Near Optimality and Tractability in Stochastic Nonlinear Control

We consider the problem of nonlinear stochastic optimal control. This is fundamentally intractable owing to Bellman's infamous "curse of dimensionality". We present a "decoupling principle" for the tractable feedback design for such problems, wherein, first, a nominal open-loop problem is solved, followed by a suitable linear feedback design around the open-loop. The performance of the resulting feedback law is shown to be asymptotically close to the true stochastic feedback law to fourth order in a small noise parameter $\epsilon$. The decoupling theory is empirically tested on robotic planning problems under uncertainty.