The ability to certify systems driven by neural networks is crucial for future rollouts of machine learning technologies in aerospace applications. In this study, the neural networks are used to represent a fuel-optimal feedback controller for two different 3-degree-of-freedom pinpoint landing problems. It is shown that the standard sum-ofsquares Lyapunov candidate is too restrictive to assess the stability of systems with fuel-optimal control profiles. Instead, a parametric Lyapunov candidate (i.e. a neural network) can be trained to sufficiently evaluate the closed-loop stability of fuel-optimal control profiles. Then, a stability-constrained imitation learning method is applied, which simultaneously trains a neural network policy and neural network Lyapunov function such that feedback-optimal control is achieved, and Lyapunov stability is verified. Phase-space plots of the Lyapunov derivative show the improvement in stability assessment provided by the neural network Lyapunov function, and Monte Carlo simulations demonstrate the stable, feedback-optimal control provided by the policy
