Convergence of numerical optimal feedback policies for deterministic optimal control problems

We consider a Markov chain based numerical approximation method for a class of deterministic nonlinear optimal control problems. It is known that methods of this type yield convergent approximations to the value function on the entire domain. These results do not easily extend to the optimal control, which need not be uniquely defined on the entire domain. There are, however, regions of strong regularity on which the optimal control is well defined and smooth. Typically, the union of these regions is open and dense in the domain. Using probabilistic methods, we prove that on the regions of strong regularity, the Markov chain method yields a convergent sequence of approximations to the optimal feedback control. The result is illustrated with several examples.