Shooting and Numerical Continuation Methods for Computing Time-minimal and Energy-minimal Trajectories in the Earth-moon System Using Low Propulsion

In this article we describe the principle of computations of optimal transfers between quasi-Keplerian orbits in the Earth-Moon system using low-propulsion. The spacecraft’s motion is modeled by the equations of the control restricted 3-body problem and we base our work on previous studies concerning the orbit transfer in the two-body problem where geometric and numeric methods were developped to compute optimal solutions. Using numerical simple shooting and continuation methods connected with fundamental results from control theory, as the Pontryagin Maximum Principle and the second order optimality conditions related to the concept of conjugate points, we compute time-minimal and energy-minimal trajectories between the geostationary initial orbit and a final circular one around the Moon, passing through the neighborhood of the libration point L1. Our computations give simple trajectories, obtained by referring to numerical values of the SMART-1 mission.