Optimal Control for the Moon Lander: The Classical Problem and Variants

These notes concern a classical optimal control problem in aerospace engineering, how to land safely a spacecraft on the moon. Starting from an analysis of the original problem proposed by Miele in 1962, consisting in minimizing the consumption of fuel used for landing, we study a variant of the problem consisting in minimizing the landing time with a different thrust control. We prove that the optimal control (if it exists) is piecewise constant and provides the exact landing strategy, in dependence of the initial data (height, velocity, and fuel of the lander).