Indirect Methods for the Optimization of Spacecraft Trajectories

In this chapter, a general methodology to apply the theory of optimal control to spacecraft trajectories is outlined. This peculiar procedure allows for an almost mechanical derivation of the boundary conditions which must be satisfied by an optimal trajectory, depending on the specific constraints of the problem under analysis. The general way of posing the optimal control problem makes this indirect approach suitable to manage many specific features of the space missions, such as, impulsive and/or low-thrust engines, planetary flybys, atmospheric flight, and so on. Peculiarities of the problem simply modify the set of differential equations and boundary conditions in the context of the same theoretical frame. Examples will show that the indirect approach can deal efficiently with complex problems of space trajectory optimization. As in the case of direct methods, the indirect approach requires a tentative solution, and convergence to the optimum is typically obtained if the tentative solution is sufficiently close to the optimal one. Suitable procedures to find tentative guesses for the considered problems are described.