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Calculus of Variations with Fractional
and Classical Derivatives
Tatiana Odzijewicz∗ Delfim F. M. Torres∗∗
∗ Department of Mathematics, University of Aveiro
3810-193 Aveiro, Portugal (e-mail: tatianao@ua.pt)
∗∗ Department of Mathematics, University of Aveiro
3810-193 Aveiro, Portugal (e-mail: delfim@ua.pt)
Abstract: Wegiveaproperfractionalextensionoftheclassicalcalculus of variations. Necessary
optimality conditions of Euler-Lagrange type for variational problems containing both fractional
and classical derivatives are proved. The fundamental problem of the calculus of variations with
mixed integer and fractional order derivatives as well as isoperimetric problems are considered.
Keywords: variational analysis; optimality; Riemann-Liouville fractional operators; fractional
differentiation; isoperimetric problems.
1. INTRODUCTION anditsapplications have been written—see, e.g., [Agrawal,
2002, Agrawal and Baleanu, 2007, Almeida and Torres,
One of the classical problems of mathematics consists in 2009a, 2010, Atanackovi´c et al., 2008, Baleanu, 2008, El-
finding a closed plane curve of a given length that encloses Nabulsi and Torres, 2007, Frederico and Torres, 2007,
the greatest area: the isoperimetric problem. The legend 2008, Klimek, 2002, Malinowska and Torres, 2010] and
says that the first person who solved the isoperimetric references therein. For the study of fractional isoperimetric
problem was Dido, the queen of Carthage, who was offered problems see [Almeida et al., 2009].
as much land as she could surround with the skin of a In the pioneering paper [Agrawal, 2002], and others that
bull. Dido’s problem is nowadays part of the calculus of followed, the fractional necessary optimality conditions are
variations [Gelfand and Fomin, 1963, van Brunt, 2004]. proved under the hypothesis that admissible functions y
Fractional calculus is a generalization of (integer) differ- have continuous left and right fractional derivatives on the
ential calculus, allowing to define derivatives (and inte- closed interval [a,b]. By considering that the admissible
grals) of real or complex order [Kilbas et al., 2006, Miller functions y have continuous left fractional derivatives on
and Ross, 1993, Podlubny, 1999]. The first application of the whole interval, then necessarily y(a) = 0; by consider-
fractional calculus belongs to Niels Henrik Abel (1802– ing that the admissible functions y have continuous right
1829) and goes back to 1823 [Abel, 1965]. Abel applied the fractional derivatives, then necessarily y(b) = 0. This fact
fractional calculus to the solution of an integral equation has been independently remarked, in different contexts, at
which arises in the formulation of the tautochrone problem. least in [Almeida et al., 2009, Almeida and Torres, 2010,
Thisproblem,sometimesalsocalledtheisochrone problem, Atanackovi´c et al., 2008, Jelicic and Petrovacki, 2009].
is that of finding the shape of a frictionless wire lying in a In our work we want to be able to consider arbitrarily
vertical plane such that the time of a bead placed on the given boundary conditions y(a) = y and y(b) = y (and
a b
wire slides to the lowest point of the wire in the same time isoperimetric constraints). For that we consider variational
regardless of where the bead is placed. The cycloid is the functionals with integrands involving not only a fractional
isochrone as well as the brachistochrone curve: it gives the derivative of order α ∈ (0,1) of the unknown function
shortest time of slide and marks the born of the calculus y, but also the classical derivative y′. More precisely, we
of variations. consider dependence of the integrands on the independent
′ α
variable t, unknown function y, and y + kaD y with k
The study of fractional problems of the calculus of vari- t
ations and respective Euler-Lagrange type equations is a real parameter. As a consequence, one gets a proper
a subject of current strong research due to its many extension of the classical calculus of variations, in the sense
applications in science and engineering, including me- that the classical theory is recovered with the particular
chanics, chemistry, biology, economics, and control the- situation k = 0. We remark that this is not the case with
ory. In 1996-1997 Riewe obtained a version of the Euler– all the previous literature on the fractional variational
Lagrange equations for fractional variational problems calculus, where the classical theory is not included as a
combining the conservative and nonconservative cases particular case and only as a limit, when α → 1.
[Riewe, 1996, 1997]. Since then, numerous works on the The text is organized as follows. In Section 2 we briefly
fractional calculus of variations, fractional optimal control recall the necessary definitions and properties of the frac-
⋆ tional calculus in the sense of Riemann-Liouville. Our
This work is part of the first author’s PhD project, which is carried results are stated, proved, and illustrated through an ex-
out at the University of Aveiro under the Doctoral Programme ample, in Section 3. We end with Section 4 of conclusion.
Mathematics and Applications of Universities of Aveiro and Minho.
2. PRELIMINARIES hasanextremum.Weassumethatkisafixedrealnumber,
F ∈ C2([a,b] × R2;R), and ∂ F (the partial derivative
3
In this section some basic definitions and properties of of F(·,·,·) with respect to its third argument) has a
fractional calculus are presented. For more on the subject continuous right Riemann-Liouville fractional derivative of
werefer the reader to the books [Kilbas et al., 2006, Miller order α.
and Ross, 1993, Podlubny, 1999]. Definition 5. A function y ∈ C1[a,b] that satisfies the
Definition 1. (Left and right Riemann-Liouville derivatives). given boundary conditions (3) is said to be admissible for
α problem (2)-(3).
Let f be a function defined on [a,b]. The operator aD ,
Z t
1 t
α n n−α−1 α
D f(t) = D (t − τ) f(τ)dτ , For simplicity of notation we introduce the operator [y]
a t Γ(n−α) defined by k
a
is called the left Riemann-Liouville fractional derivative of α ′ α
[y] (t) = (t,y(t),y (t) + kaD y(t)) .
order α, and the operator Dα, k t
t b With this notation we can write (2) simply as
α −1 nZ b n−α−1 Z b
tD f(t) = D (τ −t) f(τ)dτ , α
b Γ(n−α) J(y) = F[y] (t)dt.
t k
is called the right Riemann-Liouville fractional derivative a
of order α, where α ∈ R+ is the order of the derivatives Theorem 6. (The fractional Euler-Lagrange equation). Ify
and the integer number n is such that n−1 ≤ α < n. is an extremizer (minimizer or maximizer) of problem (2)-
Definition 2. (Mittag-Leffler function). Let α,β > 0. The (3), then y satisfies the Euler-Lagrange equation
Mittag-Leffler function is defined by α d α α α
∂ F[y] (t)− ∂ F[y] (t)+k D ∂ F[y] (t) = 0 (4)
2 k dt 3 k t b 3 k
∞ k
E (z)=X z . for all t ∈ [a,b].
α,β Γ(αk+β)
k=0 Proof. Suppose that y is a solution of (2)-(3). Note
Theorem 3. (Integration by parts). If f,g and the frac- that admissible functions yˆ can be written in the form
α α
tional derivatives aD g and tD f are continuous at every 1
t b yˆ(t) = y(t)+ǫη(t), where η ∈ C [a,b], η(a) = η(b) = 0, and
point t ∈ [a,b], then ǫ ∈ R. Let J(ǫ) = RbF(t,y(t) + ǫη(t), d (y(t)+ǫη(t)) +
Z b α Z b α α a α dt
f(t) D g(t)dt = g(t) D f(t)dt (1) kaDt (y(t)+ǫη(t)))dt. Since aDt is a linear operator, we
a t t b know that
a a
for any 0 < α < 1. α α α
D (y(t)+ǫη(t)) = D y(t)+ǫ D η(t).
a t a t a t
Remark 4. If f(a) 6= 0, then Dαf(t)| =∞.Similarly,
α a t t=a Onthe other hand,
if f(b) 6= 0, then tDb f(t)| = ∞. Thus, if f possesses Z b
t=b dJ d
continuous left and right Riemann-Liouville fractional = F[yˆ]α(t)dt
k
derivatives at every point t ∈ [a,b], then f(a) = f(b) = 0. dǫ ǫ=0 a dǫ
b Z ǫ=0
This explains why the usual term f(t)g(t)| does not b
a α α dη(t) (5)
appear on the right-hand side of (1). = ∂ F[y] (t)·η(t)+∂ F[y] (t)
2 k 3 k dt
a
3. MAIN RESULTS α α
+k∂ F[y] (t) D η(t) dt.
3 k a t
Following [Jelicic and Petrovacki, 2009], we prove opti- Using integration by parts we get
mality conditions of Euler-Lagrange type for variational Z b dη b Z b d
∂ F dt = ∂ Fη| − (η ∂ F)dt (6)
problems containing classical and fractional derivatives 3 dt 3 a dt 3
simultaneously. In Section 3.1 the fundamental variational a a
and Z Z
problem is considered, while in Section 3.2 we study the b b
α
isoperimetric problem. Our results cover fractional vari- k ∂ F D ηdt = η D ∂ Fdt. (7)
3 a t t b 3
ational problems subject to arbitrarily given boundary a a
conditions. This is in contrast with [Agrawal, 2002, 2008, Substituting (6) and (7) into (5), and having in mind that
Agrawal and Baleanu, 2007, Baleanu et al., 2009], where η(a) = η(b) = 0, it follows that
the necessary optimality conditions are valid for appropri- Z b
dJ α d α
ate zero valued boundary conditions (cf. Remark 4). For a = η(t) ∂ F[y] (t)− ∂ F[y] (t)
dǫ 2 k dt 3 k
discussion on this matter see [Almeida and Torres, 2010, ǫ=0 a
Atanackovi´c et al., 2008, Jelicic and Petrovacki, 2009]. α α
+k D ∂ F[y] (t) dt.
t b 3 k
dJ
3.1 The Euler-Lagrange equation Anecessary optimality condition is given by dǫ =0.
Hence, ǫ=0
Let 0 < α < 1. Consider the following problem: find a Z b d
function y ∈ C1[a,b] for which the functional α α
η(t) ∂ F[y] (t)− ∂ F[y] (t)
Z 2 k dt 3 k
b a
′ α α α
J(y) = F(t,y(t),y (t) +kaD y(t))dt (2) +k D ∂ F[y] (t) dt = 0. (8)
t t b 3 k
a
subject to given boundary conditions We obtain (4) applying the fundamental lemma of the
y(a) = y , y(b) = y , (3) calculus of variations to (8).
a b
Remark 7. Note that for k = 0 our necessary optimality By the implicit function theorem, there exists a function
condition (4) reduces to the classical Euler-Lagrange equa- ǫ (·) defined in a neighborhood of (0,0) such that
tion [Gelfand and Fomin, 1963, van Brunt, 2004]. 2
ˆ
I(ǫ ,ǫ (ǫ )) = 0.
1 2 1
3.2 The fractional isoperimetric problem ˆ
Let J(ǫ ,ǫ ) = J(yˆ). Then, by the Lagrange multiplier
1 2
rule, there exists a real λ such that
As before, let 0 < α < 1. We now consider the problem of ˆ ˆ
extremizing a functional ∇(J(0,0)−λI(0,0)) = 0.
Z b ′ α Because Z
J(y) = F(t,y(t),y (t) +kaD y(t))dt (9) ˆ b
t ∂J = η ∂ F − d ∂ F +k Dα∂ F dt
a 1 2 3 t b 3
∂ǫ dt
in the class y ∈ C1[a,b] when subject to given boundary 1 (0,0) a
conditions and
y(a) = y , y(b) = y , (10) Z
a b ˆ b
and an isoperimetric constraint ∂I d α
= η ∂ G− ∂ G+kD ∂ G dt,
Z ∂ǫ 1 2 dt 3 t b 3
b 1 (0,0) a
′ α
I(y) = G(t,y(t),y (t) + kaD y(t))dt = ξ. (11)
t one has
a Z "
We assume that k and ξ are fixed real numbers, F,G ∈ b d
C2([a,b]×R2;R), and ∂ F and ∂ G have continuous right η ∂ F − ∂ F +k Dα∂ F
3 3 1 2 dt 3 t b 3
Riemann-Liouville fractional derivatives of order α. a #
Definition 8. A function y ∈ C1[a,b] that satisfies the d α
−λ ∂ G− ∂ G+kD ∂ G dt=0.
given boundary conditions (10) and isoperimetric con- 2 dt 3 t b 3
straint (11) is said to be admissible for problem (9)-(11).
Since η is an arbitrary function, (12) follows from the
Definition 9. An admissible function y is an extremal for 1
I if it satisfies the fractional Euler-Lagrange equation fundamental lemma of the calculus of variations.
α d α α α
∂ G[y] (t)− ∂ G[y] (t)+k D ∂ G[y] (t) = 0
2 k dt 3 k t b 3 k 3.3 An example
for all t ∈ [a,b]. Let α ∈ (0,1) and k,ξ ∈ R. Consider the following
The next theorem gives a necessary optimality condition fractional isoperimetric problem:
for the fractional isoperimetric problem (9)-(11). Z 1 ′ α 2
J(y) = (y +k0D y) dt−→min
Theorem 10. Let y be an extremizer to the functional t
(9) subject to the boundary conditions (10) and the 0Z
1
isoperimetric constraint (11). If y is not an extremal for I, ′ α (14)
I(y) = (y +k0D y)dt=ξ
then there exists a constant λ such that t
0
α d α α α Z 1 1−α
∂ H[y] (t)− ∂ H[y] (t)+k D ∂ H[y] (t) = 0 (12)
2 k 3 k t b 3 k y(0) = 0, y(1) = E −k(1−τ) ξdτ.
dt 0 1−α,1
for all t ∈ [a,b], where H(t,y,v) = F(t,y,v) − λG(t,y,v). In this case the augmented Lagrangian H of Theorem 10
Proof. We introduce the two parameter family is given by H(t,y,v) = v2 −λv. One can easily check that
yˆ = y + ǫ η +ǫ η , (13) Z t 1−α
1 1 2 2 y(t) = E −k(t−τ) ξdτ (15)
in which η and η are such that η ,η ∈ C1[a,b] and they 0 1−α,1
1 2 1 2
have continuous left and right fractional derivatives. We • is not an extremal for I;
also require that ′ α
• satisfies y +k0D y = ξ (see, e.g., [Kilbas et al., 2006,
η (a) = η (b) = 0 = η (a) = η (b). t
1 1 2 2 p. 297, Theorem 5.5]).
First we need to show that in the family (13) there are Moreover, (15) satisfies (12) for λ = 2ξ, i.e.,
curves such that yˆ satisfies (11). Substituting y by yˆ in
(11), I(yˆ) becomes a function of two parameters ǫ ,ǫ . d ′ α
1 2 − (2(y +k0D y)−2ξ)
Let dt t
Z b α ′ α
′ α +ktD (2(y +k0D y)−2ξ)=0.
ˆ 1 t
I(ǫ ,ǫ ) = G(t,yˆ,yˆ + k D yˆ)dt − ξ.
1 2 a t Weconclude that (15) is the extremal for problem (14).
ˆ a
Then, I(0,0) = 0 and Example 11. Choose k = 0. In this case the isoperimetric
Z
ˆ b constraint is trivially satisfied, (14) is reduced to the
∂I d α classical problem of the calculus of variations
= η ∂ G− ∂ G+kD ∂ G dt.
∂ǫ 2 2 dt 3 t b 3 Z
2 (0,0) a 1
Since y is not an extremal for I, by the fundamental lemma J(y) = (y′(t))2dt −→ min (16)
of the calculus of variations there is a function η such that 0
2 y(0) = 0, y(1) = ξ,
ˆ
∂I 6=0. and our general extremal (15) simplifies to the well-known
∂ǫ minimizer y(t) = ξt of (16).
2 (0,0)
Example 12. When α → 1 the isoperimetric constraint is ACKNOWLEDGEMENTS
redundant with the boundary conditions, and the frac-
tional problem (14) simplifies to the classical variational The first author is supported by the Portuguese Founda-
problem tion for Science and Technology (FCT) through the PhD
2 Z 1 ′ 2 fellowship SFRH/BD/33865/2009; the second author by
J(y) = (k +1) 0 y (t) dt −→ min (17) FCT through the Center for Research and Development
in Mathematics and Applications (CIDMA).
y(0) = 0, y(1) = ξ .
k+1
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