Abstract

We define a new operational matrix of fractional derivative in the Caputo type and apply a spectral method to solve a two-dimensional fractional optimal control problem (2D-FOCP). To acquire this aim, first we expand the state and control variables based on the fractional order of Bernstein functions. Then we reduce the constraints of 2D-FOCP to a system of algebraic equations through the operational matrix. Now, one can solve straightforward the problem and drive the approximate solution of state and control variables. The convergence of the method in approximating the 2D-FOCP is proved. We demonstrate the efficiency and superiority of the method by comparing the results obtained by the presented method with the results of previous methods in some examples.

中文翻译：

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利用广义分数阶Bernstein函数求解二维分数最优控制问题的有效近似方法

摘要

我们定义了Caputo类型的分数导数的新运算矩阵，并应用频谱方法来解决二维分数最优控制问题（2D-FOCP）。为了达到这个目的，首先我们根据伯恩斯坦函数的分数阶扩展状态和控制变量。然后，我们通过运算矩阵将2D-FOCP的约束减少到一个代数方程组。现在，人们可以直接解决问题，并驱动状态和控制变量的近似解。证明了近似二维FOCP方法的收敛性。在某些示例中，我们通过比较所提出的方法获得的结果与先前方法的结果来证明该方法的效率和优越性。