Page 1

arXiv:1110.4141v1 [math.OC] 18 Oct 2011

Submitted 26-Sept-2011; accepted 18-Oct-2011;

to Advances in Harmonic Analysis and Operator Theory,

The Stefan Samko Anniversary Volume

(Eds: A. Almeida, L. Castro, F.-O. Speck),

Operator Theory: Advances and Applications, Birkh¨ auser Verlag.

http://www.springer.com/series/4850

Fractional variational calculus of variable order

T. Odzijewicz, A. B. Malinowska and D. F. M. Torres

To Professor Stefan Samko on the occasion of his 70th birthday

Abstract. We study the fundamental problem of the calculus of variations

with variable order fractional operators. Fractional integrals are considered in

the sense of Riemann–Liouville while derivatives are of Caputo type.

Mathematics Subject Classification (2010). 26A33; 34A08; 49K05.

Keywords. Fractional operators; fractional integration and differentiation of

variable order; fractional variational analysis; Euler–Lagrange equations.

1. Introduction

Fractional calculus is a discipline that studies integrals and derivatives of non-

integer (real or complex) order [13,16,23,30]. The field was born in 1695 when

Leibniz asked in his letter to L’Hˆ opital what should be the meaning of a derivative

of non-integer order. After the letters between Leibniz and L’Hˆ opital, fractional

calculus became an ongoing topic with many well-known mathematicians con-

tributing to its theory [32]. The subject is nowadays very active due to its many

applications in mechanics, chemistry, biology, economics, and control theory [33].

In 1993, Samko and Ross proposed an interesting generalization to fractional

calculus [31] (see also Samko’s paper [29] of 1995). They introduced the study

of fractional integration and differentiation when the order is not a constant but

a function. More precisely, they considered an extension of Riemann–Liouville

and Fourier definitions [28,29, 31]. Afterwards, several works were dedicated to

variable order fractional operators, their applications and interpretations (see, e.g.,

[1,7,17]). In particular, Samko’s variable order fractional calculus turns out to be

very useful in mechanics and in the theory of viscous flows [7,9,17,22,24,25].

In this note we develop the fractional calculus of variations via Samko’s vari-

able order approach. The fractional variational calculus was born in 1996-1997

Part of the first author’s Ph.D., which is carried out at the University of Aveiro under the

Doctoral Programme in Mathematics and Applications of Universities of Aveiro and Minho.

Page 2

2T. Odzijewicz, A. B. Malinowska and D. F. M. Torres

with the works of Riewe in mechanics [26,27], and is now under strong current re-

search (see [2,3,5,6,8,10,11,14,15,18–21] and references therein). However, to the

best of the authors knowledge, results for the variable order case are a rarity and

reduce to those in [4]. Motivated by the advancements of [3,21], and in contrast

with [4], we consider here fractional problems of the calculus of variations where

the Lagrangian depends on classical integer order derivatives and both on variable

order fractional derivatives and integrals.

The paper is organized as follows. In §2 a brief review to the variable order

fractional calculus is given. Our results are then formulated and proved in §3: we

show the boundedness of the variable order Riemann–Liouville fractional integral

in the space L1[a,b] (Theorem 3.1); integration by parts formulas for variable order

fractional operators (Theorems 3.2 and 3.3); and a necessary optimality condition

for our general fundamental problem of the variable order fractional variational

calculus (Theorem 3.5). An illustrative example is discussed in §4.

2. Preliminaries

The reference book for fractional analysis and its applications is [30]. Here we recall

the necessary definitions for the variable order fractional calculus. Throughout the

text α denotes a given absolutely continuous function.

Definition 2.1 (Left and right Riemann–Liouville integrals of variable order). Let

0 < α(t) < 1 and f ∈ L1[a,b]. Then,

aIα(t)

t

f(t) =

1

Γ(α(t))

t ?

a

(t − τ)α(t)−1f(τ)dτ

(t > a)

is called the left Riemann–Liouville integral of variable fractional order α(t), while

tIα(t)

b

f(t) =

1

Γ(α(t))

b

?

t

(τ − t)α(t)−1f(τ)dτ

(t < b)

denotes the right Riemann–Liouville integral of variable fractional order α(t).

Definition 2.2 (Left and right Riemann–Liouville derivatives of variable order). Let

0 < α(t) < 1. IfaI1−α(t)

t

f ∈ AC[a,b], then the left Riemann–Liouville derivative

of variable fractional order α(t) is defined as

aDα(t)

t

f(t) =

1

Γ(1 − α(t))

d

dt

t ?

a

(t − τ)−α(t)f(τ)dτ

(t > a)

Page 3

Fractional variational calculus of variable order3

while the right Riemann–Liouville derivative of variable fractional order α(t) is

defined for functions f such thattI1−α(t)

b

f ∈ AC[a,b] by

tDα(t)

b

f(t) =

−1

Γ(1 − α(t))

d

dt

b

?

t

(τ − t)−α(t)f(τ)dτ

(t < b).

Definition 2.3 (Left and right Caputo derivatives of variable fractional order). Let

0 < α(t) < 1. If f ∈ AC[a,b], then the left Caputo derivative of variable fractional

order α(t) is defined as

C

aDα(t)

t

f(t) =

1

Γ(1 − α(t))

t ?

a

(t − τ)−α(t)d

dτf(τ)dτ

(t > a)

while the right Caputo derivative of variable fractional order α(t) is given by

C

tDα(t)

b

f(t) =

−1

Γ(1 − α(t))

b

?

t

(τ − t)−α(t)d

dτf(τ)dτ

(t < b).

The following result will be useful in the proof of Theorems 3.1 and 3.2.

Theorem 2.4 (cf. [12]). If x ∈ [0,1], the Gamma function satisfies the inequalities

x2+ 1

x + 1

≤ Γ(x + 1) ≤x2+ 2

x + 2.

(2.1)

3. Main results

In §3.1 we prove that the variable fractional order Riemann–Liouville integral

aIα(t)

t

is bounded on the space L1[a,b] (Theorem 3.1). In §3.2 we obtain a formula

of integration by parts for Riemann–Liouville integrals of variable order (Theo-

rem 3.2) and a formula of integration by parts for derivatives of variable fractional

order (Theorem 3.3). Finally, in §3.3 we use the obtained formulas of integration

by parts to derive a necessary optimality condition for a general problem of the

calculus of variations involving variable order fractional operators (Theorem 3.5).

3.1. Boundedness

The following result allow us to consider problems of the calculus of variations with

a Lagrangian depending on left Riemann–Liouville integrals of variable order.

Theorem 3.1. Let

and for all t ∈ [a,b]. The Riemann–Liouville integralaIα(t)

variable fractional order α(t) is a linear and bounded operator.

1

n< α(t) < 1 for a certain n ∈ N greater or equal than two

t

: L1[a,b] → L1[a,b] of

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4T. Odzijewicz, A. B. Malinowska and D. F. M. Torres

Proof. The operator is obviously linear. Let1

n< α(t) < 1, f ∈ L1[a,b], and define

F(τ,t) :=

????

1

Γ(α(t))(t − τ)α(t)−1??? · |f(τ)|

1

n< α(t) < 1, then

if τ ≤ t;

0 if τ > t

for all (τ,t) ∈ ∆ = [a,b] × [a,b]. Since

1. for τ + 1 ≤ t we have ln(t − τ) ≥ 0 and (t − τ)α(t)−1< 1;

2. for τ ≤ t < τ + 1 we have ln(t − τ) < 0 and (t − τ)α(t)−1< (t − τ)

1

n−1.

Therefore,

?b

a

??b

?b

?b

a

F(τ,t)dt

?

dτ =

?b

1

a

|f(τ)|

??b

τ

????

1

Γ(α(t))(t − τ)α(t)−1

????dt

?

dτ

=

a

|f(τ)|

??τ+1

??τ+1

τ

????

????

Γ(α(t))(t − τ)α(t)−1

????dt +

????dt +

?b

τ+1

????

1

Γ(α(t))(t − τ)α(t)−1

????dt

?

dτ

<

a

|f(τ)|

τ

1

Γ(α(t))(t − τ)

1

n−1

?b

τ+1

????

1

Γ(α(t))

????dt

?

dτ.

Moreover, by inequality (2.1), one has

?b

a

|f(τ)|

??τ+1

??τ+1

??τ+1

τ

????

1

Γ(α(t))(t − τ)

1

n−1

????dt +

?b

τ+1

????

1

Γ(α(t))

????dt

(α(t) − 1)2+ 1dt

?

dτ

<

?b

?b

a

|f(τ)|

τ

α(t)

(α(t) − 1)2+ 1(t − τ)

1

n−1dt +

?b

?b

τ+1

α(t)

?

dτ

<

a

|f(τ)|

τ

(t − τ)

1

n−1dt + b − τ − 1

?

dτ =

a

|f(τ)|(n + b − τ − 1)dτ

< (n + b − a)?f? < ∞.

It follows from Fubini’s theorem that F is integrable in the square ∆ and

???aIα(t)

t

f

??? =

?b

?b

?b

a

????

1

Γ(α(t))

??t

??b

?t

1

a

(t − τ)α(t)−1f(τ)dτ

????dt

≤

a

a

????

F(τ,t)dτ

Γ(α(t))(t − τ)α(t)−1f(τ)

?

< (n + b − a)?f?.

????dτ

?

dt

=

aa

dt

Therefore,aIα(t)

t

: L1[a,b] → L1[a,b] and

???aIα(t)

t

??? < n + b − a.

?

Page 5

Fractional variational calculus of variable order5

3.2. Integration by parts formulas

The integration by parts formulas we now obtain have an important role in the

proof of the generalized Euler–Lagrange equation (3.5). We note that in Theo-

rem 3.2 the left-hand side of (3.1) involves a left integral of variable order while

on the right-hand side it appears a right integral.

Theorem 3.2. Let 0 < α(t) < 1 and f,g ∈ C[a,b]. Then,

?b

a

g(t)aIα(t)

t

f(t)dt =

?b

a

f(t)tIα(t)

b

g(t)dt.

(3.1)

Proof. Let 0 < α(t) < 1, f,g ∈ C[a,b], and define

F(τ,t) :=

? ???

1

Γ(α(t))(t − τ)α(t)−1g(t)f(τ)

0

???

if τ ≤ t;

if τ > t

for all (τ,t) ∈ ∆ = [a,b] × [a,b]. Since f and g are continuous functions on [a,b],

they are bounded on [a,b], i.e., there exist C1,C2> 0 such that |g(t)| ≤ C1and

|f(t)| ≤ C2, t ∈ [a,b]. Therefore,

?b

a

??b

a

F(τ,t)dτ

?

dt =

?b

a

??t

?b

?b

a

????

1

Γ(α(t))(t − τ)α(t)−1g(t)f(τ)

????dτ

?

dt

≤ C1C2

a

??t

??t

a

????

Γ(α(t))(t − τ)α(t)−1dτ

1

Γ(α(t))(t − τ)α(t)−1

????dτ

?

dt

= C1C2

a

a

1

?

dt.

Moreover,

?t

a

(t − τ)α(t)−1dτ = (t − a)α(t) 1

α(t).

Then, by (2.1), one has

C1C2

?b

a

1

Γ(α(t) + 1)(t − a)α(t)dt ≤ C1C2

?b

?b

a

α(t) + 1

α(t)2+ 1(t − a)α(t)dt

< C1C2

a

(t − a)α(t)dt.

Since 0 < α(t) < 1, then

1. for 1 ≤ t − a we have ln(t − a) ≥ 0 and (t − a)α(t)< (t − a);

2. for 1 > t − a we have ln(t − a) < 0 and (t − a)α(t)< 1.

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6T. Odzijewicz, A. B. Malinowska and D. F. M. Torres

Therefore,

C1C2

?b

a

(t − a)α(t)dt = C1C2

??a+1

a

(t − a)α(t)dt +

?b

= C1C21 + (b − a)2

a+1

(t − a)α(t)dt

?

< C1C2

??a+1

a

dt +

?b

a+1

(t − a)dt

?

2

< ∞.

Hence, one can use the Fubini theorem to change the order of integration:

?b

a

g(t)aIα(t)

t

f(t)dt =

?b

a

??t

a

g(t)f(τ)

1

Γ(α(t))(t − τ)α(t)−1dτ

?

?

dt

=

?b

a

??b

τ

g(t)f(τ)

1

Γ(α(t))(t − τ)α(t)−1dtdτ =

?b

a

f(τ)τIα(t)

b

g(τ)dτ.

?

In our second formula (3.2) of fractional integration by parts, the left-hand

side contains a left Caputo derivative of variable fractional order α(t), while on

the right-hand side it appears right Riemann–Liouville integrals of variable order

1 − α(t) and a right Riemann–Liouville derivative of variable order α(t).

Theorem 3.3. Let 0 < α(t) < 1. If α, f andtI1−α(t)

b

g ∈ AC[a,b], then

?b

a

g(t)C

aDα(t)

t

f(t)dt = f(t)tI1−α(t)

b

g(t)

???

b

a

−

?b

a

f(t)

?

α′(t)ψ(1 − α(t))tI1−α(t)

b

g(t) −tDα(t)

b

g(t)

?

dt,

(3.2)

where ψ is the digamma function, i.e., ψ(x) =

d

dxlnΓ(x) =Γ′(x)

Γ(x).

Proof. By Definition 2.3, it follows thatC

Theorem 3.2 and integration by parts for classical (integer) derivatives, we obtain

aDα(t)

t

f(t) = aI1−α(t)

t

d

dtf(t). Applying

?b

a

g(t)C

aDα(t)

t

f(t)dt =

?b

b

a−

a

g(t)aI1−α(t)

t

d

dtf(t)dt =

?b

a

d

dtf(t)tI1−α(t)

b

g(t)dt

= f(t)tI1−α(t)

b

g(t)

???

???

?b

?b

a

f(t)d

dttI1−α(t)

b

g(t)dt

= f(t)tI1−α(t)

b

g(t)

b

a−

a

f(t)

?

α′(t)ψ(1 − α(t))tI1−α(t)

b

g(t) −tDα(t)

b

g(t)

?

dt.

?

Page 7

Fractional variational calculus of variable order7

3.3. A fundamental variational problem of variable fractional order

We consider the problem of extremizing (minimizing or maximizing) the functional

J[y] =

b

?

a

F

?

t,y(t),y′(t),C

aDα(t)

t

y(t),aIβ(t)

t

y(t)

?

dt

(3.3)

subject to the boundary conditions

y(a) = ya,y(b) = yb,

(3.4)

where α and β are absolutely continuous functions of t ∈ [a,b] taking values in

[0,1]. For simplicity of notation, we introduce the operator {·,·,·} defined by

{y,α,β}(t) =

?

t,y(t),y′(t),C

aDα(t)

t

y(t),aIβ(t)

t

y(t)

?

.

We assume that F ∈ C1?[a,b] × R4;R?, t ?→ ∂4F {y,α,β}(t) has absolutely con-

b

continuous variable order fractional integraltIβ(t)

continuous usual derivative

dt.

tinuous integraltI1−α(t)

and continuous derivativetDα(t)

b

, t ?→ ∂5F {y,α,β}(t) has

, and t ?→ ∂3F {y,α,β}(t) has

b

d

Definition 3.4. A Lipshitz function y ∈ Lip([a,b];R) is said to be admissible for

the variational problem (3.3)–(3.4), ifC

on the interval [a,b], and y satisfies the given boundary conditions (3.4).

aDα(t)

t

y andaIβ(t)

t

y exist and are continuous

Theorem 3.5. Let y be a solution to problem (3.3)–(3.4). Then, y satisfies the

generalized Euler–Lagrange equation

∂2F {y,α,β}(t) −d

dt∂3F {y,α,β}(t) +tIβ(t)

∂4F {y,α,β}(t) +tDα(t)

b

∂5F {y,α,β}(t)

− α′(t)ψ(1 − α(t))tI1−α(t)

bb

∂4F {y,α,β}(t) = 0 (3.5)

for all t ∈ [a,b].

Proof. Suppose that y is an extremizer of J. Consider the value of J at a nearby

function ˆ y(t) = y(t)+εη(t), where ε ∈ R is a small parameter and η ∈ Lip([a,b];R)

is an arbitrary function satisfying η(a) = η(b) = 0 and such thatC

aIβ(t)

t

ˆ y(t) are continuous. Let

?b

A necessary condition for y to be an extremizer is given by

aDα(t)

t

ˆ y(t) and

J(ε) = J[ˆ y] =

a

F {ˆ y,α,β}(t)dt.

dJ

dε

????

ε=0

= 0 ⇔

b

?

a

?

∂2F {y,α,β}(t) · η(t) + ∂3F {y,α,β}(t)d

dtη(t)

+ ∂4F {y,α,β}(t)C

aDα(t)

t

η(t) + ∂5F {y,α,β}(t) ·aIβ(t)

t

η(t)

?

dt = 0.

(3.6)

Page 8

8T. Odzijewicz, A. B. Malinowska and D. F. M. Torres

Using the classical and the generalized fractional integration by parts formulas of

Theorems 3.2 and 3.3, we obtain

?b

a

∂3Fdη

dtdt = ∂3Fη|b

a−

?b

a

?

ηd

dt∂3F

?

dt,

b

?

a

∂4FC

aDα(t)

t

ηdt = ηtI1−α(t)

b

∂4F

???

b

a

−

b

?

a

η

?

α′(t)ψ(1 − α(t))tI1−α(t)

b

∂4F −tDα(t)

b

∂4F

?

dt,

and

b

?

a

∂5FaIβ(t)

t

ηdt =

b

?

a

ηtIβ(t)

b

∂5Fdt.

Because η(a) = η(b) = 0, (3.6) simplifies to

?b

a

η(t)

?

∂2F {y,α,β}(t) −d

dt∂3F {y,α,β}(t)

− α′(t)ψ(1 − α(t))tI1−α(t)

b

∂4F {y,α,β}(t) +tDα(t)

b

∂4F {y,α,β}(t)

+tIβ(t)

b

∂5F {y,α,β}(t)

?

dt = 0.

One obtains (3.5) by the fundamental lemma of the calculus of variations (see,

e.g., [34]).

?

4. An illustrative example

Let 0 < α(t) < 1 and γ > −1. In Example 1 we make use of the identity

aIα(t)

t

[(t − a)γ] =Γ(γ + 1)(t − a)γ+α(t)

Γ(γ + α(t) + 1)

(4.1)

that one can find in the Samko and Ross paper [31].

Example 1. Let J be the functional defined by

J[y] =

b

?

a

?

1 +

Γ(α(t) + 3)

2Γ(3)(t − a)2+α(t)

?

aIα(t)

t

y(t))

?2

−aIα(t)

t

y(t)dt

Page 9

Fractional variational calculus of variable order9

with endpoint conditions y(a) = 0 and y(b) = (b − a)2. If y is an extremal for J,

then the following necessary optimality condition is satisfied:

2 1 +

2Γ(3)(t−a)2+α(t)

tIα(t)

b

Γ(α(t)+3)

Γ(3)(t−a)2+α(t) aIα(t)

t

y(t) − 1

?

Γ(α(t)+3)

?

aIα(t)

t

y(t)

?2

−aIα(t)

t

y(t)

= 0.

(4.2)

By identity (4.1), function

y(t) = (t − a)2

(4.3)

is a solution to the variable order fractional differential equation

y(t) =Γ(3)(t − a)2+α(t)

aIα(t)

t

Γ(α(t) + 3)

.

Therefore, it is a solution to the Euler–Lagrange equation (4.2). Note that (4.3) is

a locally Lipschitz function. Indeed,

??(t1− a)2− (t2− a)2??= |(t1− a) + (t2− a)| · |(t1− a) − (t2− a)|

Acknowledgments

Work supported by FEDER funds through COMPETE — Operational Programme

Factors of Competitiveness (“Programa Operacional Factores de Competitivi-

dade”) and by Portuguese funds through the Center for Research and Develop-

ment in Mathematics and Applications (University of Aveiro) and the Portuguese

Foundation for Science and Technology (“FCT — Funda¸ c˜ ao para a Ciˆ encia e a

Tecnologia”), within project PEst-C/MAT/UI4106/2011with COMPETE number

FCOMP-01-0124-FEDER-022690.Odzijewicz was also supported by FCT through

the Ph.D. fellowship SFRH/BD/33865/2009; Malinowska by Bia? lystok Univer-

sity of Technology grant S/WI/02/2011; and Torres by FCT through the project

PTDC/MAT/113470/2009.

= |t1+ t2− 2a| · |t1− t2| ≤ (|t1| + |t2| + 2|a|)|t1− t2| ≤ 2(|a| + |b|)|t1− t2|.

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T. Odzijewicz

Center for Research and Development in Mathematics and Applications

Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

e-mail: tatianao@ua.pt

A. B. Malinowska

Faculty of Computer Science, Bia? lystok University of Technology

15-351 Bia? lystok, Poland

e-mail: a.malinowska@pb.edu.pl

D. F. M. Torres

Center for Research and Development in Mathematics and Applications

Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

e-mail: delfim@ua.pt