Computational Methods in the Fractional Calculus of Variations and Optimal Control
Universidade de Aveiro（阿威罗大学）；University of Minho（米尼奥大学）
Shakoor Pooseh
备注：
PhD thesis, Doctoral Programme in Mathematics and Applications (PDMA), University of Aveiro and University of Minho, 2013. Supervisor: Delfim F. M. Torres; co-supervisor: Ricardo Almeida. Defended and approved 18-Sept-2013. See http://hdl.handle.net/10773/11510. University of Aveiro, PhD thesis, 2013. RIA:hdl.handle.net/10773/11510
The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the lack of analytic methods to solve such fractional problems, numerical techniques are developed. Here, we mainly investigate the approximation of fractional operators by means of series of integer-order derivatives and generalized finite differences. We give upper bounds for the error of proposed approximations and study their efficiency. Direct and indirect methods in solving fractional variational problems are studied in detail. Furthermore, optimality conditions are discussed for different types of unconstrained and constrained variational problems and for fractional optimal control problems. The introduced numerical methods are employed to solve some illustrative examples.
The Variable-Order Fractional Calculus of Variations
这本书打算深化分数微积分的研究，特别强调变阶算子。报告分为两部分，内容如下。在第一部分中，我们回顾了分数阶微积分（第1章）和分数阶变分微积分（第2章）的基本概念。在第一章中，我们首先对分数阶微积分做了一个简单的概述，并介绍了分数阶微积分中的一些特殊函数的理论。然后，我们回顾了几个分数阶算子（积分和导数）的定义和所考虑的分数阶导数和积分的一些性质。在本章的最后，我们回顾了不同算子的分部积分法公式。第二章简要介绍了经典的变分法，并回顾了不同的变分问题，如等周问题或端点可变的问题。在本章的最后，我们介绍了分数阶变分法的理论和几个变阶分数阶变分问题。在第二部分中，我们系统化了（Tavares，Almeida and Torres，2015，2016，2017，2018）中关于变阶分数阶微积分的一些最新结果。在第三章中，我们考虑了三种不同阶数的分数阶Caputo导数，给出了这三种分数阶Caputo导数的逼近公式，并证明了误差的上界公式。第四章介绍了变阶算子的组合Caputo分数阶导数和相应的高阶算子。并给出了一些性质。然后，我们证明了几种类型的分数阶变分问题的分数阶欧拉-拉格朗日方程，有或无约束。
Fractional $h$-difference equations arising from the calculus of variations
本文介绍了[N. R. O.巴塞河A. C. Ferreira，D. F. M. Torres：离散时间分数变分问题，信号处理。91（2011），no. 3，513- 524]，其丰富了用于涉及左和右分数差分算子的离散方程的显式求解的有用工具。证明了右分数$h$和的新结果。算例表明了所得结果在求解分数阶离散Euler-Lagrange方程中的有效性。
