Definition and analysis of properties of optimal three-impulse point-to-orbit transfers under time constraint

This paper deals with energetically optimal multi-impulse transfers of a spacecraft in the central Newtonian gravitational field near a planet. The transfer from a point on initial orbit to the final orbit with the given angular momentum and energy constants is considered. The transfer time is bounded above.With the distance from spacecraft to planet limited and the time free, such parameters of given orbits are chosen that the 3-impulse apsidal transfer T_r is optimal with an intermediate impulse at the maximum distance. On the basis of necessary optimality conditions an algorithm is developed to numerically determine the desired optimal transfer trajectory T_t under time constraint, the apsidal trajectory T_r being taken as initial approach. From the geometry and energy viewpoints, both trajectories T_t and T_r are close to each other. The trajectory T_t is also 3-impulsive, all impulses on it are nonapsidal. The distance from the planet is larger and the sum of impulses is less for this trajectory than for the initial trajectory T_r with the same transfer time.The simplified solution of the problem is constructed producing good approximation to the exact numerical optimization results. The solution asymptotics is found when the transfer time tends to infinity.