp 1033ndash1051 2010
[10] G Arioli and H Koch ldquoIntegration of dissipative partialdifferential equations a case studyrdquo SIAM Journal on AppliedDynamical Systems vol 9 no 3 pp 1119ndash1133 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 5
[2] G Arioli and M Gamba ldquoAn algorithm for the studyof parameter dependence for hyperbolic systemsrdquo MOX-Report 052011 httpmoxpolimiititprogettipubblicazioniquaderni05-2011pdf
[3] G Arioli and M Gamba ldquoAutomatic computation of Cheby-shev polynomials for the study of parameter dependencefor hyperbolic systemsrdquo MOX-Report 072011 httpmoxpolimiititprogettipubblicazioniquaderni07-2011pdf
[4] K Makino and M Berz ldquoHigher order verified inclusions ofmultidimensional systems by Taylor modelsrdquo in Proceedings ofthe 3rd World Congress of Nonlinear Analysts part 5 CataniaItaly July 2000
[5] K Makino and M Berz ldquoHigher order verified inclusionsofmultidimensional systems by TaylormodelsrdquoNonlinear Anal-ysis Theory Methods and Applications vol 47 no 5 pp 3503ndash3514 2001
[6] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 6 no 3 pp 239ndash316 2003
[7] M Berz and K Makino ldquoTaylor models and other validatedfunctional inclusionmethodsrdquo International Journal of Pure andApplied Mathematics vol 4 no 4 pp 379ndash456 2003
[8] M Berz and K Makino ldquoHigher order multivariate automaticdifferentiation validated computation of remainder boundsrdquoTransactions on Mathematics vol 3 no 1 pp 37ndash44 2004
[9] G Arioli and H Koch ldquoComputer-assisted methods for thestudy of stationary solutions in dissipative systems appliedto the Kuramoto-Sivashinski equationrdquo Archive for RationalMechanics and Analysis vol 197 no 3 pp 1033ndash1051 2010
[10] G Arioli and H Koch ldquoIntegration of dissipative partialdifferential equations a case studyrdquo SIAM Journal on AppliedDynamical Systems vol 9 no 3 pp 1119ndash1133 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of