3D geosynchronous transfer of a satellite: Continuation on the thrust

The minimum-time transfer of a satellite from a low and eccentric initial orbit toward a high geostationary orbit is considered. This study is preliminary to the analysis of similar transfer cases with more complicated performance indexes (maximization of payload, for instance). The orbital inclination of the spacecraft is taken into account (3D model), and the thrust available is assumed to be very small ( e. g. 0.3 Newton for an initial mass of 1500 kg). For this reason, many revolutions are required to achieve the transfer and the problem becomes very oscillatory. In order to solve it numerically, an optimal control model is investigated and a homotopic procedure is introduced, namely continuation on the maximum modulus of the thrust: the solution for a given thrust is used to initiate the solution for a lower thrust. Continuous dependence of the value function on the essential bound of the control is first studied. Then, in the framework of parametric optimal control, the question of differentiability of the transfer time with respect to the thrust is addressed: under specific assumptions, the derivative of the value function is given in closed form as a first step toward a better understanding of the relation between the minimum transfer time and the maximum thrust. Numerical results obtained by coupling the continuation technique with a single-shooting procedure are detailed.