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This paper provides the design and implementation of an L1-optimal control of a quadrotor unmanned aerial vehicle (UAV). The quadrotor UAV is an underactuated rigid body with four propellers that generate forces along the rotor axes. These four forces are used to achieve asymptotic tracking of four outputs, namely the position of the center of mass of the UAV and the heading. With perfect knowledge of plant parameters and no measurement noise, the magnitudes of the errors are shown to exponentially converge to zero. In the case of parametric uncertainty and measurement noise, the controller yields an exponential decrease of the magnitude of the errors in an L1-optimal sense. In other words, the controller is designed so that it minimizes the L∞-gain of the plant with respect to disturbances. The performance of the controller is evaluated in experiments and compared with that of a related robust nonlinear controller in the literature. The experimental data shows that the proposed controller rejects persistent disturbances, which is quantified by a very small magnitude of the mean error.