This paper introduces a new accurate model for periodic fractional optimal control problems (PFOCPs) using Riemann-Liouville (RL) and Caputo fractional derivatives (FDs) with sliding fixed memory lengths. The paper also provides a novel numerical method for solving PFOCPs using Fourier and Gegenbauer pseudospectral methods. By employing Fourier collocation at equally spaced nodes and Fourier and Gegenbauer quadratures, the method transforms the PFOCP into a simple constrained nonlinear programming problem (NLP) that can be treated easily using standard NLP solvers. We propose a new transformation that largely simplifies the problem of calculating the periodic FDs of periodic functions to the problem of evaluating the integral of the first derivatives of their trigonometric Lagrange interpolating polynomials, which can be treated accurately and efficiently using Gegenbauer quadratures. We introduce the notion of the {\alpha}th-order fractional integration matrix with index L based on Fourier and Gegenbauer pseudospectral approximations, which proves to be very effective in computing periodic FDs. We also provide a rigorous priori error analysis to predict the quality of the Fourier-Gegenbauer-based approximations to FDs. The numerical results of the benchmark PFOCP demonstrate the performance of the proposed pseudospectral method.

翻译：本文针对具有滑动固定记忆长度的黎曼-刘维尔（RL）和卡普托分数阶导数（FDs），提出了一种新的周期性分数阶最优控制问题（PFOCPs）精确模型。同时，本文提供了一种利用傅里叶和盖根鲍尔伪谱法求解PFOCPs的新型数值方法。通过在等距节点上采用傅里叶配置法以及傅里叶和盖根鲍尔求积法，该方法将PFOCP转化为一个简单的约束非线性规划问题（NLP），便于使用标准NLP求解器处理。我们提出了一种新的变换，将周期函数的周期性分数阶导数计算问题大幅简化为对其三角拉格朗日插值多项式一阶导数进行积分评估的问题，而该积分可通过盖根鲍尔求积法精确高效地处理。基于傅里叶和盖根鲍尔伪谱近似，我们引入了带指标L的α阶分数阶积分矩阵概念，该矩阵在计算周期性分数阶导数时表现出极高的有效性。此外，我们提供了严格的先验误差分析，以评估基于傅里叶-盖根鲍尔近似的分数阶导数计算精度。基准PFOCP的数值结果验证了所提伪谱法的性能。