Optimality Conditions for Fractional Variational Problems with Dependence on a Combined Caputo Derivative of Variable Order

0.15 MB·English

by  Dina Tavares

This is a preprint of a paper whose final and definite form will be published in
Optimization, ISSN 0233-1934(Print), 1029-4945(Online).
Submitted 05-Feb-2014;revised 16-May, 14-July and 16-Dec-2014;accepted 08-Jan-2015.
RESEARCH ARTICLE
Optimality Conditions for Fractional Variational Problems with
Dependence on a Combined Caputo Derivative of Variable Order
ab b∗ b
Dina Tavares , Ricardo Almeida and Delfim F. M. Torres
5
1 a b
Polytechnic Institute of Leiria, 2410–272 Leiria, Portugal; Center for Research and
0
2 Development in Mathematics and Applications (CIDMA), Department of Mathematics,
University of Aveiro, 3810–193 Aveiro, Portugal
n
a
J (Submitted Feb 5, 2014; Revised May 16, July 14 and Dec 16, 2014; Accepted Jan 8, 2015)

9
Weestablishnecessaryoptimalityconditions forvariational problemswithaLagrangiande-

] pending on a combined Caputo derivative of variable fractional order. The endpoint of the
C integral is free, and thus transversality conditions are proved. Several particular cases are
O consideredillustratingthenewresults.
h. Keywords:dynamicoptimization; fractionalcalculus;variablefractionalorder;fractional
calculusofvariations.
t
a
m AMS Subject Classification:26A33; 34A08;49K05.
[

1 1. Introduction
v
2
8 The fractional calculus of variations deals with optimization of functionals that
0 depend on some fractional operator [1, 2]. This is a fast growing subject, and
2
different approaches have been developed by considering different types of La-
0
grangians, e.g., depending on Riemann–Liouville or Caputo fractional derivatives,
.
1
fractional integrals, and mixed integer-fractional order operators [3–9]. The com-
0
5 mon procedure to address such fractional variational problems consists in solving
1 a fractional differential equation, called the Euler–Lagrange equation, which every
:
v minimizer/maximizer of thefunctionalmustsatisfy.With thehelpof theboundary
i conditions imposed on the problem at the initial time t = a and at the terminal
X
time t = b, one solves, often with the help of some numerical procedure, the frac-
r
a tional differential equation and obtain the possiblesolution to theproblem [10–15].
Whenoneof theboundaryconditions ismissing,thatis,whenthesetof admissible
functionsmay take any value atoneof the boundaries,then anauxiliary condition,
knownasatransversality condition, needstobeobtained inordertosolve thefrac-
tional equation [16]. In this paper we do not only assume that x(b) is free, but the
endpoint b is also variable. Thus, we are not only interested in finding an optimal
curvex(·)butalsotheendpointofthevariationalintegral, denotedinthesequelby
T. This is known in the literature as a free-time problem, and is already well stud-
ied for integer-order derivatives (see, e.g., [17]). Moreover, we consider a combined
Caputo fractional derivative, that involves the left and the right Caputo fractional
derivative [18–20]. Also, the order of the fractional operator is not a constant, and
∗Correspondingauthor.Email:[email protected]
1
may depend on time [21–23]. Very useful physical applications have given birth to
the variable order fractional calculus, for example in modeling mechanical behav-
iors [24–26]. Nowadays, variable order fractional calculus is particularly recognized
as ausefuland promisingapproach in the modelling of diffusion processes,in order
tocharacterize time-dependentorconcentration-dependentanomalousdiffusion,or
diffusion processes in inhomogeneous porous media [25]. The importance of such
variable order operators is due to the fact that the behaviors of dynamical systems
often change with their own evolution [26]. For numerical modeling of time frac-
tional diffusion equations we refer the reader to [24]. Our variational problem may
describe better some nonconservative physical phenomena, since fractional opera-
tors are nonlocal and contain memory of the process [27–29]. We believe that it is
reasonablethattheorderofthefractional operatorsisinfluencedalsobytheacting
dynamics, which varies with time, and we trust that the results now obtained can
bemoreeffective toprovideamathematical frameworktosomecomplexdynamical
problems [30–34].
The paper is organized in the following way. In Section 2 we introduce the basic
concepts of the combined variable-order fractional calculus, and we recall two for-
mulas of fractional integration by parts (Theorem 2.1). The problem is then stated
in Section 3, consisting of the variational functional
T
J(x,T) = L t,x(t),CDα(·,·),β(·,·)x(t) dt+φ(T,x(T)),
γ
Za (cid:16) (cid:17)
where CDα(·,·),β(·,·)x(t) stands for the combined Caputo fractional derivative of
γ
variable fractional order (Definition 1), subject to the boundary condition x(a) =
x . Our goal is to findnecessary optimality conditions that every extremizer (x,T)
a
mustsatisfy.Themainresultsof thepaperprovidenecessary optimality conditions
of Euler–Lagrange type, described by fractional differential equations of variable
order, and different transversality optimality conditions (see Theorems 3.1 and
3.2). Some particular cases of interest are considered in Section 4. We end with
two illustrative examples (Section 5).
2. Fractional calculus of variable order
In this section we present the fundamental notions of the fractional calculus of
variable order. We consider the order of the derivative and of the integral to be a
continuous function α(·,·) with domain [a,b]2, taking values on the open interval
(0,1). Following [22], given a function x :[a,b] → R we define:
• the left and right Riemann–Liouville fractional integrals of order α(·,·) by
t 1
Iα(·,·)x(t) = (t−τ)α(t,τ)−1x(τ)dτ
a t Γ(α(t,τ))
Za
and
b 1
Iα(·,·)x(t) = (τ −t)α(τ,t)−1x(τ)dτ,
t b Γ(α(τ,t))
Zt
respectively;
2
• the left and right Riemann–Liouville fractional derivatives of order α(·,·) by
d t 1
Dα(·,·)x(t) = (t−τ)−α(t,τ)x(τ)dτ
a t dt Γ(1−α(t,τ))
Za
and
d b −1
Dα(·,·)x(t) = (τ −t)−α(τ,t)x(τ)dτ,
t b dt Γ(1−α(τ,t))
Zt
respectively;
• the left and right Caputo fractional derivatives of order α(·,·) by
t 1
CDα(·,·)x(t) = (t−τ)−α(t,τ)x(1)(τ)dτ
a t Γ(1−α(t,τ))
Za
and
b −1
CDα(·,·)x(t) = (τ −t)−α(τ,t)x(1)(τ)dτ,
t b Γ(1−α(τ,t))
Zt
respectively.
Motivated by thecombined fractional Caputoand Riemann–Liouville definitions
[18–20], we propose the following definitions.
Definition 1 Let α, β : [a,b]2 → (0,1) and γ = (γ ,γ ) ∈ [0,1]2. The combined
1 2
α(·,·),β(·,·)
Riemann–Liouville fractional derivative operator D is defined by
γ
Dα(·,·),β(·,·) = γ Dα(·,·)+γ Dβ(·,·),
γ 1a t 2t b
acting on x ∈ C([a,b]) in the following way:
Dα(·,·),β(·,·)x(t) = γ Dα(·,·)x(t)+γ Dβ(·,·)x(t).
γ 1a t 2t b
Analogously, the combined Caputo fractional derivative operator, denoted by
CDα(·,·),β(·,·), is defined by
γ
CDα(·,·),β(·,·) = γ CDα(·,·)+γ CDβ(·,·),
γ 1a t 2t b
acting on x ∈ C1([a,b]) as expected:
CDα(·,·),β(·,·)x(t) = γ CDα(·,·)x(t)+γ CDβ(·,·)x(t).
γ 1a t 2t b
The following theorem is proved in [35] and is a generalization of the standard
fractional formula of integration by parts for a constant α (see, e.g., formula (2) of
[16]).
3
Theorem 2.1 (Theorem 3.2 of [35]) If x,y ∈ C1[a,b], then the fractional integra-
tion by parts formulas
b b t=b
y(t)CDα(·,·)x(t)dt = x(t) Dα(·,·)y(t)dt+ x(t) I1−α(·,·)y(t)
a t t b t b
Za Za h it=a
and
b b t=b
y(t)CDα(·,·)x(t)dt = x(t) Dα(·,·)y(t)dt− x(t) I1−α(·,·)y(t)
t b a t a t
Za Za h it=a
hold.
3. Necessary optimality conditions
Consider the norm defined on the linear space C1([a,b])×R by
k(x,t)k := max |x(t)|+ max CDα(·,·),β(·,·)x(t) +|t|.
γ
a≤t≤b a≤t≤b
(cid:12) (cid:12)
(cid:12) (cid:12)
Let D denote the subsetC1([a,b])×[a,b] end(cid:12)owed with theno(cid:12)rm k(·,·)k, such that
CDα(·,·),β(·,·)x(t) exists and is continuous on the interval [a,b].
γ
Definition 2 We say that (x⋆,T⋆) ∈ D is a local minimizer to the functional J :
D → R if there exists some ǫ > 0 such that
∀(x,T)∈ D : k(x⋆,T⋆)−(x,T)k < ǫ ⇒ J(x⋆,T⋆) ≤ J(x,T).
Along the work, we denote by ∂ z, i ∈ {1,2,3}, the partial derivative of a func-
i
tion z : R3 → R with respect to its ith argument, and by L the Lagrangian
L : C1 [a,b]×R2 → R. For simplicity of notation, we introduce the operator
α,β
[·] defined by
γ (cid:0) (cid:1)
[x]α,β(t) = t,x(t),CDα(·,·),β(·,·)x(t) .
γ γ
(cid:16) (cid:17)
Consider the following problem of the calculus of variations: find the local mini-
mizers of the functional J : D → R, with
T
J(x,T) = L[x]α,β(t)dt+φ(T,x(T)), (1)
γ
Za
over all (x,T) ∈D satisfying theboundarycondition x(a) = x ,for afixed x ∈ R.
a a
The terminal time T and terminal state x(T) are free. The terminal cost function
φ: [a,b]×R → R is at least of class C1.
The next theorem gives fractional necessary optimality conditions to problem
(1). In the sequel, we need the auxiliary notation of the dual fractional derivative:
β(·,·),α(·,·) β(·,·) α(·,·)
D = γ D +γ D , where γ = (γ ,γ ).
γ 2a t 1t T 2 1
4
Theorem 3.1 Suppose that (x,T) is a local minimizer to the functional (1) on D.
Then, (x,T) satisfies the fractional Euler–Lagrange equations
∂ L[x]α,β(t)+Dβ(·,·),α(·,·)∂ L[x]α,β(t) = 0, (2)
2 γ γ 3 γ
on the interval [a,T], and
γ Dβ(·,·)∂ L[x]α,β(t)− Dβ(·,·)∂ L[x]α,β(t) = 0, (3)
2 a t 3 γ T t 3 γ
(cid:16) (cid:17)
on the interval [T,b]. Moreover, (x,T) satisfies the transversality conditions
L[x]α,β(T)+∂ φ(T,x(T))+∂ φ(T,x(T))x′(T) =0,
γ 1 2
1−α(·,·) α,β 1−β(·,·) α,β
γ I ∂ L[x] (t)−γ I ∂ L[x] (t) +∂ φ(T,x(T)) = 0,
 1t T 3 γ 2T t 3 γ 2
t=T
γh I1−β(·,·)∂ L[x]α,β(t)− I1−β(·,·)∂ L[x]α,β(t) i = 0.
2 T t 3 γ a t 3 γ
t=b
 h i (4)


Proof. Let (x,T) bea solution to the problem and (x+ǫh,T +ǫ∆T)bean admis-
sible variation, where h ∈ C1([a,b]) is a perturbing curve, △T ∈ R represents an
arbitrarily chosen small change in T and ǫ ∈ R represents a small number (ǫ → 0).
The constraint x(a) = x implies that all admissible variations must fulfill the
a
condition h(a) = 0. Define j(·) on a neighborhood of zero by
j(ǫ) = J(x+ǫh,T +ǫ△T)
T+ǫ△T
= L[x+ǫh]α,β(t)dt+φ(T +ǫ△T,(x+ǫh)(T +ǫ△T)).
γ
Za
The derivative j′(ǫ) is
T+ǫ△T
j′(ǫ) = ∂ L[x+ǫh]α,β(t)h(t)+∂ L[x+ǫh]α,β(t)CDα(·,·),β(·,·)h(t) dt
2 γ 3 γ γ
Za (cid:16) (cid:17)
+L[x+ǫh]α,β(T +ǫ∆T)∆T +∂ φ(T +ǫ△T,(x+ǫh)(T +ǫ△T)) ∆T
γ 1
+∂ φ(T +ǫ△T,(x+ǫh)(T +ǫ△T)) (x+ǫh)′(T +ǫ△T).
2
Considering the differentiability properties of j, a necessary condition for (x,T) to
be a local extremizer is given by j′(ǫ)| =0, that is,
ǫ=0
T
∂ L[x]α,β(t)h(t)+∂ L[x]α,β(t)CDα(·,·),β(·,·)h(t) dt+L[x]α,β(T)∆T
2 γ 3 γ γ γ
Za (cid:16) (cid:17)
+∂ φ(T,x(T))∆T +∂ φ(T,x(T)) h(t)+x′(T)△T = 0. (5)
1 2
(cid:2) (cid:3)
The second addend of the integral function (5),
T
∂ L[x]α,β(t)CDα(·,·),β(·,·)h(t)dt, (6)
3 γ γ
Za
5
can be written, using the definition of combined Caputo fractional derivative, as
T
∂ L[x]α,β(t)CDα(·,·),β(·,·)h(t)dt
3 γ γ
Za
T
= ∂ L[x]α,β(t) γ CDα(·,·)h(t)+γ CDβ(·,·)h(t) dt
3 γ 1a t 2t b
Za h i
T
= γ ∂ L[x]α,β(t)CDα(·,·)h(t)dt
1 3 γ a t
Za
b b
+γ ∂ L[x]α,β(t)CDβ(·,·)h(t)dt− ∂ L[x]α,β(t)CDβ(·,·)h(t)dt .
2 3 γ t b 3 γ t b
(cid:20)Za ZT (cid:21)
Integrating by parts (see Theorem 2.1), and since h(a) = 0, the term (6) can be
written as
T
γ h(t) Dα(·,·)∂ L[x]α,β(t)dt+ h(t) I1−α(·,·)∂ L[x]α,β(t)
1 t T 3 γ t T 3 γ
(cid:20)Za h it=T(cid:21)
b
+γ h(t) Dβ(·,·)∂ L[x]α,β(t)dt− h(t) I1−β(·,·)∂ L[x]α,β(t)
2 a t 3 γ a t 3 γ
"Za h it=b
b
− h(t) Dβ(·,·)∂ L[x]α,β(t)dt− h(t) I1−β(·,·)∂ L[x]α,β(t)
T t 3 γ T t 3 γ
ZT h it=b
+ h(t) I1−β(·,·)∂ L[x]α,β(t) .
T t 3 γ
t=T!#
h i
β(·,·),α(·,·)
Unfolding these integrals, and considering the fractional operator D with
γ
γ = (γ ,γ ), then (6) is equivalent to
2 1
T
h(t)Dβ(·,·),α(·,·)∂ L[x]α,β(t)dt
γ 3 γ
Za
b
+ γ h(t) Dβ(·,·)∂ L[x]α,β(t)− Dβ(·,·)∂ L[x]α,β(t) dt
2 a t 3 γ T t 3 γ
ZT h i
+ h(t) γ I1−α(·,·)∂ L[x]α,β(t)−γ I1−β(·,·)∂ L[x]α,β(t)
1t T 3 γ 2T t 3 γ
t=T
h (cid:16) (cid:17)i
+ h(t)γ I1−β(·,·)∂ L[x]α,β(t)− I1−β(·,·)∂ L[x]α,β(t) .
2 T t 3 γ a t 3 γ
t=b
h (cid:16) (cid:17)i
6
Substituting these relations into equation (5), we obtain
T
0= h(t) ∂ L[x]α,β(t)+Dβ(·,·),α(·,·)∂ L[x]α,β(t) dt
2 γ γ 3 γ
Za h i
b
+ γ h(t) Dβ(·,·)∂ L[x]α,β(t)− Dβ(·,·)∂ L[x]α,β(t) dt
2 a t 3 γ T t 3 γ
ZT h i
+h(T) γ I1−α(·,·)∂ L[x]α,β(t)−γ I1−β(·,·)∂ L[x]α,β(t)+∂ φ(t,x(t))
1t T 3 γ 2T t 3 γ 2
t=T
h i
+∆T L[x]α,β(t)+∂ φ(t,x(t))+∂ φ(t,x(t))x′(t)
γ 1 2
t=T
h i
+h(b) γ I1−β(·,·)∂ L[x]α,β(t)− I1−β(·,·)∂ L[x]α,β(t) .
2 T t 3 γ a t 3 γ
t=b
h (cid:16) (cid:17)i (7)
As h and △T are arbitrary, we can choose △T = 0 and h(t) = 0, for all t ∈ [T,b],
buth is arbitrary in t ∈ [a,T). Then,for all t ∈ [a,T], we obtain the firstnecessary
condition (2):
∂ L[x]α,β(t)+Dβ(·,·),α(·,·)∂ L[x]α,β(t) = 0.
2 γ γ 3 γ
Analogously, considering △T = 0, h(t) = 0, for all t ∈ [a,T], and h arbitrary on
(T,b], we obtain the second necessary condition (3):
γ Dβ(·,·)∂ L[x]α,β(t)− Dβ(·,·)∂ L[x]α,β(t) = 0.
2 a t 3 γ T t 3 γ
(cid:16) (cid:17)
As (x,T) is a solution to the necessary conditions (2) and (3), then equation (7)
takes the form
0 =h(T) γ I1−α(·,·)∂ L[x]α,β(t)−γ I1−β(·,·)∂ L[x]α,β(t)+∂ φ(t,x(t))
1t T 3 γ 2T t 3 γ 2
t=T
h i
+∆T L[x]α,β(t)+∂ φ(t,x(t))+∂ φ(t,x(t))x′(t)
γ 1 2
t=T
h i
+h(b) γ I1−β(·,·)∂ L[x]α,β(t)− I1−β(·,·)∂ L[x](t) .
2 T t 3 γ a t 3
t=b
h (cid:16) (cid:17)i (8)
Transversality conditions(4)areobtainedforappropriatechoices ofvariations.
In the next theorem, considering the same problem (1), we rewrite the transver-
sality conditions (4) in terms of the increment on time ∆T and on the consequent
increment on x and ∆x , given by
T
∆x = (x+h)(T +∆T)−x(T). (9)
T
Theorem 3.2 Let (x,T) be a local minimizer to the functional (1) on D. Then,
the fractional Euler–Lagrange equations (2) and (3) are satisfied together with the
7
following transversality conditions:
α,β
L[x] (T)+∂ φ(T,x(T))
γ 1
+x′(T) γ I1−β(·,·)∂ L[x]α,β(t)−γ I1−α(·,·)∂ L[x]α,β(t) = 0,
 2T t 3 γ 1t T 3 γ
t=T
 γ1tIT1−α(·,·)∂h3L[x]αγ,β(t)−γ2TIt1−β(·,·)∂3L[x]αγ,β(t) +∂2φ(iT,x(T)) = 0,
 t=T
γh I1−β(·,·)∂ L[x]α,β(t)− I1−β(·,·)∂ L[x]α,β(t) i = 0.
2 T t 3 γ a t 3 γ
 t=b
 h i (10)


Proof. The Euler–Lagrange equations are deduced following similar arguments as
the ones presented in Theorem 3.1. We now focus our attention on the proof of
the transversality conditions. Using Taylor’s expansion up to first order for a small
∆T, and restricting the set of variations to those for which h′(T)= 0, we obtain
(x+h)(T +∆T)= (x+h)(T)+x′(T)∆T +O(∆T)2.
Rearranging the relation (9) allows us to express h(T) in terms of ∆T and ∆x :
T
h(T) = ∆x −x′(T)∆T +O(∆T)2.
T
Substitution of this expression into (8) gives us
0 =∆x γ I1−α(·,·)∂ L[x]α,β(t)−γ I1−β(·,·)∂ L[x]α,β(t)+∂ φ(t,x(t))
T 1t T 3 γ 2T t 3 γ 2
t=T
h i
+∆T L[x]α,β(t)+∂ φ(t,x(t))
γ 1
h
−x′(t) γ I1−α(·,·)∂ L[x]α,β(t)−γ I1−β(·,·)∂ L[x]α,β(t)
1t T 3 γ 2T t 3 γ
t=T
(cid:16) (cid:17)i
+h(b) γ I1−β(·,·)∂ L[x]α,β(t)− I1−β(·,·)∂ L[x]α,β(t) +O(∆T)2.
2 T t 3 γ a t 3 γ
t=b
h (cid:16) (cid:17)i
Transversality conditions (10) are obtained using appropriate choices of variations.
4. Particular cases
Now, we specify our results to three particular cases of variable terminal points.
4.1. Vertical terminal line
This case involves a fixed upper bound T. Thus, ∆T = 0 and, consequently, the
second term in (8) drops out. Since ∆x is arbitrary, we obtain the following
T
transversality conditions: if T < b, then
1−α(·,·) α,β 1−β(·,·) α,β
γ I ∂ L[x] (t)−γ I ∂ L[x] (t) +∂ φ(T,x(T)) = 0,
1t T 3 γ 2T t 3 γ 2
t=T
γh I1−β(·,·)∂ L[x]α,β(t)− I1−β(·,·)∂ L[x]α,β(t) i = 0;
 2 T t 3 γ a t 3 γ t=b
h i

8
if T = b, then ∆x = h(b) and the transversality conditions reduce to
T
γ I1−α(·,·)∂ L[x]α,β(t)−γ I1−β(·,·)∂ L[x]α,β(t) +∂ φ(b,x(b)) = 0.
1t T 3 γ 2t T 3 γ 2
t=b
h i
4.2. Horizontal terminal line
In this situation, we have ∆x = 0 but ∆T is arbitrary. Thus, the transversality
T
conditions are
α,β
L[x] (T)+∂ φ(T,x(T))
γ 1
+x′(T) γ I1−β(·,·)∂ L[x]α,β(t)−γ I1−α(·,·)∂ L[x]α,β(t) = 0,
 2T t 3 γ 1t T 3 γ
t=T
γ I1−β(·,·)h∂ L[x]α,β(t)− I1−β(·,·)∂ L[x]α,β(t) = 0. i
2 T t 3 γ a t 3 γ
t=b
 h i


4.3. Terminal curve
Now the terminal point is described by a given curve ψ : C1([a,b]) → R, in the
sense that x(T)= ψ(T). From Taylor’s formula, for a small arbitrary ∆T, one has
∆x(T)= ψ′(T)∆T +O(∆T)2.
Hence, the transversality conditions are presented in the form
L[x]α,β(T)+∂ φ(T,x(T))+∂ φ(T,x(T))ψ′(T)
γ 1 2
+(x′(T)−ψ′(T)) γ I1−β(·,·)∂ L[x]α,β(t)−γ I1−α(·,·)∂ L[x]α,β(t) = 0,
 2T t 3 γ 1t T 3 γ
t=T
γ I1−β(·,·)∂ L[x]αh,β(t)− I1−β(·,·)∂ L[x]α,β(t) = 0. i
2 T t 3 γ a t 3 γ
t=b
 h i


5. Examples
Inthis section weshowtwoexamples forthemainresultofthepaper.Letα(t,τ) =
α(t) and β(t,τ) = β(τ) be two functions depending on a variable t and τ only,
respectively. Consider the following fractional variational problem: to minimize the
functional
T t1−α(t) (10−t)1−β(t) 2
J(x,T) = 2α(t)−1+ CDα(·),β(·)x(t)− − dt
γ 2Γ(2−α(t)) 2Γ(2−β(t))
Z0 " (cid:18) (cid:19) #
for t ∈ [0,10], subject to the initial condition x(0) = 0 and where γ = (γ ,γ ) =
1 2
(1/2,1/2). Simple computations show that for x(t) = t, with t ∈ [0,10], we have
t1−α(t) (10−t)1−β(t)
CDα(·,·),β(·,·)x(t)= + .
γ 2Γ(2−α(t)) 2Γ(2−β(t))
For x(t) = t the functional reduces to
T
J(x,T)= (2α(t)−1)dt.
Z0
9
InordertodeterminetheoptimaltimeT,wehave tosolve theequation2α(T) = 1.
For example, let α(t) = t2/2. In this case, since J(x,T) ≥ −2/3 for all pairs (x,T)
andJ(x,1) = −2/3,weconcludethatthe(global)minimumvalueofthefunctional
is −2/3, obtained for x and T = 1. It is obvious that the two Euler–Lagrange
equations (2) and (3) are satisfied when x = x, since
∂ L[x]α,β(t)= 0 for all t ∈ [0,10].
3 γ
Using this relation, together with
L[x]α,β(1) = 0,
γ
the transversality conditions (4) are also verified.
For our last example, consider the functional
T t1−α(t) (10−t)1−β(t) 3
J(x,T) = 2α(t)−1+ CDα(·),β(·)x(t)− − dt,
γ 2Γ(2−α(t)) 2Γ(2−β(t))
Z0 " (cid:18) (cid:19) #
where the remaining assumptions and conditions are as in the previous example.
For this case, x(t) = t and T = 1 still satisfy the necessary optimality conditions.
However, it is now not obvious that (x,1) is a local minimizer to the problem.
Acknowledgments
This work is part of first author’s Ph.D., which is carried out at the University
of Aveiro under the Doctoral Programme Mathematics and Applications of Uni-
versities of Aveiro and Minho. It was supported by Portuguese funds through the
Center for Research and Development in Mathematics and Applications (CIDMA),
andThe Portuguese Foundation for Science and Technology (FCT), withinproject
PEst-OE/MAT/UI4106/2014. Tavares was also supported by FCT through the
Ph.D. fellowship SFRH/BD/42557/2007; Torres by EU funding under the 7th
FrameworkProgrammeFP7-PEOPLE-2010-ITN,grantagreementnumber264735-
SADCO, and by project PTDC/EEI-AUT/1450/2012, co-financed by FEDER un-
der POFC-QREN with COMPETE reference FCOMP-01-0124-FEDER-028894.
The authors are grateful to three referees for valuable remarks and comments,
which significantly contributed to the quality of the paper.
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