Optimizing Flight Trajectories for Space Vehicles with an Additional Fuel Tank. II

We solve the optimization problem for space trajectories of spacecraft flights with an auxiliary fuel tank from a low round orbit of a man-made Earth satellite to a geotransitional orbit. Control over the spacecraft motion is performed with a jet engine of bounded thrust. To discard the auxiliary tank, one has to turn off the engine, which takes some known time. The mass of the discarded tank is assumed to be proportional to the mass of fuel spent, and the mass of the engine and additional constructions is proportional to the thrust-to-weight ratio. We minimize the value of injection impulse needed to transfer to the geostationary orbit for a given useful mass. In the second part of the paper the problem at hand is formalized as an optimal control problem for a collection of dynamical systems and is solved based on the corresponding maximum principle. In this work we solve boundary problems of the maximum principle numerically with the shooting method. As a result of solving the problem, we construct one- and two-revolution Pontryagin extremals. We perform a series of parametric computations that are used to determine optimal parameters of the spacecraft construction: the best thrust-to-weight ratio and the best distribution of fuel among the tanks.