Unmanned aerial vehicles (UAVs) are making increasingly long flights today with significantly longer mission times. This requires the UAVs to have long endurance as well as have long range capabilities. Motivated by locomotory patterns in birds and marine animals which demonstrate a powered-coasting-powered periodic locomotory behavior, an optimal control problem is formulated to study UAV trajectory planning. The concept of differential flatness is used to reformulate the optimal control problem as a nonlinear programing problem where the flat outputs are parameterized using Fourier series. The Π test is also used to verify the existence of a periodic solution which outperforms the steady-state motion. An example of an Aerosonde UAV is used to illustrate the improvement in endurance and range costs of the periodic control solutions relative to the equilibrium flight.