THE DARBOUX POINT AND THE CONJUGATE POINT ON OPTIMAL DEORBIT FOR REENTRY TRAJECTORIES+

Abstract--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 fl)r optimal switching at a comer 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 p anar and noncoplanar dcorbit 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. In optimal control the appearance of a conjugate point on an extremal indicates the loss of local optimality. It has been noted in several papersl 1,2] that in some cases an extremal oses global optimality at a particular point