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This brief addresses the control problem of linear time-invariant discrete-time systems with delays. The control is under positivity constraint, which means that the resulting closed-loop systems are not only stable, but also positive. The contribution lies in three aspects. First, a necessary and sufficient condition is established for the existence of such controllers for discrete-time delayed systems. Second, a sufficient condition is provided under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Third, sufficient conditions are proposed for discrete-time delayed systems with uncertainties, whether or not bounded control is considered. All the results are formulated as linear programming problems, hence easy to be verified. And the controllers are explicitly constructed if existent.