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Higher order fractional variational optimal control problems with delayed arguments
Applied Mathematics and Computation 218 (2012) 9234–9240
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Higher order fractional variational optimal control problems with delayed arguments Fahd Jarad ⇑, Thabet Abdeljawad (Maraaba), Dumitru Baleanu 1 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey
a r t i c l e
i n f o
Keywords: Fractional derivatives Delay
a b s t r a c t This article deals with higher order Caputo fractional variational problems in the presence of delay in the state variables and their integer higher order derivatives. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Lagrangian theories involving higher-order derivatives appear of great interest as an imbedding for field theories with fields of subcanonical dimension, e.g. the Heisenberg nonlinear spinor theory, for which local interactions are less singular than in the canonical case. The calculus of variation has a long history of communications with other fields of mathematics such as geometry and differential equations, and with physics. Recently the calculus of variations has found applications in economics and some branches of engineering such as electrical engineering. Optimal control which is a rapidly expanded field can be regarded as a part of the calculus of variations. Recently, the fractional calculus which is as old as the classical calculus has become a candidate in solving problems of complex systems which appear in various fields of science [1–5,8–10,14]. Several authors reported interesting results when they used the fractional calculus in control theory [11,12]. From an experimental point of view, the combination of delay with the fractional calculus may give better results. Optimal control problems with time delay in calculus of variations were discussed in [13]. Variational optimal control problems within fractional derivatives were considered in [15]. Fractional variational problems in the presence of delay were studied in [6,7]. The aim of this paper is to deal with optimal control fractional variational problems in the presence of delay in the state variable and its higher order derivatives. The structure of the paper is as follows: In Section 2 the necessary definitions in the fractional calculus used in this manuscript are reviewed. In Section 3 the unconstrained fractional Euler–Lagrange equations with delay are discussed. The fractional control problem is presented in Section 5. 2. Basic definitions We present in this section some basic definitions related to fractional derivatives. The Left Riemann–Liouville fractional integral and The Right Riemann–Liouville fractional integral are defined respectively by ⇑ Corresponding author. 1
E-mail addresses: [email protected] (F. Jarad), [email protected], [email protected] (D. Baleanu). On leave of absence from Institute of Space Sciences, P.O. Box MG-23, Magurele-Bucharest R 76900, Romania.
0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.02.080
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
aI
a
1 CðaÞ
f ðtÞ ¼
Iab f ðtÞ ¼
1 CðaÞ
Z
t
9235
ðt sÞa1 f ðsÞds;
ð1Þ
ðs tÞa1 f ðsÞds;
ð2Þ
a
Z
b t
where a > 0; n 1 < a < n. Here and in the following CðaÞ represents the Gamma function. The Left Riemann–Liouville fractional derivative is defined by aD
a
f ðtÞ ¼
n Z t 1 d ðt sÞna1 f ðsÞds: Cðn aÞ dt a
ð3Þ
The Right Riemann–Liouville fractional derivative is defined by
Dab f ðtÞ ¼
n Z b 1 d ðs tÞna1 f ðsÞds; dt Cðn aÞ t
ð4Þ
The fractional derivative of a constant takes the form aD
a
C¼C
ðt aÞa Cð1 aÞ
ð5Þ
and the fractional derivative of a power of t has the following form aD
a
ðt aÞb ¼
Cða þ 1Þðt aÞba Cðb a þ 1Þ
ð6Þ
for b > 1; a P 0. The Left Caputo fractional derivative is defined by C a a D f ðtÞ
¼
Z
1 Cðn aÞ
t
ðt sÞna1
a
d ds
n
f ðsÞds
ð7Þ
and the Right Caputo fractional derivative C
Db a f ðtÞ ¼
1 Cðn aÞ
Z
b t
n d ðs tÞna1 f ðsÞds; ds
ð8Þ
where a represents the order of the derivative such that n 1 < a < n. By definition the Caputo fractional derivative of a constant is zero. The Riemann–Liouville fractional derivatives and Caputo fractional derivatives are connected with each other by the following relations: C a a D f ðtÞ
C
¼ a Da f ðtÞ
Dab f ðtÞ ¼ Dab f ðtÞ
f ðkÞ ðaÞ
n1 P
k¼0 Cðk a þ 1Þ
ðt aÞka ;
ð1Þk f ðkÞ ðbÞ ðb tÞka : k¼0 Cðk a þ 1Þ
n1 P
ð9Þ
ð10Þ
In [1], a formula for the fractional integration by parts on the whole interval ½a; b was given by the following lemma. Lemma 2.1. Let a > 0; p; q P 1, and 1p þ 1q 6 1 þ a ðp – 1 and q – 1 in the case when 1p þ 1q ¼ 1 þ aÞ. (a) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
b
uðtÞða Ia wÞðtÞdt ¼
a
Z
b
a
wðtÞ Iab u ðtÞdt:
ð11Þ
(b) If g 2 Iab ðLp Þ and f 2a Ia ðLq Þ, then
Z a
b
gðtÞ a Da f ðtÞdt ¼
Z a
b
f ðtÞ Dab g ðtÞdt;
ð12Þ
where a Ia ðLp Þ :¼ ff : f ¼a Ia g; g 2 Lp ða; bÞg and Iab ðLp Þ :¼ ff : f ¼ Iab g; g 2 Lp ða; bÞg. In [6,7], other formulas for the fractional integration by parts on the subintervals ½a; r and ½r; b were given by the following lemmas
9236
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
Lemma 2.2. Let a > 0; p; q P 1; r 2 ða; bÞ and 1p þ 1q 6 1 þ a (p – 1 and q – 1 in the case when 1p þ 1q ¼ 1 þ a). (a) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
r
uðtÞ a Ia w ðtÞdt ¼
Z
a
r
a
wðtÞ Iar u ðtÞdt;
ð13Þ
and thus if g 2 Iar ðLp Þ and f 2 a Ia ðLq Þ, then
Z
r
Z
gðtÞ a Da f ðtÞdt ¼
a
r
f ðtÞ Dar g ðtÞdt:
a
ð14Þ
(b) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
b
uðtÞða Ia wÞðtÞdt ¼
Z
r
b
r
1 wðtÞ Iab u ðtÞdt þ CðaÞ
Z
Z
r
!
b
uðsÞðs tÞa1 ds dt
wðtÞ a
ð15Þ
r
and hence if g 2 Iab ðLp Þ and f 2 a Ia ðLq Þ, then
Z
b
gðtÞ a Da f ðtÞdt ¼
Z
r
b r
1 f ðtÞ Dab g ðtÞdt CðaÞ
Z
r
Z
a a D f ðtÞ
a
b
r
! Dab g ðsÞðs tÞa1 ds dt:
ð16Þ
That is
Z
b
a
gðtÞ a D f ðtÞdt ¼
Z
r
b r
1 f ðtÞ Db g ðtÞdt CðaÞ a
Z
r
Z
a
f ðtÞDr
a
b
r
a
!
Db g ðsÞðs tÞ
a1
ds dt:
ð17Þ
Lemma 2.3. Let a > 0; p; q P 1; r 2 ða; bÞ and 1p þ 1q 6 1 þ a ðp – 1 and q – 1 in the case when 1p þ 1q ¼ 1 þ aÞ). (a) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
b
r
Z
uðtÞ Iab w ðtÞdt ¼
b
wðtÞ r Ia u ðtÞdt
ð18Þ
r
and thus if g 2r Ia ðLp Þ and f 2 Iab ðLq Þ, then
Z
b
r
gðtÞ Dab f ðtÞdt ¼
Z
b
f ðtÞ r Da g ðtÞdt:
ð19Þ
r
(b) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
r
uðtÞðb Ia wÞðtÞdt ¼
Z
a
r
a
1 wðtÞ Iaa u ðtÞdt þ CðaÞ
Z
Z
b
r
uðsÞðt sÞa1 ds dt
ð20Þ
Z r Dab f ðtÞ ða Da gÞðsÞðt sÞa1 ds dt:
ð21Þ
wðtÞ r
a
and hence if g 2 a Ia ðLp Þ and f 2 Iab ðLq Þ, then
Z
r
a
gðtÞ Dab f ðtÞdt ¼
Z
r
a
1 f ðtÞ a Da g ðtÞdt CðaÞ
Z
b
r
a
That is
Z a
r
gðtÞ Dab f ðtÞdt ¼
Z a
r
1 f ðtÞ a Da g ðtÞdt CðaÞ
Z
b
f ðtÞr Da
Z
r
r
aD
a
g ðsÞðt sÞa1 ds dt:
ð22Þ
a
3. The unconstrained Caputo fractional variation with delay Before we consider the fractional control problem, let us consider the following fractional variational problem with delay. Minimize
JðyÞ ¼
Z
h L t; a C Da1 yðtÞ; a C Da2 yðtÞ; . . . ; a C Dan yðtÞ; C Dbb1 yðtÞ; C Dbb2 yðtÞ; . . . ; C Dbbm yðtÞ; yðtÞ; y0 ðtÞ; . . . ; yðkÞ ðtÞ; yðt sÞ; a y0 ðt sÞ; . . . ; yðkÞ ðt sÞ dt; b
ð23Þ
9237
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
such that
amax ¼ maxfai ; bj g16i6n;16j6m ; yðlÞ ðbÞ ¼ cl ; l ¼ 0; 1; 2; . . . ; k 1; yðtÞ ¼ /ðtÞ t 2 ½a s; a; s > 0; s < b a; ai ; bj 2 R 8i ¼ 1; 2; . . . ; n; 8j ¼ 1; 2; . . . ; m;
k 1 6 amax < k;
ð24Þ
where cl are constant, /ðtÞ is a smooth function and L is a function with continuous first and second partial derivatives with respect to all of its arguments. If the above variational problem (23) has a minimum at y0 ðtÞ and gðtÞ 2 R is an admissible function such that gðtÞ 0 in the interval ½a s; a and gðlÞ ðbÞ ¼ 0 then the function
nðtÞ ¼ Jðy0 þ gÞ; where
ð25Þ
2 R admits a minimum at ¼ 0. Hence Z
b
"
a
n P
ai
C
@ iþ1 LðtÞa D gðtÞ þ
i¼1
m P
@ nþjþ1 LðtÞ
C
j¼1
b D bj
k P
gðtÞ þ
k P
ðpÞ
@ mþnþpþ2 LðtÞg ðtÞ þ
p¼0
# ðpÞ
@ mþnþkþ3 LðtÞg ðt sÞ dt ¼ 0;
ð26Þ
p¼0
where @ i L is the partial derivative of L with respect to its ith argument. On using the connection formulas (9), (10) and (26) reads
Z
b
"
a
n P
@ iþ1 LðtÞaDai gðtÞ þ
i¼1
m P
k P
b
@ nþjþ1 LðtÞDbj gðtÞ þ
j¼1
@ mþnþpþ2 LðtÞgðpÞ ðtÞ þ
p¼0
k P
# @ mþnþkþ3 LðtÞgðpÞ ðt sÞ dt ¼ 0:
ð27Þ
p¼0
Now if one splits the integral, makes the change of variables for t s and uses the fact that g 0 in ½a s; a, (27) becomes
Z
bs
"
a
þ
n P
ai
@ iþ1 LðtÞaD gðtÞ þ
i¼1
Z
"
b bs
n P
m P
b @ nþjþ1 LðtÞDbj j¼1
@ iþ1 LðtÞaDai gðtÞ þ
i¼1
m P
gðtÞ þ
k P
ðpÞ
@ mþnþpþ2 LðtÞg ðtÞ þ
p¼0 b
@ nþjþ1 LðtÞDbj gðtÞ þ
j¼1
k P
#
k P
# ðpÞ
@ mþnþkþ3 Lðt þ sÞg ðtÞ dt
p¼0
@ mþnþpþ2 LðtÞgðpÞ ðtÞ dt ¼ 0:
ð28Þ
p¼0
By using the integration by parts formulas in the mentioned above Lemmas 2.1–2.3 and the usual integration by parts formula, one obtains the following
Z
bs
a
"
n P
i¼1
a
p p k P d d ð1Þp p ð@ mþnþkþpþ3 LÞðt þ sÞ p ð@ mþnþpþ2 LÞðtÞ þ dt dt p¼0 p¼0 j¼1 !# Z b "n m ai P ai P Db @ iþ1 L ðsÞðs tÞai 1 ds gðtÞdt þ Db ð@ iþ1 LÞðtÞ þ bs Dbj ð@ nþjþ1 LÞðtÞ
Dbi s ð@ iþ1 LÞðtÞ þ
n P
1 a Dbi s i¼1 Cðai Þ m P 1 Dbj Cðbj Þbs
Z
b
bs
Z
bs
m P
aD
bj
k P
ð@ nþjþ1 LÞðtÞ þ
ð1Þp
!
bs
k P
i¼1 p
j¼1
#
d p ð@ mþnþpþ2 LÞðtÞ gðtÞdt dt bs bs q q k p1 k p1 P P P d d m ð1Þq q ð@ mþnþpþ2 LÞðtÞgpq1 ðtÞ þ ð1Þq q ð@ mþnþkþ3 LÞðt þ sÞgpq1 ðtÞ þ dt dt p¼1q¼0 p¼1q¼0 a a b q k p1 P P d þ ð1Þq q ð@ mþnþpþ2 LÞðtÞgpq1 ðtÞ ¼ 0: dt p¼1q¼0
ða Dbj @ nþjþ1 LÞðsÞðt sÞbj 1 ds þ
a
j¼1
ð1Þp
p¼0
ð29Þ
bs
In Eq. (29) if one chooses g such that gðaÞ ¼ 0 and g 0 on ½b s; b, one gets n P i¼1
m P
a
bj a D ð@ nþjþ1 LÞðtÞ j¼1
Dbi s ð@ iþ1 LÞðtÞ þ n P
1 a Di C ð ai Þ bs i¼1
Z
b
bs
k P
þ
p p k P d d ð1Þp p ð@ mþnþkþpþ3 LÞðt þ sÞ p ð@ mþnþpþ2 LÞðtÞ þ dt dt p¼0 !
ð1Þp
p¼0
a Db i @ iþ1 L ðsÞðs tÞai 1 ds
¼ 0:
ð30Þ
In Eq. (29) if one chooses g such that gðlÞ ðbÞ ¼ 0 and g 0 on ½a; b s, one gets n P i¼1
a
Db i ð@ iþ1 LÞðtÞ þ k P
m P
bj bs D ð@ nþjþ1 LÞðtÞ
j¼1
m P
1 bj bs D j¼1 Cðbj Þ
Z
bs
! ða Dbj @ nþjþ1 LÞðsÞðt sÞbj 1 ds
a
p
d þ ð1Þp p ð@ mþnþpþ2 LÞðtÞ ¼ 0: dt p¼0
ð31Þ
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F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
Now since both integrals in (29) are now zero, one gets
bs bs q q k p1 P P d q d pq1 pq1 ð1Þ ðtÞ þ ð1Þ q ð@ mþnþkþ3 LÞðtÞg ðt þ sÞ q ð@ mþnþpþ2 LÞðtÞg dt dt p¼1q¼0 p¼1q¼0 a a b q k p1 P P d þ ð1Þq q ð@ mþnþpþ2 LÞðtÞgpq1 ðtÞ ¼ 0: dt p¼1q¼0 k p1 P P
q
ð32Þ
bs
Thus one can state the following theorem Theorem 3.1. Let JðyÞ be a functional of the form Eq. (23) defined on a set of continuous functions yðtÞwhich have continuous Caputo fractional left order derivatives of orders ai and right derivative of order bj in ½a; b and satisfy the conditions in Eq. (24). Let L : ½a s; b Rmþnþ2kþ2 ! R have continuous first and second partial derivatives with respect to all of its arguments. Then the necessary conditions that JðyÞ possesses a minimum at yðxÞ are the Euler–Lagrange equations
! p p m k k P P P @L @L @L @L p d p d bj ðtÞ þ ðtÞ þ ðt þ sÞ ðtÞ þ Dbs D ð1Þ ð1Þ a a p p b @aC D i yðtÞ dt @yðpÞ ðtÞ dt @yðpÞ ðt sÞ p¼0 p¼0 i¼1 j¼1 @ C Dbj yðtÞ ! Z b n P 1 @L a a ai 1 Db i ds ¼0 ðsÞðs tÞ Dbi s @ a C Dai yðtÞ bs i¼1 Cðai Þ
n P
ai
ð33Þ
for a 6 t 6 b s,
! p m k P P @L @L @L bj p d ðtÞ þ D ð1Þ ðtÞ þ ðtÞ p bs b @aC Dai yðtÞ dt @yðpÞ ðtÞ p¼0 i¼1 j¼1 @ C Dbj yðtÞ ! !! Z bs m P 1 @L bj 1 bj bj ðsÞðt sÞ ds D ¼0 aD b a j¼1Cðbj Þbs @ C Dbj yðtÞ
n P
a
Db i
ð34Þ
for b s 6 t 6 b, and the transversality conditions
bs bs q q k p1 P P d @L @L q d pq1 pq1 ð1Þ g ðtÞ þ ð1Þ s Þ g ðt þ s Þ ðtÞ ðt þ q q dt @yðpÞ ðtÞ dt @yðpÞ ðt sÞ p¼1q¼0 p¼1q¼0 a a b q p1 k PP d @L þ ð1Þq q ¼0 ðtÞgpq1 ðtÞ dt @yðpÞ ðtÞ p¼1q¼0 k p1 P P
q
ð35Þ
bs
for any admissible function g satisfying gðtÞ 0 t 2 ½a s; a, gðlÞ ðbÞ ¼ 0; l ¼ 0; 1; 2; . . . ; k 1. Theorem 3.1 can be generalized as follows. Theorem 3.2. Consider the functional of the form
Jðy1 ; y2 ; . . . ; yd Þ ¼ a C
C
Z
a
b
L t; a C Da1 y1 ðtÞ; a C Da2 y1 ðtÞ; . . . ; a C Dan y1 ðtÞ;
Da1 y2 ðtÞ; a C Da2 y2 ðtÞ; . . . ; a C Dan y2 ðtÞ; . . . ; a C Da1 yd ðtÞ; a C Da2 yd ðtÞ; . . . ; a C Dan yd ðtÞ; C Dbb1 y1 ðtÞ; C Dbb2 y1 ðtÞ; . . . ; C Dbbm y1 ðtÞ; ðkÞ
Dbb1 y2 ðtÞ; C Dbb2 y2 ðtÞ; . . . ; C Dbbm y2 ðtÞ; . . . ; C Dbb1 yd ðtÞ; C Dbb2 yd ðtÞ; . . . ; C Dbbm yd ðtÞ; y1 ðtÞ; y01 ðtÞ; . . . ; y1 ðtÞ;
ðkÞ ðkÞ y2 ðtÞ; y02 ðtÞ; . . . ; y2 ðtÞ; . . . ; yd ðtÞ; y0d ðtÞ; . . . ; yd ðtÞ; y1 ðt
y2 ðt sÞ; y02 ðt s
ðkÞ Þ; . . . ; y2 ðt
sÞ; y01 ðt s
ðkÞ Þ; . . . ; y1 ðt
i ðkÞ sÞ; . . . ; yd ðt sÞ; y0d ðt sÞ; . . . ; yd ðt sÞ dt
sÞ; ð36Þ
defined on sets of continuous functions yi ðxÞ; i ¼ 1; 2; . . . ; d that have left Caputo fractional derivatives of order ai 2 R; i ¼ 1; 2; . . . ; n and right Caputo fractional derivatives of order bj 2 R; j ¼ 1; 2; . . . m in the interval ½a; b and satisfy the conditions ðlÞ
yi ðbÞ ¼ cil ;
l ¼ 0; 1; . . . ; k;
yi ðtÞ ¼ /i ðtÞ i ¼ 1; 2 . . . ; d; t 2 ½a s; a a < b;
s > 0; s < b a;
ð37Þ
where k 1 6 amax < k; amax ¼ maxfai ; bj g16i6n;16j6m ; cil ’s are constant and /i ’s are smooth functions and L : ½a s; b Rdðmþnþ2kþ2Þ ! R is a function with continuous first and second partial derivatives with respect to all of its arguments. For yi ðxÞ; i ¼ 1; 2; . . . ; d, satisfying (37) to be a minimum of (36), it is necessary that
! ! ! p p m k k P P P @L @L @L @L p d p d bj ðtÞ þ D ð1Þ ð1Þ ðtÞ þ ðt þ sÞ ðtÞ þ a p p b @aC Dai yz ðtÞ dt @yðpÞ dt @yðpÞ p¼0 p¼0 i¼1 j¼1 @ C Dbj yz ðtÞ z ðtÞ z ðt sÞ ! Z b n P 1 @L a a ai 1 ðsÞðs tÞ Db i ds Dbi s ¼0 ð38Þ @ a C Dai yz ðtÞ bs i¼1 Cðai Þ
n P
a
Dbi s
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
9239
for a 6 t 6 b s; z ¼ 1; 2; . . . ; d
! ! p m k P P @L @L @L bj p d ðtÞ þ ðtÞ þ ðtÞ D ð1Þ p bs b @aC Dai yz ðtÞ dt @yðpÞ p¼0 i¼1 j¼1 @ C Dbj yz ðtÞ z ðtÞ ! !! Z bs m P 1 @L bj 1 bj bj ðsÞðt sÞ ds ¼0 aD bs D b a j¼1 Cðbj Þ @ C Dbj yz ðtÞ n P
a
Db i
ð39Þ
for b s 6 t 6 b; z ¼ 1; 2; . . . ; d and the transversality conditions
bs bs ! ! q q k p1 P P d @L @L q d pq1 pq1 ð1Þ q ðtÞ þ ð1Þ q ðt þ sÞ ðtÞgz ðt þ sÞgz ðpÞ ðpÞ dt @yz ðtÞ dt ðt s Þ @y p¼1q¼0 p¼1q¼0 z a a b ! q k p1 P P d @L ðtÞgpq1 þ ð1Þq q ðtÞ ¼0 z dt @yðpÞ p¼1q¼0 z ðtÞ k p1 P P
q
ð40Þ
bs
for any admissible vector function ðg ¼ g1 ; g2 ; . . . ; gd Þ satisfying gðtÞ ð0; 0; . . . ; 0Þ t 2 ½a s; a; gðlÞ ðbÞ ¼ ð0; 0; . . . ; 0Þ; d ¼ 0; 1; 2; . . . ; k 1. 4. The fractional optimal control problem Find the optimal control variable uðtÞ which minimizes the performance index
Jðy; uÞ ¼
Z
h F t; uðtÞ; a C Da1 yðtÞ; a C Da2 yðtÞ; . . . ; a C Dan yðtÞ; C Dbb1 yðtÞ; C Dbb2 yðtÞ; . . . ; C Dbbm yðtÞ; yðtÞ; y0 ðtÞ; . . . ; yðkÞ ðtÞ; yðt sÞ; a i ð41Þ y0 ðt sÞ; . . . ; yðkÞ ðt sÞ dt b
subject to the constraint
h G t; uðtÞ; a C Da1 yðtÞ; a C Da2 yðtÞ; . . . ; a C Dan yðtÞ; C Dbb1 yðtÞ; C Dbb2 yðtÞ; . . . ; C Dbbm yðtÞ; yðtÞ; y0 ðtÞ; . . . ; yðkÞ ðtÞ; yðt sÞ; i y0 ðt sÞ; . . . ; yðkÞ ðt sÞ ¼ 0
ð42Þ
such that
yðlÞ ðbÞ ¼ cl ;
l ¼ 0; 1; 2; . . . ; k 1;
i ¼ 1; 2; . . . ; n;
yðtÞ ¼ /ðtÞ t 2 ½a s; a; a < b;
s > 0 s < b a; ai 2 R;
bj 2 R; j ¼ 1; 2; . . . ; m;
ð43Þ
where cl are constant and F and G are functions ½a s; b Rnþmþ2kþ3 ! R with continuous first and second partial derivatives with respect to all of their arguments arguments. To find the optimal control, one defines a modified performance index as
bJðy; uÞ ¼
Z
b
fF½ þ kðtÞG½gdt;
ð44Þ
a
where k is a Lagrange multiplier or an adjoint variable. Using the conditions (38)–(40) in Theorem 3.2, the following necessary equations for optimal control are found: Euler–Lagrange equations
! ! n m m P P P @F @G @F @G ai bj bj ðtÞ þ ðtÞ þ D k D D k ðtÞ þ ðtÞ a a bs b b @aC Dai yðtÞ @aC Dai yðtÞ i¼1 i¼1 j¼1 j¼1 @ C Dbj yðtÞ @ C Dbj yðtÞ p p p k k k P P P d @F @G @F p d p d ðtÞ þ ðtÞ þ ðt þ sÞ ð1Þ k ð1Þ þ ð1Þp p p p @yðpÞ ðtÞ dt @yðpÞ ðtÞ dt dt @yðpÞ ðt sÞ p¼0 p¼0 p¼0 ! ! Z b p k n P P d @G 1 @F a a ðsÞðs tÞai 1 ds þ ð1Þp p k ðpÞ Db i C a ðt þ sÞ Dbi s @y ðt sÞ dt @ a D i yðtÞ bs p¼0 i¼1 Cðai Þ !! ! Z b n P 1 @G @F @G a a ðtÞ þ kðtÞ ðtÞ ¼ 0 Dbi s Db i k C a þ ðsÞðs tÞai 1 ds @uðtÞ @uðtÞ @ a D i yðtÞ bs i¼1 Cðai Þ n P
a
Dbi s
for a 6 t 6 b s,
ð45Þ
9240
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
! ! n m m P P P @F @G @F @G bj bj ai ðtÞ þ ðtÞ D k D D k ðtÞ þ ðtÞ þ b bs bs b b @aC Dai yðtÞ @aC Dai yðtÞ i¼1 i¼1 j¼1 j¼1 @ C Dbj yðtÞ @ C Dbj yðtÞ ! !! Z bs p p k k m P P P @F @G 1 @F p d p d bj 1 bj bj þ ð1Þ ðsÞðt sÞ ds D ðtÞ þ ð1Þ ðtÞ þ aD p p k b @yðpÞ ðtÞ dt @yðpÞ ðtÞ dt a p¼0 p¼0 j¼1Cðbj Þbs @ C Dbj yðtÞ ! ! ! Z bs m P 1 @L @F @G bj ðtÞ þ kðtÞ ðtÞ ¼ 0 ð46Þ D bj k þ ðsÞðt sÞbj 1 ds þ aD bj C @uðtÞ @uðtÞ C ðb Þ a j bs j¼1 @ Db yðtÞ
n P
a
Db i
for b s 6 t 6 b, and the transversality conditions.
bs bs q q k p1 P P d @F @G q d pq1 pq1 g ðtÞ þ ð1Þ k g ðtÞ ðtÞ ðtÞ q q ðpÞ ðpÞ @y ðtÞ dt @y ðtÞ dt p¼1q¼0 p¼1q¼0 a a bs bs q q p1 p1 k k PP PP @F @G q d q d pq1 pq1 þ ð1Þ ðt þ sÞ þ ð1Þ ðt þ sÞ ðt þ sÞg ðt þ sÞg q q k ðpÞ ðpÞ @y ðt sÞ dt @y ðt sÞ dt p¼1q¼0 p¼1q¼0 a a q q b P b k p1 k p1 P P P d @F d @G ðtÞgpq1 ðtÞ ðtÞgpq1 ðtÞ þ ð1Þq q ð1Þq q k ðpÞ ¼ 0; p¼1q¼0 @y ðtÞ dt @yðpÞ ðtÞ dt p¼1q¼0 bs bs k p1 P P
ð1Þq
ð47Þ
where g is any admissible function satisfying gðtÞ 0 t 2 ½a s; a; gðlÞ ðbÞ ¼ 0; l ¼ 0; 1; 2; . . . ; k 1. 5. Conclusion In this manuscript we have developed a fractional control problem in the presence of both left and right Caputo fractional derivatives of any order and delay in the state variables and their derivatives. The results were applied in order to find the necessary conditions for the optimal control problems. When ai ! 1; 8i ¼ 1; . . . ; n and bj ! 1; 8j ¼ 1; . . . ; m the classical problem is recovered. Acknowledgments This work is partially supported by the Scientific and Technical Research Council of Turkey. The authors thank the referees for their valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
A.A. Kilbas, H.H. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives – Theory and Applications, Gordon and Breach, Linghorne, P.A., 1993. R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Inc, Connecticut, 2006. N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives, Rheol. Acta 45 (2006) 765–771. I.S. Jesus, J.A.T. Machado, Fractional control of heat diffusion system, Nonlinear Dynam. 54 (3) (2008) 263–282. D. Baleanu, T. Maraaba, F. Jarad, Fractional principles with delay, J. Phys. A: Math. Theor. 41 (2008), http://dx.doi.org/10.1088/1751-8113/41/31/ 315403. F. Jarad, T. Maraaba, D. Baleanu, Fractional variational principles with delay within Caputo derivatives, Rep. Math. Phys. 65 (1) (2010) 17–28. F. Mainardi, Y. Luchko, G. Pagnini, The fundamental solution of the space-time fractional the space-time fractional diffusion equation, Frac. Calc. Appl. Anal. 4 (2) (2001) 153–192. E. Scalas, R. Gorenflo, F. Mainardi, Uncoupled continuous-time random walks: solution and limiting behavior of the master equation, Phys. Rev. E 69 (2004) 011107-1. C.P. li, Z.G. Zhao, Introduction to fractional integrability and differentiability, Eur. Phys. J. Special Topics 193 (2011) 5–26. O.P. Agrawal, D. Baleanu, Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, J. Vib. Control 13 (9–10) (2007) 1269–1281. Y.Q. Chen, B.M. Vinagre, I. Podlubny, Continued fraction expansion approaches to discretizing fractional order derivatives-an expository review, Nonlinear Dynam. 38 (1–4) (2004) 155–170. J.F. Rosenblueth, Systems with time delay in the calculus of variations: a variational approach, J. Math. Control Inf. 5 (2) (1988) 125–145. V.E. Tarasov, G.M. Zaslavsky, Nonholonomic constraints with fractional derivatives, J. Phys. A: Math. Gen. 39 (31) (2006) 9797–9815. O.P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam. 38 (2004) 323–337.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Higher order fractional variational optimal control problems with delayed arguments Fahd Jarad ⇑, Thabet Abdeljawad (Maraaba), Dumitru Baleanu 1 Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey
a r t i c l e
i n f o
Keywords: Fractional derivatives Delay
a b s t r a c t This article deals with higher order Caputo fractional variational problems in the presence of delay in the state variables and their integer higher order derivatives. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Lagrangian theories involving higher-order derivatives appear of great interest as an imbedding for field theories with fields of subcanonical dimension, e.g. the Heisenberg nonlinear spinor theory, for which local interactions are less singular than in the canonical case. The calculus of variation has a long history of communications with other fields of mathematics such as geometry and differential equations, and with physics. Recently the calculus of variations has found applications in economics and some branches of engineering such as electrical engineering. Optimal control which is a rapidly expanded field can be regarded as a part of the calculus of variations. Recently, the fractional calculus which is as old as the classical calculus has become a candidate in solving problems of complex systems which appear in various fields of science [1–5,8–10,14]. Several authors reported interesting results when they used the fractional calculus in control theory [11,12]. From an experimental point of view, the combination of delay with the fractional calculus may give better results. Optimal control problems with time delay in calculus of variations were discussed in [13]. Variational optimal control problems within fractional derivatives were considered in [15]. Fractional variational problems in the presence of delay were studied in [6,7]. The aim of this paper is to deal with optimal control fractional variational problems in the presence of delay in the state variable and its higher order derivatives. The structure of the paper is as follows: In Section 2 the necessary definitions in the fractional calculus used in this manuscript are reviewed. In Section 3 the unconstrained fractional Euler–Lagrange equations with delay are discussed. The fractional control problem is presented in Section 5. 2. Basic definitions We present in this section some basic definitions related to fractional derivatives. The Left Riemann–Liouville fractional integral and The Right Riemann–Liouville fractional integral are defined respectively by ⇑ Corresponding author. 1
E-mail addresses: [email protected] (F. Jarad), [email protected], [email protected] (D. Baleanu). On leave of absence from Institute of Space Sciences, P.O. Box MG-23, Magurele-Bucharest R 76900, Romania.
0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.02.080
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
aI
a
1 CðaÞ
f ðtÞ ¼
Iab f ðtÞ ¼
1 CðaÞ
Z
t
9235
ðt sÞa1 f ðsÞds;
ð1Þ
ðs tÞa1 f ðsÞds;
ð2Þ
a
Z
b t
where a > 0; n 1 < a < n. Here and in the following CðaÞ represents the Gamma function. The Left Riemann–Liouville fractional derivative is defined by aD
a
f ðtÞ ¼
n Z t 1 d ðt sÞna1 f ðsÞds: Cðn aÞ dt a
ð3Þ
The Right Riemann–Liouville fractional derivative is defined by
Dab f ðtÞ ¼
n Z b 1 d ðs tÞna1 f ðsÞds; dt Cðn aÞ t
ð4Þ
The fractional derivative of a constant takes the form aD
a
C¼C
ðt aÞa Cð1 aÞ
ð5Þ
and the fractional derivative of a power of t has the following form aD
a
ðt aÞb ¼
Cða þ 1Þðt aÞba Cðb a þ 1Þ
ð6Þ
for b > 1; a P 0. The Left Caputo fractional derivative is defined by C a a D f ðtÞ
¼
Z
1 Cðn aÞ
t
ðt sÞna1
a
d ds
n
f ðsÞds
ð7Þ
and the Right Caputo fractional derivative C
Db a f ðtÞ ¼
1 Cðn aÞ
Z
b t
n d ðs tÞna1 f ðsÞds; ds
ð8Þ
where a represents the order of the derivative such that n 1 < a < n. By definition the Caputo fractional derivative of a constant is zero. The Riemann–Liouville fractional derivatives and Caputo fractional derivatives are connected with each other by the following relations: C a a D f ðtÞ
C
¼ a Da f ðtÞ
Dab f ðtÞ ¼ Dab f ðtÞ
f ðkÞ ðaÞ
n1 P
k¼0 Cðk a þ 1Þ
ðt aÞka ;
ð1Þk f ðkÞ ðbÞ ðb tÞka : k¼0 Cðk a þ 1Þ
n1 P
ð9Þ
ð10Þ
In [1], a formula for the fractional integration by parts on the whole interval ½a; b was given by the following lemma. Lemma 2.1. Let a > 0; p; q P 1, and 1p þ 1q 6 1 þ a ðp – 1 and q – 1 in the case when 1p þ 1q ¼ 1 þ aÞ. (a) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
b
uðtÞða Ia wÞðtÞdt ¼
a
Z
b
a
wðtÞ Iab u ðtÞdt:
ð11Þ
(b) If g 2 Iab ðLp Þ and f 2a Ia ðLq Þ, then
Z a
b
gðtÞ a Da f ðtÞdt ¼
Z a
b
f ðtÞ Dab g ðtÞdt;
ð12Þ
where a Ia ðLp Þ :¼ ff : f ¼a Ia g; g 2 Lp ða; bÞg and Iab ðLp Þ :¼ ff : f ¼ Iab g; g 2 Lp ða; bÞg. In [6,7], other formulas for the fractional integration by parts on the subintervals ½a; r and ½r; b were given by the following lemmas
9236
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
Lemma 2.2. Let a > 0; p; q P 1; r 2 ða; bÞ and 1p þ 1q 6 1 þ a (p – 1 and q – 1 in the case when 1p þ 1q ¼ 1 þ a). (a) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
r
uðtÞ a Ia w ðtÞdt ¼
Z
a
r
a
wðtÞ Iar u ðtÞdt;
ð13Þ
and thus if g 2 Iar ðLp Þ and f 2 a Ia ðLq Þ, then
Z
r
Z
gðtÞ a Da f ðtÞdt ¼
a
r
f ðtÞ Dar g ðtÞdt:
a
ð14Þ
(b) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
b
uðtÞða Ia wÞðtÞdt ¼
Z
r
b
r
1 wðtÞ Iab u ðtÞdt þ CðaÞ
Z
Z
r
!
b
uðsÞðs tÞa1 ds dt
wðtÞ a
ð15Þ
r
and hence if g 2 Iab ðLp Þ and f 2 a Ia ðLq Þ, then
Z
b
gðtÞ a Da f ðtÞdt ¼
Z
r
b r
1 f ðtÞ Dab g ðtÞdt CðaÞ
Z
r
Z
a a D f ðtÞ
a
b
r
! Dab g ðsÞðs tÞa1 ds dt:
ð16Þ
That is
Z
b
a
gðtÞ a D f ðtÞdt ¼
Z
r
b r
1 f ðtÞ Db g ðtÞdt CðaÞ a
Z
r
Z
a
f ðtÞDr
a
b
r
a
!
Db g ðsÞðs tÞ
a1
ds dt:
ð17Þ
Lemma 2.3. Let a > 0; p; q P 1; r 2 ða; bÞ and 1p þ 1q 6 1 þ a ðp – 1 and q – 1 in the case when 1p þ 1q ¼ 1 þ aÞ). (a) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
b
r
Z
uðtÞ Iab w ðtÞdt ¼
b
wðtÞ r Ia u ðtÞdt
ð18Þ
r
and thus if g 2r Ia ðLp Þ and f 2 Iab ðLq Þ, then
Z
b
r
gðtÞ Dab f ðtÞdt ¼
Z
b
f ðtÞ r Da g ðtÞdt:
ð19Þ
r
(b) If u 2 Lp ða; bÞ and w 2 Lq ða; bÞ, then
Z
r
uðtÞðb Ia wÞðtÞdt ¼
Z
a
r
a
1 wðtÞ Iaa u ðtÞdt þ CðaÞ
Z
Z
b
r
uðsÞðt sÞa1 ds dt
ð20Þ
Z r Dab f ðtÞ ða Da gÞðsÞðt sÞa1 ds dt:
ð21Þ
wðtÞ r
a
and hence if g 2 a Ia ðLp Þ and f 2 Iab ðLq Þ, then
Z
r
a
gðtÞ Dab f ðtÞdt ¼
Z
r
a
1 f ðtÞ a Da g ðtÞdt CðaÞ
Z
b
r
a
That is
Z a
r
gðtÞ Dab f ðtÞdt ¼
Z a
r
1 f ðtÞ a Da g ðtÞdt CðaÞ
Z
b
f ðtÞr Da
Z
r
r
aD
a
g ðsÞðt sÞa1 ds dt:
ð22Þ
a
3. The unconstrained Caputo fractional variation with delay Before we consider the fractional control problem, let us consider the following fractional variational problem with delay. Minimize
JðyÞ ¼
Z
h L t; a C Da1 yðtÞ; a C Da2 yðtÞ; . . . ; a C Dan yðtÞ; C Dbb1 yðtÞ; C Dbb2 yðtÞ; . . . ; C Dbbm yðtÞ; yðtÞ; y0 ðtÞ; . . . ; yðkÞ ðtÞ; yðt sÞ; a y0 ðt sÞ; . . . ; yðkÞ ðt sÞ dt; b
ð23Þ
9237
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
such that
amax ¼ maxfai ; bj g16i6n;16j6m ; yðlÞ ðbÞ ¼ cl ; l ¼ 0; 1; 2; . . . ; k 1; yðtÞ ¼ /ðtÞ t 2 ½a s; a; s > 0; s < b a; ai ; bj 2 R 8i ¼ 1; 2; . . . ; n; 8j ¼ 1; 2; . . . ; m;
k 1 6 amax < k;
ð24Þ
where cl are constant, /ðtÞ is a smooth function and L is a function with continuous first and second partial derivatives with respect to all of its arguments. If the above variational problem (23) has a minimum at y0 ðtÞ and gðtÞ 2 R is an admissible function such that gðtÞ 0 in the interval ½a s; a and gðlÞ ðbÞ ¼ 0 then the function
nðtÞ ¼ Jðy0 þ gÞ; where
ð25Þ
2 R admits a minimum at ¼ 0. Hence Z
b
"
a
n P
ai
C
@ iþ1 LðtÞa D gðtÞ þ
i¼1
m P
@ nþjþ1 LðtÞ
C
j¼1
b D bj
k P
gðtÞ þ
k P
ðpÞ
@ mþnþpþ2 LðtÞg ðtÞ þ
p¼0
# ðpÞ
@ mþnþkþ3 LðtÞg ðt sÞ dt ¼ 0;
ð26Þ
p¼0
where @ i L is the partial derivative of L with respect to its ith argument. On using the connection formulas (9), (10) and (26) reads
Z
b
"
a
n P
@ iþ1 LðtÞaDai gðtÞ þ
i¼1
m P
k P
b
@ nþjþ1 LðtÞDbj gðtÞ þ
j¼1
@ mþnþpþ2 LðtÞgðpÞ ðtÞ þ
p¼0
k P
# @ mþnþkþ3 LðtÞgðpÞ ðt sÞ dt ¼ 0:
ð27Þ
p¼0
Now if one splits the integral, makes the change of variables for t s and uses the fact that g 0 in ½a s; a, (27) becomes
Z
bs
"
a
þ
n P
ai
@ iþ1 LðtÞaD gðtÞ þ
i¼1
Z
"
b bs
n P
m P
b @ nþjþ1 LðtÞDbj j¼1
@ iþ1 LðtÞaDai gðtÞ þ
i¼1
m P
gðtÞ þ
k P
ðpÞ
@ mþnþpþ2 LðtÞg ðtÞ þ
p¼0 b
@ nþjþ1 LðtÞDbj gðtÞ þ
j¼1
k P
#
k P
# ðpÞ
@ mþnþkþ3 Lðt þ sÞg ðtÞ dt
p¼0
@ mþnþpþ2 LðtÞgðpÞ ðtÞ dt ¼ 0:
ð28Þ
p¼0
By using the integration by parts formulas in the mentioned above Lemmas 2.1–2.3 and the usual integration by parts formula, one obtains the following
Z
bs
a
"
n P
i¼1
a
p p k P d d ð1Þp p ð@ mþnþkþpþ3 LÞðt þ sÞ p ð@ mþnþpþ2 LÞðtÞ þ dt dt p¼0 p¼0 j¼1 !# Z b "n m ai P ai P Db @ iþ1 L ðsÞðs tÞai 1 ds gðtÞdt þ Db ð@ iþ1 LÞðtÞ þ bs Dbj ð@ nþjþ1 LÞðtÞ
Dbi s ð@ iþ1 LÞðtÞ þ
n P
1 a Dbi s i¼1 Cðai Þ m P 1 Dbj Cðbj Þbs
Z
b
bs
Z
bs
m P
aD
bj
k P
ð@ nþjþ1 LÞðtÞ þ
ð1Þp
!
bs
k P
i¼1 p
j¼1
#
d p ð@ mþnþpþ2 LÞðtÞ gðtÞdt dt bs bs q q k p1 k p1 P P P d d m ð1Þq q ð@ mþnþpþ2 LÞðtÞgpq1 ðtÞ þ ð1Þq q ð@ mþnþkþ3 LÞðt þ sÞgpq1 ðtÞ þ dt dt p¼1q¼0 p¼1q¼0 a a b q k p1 P P d þ ð1Þq q ð@ mþnþpþ2 LÞðtÞgpq1 ðtÞ ¼ 0: dt p¼1q¼0
ða Dbj @ nþjþ1 LÞðsÞðt sÞbj 1 ds þ
a
j¼1
ð1Þp
p¼0
ð29Þ
bs
In Eq. (29) if one chooses g such that gðaÞ ¼ 0 and g 0 on ½b s; b, one gets n P i¼1
m P
a
bj a D ð@ nþjþ1 LÞðtÞ j¼1
Dbi s ð@ iþ1 LÞðtÞ þ n P
1 a Di C ð ai Þ bs i¼1
Z
b
bs
k P
þ
p p k P d d ð1Þp p ð@ mþnþkþpþ3 LÞðt þ sÞ p ð@ mþnþpþ2 LÞðtÞ þ dt dt p¼0 !
ð1Þp
p¼0
a Db i @ iþ1 L ðsÞðs tÞai 1 ds
¼ 0:
ð30Þ
In Eq. (29) if one chooses g such that gðlÞ ðbÞ ¼ 0 and g 0 on ½a; b s, one gets n P i¼1
a
Db i ð@ iþ1 LÞðtÞ þ k P
m P
bj bs D ð@ nþjþ1 LÞðtÞ
j¼1
m P
1 bj bs D j¼1 Cðbj Þ
Z
bs
! ða Dbj @ nþjþ1 LÞðsÞðt sÞbj 1 ds
a
p
d þ ð1Þp p ð@ mþnþpþ2 LÞðtÞ ¼ 0: dt p¼0
ð31Þ
9238
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
Now since both integrals in (29) are now zero, one gets
bs bs q q k p1 P P d q d pq1 pq1 ð1Þ ðtÞ þ ð1Þ q ð@ mþnþkþ3 LÞðtÞg ðt þ sÞ q ð@ mþnþpþ2 LÞðtÞg dt dt p¼1q¼0 p¼1q¼0 a a b q k p1 P P d þ ð1Þq q ð@ mþnþpþ2 LÞðtÞgpq1 ðtÞ ¼ 0: dt p¼1q¼0 k p1 P P
q
ð32Þ
bs
Thus one can state the following theorem Theorem 3.1. Let JðyÞ be a functional of the form Eq. (23) defined on a set of continuous functions yðtÞwhich have continuous Caputo fractional left order derivatives of orders ai and right derivative of order bj in ½a; b and satisfy the conditions in Eq. (24). Let L : ½a s; b Rmþnþ2kþ2 ! R have continuous first and second partial derivatives with respect to all of its arguments. Then the necessary conditions that JðyÞ possesses a minimum at yðxÞ are the Euler–Lagrange equations
! p p m k k P P P @L @L @L @L p d p d bj ðtÞ þ ðtÞ þ ðt þ sÞ ðtÞ þ Dbs D ð1Þ ð1Þ a a p p b @aC D i yðtÞ dt @yðpÞ ðtÞ dt @yðpÞ ðt sÞ p¼0 p¼0 i¼1 j¼1 @ C Dbj yðtÞ ! Z b n P 1 @L a a ai 1 Db i ds ¼0 ðsÞðs tÞ Dbi s @ a C Dai yðtÞ bs i¼1 Cðai Þ
n P
ai
ð33Þ
for a 6 t 6 b s,
! p m k P P @L @L @L bj p d ðtÞ þ D ð1Þ ðtÞ þ ðtÞ p bs b @aC Dai yðtÞ dt @yðpÞ ðtÞ p¼0 i¼1 j¼1 @ C Dbj yðtÞ ! !! Z bs m P 1 @L bj 1 bj bj ðsÞðt sÞ ds D ¼0 aD b a j¼1Cðbj Þbs @ C Dbj yðtÞ
n P
a
Db i
ð34Þ
for b s 6 t 6 b, and the transversality conditions
bs bs q q k p1 P P d @L @L q d pq1 pq1 ð1Þ g ðtÞ þ ð1Þ s Þ g ðt þ s Þ ðtÞ ðt þ q q dt @yðpÞ ðtÞ dt @yðpÞ ðt sÞ p¼1q¼0 p¼1q¼0 a a b q p1 k PP d @L þ ð1Þq q ¼0 ðtÞgpq1 ðtÞ dt @yðpÞ ðtÞ p¼1q¼0 k p1 P P
q
ð35Þ
bs
for any admissible function g satisfying gðtÞ 0 t 2 ½a s; a, gðlÞ ðbÞ ¼ 0; l ¼ 0; 1; 2; . . . ; k 1. Theorem 3.1 can be generalized as follows. Theorem 3.2. Consider the functional of the form
Jðy1 ; y2 ; . . . ; yd Þ ¼ a C
C
Z
a
b
L t; a C Da1 y1 ðtÞ; a C Da2 y1 ðtÞ; . . . ; a C Dan y1 ðtÞ;
Da1 y2 ðtÞ; a C Da2 y2 ðtÞ; . . . ; a C Dan y2 ðtÞ; . . . ; a C Da1 yd ðtÞ; a C Da2 yd ðtÞ; . . . ; a C Dan yd ðtÞ; C Dbb1 y1 ðtÞ; C Dbb2 y1 ðtÞ; . . . ; C Dbbm y1 ðtÞ; ðkÞ
Dbb1 y2 ðtÞ; C Dbb2 y2 ðtÞ; . . . ; C Dbbm y2 ðtÞ; . . . ; C Dbb1 yd ðtÞ; C Dbb2 yd ðtÞ; . . . ; C Dbbm yd ðtÞ; y1 ðtÞ; y01 ðtÞ; . . . ; y1 ðtÞ;
ðkÞ ðkÞ y2 ðtÞ; y02 ðtÞ; . . . ; y2 ðtÞ; . . . ; yd ðtÞ; y0d ðtÞ; . . . ; yd ðtÞ; y1 ðt
y2 ðt sÞ; y02 ðt s
ðkÞ Þ; . . . ; y2 ðt
sÞ; y01 ðt s
ðkÞ Þ; . . . ; y1 ðt
i ðkÞ sÞ; . . . ; yd ðt sÞ; y0d ðt sÞ; . . . ; yd ðt sÞ dt
sÞ; ð36Þ
defined on sets of continuous functions yi ðxÞ; i ¼ 1; 2; . . . ; d that have left Caputo fractional derivatives of order ai 2 R; i ¼ 1; 2; . . . ; n and right Caputo fractional derivatives of order bj 2 R; j ¼ 1; 2; . . . m in the interval ½a; b and satisfy the conditions ðlÞ
yi ðbÞ ¼ cil ;
l ¼ 0; 1; . . . ; k;
yi ðtÞ ¼ /i ðtÞ i ¼ 1; 2 . . . ; d; t 2 ½a s; a a < b;
s > 0; s < b a;
ð37Þ
where k 1 6 amax < k; amax ¼ maxfai ; bj g16i6n;16j6m ; cil ’s are constant and /i ’s are smooth functions and L : ½a s; b Rdðmþnþ2kþ2Þ ! R is a function with continuous first and second partial derivatives with respect to all of its arguments. For yi ðxÞ; i ¼ 1; 2; . . . ; d, satisfying (37) to be a minimum of (36), it is necessary that
! ! ! p p m k k P P P @L @L @L @L p d p d bj ðtÞ þ D ð1Þ ð1Þ ðtÞ þ ðt þ sÞ ðtÞ þ a p p b @aC Dai yz ðtÞ dt @yðpÞ dt @yðpÞ p¼0 p¼0 i¼1 j¼1 @ C Dbj yz ðtÞ z ðtÞ z ðt sÞ ! Z b n P 1 @L a a ai 1 ðsÞðs tÞ Db i ds Dbi s ¼0 ð38Þ @ a C Dai yz ðtÞ bs i¼1 Cðai Þ
n P
a
Dbi s
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
9239
for a 6 t 6 b s; z ¼ 1; 2; . . . ; d
! ! p m k P P @L @L @L bj p d ðtÞ þ ðtÞ þ ðtÞ D ð1Þ p bs b @aC Dai yz ðtÞ dt @yðpÞ p¼0 i¼1 j¼1 @ C Dbj yz ðtÞ z ðtÞ ! !! Z bs m P 1 @L bj 1 bj bj ðsÞðt sÞ ds ¼0 aD bs D b a j¼1 Cðbj Þ @ C Dbj yz ðtÞ n P
a
Db i
ð39Þ
for b s 6 t 6 b; z ¼ 1; 2; . . . ; d and the transversality conditions
bs bs ! ! q q k p1 P P d @L @L q d pq1 pq1 ð1Þ q ðtÞ þ ð1Þ q ðt þ sÞ ðtÞgz ðt þ sÞgz ðpÞ ðpÞ dt @yz ðtÞ dt ðt s Þ @y p¼1q¼0 p¼1q¼0 z a a b ! q k p1 P P d @L ðtÞgpq1 þ ð1Þq q ðtÞ ¼0 z dt @yðpÞ p¼1q¼0 z ðtÞ k p1 P P
q
ð40Þ
bs
for any admissible vector function ðg ¼ g1 ; g2 ; . . . ; gd Þ satisfying gðtÞ ð0; 0; . . . ; 0Þ t 2 ½a s; a; gðlÞ ðbÞ ¼ ð0; 0; . . . ; 0Þ; d ¼ 0; 1; 2; . . . ; k 1. 4. The fractional optimal control problem Find the optimal control variable uðtÞ which minimizes the performance index
Jðy; uÞ ¼
Z
h F t; uðtÞ; a C Da1 yðtÞ; a C Da2 yðtÞ; . . . ; a C Dan yðtÞ; C Dbb1 yðtÞ; C Dbb2 yðtÞ; . . . ; C Dbbm yðtÞ; yðtÞ; y0 ðtÞ; . . . ; yðkÞ ðtÞ; yðt sÞ; a i ð41Þ y0 ðt sÞ; . . . ; yðkÞ ðt sÞ dt b
subject to the constraint
h G t; uðtÞ; a C Da1 yðtÞ; a C Da2 yðtÞ; . . . ; a C Dan yðtÞ; C Dbb1 yðtÞ; C Dbb2 yðtÞ; . . . ; C Dbbm yðtÞ; yðtÞ; y0 ðtÞ; . . . ; yðkÞ ðtÞ; yðt sÞ; i y0 ðt sÞ; . . . ; yðkÞ ðt sÞ ¼ 0
ð42Þ
such that
yðlÞ ðbÞ ¼ cl ;
l ¼ 0; 1; 2; . . . ; k 1;
i ¼ 1; 2; . . . ; n;
yðtÞ ¼ /ðtÞ t 2 ½a s; a; a < b;
s > 0 s < b a; ai 2 R;
bj 2 R; j ¼ 1; 2; . . . ; m;
ð43Þ
where cl are constant and F and G are functions ½a s; b Rnþmþ2kþ3 ! R with continuous first and second partial derivatives with respect to all of their arguments arguments. To find the optimal control, one defines a modified performance index as
bJðy; uÞ ¼
Z
b
fF½ þ kðtÞG½gdt;
ð44Þ
a
where k is a Lagrange multiplier or an adjoint variable. Using the conditions (38)–(40) in Theorem 3.2, the following necessary equations for optimal control are found: Euler–Lagrange equations
! ! n m m P P P @F @G @F @G ai bj bj ðtÞ þ ðtÞ þ D k D D k ðtÞ þ ðtÞ a a bs b b @aC Dai yðtÞ @aC Dai yðtÞ i¼1 i¼1 j¼1 j¼1 @ C Dbj yðtÞ @ C Dbj yðtÞ p p p k k k P P P d @F @G @F p d p d ðtÞ þ ðtÞ þ ðt þ sÞ ð1Þ k ð1Þ þ ð1Þp p p p @yðpÞ ðtÞ dt @yðpÞ ðtÞ dt dt @yðpÞ ðt sÞ p¼0 p¼0 p¼0 ! ! Z b p k n P P d @G 1 @F a a ðsÞðs tÞai 1 ds þ ð1Þp p k ðpÞ Db i C a ðt þ sÞ Dbi s @y ðt sÞ dt @ a D i yðtÞ bs p¼0 i¼1 Cðai Þ !! ! Z b n P 1 @G @F @G a a ðtÞ þ kðtÞ ðtÞ ¼ 0 Dbi s Db i k C a þ ðsÞðs tÞai 1 ds @uðtÞ @uðtÞ @ a D i yðtÞ bs i¼1 Cðai Þ n P
a
Dbi s
for a 6 t 6 b s,
ð45Þ
9240
F. Jarad et al. / Applied Mathematics and Computation 218 (2012) 9234–9240
! ! n m m P P P @F @G @F @G bj bj ai ðtÞ þ ðtÞ D k D D k ðtÞ þ ðtÞ þ b bs bs b b @aC Dai yðtÞ @aC Dai yðtÞ i¼1 i¼1 j¼1 j¼1 @ C Dbj yðtÞ @ C Dbj yðtÞ ! !! Z bs p p k k m P P P @F @G 1 @F p d p d bj 1 bj bj þ ð1Þ ðsÞðt sÞ ds D ðtÞ þ ð1Þ ðtÞ þ aD p p k b @yðpÞ ðtÞ dt @yðpÞ ðtÞ dt a p¼0 p¼0 j¼1Cðbj Þbs @ C Dbj yðtÞ ! ! ! Z bs m P 1 @L @F @G bj ðtÞ þ kðtÞ ðtÞ ¼ 0 ð46Þ D bj k þ ðsÞðt sÞbj 1 ds þ aD bj C @uðtÞ @uðtÞ C ðb Þ a j bs j¼1 @ Db yðtÞ
n P
a
Db i
for b s 6 t 6 b, and the transversality conditions.
bs bs q q k p1 P P d @F @G q d pq1 pq1 g ðtÞ þ ð1Þ k g ðtÞ ðtÞ ðtÞ q q ðpÞ ðpÞ @y ðtÞ dt @y ðtÞ dt p¼1q¼0 p¼1q¼0 a a bs bs q q p1 p1 k k PP PP @F @G q d q d pq1 pq1 þ ð1Þ ðt þ sÞ þ ð1Þ ðt þ sÞ ðt þ sÞg ðt þ sÞg q q k ðpÞ ðpÞ @y ðt sÞ dt @y ðt sÞ dt p¼1q¼0 p¼1q¼0 a a q q b P b k p1 k p1 P P P d @F d @G ðtÞgpq1 ðtÞ ðtÞgpq1 ðtÞ þ ð1Þq q ð1Þq q k ðpÞ ¼ 0; p¼1q¼0 @y ðtÞ dt @yðpÞ ðtÞ dt p¼1q¼0 bs bs k p1 P P
ð1Þq
ð47Þ
where g is any admissible function satisfying gðtÞ 0 t 2 ½a s; a; gðlÞ ðbÞ ¼ 0; l ¼ 0; 1; 2; . . . ; k 1. 5. Conclusion In this manuscript we have developed a fractional control problem in the presence of both left and right Caputo fractional derivatives of any order and delay in the state variables and their derivatives. The results were applied in order to find the necessary conditions for the optimal control problems. When ai ! 1; 8i ¼ 1; . . . ; n and bj ! 1; 8j ¼ 1; . . . ; m the classical problem is recovered. Acknowledgments This work is partially supported by the Scientific and Technical Research Council of Turkey. The authors thank the referees for their valuable comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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