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：本文针对使用Riemann-Liouville (RL)与Caputo分数阶导数(固定滑动记忆长度)的周期分数阶最优控制问题(PFOCPs)，提出了一种新的精确模型。同时，文章提供了一种基于Fourier和Gegenbauer伪谱方法求解PFOCPs的新数值方法。该方法通过在等距节点处采用Fourier配点法，结合Fourier和Gegenbauer求积公式，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，将计算周期函数的周期分数阶导数问题大幅简化为评估其三角拉格朗日插值多项式一阶导数的积分问题，该积分可利用Gegenbauer求积法精确高效地处理。我们引入了基于Fourier与Gegenbauer伪谱近似的α阶分数阶积分矩阵(指数为L)概念，证明了该矩阵在计算周期分数阶导数方面极为有效。此外，我们提供了严格的先验误差分析，以预测基于Fourier-Gegenbauer的分数阶导数近似质量。基准PFOCP的数值结果验证了所提伪谱方法的性能。