Optimal control of linear systems with random inputs over a finite operating time is studied using dynamic programming. A quadratic cost functional is used because an analytic solution can be obtained. Minimizing the cost functional over the finite time operation lends to a mode of control towards the end of the interval which is unsatisfactory. This does not occur if our mode of control during the finite interval is the same as it would have boon if that interval was just the first part of a much longer period of operation during which the quadratic cost functional was minimized.