Fractional-Order Legendre Functions and Their Application to Solve Fractional Optimal Control of Systems Described by Integro-differential Equations
Fractional-Order Legendre Functions and Their Application to Solve Fractional Optimal Control of Systems Described by Integro-differential Equations
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DOI：
10.1007/s10440-018-0175-0
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发表时间：
2018-03
影响因子：
1.6
通讯作者：
K. Rabiei;Y. Ordokhani;E. Babolian
中科院分区：
文献类型：
--
作者：
K. Rabiei;Y. Ordokhani;E. Babolian
本文引入一组称为分数阶勒让德函数的函数来求解线性和非线性分数阶积分-微分方程组的最优控制问题。我们考虑这些函数的性质来构造分数次积分的运算矩阵。并首次得到了这些函数乘法运算矩阵的一般公式，从而解决了非线性问题。然后利用这些矩阵将上述分式最优控制问题归结为一个代数方程组。实际上，问题的函数是用约束方程、性能指标和条件中系数未知的分数阶勒让德函数来逼近的。因此，分数次最优控制问题转化为优化问题，然后可以用数值方法求解。讨论了该方法的收敛问题，并给出了数值算例，验证了该方法的有效性和准确性。
In this paper, we introduce a set of functions called fractional-order Legendre functions (FLFs) to obtain the numerical solution of optimal control problems subject to the linear and nonlinear fractional integro-differential equations. We consider the properties of these functions to construct the operational matrix of the fractional integration. Also, we achieved a general formulation for operational matrix of multiplication of these functions to solve the nonlinear problems for the first time. Then by using these matrices the mentioned fractional optimal control problem is reduced to a system of algebraic equations. In fact the functions of the problem are approximated by fractional-order Legendre functions with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem converts to an optimization problem, which can then be solved numerically. The convergence of the method is discussed and finally, some numerical examples are presented to show the efficiency and accuracy of the method.