The Goddard Problem
The Goddard problem is the name for a simplified sounding rocket optimization problem. The problem is to find the optimal thrust curve for a vertical 1D rocket trajectory with a flat Earth, non varying gravity and simple v2 drag model with exponential decaying atmosphere in the form e-kh.
There are 3 state variables: distance from the center of the Earth r, velocity v, and mass m. We constrain ourselves to a line radial with Earth (along r) such that the problem becomes spatially one dimensional.
Figure 1 — State Variables