An analytic solution for constant-thrust, optimal-coast, minimum-propellant space trajectories

The variational formulation of the minimum-propellant control of a space vehicle in an inverse-square gravitational field leads to a nonlinear, two-point, boundary-value problem. Although for the case of unbounded thrust-magnitude (impulsive propulsion) solutions are easily found, numerical solutions for the case of bounded thrust are not so readily obtainable. This paper gives an analytic method for determining the optimal control of a thrust-limited, rocket-powered vehicle in terms of an expansion about the optimal impulsive solution. The expansion is in terms of two independent vehicle parameters, the initial thrust-acceleration and the rocket jet exhaust velocity (a measure of the specific impulse). The resulting solution is valid over a wide range of vehicle parameters of interest and gives the thrust-vector-time history on the optimal bounded-thrust trajectory. The solution is easy to implement on a high-speed digital computer and requires very little computing time.