The darboux point and the conjugate point on optimal deorbit for reentry trajectories

The concept of the Darboux point at which an extremal loses its global optimality is extended to the case of discontinuous control. Using Contensou's domain of maneuverability, the condition for optimal switching at a corner is derived and the optimality of the trajectory in the neighborhood of a Darboux point is analyzed. The theory is applied to the problems of minimum-fuel planar and noncoplanar deorbit from elliptical orbits for atmospheric entry at a prescribed angle. In each case, the global optimal trajectory is assessed and it is found that in these nonlinear problems the Darboux point and the conjugate point are distinct. The global optimality is always lost before local optimality.