This paper extends the study of the Powered Descent Guidance (PDG) problem for soft planetary pinpoint landing [A, Lu]. This optimal control problem seeks the best trajectory for propulsively landing a vehicle at a given point while consuming as little propellant as possible and satisfying constraints on the control and the state. As such, it is very relevant to applications to scientific deep-space missions and reusable launchers, whose technical feasibility and economic interest have now been fully demonstrated by SpaceX and Blue Origin.

The problem and its variants have been extensively studied using finite-dimensional optimization tech- niques (e.g., convex solvers). But even the variant with the simplest dynamics is not yet completely under- stood mathematically, as highlighted by the recent study of Leparoux, Hérissé and Jean in 2022 [L], where the Max-Min-Max structure of the optimal trajectories is demonstrated in the following setting: constant gravity, negligible drag, lower and upper bounds on the thrust magnitude, a pointing constraint that asks for a close-to-vertical thrust direction, and finally a glide-slope state constraint that requires the lander to remain above an inverted cone with apex at the target landing position (this models the avoidance of terrain hazards).

In the present work, we seek a better understanding, in the same setting as above, of the possible mathematical structures of the optimal landing trajectories. We use an ad hoc version of the maximum principle (see Vinter [V]) that takes account of state constraints. Our first contribution is the complete derivation of the state and costate dynamics along a glide-slope state-constrained arc of positive duration: by successive differentiation of this state constraint up to the fourth order and substitution of the costate dynamics, we obtain an explicit expression for the non-negative constraint multiplier, that we can use to integrate the state and costate along state-constrained arcs. In the process, we are able to prove, under a mild assumption, that the the pointing constraint can not become active inside a positive-length glide arc.

Our second contribution is the development of a robust hybrid direct/indirect method, for computing exact solution trajectories to the PDG problem. A direct transcription of the problem into a nonlinear program (Crank–Nicolson discretization with adaptive multigrid warm-start modeled with JuMP and solved by Ipopt) provides a feasible primal-dual solution for random initial conditions. A careful analysis of this solution lets us reliably detects all junction types (pointing activation/deactivation, throttle switches, glide- slope touch points, and start/stop of sustained glide arcs). An indirect shooting method is then defined thanks to the inferred structure, and initialized by the direct transcription solution.

To improve the convergence rate of the shooting method, special attention is devoted to glide touch points and constrained arcs, where the piecewise-linear nature of the velocity costate is exploited while discontinuities in the position costate are carefully taken care of.

Extensive randomized experiments demonstrate that the method correctly identifies and solves problems exhibiting pointing-constraint activation (along a positive-length arc), throttle switches, instantaneous touch- and-go events, and sustained glide arcs (including glide-to-land terminal segments). The resulting solver reliably converges from random initial states typical of the terminal phase of planetary landing, providing certified optimal trajectories that respect all operational constraints. The explicit constrained-arc control law and the automated structure-detection/shooting framework significantly broaden the range of solvable PDG instances and offer new insight into the interplay between non-convex control bounds and state constraintsin aerospace guidance problems.

[L] C. Leparoux, B. Hérissé and F. Jean, “Structure of optimal control for planetary landing with control and state constraints,” ESAIM: Control, Optimisation and Calculus of Variations, vol. 28, 2022.

[V] R. Vinter, “Optimal Control,” Birkhau ?ser, 2010.

[A] B. Aç?kme?e and S. R. Ploen, “Convex Programming Approach to Powered Descent Guidance for Mars Landing,” Journal of Guidance, Control and Dynamics, vol. 30, no. 5, 2007.

[Lu] P. Lu, “Propellant-optimal powered descent guidance,” Journal of Guidance, Control, and Dynamics, vol. 41, no. 4, 2017.