Continuation with Stationary Condition-Based Termination for Optimal Orbit Transfer Problems

Indirect methods for solving optimal control problems often suffer from numerical issues, namely, the narrow convergence domain of initial guesses for the unknown variables. Continuation methods provide a systematic means to address this issue, but many existing formulations remain somewhat problem-specific and can fail in particular cases. In this work, a broadly applicable method to solve optimal spacecraft transfer problems with one or more free parameters is proposed. These free parameters are fixed in an initial (easier-to-solve) problem and gradually freed via continuation in one or more stages (freeing one parameter per stage). Starting from a solution to the initial problem, a temporarily prescribed parameter is selected as the continuation parameter and varied in the direction of decreasing cost until the corresponding stationary condition is satisfied, with termination determined using an event-detection scheme similar to that employed in initial-value problem solvers. This process is then repeated until all remaining temporarily prescribed parameters have been freed. The proposed method is demonstrated on two distinct orbital transfer scenarios: a fuel-optimal near-Earth station-keeping scenario and a fuel-optimal transfer between halo orbits.