Optimizing flight trajectories for space vehicles with an additional fuel tank. I

We consider optimization problems for spatial trajectories of space flights for space vehicles with an additional fuel tank from a low circular orbit of a man-made Earth satellite to a geotransitional orbit. The motion of the space vehicle is controlled through a jet engine with bounded thrust. To discard the additional tank, the vehicle needs to switch off its engine and spend a certain time. The mass of the discarded tank is assumed to be proportional to the mass of the expended fuel, and the mass of the engine and additional constructions is proportional to the thrust. We minimize the value of the impulse needed to transfer to the geostationary orbit for a given useful mass, or, which is the same, maximize the useful mass for a given value of this impulse. In the first part of the paper, we consider in detail the history of this problem and computational schemes that can be used to solve this kind of problems.