Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 890396, 15 pages
doi:10.1155/2012/890396
Research Article
On Riesz-Caputo Formulation for
Sequential Fractional Variational Principles
Fahd Jarad,1 Thabet Abdeljawad,1 and Dumitru Baleanu1, 2
1
Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University,
06530 Ankara, Turkey
2
Institute for Space Sciences, P.O. Box MG-23, 76900 Magurele-Bucharest, Romania
Correspondence should be addressed to Dumitru Baleanu, dumitru@cankaya.edu.tr
Received 6 October 2011; Accepted 23 December 2011
Academic Editor: Wing-Sum Cheung
Copyright q 2012 Fahd Jarad et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
This paper deals with sequential Riesz-Caputo fractional variational problems with and without
the presence of delay in the state variables and their derivatives. In both cases the necessary
conditions for the optimal control are reported.
1. Introduction
Fractional calculus which is as old as the classical calculus has become a candidate in solving
problems of complex systems which appear in various branches of science 1–10.
In the theory of time-delay systems the correct formulation of some initial value
problems together with the representation of solutions as well as the asymptotic behavior
of solutions are some of the general problems to be discussed. From the point of view of
the fractional calculus we have almost the same problems to be discussed. The combination
of these two main tools will lead us to a better description of the dynamics of the complex
systems. In 10, fractional order modeling of delay dynamics was used to better characterize
the delay behavior. Recently, it was shown that the delay parameter modifies the time
window for the fractional order time derivative by allowing fading memory of an earlier
time and introduces specific information about initial conditions see 11 and the references
therein.
Optimal control problems with time delay in calculus of variations were investigated
in 12–14. Variational optimal control problems within the framework of fractional calculus
were considered in 15. Very recently, the fractional variational problems in the presence of
delay were discussed in 16, 17.
2
Abstract and Applied Analysis
The aim of this paper is to find necessary conditions for sequential Riesz-Caputo
fractional variational and optimal control problems with or without delay.
The structure of the paper is as follows: in Section 2 the basic definitions in fractional
calculus are presented. Section 3 is devoted to the variational and optimal control problem
within sequential Riesz-Caputo fractional derivatives in the absence of delay. In Section 4 the
Riesz-Caputo variational principles with delay are discussed.
2. Basic Definitions
In this section, some basic definitions related to fractional derivatives are presented.
The left Riemann-Liouville fractional integral and the right Riemann-Liouville
fractional integral are defined, respectively, by
t
1
t − τα−1 fτdτ,
a I ft Γα a
b
1
α
Ib ft τ − tα−1 fτdτ,
Γα t
α
2.1
where f ∈ L1 a, b. Here and in the following Γα represents the gamma function.
The left Riemann-Liouville fractional derivative and the right Riemann-Liouville
fractional derivatives are defined, respectively, by
n t
d
1
t − τn−α−1 fτdτ,
a D ft Γn − α dt
a
n b
d
1
Dbα ft −
τ − tn−α−1 fτdτ,
Γn − α
dt
t
α
2.2
where α > 0, n α 1, and f ∈ ACn a, b the space of complex-valued functions ft
having continuous derivatives up to n − 1 on a, b and f n−1 x ∈ ACa, b the space of
absolutely continuous functions on a, b. In particular,
aD
aD
n
0
ft Db0 ft ft,
ft f n t,
Dbn ft −1n f n t.
2.3
The fractional derivative of a constant takes the form
aD
α
C C
t − a−α
Γ1 − α
2.4
and the fractional derivative of a power of t has the following form:
β
α
a D t − a for β > −1, α ≥ 0.
Γα 1t − aβ−α
,
Γ β−α1
2.5
Abstract and Applied Analysis
3
The left Caputo fractional derivative is
C α
a D ft
1
Γn − α
t
t − τn−α−1
a
d
dτ
n
fτdτ,
2.6
while the right Caputo fractional derivative is
C
Dbα ft
1
Γn − α
b
t
d n
fτdτ,
τ − tn−α−1 −
dτ
2.7
where f ∈ ACn a, b and α represents the order of the derivative such that n α 1. 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 α
a D ft
a Dα ft −
C
Dbα ft
Dbα ft
−
n−1
f k a
t − ak−α ,
Γk
−
α
1
k0
n−1
−1k f k b
k0
Γk − α 1
b − t
k−α
2.8
.
In 18, 19, the Riesz Riemann-Liouville fractional derivative or simply the Riesz
fractional derivative and the Caputo Riesz fractional derivative are, respectively, represented
as
1 α
n
α
a D ft −1 D b ft ,
2
1 C α
nC α
RC α
Db ft .
a Db ft a D ft −1
2
R α
a Db ft
2.9
2.10
In 1, a formula for the fractional integration by parts on the whole interval a, b was
given by the following lemma.
Lemma 2.1. Let α > 0, p, q ≥ 1, and 1/p 1/q ≤ 1 α p / 1 and q / 1 in the case when 1/p 1/q 1 α.
a If ϕ ∈ Lp a, b and ψ ∈ Lq a, b, then
b
ϕt a I α ψ tdt a
b
a
ψt Ibα ϕ tdt.
2.11
ft Dbα g tdt,
2.12
b If g ∈ Ibα Lp and f ∈ a I α Lq , then
b
a
gt a Dα f tdt b
a
4
Abstract and Applied Analysis
where a I α LP {f : f a I α ϕ, ϕ ∈ Lp a, b} and Ibα LP {f : f Ibα φ, φ ∈
Lp a, b}.
Lemma 2.1 was generalized in 16, Lemma 2 and 17, Lemma 2, with the same
conditions, to be applicable to subintervals a, r and r, b where r ∈ a, b.
3. The Sequential Riesz-Caputo Fractional Variational Problem
In this section we consider the following variational problem.
Minimize
J y b α
RC 2α
RC kα
L t, yt, RC
a Db yt, a Db yt, . . . , a Db yt dt,
3.1
a
where y ∈ AC2r a, b, 0 < α ≤ 1, r − 1 < kα ≤ r, r ∈ N yi a ci , yi b di , i 0, 1, . . . , r − 1,
ci ’s and di ’s are constant, and L : a, b × Rk1 → R has first and second partial derivatives
with respect to all of its variables.
Theorem 3.1. Let Jy be the functional of the form 3.1 defined on the set of functions yt which
have continuous Riesz-Caputo derivatives of order iα, i 1, . . . , k and which satisfy the boundary
conditions yi a ci , yi b di . Then a necessary condition for Jy to have a minimum for a
given yt is that yt satisfy the following Euler-Lagrange equation:
k
∂L −1i Ra Dbiα
∂y i1
∂L
iα
∂ RC
a Db y
0.
3.2
Proof. Assume that Jy has a minimum for y∗ t. Define a family of curves yt y∗ tηt
where ∈ R and η is a function in AC2r a, b satisfying ηi a ηi b 0, i 0, 1, . . . , r − 1.
The function
γ J y∗ t ηt
3.3
admits a minimum when γ 0 0. That is,
b
a
k
∂L
∂L
RC iα
ηt a Db ηtdt 0.
RC iα
∂y∗
∂
D
y∗
i1 a
b
3.4
Using 2.10, 2.8, and 1, Lemma 2.7, 3.4 becomes
b
a
k
∂L −1i Ra Dbiα
∂y∗ i1
∂L
iα
∂ RC
a Db y∗
ηtdt 0.
3.5
Abstract and Applied Analysis
5
Since η is arbitrary, one has
k
∂L −1i Ra Dbiα
∗
∂y
i1
∂L
iα ∗
∂ RC
a Db y
0.
3.6
When α 1, the results in Theorem 3.1 coincide with the results in 20, p. 59.
Example 3.2. Let us consider the following action:
b
a
1 2
RC α
y ft a Db y dt,
2
3.7
where ft ∈ ACa, b, 0 < α ≤ 1, ya c1 , and yb d1 . Then by Theorem 3.1, the
necessary condition for optimality is
y Ra Dbα ft.
3.8
One can generalize Theorem 3.1 as follows.
Corollary 3.3. Consider the functional of the form
J y1 , y2 , . . . , yl
b α
RC 2α
RC kα
L t, y1 t, RC
a Db y1 t, a Db y1 t, . . . , a Db y1 t,
a
α
y2 t, RC
a Db y2 t,
RC 2α
RC kα
a Db y2 t, . . . , a D b y2 t, . . . ,
3.9
α
RC 2α
RC kα
dt,
D
y
D
y
D
y
yl t, RC
.
.
.
,
t,
t,
t
l
l
l
a
a
a
b
b
b
defined on sets of continuous functions yj , j 1, 2 . . . , l that have Riesz Caputo fractional derivatives
of order iα, i 1, 2, . . . , k, 0 < α ≤ 1, r −1 < kα ≤ r in the interval a, b and satisfy the conditions
i
yj a cij ,
i
yj b dij ,
i 0, 1, . . . , r − 1; j 1, 2, ..l,
3.10
where aij , dij are constant and L : a, b × Rlk1 → R is a function with continuous first and second
partial derivatives with respect to all of its arguments. For yj , j 1, 2, . . . , l to be a minimum of 3.9,
it is necessary that yj x satisfy the following Euler-Lagrange equations:
k
∂L −1i Ra Dbiα
∂yj i1
∂L
iα
∂ RC
a Db yj
0,
j 1, 2, . . . , l.
3.11
6
Abstract and Applied Analysis
3.1. The Optimal Control Case
The optimal control problem we consider is to find ut that minimizes the performance index
J y, u b α
RC 2α
RC kα
F t, ut, yt, RC
a Db yt, a Db yt, . . . , a Db yt dt,
3.12
a
subject to the constraint
α
RC 2α
RC kα
G t, ut, yt, RC
a Db yt, a Db yt, . . . , a Db yt 0,
3.13
such that
yi a ci ,
yi b di ,
i 0, 1, . . . , r − 1,
3.14
where ci , di are constant, 0 < α ≤ 1, r − 1 < kα ≤ r, and F, G : a, b × Rk2 → R are functions
with continuous first and second partial derivatives with respect to their arguments.
To find the optimal control, one defines the modified performance index as
J y, u b
F· λtG·dt,
3.15
a
where λt is called a Lagrange multiplier or adjoint variable. Using conditions 3.11, the
following necessary conditions for optimal control are found:
∂G
∂F
λ
0,
∂u
∂u
k
∂F
∂F
∂G ∂G
i R iα
R iα
λ
−1 a Db
a Db λ RC iα
0.
iα
∂y
∂y i1
∂ RC
∂ a Db y
a Db y
3.16
When α 1, k 1, 3.16 coincides with the results in 20, Theorem 6.2.2.
4. The Riesz-Caputo Sequential Fractional Variational
Principles with Delay
In this section the following problem with delay is considered.
Minimize
J y b α
RC 2α
RC kα
L t, RC
a Db yt, a Db yt, . . . , a Db yt ,
a
r
r
yt, y t, . . . , y t, yt − τ, y t − τ, . . . , y t − τ dt
4.1
Abstract and Applied Analysis
7
such that
yt ft,
yi b di ,
a − τ ≤ t ≤ a,
i 0, 1, 2, . . . , r − 1, r − 1 < kα ≤ r,
4.2
where di are constant, y ∈ AC2r a, b, f is a smooth function, and L : a − τ, b × Rk2r2 → R
is a function with continuous first- and second-order partial derivatives with respect to its
arguments.
Theorem 4.1. If yt satisfying 4.2 is a minimum of 4.1, then it is necessary that it satisfies the
Euler-Lagrange equations
∂L
∂L
t τ t
∂yt − τ
∂yt
r
i
i
∂L
∂L
i d
i d
t τ −1 i
t
−1 i
dt ∂yi t − τ
dt ∂yi t
i1
k
∂L
1 1
i R iα
Diα
t −
−1 a Db−τ
RC iα
2 Γiα b−τ
∂ a Db yt
i1
b
∂L
iα−1
iα
×
Db
0,
ss − t ds
iα
∂ RC
b−τ
a Db yt
4.3
for a ≤ t ≤ b − τ
∂L
t
∂yi t
∂L
iα
r
∂L
di
t −1i i
∂yt
dt
i1
k
−1i
i1
1 1
iα
t −
b−τ D
iα
2
Γiα
∂ RC
D
yt
a
b
b−τ
∂L
iα−1
iα
×
0,
st − s ds
aD
iα
∂ RC
a
a Db yt
R
b−τ Db
4.4
for b − τ ≤ t ≤ b, and the transversality condition
i−1
r dl
−1l l
dt
i1 l0
∂L
∂yi t
i−1
r dl
−1l l
dt
i1 l0
r i−1
dl
−1l l
dt
i1 l0
tη
∂L
∂yi t
r−l−1
b−τ
t
a
b
tηr−l−1 t
∂L
i
∂y t − τ
4.5
b−τ
t τη
r−l−1
b−τ
t
0,
a
8
Abstract and Applied Analysis
where η ∈ AC2r a, b is any arbitrary function such that ηi b 0, i 0, 1, . . . , r −1, η ≡ 0, a−τ ≤
t ≤ a.
Proof. Assume that Jy has a minimum for y0 t. Define a family of curves yt y0 tηt
where ∈ R and η is a function in AC2r a, b satisfying ηi b 0, i 0, 1, . . . , r − 1, η ≡
0, a − τ ≤ t ≤ a. The function
γ J y0 t ηt
4.6
admits a minimum when γ 0 0. That is, at y0 , one has
b
k
∂L
∂L
∂L
iα
tηt tηt − τ t RC
a Db ηt
RC
iα
∂yt
∂yt − τ
i1 ∂ a Db yt
r
∂L
∂L
i
i
tη t tη t − τ dt 0.
i
∂yi t − τ
i1 ∂y t
a
4.7
Using 2.8, 2.10, and the fact that ηt ≡ 0, a − τ ≤ t ≤ a and ηi b 0, i 0, 1, . . . , r − 1, one gets
b
a
k
∂L
∂L
∂L
tηt tηt − τ t Ra Dbiα ηt
RC
iα
∂yt
∂yt − τ
∂
D
yt
i1 a
b
r
∂L
∂L
i
i
tη t tη t − τ dt 0.
i
∂yi t − τ
i1 ∂y t
4.8
Splitting the integrals, making the change of variables t − τ ↔ t and keeping in mind that
ηt ≡ 0, a − τ ≤ t ≤ a, 4.8 becomes
b−τ a
k
∂L
∂L
∂L
tηt t τηt t Ra Dbiα ηt
RC
iα
∂yt
∂yt − τ
∂
D
yt
i1 a
b
r
∂L
∂L
i
i
tη t t τη t dt
i
∂yi t − τ
i1 ∂y t
b k
r
∂L
∂L
∂L
R iα
i
tηt t a Db ηt tη t dt 0.
RC iα
i
b−τ ∂yt
i1 ∂ a Db yt
i1 ∂y t
4.9
Abstract and Applied Analysis
9
If one uses the usual integration by parts formula, the integration by parts formulas in 16,
Lemma 2 and 17, Lemma 2, and the properties of the function η, one obtains
b−τ ∂L
∂L
t τ t
∂yt − τ
∂yt
r
i
i
∂L
∂L
i d
i d
t τ−1 i
t
−1 i
dt ∂yi t − τ
dt ∂yi t
i1
k
∂L
1 1
i R iα
iα
Db−τ
t −
−1 a Db−τ
RC iα
2
Γiα
∂¡
D
yt
a
i1
b
b
∂L
iα−1
iα
×
Db
ηtdt
ss − t ds
iα
∂ RC
b−τ
a Db yt
a
b b−τ
r
∂L
di
t −1i i
∂yt
dt
i1
k
−1
i
R
iα
b−τ Db
i1
×
∂L
∂yi t
iα
∂ RC
a Db yt
aD
i−1
r dl
−1l l
dt
i1 l0
i−1
r dl
−1l l
dt
i1 l0
∂L
∂yi t
∂L
∂yi t
iα
∂ RC
a Db yt
tη
r−l−1
4.10
1 1
iα
b−τ D
2 Γiα
t −
∂L
iα
a
r i−1
dl
−1l l
dt
i1 l0
t
∂L
b−τ
st − s
iα−1
ds
ηtdt
b−τ
t
a
tη
∂L
∂yi t − τ
r−l−1
b
t
b−τ
t τη
r−l−1
b−τ
0.
t
a
If one chooses η such that ηa 0 and η ≡ 0 on b − τ, b, 4.3 holds. If one chooses
ηb 0 and η ≡ 0 on a, b − τ, 4.4 holds. Then since the first and second integrals in 4.10
become zero, 4.5 holds.
The result in Theorem 4.1 coincides with the results in 14, Corollaries 2.1 and 2.2.
Example 4.2. Consider
2
0
1 2
1 2
RC α
y t − 1 y t ft 0 D2 yt dt,
2
2
4.11
10
Abstract and Applied Analysis
where 0 < α ≤ 1, y2 constant, and yt gt for − 1 ≤ t ≤ 0. The necessary conditions
for optimality according to Theorem 4.1 are
1 R α 1
Dα
D f t yt 2 0 1
4Γα 1
2
1
D2α f
α−1
α−1
ss − t
t,
4.12
for 0 < t < 1 and
yt R α
1 D2 f
1
α
t −
1D
2Γα
1
0D
α
f st − s
t,
4.13
0
for 1 < t < 2.
Theorem 4.1 can be generalized as follows
Corollary 4.3. Consider the functional of the form
J y1 , y2 , . . . , ym
b α
RC 2α
RC kα
L t, RC
a Db y1 t, a Db y1 t, . . . , a Db y1 t,
a
r
r
y 1 t, y1 t, . . . , y1 t, y1 t − τ, y1 t − τ, . . . , y1 t − τ,
RC α
RC 2α
RC kα
a Db y2 t, a Db y2 t, . . . , a Db y2 t, y2 t, y2 t, . . . ,
r
y2 t, y2 t
− τ, y2 t −
r
τ, . . . , y2 t
4.14
− τ, . . . ,
RC α
RC 2α
RC kα
a Db ym t, a Db ym t, . . . , a Db ym t, ym t,
r
r
ym
t, . . . , ym t, ym t − τ, ym
t − τ, . . . , ym t − τ dt,
defined on sets of functions yj t ∈ AC2r a, b, j 1, 2, . . . , m that have Riesz-Caputo fractional
derivatives of order iα, i 1, 2, . . . , k, 0 < α ≤ 1, r − 1 < kα ≤ r, and satisfy the conditions
i
yj dij ,
yj t fj t,
t ∈ a − τ, a, τ < b − a, a < b,
4.15
where dij are constant, fj t are smooth functions and L : a − τ, b × Rmk2r2 → R is a function
having first-and second-order partial derivatives with respect to its arguments. For yj , j 1, 2, . . . , m
Abstract and Applied Analysis
11
satisfying 4.15 to be a minimum of 4.14, it is necessary that yj satisfy the Euler-Lagrange
equations
∂L
∂L
t
t τ ∂yj t − τ
∂yj t
⎞
⎞ ⎤
⎛
⎛
⎡
r
i
i
∂L
∂L
d
d
i
i
⎠t τ −1
⎠t⎦
⎝
⎣−1 i ⎝ i
dt
dti ∂yi t
∂yj t − τ
i1
j
k
iα
−1i Ra Db−τ
i1
×
b
b−τ
∂L
iα
∂ RC
a Db yj t
Dbiα
t −
∂L
iα
∂ RC
a Db yj t
4.16
1 1
Diα
2 Γiα b−τ
ss − tiα−1 ds
0,
for a ≤ t ≤ b − τ, j 1, 2 . . . , m
⎞
⎛
r
i
∂L
∂L
d
t −1i i ⎝ i ⎠t
∂yj t
dt
∂yj t
i1
k
−1
i
R
iα
b−τ Db
i1
×
∂L
iα
∂ RC
a Db yj t
b−τ
aD
iα
a
1 1
iα
b−τ D
2 Γiα
t −
∂L
iα
∂ RC
a Db yj t
4.17
st − s
iα−1
ds
0,
for b − τ ≤ t ≤ b, j 1, 2 . . . , m, and the transversality condition
b−τ
⎞
⎛
r i−1
l
∂L
d
r−l−1
−1l l ⎝ i ⎠tηj
t
dt
∂y t
i1 l0
j
a
b
dl ⎝ ∂L ⎠
r−l−1
tηj
−1 l
t
i
dt
∂yj t
i1 l0
i−1
r ⎛
⎞
4.18
l
b−τ
b−τ
∂L
d
⎠t τηr−l−1 t
0,
−1l l ⎝ i
j
dt
∂yj t − τ
i1 l0
i−1
r ⎛
⎞
l
a
where η η1 , η2 , . . . , ηm is an an arbitrary vector-valued function satisfying ηt
0, 0, . . . , 0, t ∈ a − τ, a, and ηi b 0, 0, . . . , 0, i 0, 1, . . . , r − 1.
12
Abstract and Applied Analysis
4.1. The Optimal Control Case with Delay
Find the optimal control variable u such that u minimizes the performance index
b α
RC 2α
RC kα
J y, u F t, ut, RC
a Db yt, a Db yt, . . . , a Db yt ,
a
yt, y t, . . . , y
k
t, yt − τ, y t − τ, . . . , y
k
4.19
t − τ dt
subject to the constraint
α
RC 2α
RC kα
G t, ut, RC
a Db yt, a Db yt, . . . , a Db yt ,
yt, y t, . . . , y
k
t, yt − τ, y t − τ, . . . , y
k
4.20
t − τ 0
such that
yt ft,
a − τ ≤ t ≤ a,
yi di ,
i 0, 1, 2, . . . , r − 1, 0 < α ≤ 1,
4.21
where r − 1 < kα ≤ r di are constant, y ∈ AC2r a, b, f is a smooth function and F, G :
a−τ, b×Rk2r3 → R are functions with continuous first and second order partial derivatives
with respect to their arguments.
If one defines a modified performance index as
J y, u b
F· λtG·dt,
a
applying Corollary 4.3, it is necessary that
∂F
∂F
t τ t
∂yt − τ
∂yt
r
i
i
∂F
∂F
i d
i d
t τ−1 i
t
−1 i
dt ∂yi t − τ
dt ∂yi t
i1
k
∂F
1 1
i R iα
Diα
t −
−1 a Db−τ
RC iα
2 Γiα b−τ
∂ a Db yt
i1
b
∂F
∂G
iα−1
iα
−
t
×
Db
ds
λ
t τ
ss
iα
∂yt − τ
∂ RC
b−τ
a Db yt
4.22
Abstract and Applied Analysis
13
r
i
i
∂G
∂G
∂G
i d
i d
λ
t t τ−1 i λ i
t
−1 i λ i
∂yt
dt
dt
∂y t − τ
∂y t
i1
k
iα
−1i Ra Db−τ
i1
×
b
b−τ
λ
Dbiα
λ
∂G
iα
∂ RC
a Db yt
∂G
iα
∂ RC
a Db yt
t −
1 1
Diα
2 Γiα b−τ
ss − tiα−1 ds
0,
∂G
∂F
t λt
t 0,
∂ut
∂ut
4.23
for a ≤ t ≤ b − τ,
r
i
∂F
∂F
i d
t −1 i
t
∂yt
dt ∂yi t
i1
k
∂F
1 1
i R
iα
iα
−1 b−τ Db
t −
b−τ D
RC iα
2 Γiα
∂ a Db yt
i1
b−τ
∂F
iα−1
iα
×
st − s ds ,
aD
iα
∂ RC
a
a Db yt
r
i
∂G
∂G
i d
λ
t −1 i λ i
t
∂yt
dt
∂y t
i1
k
∂G
1 1
i R
iα
iα
−1 b−τ Db λ RC iα
t −
b−τ D
2 Γiα
∂ a Db yt
i1
b−τ
∂F
iα−1
iα
λ RC iα
×
0,
st − s ds
aD
∂ a Db yt
a
∂G
∂F
t λt
t 0,
∂ut
∂ut
4.24
for b − τ < t < b, and the transversality condition
b−τ
∂F
r−l−1
tη
t
i
∂y t
a
b
i−1
k
l
∂F
l d
r−l−1
tη
−1 l
t
i
dt ∂y t
i1 l0
b−τ
b−τ
k
i−1
l
∂F
l d
r−l−1
0,
τη
t
−1 l
t
i
dt ∂y t − τ
i1 l0
a
k i−1
dl
−1l l
dt
i1 l0
4.25
14
Abstract and Applied Analysis
where η is any arbitrary function satisfying ηt 0, t ∈ a − τ, t, and ηi b 0, i 0, 1, 2, . . . , r − 1.
5. Conclusion
Fractional variational principles started to be used in several branches of science and
engineering. However the delay is present in various phenomena having great impact in
science and engineering. One of the main questions is to combine in an optimal way the
properties of the fractional calculus and those of the delay having in mind to obtain a
more general variational principles. In this paper we have used the fractional Riesz-Caputo
derivative and the delay in the state variable. We mention that the definition of the RieszCaputo derivative contains both left and right fractional derivative. The necessary conditions
for the optimal control were obtained. When α → 1 or the delay is absent the classical results
are obtained.
Acknowledgment
This work is partially supported by the Scientific and Technical Research Council of Turkey.
References
1 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential
Equations, vol. 204, Elsevier Science, Amsterdam, The Netherlands, 2006.
2 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach
Science Publishers, Linghorne, Pa, USA, 1993, Theory and Applications.
3 R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, Conn, USA, 2006.
4 N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential
equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, pp. 765–771, 2006.
5 I. S. Jesus and J. A. Tenreiro Machado, “Fractional control of heat diffusion systems,” Nonlinear
Dynamics, vol. 54, no. 3, pp. 263–282, 2008.
6 F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional
diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001.
7 E. Scalas, R. Gorenflo, and F. Mainardi, “Uncoupled continuous-time random walks: solution and
limiting behavior of the master equation,” Physical Review E, vol. 69, no. 1, artice 011107, 2004.
8 O. P. Agrawal and D. Baleanu, “A Hamiltonian formulation and a direct numerical scheme for
fractional optimal control problems,” Journal of Vibration and Control, vol. 13, no. 9-10, pp. 1269–1281,
2007.
9 V. E. Tarasov and G. M. Zaslavsky, “Nonholonomic constraints with fractional derivatives,” Journal of
Physics A, vol. 39, no. 31, pp. 9797–9815, 2006.
10 Y. Chen, “Fractional calculus, delay dynamics and networked control systems,” in Proceedings of the
3rd International Symposium on Resilient Control Systems (ISRCS ’10), pp. 58–63, Idaho Falls, Idhao,
USA, August 2010.
11 S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,”
Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011.
12 J. F. Rosenblueth, “Systems with time delay in the calculus of variations: a variational approach,”
Journal of Mathematical Control and Information, vol. 5, no. 2, pp. 125–145, 1988.
13 T. Abdeljawad, F. Jarad, and D. Baleanu, “Variational optimal-control problems with delayed
arguments on time scales,” Advances in Difference Equations, vol. 2009, Article ID 840386, 15 pages,
2009.
14 D. K. Hughes, “Variational and optimal control problems with delayed argument,” Journal of
Optimization Theory and Applications, vol. 2, no. 1, pp. 1–15, 1968.
15 O. P. Agrawal, “A general formulation and solution scheme for fractional optimal control problems,”
Nonlinear Dynamics, vol. 38, no. 1–4, pp. 323–337, 2004.
Abstract and Applied Analysis
15
16 D. Baleanu, T. Maaraba, and F. Jarad, “Fractional variational principles with delay,” Journal of Physics.
A, vol. 41, no. 31, 2008.
17 F. Jarad, T. Abdeljawad, and D. Baleanu, “Fractional variational principles with delay within Caputo
derivatives,” Reports on Mathematical Physics, vol. 65, no. 1, pp. 17–28, 2010.
18 I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
19 O. P. Agrawal, “Fractional variational calculus in terms of Riesz fractional derivatives,” Journal of
Physics, vol. 40, no. 24, pp. 6287–6303, 2007.
20 B. van Brunt, The Calculus of Variations, Springer, New York, NY, USA, 2004.
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