Optimal injection point for launch trajectories with parametric thrust profile

The problem of finding the optimal thrust profile of a launcher upper stage is analyzed. The engine is non-re-ignitable and it is continuously thrusting, following either a linear or a bilevel parametric profile, until reaching the targeted coplanar orbit. This problem differs from the classical rocket problem where the thrust level is a time-dependent function varying freely between prescribed bounds. Applying the maximum principle yields an analytical closed-loop solution for the thrust direction. Furthermore the final point is found to be necessarily at an apsis, reached from above in the case of a perigee injection. The optimal control problem reduces to a nonlinear problem with only the thrust profile parameters as unknowns. This formulation eases preliminary design studies aiming at defining the optimum upper stage thrust profile. An application case targeting a geostationary transfer orbit illustrates the solution method.