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In this paper, we propose a two-step controller design method with control Lyapunov functions (CLFs) for nonlinear systems with convex input constraints. In the first step, we derive an input which minimizes the time derivative of a local CLF via nonlinear convex optimization. According to the Karush-Kuhn-Tucker condition (KKT-condition), we clarify the necessary and sufficient condition for the minimizing input. Then, we discuss the continuity of the minimizing input. We also consider the relation between the minimizing input and the asymptotically stabilizable domain. In the second step, we design a continuous asymptotically stabilizing controller based on the derived minimizing input for the system. Finally, we confirm the effectiveness of the proposed method through an example.