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A mobile robot's motion is directly affected by the maximum velocity its actuators can provide. This problem becomes more serious when the robot tracks a moving target which imposes demanding requirements on the robot's driving capabilities. To find a solution to this problem, this paper extends the conventional Lyapunov-based motion controller while taking the limit on the robot's velocity into consideration. The effects of the target's motion are explicitly addressed in the controller design. The control outputs are attenuated by selected sinusoidal functions of the states without compromising the asymptotic stability of the controlled system. Next, a new Lyapunov function reflecting the priority of the tracking objectives is chosen to develop the controller which further reduces the surges and fluctuations of the controller outputs. The smoothness of the robot motion is improved, the reasonable magnitudes of the robot velocities and a better target tracking performance are achieved. Extensive simulation and experimental results in various situations are provided to verify the effectiveness of the proposed approaches.