In this study, a numerical method based on Hermite polynomial approximation for solving a class of fractional optimal control problems is presented. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Operational matrices of integration by using such known formulas as Caputo and Riemann–Liouville operators for computing fractional derivatives and integration of polynomials is introduced and used to reduce the problem of a system of algebraic equations. The convergence of the proposed method is analyzed, and the error upper bound for the operational matrix of the fractional integration is obtained. To confirm the validity and accuracy of the proposed numerical method, three numerical examples are presented along with a comparison between our numerical results and those obtained using Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

本研究提出了一种基于Hermite多项式逼近的数值方法来解决一类分数最优控制问题。分数导数的阶数应小于1，并以Caputo的意义进行描述。通过使用诸如Caputo和Riemann-Liouville运算符之类的已知公式来计算分数导数和多项式积分的积分运算矩阵被引入并用于减少代数方程组的问题。分析了所提方法的收敛性，得到了分数积分运算矩阵的误差上限。为了确认所提出数值方法的有效性和准确性，给出了三个数值示例，并对我们的数值结果与使用Legendre多项式获得的数值进行了比较。包括说明性示例以证明新技术的有效性和适用性。