Numerical solution of some types of fractional optimal control problems

Transcription

1 Numerical Analysis and Scienific mpuing Preprin Seria Numerical sluin f sme ypes f fracinal pimal cnrl prblems N.H. Sweilam T.M. Al-Ajmi R.H.W. Hppe Preprin #23 Deparmen f Mahemaics Universiy f Husn Nvember 2013

2 NUMERIAL SOLUTION OF SOME TYPES OF FRATIONAL OPTIMAL ONTROL PROBLEMS N.H. SWEILAM, T.M. AL-AJMI, AND R.H.W. HOPPE Absrac. We presen w differen appraches fr he numerical sluin f fracinal pimal cnrl prblems (FOPs) based n a specral mehd using hebyshev plynmials. The fracinal derivaive is described in he apu sense. The firs apprach fllws he paradigm pimize firs, hen discreize and relies n he apprimain f he necessary pimaliy cndiins in erms f he assciaed Hamilnian. In he secnd apprach, he sae equain is discreized firs using he lenshaw and uris scheme fr he numerical inegrain f nn-singular funcins fllwed by he Rayleigh-Riz mehd evaluae bh he sae and cnrl variables. Tw illusraive eamples are included demnsrae he validiy and applicabiliy f he suggesed appraches. Key wrds. specral mehd fracinal calculus, fracinal pimal cnrl, numerical sluin, hebyshev AMS subjec classificains. 65K10, 26A33, 49K15 1. Inrducin. FOP refers he minimizain f an bjecive funcinal subjec dynamical cnsrains n he sae and he cnrl which have fracinal rder mdels. Fracinal rder mdels are smeimes mre apprpriae han cnveninal ineger rder mdels describe physical sysems ([3],[7],[11], [17]). Fr eample, i has been shwn ha maerials wih memry and herediary effecs and dynamical prcesses including gas diffusin and hea cnducin in fracal prus media can be mre adequaely mdeled by fracinal rder mdels [18]. Numerical mehds fr slving FOPs have been suggesed in ([1],[8],[12], and [16]). This paper presens w numerical mehds fr slving sme ypes f FOPs where fracinal derivaives are inrduced in he apu sense. These numerical mehds rely n he specral mehd where hebyshev plynmials are used apprimae he unknwn funcins. hebyshev plynmials are widely used in numerical cmpuain ([9], [15]). Fr he firs numerical mehd, we fllw he apprach pimize firs, hen discreize and derive he necessary pimaliy cndiins in erms f he assciaed Hamilnian. The necessary pimaliy cndiins give rise fracinal bundary value prblems ha have lef apu and righ Riemann Liuville fracinal derivaives. We cnsruc an apprimain f he righ Riemann Liuville fracinal derivaives and slve he fracinal bundary value prblems by he specral mehd. The secnd mehd relies n he sraegy discreize firs, hen pimize. The lenshaw and uris scheme [4] is used fr he discreizain f he sae equain and he bjecive funcinal. The Rayleigh-Riz prvides he pimaliy cndiins in he discree regime. Deparmen f Mahemaics, Faculy f Science, air Universiy, Giza 12613, Egyp (nsweilam@sci.cu.edu.eg). Deparmen f Mahemaics, Faculy f Science, air Universiy, Giza 12613, Egyp (mamer@sci.cu.edu.eg). Insiue f Mahemaics, Universiy f Augsburg, D-86159, Augsburg, Germany, and Deparmen f Mahemaics, Universiy f Husn, Husn, TX , USA (rhp@mah.uh.edu). The auhr has been suppred by he DFG Pririy Prgrams SPP 1253 and SPP 1506, by he NSF grans DMS , DMS , and by he Eurpean Science Fundain wihin he Newrking Prgramme OPTPDE. 1

3 The paper is rganized as fllws: In secin 2, sme basic nains and preliminaries as well as prperies f he shifed hebyshev plynmials are inrduced. Secin 3 cnains he necessary pimaliy cndiins f he FOP mdel. Secin 4 is deved he apprimains f he fracinal derivaives. In secin 5, we develp he w numerical schemes and presen w illusraive eamples demnsrae he validiy and applicabiliy f he suggesed appraches. Finally, in secin 6, we prvide a brief cnclusin and sme final remarks. 2. Basic Nains and Preliminaries Fracinal Derivaives and Inegrals. Definiin 2.1. Le : [a, b] R be a funcin, α > 0 a real number, and n = α, where α denes he smalles ineger greaer han r equal α. The lef (lef RLFI) and righ (righ RLFI) Riemann Liuville fracinal inegrals are defined by ai α () = 1 Γ(α) a ( τ) α 1 (τ)dτ (lef RLFI), I α b () = 1 Γ(α) b (τ ) α 1 (τ)dτ (righ RLFI). The lef (lef RLFD) and righ (righ RLFD) Riemann Liuville fracinal derivaives are given accrding ad α 1 () = Γ(n α) d n d n a ( τ) n α 1 (τ)dτ (lef RLFD), d n Db α () = ( 1)n Γ(n α) d n b (τ ) n α 1 (τ)dτ (righ RLFD). (2.1) Mrever, he lef (lef FD) and righ (righ FD) apu fracinal derivaives are defined by means f a D α 1 () = Γ(n α) a ( τ) n α 1 (n) (τ)dτ (lef FD), Db α () = ( 1)n Γ(n α) b (τ ) n α 1 (n) (τ)dτ (righ FD). (2.2) The relain beween he righ RLFD and he righ FD is as fllws [2]: n 1 Db α () = Db α (k) (b) () Γ(k α + 1) (b )k α, (2.3) k=0 2

4 Furher, i hlds and 0 D α c = 0, where c is a cnsan, (2.4) 0 D α n = { 0, fr n N0 and n < α Γ(n+1) Γ(n+1 α) n α, fr n N 0 and n α, (2.5) where N 0 = {0, 1, 2,...}. We recall ha fr α N he apu differenial perar cincides wih he usual differenial perar f ineger rder. Fr mre deails n he fracinal derivaives definiins and is prperies we refer [10]-[13] Shifed hebyshev Plynmials. The well-knwn hebyshev plynmials are defined n he inerval [ 1, 1] and can be deermined by he fllwing recurrence frmula [14]: T n+1 (z) = 2z T n (z) T n 1 (z), T 0 (z) = 1, T 1 (z) = z, n = 1, 2,.... The analyic frm f he hebyshev plynmials T n (z) f degree n is as fllws T n (z) = n/2 ( 1) i n 2 i 1 n (n i 1)! 2 (i)! (n 2 i)! zn 2 i, (2.6) i=0 where n denes he bigges ineger less han r equal n. cndiin reads 1 T i (z) T j (z) π, fr i = j = 0; π dz = 1 1 z 2 2, fr i = j 0; 0, fr i j. The rhgnaliy (2.7) In rder use hese plynmials n he inerval [0, L], we use he s-called shifed hebyshev plynmials by inrducing he change f variable z = 2 L 1. The shifed hebyshev plynmials are defined accrding T n() = T n ( 2 L 1) where T 0 () = 1 T 1 () = 2 L 1. Their analyic frm is given by T n() = n n ( 1) n k 22k (n + k 1)! (2k)! (n k)!l k k, n = 1, 2,..., (2.8) k=0 We ne ha frm (2.8) implies T n(0) = ( 1) n, T n(l) = 1. Furher, i is easy see ha he rhgnaliy cndiin reads L 0 T j ()T k ()w()d = δ jk h k, (2.9) wih he weigh funcin w() = 1 L 2, h k = b k 2 π, b 0 = 2, b k = 1 fr k 1. A funcin y L 2 ([0, L]) can be epressed in erms f shifed hebyshev plynmials as y() = c n Tn(), j=0 3

5 where he cefficiens c n are given by c n = 1 h n L 0 y()t n()w()d, n = 0, 1,.... (2.10) 3. Necessary Opimaliy ndiins. Le α (0, 1) and le L, f : [a, + [ R 2 R be w differeniable funcins. We cnsider he fllwing FOP [12]: minimize J(, u, T ) = subjec he dynamical sysem T a L(, (), u())d, M 1 ẋ() + M 2 a D α () = f(, (), u()), (a) = a, (T ) = T, (3.1a) (3.1b) (3.1c) where M 1, M 2 0, T, a and T are fied real numbers. Therem 3.1. [12] If (, u, T ) is a minimizer f (3.1a)-(3.1c), hen here eiss an adjin sae λ fr which he riple (, u, λ) saisfies he pimaliy cndiins M 1 ẋ() + M 2 a D α () = H (, (), u(), λ()), λ (3.2a) M 1 λ() M2 DT α λ() = H (, (), u(), λ()), (3.2b) H (, (), u(), λ()) = 0, u (3.2c) fr all [a, T ], where he Hamilnian H is defined by H(,, u, λ) = L(,, u) + λf(,, u). Remark 3.1. Under sme addiinal assumpins n he bjecive funcinal L and he righ-hand side f, e.g., cnveiy f L and lineariy f f in and u, he pimaliy cndiins (3.2a)-(3.2c) are als sufficien. 4. Numerical Apprimains. In his secin, we prvide numerical apprimains f he lef FD and he righ RLFD using hebyshev plynmials. We chse he grid pins be he hebyshev-gauss-lba pins assciaed wih he inerval [0, L], i.e., r = L 2 L 2 cs(πr ), r = 0, 1,..., N. N lenshaw and uris [4] inrduced an apprimain y N f he funcin y. We refrmulae i be used wih respec he shifed hebyshev plynmials as fllws y N () = n=0 a n T n(), a n = 2 N y( r )T n( r ). (4.1) Here, he summain symbl wih duble primes denes a sum wih bh firs and las erms halved. 4

6 4.1. Apprimain f he Lef FD. In he sequel, sme basic resuls fr he apprimain f he fracinal derivaive 0 D α y() are given. Therem 4.1. [6] An apprimain f he fracinal derivaive f rder α in he apu sense f he funcin y a s is given by where d α s,r = 4θ r N n= α j=0 k= α 0 D α y N ( s ) = n y( r )d α s,r, α > 0, (4.2) nθ n ( 1) n k (n + k 1)!Γ(k α )T n( r )Tj ( s) b j L α Γ(k )(n k)!γ(k α j + 1)Γ(k α + j + 1), (4.3) and s, r = 0, 1,..., N wih θ 0 = θ N = 1 2, θ i = 1 i = 1, 2,..., N 1. An upper bund fr he errr in he apprimain f he fracinal derivaive 0 D α f he funcin y is given as fllws: Therem 4.2. [5] Le 0 D α y N () be he apprimain f he fracinal derivaive 0 D α f he funcin y as given by (4.2). Then i hlds 0 D α y() 0 D α y N () 2 n=0 a n Ω n ( G(k α;t 0,...,T N ) G(T 0,..., T N ) ) 1 2 (4.4) where n ( 1) n k 2n(n + k 1)!Γ(k α Ω n = ) b j L α Γ(k + 1 k= α 2 )(n k)!γ(k α j + 1)Γ(k α + j + 1)T j (), (4.5) <, > <, y 1 > <, y n > < y 1, > < y 1, y 1 > < y 1, y n > G(; y 1, y 2,..., y n ) = < y n, > < y n, y 1 > < y n, y n > 4.2. Apprimain f he Righ RLFD. Le f be a sufficienly smh funcin in [0, b] and le J(s; f) be defined as fllws J(s; f) = Frm (2.2) and (2.3) we deduce sd α b f(s) = b s ( s) α f ()d, 0 < s < b. (4.6) f(b) Γ(1 α) (b J(s; f) s) α + Γ(1 α). We apprimae f(), 0 b, by a sum f shifed hebyshev plynmials T k ( 2 b 1) accrding f() p N () = k=0 a k T k ( 2 b 1), a k = 2 N 5 j=0 f( j )T k ( 2 j b 1), (4.7)

7 where j = b 2 b 2 πj cs( N ), j = 0,..., N, and bain J(s; f) J(s; p N ) = b s p N ()( s) α d. (4.8) Lemma 4.1. Le p N be he plynmial f degree N as given by (4.7). Then here eiss a plynmial F N 1 f degree N 1 such ha s [p N() p N(s)]( s) α d = [F N 1 () F N 1 (s)]( s) 1 α. (4.9) Prf. Le p N () p N (s) be epanded in a Taylr series a = s: Then, p N () p N (s) = N 1 k=1 A k (s)( s) k. s [p N() p N (s)]( s) α d = The asserin fllws, if we chse F N 1 () = wih an arbirary cnsan A 0 (s). In view f (4.9) we have J(s; p N ) = b s N 1 k=1 A k (s) N 1 = [( s) 1 α N 1 k=0 s k=1 A k (s)( s) k, k α + 1 ( s) k α d A k (s)( s) k k α + 1 ] s. p N ()( s) α d = [ p N (s) 1 α + F N 1(b) F N 1 (s)](b s) 1 α. (4.10) Mrever, s Db α f(s) can be apprimaed by means f sd α b f(s) f(b) Γ(1 α) (b s) α + J(s; p N ) Γ(1 α). (4.11) We epress F N 1 () in (4.10) by a sum f hebyshev plynmials and prvide he recurrence relain saisfied by he hebyshev cefficiens. Differeniaing bh sides f (4.9) wih respec yields {p N() p N(s)}( s) α = F N 1()( s) 1 α + {F N 1 () F N 1 (s)}(1 α)( s) α, whence p N() p N(s) = F N 1()( s) + {F N 1 () F N 1 (s)}(1 α). (4.12) 6

8 T evaluae F N 1 (s) in (4.10), we epand F N 1 () in erms f he shifed hebyshev plynmials F N 1() = N 2 k=0 b k T k ( 2 b 1), 0 b, (4.13) where he summain symbl wih ne prime denes a sum wih he firs erm halved. Inegraing bh sides f (4.13) gives F N 1 () F N 1 (s) = b 4 N 1 k=1 where b N 1 = b N = 0. On he her hand, we have ( s)f N 1() = b 2 F N 1(){( 2 b b k 1 b k+1 {T k ( 2 k b 1) T k( 2s 1)}, (4.14) b 1) (2s b 1)}. By using he relain T k+1 (u) + T k 1 (u) = 2uT k (u) and (4.13), i fllws ha ( s)f N 1() = b 4 where b 1 = b 1. Le N 1 k=0 {b k+1 2( 2s b 1)b k + b k 1 }T k ( 2 b 1), (4.15) p N () = N 1 k=0 c k T k ( 2 b 1). (4.16) Insering F N 1 () F N 1 (s) and ( s)f N 1 () as given by (4.14) and (4.15) in (4.12) and aking (4.16) in accun, we ge {1 1 α }b k+1 2( 2s k b 1)b k + {1 + 1 α }b k 1 = 4 k b c k, 1 k. (4.17) The hebyshev cefficiens c k f p N () as given by (4.16) can be evaluaed by inegraing (4.16) and cmparing i wih (4.7): c k 1 = c k+1 + 4k b a k, k = N, N 1,..., 1, (4.18) wih saring values c N = c N+1 = 0, where a k are he hebyshev cefficiens f p N (). 5. Numerical Resuls. In his secin, we develp w algrihms (Algrihm I and Algrihm II) fr he numerical sluin f FOPs and apply hem w illusraive eamples Eample 1. We cnsider he fllwing FOP frm [12]: min J(, u) = subjec he dynamical sysem 1 0 (u() (α + 2)()) 2 d, ẋ() + 0 D α () = u() + 2, (5.1a) (5.1b) 7

9 and he bundary cndiins The eac sluin is given by (0) = 0, (1) = 2 Γ(3 + α). (5.1c) 2 α+2 ( (), ū()) = ( Γ(α + 3), 2 α+1 ). (5.2) Γ(α + 2) Algrihm I. The firs algrihm fr he sluin f (5.1a)-(5.1c) fllws he pimize firs, hen discreize apprach. I is based n he necessary pimaliy cndiins frm Therem 3.1 and implemens he fllwing seps: Sep 1: mpue he Hamilnian H = (u() (α + 2)()) 2 + λ(u() + 2 ). (5.3) Sep 2: Derive he necessary pimaliy cndiins frm Therem 3.1: λ() D1 α λ() = H = 2(α + 2)(u() (α + 2)()), (5.4a) ẋ() + 0 D α () = H λ = u() + 2, 0 = H = 2(u() (α + 2)()) + λ. u (5.4c) Use (5.4c) in (5.4a) and (5.4b) bain (5.4b) λ() + D1 α (α + 2) λ() = λ(), ẋ() + 0 D α () = λ (α + 2) + () (5.5a) (5.5b) Sep 3: By using hebyshev epansin, ge an apprimae sluin f he cupled sysem (5.5a),(5.5b) under he bundary cndiins (5.1c): Sep 3a: In rder slve (5.5a) by he hebyshev epansin mehd, use (4.1) apprimae λ. A cllcain scheme is defined by subsiuing (4.1),(4.2), and (4.11) in (5.5a) and evaluaing he resuls a he shifed Gauss-Lba ndes s, s = 1, 2,..., N 1. This gives: d 1 s,rλ( r ) + λ(1) Γ(1 α) (1 s) α + J( s; p n ) Γ(1 α) = α + 2 λ( s ), (5.6) s s = 1, 2,..., N 1, where d 1 s,r is defined in (4.3). The sysem (5.6) represens N 1 algebraic equains which can be slved fr he unknwn cefficiens λ( 1 ), λ( 2 ),..., λ( N 1 ). nsequenly, i remains cmpue he w unknwns λ( 0 ), λ( N ). This can be dne by using any w pins a, b ]0, 1[ which differ frm he Gauss-Lba ndes and saisfy (5.5a). We end up wih w equains in w unknwns: λ( a ) + D α 1 λ( a ) = α + 2 a λ( a ), λ(b ) + D α 1 λ( b ) = α + 2 b λ( b ). 8

10 Sep 3b: In rder slve (5.5b) by he hebyshev epansin mehd, we use (4.1) apprimae. A cllcain scheme is defined by subsiuing (4.1), (4.2) and he cmpued λ in (5.5b) and evaluaing he resuls a he shifed Gauss-Lba ndes s, s = 1, 2,..., N 1. This resuls in: d 1 s,r( r ) + d α s,r( r ) = λ( s) 2 2 s + α + 2 s ( s ) + 2 s, s = 1, 2,..., N 1, (5.7) where d 1 s,r and d α s,r are defined in (4.3). By using he bundary cndiins, we have ( 0 ) = 0 and ( N ) = 2 Γ(3+α). The sysem (5.7) represens N 1 algebraic equains which can be slved fr he unknwn cefficiens ( 1 ), ( 2 ),..., ( N 1 ). Figures display he eac and apprimae sae and he eac and apprimae cnrl u fr α = 1 2 and N = 2, App.,Α 1 2, N 2 Eac,Α 1 2 u App.,Α 1 2, N 2 Eac,Α Fig Eac and apprimae sae. Fig Eac and apprimae cnrl App.,Α 1 2, N 3 Eac,Α 1 2 u App.,Α 1 2, N 3 Eac,Α 1 2 Fig Eac and apprimae sae. Fig Eac and apprimae cnrl. Table 5.1 cnains he maimum errrs in he sae and in he cnrl u fr N = 2, N = 3 and N = 5. 9

11 Table 5.1 Maimum errrs in he sae and in he cnrl u fr differen values f N. N = 2 N = 3 N = 5 Ma. errr in E E E 4 Ma. errr in u E E E Algrihm II. The secnd algrihm fllws he discreize firs, hen pimize apprach and prceeds accrding he fllwing seps: Sep 1: Subsiue (5.1b) in (5.1a) bain min J = 1 0 ([ẋ() + 0 D α () 2 ] (α + 2)()) 2 d. (5.8) Sep 2: Apprimae using he lenshaw and uris frmula (4.1) and apprimae he apu fracinal derivaive 0 D α and ẋ using (4.2). Then, (5.8) akes he frm min J = 1 ([ d 1,r( r ) + d α,r( r ) 2 ] (α + 2) a n Tn()) 2 d, (5.9) 0 where d α,r is defined as in (4.3) replacing s by. Sep 3: Use = 1 2 (η + 1) ransfrm (5.9) : min J = n=0 ( 1 N 2 (η + 1)[ d 1 η,r(η r ) + d α η,r(η r ) ( 1 2 (η + 1))2 ] (α + 2) n=0 Sep 4: Use he lenshaw and uris frmula [4] where 1 1 F (η)dη = 2 m a n T n(η)) 2 dη. (5.10) m m s=0 i=0 θ s F (η s ) 2i + 1 [T s (η 2i ) T s (η 2i+2 )], (5.11) θ 0 = θ m = 1 2, θ s = 1 s = 1, 2,..., m 1, η i = cs[ (πi) m ] i < m, η i = 1 i > m, apprimae he inegral (5.10) as a finie sum f shifed hebyshev plynmials as fllws min J = 1 m m θ s m 2i + 1 (1 2 (η s + 1)[ d 1 η s,r(η r ) + d α η s,r(η r ) ( 1 2 (η s + 1)) 2 ] s=0 i=0 (α + 2) n=0 a n T n(η s )) 2 [T s (η 2i ) T s (η 2i+2 )]. (5.12) Sep 5: Accrding he Rayleigh-Riz mehd, he criical pins f he bjecive funcinal (5.1a) are given by J ( 1 ) = 0, J ( 2 ) = 0,..., J ( N ) = 0, 10

12 which leads a sysem f nnlinear algebraic equains. Slve his sysem by Newn s mehd bain ( 1 ), ( 2 ),..., ( N 1 ) and use he bundary cndiins ge ( 0 ), ( N ). Then, he pair (, u) which slves he FOP has he frm () = 2 N n=0 u() = ẋ() + 0 D α () 2. ( r )T n( r )T n(), (5.13a) (5.13b) Figures display he eac and apprimae sae and he eac and apprimae cnrl u fr α = 1 2 and N = m = 2, App.,Α 1 2, N m 2 Eac,Α 1 2 u App.,Α 1 2, N m 2 Eac,Α 1 2 Fig Eac and apprimae sae Fig Eac and apprimae cnrl App.,Α 1 2, N m 3 Eac,Α u App.,Α 1 2, N m 3 Eac,Α 1 2 Fig Eac and apprimae sae Fig Eac and apprimae cnrl Table 5.2 cnains he maimum errrs in he sae and in he cnrl u fr N = m = 2, N = m = 3 and N = m = 5. Table 5.2 Maimum errrs in he sae and in he cnrl u fr differen values f N. N = m = 2 N = m = 3 N = m = 5 Ma. errr in E E E 4 Ma. errr in u E E E 3 11

13 A cmparisn f Table 1 and Table 2 reveals ha bh algrihms yield cmparable numerical resuls which are mre accurae han hse bained by he algrihm used in [12] Eample 2. We cnsider he fllwing linear-quadraic pimal cnrl prblem: min J(, u) = 1 0 (u() ()) 2 d, (5.14a) subjec he dynamical sysem ẋ() + 0 D α () = u() () + 6α+2 Γ(α + 3) + 3, (5.14b) and he bundary cndiins (0) = 0, (1) = 6 Γ(α + 4). (5.14c) The eac sluin is given by 6 α+3 ( (), ū()) = ( Γ(α + 4), 6 α+3 ). (5.15) Γ(α + 4) We ne ha fr Eample 2 he pimaliy cndiins saed in Therem 3.1 are als sufficien (cf. Remark 3.1). Table 3 cnains a cmparisn beween he maimum errr in he sae and in he cnrl u fr Algrihm I and Algrihm II. Alg. I, N = 3 Alg. II, N = m = 3 ma. errr in E E 2 ma. errr in u E E 1 Alg. I, N = 5 Alg. II, N = m = 5 ma. errr in E E 4 ma. errr in u E E 3 As ppsed Eample 1, in his case Algrihm I perfrms subsanially beer han Algrihm II. 6. nclusins. In his aricle, we have presened w algrihms fr he numerical sluin f a wide class f fracinal pimal cnrl prblems, ne based n he pimize firs, hen discreize apprach and he her ne n he discreize firs, hen pimize sraegy. In bh algrihms, he sluin is apprimaed by N-erm runcaed hebyshev series. Numerical resuls fr w illusraive eamples shw ha he algrihms cnverge as he number f erms is increased and ha he firs algrihm is mre accurae han he secnd ne. REFERENES [1] O. P. Agrawal, A general frmulain and sluin scheme fr fracinal pimal cnrl prblems, Nnlinear Dynamics, 38(1), (2004). 12

14 [2] R. Almeida and D. F. M. Trres, Necessary and sufficien cndiins fr he fracinal calculus f variains wih apu derivaives, mmun. Nnlinear Sci. Numer. Simula., 16, (2011). [3] R. L. Bagley and P. J. Trvik., On he appearance f he fracinal derivaive in he behavir f real maerials, Appl. Mech., 51, , (1984). [4]. lenshaw and A. uris, A mehd fr numerical inegrain n an aumaic cmpuer, Numer. Mah., 2, (1960). [5] M. M. Khader and A. S. Hendy, An efficien numerical scheme fr slving fracinal pimal cnrl prblems, Inernainal Jurnal f Nnlinear Science, (14) 3, (2012) [6] M. M. Khader and A. S. Hendy, Fracinal hebyshev Finie Difference Mehd fr Slving he Fracinal BVPs, Appl. Mah. and Infrmaics, 31, (2012). [7] M. M. Khader, N. H. Sweilam and A. M. S. Mahdy, An efficien numerical mehd fr slving he fracinal diffusin equain, J. f Applied Mah. and Biinfrmaics, 1, 1-12 (2011). [8] M. M. Khader, N. H. Sweilam, and A. M. S. Mahdy, Numerical sudy fr he fracinal differenial equains generaed by pimizain prblem using hebyshev cllcain mehd and FDM, Appl. Mah. Inf. Sci., 7 (5), (2013). [9] A. K. Khalifa, E. M. E. Elbarbary, and M. A. Abd-Elrazek, hebyshev epansin mehd fr slving secnd and furh-rder ellipic equains, Applied Mahemaics and mpuain, 135, (2003). [10] K. B. Oldham and J. Spanier, The Fracinal alculus, Academic Press, New Yrk [11] A. Ousalup, F. Levrn, B. Mahieu, and F. M. Nan, Frequency-band cmple nnineger differeniar: characerizain and synhesis, IEEE Transacins n ircuis and Sysems, 47, (2000). [12] S. Pseh, R. Almeida, and D. F. M. Trres, Fracinal rder pimal cnrl prblems wih free erminal ime, Jurnal f Indusrial and Managemen Opimizain, in press, hp://ariv.rg/abs/ [13] S. Samk, A. Kilbas, and O. Marichev, Fracinal Inegrals and Derivaives: Thery and Applicains, Grdn and Breach, Lndn (1993). [14] M. A. Snyder, hebyshev Mehds in Numerical Apprimain, Prenice-Hall, Inc. Englewd liffs, N. J [15] N. H. Sweilam and M. M. Khader, A hebyshev pseud-specral mehd fr slving fracinal rder inegr-differenial equains, ANZIAM, 51, (2010). [16] N. H. Sweilam, M. M. Khader, and A. M. S. Mahdy, mpuinal mehds fr fracinal differenial equains generaed by pimizain prblem, Jurnal f Fracinal alculus and Applicains, 3(S), 1-12 (2012). [17]. Tricaud and Y.-Q. hen, An apprimain mehd fr numerically slving fracinal rder pimal cnrl prblems f general frm, mpuers and Mahemaics wih Applicains, 59, (2010). [18] M. Zamani, M. Karimi-Gharemani, and N. Sadai, FOPID cnrller design fr rbus perfrmance using paricle swarm pimizain, Jurnal f Fracinal alculus and Applied Analysis (FAA), 10(2), (2007). 13