Nonholonomic wheeled mobile robots are often required to implement control algorithms designed for holonomic kinematic systems. This creates a velocity tracking problem for an actual wheeled mobile robot. In this paper, we investigate the issue of tracking a desired velocity in the least amount of time, for a differential drive nonholonomic wheeled mobile robot with torque inputs. The Pontryagin Maximum Principle provides time-optimal controls that must be implemented as open-loop commands to the motors. We propose two discontinuous state-based feedback control laws, such that the associated closed-loop systems track a desired velocity in minimum time. The feedback control laws are rigorously shown to produce only time-optimal trajectories, by constructing a regular synthesis for each control law. The availability of these time-optimal feedback control laws makes re-computation of open-loop time-optimal controls (due to changes in the desired velocity or input disturbances) unnecessary. (C) 2017 Published by Elsevier Ltd.