An analysis of optimality of trajectories for mission Earth–asteroid–Earth

Energy optimal impulsive trajectories for space flight from the Earth to the asteroid and following return to the Earth are analyzed. The trajectories are assumed to be obtained by direct methods. There are constructed adjoint functions for heliocentric space trajectories with applying, in a nominal case, three velocity impulses in the celestial body’s spheres of influence. It allows investigating the optimality in a class of multi-impulsive flights. The derivatives of functional with respect to the trajectory boundary times are obtained. They allow testing the transversality conditions for the mission trajectory. Numerical examples are given for the mission Earth–asteroid Apophis–Earth.