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This paper considers the problem of constrained nonlinear trajectory tracking control for unmanned air vehicles (UAVs). We assume that the UAV is equipped with longitudinal and lateral autopilots which reduces the 12-state model to a six-state model with altitude, heading, and velocity command inputs. One of the novel features of our approach is that we explicitly account for heading rate and velocity input constraints. For a UAV, the velocity is constrained to lie between two positive constants, and therefore presents particular challenges for the control design. We propose a control Lyapunov function (CLF) approach. We first introduce a CLF for the input constrained case, and then construct the set of all constrained inputs that are feasible with respect to this CLF. The control input is then selected from this "feasible" set. The proposed approach is applied to a simulation scenario, where the UAV is assigned to transition through several targets in the presence of multiple dynamic threats.