4. Conclusion

This paper presented the solution of a hybrid optimal control problem involving simultaneously the trajectory and thrust profile of a launch vehicle upper stage. The trajectory is assumed to be flat and takes place outside the atmosphere. The objective is to reach a given elliptical orbit using as little propellant as possible, with a linear or two-stage thrust profile. These profiles correspond to the operation of most thrusters, with thrust varying between 50% and 100% of a maximum level, depending on the technology employed.

Unlike the classical problem, which assumes that the thrust level varies freely between bounds, this hybrid problem admits an analytical solution in the form of a state feedback control. We also establish that the optimal injection point is the perigee of the target orbit, reached by a descending arc. These properties allow us to reduce the...