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The subject of this paper is the optimal control of linear plants whose coefficients and whose reference signals are random processes. The discrete-time as well as the continuous-time problems are treated and some fairly deep-seated distinctions between the two are pointed out. Special emphasis is placed on control over long periods of time. A set of assumptions is laid down under which the transition to infinitely long periods of operation is tractable. It is shown that a discrete-time plant is controllable, in the sense that the rate of the mean-squared error remains finite, only when certain necessary and sufficient conditions are fulfilled by the statistics of the plant parameters.