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：本文针对周期分数阶最优控制问题（PFOCPs），利用具有滑动固定记忆长度的Riemann-Liouville（RL）和Caputo分数阶导数（FDs），提出了一种新的精确模型。同时，本文还提供了一种基于傅里叶和盖根鲍尔伪谱法的PFOCP数值求解新方法。通过在等距节点处采用傅里叶配点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），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积分的问题，该问题可通过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（索引为L）概念，该矩阵在计算周期分数阶导数时表现出极高有效性。我们还提供了严格的先验误差分析，以预测基于傅里叶-盖根鲍尔分数阶导数近似的质量。基准周期分数阶最优控制问题的数值结果验证了所提伪谱法的性能。