On the Optimal Feedback Law in Stochastic Optimal Nonlinear Control

We consider the problem of nonlinear stochastic optimal control. This problem is thought to be fundamentally intractable owing to Bellman's infamous "curse of dimensionality". We present a result that shows that repeatedly solving an open-loop deterministic problem from the current state, similar to Model Predictive Control (MPC), results in a feedback policy that is $O(\epsilon^4)$ near to the true global stochastic optimal policy. Furthermore, empirical results show that solving the Stochastic Dynamic Programming (DP) problem is highly susceptible to noise, even when tractable, and in practice, the MPC-type feedback law offers superior performance even for stochastic systems.