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In this paper, we investigate the problem of controller design for positive systems with the use of linear Lyapunov function. We first present an analytical method to compute the exact value of the l1-induced norm. Then, we propose a novel characterization under which discrete positive system is asymptotically stable with a prescribed l1-induced performance. Based on the characterization, a necessary and sufficient condition for the existence of desired controllers is presented, and an iterative linear matrix inequality approach is developed to solve the design condition. Finally, a numerical example is given to illustrate the effectiveness of the proposed theoretical results.