We study two different diffusion control problems arising from asymptotic analysis of controlled stochastic networks in heavy traffic.  

The first concerns ergodic drift control for reflected diffusions in polyhedral domains and the second is a singular control problem with state constraints. These arise as formal diffusion approximations of arrival/service rate control and scheduling control problems, respectively, for critically loaded stochastic processing networks.

From prior works it is known that under suitable conditions, value functions of appropriately scaled stochastic processing systems are asymptotically bounded below by those of the corresponding diffusion control problems. In this work we show that under broad conditions the reverse inequality holds, thus establishing convergence of value functions. The result provides a mathematical justification for the use of diffusion control problems as approximating models for such processing systems.