Analytical guidance in the neighborhood of optimal multi-impulse trajectories

A first-order minimum propellant guidance law is developed for multi-impulse trajectories in an inverse-square gravitational field. A second-order variational analysis is used to formulate the guidance problem as an accessory minimum problem, i.e. minimize a quadratic form (second-variation in propellant consumption) subject to linear constraints (variational equations of motion and deterministic boundary conditions). Solution of the accessory minimum problem provides the optimal guidance law in feedback form. It is emphasized that this analysis takes into account the nominal impulse programme when calculating the optimal guidance corrections. It is shown that for multi-impulse transfers it is in general, non-optimal to add impulses. All corrections to the trajectory should be made by a combination of small changes in timing, magnitude and direction of the nominal impulses.