Extended Necessary Conditions for Multi-Arc Aerospace Trajectory Optimization

A variety of aerospace trajectory optimization problems are subject to either discontinuities or constraints at intermediate times, which define multiple arcs with distinctive governing equations. This work addresses multi-arc optimal control problems, with special interest regarding aerospace trajectories, and specifically focuses on the multipoint corner conditions that belong to the complete set of necessary conditions for an extremal, in the context of a general formulation. This includes intermediate times and states in the objective functional, together with unknown time-independent parameters. This study shows that 16 cases can occur for the multipoint corner conditions and groups them into three classes. Explicit, closed-form solutions of the multipoint corner relations are identified in each class, if certain conditions are met. In an indirect solution approach, these explicit expressions can be employed sequentially, thus reducing the number of unknowns of multi-arc problems to the same number of single-arc optimal control problems. This is extremely useful in the presence of a large number of arcs. Two challenging aerospace trajectory optimization problems are analyzed as illustrative examples, i.e., (i) the minimum-fuel ascent path of a multistage launch vehicle and (ii) minimum-time low-thrust orbit transfers with eclipse constraints on the available thrust.