This work examines the problem of finding optimal periodic solutions to minimize the fuel consumption in Unmanned Aerial Vehicles (UAV) to enhance endurance. Previous work examined this problem using the concept of differential flatness. This work examines the formal optimal control derivation of the periodic control of the UAV problem. The solution obtained from the optimal control derivation is compared to that derived from the differential flatness based formulation. In this work necessary conditions for optimality and insights are provided to the structure of the optimal periodic problem. In conclusion, the challenges and possible simplifications of solving this problem is discussed.