Variational Problems Involving a Caputo-Type Fractional Derivative

0.17 MB·

by  Ricardo Almeida

Variational Problems Involving a Caputo-Type Fractional
Derivative
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Ricardo Almeida
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[email protected]
b
e
F Center for Research and Development in Mathematics and Applications (CIDMA)
Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal
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] Abstract
C
O We study calculus of variations problems, where the Lagrange function depends on the
Caputo-Katugampolafractionalderivative. Thistypeoffractionaloperatorisageneralization
.
h of the Caputo and the Caputo–Hadamard fractional derivatives, with dependence on a real
at parameter ρ. We present sufficient and necessary conditions of first and second order to
m determine the extremizers of a functional. The cases of integral and holomonic constraints
are also considered.
[

Mathematics Subject Classification 2010: 26A33,34A08,34K28.
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v Keywords: fractionalcalculus,Caputo–Katugampolafractionalderivative,variationalproblems.
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1 Introduction
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0 Fractionalcalculus of variations was first studied by Riewe in [23], where he showedthat noncon-
.
1 servative forces such as friction are modeled by non-integer order derivatives. In fact, although
0 mostknownmethodsdealwithconservativesystems,naturalprocessesarenonconservativeandso
6 the usual Lagrange formulation is inadequate to characterize these phenomena. It turns out that
1
fractionalcalculus, due to its non-local character,may better describe the behavior of the certain
:
v realprocesses. For this reasonnowadaysit is animportantresearcharea,whichhas attractedthe
i
X attention not only of mathematicians, but also of physicists and engineers. Generally speaking,
fractionalcalculusdealswithintegralsandderivativesofarbitraryrealorder,anditwasconsidered
r
a sincetheverybeginningofcalculus,butonlyonthepastdecadesithasproventobeapplicablein
real problems. We can find severaldefinitions for fractional operators,and to decide which one is
more efficient to model the problemis a question whose answer depends onthe system. Thus, we
find several works that deal with similar subjects, for different kinds of fractional operators (see
e.g. [2, 4, 5, 6, 7, 8, 9, 10, 13, 14, 19, 20, 21, 22] and references therein). In this paper we intend
to present a more generaltheory, that includes the Caputo and the Caputo–Hadamardfractional
derivatives, following the work started in [1].
The text is organized as follows. In Section 2 we review the necessary definitions and results
for the Caputo–Katugampola fractional derivative. The main results of the paper are presented
in Section 3. First, in §3.1 we consider the fundamental problem, where we present necessary
and sufficient conditions that every minimizer of the functional must fulfill. In §3.2 we prove
a second order condition to determine if the extremals are in fact minimizers for the functional.
Then,weconsidervariationalproblemssubjecttoanintegralconstraintin§3.3andtoaholomonic
constraintin§3.4. Weendwithageneralizationofthevariationalproblem,knownintheliterature
as Herglotz problem (§3.5).
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2 Caputo–Katugampola fractional derivative
We find several definitions for fractional derivatives, each of them presenting its advantages and
disadvantages. One of those, considered mainly by engineers, is the Caputo derivative exhibiting
two important features: the derivative of a constant is zero and the Laplace transform depends
only oninteger-orderderivatives. Givena functionx:[a,b]→R,the Caputo fractionalderivative
of x of order α∈(0,1) is defined by [17]
1 d t 1
CDα x(t)= [x(τ)−x(a)]dτ,
a+ Γ(1−α)dtZ (t−τ)α
a
where Γ denotes the well-known Gamma function,
∞
Γ(z)= tz−1exp(−t)dt, z >0,
Z
0
and if x is of class C1, then we have the equivalent form
1 t 1
CDα x(t)= x′(τ)dτ.
a+ Γ(1−α)Z (t−τ)α
a
From the definition, it is clear that if x is a constant function, then CDα x(t) = 0 and that, if
a+
x(a) = 0, then the Caputo fractional derivative coincides with the Riemann–Liouville fractional
derivative. The Caputo–Hadamard derivative is a very recent concept [11, 12], and it combines
the Caputo derivative with the Hadamard fractional operator. Given a function x, the Caputo–
Hadamard fractional derivative of order α is defined as
1 d t t −α x(τ)−x(a)
HDα x(t)= t ln dτ,
a+ Γ(1−α) dtZ (cid:18) τ(cid:19) τ
a
and again if x is of class C1, then
1 t t −α
HDα x(t)= ln x′(τ)dτ.
a+ Γ(1−α)Z (cid:18) τ(cid:19)
a
In [1], a new type of operator is presented, that generalizes the two previous operators, by
introducing a new parameter ρ > 0 in the definition. The same idea has already been done
in [15, 16], where a new operator is defined which generalizes the Riemann–Liouville and the
Hadamard fractional derivatives.
Definition 1. Let 0 < a < b < ∞, x : [a,b] → R be a function, and α ∈ (0,1) and ρ > 0 two
fixed reals. The Caputo–Katugampola fractional derivative of order α is defined as
ρα d t τρ−1
CDα,ρx(t)= t1−ρ [x(τ)−x(a)]dτ.
a+ Γ(1−α) dtZ (tρ−τρ)α
a
This was motivated by the recent notion due to Katugampola in [15], where a new fractional
integral operator Iα,ρx(t) is presented,
a+
ρ1−α t τρ−1
Iα,ρx(t)= x(τ)dτ.
a+ Γ(α) Z (tρ−τρ)1−α
a
When α∈N, the fractional integral reduces to a n-fold integral of the form
t τ1 τn−1
τρ−1dτ τρ−1dτ ... τρ−1x(τ )dτ .
Z 1 1 Z 2 2 Z n n n
a a a
2
If x is continuously differentiable, then the fractionaloperator can be written in an equivalent
way [1]:
ρα t 1
CDα,ρx(t)= x′(τ)dτ,
a+ Γ(1−α)Z (tρ−τρ)α
a
that is, we have
d
CDα,ρx(t)=I1−α,ρ τ1−ρ x (t).
a+ a+ (cid:18) dτ (cid:19)
The main results of [1] are:
1. the operator is linear and bounded from C([a,b]) to C([a,b]),
2. if x∈C1[a,b], then the map t7→CDα,ρx(t) is continuous in [a,b] and CDα,ρx(a)=0,
a+ a+
3. if x is continuous, then CDα,ρIα,ρx(t)=x(t),
a+ a+
4. if x∈C1[a,b], then Iα,ρCDα,ρx(t)=x(t)−x(a).
a+ a+
We remark that, taking ρ=1, we obtain the Caputo fractional derivative,
CDα x(t)=CDα,1x(t),
a+ a+
and doing ρ→0+, then we get the Caputo–Hadamardfractional derivative:
HDα x(t)=CDα,0+x(t).
a+ a+
One crucial result for our present work is an integration by parts formula. For that, we need
the two following auxiliary operators, a fractional integral type
ρ1−α b 1
Iα,ρx(t)= x(τ)dτ,
b− Γ(α) Z (τρ−tρ)1−α
t
and a fractional differential type
ρα d b 1
Dα,ρx(t)= x(τ)dτ.
b− Γ(1−α)dtZ (τρ−tρ)α
t
Theorem 1. Let x be a continuousfunction andy be a functionof classC1. Then, the following
equality holds:
bx(t)CDα,ρy(t)dt= y(t)I1−α,ρx(t) t=b − by(t)Dα,ρx(t)dt.
Za a+ h b− it=a Za b−
Proof. Starting with the definition, we have
b ρα b t 1
x(t)CDα,ρy(t)dt= x(t) y′(τ)dτdt.
Z a+ Γ(1−α)Z Z (tρ−τρ)α
a a a
Using the Dirichlet’s formula, we get
b t 1 b b 1
x(t) y′(τ)dτdt= x(τ) y′(t)dτdt.
Z Z (tρ−τρ)α Z Z (τρ−tρ)α
a a a t
Integrating by parts, considering
b 1
u′(t)=y′(t) and v(t)= x(τ) dτ,
Z (τρ −tρ)α
t
we obtain the desired formula.
3
3 The variational problem
Fractional calculus of variations appeared in 1996, with the work ok Riewe [23], since as he ex-
plained,”TraditionalLagrangianandHamiltonianmechanicscannotbeusedwithnonconservative
forces suchas friction”. Since then, severalstudies have appeared,for different types of fractional
derivatives and/or fractional integrals, namely to determine necessary and sufficient conditions
that any extremal for the variational functional must satisfy.
To start, we present the concept of minimizer for a given functional. On the space C1[a,b],
consider the norm k·k given by
kxk= max |x(t)|+ max CDα,ρx(t) .
a+
t∈[a,b] t∈[a,b]
(cid:12) (cid:12)
(cid:12) (cid:12)
Let D ⊆ C1[a,b] be a nonempty set and J a functional defined on D. We say that x is a local
minimizer of J in the set D if there exists a neighborhoodV (x) such that for all x∗ ∈V (x)∩D,
δ δ
we have J(x) ≤ J(x∗). Note that any function x∗ ∈ V (x)∩D can be represent in the form
δ
x∗ =x+ǫh, where |ǫ|≪1 and h is such that x+ǫh∈D.
The purpose of this work is to study fractional calculus of variations problems, where the
integralfunctionaldependsontheCaputo–Katugampolafractionalderivative. Givenx∈C1[a,b],
consider the functional
b
J(x)= L(t,x(t),CDα,ρx(t))dt, (1)
Z a+
a
with the following assumptions:
1. L : [a,b] × R2 → R is continuously differentiable with respect to the second and third
arguments;
2. given any function x, the map t7→Dα,ρ(∂ L(t,x(t),CDα,ρx(t))) is continuous.
b− 3 a+
Here, and along the work, given a function with several independent variables f : A ⊆ Rn → R,
we denote
∂f
∂ f(x ,...,x ):= (x ,...,x ).
i 1 n ∂x 1 n
i
We remarkthat x may be fixed or free att=a and t=b. Bothcases will be consideredlater.
3.1 The fundamental problem
We seek necessaryandsufficientconditionsthat eachextremizersofthe functionalmustfulfill. In
order to obtain such equations we consider variations of the solutions and use the fact that the
first variation of the functional must vanish at the minimizer.
Theorem 2. Let x be a minimizer of the functional J as in (1), defined on the subspace
U = x∈C1[a,b] : x(a)=x and x(b)=x ,
a b
(cid:8) (cid:9)
where x ,x ∈R are fixed. Then, x is a solution for the fractional differential equation
a b
∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) =0 (2)
2 a+ b− 3 a+
(cid:0) (cid:1)
on [a,b].
Proof. Let x+ǫh be a variation of x, with |ǫ| ≪ 1 and h ∈ C1[a,b]. Since x+ǫh must belong
to the set U, the boundary conditions h(a) = 0 = h(b) must hold. Define the function j in a
neighborhood of zero as
j(ǫ)=J(x+ǫh).
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Since x is a minimizer of J, then ǫ=0 is a minimizer of j and so j′(0)=0. Computing j′(0) and
using Theorem 1, we get
b
∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) h(t)dt
Z 2 a+ b− 3 a+
a (cid:2) (cid:0) (cid:1)(cid:3)
+ h(t)I1−α,ρ ∂ L(t,x(t),CDα,ρx(t)) t=b =0. (3)
b− 3 a+
h (cid:0) (cid:1)it=a
Since h(a)=0=h(b) and h is arbitrary elsewhere, we conclude that
∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) =0, ∀t∈[a,b].
2 a+ b− 3 a+
(cid:0) (cid:1)
Definition 2. Eq. (2) is called the Euler–Lagrange equation associated to functional J. The
solutions of this fractional differential equation are called extremals of J.
Remark 1. The case of several dependent variables is similar, and the Euler–Lagrange equations
are easily deduced. Let
b
J(x)= L(t,x(t),CDα,ρx(t))dt,
Z a+
a
where
1. x(t)=(x (t),...,x (t)) and CDα,ρx(t)=(CDα,ρx (t),...,CDα,ρx (t));
1 n a+ a+ 1 a+ n
2. L : [a,b]×R2n → R is continuously differentiable with respect to its ith argument, for all
i∈{2,...,2n+1};
3. given any function x, the maps t 7→ Dα,ρ(∂ L(t,x(t),CDα,ρx(t))) is continuous, for all
b− n+i a+
i∈{2,...,n+1}.
In this case, if x is a minimizer of the functional J, subject to the restrictions x(a) = x and
a
x(b)=x , where x ,x ∈Rn are fixed, then x is a solution of
b a b
∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) =0,
i a+ b− n+i a+
(cid:0) (cid:1)
for all i∈{2,...,n+1}.
We remark that Eq. (2) gives only a necessary condition. To deduce a sufficient condition, we
recall the notion of convex function. Given a function L(t,x,y) continuously differentiable with
respect to the second and third arguments, we say that L is convex in S ⊆R3 if condition
L(t,x+x ,y+y )−L(t,x,y)≥∂ L(t,x,y)x +∂ L(t,x,y)y
1 1 2 1 3 1
holds for all (t,x,y),(t,x+x ,y+y )∈S.
1 1
Theorem 3. IfthefunctionLasin(1)isconvexin[a,b]×R2,theneachsolutionofthefractional
Euler–Lagrangeequation (2) minimizes J in U.
Proof. Let x be a solutionof Eq. (2) and x+ǫh be a variationof x, with |ǫ|≪1 and h∈C1[a,b]
with h(a)=0=h(b). Then
b
J(x+ǫh)−J(x) = L(t,x(t)+ǫh(t),CDα,ρx(t)+ǫCDα,ρh(t))−L(t,x(t),CDα,ρx(t)) dt
Z a+ a+ a+
ab(cid:2) (cid:3)
≥ ∂ L(t,x(t),CDα,ρx(t))ǫh(t)+∂ L(t,x(t),CDα,ρx(t))ǫCDα,ρh(h) dt
Z 2 a+ 3 a+ a+
ab(cid:2) (cid:3)
= ∂ L(t,x(t),CDα,ρx(t))−Dα,ρ(∂ L(t,x(t),CDα,ρx(t))) ǫh(t)dt
Z 2 a+ b− 3 a+
a (cid:2) (cid:3)
=0
since x is a solution of (2). Therefore, x is a local minimizer of J.
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Ifwedonotimposeanyrestrictionsontheboundaries,weobtaintwotransversalityconditions.
Theorem 4. Let x be a minimizer of the functional J as in (1). Then, x is a solution for the
fractional differential equation
∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) =0, t∈[a,b].
2 a+ b− 3 a+
(cid:0) (cid:1)
If x(a) is free, then
I1−α,ρ ∂ L(t,x(t),CDα,ρx(t)) at t=a.
b− 3 a+
(cid:0) (cid:1)
If x(b) is free, then
I1−α,ρ ∂ L(t,x(t),CDα,ρx(t)) at t=b.
b− 3 a+
(cid:0) (cid:1)
Proof. Since x is a minimizer, then
∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) =0,
2 a+ b− 3 a+
(cid:0) (cid:1)
for all t∈[a,b]. Therefore, using Eq. (3), we obtain
h(t)I1−α,ρ ∂ L(t,x(t),CDα,ρx(t)) t=b =0.
b− 3 a+
h (cid:0) (cid:1)it=a
If x(a) is free, then h(a) is also free and taking h(a)6=0 and h(b)=0, we get
I1−α,ρ ∂ L(t,x(t),CDα,ρx(t)) at t=a.
b− 3 a+
(cid:0) (cid:1)
The second case is similar.
Observe that in the previous results, the initial point of the cost functional coincides with the
initial point of the fractional derivative. Next we consider a more general type of problems, by
considering A∈(a,b) and the functional
b
J(x)= L(t,x(t),CDα,ρx(t))dt, (4)
Z a+
A
with the same assumptions on L as previously, defined on the set
U = x∈C1[a,b] : x(A)=x and x(b)=x .
A A b
(cid:8) (cid:9)
Theorem 5. If x is a minimizer of the functional J as in (4), then x satisfies the fractional
differential equations
Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) −Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) =0 on[a,A],
A− 3 a+ b− 3 a+
(cid:0) (cid:1) (cid:0) (cid:1)
∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) =0 on[A,b],
2 a+ b− 3 a+
(cid:0) (cid:1)
and the transversality condition
I1−α,ρ ∂ L(t,x(t),CDα,ρx(t)) −I1−α,ρ ∂ L(t,x(t),CDα,ρx(t)) =0 at t=a.
A− 3 a+ b− 3 a+
(cid:0) (cid:1) (cid:0) (cid:1)
Proof. Let x+ǫh be a variation of x, with |ǫ| ≪ 1, and h ∈ C1[a,b] with h(A) = 0 = h(b).
Computing the first variation of J, we deduce the following
b
0 = ∂ L(t,x(t),CDα,ρx(t))h(t)+∂ L(t,x(t),CDα,ρx(t))CDα,ρh(t) dt
Z 2 a+ 3 a+ a+
Ab(cid:2) (cid:3)
= ∂ L(t,x(t),CDα,ρx(t))h(t)+∂ L(t,x(t),CDα,ρx(t))CDα,ρh(t) dt
Z 2 a+ 3 a+ a+
a (cid:2)A (cid:3)
− ∂ L(t,x(t),CDα,ρx(t))h(t)+∂ L(t,x(t),CDα,ρx(t))CDα,ρh(t) dt.
Z 2 a+ 3 a+ a+
a (cid:2) (cid:3)
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Integrating by parts, and using the fact that h(A)=0=h(b), we arrive at
A
Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) −Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) h(t)dt
Z A− 3 a+ b− 3 a+
a (cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
b
+ ∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) h(t)dt
Z 2 a+ b− 3 a+
A (cid:2) (cid:0) (cid:1)(cid:3)
+h(a) I1−α,ρ ∂ L(t,x(t),CDα,ρx(t)) −I1−α,ρ ∂ L(t,x(t),CDα,ρx(t)) t=a =0.
A− 3 a+ b− 3 a+
h (cid:0) (cid:1) (cid:0) (cid:1)i
Since h is an arbitrary function, we obtain the three necessary conditions.
One interesting question is to determine the best type of fractional derivative for which the
functional attains the minimum possible value. The Caputo–Katugampola fractional derivative
depends onan extra parameterρ, and we canobtain e.g. the Caputo and the Caputo–Hadamard
fractional derivatives when ρ = 1 and ρ → 0+, respectively. Thus, we are interested now to
determinenotonlythe minimizerxbutalsothe valueofρ forwhichJ attainsitsminimumvalue.
Theorem 6. Let (x,ρ) be a minimizer of the functional J given by
ρ
b
J (x,ρ)= L(t,x(t),CDα,ρx(t))dt,
ρ Z a+
a
defined on the subspace U ×R+, where
U = x∈C1[a,b] : x(a)=x and x(b)=x .
a b
(cid:8) (cid:9)
Then, (x,ρ) is a solution for the fractional differential equation
∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) =0
2 a+ b− 3 a+
(cid:0) (cid:1)
on [a,b], and satisfies the integral equation
b
∂ L(t,x(t),CDα,ρx(t))ψ′(ρ)dt=0,
Z 3 a+
a
where ψ(ρ)=CDα,ρx(t).
a+
Proof. Let(x+ǫh,ρ+ǫρ )be avariationof(x,ρ), with|ǫ|≪1, h∈C1[a,b]withh(a)=0=h(b)
0
and ρ ∈R. If we define j as
0
j(ǫ)=J(x+ǫh,ρ+ǫρ ),
0
then
b
0=j′(0)= ∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) h(t)dt
Z 2 a+ b− 3 a+
a (cid:2) (cid:0) (cid:1)(cid:3)
b
+ ∂ L(t,x(t),CDα,ρx(t))ψ′(ρ)ρ dt.
Z 3 a+ 0
a
If we consider ρ =0, by the arbitrariness of h on (a,b), we conclude that
0
∂ L(t,x(t),CDα,ρx(t))−Dα,ρ ∂ L(t,x(t),CDα,ρx(t)) =0, ∀t∈[a,b].
2 a+ b− 3 a+
(cid:0) (cid:1)
So,
b
∂ L(t,x(t),CDα,ρx(t))ψ′(ρ)ρ dt=0.
Z 3 a+ 0
a
Taking ρ =1, we get the second necessary condition.
0
7
3.2 The Legendre condition
The Legendre condition is a second-order condition which an extremal must fulfill in order to be
a minimizer of the functional. We suggest [18] where a similar problem is solved for functionals
depending onthe Riemann–Liouville fractionalderivative. To simplify notation,we introduce the
following
∂2f
∂2f(x ,...,x ):= (x ,...,x ).
ij 1 n ∂x ∂x 1 n
i j
Theorem 7. Let x be a minimizer of the functional J as in (1), defined on the subspace U. If
∂2L exists and is continuous for i,j ∈{2,3}, then x satisfies
ij
∂2 L(t,x(t),CDα,ρx(t))≥0 (5)
33 a+
on [a,b].
Proof. Let x+ǫh be a variation of x, with h∈C1[a,b] such that h(a)=0=h(b). If we define
j(ǫ)=J(x+ǫh),
then we must have j′′(ǫ)≥0, that is
b
∂2 L(t,x(t),CDα,ρx(t))h2(t)+2∂2 L(t,x(t),CDα,ρx(t))h(t)CDα,ρh(t)
Z 22 a+ 23 a+ a+
a (cid:2)
+∂2 L(t,x(t),CDα,ρx(t))(CDα,ρh(t))2 dt≥0. (6)
33 a+ a+
(cid:3)
Assume that there exists some t ∈[a,b] for which
0
∂2 L(t ,x(t ),CDα,ρx(t ))<0.
33 0 0 a+ 0
Thus, there exists a subinterval[c,d]⊆[a,b]and three realconstants C ,C ,C with C <0 such
1 2 3 3
that
∂2 L(t,x(t),CDα,ρx(t))<C , ∂2 L(t,x(t),CDα,ρx(t))<C , ∂2 L(t,x(t),CDα,ρx(t))<C ,
22 a+ 1 23 a+ 2 33 a+ 3
(7)
for all t∈[c,d]. Define the function h:[c,d]→R by
α+4 α+10 4
h(t)=(α+2)(tρ−cρ)1+α−2 (tρ−cρ)2+α+ (tρ−cρ)3+α− (tρ−cρ)4+α.
dρ−cρ (dρ−cρ)2 (dρ−cρ)3
Then, h is of class C1, h(c)=0=h(d), h′(c)=0=h′(d) and CDα,ρh(c)=0=CDα,ρh(d). Also,
c+ c+
we have the following upper bounds:
h(t)≤28(dρ−cρ)1+α and CDα,ρh(t)≤C(dρ−cρ), C =50ραΓ(2+α),
c+
for all t∈[c,d]. Thus, the function h:[a,b]→R defined by
h(t) if t∈[c,d]
h(t)=
(cid:26) 0 otherwise
is of class C1, h(a)=0=h(b) and its fractional derivative
CDα,ρh(t) if t∈[c,d]
CDα,ρh(t)= c+
a+ (cid:26) 0 otherwise
is continuous. Inserting this variation h into the second order condition (6), and using relations
(7), we obtain
b
∂2 L(t,x(t),CDα,ρx(t))h2(t)+2∂2 L(t,x(t),CDα,ρx(t))h(t)CDα,ρh(t)
Z 22 a+ 23 a+ a+
a (cid:2)
8
+∂2 L(t,x(t),CDα,ρx(t))(CDα,ρh(t))2 dt
33 a+ a+
(cid:3)
d
= ∂2 L(t,x(t),CDα,ρx(t))h2(t)+2∂2 L(t,x(t),CDα,ρx(t))h(t)CDα,ρh(t)
Z 22 a+ 23 a+ a+
c (cid:2)
+∂2 L(t,x(t),CDα,ρx(t))(CDα,ρh(t))2 dt
33 a+ a+
(cid:3)
d
≤ C 282(dρ−cρ)2+2α+C 56C(dρ−cρ)2+α+C C2(dρ−cρ)2 dt
1 2 3
Z
c (cid:2) (cid:3)
=(dρ−cρ)2(d−c) C 282(dρ−cρ)2α+C 56C(dρ−cρ)α+C C2 ,
1 2 3
(cid:2) (cid:3)
which is negative if dρ−cρ is chosen arbitrarily small, and thus we obtain a contradiction.
3.3 The isoperimetric problem
The isoperimetric problem is an old question, and was considered first by the ancient Greeks. It
is relatedto finding a closedplane curvewitha fixedperimeterl whichenclosesthe greatestarea,
that is, if we consider the place curve with parametric equations (x(t),y(t)), t ∈ [a,b], then we
wish to maximize the functional
1 b
J(x,y)= (x(t)y′(t)−x′(t)y(t))dt,
2Z
a
under the boundary restrictions
x(a)=x(b) and y(a)=y(b),
and the isoperimetric constraint
b
(x′)2(t)+(y′)2(t)dt=l.
Z
a p
Only in 1841, a rigorous proof that the solution is a circle was obtained by Steiner.
Nowadays,anyvariationalproblemthatinvolvesanintegralconstraintiscalledanisoperimetric
problem. For the following, let l ∈ R be fixed, g : [a,b]×R2 → R be continuously differentiable
with respect to the second and third arguments such that, for any function x∈C1[a,b], the map
t 7→ Dα,ρ(∂ g(t,x(t),CDα,ρx(t))) is continuous. The integral constraint that we will consider is
b− 3 a+
the following:
b
I(x)= g(t,x(t),CDα,ρx(t))dt =l. (8)
Z a+
a
Theorem 8. Let x be a minimizer of the functional J as in (1), defined on the subspace
U = x∈C1[a,b] : x(a)=x and x(b)=x ,
a b
(cid:8) (cid:9)
subject to the additional restriction (8). If x is not an extremal of I, then there exists a real λ
suchthat, defining the functionK :[a,b]×R2→R byK =L+λg,xis asolutionofthe equation
∂ K(t,x(t),CDα,ρx(t))−Dα,ρ ∂ K(t,x(t),CDα,ρx(t)) =0 (9)
2 a+ b− 3 a+
(cid:0) (cid:1)
on [a,b].
Proof. Consideravariationofxwithtwoparametersx+ǫ h +ǫ h ,with|ǫ |≪1andh ∈C1[a,b]
1 1 2 2 i i
satisfying h (a)=0=h (b), for i=1,2. Define the functions i and j with two parameters(ǫ ,ǫ )
i i 1 2
in a neighborhood of zero as
i(ǫ ,ǫ )=I(x+ǫ h +ǫ h )−l,
1 2 1 1 2 2
9
and
j(ǫ ,ǫ )=L(x+ǫ h +ǫ h ).
1 2 1 1 2 2
Using Theorem 1, we get
∂i b
(0,0)= ∂ g(t,x(t),CDα,ρx(t))−Dα,ρ ∂ g(t,x(t),CDα,ρx(t)) h (t)dt
∂ǫ Z 2 a+ b− 3 a+ 2
2 a (cid:2) (cid:0) (cid:1)(cid:3)
+ h (t)I1−α,ρ ∂ g(t,x(t),CDα,ρx(t)) t=b =0.
2 b− 3 a+
h (cid:0) (cid:1)it=a
Since h (a)=0=h (b) and x is not an extremal of I, there exists some function h such that
2 2 2
∂i
(0,0)6=0.
∂ǫ
2
Thus, by the Implicit Function Theorem, there exists an unique function ǫ (·) defined in a neigh-
2
borhood of zero such that i(ǫ ,ǫ (ǫ )) = 0, that is, there exists a subfamily of variations that
1 2 1
satisfy the isoperimetric constraint (8).
On the other hand, (0,0) is a minimizer of j, under the restriction i(·,·) = 0, and we just
proved that ∇i(0,0) 6= 0. Appealing to the Lagrange Multiplier Rule, there exists a real λ such
that ∇(j+λi)(0,0)=0. Differentiating the map
ǫ 7→j(ǫ ,ǫ )+λi(ǫ ,ǫ ),
1 1 2 1 2
and putting (ǫ ,ǫ )=(0,0), we get
1 2
b
∂ K(t,x(t),CDα,ρx(t))−Dα,ρ ∂ K(t,x(t),CDα,ρx(t)) h (t)dt
Z 2 a+ b− 3 a+ 1
a (cid:2) (cid:0) (cid:1)(cid:3)
+ h (t)I1−α,ρ ∂ K(t,x(t),CDα,ρx(t)) t=b =0.
1 b− 3 a+
h (cid:0) (cid:1)it=a
Using the boundary conditions h (a)=0=h (b), we prove the desired.
1 1
Remark 2. WecanincludethecasewherexisanextremalofI. Inthis case,weapplythegeneral
formof the LagrangeMultiplier Rule, that is, there exist two realsλ andλ, not both zeros,such
0
that if we define the function K : [a,b]×R2 → R by K = λ L+λg, x is a solution of the
0 0 0
equation
∂ K (t,x(t),CDα,ρx(t))−Dα,ρ ∂ K (t,x(t),CDα,ρx(t)) =0
2 0 a+ b− 3 0 a+
(cid:0) (cid:1)
on [a,b].
Theorem 9. Suppose that the functions L and g as in (1) and (8) are convex in [a,b]×R2, and
let λ ≥ 0 be a real. Define the function K = L+λg. Then, each solution x of the fractional
Euler–Lagrangeequation (9) minimizes J in U, subject to the integral constraint (8).
Proof. First, observe that the function K is convex. So, by Theorem 3, we conclude that x
minimizes K, that is, for all variations x+ǫh, we have
b b
L(t,x(t)+ǫh(t),CDα,ρx(t)+ǫCDα,ρh(t))dt+ λg(t,x(t)+ǫh(t),CDα,ρx(t)+ǫCDα,ρh(h))dt
Z a+ a+ Z a+ a+
a a
b b
≥ L(t,x(t),CDα,ρx(t))dt+ λg(t,x(t),CDα,ρx(t))dt.
Z a+ Z a+
a a
Using the integral constraint, we obtain
b b
L(t,x(t)+ǫh(t),CDα,ρx(t)+ǫCDα,ρh(t))dt+l≥ L(t,x(t),CDα,ρx(t))dt+l,
Z a+ a+ Z a+
a a
10