Computational Methods in the Fractional Calculus of Variations

The following sections are included:

• Preface
• Contents

This book is devoted to the study of numerical methods in the calculus of variations and optimal control in the presence of fractional derivatives and/or integrals. A fractional problem of the calculus of variations and optimal control consists in the study of an optimization problem in which the objective functional or constraints depend on derivatives and integrals of arbitrary, real or complex, orders. This is a generalization of the classical theory, where derivatives and integrals can only appear in integer orders. Throughout the book, we call the problems in the calculus of variations and optimal control variational problems. If at least one fractional term exists in the formulation, it is called a fractional variational problem…

In this and the next two chapters we review the basic concepts that have an essential role in the understanding of the book. Chapter 2 starts with the notion of the calculus of variations and, without going into details, also recalls the optimal control theory, pointing out the variational approach together with main concepts, definitions, and some important results from the classical theory. A brief historical introduction to the fractional calculus is given afterwards, in Chapter 3. At the same time, we introduce the theoretical framework of the whole work, fixing notations and nomenclature. In Chapter 4, the calculus of variations and optimal control problems involving fractional operators are discussed as fractional variational problems.

In the early ages of modern differential calculus, right after the introduction of for the first derivative, in a letter dated 1695, l'Hopital asked Leibniz the meaning of , the derivative of order (Miller and Ross, 1993). The appearance of as a fraction gave the name fractional calculus to the study of derivatives, and integrals, of any order, real or complex…

A fractional problem of the calculus of variations and optimal control consists in the study of an optimization problem, in which the objective functional or constraints depend on derivatives and/or integrals of arbitrary, real or complex, orders. This is a generalization of the classical theory, where derivatives and integrals can only appear in integer orders. The aim of this chapter is to present necessary conditions that every extremizer of a functional must fulfill. Usually such equations involve fractional operators, and are called (fractional) Euler–Lagrange equations. Solving such fractional equations we obtain a list of possible solutions for the problem. In the literature we can find several distinct types of functionals, involving distinct fractional operators, and thus different Euler–Lagrange equations are deduced (Malinowska and Torres, 2012). This chapter is essentially based on the original results published in (Almeida, 2012, 2013).

In this chapter we give a short survey on the numerical methods for solving fractional variational problems. As mentioned earlier, the fractional calculus of variations started with the works of Riewe (1996, 1997), in the last years of the nineties. Later, the notion of fractional optimal control appeared in the works of Agrawal (2004) and Frederico and Torres (2008a). It is not surprising that the numerical achievements in these fields is at an early stage. In this chapter we shall review some recent papers which can be classified as direct or indirect methods…

This chapter is devoted to two approximations for the Riemann–Liouville, Caputo and Hadamard derivatives that are referred to as fractional operators afterwards. We introduce the expansions of fractional operators in terms of infinite sums involving only integer-order derivatives. These expansions are then used to approximate fractional operators in problems like fractional differential equations, fractional calculus of variations, fractional optimal control, etc. In this way, one can transform such problems into classical problems. Hereafter, a suitable method, that can be found in the classical literature, is employed to find an approximate solution for the original fractional problem. Here we focus mainly on the left derivatives and the details of extracting corresponding expansions. Right derivatives are given whenever it is needed to apply new techniques.

We obtain a new decomposition of the Riemann–Liouville operators of fractional integration as a series involving derivatives (of integer-order). The new formulas are valid for functions of class Cⁿ, n ∈ ℕ, and allow us to develop suitable numerical approximations with known estimations for the error. The usefulness of the obtained results, in solving fractional integral equations, is illustrated in (Pooseh, Almeida and Torres, 2012a).

In the presence of fractional operators, the same ideas that were discussed in Section 2.1.3, are applied to discretize the problem. Many works can be found in the literature that use different types of basis functions to establish Ritz-like methods for the fractional calculus of variations and optimal control. Nevertheless, finite differences have received less interest. A brief introduction of using finite differences has been made in (Riewe, 1996), which can be regarded as a predecessor to what we call here an Euler-like direct method. A generalization of Leitmann's direct method can be found in (Almeida and Torres, 2010), while (Lotfi, Dehghan and Yousefi, 2011) discusses the Ritz direct method for optimal control problems that can easily be reduced to a problem of the calculus of variations.

As in the classical case, indirect methods in fractional sense provide necessary conditions of optimality using the first variation. Fractional Euler-Lagrange equations are now a well-known and well-studied subject in fractional calculus. For a simple problem of the form (4.1), following (Agrawal, 2002), a necessary condition implies that the solution must satisfy a fractional boundary value differential equation…

This chapter is devoted to fractional-order optimal control problems in which the dynamic control system involves integer and fractional-order derivatives and the terminal time is free. Necessary conditions for a state/control/terminal-time triplet to be optimal are obtained. Situations with constraints present at the end time are also considered. Under appropriate assumptions, it is shown that the obtained necessary optimality conditions become sufficient. Numerical methods to solve the problems are presented, and some computational simulations are discussed in detail (Pooseh, Almeida and Torres, 2014).

We now obtain approximation formulas for fractional integrals and derivatives of Riemann–Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type.

In this chapter we introduce a discrete-time fractional calculus of variations on the time scales ℤ and (hℤ)_a. First and second-order necessary optimality conditions are established. Some numerical examples illustrating the use of the new Euler–Lagrange and Legendre type conditions are given.

The realm of numerical methods in scientific fields is vastly growing due to the very fast progresses in computational sciences and technologies. Nevertheless, the intrinsic complexity of fractional calculus, caused partially by non-local properties of fractional derivatives and integrals, makes it rather difficult to find efficient numerical methods in this field. It seems enough to mention here that, up to the time of this book, and to the best of our knowledge, there is no routine available for solving a fractional differential equation as there is the Runge–Kutta method for ordinary ones. Despite this fact, however, the literature exhibits a growing interest and improving achievements in numerical methods for fractional calculus in general and fractional variational problems specifically…

The following sections are included:

• MATLAB® code
• Bibliography
• Index