On the Existence and Uniqueness of Minimum Time Optimal Trajectory for a Micro Air Vehicle under Wind Conditions

In this paper, we show that a minimum time solution for the trajectory planning problem in the presence of wind for a micro air vehicle with a minimum turn radius constraint exists. For a non-zero, time-varying wind vector field, we provide easily checked sufficient conditions under which a minimum-time solution exists. We also prove uniqueness for almost every combination of initial and final conditions in the case of a wind vector field that varies with time but is constant in the spatial variable for each time instant. We note here that the well known Pontryagin’s Maximum Principle [11] only yields necessary conditions for the optimal solution but does not yield its existence. We use Filippov’s Theorem [13] to conclude existence of the minimum time solution, after demonstrating the existence of one solution, provided certain conditions are satisfied by the wind vector field.