This study aims to generate a novel set of basis functions called the orthonormal piecewise Chelyshkov functions to solve a certain category of optimal control problems whose dynamical system is governed by a nonlinear fractional differential equation. A new fractional integral matrix associated with these basis functions is derived. This matrix significantly reduces the computations in solving such problems. The proposed approach transforms the original problem into a nonlinear programming one by expanding the control and state variables in terms of the orthonormal piecewise Chelyshkov functions and employing the derived fractional integral matrix. Some numerical problems are examined for verification of the proposed method.

本研究旨在生成一组称为正交分段 Chelyshkov 函数的新基函数，以解决其动力系统由非线性分数微分方程控制的某一类最优控制问题。推导出与这些基函数相关的新分数积分矩阵。该矩阵显着减少了解决此类问题的计算量。所提出的方法通过根据正交分段 Chelyshkov 函数扩展控制和状态变量并使用导出的分数积分矩阵，将原始问题转换为非线性规划问题。检查了一些数值问题以验证所提出的方法。