In this paper, the Legendre spectral-collocation method was applied to obtain approximate solutions for some types of fractional optimal control problems (FOCPs). The fractional derivative was described in the Caputo sense. Two different approaches were presented, in the first approach, necessary optimality conditions in terms of the associated Hamiltonian were approximated. In the second approach, the state equation was discretized first using the trapezoidal rule for the numerical integration followed by the Rayleigh–Ritz method to evaluate both the state and control variables. Illustrative examples were included to demonstrate the validity and applicability of the proposed techniques.
Journal of Advanced Research (2015) 6, 393–403 Cairo University Journal of Advanced Research ORIGINAL ARTICLE Legendre spectral-collocation method for solving some types of fractional optimal control problems Nasser H Sweilam *, Tamer M Al-Ajami Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt A R T I C L E I N F O Article history: Received 26 March 2014 Received in revised form 30 April 2014 Accepted 13 May 2014 Available online 22 May 2014 Keywords: Legendre spectral-collocation method Fractional order differential equations Pontryagin’s maximum principle Necessary optimality conditions Rayleigh–Ritz method A B S T R A C T In this paper, the Legendre spectral-collocation method was applied to obtain approximate solutions for some types of fractional optimal control problems (FOCPs) The fractional derivative was described in the Caputo sense Two different approaches were presented, in the first approach, necessary optimality conditions in terms of the associated Hamiltonian were approximated In the second approach, the state equation was discretized first using the trapezoidal rule for the numerical integration followed by the Rayleigh–Ritz method to evaluate both the state and control variables Illustrative examples were included to demonstrate the validity and applicability of the proposed techniques ª 2014 Production and hosting by Elsevier B.V on behalf of Cairo University Introduction Differential Equations (DEs) play a major role in mathematical modeling of real-life models in engineering, science and many other fields Generally speaking the analytical methods are not suitable for large scale problems with complex solution regions Numerical methods are commonly used to get an approximate solution for the DEs which are non-linear or the derivation of the analytical methods is difficult Numerical * Corresponding author Tel.: +20 1003543201 E-mail address: nsweilam@sci.cu.edu.eg (N.H Sweilam) Peer review under responsibility of Cairo University Production and hosting by Elsevier methods for DEs have been explored rapidly with the development of digital computers Optimal control deals with the problem of finding a control law for a given dynamical system An optimal control problem is a set of DEs describing the paths of the control variables that minimize a function of state and control variables A necessary condition for an optimal control problem can be derived using Pontryagin’s maximum principle and a sufficient condition can be obtained using Hamilton–Jacobi–Bellman equation Fractional order DEs have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, and engineering Fractional order models are more appropriate than conventional integer order to describe physical systems [1–4] For example, it has been illustrated that the so-called fractional Cable equation, which is similar to the traditional Cable equation except that the order of derivative with respect to the space and/or time is 2090-1232 ª 2014 Production and hosting by Elsevier B.V on behalf of Cairo University http://dx.doi.org/10.1016/j.jare.2014.05.004 394 N.H Sweilam and T.M Al-Ajami fractional, can be more adequately modeled by fractional order models than integer order models [5] In the recent years, the dynamic behaviors of fractionalorder differential systems have received increasing attention FOCP refers to the minimization of an objective functional subject to dynamic constraints, on state and control variables, which have fractional order models Some numerical methods for solving some types of FOCPs were recorded [6–10] and the references cited therein This paper is a continuation of the authors work in this area of research [9,10] The main aim of this work was to use the advantage of the Legender spectral-collocation method to study FOCPs, two efficient numerical methods for solving some types of FOCPs are presented where fractional derivatives are introduced in the Caputo sense These numerical methods depend upon the spectral method where the Legendre polynomials are used to approximate the unknown functions Legendre polynomials are well known family of orthogonal polynomials on the interval ½À1; 1 that have many applications [11] They are widely used because of their good properties in the approximation of functions The structure of this paper was arranged in the following way: In Section ‘Preliminaries and notations’, preliminaries, notations and properties of the shifted Legendre polynomials were introduced In Section ‘Necessary optimality conditions’, necessary optimality conditions of the FOCP model were given In Section ‘Numerical approximation’, the basic formulation of the proposed approximate formulas of the fractional derivatives was obtained In Section ‘Error estimates’, error estimates for the approximated fractional derivatives were given In Section ‘Numerical results’, illustrative examples were included to demonstrate the validity and applicability of the proposed technique Finally, in Section ‘Conclusions’, this paper ends with a brief conclusion and some remarks C a a Dt xtị ẳ C a t Db xtị ẳ Cðn À aÞ ðÀ1Þn Cðn À aÞ Fractional derivatives and integrals a t Ib xtị ẳ Caị a Rb t s tịa1 xsịds a t Db xtị ẳ dn CðnÀaÞ dt n Z The relation between right RLFD and right CFD [12]: C a t Db xtị ẳ t Dab xtị 1ịn d Cnaị dtn ðtÀsÞnÀaÀ1 xðsÞds ðleft RLFDÞ; ð1Þ ðsÀtÞ nÀ1 X xðkÞ ðbÞ b tịka ; Ck a ỵ 1ị kẳ0 3ị C a Dt C ẳ 0; where C is a constant; ð4Þ ( C a n Dt t ẳ 0; Cnỵ1ị na t ; Cnỵ1aị for for n N0 and n < dae; n N0 and n P dae: 5ị where N0 ẳ f0; 1; 2; g Recall that for a N, the Caputo differential operator coincides with the usual differential operator of integer order For more details on the fractional derivatives definitions and its properties see [13,14] The shifted Legendre polynomials The well known Legendre polynomials are defined on the interval ½À1; 1 and can be determined with the aid of the following recurrence formula [15]: 2n ỵ n zLn zị Ln1 zị; nỵ1 nỵ1 ẳ z; n ẳ 1; 2; : Lnỵ1 zị ¼ bn=2c X nÀaÀ1 ðÀ1Þm L0 ðzÞ ¼ 1; L1 ðzÞ ð2n À 2mÞ! znÀ2m ; m!ðn À mÞ!ðn À 2mÞ! n ð6Þ where bnc denotes the biggest integer less than or equal to n Moreover, we have [16]: nn ỵ 1ị jLn xịj 1; and L0n xị ; 8x ẵ1;1;n P 0: 2n ỵ 1ịLn xị ẳ L0nỵ1 xị L0n1 xị; n P 1; ðright RLFIÞ: t a n Z b ðs À tÞnÀaÀ1 xðnÞ ðsÞds ðright CFDÞ: t ð7Þ and ðt À sÞaÀ1 xðsÞds ðleft RLFIÞ; The left (left RLFD) and right (right RLFD) Riemann– Liouville fractional derivatives are defined, respectively, by: a a Dt xtị ẳ b In the following some basic properties are presented: Ln zị ẳ Denition Let x : ½a; b ! R be a function, a > a real number, and n ¼ dae, where dae denotes the smallest integer greater than or equal to a The left (left RLFI) and right (right RLFI) Riemann–Liouville fractional integrals are defined, respectively, by: Rt ðt sịna1 xnị sịds left CFDị; a 2ị mẳ0 ¼ CðaÞ Z t The analytic form of the Legendre polynomials Ln ðzÞ of degree n is given by Preliminaries and notations a a It xðtÞ Z xðsÞds ðright RLFDÞ: t The left (left CFD) and right (right CFD) Caputo fractional derivatives are defined respectively, by: ð8Þ In order to use these polynomials on the interval ½0; L we use the so-called shifted Legendre polynomials by introducing the change of variable z ¼ 2tL À The shifted Legendre polynomials are defined as follows: 2t 2t where P0 tị ẳ P1 tị ẳ 1: Pn tị ¼ Ln À1 L L The analytic form of the shifted Legendre polynomials Pn ðtÞ of degree n is given by: Pn tị ẳ n X 1ịnỵm mẳ0 n ỵ mÞ!tm L ðn À mÞ!ðm!Þ2 m : ð9Þ On the fractional optimal control problems 395 Note that from Eq (9), we can see that Pn 0ị ẳ 1ịn ; Pn Lị ẳ The function ytị which belongs to the space of square integrable in ½0; L, may be expressed in terms of shifted Legendre polynomials as ytị ẳ X cm Pm tị; fN tị ẳ where the coefcients cm are given by: Z 2m ỵ L cm ¼ yðtÞpm ðtÞ dt; m ¼ 0; 1; : L 10ị 18ị mẳ0 where fN tị is an approximation of fðtÞ If fN ðtÞ is the interpolation of ftị on the LegendreGaussLobatto points ftm gN mẳ0 , then am can be determined by am ¼ m¼0 N X am Pm ðtÞ; N X fðtk ÞPm tk ịxk ; cm kẳ0 19ị L where cm ẳ 2mỵ1 for m N 1; cN ¼ NL , and fxk gN k¼0 are the corresponding quadrature weights [17,18] In the following, approximation of the fractional derivative C a Dt fðtÞ is given Necessary optimality conditions Let a ð0; 1Þ and let L; f : ẵa; ỵ1ẵR ! R be two differentiable functions Consider the following FOCP [8]: Z T minimize Jðx; u; Tị ẳ Lt; xtị; utịị dt; 11ị Theorem [9] let fðtÞ be approximated by shifted Legendre polynomials as (18) and (19) and also a > 0, then C a Dt fN ðtÞ N X N X di;ka tka ; 20ị iẳdaekẳdae where di;ka is given by: a subject to the dynamic system: _ ỵ M2 aC Dat xtị ẳ ft; xtị; utịị; M1 xtị % 1ịiỵkị i þ kÞ! : L ði À kÞ!ðkÞ!Cðk þ À aị 12ị di;ka ẳ 13ị Approximation of right RLFD k 21ị where the boundary conditions are as follows: xaị ẳ xa ; where M1 ; M2 – 0; T; xa are fixed real numbers Theorem [8] If ðx; u; TÞ is a minimizer of (11)–(13), then there exists an adjoint state k for which the triple ðx; u; kÞ satisfies the optimality conditions Let fðsÞ be a sufficiently smooth function in ½0; b; < s < b and wðs; fÞ be defined as follows: Z b wðs; fÞ ¼ ðt À sÞÀa f0 ðtÞdt; ð22Þ s @H _ ỵ M2 Ca Dat xtị ẳ M1 xtị t; xtị; utị; ktịị; @k _ M2 t Da ktị ẳ À @H ðt; xðtÞ; uðtÞ; kðtÞÞ; M1 kðtÞ T @x @H t; xtị; utị; ktịị ẳ 0; @u 14ị 15ị 16ị for all t ẵa; T, and the transversality condition: M1 ktị ỵ M2 t I1a T ktị tẳT ẳ 0; from (2) and (3), we have: a s Db fsị ẳ fbị ws; fị b sÞÀa À : Cð1 À aÞ Cð1 À aÞ let fðxÞ be approximated by shifted Legendre polynomials as (18) and (19) Then we claim: Z b wðs; fÞ % wðs; fN ị ẳ f0N tịt sịa dt: 23ị s ð17Þ where the Hamiltonian H is defined by Hðt; x; u; kị ẳ Lt; x; uị ỵ kft; x; uị: If xðTÞ is fixed, there is no transversality condition Remark Under some additional assumptions on the objective functional L and the right-hand side f, e.g., convexity of L and linearity of f in x and u, the optimality conditions (14)–(16) are also sufficient Numerical approximation In this section, numerical approximations for the left CFD and the right RLFD using Legendre polynomials are presented Let fðtÞ be a function defined on the interval ½0; L, and N be positive integer Denote by Lemma Let fN ðtÞ be a polynomial of degree N given by (18) Then there exists a polynomial FNÀ1 ðtÞ of degree N À such that Z x  à fN ðtÞ À fN0 ðsÞ ðt À sÞÀa dt s ðxÞ À FNÀ1 ðsÞðx sị1a : ẳ ẵFN1 24ị Proof Let fN0 tị À fN0 ðsÞ be expanded in Taylor series at t ¼ s as follows: fN0 ðtÞ À fN0 ðsÞ ¼ NÀ1 X Ak ðsÞðt À sÞk ; where Ak ðsÞ ẳ f kỵ1ị sị k! kẳ1 396 N.H Sweilam and T.M Al-Ajami By inserting FNÀ1 ðxÞ À FNÀ1 ðsÞ and ðx À sÞFNÀ1 ðxÞ given by (29) and (30), respectively, into (27), and from (31), we have: kaỵ1 2s kỵa bk1 bk bkỵ1 ẳ ck ; k: 32ị 2k b 2k ỵ b Then, Z x  NÀ1 X à fN0 ðtÞ À fN0 ðsÞ ðt sịa dt ẳ Ak sị s Z x t sịka dt: s kẳ1 Then, " #x Z x NÀ1 X  à Ak ðsÞðt À sÞk Àa 1Àa fN ðtÞ À fN ðsÞ ðt À sÞ dt ẳ t sị : kaỵ1 s kẳ1 s We have (24) if we choose FN1 xị ẳ N1 X Ak sịx sịk kẳ0 kaỵ1 with starting values cN ẳ cNỵ1 ẳ , where ak are the Legendre coefficients of fN ðxÞ ; with an arbitrary constant A0 ðsÞ The Legendre coefficients ck of f0N ðxÞ given by (31) can be evaluated by integrating (31) and comparing it with (18) and (19) & ' ckỵ1 33ị ck1 ẳ 2k 1ị ỵ ak ; k ẳ N; N À 1; ; 1; 2k þ b Error estimates h From (24) we have: Z b ws; fN ị ẳ fN0 tịt sịa dt In the following, we give an upper bound for the coefficients am of Legendre expansion of a function f on ẵ0; s ! fN0 sị ỵ FN1 bị FN1 sị b sị1a ; 1a ẳ 25ị Lemma If f; f0 ; ; fkị are absolutely continuous on ẵ0; and if jf kỵ1ị tịj Wk < 1; 8t ẵ0; for some k P 1, then for each m P k, ð26Þ jam j and s Dab fðsÞ can be approximated as follows, a s Db fðsÞ % fðbÞ wðs; fN Þ : ðb À sÞÀa À Cð1 À aÞ Cð1 À aÞ Now, we express FNÀ1 ðtÞ in (25) by a sum of the Legendre polynomials and show the recurrence relation satisfied by the Legendre coefficients Differentiating both sides of (24) with respect to x yields È ẫ fN xị fN0 sị x sịa ẳ F0N1 xịx sị1a ỵ fFN1 xị a pWk : 2ð2m À 1Þð2m À 3Þ ð2m À 2k ỵ 1ị Proof We have: am ẳ 2m ỵ 1ị Z F0N1 xị ẳ N2 X bk Pk xị; 28ị kẳ0 Integrating both sides of (28) gives N1 bX bk1 bkỵ1 FN1 xị FN1 sị ẳ fPk xị Pk sịg; kẳ1 2k 2k ỵ 29ị & ' 2x 2s : b b 2x xịỵkPk ðxÞ Then, by using the relation b À Pk xị ẳ kỵ1ịPkỵ1 2kỵ1 and Eq (28), we have: b 0 x sịFN1 xị ẳ FN1 xị ' N1 & bX kbk1 k ỵ 1ịbkỵ1 2s ỵ bk Pk xị; 30ị kẳ0 2k b 2k ỵ where b1 ¼ b1 Let f0N ðxÞ ¼ NÀ1 X ck Pk xị: kẳ0 p f 1 þ cos hÞ Lm ðcos hÞ sin hdh Z p f0 1 ỵ cos hị Lm1 cos hị Lmỵ1 cos hịị  sin hdh: Again, integrating by parts, where bNÀ1 ¼ bN ¼ On the other hand, we have ðx sịF N1 xị ẳ Z Integrating by parts, using Eq (8), am ¼ x b; 2m ỵ 1ị 27ị To evaluate FN1 sị in (25) we expand F0NÀ1 ðxÞ in terms of the shifted Legendre polynomials fxịPm xịdx: Using the substitution x ẳ 12 ỵ cos hị, we have: am ẳ fN0 xị f0N sị ẳ F0N1 xịx sị ỵ fFN1 ðxÞ À FNÀ1 ðsÞgð1 À aÞ: À FNÀ1 ðsÞgð1 À aÞðx À sÞ : Then, ð34Þ ð31Þ Lmỵ2 cos hị Lm cos hị f þ cos hÞ 2m þ Lm ðcos hÞ À LmÀ2 ðcos hÞ sin hdh: À 2m À 1 am ¼ Z p 00 For k ¼ 1, to keep the formula simple, we not keep track of these different denominators but weaken the inequality slightly by replacing them with 2m À 1, jam j 00 Lmỵ2 cos hị Lm cos hị f þ cos hÞ 2m þ Lm ðcos hÞ À LmÀ2 ðcos hÞ pW1 À j sin hjdh 2ð2m À 1Þ ; 2m À 1 Z p since jLm j 1; 8m and j sin hj Further integrations by parts, The result is Eq (34) h On the fractional optimal control problems 397 Lemma Suppose that f satisfies hypotheses of Lemma Let fN be the truncated Legendre expansion of f Then for k > 3; 8x ½0; 1 and N ! k, f xị f xị N N2 ỵ NịpWk : 2kỵ2 N2 3N ỵ 2ịk 3ịN 3ịN 4ị N k ỵ 1Þ ð35Þ Proof We have: X N X X 0 f ðxÞ À f xị ẳ aj P0 xị aj Pj xị ¼ aj Pj ðxÞ N j jẳ1 jẳ1 jẳNỵ1 X X jaj jjP0j xịj jẳNỵ1 jẳNỵ1 jaj j fxị f~N xị ỵ KNịịkfxị p xịk ; where pà is the best approximation of f and KðNÞ is the Lebesgue constant ÀpffiffiffiffiÁ for which the following estimate holds, KNị ẳ O N on ẵ1; [20], and from Eq (7) and Lemma 4, we have X pWk aj Pj xị kỵ1 jẳNỵ1 k 1ịN À 1ÞðN À 2Þ ðN À k þ 1Þ It is well known that the truncated Chebyshev expansion is very close to the best polynomial approximation [21] Therefore, from [22] (we reformulate the Chebyshev error bound on ẵ0; 1ị, jj ỵ 1ị ; Wk kNN 1ịN 2ị N k ỵ 1ị pWk þ kþ1 ðk À 1ÞðN À 1ÞðN À 2Þ N k ỵ 1ị jfN xị f~N xịj ỵ KNịị since jP0j xịj jjỵ1ị Eq (7) Then, from Lemma 4, f xị f xị N X jẳNỵ1 ẳ 6 pWk jj ỵ 1ị 22j 1ị2j 3ị 2j 2k ỵ 1ị X jẳNỵ1 kỵ2 ẳ Hence, pWk jj ỵ 1ị j 12 j À 32 j À 2kÀ1 f ðxÞ À f~0 ðxÞ N X pWk jj ỵ 1ị kỵ2 j 1ịj 2ị j kị jẳNỵ1 X pWk NN ỵ 1ị jẳNỵ1 kỵ2 X jẳNỵ1 kỵ2 Now, in order to estimate the error of the approximated fractional derivatives, we have to estimate the error of the first derivative of the LGL interpolation as the following Suppose that f satisfies hypotheses of Lemma Let f~N be LGL interpolation of f Assume that k > and x ½0; 1 We have for : f ðxÞ À f~0 ðxÞ N ẳ f xị fN0 xị þ fN0 ðxÞ À f~N0 ðxÞ f xị fN0 xị ỵ fN0 xị f~N0 xị pWk NN ỵ 1ị N 3N ỵ 2ịk 3ịN 3ịN 4ị N k ỵ 1ị ỵ f0N xị f~0N xị: kỵ2 Markovs inequality asserts that max jP0 ðxÞj 2n2 max jPðxÞj 06x61 06x61 X ~ fN ðxÞ f~N xị ẳ fxị aj Pj xị fN xị jẳNỵ1 X aj Pj ðxÞ: fðxÞ À f~N xị ỵ jẳNỵ1 Since in [16]: Numerical results In this section, we develop two algorithms (Algorithms and 2) for the numerical solution of FOCPs and apply them to two illustrative examples For the first Algorithm, we follow the approach ‘‘optimize first, then discretize’’ and derive the necessary optimality conditions in terms of the associated Hamiltonian The necessary optimality conditions give rise to fractional boundary value problems We solve the fractional boundary value problems by the spectral method The second Algorithm relies on the strategy ‘‘discretize first, then optimize’’ The Rayleigh–Ritz method provides the optimality conditions in the discrete regime Example We consider the following FOCP from [8,10]: 06x61 for all polynomials of degree at most n with real coefficients [19], so f ðxÞ À f~0 ðxÞ 2N2 max jfN ðxÞ À f~N xịj: N N But pWk NN ỵ 1ị 2kỵ2 N2 3N ỵ 2ịk 3ịN 3ịN 4ị N k ỵ 1ị Wk þ 2N2 ðð1 þ KðNÞÞ k kNðN À 1ÞðN 2ị N k ỵ 1ị pWk ỵ kỵ1 ị: k 1ịN 1ịN 2ị N k ỵ 1ị N2 3N ỵ 2ịj 3ịj 4ị j kị pWk NN ỵ 1ị N2 3N ỵ 2ịk 3ịN 3ịN 4ị N k ỵ 1ị k Jx; uị ẳ Z tutị a ỵ 2ÞxðtÞÞ2 dt; ð36Þ subject to the dynamical system _ ỵ C0 Dat xtị ẳ utị ỵ t2 ; xtị 37ị and the boundary conditions x0ị ẳ 0; x1ị ẳ : C3 ỵ aị The exact solution is given by 2taỵ2 2taỵ1 : ; xtị; utịị ẳ Ca ỵ 3ị Ca ỵ 2ị 38ị 39ị 398 N.H Sweilam and T.M Al-Ajami Algorithm The first algorithm for the solution of (36)–(38) follows the ‘‘optimize first, then discretize’’ approach It is based on the necessary optimality conditions from Theorem and implements the following steps: Step 1: Compute the Hamiltonian H ẳ tutị a ỵ 2ịxtịị2 þ kðuðtÞ þ t2 Þ: ð40Þ Step 2: Derive the necessary optimality conditions from Theorem 1: _ À t Da ktị ẳ @H ẳ 2a ỵ 2ịtutị a þ 2ÞxðtÞÞ; kðtÞ @x @H C a _ þ Dt xtị ẳ ẳ utị ỵ t2 ; xtị @k @H 0ẳ ẳ 2ttutị a ỵ 2ịxtịị ỵ k: @u ð41Þ ð42Þ ð43Þ Fig 1b Exact and approximate control Use (43) in (41) and (42) to obtain _ ỵ t Da ktị ẳ a ỵ 2ị ktị; ktị t k a ỵ 2ị C a _ ỵ Dt xtị ẳ ỵ xtị ỵ t2 : xðtÞ 2t t ð44Þ ð45Þ Step 3: By using Legendre expansion, get an approximate solution of the coupled system (44) and (45) under the boundary conditions (38): Step 3a: In order to solve (44) by the Legendre expansion method, use (18) and (19) to approximate k A collocation scheme is defined by substituting (18), (19), (20) and (26) into (44) and evaluating the results at the shifted LegenÀ1 dre–Gauss–Lobatto nodes ftk gNk¼1 This gives: À N X i X d1i;k tk1 ỵ s iẳ1 kẳ1 ẳ kð1Þ wðts ; kn Þ ð1 À ts ÞÀa À C1 aị C1 aị aỵ2 kts ị; ts Fig 1c Exact and approximate state 46ị s ẳ 1; 2; ; N À 1, where d1i;k is defined in (21) The system (46) represents N À algebraic equations which can be solved for the unknown coefficients kðt1 Þ; kðt2 Þ; ; kðtNÀ1 Þ.Consequently, it remains to compute the two unknowns kðt0 Þ; kðtN Þ This can be done by using any two points ta ; tb 20; 1½ which differ from the Legendre–Gauss–Lobatto nodes and satisfy (44) We end up with two equations in two unknowns: Fig 1d Exact and approximate control Table Maximum errors in the state x and in the control u for different values of N Max error in x Max error in u Fig 1a Exact and approximate state N¼2 N¼3 N¼5 3:1055E À 2:0410E À 4:0702E À 4:5860E À 3:5526E À 9:1353E À On the fractional optimal control problems 399 _ a ị ỵ t Da kta ị ẳ aỵ2 kta ị; kt ta _ b ị ỵ t Da ktb ị ẳ aỵ2 ktb ị: kt tb Step 3b: In order to solve (45) by the Legendre expansion method, we use (18) and (19) to approximate the state x A collocation scheme is defined by substituting (18)–(20) and then computed k into (45) and evaluating the results À1 at the shifted Legendre–Gauss–Lobatto nodes ftk gNk¼1 This results in N À system of algebraic equations which can be solved for the unknown coefficients xðt1 Þ; xðt2 Þ; ; xðtN À1 Þ By using the boundary conditions, we have xt0 ị ẳ and Figs 1a,1b,1c and 1d display the exact and xtN ị ẳ C3ỵaị approximate state x and the exact and approximate control u for a ¼ 12 and N ¼ 2, Table contains the maximum errors in the state x and in the control u for N ¼ 2; N ¼ and N ¼ ( ) N N X X xtị ẳ xtk ịPm tk ịxk Pm tị; cm kẳ0 mẳ0 49ị _ ỵ C0 Dat xtị t2 : utị ẳ xtị 50ị Figs 1e,1f,1g and 1h display the exact and approximate state x and the exact and approximate control u for a ¼ 12 , N ¼ and N ¼ Table contains the maximum errors in the state x and in the control u for N ¼ 2; N ¼ and N ¼ 5.A comparison of Tables and reveals that both algorithms yield comparable numerical results which are more accurate than those obtained by the algorithm used in [8] Example We consider the following linear-quadratic optimal control problem [10]: Jx; uị ẳ Algorithm The second algorithm follows the ‘‘discretize first, then optimize’’ approach and proceeds according to the following steps: Z ðuðtÞ xtịị2 dt; subject to the dynamical system _ ỵ C0 Dat xtị ẳ utị xtị ỵ xtị Step 1: Substitute (37) into (36) to obtain Z À _ ỵ C0 Dat xtị t2 a ỵ 2ịxtị dt: J ẳ t xtị x0ị ¼ 0; xð1Þ ¼ Step 2: Approximate x using the Legendre expansion (18) and (19) and approximate the Caputo fractional derivative C a _ using (20) on the Legendre–Gauss–Lobatto Dt x and x nodes Then, (47) takes the form # Z "X N X i N X i X kÀ1 a kÀa minJ ¼ t di;k t ỵ di;k t t iẳ1 kẳ1 N X a ỵ 2ị an Pn tị 6taỵ2 ỵ t3 ; Ca ỵ 3ị : Ca ỵ 4ị iẳdaekẳdae !2 48ị dt; nẳ0 a where di;k is dened as in (21) Step 3: Define " # N X i N X i X X kÀ1 a kÀa Xtị ẳ t di;k t ỵ di;k t t iẳ1 kẳ1 a ỵ 2ị iẳdaekẳdae N X !2 Fig 1e an Pn ðtÞ Exact and approximate state n¼0 Using the composite trapezoidal integration technique, ! NÀ1 X Jẳ Xt0 ị ỵ XtN ị ỵ Xtk Þ : 2N k¼1 Step 4: The extremal values of functionals of the general form (6.1), according to Rayleigh–Ritz method give @J ẳ 0; @xt1 ị @J ẳ 0; @xt2 Þ .; ð52Þ and the boundary conditions ð47Þ 0 51ị @J ẳ 0; @xtN ị so, after using the boundary conditions, we obtain a system of algebraic equations Step 5: Solve the algebraic system by using the Newton– Raphson method to obtain xðt1 Þ; xðt2 Þ; ; xðtN À1 Þ and using the boundary conditions to get xðt0 Þ; xðtN Þ, then the function xðtÞ which extremes FOCPs has the following form: Fig 1f Exact and approximate control ð53Þ 400 N.H Sweilam and T.M Al-Ajami Table Maximum errors in the state x and in the control u for different values of N N¼3 Alg Alg Max error in x Max error in u 8:8025E À 8:8025E À 5:1966E À 4:3260E À N¼5 Alg Alg Max error in x Max error in u 1:0903E À 1:0903E À 4:5321E À 6:3134E À 1 _ ỵ xtị 2 Fig 1g C a Dt xtị ẳ xtị ỵ utị; and the boundary conditions p p x0ị ẳ 1; x1ị ẳ cosh ỵ b sinh ; Exact and approximate state ð56Þ ð57Þ where p p p cosh ỵ sinh b ¼ À pffiffiffi ÀpffiffiffiÁ ÀpffiffiffiÁ ffi À0:98 cosh ỵ sinh For this problem we have the exact solution in the case of a ¼ as follows [24]: p p xtị ẳ cosh 2t þ b sinh 2t ; pffiffiffi Á À Àpffiffiffi Á p p utị ẳ ỵ 2b cosh 2t ỵ ỵ b sinh 2t : Exact and approximate control Fig 1h Table Maximum errors in the state x and in the control u for different values of N Max error in x Max error in u N¼2 N¼3 N¼5 2:7313E À 2:5699E À 2:2570E À 4:4538E À 1:6006E À 8:2254E À Fig 2a The exact solution is given by 6taỵ3 6taỵ3 : ; Ca ỵ 4ị Ca ỵ 4ị Exact and Algorithm approximate, state ð xðtÞ; uðtÞÞ ¼ ð54Þ We note that for Example the optimality conditions stated in Theorem are also sufficient (cf Remark 1) Table contains a comparison between the maximum error in the state x and in the control u for Algorithms and The next two examples are modifications of the problems presented in [23,24] Example Consider the following time invariant problem: Jx; uị ẳ Z x2 tị ỵ u2 tịịdt; 55ị subject to the dynamical system Fig 2b Exact and Algorithm approximate, control On the fractional optimal control problems 401 Figs 2a and 2b display Algorithm approximate solutions of xðtÞ and utị for N ẳ and a ẳ 0:8, 0.9, 0.99, and exact solution for a ¼ Figs 2c and 2d display Algorithm approximate solutions of xðtÞ and utị for N ẳ 3, and a ẳ 0:9, and exact solution for a ¼ Figs 2e and 2f display Algorithm approximate solutions of xðtÞ and utị for N ẳ and a ẳ 0:8, 0.9, 0.99 and exact solution for a ¼ Figs 2g and 2h display Algorithm approximate solutions of xðtÞ and utị for N ẳ 3, and a ẳ 0:9 and exact solution for a ¼ Figs 2b, 2d, 2f and 2h illustrate that the approximate control converges better to the exact solution in Algorithm than Algorithm Fig 2e Exact and Algorithm approximate, state Table contains a comparison between approximate J in Algorithms and for ‘‘N ¼ with different values of a’’ and ‘‘N ¼ with a ¼ 0:9’’ where the exact is ‘‘J ¼ 0:192909 for a ¼ 1’’ Example Consider the following time variant problem: Jðx; uị ẳ Z x2 tị ỵ u2 tịịdt; ð58Þ Fig 2f Fig 2c Exact and Algorithm approximate, control Exact and Algorithm approximate, state Fig 2g Exact and Algorithm approximate, state subject to the dynamical system, 1 _ ỵ xtị 2 C a Dt xtị ẳ txtị ỵ utị; 59ị and the initial condition, x0ị ẳ 1: Fig 2d Exact and Algorithm approximate, control ð60Þ Algorithm has a modification to step 3a and step 3b where we have x0ị ẳ and k1ị ẳ and we use any two point 402 N.H Sweilam and T.M Al-Ajami Fig 2h Table Fig 3c Exact and Algorithm approximate, control Algorithm approximate , state Approximate J for Algorithms and N¼3 J, Alg J, Alg a ¼ 0:8 a ¼ 0:9 a ¼ 0:99 0:193035 0:193929 0:195687 0:185312 0:196629 0:212169 N¼5 J, Alg J, Alg a ¼ 0:9 0:187676 0:19636 Fig 3d Table Fig 3a Algorithm approximate, state Algorithm approximate, control Approximate J for Algorithms and N¼3 J, Alg J, Alg a ¼ 0:8 a ¼ 0:9 a ¼ 0:99 0:488123 0:487306 0:484141 0:481819 0:487719 0:497106 ta ; tb 20; 1½ which differ from LGL nodes and satisfy the necessary equation like (44) or (44) to determine xð1Þ and kð0Þ Also in Algorithm , there is a modification to step where we solve the non-linear algebraic system of equations to obtain xðt1 Þ; xðt2 Þ; ; xðtN Þ and use the initial condition to get xðt0 Þ Figs 3a and 3b display Algorithm approximate solutions of xtị and utị for N ẳ and a ¼ 0:8; 0:9; 0:99 Figs 3c and 3d display Algorithm approximate solutions of xðtÞ and uðtÞ for N ¼ and a ¼ 0:8; 0:9; 0:99 Table contains a comparison between approximate J in Algorithms and for different values of a and N ¼ Conclusions Fig 3b Algorithm approximate, control In this work, Legendre spectral-collocation method is used to study some types of fractional optimal control problems Two On the fractional optimal control problems efficient algorithms for the numerical solution of a wide class of fractional optimal control problems are presented In the first algorithm we derive the necessary optimality conditions in terms of the associated Hamiltonian The necessary optimality conditions give rise to fractional boundary value problems that have left Caputo and right Riemann–Liouville fractional derivatives We drive an approximation of right Riemann– Liouville fractional derivatives and solve these fractional boundary value problems using the spectral method In the second algorithm, the state equation is adjoined to the objective functional which discretized and then the composite trapezoidal integration technique and the Rayleigh–Ritz method are used to evaluate both the state and control variables In both algorithms, the solution is approximated by N-term truncated Legendre series Numerical results show that the two algorithms converge as the number of terms increase For the first example, it is noted that Algorithm is more accurate than Algorithm but in the second one Algorithm is better in finding the control variable Also Examples and show that Algorithm is preferable than Algorithm In general, the two algorithms are efficient and give the optimum solution Conflict of interest The authors have declared no conflict of interest Compliance with Ethics Requirements This article does not contain any studies with human or animal subjects References [1] Bagley RL, Torvik PJ On the appearance of the fractional derivative in the behavior of real materials Appl Mech 1984;51: 294–8 [2] Khader MM, Sweilam NH, Mahdy AMS An efficient numerical method for solving the fractional diffusion equation J Appl Math Bioinf 2011;1:1–12 [3] Oustaloup A, Levron F, Mathieu B, Nanot FM Frequencyband complex noninteger differentiator: characterization and synthesis IEEE Trans Circ Syst 2000;47:25–39 [4] Tricaud C, Chen Y-Q An approximation method for numerically solving fractional order optimal control problems of general form Comput Math Appl 2010;59:1644–55 [5] Sweilam NH, Khader MM, Adel M Numerical simulation of fractional Cable equation of spiny neuronal dendrites J Adv Res 2014;5(2):253–9 403 [6] Agrawal OP A general formulation and solution scheme for fractional optimal control problems Nonlinear Dyn 2004;38(1): 323–37 [7] Khader MM, Sweilam NH, Mahdy AMS Numerical study for the fractional differential equations generated by optimization problem using Chebyshev collocation method and FDM Appl Math Inf Sci 2013;7(5):2011–8 [8] Pooseh S, Almeida R, Torres DFM A numerical scheme to solve fractional optimal control problems In: Conference Papers in Mathematics, 2013; 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