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：本文提出了一种精确模型，用于描述具有滑动固定记忆长度的黎曼-刘维尔和卡普托分数阶导数下的周期分数阶最优控制问题。同时，本文提供了一种基于傅里叶和盖根鲍尔伪谱方法求解该问题的新型数值方法。通过采用等距节点的傅里叶配置以及傅里叶-盖根鲍尔求积，该方法将周期分数阶最优控制问题转化为一个简单的约束非线性规划问题，该问题可利用标准非线性规划求解器轻松处理。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的复杂问题极大地简化为对其三角拉格朗日插值多项式一阶导数的积分进行评估，该积分可通过盖根鲍尔求积精确高效地处理。基于傅里叶和盖根鲍尔伪谱近似，我们引入了带有指数L的α阶分数阶积分矩阵概念，该矩阵在计算周期分数阶导数时极为有效。此外，我们提供了严格的先验误差分析，以预测基于傅里叶-盖根鲍尔近似的分数阶导数计算质量。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方法的性能。