A comment on stabilizing reinforcement learning

This is a short comment on the paper "Asymptotically Stable Adaptive-Optimal Control Algorithm With Saturating Actuators and Relaxed Persistence of Excitation" by Vamvoudakis et al. The question of stability of reinforcement learning (RL) agents remains hard and the said work suggested an on-policy approach with a suitable stability property using a technique from adaptive control - a robustifying term to be added to the action. However, there is an issue with this approach to stabilizing RL, which we will explain in this note. Furthermore, Vamvoudakis et al. seems to have made a fallacious assumption on the Hamiltonian under a generic policy. To provide a positive result, we will not only indicate this mistake, but show critic neural network weight convergence under a stochastic, continuous-time environment, provided certain conditions on the behavior policy hold.