Near optimality of Lipschitz and smooth policies in controlled diffusions

For optimal control of diffusions under several criteria, due to computational or analytical reasons, many studies have a apriori assumed control policies to be Lipschitz or smooth, often with no rigorous analysis on whether this restriction entails loss. While optimality of Markov/stationary Markov policies for expected finite horizon/infinite horizon (discounted/ergodic) cost and cost-up-to-exit time optimal control problems can be established under certain technical conditions, an optimal solution is typically only measurable in the state (and time, if the horizon is finite) with no apriori additional structural properties. In this paper, building on our recent work (Pradhan and Yüksel, 2024) establishing the regularity of optimal cost on the space of control policies under the Borkar control topology for a general class of controlled diffusions in $R^d$, we establish near optimality of smooth or Lipschitz continuous policies for optimal control under expected finite horizon, infinite horizon discounted, infinite horizon average, and up-to-exit time cost criteria. Under mild assumptions, we first show that smooth/Lipschitz continuous policies are dense in the space of Markov/stationary Markov policies under the Borkar topology. Then utilizing the continuity of optimal costs as a function of policies on the space of Markov/stationary policies under the Borkar topology, we establish that optimal policies can be approximated by smooth/Lipschitz continuous policies with arbitrary precision. While our results are extensions of our recent work, the practical significance of an explicit statement and accessible presentation dedicated to Lipschitz and smooth policies, given their prominence in the literature, motivates our current paper.