Adjoints estimation methods for impulsive Moon-to-Earth trajectories in the restricted three-body problem

A study of optimal impulsive Moon‐to‐Earth trajectories is presented in a planar circular restricted three‐body framework. Two‐dimensional return trajectories from circular lunar orbits are considered, and the optimization criterion is the total velocity change. The optimal conditions are provided by the optimal control theory. The boundary value problem that arises from the application of the theory of optimal control is solved using a procedure based on Newton's method. Motivated by the difficulty of obtaining convergence, the search for the initial adjoints is carried out by means of two different techniques: homotopic approach and adjoint control transformation. Numerical results demonstrate that both initial adjoints estimation methods are effective and efficient to find the optimal solution and allow exploring the fundamental tradeoff between the time of flight and required ΔV.