Space Trajectories. Orbit Injection at Optimum Thrust Level

The problem of finding the optimum thrust level for a rocket engine has been studied intensively since Goddard's seminal work in the early 20th century. The classical formulation assumes that the thrust level varies freely between zero and a given maximum value. The optimal solution is then composed of maximum thrust arcs and ballistic arcs.

This article presents an alternative hypothesis consisting of a parametric thrust profile, either linear or two-stage. This model is representative of the operation of most engines. The aim is to simultaneously optimize the thrust profile and the trajectory of the last stage to reach the target orbit while minimizing propellant consumption. Initial conditions are given and result from the flight of the lower stages. The target orbit is an ellipse of given apogee and perigee. The trajectory of the last stage is assumed to be flat, which is the case in most practical applications.

This hybrid optimal control problem including thrust profile is of great practical importance when designing new engines. The study of optimality conditions shows that this problem admits a state feedback analytical solution, with the optimal thrust direction expressed as a function of current kinematic conditions (position, velocity). This study also shows that the optimal injection point is the perigee of the target orbit, and that the orbit will be reached by a descending phase. This theoretical result explains the initially surprising shape of many launch trajectories, first ascending, then descending. It turns out that this shape is in fact associated with a choice of thrust level close to the optimum. These properties make it possible to reduce the optimal control problem to a two-variable nonlinear problem, which is easy to solve numerically.

The theoretical study also provides an analytical expression for the adjoint vector as a function of position and velocity conditions. Initialization of the adjoint vector is a key issue in space-optimized trajectory problems. It turns out that this analytical initialization, obtained when the thrust level is optimized, is effective in solving various problems, particularly when the engine thrust level is fixed.

This article presents the formulation of the optimal control problem and the theoretical study of optimality conditions. The various properties of the optimal trajectory are established, then used to solve practical applications.