For a class of subsonic aircraft, endurance can be improved signiflcantly by ∞ying in a periodic path rather than in steady state. The optimal periodic endurance problem is formulated where the performance criterion for endurance, fuel used over ∞ight time, is minimized and the aircraft is constrained to ∞y a periodic path while circles above a target on the ground. The optimal steady state endurance problem is formulated where the instantaneous fuel rate is minimized subject to the aircraft dynamics being in equilibrium. Both optimization problems are solved numerically. An example shows for an aircraft with maximal lift-to-drag ratio of 17.4 and thrust-to-weight ratio of 0.5, the endurance of the optimal periodic ∞ight is over three times of the endurance of the optimal steady state ∞ight. In order to mechanize the optimal periodic ∞ight, a periodic guidance law is developed.