A minimal time optimal control for a drone landing problem

Abstract

We study a variant of the classical safe landing optimal control problem in aerospace engineering, introduced by Miele (1962), where the target was to land a spacecraft on the moon by minimizing the consumption of fuel. A more modern model consists in replacing the spacecraft by a hybrid gas-electric drone. Assuming that the drone has a failure and that the thrust (representing the control) can act in both vertical directions, the new target is to land safely by minimizing time, no matter of what the consumption is. In dependence of the initial data (height, velocity, and fuel), we prove that the optimal control can be of four different kinds, all being piecewise constant. Our analysis covers all possible situations, including the nonexistence of a safe landing strategy due to the lack of fuel or for heights/velocities for which also a total braking is insufficient to stop the drone.