Fractional EulerLagrange differential equations via Caputo derivatives^{†}^{†}thanks: This is a preprint of a paper whose final and definite form will appear as Chapter 9 of the book Fractional Dynamics and Control, D. Baleanu et al. (eds.), Springer New York, 2012, DOI:10.1007/9781461404576_9, in press.
Abstract
We review some recent
results of the fractional variational calculus.
Necessary optimality conditions of Euler–Lagrange
type for functionals with a Lagrangian
containing left and right Caputo derivatives are given.
Several problems are considered: with fixed or free
boundary conditions, and in presence of
integral constraints that also depend on Caputo derivatives.
MSC 2010: 26A33; 34K37; 49K05; 49K21.
1 Introduction
Fractional calculus plays an important role in many different areas, and has proven to be a truly multidisciplinary subject Kilbas ; Podlubny . It is a mathematical field as old as the calculus itself. In a letter dated 30th September 1695, Leibniz posed the following question to L’Hopital: “Can the meaning of derivative be generalized to derivatives of noninteger order?” Since then, several mathematicians had investigated Leibniz’s challenge, prominent among them were Liouville, Riemann, Weyl, and Letnikov. There are many applications of fractional calculus, e.g., in viscoelasticity, electrochemistry, diffusion processes, control theory, heat conduction, electricity, mechanics, chaos and fractals, and signals and systems MR2605606 ; Magin:et:all .
Several methods to solve fractional differential equations are available, using Laplace and Fourier transforms, truncated Taylor series, and numerical approximations. In Almeida4 a new direct method to find exact solutions of fractional variational problems is proposed, based on a simple but powerful idea introduced by Leitmann, that does not involve solving (fractional) differential equations Tor:Leit . By an appropriate coordinate transformation, we rewrite the initial problem to an equivalent simpler one; knowing the solution for the new equivalent problem, and since there exists an onetoone correspondence between the minimizers (or maximizers) of the new problem with the ones of the original, we determine the desired solution. For a modern account on Leitmann’s direct method see MyID:183 ; MyID:187 .
The calculus of variations is a field of mathematics that deals with extremizing functionals vanBrunt . The variational functionals are often formed as definite integrals involving unknown functions and their derivatives. The fundamental problem consists to find functions , , that extremize a given functional when subject to boundary conditions and . Since this can be a hard task, one wishes to study necessary and sufficient optimality conditions. The simplest example is the following one: what is the shape of the curve , , joining two fixed points and , that has the minimum possible length? The answer is obviously the straight line joining and . One can obtain it solving the corresponding Euler–Lagrange necessary optimality condition. If the boundary condition is not fixed, i.e., if we are only interested in the minimum length, the answer is the horizontal straight line , (free endpoint problem). In this case we need to complement the Euler–Lagrange equation with an appropriate natural boundary condition. For a general account on Euler–Lagrange equations and natural boundary conditions, we refer the reader to MyID:141 ; MyID:169 and references therein. Another important family of variational problems is the isoperimetric one MyID:136 . The classical isoperimetric problem consists to find a continuously differentiable function , , satisfying given boundary conditions and , which minimizes (or maximizes) a functional
subject to the constraint
The most famous isoperimetric problem can be posed as follows. Amongst all closed curves with a given length, which one encloses the largest area? The answer, as we know, is the circle. The general method to solve such problems involves an Euler–Lagrange equation obtained via the concept of Lagrange multiplier (see, e.g., MyID:131 ).
The fractional calculus of variations is a recent field, initiated in 1997, where classical variational problems are considered but in presence of some fractional derivative or fractional integral Riewe:1997 . In the past few years an increasing of interest has been put on finding necessary conditions of optimality for variational problems with Lagrangians involving fractional derivatives Agrawal ; Ata:et:al ; Baleanu1 ; Baleanu2 ; ElNabulsi:Torres ; Frederico:Torres ; MyID:089 ; MyID:149 ; MyID:163 ; MyID:181 , fractional derivatives and fractional integrals MyID:182 ; Almeida1 ; MyID:085 , classical and fractional derivatives MyID:207 , as well as fractional difference operators MyID:152 ; MyID:179 . A good introduction to the subject is given in the monograph Klimek . Here we consider unconstrained and constrained fractional variational problems via Caputo operators.
2 Preliminaries and notations
There exist several definitions of fractional derivatives and fractional integrals, e.g., Riemann–Liouville, Caputo, Riesz, Riesz–Caputo, Weyl, Grunwald–Letnikov, Hadamard, and Chen. Here we review only some basic features of Caputo’s fractional derivative. For proofs and more on the subject, we refer the reader to Kilbas ; Podlubny .
Let be an integrable function, , and be the Euler gamma function. The left and right Riemann–Liouville fractional integral operators of order are defined by^{1}^{1}1Along the work we use round brackets for the arguments of functions, and square brackets for the arguments of operators. By definition, an operator receives a function and returns another function.
and
respectively. The left and right Riemann–Liouville fractional derivative operators of order are, respectively, defined by
and
where . Interchanging the composition of operators in the definition of Riemann–Liouville fractional derivatives, we obtain the left and right Caputo fractional derivatives of order :
and
Theorem 2.1
Assume that is of class on . Then its left and right Caputo derivatives are continuous on the closed interval .
One of the most important results for the proof of necessary optimality conditions, is the integration by parts formula. For Caputo derivatives the following relations hold.
Theorem 2.2
Let , and be functions. Then,
and
where and whenever .
In the particular case when , we get from Theorem 2.2 that
and
In addition, if is such that , then
and
Along the work, we denote by , (), the partial derivative of function with respect to its th argument. For convenience of notation, we introduce the operator defined by
where .
3 Euler–Lagrange equations
The fundamental problem of the fractional calculus of variations is addressed in the following way: find functions ,
that maximize or minimize the functional
(1) 
As usual, the Lagrange function is assumed to be of class on all its arguments. We also assume that has continuous right Riemann–Liouville fractional derivative of order and has continuous left Riemann–Liouville fractional derivative of order for .
In Agrawal a necessary condition of optimality for such functionals is proved. We remark that although functional (1) contains only Caputo fractional derivatives, the fractional Euler–Lagrange equation also contains Riemann–Liouville fractional derivatives.
Theorem 3.1 (Euler–Lagrange equation for (1))
If is a minimizer or a maximizer of on , then is a solution of the fractional differential equation
(2) 
for all .
Proof
4 The isoperimetric problem
The fractional isoperimetric problem is stated in the following way: find the minimizers or maximizers of functional as in (1), over all functions satisfying the fractional integral constraint
Similarly as , is assumed to be of class with respect to all its arguments, function is assumed to have continuous right Riemann–Liouville fractional derivative of order and continuous left Riemann–Liouville fractional derivative of order for . A necessary optimality condition for the fractional isoperimetric problem is given in Almeida3 .
Theorem 4.1
Let be a minimizer or maximizer of on , when restricted to the set of functions such that . In addition, assume that is not an extremal of . Then, there exists a constant such that is a solution of
(3) 
for all , where .
Proof
Given , and , consider
and
Since is not an extremal for , there exists a function such that
and by the implicit function theorem, there exists a function , defined in some neighborhood of zero, such that
Applying the Lagrange multiplier rule (see, e.g., (vanBrunt, , Theorem 4.1.1)) there exists a constant such that
Differentiating and at , and integrating by parts, we prove the theorem.
Example 1
The case when is an extremal of is also included in the results of Almeida3 .
Theorem 4.2
If is a minimizer or a maximizer of on , subject to the isoperimetric constraint , then there exist two constants and , not both zero, such that
for all , where .
5 Transversality conditions
We now give the natural boundary conditions (also known as transversality conditions) for problems with the terminal point of integration free as well as .
Let
The type of functional we consider now is
(4) 
where the operator is defined by
These problems are investigated in Agrawal and more general cases in Almeida2 .
Theorem 5.1
Suppose that minimizes or maximizes defined by (4) on . Then
(5) 
for all . Moreover, the following transversality conditions hold:
Proof
The result is obtained by considering variations of function and variations of as well, and then applying the Fermat theorem, integration by parts, Leibniz’s rule, and using the arbitrariness of and .
Transversality conditions for several other situations can be easily obtained. Some important examples are:

If is fixed but is free, then besides the Euler–Lagrange equation (5) one obtains the transversality condition

If is given but is free, then the transversality condition is

If is not given but is restricted to take values on a certain given curve , i.e., , then
Acknowledgements.
Work supported by FEDER funds through COMPETE — Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT — Fundação para a Ciência e a Tecnologia”), within project PEstC/MAT/UI4106/2011 with COMPETE number FCOMP010124FEDER022690. Agnieszka Malinowska is also supported by Białystok University of Technology grant S/WI/2/2011.Bibliography
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