Generalized transversality conditions in fractional calculus of variations

$Page 1: Generalized transversality conditions in fractional calculus of variations$
Commun Nonlinear Sci Numer Simulat 18 (2013) 443–452

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Commun Nonlinear Sci Numer Simulat

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Generalized transversality conditions in fractional calculus of variations

Ricardo Almeida a, Agnieszka B. Malinowska b,⇑a Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugalb Faculty of Computer Science, Białystok University of Technology, 15-351 Białystok, Poland

a r t i c l e i n f o

Article history:Received 27 December 2011Received in revised form 3 June 2012Accepted 15 July 2012Available online 3 August 2012

Keywords:Calculus of variationsFractional calculusFractional Euler–Lagrange equationTransversality conditionsCaputo fractional derivative

1007-5704/$ - see front matter � 2012 Elsevier B.Vhttp://dx.doi.org/10.1016/j.cnsns.2012.07.009

⇑ Corresponding author.E-mail addresses: [email protected] (R. Alm

a b s t r a c t

Problems of calculus of variations with variable endpoints cannot be solved without trans-versality conditions. Here, we establish such type of conditions for fractional variationalproblems with the Caputo derivative. We consider: the Bolza-type fractional variationalproblem, the fractional variational problem with a Lagrangian that may also depend onthe unspecified end-point uðbÞ, where x ¼ uðtÞ is a given curve, and the infinite horizonfractional variational problem.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The calculus of variations is concerned with the problem of extremizing functionals. It has many applications in physics,geometry, engineering, dynamics, control theory, and economics. The formulation of a problem of the calculus of variationsrequires two steps: the specification of a performance criterion; and then, the statement of physical constraints that shouldbe satisfied. The basic problem is stated as follows: among all differentiable functions x : ½a; b� ! R such that xðaÞ ¼ xa andxðbÞ ¼ xb, with xa; xb fixed reals, find the ones that minimize (or maximize) the functional

JðxÞ ¼Z b

aLðt; xðtÞ; x0ðtÞÞdt:

One way to deal with this problem is to solve the second order differential equation

@L@x� d

dt@L@x0¼ 0

called the Euler–Lagrange equation. The two given boundary conditions provide sufficient information to determine the twoarbitrary constants. But if there are no boundary constraints, then we need to impose another conditions, called the naturalboundary conditions (see e.g. [19]),

@L@x0

� �t¼a¼ 0 and

@L@x0

� �t¼b¼ 0: ð1Þ

. All rights reserved.

eida), [email protected], [email protected] (A.B. Malinowska).

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444 R. Almeida, A.B. Malinowska / Commun Nonlinear Sci Numer Simulat 18 (2013) 443–452

Clearly, such terminal conditions are important in models, the optimal control or decision rules are not unique withoutthese conditions.

Fractional calculus deals with derivatives and integrals of a non-integer (real or complex) order. Fractional operators arenon-local, therefore they are suitable for constructing models possessing memory effect. They found numerous applicationsin various fields of science and engineering, as diffusion process, electrical science, electrochemistry, material creep, visco-elasticity, mechanics, control science, electromagnetic theory, etc. Fractional calculus is now recognized as vital mathemat-ical tool to model the behavior and to understand complex systems (see, e.g., [20,25,32,38,39,44,46,52]). TraditionalLagrangian and Hamiltonian mechanics cannot be used with nonconservative forces such as friction. Riewe [51] showed thatfractional formalism can be used when treating dissipative problems. By inserting fractional derivatives into the variationalintegrals he obtained the respective fractional Euler–Lagrange equation, combining both conservative and nonconservativecases. Nowadays the fractional calculus of variations is a subject under strong research. Investigations cover problemsdepending on Riemann–Liouville fractional derivatives (see, e.g., [1,9,13,17,29]), the Caputo fractional derivative (see, e.g.,[2,6,11,12,30,41,42]), the symmetric fractional derivative (see, e.g., [37]), the Jumarie fractional derivative (see, e.g.,[7,33,34]), and others [3,5,15,16,24,27,28].

The aim of this paper is to obtain transversality conditions for fractional variational problems with the Caputo derivative.Namely, three types of problems are considered: the first in Bolza form, the second with a Lagrangian depending on theunspecified end-point uðbÞ, where x ¼ uðtÞ is a given curve, and the third with infinite horizon. We note here, that fromthe best of our knowledge, fractional variational problems with infinite horizon have not been considered yet, and this isan open research area.

The paper is organized in the following way. Section 2 presents some preliminaries needed in the sequel. Our main resultsare stated and proved in the remaining sections. In Section 3 we consider the Bolza-type fractional variational problem anddevelop the transversality conditions in a compact form. As corollaries, we formulate conditions appropriate to various typeof variable terminal points. Section 4 provides the necessary optimality conditions for fractional variational problems with aLagrangian that may also depend on the unspecified end-point uðbÞ, where x ¼ uðtÞ is a given curve. Finally, in Section 5 wepresent the transversality condition for the infinite horizon fractional variational problem.

2. Preliminaries

In this section we present a short introduction to the fractional calculus, following [26,36,47]. In the sequel, a 2 ð0;1Þ andC represents the Gamma function:

CðzÞ ¼Z 1

0tz�1e�t dt:

Let f : ½a; b� ! R be a continuous function. Then,

1. the left and right Riemann–Liouville fractional integrals of order a are defined by

aIax f ðxÞ ¼ 1CðaÞ

Z x

aðx� tÞa�1f ðtÞdt

and

xIabf ðxÞ ¼ 1CðaÞ

Z b

xðt � xÞa�1f ðtÞdt;

respectively;2. the left and right Riemann–Liouville fractional derivatives of order a are defined by

aDax f ðxÞ ¼ 1

Cð1� aÞddx

Z x

aðx� tÞ�af ðtÞdt

and

xDabf ðxÞ ¼ �1

Cð1� aÞddx

Z b

xðt � xÞ�af ðtÞdt;

respectively.

Let f : ½a; b� ! R be a differentiable function. Then,

1. the left and right Caputo fractional derivatives of order a are defined by

aCDa

x f ðxÞ ¼ 1Cð1� aÞ

Z x

aðx� tÞ�af 0ðtÞdt

R. Almeida, A.B. Malinowska / Commun Nonlinear Sci Numer Simulat 18 (2013) 443–452 445

and

xCDa

bf ðxÞ ¼ �1Cð1� aÞ

Z b

xðt � xÞ�af 0ðtÞdt;

respectively.

Observe that if a goes to 1, then the operators aCDa

x and aDax could be replaced with d

dx and the operators xCDa

b and xDab could

be replaced with � ddx (see [47]). Moreover, we set aI0

x f ¼ xI0bf :¼ f .

Obviously, the above defined operators are linear. If f 2 C1ð½a; b�Þ, then the left and right Caputo fractional derivatives of fare continuous on ½a; b� (cf. [36, Theorem 2.2]). In the discussion to follow, we will also need the following fractional inte-grations by parts (see e.g. [3]):
Z b
agðxÞ � a

CDax f ðxÞdx ¼

Z b

af ðxÞ � xDa

bgðxÞdxþ xI1�ab gðxÞ � f ðxÞ

h ix¼b

x¼að2Þ

and
Z b
agðxÞ � x

CDab f ðxÞdx ¼

Z b

af ðxÞ � aDa

x gðxÞdx� aI1�ax gðxÞ � f ðxÞ

h ix¼b

x¼a:

Along the work, and following [14], we denote by @iL; i ¼ 1; . . . ;m (m 2 N), the partial derivative of function L : Rm ! R

with respect to its ith argument. For simplicity of notation we introduce operators ½x� and fx;ug defined by

½x�ðtÞ ¼ ðt; xðtÞ; aCDa

t xðtÞÞ;

fx;ugðt; TÞ ¼ ðt; xðtÞ; aCDa

t xðtÞ;uðTÞÞ:

3. Transversality conditions I

Let us introduce the linear space ðx; tÞ 2 C1ð½a; b�Þ � R endowed with the norm kðx; tÞk1;1 :¼maxa6t6bjxðtÞjþmaxa6t6b a

CDat xðtÞ

�� þ jtj.We consider the following type of functionals:

Jðx; TÞ ¼Z T

aL½x�ðtÞdt þ /ðT; xðTÞÞ; ð3Þ

on the set

D ¼ ðx; tÞ 2 C1ð½a; b�Þ � ½a; b� jxðaÞ ¼ xa

n o;

where the Lagrange function L : ½a; b� � R2 ! R and the terminal cost function / : ½a; b� � R! R are at least of class C1. Observethat we have a free end-point T and no constraint on xðTÞ. Therefore, they become a part of the optimal choice process. Weaddress the problem of finding a pair ðx; TÞwhich minimizes (or maximizes) the functional J on D, i.e., there exists d > 0 suchthat Jðx; TÞ 6 Jð�x; tÞ (or Jðx; TÞP Jð�x; tÞ) for all ð�x; tÞ 2 D with kð�x� x; t � TÞk1;1 < d.

Theorem 1. Consider the functional given by (3). Suppose that ðx; TÞ gives a minimum (or maximum) for functional (3) on D. Thenx is a solution of the fractional Euler–Lagrange equation

@2L½x�ðtÞ þ tDaTð@3L½x�ðtÞÞ ¼ 0; ð4Þ

on the interval ½a; T� and satisfies the transversality conditions

L½x�ðTÞ þ @1/ðT; xðTÞÞ � x0ðTÞ tI1�aT @3L½x�ðtÞ

h it¼T¼ 0;

tI1�aT @3L½x�ðtÞ

h it¼Tþ @2/ðT; xðTÞÞ ¼ 0:

8><>: ð5Þ

Proof. Let us consider a variation ðxðtÞ þ �hðtÞ; T þ �MTÞ, where h 2 C1ð½a; b�Þ;MT 2 R and � 2 R with j�j � 1. The constraintxðaÞ ¼ xa implies that all admissible variations must fulfill the condition hðaÞ ¼ 0. Define jð�Þ on a neighborhood of zero by

jð�Þ ¼ Jðxþ �h; T þ �MTÞ ¼Z Tþ�MT

aL½xþ �h�ðtÞdt þ /ðT þ �MT; ðxþ �hÞðT þ �MTÞÞ:

If ðx; TÞ minimizes (or maximizes) functional (3) on D, then j0ð0Þ ¼ 0. Therefore, one has


0 ¼Z T

a@2L½x�ðtÞhðtÞ þ @3L½x�ðtÞaCDa

t hðtÞ� �

dt þ L½x�ðTÞMT þ @1/ðT; xðTÞÞMT þ @2/ðT; xðTÞÞ hðTÞ þ x0ðTÞMT½ �:

Integrating by parts (cf. Eq. (2)), and since hðaÞ ¼ 0, we get

0 ¼Z T

a@2L½x�ðtÞ þ tD

aT ð@3L½x�ðtÞÞ

� �hðtÞdt þ tI

1�aT ð@3L½x�ðtÞÞhðtÞ

h it¼Tþ L½x�ðTÞMT þ @1/ðT; xðTÞÞMT

þ @2/ðT; xðTÞÞ hðTÞ þ x0ðTÞMT½ �

¼Z T


aT ð@3L½x�ðtÞÞ

� �hðtÞdt þ MT L½x�ðTÞ þ @1/ðT; xðTÞÞ � x0ðTÞ t I

1�aT @3L½x�ðtÞ

h it¼T

h iþ tI


h it¼Tþ @2/ðT; xðTÞÞ

h ihðTÞ þ x0ðTÞMT½ �: ð6Þ

As h and MT are arbitrary we can choose hðTÞ ¼ 0 and MT ¼ 0. Then, by the fundamental lemma of the calculus of vari-ations we deduce Eq. (4). But if x is a solution of (4), then the condition (6) takes the form

0 ¼ MT L½x�ðTÞ þ @1/ðT; xðTÞÞ � x0ðTÞ t I1�aT @3L½x�ðtÞ

h it¼T

h iþ tI



h ihðTÞ þ x0ðTÞMT½ �: ð7Þ

Restricting ourselves to those h for which hðTÞ ¼ �x0ðTÞMT we get the first equation of (5). Analogously, considering thosevariations for which MT ¼ 0 we get the second equation of (5). h

Example 1. Let

Jðx; TÞ ¼Z T

0t2 � 1þ ð0CDa

t xðtÞÞ2h i

dt; T 2 ½0;10�:

It easy to verify that a constant function xðtÞ ¼ K and the end-point T ¼ 1 satisfies the necessary conditions of optimality ofTheorem 1, with the value of K being determined by the initial-point xð0Þ.

In the case when a goes to 1, by Theorem 1 we obtain the following result.

Corollary 1 [22, Theorem 2.24]. If ðx; TÞ gives a minimum (or maximum) for

Jðx; TÞ ¼Z T

aLðt; xðtÞ; x0ðtÞÞdt þ /ðT; xðTÞÞ;

on the set

x 2 C1ð½a; b�Þ jxðaÞ ¼ xa

n o� ½a; b�;

then x is a solution of the Euler–Lagrange equation

@2Lðt; xðtÞ; x0ðtÞÞ � ddt@3Lðt; xðtÞ; x0ðtÞÞ ¼ 0;

on the interval ½a; T� and satisfies the transversality conditions

LðT; xðTÞ; x0ðTÞÞ þ @1/ðT; xðTÞÞ � x0ðTÞ@3LðT; xðTÞ; x0ðTÞÞ ¼ 0;

@3LðT; xðTÞ; x0ðTÞÞ þ @2/ðT; xðTÞÞ ¼ 0:
Now we shall rewrite the necessary conditions (5) in terms of the increment on time MT and on the consequent increment
on x; MxT . Let us fix � ¼ 1 and consider variation functions h satisfying the additional condition h0ðTÞ ¼ 0. Define the totalincrement by

MxT ¼ ðxþ hÞðT þ MTÞ � xðTÞ:

Doing Taylor’s expansion up to first order, for a small MT , we have

ðxþ hÞðT þ MTÞ � ðxþ hÞðTÞ ¼ x0ðTÞMT þ OðMTÞ2

and so we can write hðTÞ in terms of MT and MxT :

hðTÞ ¼ MxT � x0ðTÞMT þ OðMTÞ2: ð8Þ

If ðx; TÞ gives an extremum for J, then

@2L½x�ðtÞ þ tDaTð@3L½x�ðtÞÞ ¼ 0 holds for all t 2 ½a; T�:

Therefore, substituting (8) into Eq. (6) we obtain


MT L½x�ðTÞ þ @1/ðT; xðTÞÞ � x0ðTÞ t I1�aT @3L½x�ðtÞ

h it¼T

h iþ MxT tI



h iþ OðMTÞ2 ¼ 0: ð9Þ

We remark that the above equation is evaluated in one single point x ¼ T . Eq. (9) replaces the missing terminal condition inthe problem.

Let us consider five particular cases of constraints that can be specified in the optimization problem.A. Vertical terminal lineIn this case the upper bound T is fixed and consequently the variation MT ¼ 0. Therefore, Eq. (9) becomes

MxT tI1�aT @3L½x�ðtÞ


h i¼ 0

and by the arbitrariness of MxT , we deduce


h it¼Tþ @2/ðT; xðTÞÞ ¼ 0:

When / � 0, we get the natural boundary condition as obtained in [1], Eq. (25):


h it¼T¼ 0:

For / � 0 and a! 1, we get the second equation of (1).B. Horizontal terminal lineNow we have MxT ¼ 0 but MT is arbitrary. Hence, Eq. (9) implies


h it¼T¼ 0:

When / � 0 and a! 1, we get Eq. (3.11) of [23]. It is worth pointing out that the integer case has an economic interpretation(see explanation in [23, pp. 63–64]).

C. Terminal curveIn this case the terminal point is described by a given curve w, in the sense that xðTÞ ¼ wðTÞ, where w : ½a; b� ! R is a pre-

scribed differentiable curve. For a small MT , from Taylor’s formula, one has

MxT ¼ wðT þ MTÞ � wðTÞ ¼ w0ðTÞMT þ OðMTÞ2: ð10Þ

Substituting (10) into (9) yields

ðw0ðTÞ � x0ðTÞÞ t I1�aT @3L½x�ðtÞ


n oþ L½x�ðTÞ þ @1/ðT; xðTÞÞ ¼ 0:

For / � 0, we obtain Eq. (29) of [2]. For / � 0 and a! 1, we have Eq. (3.12) of [23].D. Truncated vertical terminal lineNow we consider the case where MT ¼ 0 and xðTÞP xmin. Here xmin is a minimum permissible level of x. By the Kuhn–

Tucker conditions, we obtain


h it¼Tþ @2/ðT; xðTÞÞ 6 0; xðTÞP xmin;

ðxðTÞ � xminÞ t I1�aT @3L½x�ðtÞ


n o¼ 0

for maximization problem; and


h it¼Tþ @2/ðT; xðTÞÞP 0; xðTÞP xmin;

ðxðTÞ � xminÞ t I1�aT @3L½x�ðtÞ


n o¼ 0

for minimization problem. If / � 0 and a! 1, we get Eqs. (3.17) and (3.17’) of [23].E. Truncated horizontal terminal lineIn this situation, the constraints are MxT ¼ 0 and T 6 Tmax. Therefore, as in the previous case, we obtain:


h it¼T

P 0; T 6 Tmax;

ðT � TmaxÞ L½x�ðTÞ þ @1/ðT; xðTÞÞ � x0ðTÞ tI1�aT @3L½x�ðtÞ

h it¼T

h i¼ 0

for maximization problem; and


h it¼T6 0; T 6 Tmax;


ðT � TmaxÞ L½x�ðTÞ þ @1/ðT; xðTÞÞ � x0ðTÞ tI1�aT @3L½x�ðtÞ

h it¼T

h i¼ 0

for minimization problem. Observe that for / � 0 and a! 1, we get Eqs. (3.18) and (3.18’) of [23].

4. Transversality conditions II

In this section we consider the following variational problem:Z
Jðx; TÞ ¼
T

aLfx;ugðt; TÞdt ! extr

ðx; TÞ 2 DxðTÞ ¼ uðTÞ

ð11Þ

where L : ½a; b� � R3 ! R and u : ½a; b� ! R are at least of class C1. Here x ¼ uðtÞ is a specified curve.

Theorem 2. Suppose that ðx; TÞ is a solution to problem (11). Then x is a solution of the fractional Euler–Lagrange equation

@2Lfx;ugðt; TÞ þ tDaTð@3Lfx;ugðt; TÞÞ ¼ 0; ð12Þ

on the interval ½a; T�, and satisfies the transversality condition

ðu0ðTÞ � x0ðTÞÞ t I1�aT @3Lfx;ugðt; TÞ

h it¼Tþu0ðTÞ

Z T

a@4Lfx;ugðt; TÞdt þ Lfx;ugðT; TÞ ¼ 0: ð13Þ

Proof. Suppose that ðx; TÞ is a solution to problem (11) and consider the value of J at an admissible variationðxðtÞ þ �hðtÞ; T þ �MTÞ, where � 2 R is a small parameter, MT 2 R, and h 2 C1ð½a; b�Þ with hðaÞ ¼ 0. Let

jð�Þ ¼ Jðxþ �h; T þ �MTÞ ¼Z Tþ�MT

aLfxþ �h;ugðt; T þ �MTÞdt:

Then, a necessary condition for ðx; TÞ to be a solution to problem (11) is given by

j0ð0Þ ¼ 0()Z T

a@2Lfx;ugðt; TÞhðtÞ þ @3Lfx;ugðt; TÞCa Da

t hðtÞ þ @4Lfx;ugðt; TÞu0ðTÞMTh i

dt þ Lfx;ugðT; TÞMT ¼ 0:

Integrating by parts (cf. Eq. (2)), and since hðaÞ ¼ 0, we get

0 ¼Z T

a@2Lfx;ugðt; TÞ þ tD

aTð@3Lfx;ugðt; TÞÞ

� �hðtÞdt þ I1�a

T ð@3Lfx;ugðt; TÞÞhðtÞ�t¼T þ Lfx;ugðT; TÞMT

þZ T

a@4Lfx;ugðt; TÞu0ðTÞMT½ �dt: ð14Þ

As h and MT are arbitrary, first we consider h and MT such that hðTÞ ¼ 0 and MT ¼ 0. Then, by the fundamental lemma ofthe calculus of variations we deduce Eq. (12). Therefore, in order for ðx; TÞ to be a solution to problem (11), x must be a solu-tion of the fractional Euler–Lagrange equation. But if x is a solution of (12), the first integral in expression (14) vanishes, andthen the condition (14) takes the form

t I1�aT ð@3Lfx;ugðt; TÞÞhðtÞ

h it¼Tþ Lfx;ugðT; TÞMT þ

Z T

a@4Lfx;ugðt; TÞu0ðTÞMT½ �dt ¼ 0: ð15Þ

Since the right hand point of x lies on the curve z ¼ uðtÞ, we have

xðT þ �MTÞ þ �hðT þ �MTÞ ¼ uðT þ �MTÞ:

Hence, differentiating with respect � and setting � ¼ 0 we get

hðTÞ ¼ ðu0ðTÞ � x0ðTÞÞMT: ð16Þ

Substituting (16) into (15) yields

MTðu0ðTÞ � x0ðTÞÞ t I1�aT @3Lfx;ugðt; TÞ

h it¼Tþu0ðTÞMT

Z T

a@4Lfx;ugðt; TÞdt þ Lfx;ugðT; TÞMT ¼ 0:

Since MT can take any value, we obtain condition (13). h

In the case when a goes to 1, by Theorem 2 we obtain the following result.

Corollary 2. If ðx; TÞ gives an extremum for

Jðx; TÞ ¼Z T

aLðt; xðtÞ; x0ðtÞ;uðTÞÞdt;


on the set

x 2 C1ð½a; b�Þ jxðaÞ ¼ xa

n o� ½a; b�;

then x is a solution of the Euler–Lagrange equation

@2Lðt; xðtÞ; x0ðtÞ;uðTÞÞ � ddt@3Lðt; xðtÞ; x0ðtÞ;uðTÞÞ ¼ 0; for all t 2 ½a; T�

and satisfies the following transversality condition:

ðu0ðTÞ � x0ðTÞÞ@3LðT; xðTÞ; x0ðTÞ;uðTÞÞ þu0ðTÞZ T

a@4Lðt; xðtÞ; x0ðtÞ;uðTÞÞdt þ LðT; xðTÞ; x0ðTÞ;uðTÞÞ ¼ 0:

Example 2. Consider the following problem:

Jðx; TÞ ¼Z T

0ð0CDa

t xðtÞÞ2 þu2ðTÞh i

dt !min

xðTÞ ¼ uðTÞ ¼ T;

xð0Þ ¼ 1:

ð17Þ

For this problem, the fractional Euler–Lagrange equation and the transversality condition (see Theorem 2) are given, respec-tively, by

tDaT 0

CDat xðtÞ

� �¼ 0; ð18Þ

2ð1� x0ðTÞÞ t I1�aT 0

CDat xðtÞ

h ijt¼T þ 2T2 þ ð0CDa

t xðTÞÞ2 þ T2 ¼ 0: ð19Þ

Note that it is a difficult task to solve the above fractional equations. For 0 < a < 1 a numerical or direct method should beused [10]. When a goes to 1, problem (17) becomes

Jðx; TÞ ¼Z T

0x0ðtÞÞ2 þu2ðTÞh i

dt !min

xðTÞ ¼ uðTÞ ¼ T;xð0Þ ¼ 1

ð20Þ

and Eqs. (18), (19) are replaced by

x00ðtÞ ¼ 0; ð21Þ

2 1� x0ðTÞð Þx0ðTÞ þ 2T2 þ ðx0ðTÞÞ2 þ T2 ¼ 0: ð22Þ

Solving Eqs. (21) and (22) we obtain that

~xðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�6þ 6

ffiffiffiffiffiffi13pp

� 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�6þ 6

ffiffiffiffiffiffi13pp t þ 1; T ¼ 1

6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�6þ 6

ffiffiffiffiffiffi13pq

is a candidate for minimizer to problem (20).

5. Infinite horizon fractional variational problems

Starting with the Ramsey pioneering work [50], infinite horizon variational problems have been widely used in economics(see, e.g., [18,23,35,40,45] and the references therein). One may assume that, due to some constraints of economical nature,the infinite horizon variational problem does not depend on the usual derivative but on the left Caputo fractional derivative.In this condition one has to consider the following problem:

JðxÞ ¼Z þ1

aL½x�ðtÞdt !max x 2 D1; ð23Þ

where L : ½a;þ1Þ � R2 ! R is at least of class C1 and

D1 ¼ x 2 C1ð½a;þ1ÞÞjxðaÞ ¼ xa

n o:

The integral in problem (23) may not converge. In the case where the integral diverges, there may exist more than one paththat yields an infinite value for the objective functional and it would be difficult to determine which among these paths is


optimal. To handle this and similar situations in a rigorous way, several alternative definitions of optimality for problemswith unbounded time domain have been proposed in the literature (see, e.g., [18,31,35]). Here, we follow Brock’s notionof optimality [18], i.e., a function x 2 D1 is said to be weakly maximal to problem (23) if

limT!þ1

infT 0PT

Z T 0

a½L½x�ðtÞ � L½x�ðtÞ�dt 6 0

for all x 2 D1.Using similar approach as in [40,45], we obtain the transversality condition for infinite horizon fractional variational

problems. First, a lemma that we will use in the proof of Theorem 3.

Lemma 1. If g is continuous on ½a;þ1Þ and

limT!þ1

infT 0PT

Z T 0

agðtÞhðtÞdt ¼ 0

for all continuous functions h : ½a;þ1Þ ! R with hðaÞ ¼ 0, then gðtÞ ¼ 0, for all t P a.

Proof. Can be done in a similar way as the proof of the standard fundamental lemma of the calculus of variations (see, e.g.,[19]). h

Theorem 3. Suppose that x is a weakly maximal to problem (23). Let h 2 C1ð½a;þ1ÞÞ and � 2 R. Define

Að�; T 0Þ ¼Z T 0

a

L½xþ �h�ðtÞ � L½x�ðtÞ�

dt;

Vð�; TÞ ¼ infT 0PT

Z T 0

a½L½xþ �h�ðtÞ � L½x�ðtÞ�dt;

Vð�Þ ¼ limT!þ1

Vð�; TÞ:

Suppose that

1. lim�!0Vð�;TÞ� exists for all T;

2. limT!þ1Vð�;TÞ� exists uniformly for �;

3. For every T 0 > a; T > a, and � 2 R n f0g, there exists a sequence Að�; T 0nÞ� �

n2N such that

limn!þ1

Að�; T 0nÞ ¼ infT 0PT

Að�; T 0Þ;

uniformly for �.

Then x is a solution of the fractional Euler–Lagrange equation

@2L½x�ðtÞ þ tDaTð@3L½x�ðtÞÞ ¼ 0

for all t 2 ½a;þ1Þ, and for all T > t. Moreover it satisfies the transversality condition

limT!þ1

infT 0PT

tI1�aT 0 @3L½x�ðtÞ�t¼T 0xðT

0Þ ¼ 0:

Proof. Observe that Vð�Þ 6 0 for every � 2 R, and Vð0Þ ¼ 0. Thus V 0ð0Þ ¼ 0, i.e.,

0 ¼ lim�!0

Vð�Þ�¼ lim

T!þ1infT 0PT

Z T 0

alim�!0

L½xþ �h�ðtÞ � L½x�ðtÞ�

dt ¼ limT!þ1

infT 0PT

Z T 0

a@2L½x�ðtÞhðtÞ þ @3L½x�ðtÞaCDa

t hðtÞ� �

dt:

Thus, integrating by parts and since hðaÞ ¼ 0, we get

limT!þ1

infT 0PT

Z T 0

a½@2L½x�ðtÞ þ tD

aT 0 ð@3L½x�ðtÞÞ�hðtÞdt þ ½tI1�a

T 0 @3L½x�ðtÞhðtÞ�t¼T 0

( )¼ 0: ð24Þ

By the arbitrariness of h, we may assume that hðT 0Þ ¼ 0 and so

limT!þ1

infT 0PT

Z T 0


aT 0 ð@3L½x�ðtÞÞ

� �hðtÞdt ¼ 0: ð25Þ

Applying Lemma 1 we obtain


@2L½x�ðtÞ þ tDaTð@3L½x�ðtÞÞ ¼ 0

for all t 2 ½a;þ1Þ, and for all T > t. Substituting Eq. (25) into Eq. (24) we get

limT!þ1

infT 0PT

tI1�aT 0 @3L½x�ðtÞ

h it¼T 0

hðT 0Þ ¼ 0: ð26Þ

To eliminate h from Eq. (26), consider the particular case hðtÞ ¼ vðtÞxðtÞ, where v : ½a;þ1Þ ! R is a function of class C1 suchthat vðaÞ ¼ 0 and vðtÞ ¼ const – 0, for all t > t0, for some t0 > a. We deduce that

limT!þ1

infT 0PT

tI1�aT 0 @3L½x�ðtÞ

h it¼T 0

xðT 0Þ ¼ 0: �

6. Conclusion

Transversality conditions are optimality conditions that are used along with Euler–Lagrange equations in order to find theoptimal paths (plans, programs, trajectories, etc.) of dynamical models. The importance of such conditions is well known ineconomics models and other phenomena whose effects can be spread along time, e.g., radioactive, pollution. In this paper wehave given in the compact form transversality conditions for fractional variational problems with the Caputo derivative. Thefractional variational theory is only 15 years old so there are many unanswered questions. For instance, the question of exis-tence of solutions to fractional variational problems is a complete open area of research (more about this important issue thereader can find in [21,43,53,54]). Other interesting open question is about the convergence of the objective functionalJðxÞ ¼

Rþ1a L½x�ðtÞdt. This issue should be treat carefully, especially as the functional depends on fractional type of derivatives.

Generally, in order to solve fractional Euler–Lagrange differential equations and apply transversality conditions one needs touse numerical methods. Some progress is being made already on the numerical methods for fractional variational problems[4,8,48,49] but there is still a long way to go. We believe that all pointed issues will be considered in a forthcoming papers.

Acknowledgments

This work is supported by FEDER funds through COMPETE – Operational Programme Factors of Competitiveness (‘‘Prog-rama Operacional Factores de Competitividade’’) and by Portuguese funds through the Center for Research and Developmentin Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (‘‘FCT —Fundação para a Ciência e a Tecnologia’’), within Project PEst-C/MAT/UI4106/2011 with COMPETE Number FCOMP-01–0124-FEDER-022690. Agnieszka B. Malinowska is supported by Białystok University of Technology Grant S/WI/02/2011.

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Generalized transversality conditions in fractional calculus of variations