Optimal continuous-thrust rephasing maneuver in circular orbit

The minimum-time constant-thrust circular orbit rephasing problem is studied using a curvilinear relative motion description for dynamics. The resulting optimal control problem in the thrust orientation is formulated using the indirect method and investigated both analytically and numerically. By linearizing the relative motion equations, a key nondimensional parameter is identified, which characterizes the duration of the maneuver and its qualitative structure when changing from a short-maneuver to a long-maneuver time regime, passing through a transition zone. Approximate analytical solutions are obtained for the short- and long-maneuver regimes, providing an accurate estimation of the minimum maneuver time and a clear understanding of the evolution of the optimal thrust profile, which approaches two opposite bang-bang structures in the limit cases. The nonlinear problem is investigated numerically and shown to be fully characterized by two nondimensional parameters instead of one. Finally, a comparison with different two-impulse maneuvers is conducted. The results can be used for constructing good initial guesses in more complex optimization problems and to support preliminary mission design.