Table Of ContentFractional Variational Calculus with Classical
1 and Combined Caputo Derivatives
∗
1
0
2 Tatiana Odzijewicz1 Agnieszka B. Malinowska2
n [email protected] [email protected]
a
J Delfim F. M. Torres1
5
[email protected]
1
] 1Department of Mathematics
C
University of Aveiro
O
3810-193 Aveiro, Portugal
.
h
t 2Faculty of Computer Science
a
m Bial ystok University of Technology
15-351 Bial ystok, Poland
[
1
v
2
Abstract
3
9 Wegiveaproperfractionalextensionoftheclassicalcalculusofvaria-
2 tions byconsidering variational functionals with aLagrangian depending
1. on a combined Caputo fractional derivative and the classical derivative.
0 Euler–Lagrange equations to the basic and isoperimetric problems are
1 proved,as well as transversality conditions.
1
: Keywords: fractionalderivatives;fractionalvariationalanalysis;isoperi-
v
i metricproblems;naturalboundaryconditions;Euler–Lagrangeequations.
X
r 2010 Mathematics Subject Classification: 49K05; 49K21; 26A33;
a
34A08.
1 Introduction
Fractional calculus (FC) is a generalization of (integer) differential calculus, in
the sense it deals with derivatives of real or complex order. FC was born on
30th September 1695. On that day, L’Hoˆpital wrote a letter to Leibniz, where
∗Part of the first author’s Ph.D., which is carried out at the University of Aveiro under
theDoctoralProgrammeMathematicsandApplications ofUniversitiesofAveiroandMinho.
Submitted 30-Nov-2010; accepted 14-Jan-2011; for publication in Nonlinear Analysis Series
A: Theory, Methods & Applications.
1
he asked about Leibniz’s notation of nth order derivatives of a linear function.
L’Hoˆpitalwantedtoknowtheresultforthederivativeofordern=1/2. Leibniz
repliedthat“oneday, usefulconsequences will be drawn”and,infact,hisvision
became a reality. The study of non-integer order derivatives rapidly became a
very attractive subject to mathematicians, and many different forms of frac-
tional (i.e., non-integer) derivative operators were introduced: the Grunwald–
Letnikow,Riemann–Liouville, Caputo [15, 17, 26], and the more recent notions
of Cresson [10], Jumarie [16], or Klimek [18].
The calculus of variations with fractional derivatives was born in 1996 with
the work of Riewe, to better describe nonconservative systems in mechanics
[27, 28]. It is a subject of strong current research due to its many applications
in science and engineering, including mechanics, chemistry, biology, economics,
andcontroltheory(see,e.g.,therecentpapers[2,5,6,7,8,13,19,21,23,24]).1
Following[25],weconsiderherethatthehighestderivativeintheLagrangian
is of integer order. The main advantage of our formulation, with respect to
the “pure” fractional approach adopted in the literature, is that the classical
results of variational calculus can now be obtained as particular cases. We
recall that the only possibility of obtaining the classical derivative y′ from a
fractional derivative Dαy, α (0,1), is to take the limit when α tends to one.
∈
However,ingeneralsuchalimitdoesnotexist[29]. Differentlyfrom[25],where
the fractional problems are considered in the sense of Riemann–Liouville, we
considerherecombinedCaputoderivativesCDα,β. TheoperatorCDα,β extends
γ γ
theCaputofractionalderivatives,andwasintroducedforthefirsttimein[22]as
ausefultoolinthedescriptionofsomenonconservativemodelsandmoregeneral
classes of variationalproblems. More precisely, we investigate here problems of
thecalculusofvariationswithintegrandsdependingontheindependentvariable
x,anunknownvector-functiony,itsintegerorderderivativey′,andafractional
derivative CDα,βy given as a convex combination of the left Caputo fractional
γ
derivative of order α and the right Caputo fractional derivative of order β.
The paper is organized as follows. Section 2 presents the basic definitions
and facts needed in the sequel. Our results are then stated and proved in
Section 3. We discuss the fundamental concepts of a variational calculus such
as the Euler–Lagrange equations for the basic (Theorem 13) and isoperimetric
(Theorem 18) problems, as well as transversality conditions (Theorem 15). We
end with an illustrative example of the results of the paper (Section 4).
2 Preliminaries on Fractional Calculus
In this section we present some basic necessary facts on fractional calculus.
For more on the subject and applications, we refer the reader to the books
[15, 17, 26].
1Theliteratureonfractionalvariationalcalculus isvast,andwedonottrytoprovidehere
a comprehensive review on the subject. We give only some representative references from
2010and2011. Otherreferencescanbefoundtherein.
2
Definition 1 (Riemann–Liouville fractional integrals). Let f L1([a,b]) and
∈
0 < α < 1. The left Riemann–Liouville Fractional Integral (RLFI) of order α
of a function f is defined by
1 x
Iαf(x):= (x t)α−1f(t)dt,
a x Γ(α) −
Za
and the right RLFI by
1 b
Iαf(x):= (t x)α−1f(t)dt,
x b Γ(α) −
Zx
for all x [a,b].
∈
Definition 2 (Left and right Riemann–Liouville fractional derivatives). The
left Riemann–Liouville Fractional Derivative (RLFD) of order α of a function
f, denoted by Dαf, is defined by
a x
d 1 d x
Dαf(x):= I1−αf(x)= (x t)−αf(t)dt,
a x dxa x Γ(1 α)dx −
− Za
x [a,b]. Similarly, the right RLFD of order α of a function f, denoted by
D∈αf, is defined by
x b
d 1 d b
Dαf(x):= I1−αf(x)= − (t x)−αf(t)dt,
x b −dxx b Γ(1 α)dx −
− Zx
x [a,b].
∈
Definition3(Caputofractionalderivatives). Letf AC([a,b]),whereAC([a,b])
∈
representsthespaceofabsolutelycontinuousfunctionsontheinterval[a,b]. The
left Caputo Fractional Derivative (CFD) is defined by
d 1 x d
CDαf(t):= I1−α f (x)= (x t)−α f(t)dt,
a x a x dt Γ(1 α) − dt
(cid:18) (cid:19) − Za
x [a,b], and the right CFD by
∈
d 1 b d
Dαf(x):= I1−α f (x)= − (t x)−α f(t)dt,
x b x b −dt Γ(1 α) − dt
(cid:18) (cid:19) − Zx
x [a,b], where α is the order of the derivative.
∈
Theorem 4 (Fractional integration by parts [17]). Let p 1, q 1, and
1 + 1 1+α. If g L ([a,b]) and f L ([a,b]), then the≥following≥formula
p q ≤ ∈ p ∈ q
for integration by parts holds:
b b
g(x) Iαf(x)dx= f(x) Iαg(x)dx.
a x x b
Z Z
a a
3
Definition5 (ThecombinedfractionalderivativeCDα,β [22]). Letα,β (0,1)
γ ∈
and γ [0,1]. The combined fractional derivative operator CDα,β is given by
∈ γ
CDα,β :=γCDα+(1 γ)CDβ.
γ a x − x b
Remark 6. The combined fractional derivative coincides with theright CFD in
the case γ =0, i.e., CDα,βf(x)=C Dαf(x). For γ = 1 one gets the left CFD:
0 x b
CDα,βf(x)=C Dαf(x).
1 a x
For f=[f1,...,fN]:[a,b] RN, N N, and fi AC([a,b]), i=1,...,N,
→ ∈ ∈
we put
CDγα,βf(x):= CDγα,βf1(x),...,CDγα,βfN(x) .
In the discussion to follow,(cid:2)we also need the following form(cid:3) ula for fractional
integrations by parts [22]:
b b
g(x)CDα,βf(x)dx= f(x)Dβ,αg(x)dx
γ 1−γ
Z Z
a a
x=b
+ γf(x) I1−αg(x) (1 γ)f(x) I1−βg(x) , (1)
x b − − a x x=a
h i
where Dβ,α :=(1 γ) Dβ +γ Dα.
1−γ − a x x b
3 Main Results
Consider the following functional:
b
(y)= L x,y(x),y′(x),CDα,βy(x) dx, (2)
J γ
Z
a (cid:0) (cid:1)
wherex [a,b]is the independentvariable; y(x) RN is arealvectorvariable;
y′(x) R∈N with y′ the first derivative of y; CD∈α,βy(x) RN stands for the
∈ γ ∈
combinedfractionalderivativeofyevaluatedinx;andL C1 [a,b] R3N;R .
Let D denote the set of all functions y : [a,b] RN suc∈h that y′ an×d CDα,βy
→ (cid:0) γ (cid:1)
exist and are continuous on the interval [a,b]. We endow D with the norm
y := max y(x) + max y′(x) + max CDα,βy(x) ,
k k1,∞ a≤x≤bk k a≤x≤bk k a≤x≤b γ
(cid:13) (cid:13)
where is a norm in RN. Along the work, we denote(cid:13)by ∂ K, i =(cid:13)1,...,M
i
(M Nk·)k, the partial derivative of a function K : RM R with respect to its
itha∈rgument. Letλ Rr. Forsimplicityofnotationwe→introducetheoperators
α,β α,β ∈
[] and defined by
·γ λ{·}γ
[y]α,β(x):= x,y(x),y′(x),CDα,βy(x) ,
γ γ
λ{y}αγ,β(x):= x,y(cid:0)(x),y′(x),CDγα,βy(x),λ1,(cid:1)...,λr .
(cid:0) (cid:1)
4
3.1 The Euler–Lagrange equation
We begin with the following problem of the fractional calculus of variations.
Problem 1. Find a function y D for which the functional (2), i.e.,
∈
b
α,β
(y)= L[y] (x)dx, (3)
J γ
Z
a
subject to given boundary conditions
y(a)=ya, y(b)=yb, (4)
ya,yb RN, achieves a minimum.
∈
Definition 7 (Admissible function). A function y D that satisfies all the
∈
constraints of a problem is said to be admissible to that problem. The set of
admissible functions is denoted by .
D
Remark 8. For Problem 1 the constraints mentioned in Definition 7 are the
boundary conditions (4).
We now define what is meant by minimum of on .
J D
Definition 9 (Local minimizer). A function y is said to be a local mini-
∈ D
mizer to on if there exists some δ >0 such that
J D
(y) (y) 0
J −J ≤
for all functions y with y y <δ.
∈D k − k1,∞
Similarly to the classical calculus of variations, a necessary optimality con-
dition to Problem 1 is based on the concept of variation.
Definition 10 (First variation). The first variation of at y D in the
J ∈
direction h D is defined by
∈
(y+ǫh) (y) ∂
δ (y;h):= lim J −J = J(y+ǫh) ,
J ǫ→0 ǫ ∂ǫ
(cid:12)ǫ=0
(cid:12)
provided the limit exists. (cid:12)
(cid:12)
Definition 11 (Admissible variation). An admissible variation at y for
∈D J
is a direction h D, h=0, such that
∈ 6
δ (y;h) exists; and
• J
y+ǫh for all sufficiently small ǫ.
• ∈D
Theorem 12 (see, e.g., [30]). Let be a functional defined on . Suppose
J D
that y is a local minimizer to on . Then δ (y;h)= 0 for each admissible
J D J
variation h at y.
5
We now state and prove the Euler–Lagrangeequations for Problem 1.
Theorem 13. If y = (y1,...,yN) is a local minimizer to Problem 1, then y
satisfies the system of N Euler–Lagrange equations
d
∂iL[y]αγ,β(x)− dx∂N+iL[y]αγ,β(x)+D1β−,αγ∂2N+iL[y]αγ,β(x)=0, (5)
i=2,...,N +1, for all x [a,b].
∈
Proof. Suppose that y is a solution to Problem 1 and let h be an arbitrary
admissible variation for this problem, i.e.,
h (a)=h (b)=0, i=1,...,N.
i i
According with Theorem 12, a necessary condition for y to be a minimizer is
given by
∂
(y+ǫh) =0,
∂ǫJ |ǫ=0
that is,
b N+1 N+1
d
α,β α,β
∂iL[y]γ (x)hi−1(x)+ ∂N+iL[y]γ (x)dxhi−1(x)
"
Za Xi=2 Xi=2 (6)
N+1
+ ∂2N+iL[y]αγ,β(x) CDγα,βhi−1(x) dx=0.
#
i=2
X (cid:0) (cid:1)
Usingtheintegrationbypartsformulas,fortheclassicalandCDα,β derivatives,
γ
in the second and third term of the integrand, we obtain
bN+1
d
∂iL[y]αγ,β(x)− dx∂N+iL[y]αγ,β(x)+C D1β−,αγ∂2N+iL[y]αγ,β(x) hi−1(x)dx
Za Xi=2 (cid:20) (cid:21)
N+1 x=b N+1 x=b
+ hi−1(x)∂N+iL[y]αγ,β(x) +γ hi−1(x)xIb1−α∂2N+iL[y]αγ,β(x)
" # " #
Xi=2 x=a Xi=2 x=a
N+1 x=b
−(1−γ)" hi−1(x)aIx1−β∂2N+iL[y]αγ,β(x)# =0.
Xi=2 x=a
(7)
Since h (a)=h (b)=0, i=1,...,N, we get
i i
bN+1
d
∂iL[y]αγ,β(x)− dx∂N+iL[y]αγ,β(x)+C D1β−,αγ∂2N+iL[y]αγ,β(x) hi−1(x)dx=0.
Za Xi=2 (cid:20) (cid:21)
Equalities (5) follow from the application of the fundamental lemma of the
calculus of variations (see, e.g., [30]).
6
When the Lagrangian L does not depend on fractional derivatives, then
Theorem13reducestotheclassicalresult(see,e.g.,[30]). ThefractionalEuler–
Lagrange equations via Caputo derivatives that one can find in the literature,
are also obtained as corollaries of Theorem 13. The next result is obtained
choosing a Lagrangianthat does not depend on the classical derivatives.
Corollary 14 (Theorem 6 of [22]). Let y = (y1,...,yN) be a local minimizer
to problem
b
(y)= L x,y(x),CDα,βy(x) dx min
J γ −→
Za
y((cid:0)a)=ya, y(b)=yb(cid:1),
ya, yb RN. Then, y satisfies the system of N fractional Euler–Lagrange
∈
equations
∂iL[y](x)+D1β−,αγ∂N+iL[y](x)=0, (8)
i=2,...N +1, for all x [a,b].
∈
If one considers γ = 1 (cf. Remark 6) and N = 1 in Corollary 14, then (8)
reduces to the well known Caputo fractional Euler–Lagrangeequation: if y is a
local minimizer to problem
b
(y)= L x,y(x),CDαy(x) dx min
J a x −→
Za
y(a(cid:0))=y , y(b)=y(cid:1),
a b
then y satisfies the fractional Euler–Lagrangeequation
∂2L x,y(x),CaDxαy(x) +xDbα∂3L x,y(x),CaDxαy(x) =0 (9)
for all x [a,b](cid:0)(see, e.g., [12]). (cid:1) (cid:0) (cid:1)
∈
3.2 Transversality conditions
Letl 1,...,N . AssumenowthatinProblem1theboundaryconditions(4)
∈{ }
are substituted by
y(a)=ya, y (b)=yb, i=1,...,N for i=l, and y (b) is free (10)
i i 6 l
or
y(a)=ya, y (b)=yb, i=1,...,N for i=l, and y (b) yb. (11)
i i 6 l ≤ l
Theorem 15. If y = (y1,...,yN) is a solution to Problem 1 with either (10)
or (11) as boundary conditions instead of (4), then y satisfies the system of
Euler–Lagrange equations (5). Moreover, under the boundary conditions (10)
the extra transversality condition
∂N+l+1L[y]αγ,β(x)+γxIb1−α∂2N+l+1L[y]αγ,β(x)
h −(1−γ)aIx1−β∂2N+l+1L[y]αγ,β(x) x=b =0 (12)
i
7
holds; under the boundary conditions (11) the extra transversality condition
∂N+l+1L[y]αγ,β(x)+γxIb1−α∂2N+l+1L[y]αγ,β(x)
h −(1−γ)aIx1−β∂2N+l+1L[y]αγ,β(x) x=b ≤0 (13)
i
holds, with (12) taking place if y (b)<yb.
l l
Proof. The fact that the system of Euler–Lagrange equations (5) is satisfied is
asimple consequenceoftheproofofTheorem13(onecanalwaysrestricttothe
subclass of functions h D for which h (a) = h (b) = 0, i = 1,...,N). Let
i i
∈
us assume that the boundary conditions are (10). Condition (12) follows from
(6). Suppose now that the boundary conditions are (11). Then, there are two
cases to consider. (i) If y (b)<yb, then there are admissible neighboring paths
l l
with terminal value both above and below y (b), so that h (b) can take either
l l
sign. Therefore, the transversality condition is (12). (ii) Let y (b)=yb. In this
l l
case neighboring paths with terminal value y˜ y (b) are considered. Choose
l l
≤
h such that h (b) 0. Then, ǫ 0 and the transversalitycondition, which has
l l
≥ ≤
its root in the first order condition (7), must be changed to an inequality. We
obtain (13).
WhentheLagrangiandoesnotdependonfractionalderivatives,thentheleft
handsideof(12)and(13)reducetotheclassicalexpression∂N+l+1L(x,y(x),y′(x))
(for instance, when N =1 and y(a) is fixed with y(b) free, then we get the well
knownboundary condition ∂3L(b,y(b),y′(b))=0). In the particular case when
the Lagrangian does not depend on the classical derivatives, γ = 1, N = 1,
and we have boundary conditions (10), then one obtains from Theorem 15 the
following result of [1].
Corollary 16 (cf. Theorem 1 of [1]). If y is a local minimizer to problem
b
(y)= L x,y(x),CDαy(x) dx min
J a x −→
Za
y(a)(cid:0)=y (y(b) is fre(cid:1)e),
a
then y satisfies the fractional Euler–Lagrange equation (9). Moreover,
xIb1−α∂3L(x,y(x),CaDxαy(x)) x=b =0.
(cid:2) (cid:3)
3.3 The isoperimetric problem
We now consider the following problem of the calculus of variations.
Problem 2. Minimize functional (3) subject to given boundary conditions (4)
and r isoperimetric constraints
b
j(y)= Gj[y]α,β(x)dx=ξ , j =1,...,r, (14)
G γ j
Z
a
8
where Gj C1 [a,b] R3N;R and ξ R for j =1,...,r.
j
∈ × ∈
Problems of(cid:0)the type of Pr(cid:1)oblem 2, where some integrals are to be given
a fixed value while another one is to be made a maximum or a minimum, are
called isoperimetric problems. Such variational problems have found a broad
class of important applications throughout the centuries, with numerous useful
implications in astronomy, geometry, algebra, analysis, and engineering. For
references and recent advancements on the subject, we refer the reader to [3,
4, 11, 20]. Here, in order to obtain necessary optimality conditions for the
combined fractional isoperimetric problem (Problem 2), we make use of the
following theorem.
Theorem 17 (see, e.g., Theorem 2 of [14] on p. 91). Let , 1,..., r be
J G G
functionals definedin a neighborhood of y andhaving continuousfirstvariations
in this neighborhood. Suppose that y is a local minimizer to the isoperimetric
problemgivenby (3),(4)and (14). Assumethattherearefunctionsh1,...,hr
∈
D such that
A=(a ), a :=δ k(y;hl), has maximal rank r. (15)
kl kl
G
Then there exist constants λ1,...,λr R such that the functional
∈
r
:= λ j
j
F J − G
j=1
X
satisfies
δ (y;h)=0 (16)
F
for all h D.
∈
Theorem 18. Let assumptions of Theorem 17 hold. If y is a local minimizer to
Problem 2, then y satisfies the system of N fractional Euler–Lagrange equations
d
∂iFλ{y}αγ,β(x)− dx∂N+iFλ{y}αγ,β(x)+D1β−,αγ∂2N+iFλ{y}αγ,β(x)=0, (17)
i=2,...,N +1, for all x [a,b], where function F :[a,b] R3N Rr R is
∈ × × →
defined by
r
F y α,β(x):=L[y]α,β(x) λ Gj[y]α,β(x).
λ{ }γ γ − j γ
j=1
X
Proof. UnderassumptionsofTheorem17,theequation(16)isfulfilledforevery
h D. Consider a function h such that h(a)=h(b)=0. Then,
∈
∂
0=δ (y;h)= (y+ǫh)
F ∂ǫF
(cid:12)ǫ=0
b N+1 (cid:12)(cid:12) N+1 d
=Za "i=2 ∂i Fλ{y}αγ,β(x)h(cid:12)i−1(x) + i=2 ∂N+iFλ{y}αγ,β(x)dxhi−1(x)
X X
N+1
+ ∂2N+iFλ{y}αγ,β(x)CDγα,βhi−1(x)#dx.
i=2
X
9
Using the classical and the integration by parts formula (1), and applying the
fundamentallemmaofthe calculusofvariationsinasimilarwayasinthe proof
of Theorem 13, we obtain (17).
Suppose now, that constraints (14) are characterizedby inequalities
b
(y)= Gj[y]α,β(x)dx ξ , j =1,...,r.
G γ ≤ j
Z
a
In this case we can set
b b
ξ
Gj[y]α,β(x) j dx+ (φ (x))2dx=0,
γ − b a j
Za (cid:18) − (cid:19) Za
j =1,...,r, where φ have the same continuity properties as y . Therefore, we
j i
obtain the following problem: minimize the functional
b
ˆ(y)= Lˆ x,y(x),y′(x),CDα,βy(x),φ(x) dx,
J γ
Z
a (cid:0) (cid:1)
where φ(x)=[φ1(x),...,φr(x)], subject to r isoperimetric constraints
b
ξ
Gj[y]α,β(x) j +(φ (x))2 dx=0, j =1,...,r,
γ − b a j
Za (cid:20) − (cid:21)
and boundary conditions (4). Assuming that assumptions of Theorem 18 are
satisfied, we conclude that there exist constants λ R, j =1,...,r, for which
j
∈
the system of N equations
∂iFˆ x,y(x),y′(x),CDγα,βy(x),λ1,...,λr,φ(x)
d
−(cid:0)dx∂N+iFˆ x,y(x),y′(x),CDγα,βy(x),λ1,.(cid:1)..,λr,φ(x) (18)
+D1β−,αγ∂2N+(cid:0)iFˆ x,y(x),y′(x),CDγα,βy(x),λ1,...,λr,φ(cid:1)(x) =0,
(cid:0) r (cid:1)
i=2,...,N +1, where Fˆ =Lˆ+ λ Gj ξj +φ2 and
j − b−a j
j=1
(cid:16) (cid:17)
P
λ φ (x)=0, j =1,...,r, (19)
j j
hold for all x [a,b]. Note that it is enough to assume that the regularity
∈
condition (15) holds for the constraints which are active at the local minimizer
y. Indeed,supposethatl<rconstraints,say 1,..., lforsimplicity,areactive
at the local minimizer y, and there are functGions h1,G...,hl D such that the
matrix B = (b ), b := δ k y;hj , k,j = 1,...,l < r ha∈s maximal rank l.
kj k,j
G
(cid:0) (cid:1)
10
