In [1], we inaugurated a new area of optimal control (OC) theory that we called "periodic fractional OC theory," which was developed to find optimal ways to periodically control a fractional dynamic system. The typical mathematical formulation in this area includes the class of periodic fractional OC problems (PFOCPs), which can be accurately solved numerically for a fractional order {\alpha} in the range 0 < {\alpha} < 1 using Fourier collocation at equally spaced nodes and Fourier and Gegenbauer quadratures. In this study, we extend this earlier work to cover periodic higher-order fractional OC problems (PHFOCPs) of any positive non-integer fractional order {\alpha}.

翻译：在文献[1]中，我们开创了最优控制理论的一个新领域，称之为"周期分数阶最优控制理论"，该理论旨在寻找周期性控制分数阶动态系统的最优方法。该领域的典型数学表述包含一类周期分数阶最优控制问题，针对分数阶阶次{\alpha}在0<{\alpha}<1范围内的情况，可通过等距节点上的Fourier配置法及Fourier与Gegenbauer求积法进行精确数值求解。本研究将先前工作扩展至任意正非整数分数阶{\alpha}的周期高阶分数阶最优控制问题。