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: Formulation of Euler–Lagrange equations for
fractional variational problems
爱思唯尔期刊 2001.7.27
外文文献：
Formulation of Euler–Lagrange equations
for fractional variational problems
1. Introduction
The field of calculus of variations is of significant importance in various disciplines such as science, engineering, and pure and applied mathematics. Reference [7] presents a Bliss-type multiplier rule for constrained variational problems with delay. Calculus of variations has been the starting point for various approximate numerical schemes such as Ritz, finite difference, and finite element methods (see [2,8]).
Functional minimization problems naturally occur in engineering and science where minimization of functionals, such as, Lagrangian, strain, potential, and total energy, etc. give the laws governing the systems behavior. In optimal control theory, minimization of certain functionals give control functions for optimum performance of the system.
Although many laws of the nature can be obtained using certain functionals and the theory of calculus of variations, not all laws can be obtained this way. For example, almost all systems contain internal damping, yet the traditional energy based approach cannot be used to obtain equations describing the behavior of a nonconservative system (see [9,10]). Recently, Refs. [9,10] presented a new approach to mechanics that allows one to obtain the equations for a nonconservative system using certain functionals. In these references, fractional derivative terms were introduced in functionals to obtain nonconservative terms in the desired differential equations.
Fractional derivatives, or more precisely derivatives of arbitrary orders, have played a significant role in engineering, science, and pure and applied mathematics in recent years. As [11] point out, there is hardly a field or science or engineering that has remained untouched by this field. Reference [12] provide an encyclopedic treatment of this subject. Additional background, survey, and application of this field in science, engineering, and mathematics can be found, among others, in [11–17].
Recent investigations have shown that many physical systems can be represented more accurately using fractional derivative formulations. Given this, one can imagine obtaining these
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formulations by minimizing certain functionals. These functionals will naturally contain fractional derivative terms, and mathematical tools analogous to calculus of variations will be needed to minimize these functional. However, very little work has been done in the area of fractional calculus of variations.
This paper provides some new results in the area of fractional calculus of variations. Afractional calculus of variations problem is a problem in which either the objective functional or the constraint equations or both contain at least one fractional derivative term. In this paper we will develop necessary conditions for two problems from this field, first, minimization of a functional subjected to specified boundary conditions, and second, minimization of a functional subjected to constrains and specified boundary conditions. Both functional and the constraints will be allowed to have fractional derivative terms. 2. The simplest fractional variational problem
Several definitions of a fractional derivative have been proposed. These definitions include Riemann–Liouville, Grunwald–Letnikov, Weyl, Caputo, Marchaud, and Riesz fractional derivatives. Here, we formulate the problem in terms of the left and the right Riemann–Liouville fractional derivatives, which are defined as [16].
The left Riemann–Liouville fractional derivative
1?d?Df(x)???ax?(n??)?dx??nxn???1(x??)f(?)d?， （1） ?a
and
The right Riemann–Liouville fractional derivative
1?d????xDbf(x)??(n??)?dx??nxn???1(x??)f(?)d?, （2） ?aWhere ? is the order of the derivative such that n?1???n.If?is an integer, these derivatives are defined in the usual sense, i.e.,
??d?Df(x)?d?????ax?dx?， xDbf(x)???? ??1,2,???? （3）
?dx???These derivatives will be denoted as the LRLFD and the RRLFD, respectively. Note that in the literature the Riemann–Liouville fractional derivative generally means the LRLFD. From physical point of view, ifxis considered as a time scale, the RRLFD represents an operation
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performed on the future state of the processf(x). This derivative has generally been neglected with the assumption that the present state of a process does not depend on the results of its future development. However, the derivations to follow will show that both derivatives naturally occur in a problem of fractional calculus of variations.
Using the above definitions, the first simplest fractional calculus of variations problem can be defined as follows:Let F(x,y,u,v)be a function with continuous first and second(partial) derivatives with respect to all its arguments. Then, among all functionsy(x)which have continuous LRLFD of order?and RRLFD of order?fora?x?band satisfy the boundary conditions
y(a)?ya,y(b)?yb （4） find the function for which the functional
?J?y???F(x,y,aDxy,xDb?y)dxab （5）
is an extremum, where0??,??1. The continuity requirement onFcan be given more precisely. However, these assumptions are made for simplicity. Note that (1) we have included both the LRLFD and the RRLFD for generality. (2) We first consider 0??,??1. The case of ?,??R? will be consider shortly. (3) When ????1, the above problem reduces to the simplest variational problem.
To develop the necessary conditions for the extremum, assume that y?(x)is the desired function. Let??R, and define a family of curve
y(x)?y?(x)???(x) （6）
which satisfy the boundary conditions; i.e., we require that
?a??b?0 （7）
?Since aDx and xDb?are linear operators, it follows that
a????Dxy(x)?aDxy(x)??aDx?(x)， （8a)
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????Dby(x)?xDby(x)??xDb?(x) (8b)
xSubstituting Eqs. (6) and (8) into Eq. (5), we find that for each ?(x)
????J?J?????F(x,y????,aDxy???aD??,Dy??DXxbb?)dx (9)
abis a function of ?only. Note that J??? is extremum at ??0. Differentiating Eq. (9)with respect to ?, we obtain
b??FdJ?F?F????????D??D?dx (10) ?ax?xb?d?a??y?aDxy?xDby?Equation (10) is also called the variations of J?y?at y(x)along ?(x). A necessary condition for to have an extremum is that dJd?must be zero, and this should be true for all admissible?(x). This leads to the condition that forJ?y?to have an extremum for y?y?(x)is that
??F?F?F?????D??D?dx?0 (11) ??ax?xb???y?aDxy?xDby?a?for all admissible η(x). Using the formula for fractional integration by parts, the second integral in Eq. (11) can be written as
b??F??F??D?dx?D()???dx (12) xb?ax????Dy?aDxy?a?axabbprovided that?F?aDxy or ?is zero at x?a and x?b. Using Eq. (7), this condition is satisfied, and it follows that Eq. (12) is valid. Similarly, the third integral in Eq. (11) can be written as
b??F??F??D?dx?D()???dx (13) ax?xb????Dy?Dyxba?xba?b?Substituting Eqs. (12) and (13) into Eq. (11), we get
??F?F?F????D?D?dx?0. (14) xbax??????y?aDxy?xDby?a?Since ?(x) is arbitrary, it follows from a well established result in calculus of variations that
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