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：本文提出了一种利用Riemann-Liouville（RL）和Caputo分数阶导数（FDs）结合滑动固定记忆长度的周期分数阶最优控制问题（PFOCPs）新精确模型。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。通过在等间距节点上采用傅里叶配置法以及傅里叶和盖根鲍尔求积公式，该方法将PFOCP转化为一个简单的约束非线性规划问题（NLP），可通过标准NLP求解器轻松处理。我们提出了一种新变换，将计算周期函数的周期分数阶导数的复杂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积分的问题，该问题可利用盖根鲍尔求积公式精确高效地处理。我们引入了基于傅里叶和盖根鲍尔伪谱近似的α阶分数阶积分矩阵（带索引L）概念，该矩阵在计算周期分数阶导数时具有极高有效性。同时，我们提供了严格的先验误差分析，以预测基于傅里叶-盖根鲍尔近似对分数阶导数的逼近质量。基准PFOCP的数值结果验证了所提伪谱方法的性能。