The stochastic control systemis investigated, where ƒ,gdepend on the past (XS)S≤tof the stateXt, and the controlutdepends on partial observationsy=(Yt) made at a fixed finite sequence of times and lying in a countable observation space. We show ihai for a wide class of functions ƒ,gand terminal cost there is always an optimal relaxed control, and in certain circumstances there is an optimal ordinary control. We work on a hyperfinite Loeb space (a very rich probability space constructed using Robinson's nonstandard analysis), where, with a fixed Brownian motion b we have an explicit solution to the above equation for each controlu. An optimal internal (i.e. nonstandard) control is easily obtained. We show that an internal controlUcorresponds naturally to a standard relaxed control v whose solution process is pathwise the same as that forU.