Introduction to the Fractional Calculus of Variations

Agnieszka B Malinowska, Delfim F M Torres

• Publisher: ICP
• Year: 2012
• Print ISBN:
  9781848169661
• eBook ISBN:
  9781848169685
• Topic: Mathematics
• Subtopic: Calculus
• Index: Mathematics

This invaluable book provides a broad introduction to the fascinating and beautiful subject of Fractional Calculus of Variations (FCV). In 1996, FVC evolved in order to better describe non-conservative systems in mechanics. The inclusion of non-conservatism is extremely important from the point of view of applications. Forces that do not store energy are always present in real systems. They remove energy from the systems and, as a consequence, Noether's conservation laws cease to be valid. However, it is still possible to obtain the validity of Noether's principle using FCV. The new theory provides a more realistic approach to physics, allowing us to consider non-conservative systems in a natural way. The authors prove the necessary Euler–Lagrange conditions and corresponding Noether theorems for several types of fractional variational problems, with and without constraints, using Lagrangian and Hamiltonian formalisms. Sufficient optimality conditions are also obtained under convexity, and Leitmann's direct method is discussed within the framework of FCV.

The book is self-contained and unified in presentation. It may be used as an advanced textbook by graduate students and ambitious undergraduates in mathematics and mechanics. It provides an opportunity for an introduction to FCV for experienced researchers. The explanations in the book are detailed, in order to capture the interest of the curious reader, and the book provides the necessary background material required to go further into the subject and explore the rich research literature.

Contents:

• The Classical Calculus of Variations
• Fractional Calculus of Variations via Riemann–Liouville Operators
• Fractional Calculus of Variations via Caputo Operators
• Other Approaches to the Fractional Calculus of Variations
• Towards a Combined Fractional Mechanics and Quantization

Readership: Advanced undergraduate, graduate students and researchers in mathematics, physics, operations research and applied sciences.

Chapter 1

The Classical Calculus of Variations

For the convenience of the reader, we begin with some well-known definitions and facts from the classical calculus of variations. With the exception of Section 1.5, results are given without proofs. For proofs and detailed discussions, we refer the reader to one of the many books on the subject (e.g., Giaquinta and Hildebrandt, 1996; Troutman, 1996; van Brunt, 2004). Here we follow Chachuat, 2007, which gives all the necessary background for our purposes.

1.1 Problem Statement

We are concerned with the problem of finding minima (or maxima) of a functional J : D → R, where D is a subset of a (normed) linear space D of real-valued (or real-vector-valued) functions. The formulation of a problem of the calculus of variations requires two steps: the specification of a performance criterion, and the statement of physical constraints that should be satisfied. The performance criterion J,also called cost functional (or objective), must be specified for evaluating quantitatively the performance of the system under study. The typical form of the cost is

where t ∈ [a, b] is the independent variable, usually called time; y(t) ∈ R^N, N ≥ 1, is a real vector variable, the functions y(t), a ≤ t ≤ b, are generally called trajectories or curves; y′(t) ∈ R^N stands for the derivative of y(t)with respect to time t; and L : [a, b] × R^2N → R is a real-valued function, called the Lagrangian.

Enforcing constraints in the optimization problem reduces the set of candidate functions and leads to the following definition.

Definition 1.1. A trajectory y ∈ D is said to be an admissible trajectory (or admissible function), provided it satisfies all the constraints of the problem along the interval [a, b]. The set of admissible trajectories is denoted by D.

A great variety of boundary conditions is of interest. The simplest one is to enforce both end-points fixed, e.g., y(a) = y_a and y(b) = y_b, y_a, y_b ∈ R^N. Alternatively, we may require that the trajectory y ∈ D joins a fixed point (a,y_a) to a specified curve f(t), a ≤ t ≤ T. In this case, not only the optimal trajectory y shall be determined, but also the optimal value of b. Besides boundary constraints, another type of constraints is often required,

where G^j : [a,b] × R^2N → R, j = 1,... ,r. These constraints are often referred to as isoperimetric constraints. Similar constraints with ≤ sign can be considered. More generally, constraints of the form

are called constraints of Lagrange form.

Having defined an objective functional J and constraints, one must then decide about the class of functions with respect to which the optimization shall be performed. The traditional choice in the calculus of variations is to consider the class of continuously differentiable functions, e.g., C¹([a,b]). We endow C¹ ([a, b])with a norm. The most natural choice for a norm on C¹([a,b])is

where || · || stands for the Euclidean norm in R^N. The class of functions C¹([a, b]) endowed with || · ||_1,∞ is a Banach space.

We now define what is meant by a minimum of J on D.

Definition 1.2. A trajectory ȳ ∈ D is said to be a local minimizer (resp. local maximizer) for J on D, if there exists δ > 0 such that J(ȳ) ≤ J(y) (resp. J (ȳ) ≥ J(y)) for all y ∈ D with ||y − ȳ|| _1,∞ < δ.

The concept of variation of a functional is central to the solution of problems o...

Table of contents

• Cover
• Half Title
• Title
• Copyright
• Preface
• contents
• chapter 1
• chapter 2
• chapter3
• chapter 4
• chapter 5
• Bibliography
• Index