Introduction

In this chapter, the theory and a resulting indirect method of trajectory optimization are derived and illustrated. In an indirect method, an optimal trajectory is determined by satisfying a set of necessary conditions (NC), and sufficient conditions (SC) if available. By contrast, a direct method uses the cost itself to determine an optimal solution.

Even when a direct method is used, these conditions are useful to determine whether the solution satisfies the NC for an optimal solution. If it does not, it is not an optimal solution. As an example, the best two-impulse solution obtained by a direct method is not the optimal solution if the NC indicate that three impulses are required. Thus, post-processing a direct solution using the NC (and SC if available) is essential to verify optimality.

Optimal Control, a generalization of the calculus of variations, is used to derive a set of necessary conditions for an optimal trajectory. The primer vector is a term coined by D. F. Lawden in his pioneering work in optimal trajectories. [This terminology is explained after Equation (2.24).] First-order necessary conditions for both impulsive and continuous-thrust trajectories can be expressed in terms of the primer vector. For impulsive trajectories, the primer vector determines the times and positions of the thrust impulses that minimize the propellant cost. For continuous thrust trajectories, both the optimal thrust direction and the optimal thrust magnitude as functions of time are determined by the primer vector.