Closed-Form Thrust Acceleration for Optimal Power-Limited Trajectories


A simple class of minimum-fuel power-limited trajectories has recently been discovered. The optimal thrust acceleration can be expressed in closed-form as a simple, time- varying scalar multiple of the velocity vector. The thrust direction is then always aligned with the velocity vector. This thrust law satisfies the first-order necessary conditions for an optimal trajectory in an arbitrary gravitational field. The optimal trajectory can be calculated using only the state equations, without simultaneous calculation of the primer vector.

These solutions form a subset of optimal power-limited trajectories, and have some limitations on their use. No plane change can be provided by the thrust law considered. Trajectories which have an unspecified final velocity, such as an optimal intercept, cannot be accomplished with this thrust law. However, optimal rendezvous and orbit transfer can be realized by these trajectories.

The problem of maximizing the change in total energy is examined. It is shown that thrusting along the velocity vector cannot satisfy the necessary conditions for an optimal trajectory, even though the terminal conditions dictate that the final thrust must be along the final velocity. This result is consistent with numerical results showing that optimal energy trajectories do not simply thrust along the velocity vector, even though this thrust law maximizes the instantaneous rate of change of the total energy.