The context of this work is space mechanics. More precisely, we aim at computing low thrust transfers in the Earth-Moon system modeled by the circular restricted three-body problem. The goal is to calculate the optimal steering of the spacecraft engine with respect to two optimization criteria: Final time and fuel consumption. The contributions of this thesis are of two kinds. Geometric, first, as we study the controllability of the system together with the geometry of the transfers (structure of the command) by means of geometric control tools. Numerical, then, different homotopic methods being developed. A two-three body continuation is used to compute minimum time trajectories, and then a continuation on the maximal thrust is considered to reach low thrusts. The minimum consumption problem--minimization of the L1 norm of the control--is connected by a differential continuation to the min- imization of the L2 norm of the control. The trajectories computed are then compared to those obtained using a logarithmic interior penalty. Those methods are applied to simulate the SMART-1 mission of the European Space Agency.
