  Indirect trajectory optimization methods such as Differential Dynamic
Programming (DDP) have found considerable success when only planning under
dynamic feasibility constraints. Meanwhile, nonlinear programming (NLP) has
been the state-of-the-art approach when faced with additional constraints
(e.g., control bounds, obstacle avoidance). However, a na$\"i$ve implementation
of NLP algorithms, e.g., shooting-based sequential quadratic programming (SQP),
may suffer from slow convergence -- caused from natural instabilities of the
underlying system manifesting as poor numerical stability within the
optimization. Re-interpreting the DDP closed-loop rollout policy as a
sensitivity-based correction to a second-order search direction, we demonstrate
how to compute analogous closed-loop policies (i.e., feedback gains) for
constrained problems. Our key theoretical result introduces a novel dynamic
programming-based constraint-set recursion that augments the canonical
"cost-to-go" backward pass. On the algorithmic front, we develop a hybrid-SQP
algorithm incorporating DDP-style closed-loop rollouts, enabled via efficient
parallelized computation of the feedback gains. Finally, we validate our
theoretical and algorithmic contributions on a set of increasingly challenging
benchmarks, demonstrating significant improvements in convergence speed over
standard open-loop SQP.
