Pontryagin-Bellman Differential Dynamic Programming for Low-Thrust Trajectory Optimization with Path Constraints

We present a Differential Dynamic Programming framework that parameterizes the control via Pontryagin's Minimum Principle, for constrained low-thrust trajectory optimization. This approach, dubbed Pontryagin-Bellman Differential Dynamic Programming (PDDP), optimizes the costates using a null-space trust-region method, solving a series of quadratic subproblems derived from first- and second-order sensitivities. Terminal equality constraints are handled via a general augmented Lagrangian method, while continuous-time state-path constraints are enforced using a quadratic penalty approach. The resulting solution method represents a significant improvement over classical indirect methods, which are known to face challenges for state-constrained problems. The flexibility of PDDP in handling various state-path constraints is demonstrated through minimum-radius and eclipse-avoidance trajectories in cislunar space. Finally, a trade study against indirect multiple shooting shows that PDDP achieves higher convergence rates overall, indicating improved robustness to poor initial guesses.