Optimal or approximately optimal control problem for a discrete stochastic singular system is considered. The driving wide band noise is state dependent and the dynamics is non–smooth. It is shown that the interpolated sequence converge weakly to a singularly controlled diffusion process. The optima or “nearly optima” controls of the limit diffusion are shown to be “nearly optima” for the actual system. Weak convergence analysis is used via a combination of Skorohod and pseudo path topology. Discounted cost criterion is considered