It can now be shown that
?F?F?F????T?xDb?D?l(x)?0. (38) ax???y?aDxy?xDby?yThis follows from the fact that l(x)may be selected such that m of the n equations in Eq. (38) are zero. This is true since???yhas a full rank. Rest of theη’s can be selected as independent and therefore the other n?m equations in (38) follows by using Eq. (37) and applying a theorem in calculus of variations. Note that Eq. (36) can now be obtained using Eqs. (35) and (38). Equation (38) will be called the Euler–Lagrange equation for constrained fractional variational problems.
The multiplier rule is also applicable for the case when?is also a function of the LRLFD and the RRLFD. Multiplier rule for a system containing multiple fractional derivatives can be developed in a similar manner. 5. Examples
In this section, we obtain the Euler–Lagrange equations for an unconstrained and a constrained fractional variational problems.
Example 1. As the first example, consider the following unconstrained fractional variational problem:
1?minimize J?y???(0Dxy)2dx (39)
20such that
y(0)?0 and y(1)?1. (40)
1This example with??1, for which the solution isy(x)?x, is often considered in textbooks on variational calculus. It can be shown that for this problem, the Euler–Lagrange equation is
x??D1(0Dxy)?0. (41)
It can be shown that for??12, the solution is given as
y(x)?(2??1)?dt. (42) 1??0?(1?t)(x?t)?xExample 2. As the second example, consider the following constrained fractional variational
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problem:
12minimize J?y???y12?y2dx (43)
201??such that
?D0xy1??y1?y2, (44)
y1(0)?1. (45)
This example with integral order derivative is often considered in textbooks on optimal control. It can be shown that for this problem, the Euler–Lagrange equation is
y1?l?xDl?l?0, (46)
y2?l?0. (47)
Note that in both examples both the LRLFD and the RRLFD occur in the resulting Euler–Lagrange equations even when the problems contain only LRLFDs. Such differential equations have not been studied much in the literature. A method to find solutions for such problems will be presented in a later work.
Remarks. In closing, we would like to make the following two remarks.
1. Here we have assumed that the terminal conditions are fixed and the functions meet all the smoothness requirements. The case of unspecified end conditions, unspecified end points, (the transversality conditions), and piecewise smoothness (the corner conditions) will be considered in a future work.
2. The theorems and their proofs presented here are very similar to those given in standard textbooks on calculus of variations. Thus, many of the concepts of classical calculus of variations can be extended with minor modifications to fractional calculus of variations. Given the fact that many systems are described more accurately using fractional derivative models and that nature attempts to minimize certain functionals, it is hoped that more research will continue in this field. 6. Conclusions
Euler–Lagrange equations have been presented for unconstrained and constrained fractional variational problems. The approach presented and the resulting equations are very similar to those for variational problems containing integral order derivatives. In special cases, when the
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derivatives are of integral order only, the results of fractional calculus of variations reduce to those obtained from classical calculus of variations. Given the fact that many systems can be modeled more accurately using fractional derivative models, it is hoped that future research will continue in this area.
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中文译文：
分数阶变分问题的欧拉–拉格朗日方程
1.引言
变分法领域具有重要意义在不同学科如科学、工程和纯和应用数学。文献[7]提出了一种延迟Bliss-type乘数规则约束变分问题。变分法已经成为各种近似数值方案的出发点如里兹，有限差分，有限元方法。
函数最小化问题自然发生在工程和科学最小化的泛函,如拉格朗日,应变,潜力,和总能量等。给系统行为规律。在最优控制理论,最小化的泛函,给控制功能系统的最佳性能。
虽然许多自然规律可以使用某些泛函和变分理论得到的，并不是所有的法律可以通过这种方式获得。例如,几乎所有的系统包含内部阻尼,然而,传统的基于能量的方法不能用于获得方程描述非守恒的系统的行为。最近，文献[9,10] 提出了一种新的力学方法,允许一个获得使用特定的泛函,非守恒的系统方程。在这些引用,分数导数方面介绍了泛函,得到非守恒的条件所需的微分方程。
近年来，分数阶导数，或者更准确地说是任意阶导数，在工程，科学和纯粹与应用数学中发挥了重要作用。[ 11 ]指出，几乎没有一个，一直保持不变，这一领域或科学或工程。文献[ 12 ]提供的这个问题的百科全书式的处理。额外的背景,调查,和这一领域在科学、工程和数学的应用可以在[11–17]找到。
最近的调查表明,许多物理系统可以更准确地使用分数导数公式来表示。鉴于此, 我们可以想象通过某些泛函的最小化获得这些构想。这些泛函将自然地包含分数阶导数和类似于变分法的数学工具将需要减少这些功能。然而，在分数阶微积分领域极少工作已经完成。
本文在分数微积分的变化方面提供了一些新的结果。Afractional变分法的问题是一个问题的目标函数或约束方程或者两者都包含至少一个分数导数。在本文中,我们将开发必要条件的两个问题,第一,最小化的功能被指定的边界条件,第二,最小化的功能受到约束和边界条件指定。功能和约束将被允许有分数导数的条件。 2．最简单的分数阶变分问题
一个分数阶导数的定义已经提出了几种。这些定义包括黎曼—刘维尔，格伦沃–刘维尔，外尔，卡普托, 马尔绍,和里斯分数阶导数。在这里，我们制定的左边和右边的黎曼–刘维尔分数导数项的问题，它被定义为[16]。
左黎曼–刘维尔分数阶导数
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1?d?n???1 （1） Df(x)?(x??)f(?)d???ax??(n??)dx??a?nx右黎曼–刘维尔分数阶导数
1?d??n???1 （2） Df(x)??(x??)f(?)d???xb??(n??)?dx?a是导数的次数且n，如果?是个整数，那这些导数通常定义为 ??1???n??d??Df(x)?d?????ax?1,2,???? （3） x)???? ??dx?，xDbf(dx???nx这些导数分别被表示为左导数和右导数。值得注意的是，在黎曼—刘维尔分数导数一般是指左导数。从物理的角度来看，ifxis视为一个时间尺度，右导数代表了一个操作进程f(x)的未来状态的过程。这些导数普遍被一个进程的当前状态的假设所忽视，而不依赖于其未来发展的结果。然而，推导将表明，导数自然地出现一个分数微积分的变化问题。 使用上面的定义，第一个简单的变化分数微积分问题可以定义如下：令F(x,y,u,v)是对它所有参数有一阶和二阶导数的函数。然后，在所有的函数中，当a时，y(x)都有连?x?b续的?阶左导数和?阶右导数，并且满足边界条件
y(a)?y,y(b)?y （4） ab?????JyF(x,y,Dy,Dy)dxaxxb?a （5）
b是一个极值，在这0?。对F可以给出更精确的连续性要求。然而，这些假设都?,??1很简单。注意到（1）包含一般性的左导数和右导数。（2）我们首先考虑的0。??,??1?,??R?的情况将会简明地考虑到。当????1，上述问题降低到最简单的变分问题。
为开发极值的必要条件,假设y?(x)是所需的函数。令??R，定义一组曲线 （6） y(x)?y?(x)???(x)满足边界条件；我们要求
??0 （7） a?b??因为aDx和xDb?是线性算子，它遵循
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