Necessary conditions for the optimality of singular arcs of spacecraft trajectories subject to multiple gravitational bodies

This document analyzes the optimality of intermediate thrust arcs (singular arcs) of spacecraft trajectories subject to multiple gravitational bodies. A series of necessary conditions for optimality are formally derived, including the generalized Legendre–Clebsch condition. As the order of singular optimality turns out to be two, an explicit formula for the singular optimal control is also presented. These analytical outcomes are validated by showing that they are identical to Lawden’s classical result if the equations of motion are reduced for a central gravity field. Practical utility is demonstrated by applying these analytical derivations to a candidate optimal trajectory near the Moon subject to solar and Earth perturbation. While the candidate optimal trajectory turns out to be bang-singular-bang, the intermediate thrust arc satisfies all the necessary conditions for optimality.