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The endurance of unmanned aerial vehicles (UAVs) is a critical factor in expanding the scope of their applications. Extended flight times enable UAVs to undertake longer missions, cover larger areas and perform tasks such as persistent surveillance, data collection and search and rescue operations. Optimal trajectory planning is a cost-effective method to significantly enhance UAV endurance and performance by minimizing fuel consumption. This study introduces a novel numerical optimization framework to maximize UAV endurance. Specifically, we address the problem of determining optimal thrust and cruise angle of attack for a UAV in 2D space under specific initial, periodic and boundary conditions. By normalizing the free final time optimal control problem and employing Fourier collocation and quadrature, we transform it into a nonlinear programming problem. A key contribution of this work is accurately detecting and reconstructing the thrust history, including jump discontinuities, directly from Fourier pseudospectral data without smoothing techniques. The proposed method outperforms existing approaches in solving the periodic energy-optimal path planning problem for UAVs, as it effectively reconstructs the bang-bang thrust profile, facilitating rapid and efficient thrust adjustments essential for various flight maneuvers. Furthermore, the algorithm aligns with the UAV model, ensuring seamless integration into real-world control systems. The method’s independence from prediction horizon length, due to the use of Fourier collocation on the normalized interval $[0,2\pi ]$, is a notable advantage. This characteristic offers potential for future applications in various fields involving nonsmooth optimal control problems. This research generally provides a valuable tool for researchers and engineers working on UAV design and operation, paving the way for more efficient and effective UAV systems.