Minimum-fuel powered descent for Mars pinpoint landing

Motivated by the requirement for pinpoint landing in future Mars missions, we consider the problem of minimum-fuel powered terminal descent to a prescribed landing site. The first-order necessary conditions are derived and interpreted for a point-mass model with throttle and thrust angle control and for rigid-body model with throttle and angular velocity control, clarifying the characteristics of the minimum-fuel solution in each case. The optimal thrust magnitude profile is bang-bang for both models; for the point-mass, the most general thrust magnitude profile has a maximum-minimum-maximum structure. The optimal thrust direction law for the point-mass model (alignment with the primer vector) corresponds to a singular solution for the rigid-body model. Whether the point-mass solution accurately approximates the rigid-body solution depends on the thrust direction boundary conditions imposed for the-rigid-body model. Minimum-fuel solutions, obtained numerically, illustrate the optimal strategies.