I've been wondering lately about a problem that comes from, among other places, an old video game "Lunar Lander". In the game there is a spaceship that has a small tank of fuel, and you're supposed to pilot the ship from an initial height down to the surface of the moon. You win if the landing is sufficiently soft. Your only control (at least in a one-dimensional version of the game) is a dial to control the fuel for thrust during the descent.
Finding the optimal strategy is seemingly a problem in the calculus of variations---out of all possible ways to spend the thrust during the descent, you want the one that minimizes the speed when the lander reaches the surface. Besides the game, given that we've landed real ships on the moon or Mars, I'm guessing this optimization problem is well known. Could someone point me to results in textbooks or the literature?