  We investigate variants of Goddard's problems for nonvertical trajectories.
The control is the thrust force, and the objective is to maximize a certain
final cost, typically, the final mass. In this report, performing an analysis
based on the Pontryagin Maximum Principle, we prove that optimal trajectories
may involve singular arcs (along which the norm of the thrust is neither zero
nor maximal), that are computed and characterized. Numerical simulations are
carried out, both with direct and indirect methods, demonstrating the relevance
of taking into account singular arcs in the control strategy. The indirect
method we use is based on our previous theoretical analysis and consists in
combining a shooting method with an homotopic method. The homotopic approach
leads to a quadratic regularization of the problem and is a way to tackle with
the problem of nonsmoothness of the optimal control.
