Indirect Optimization of Fuel-Optimal Many-Revolution Low-Thrust Transfers With Eclipses

An efficient indirect method is presented to determine fuel-optimal many-revolution low-thrust transfers in the presence of Earth-shadow eclipses. Specifically, the events of shadow entrance and exit are modeled as interior-point constraints. Following the observation that an ill-conditioned state transition matrix may occur when the spacecraft flies over the edge of the shadow, a two-level continuation scheme is introduced to generate many-revolution trajectories. The established computational framework integrates analytic derivatives, switching detection, and continuation with an augmented flowchart, which yields discontinuous bang-bang solutions and their gradients. Transfers from a geostationary transfer orbit to a geostationary orbit are simulated to illustrate the effectiveness and efficiency of the method developed.