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Existing adaptive optimal tracking controllers for linear continuous-time systems rely on a formulation that hinders learning control policies for general reference trajectories; generalizing approaches are currently limited to discrete-time systems. In addition, existing results usually rely on globally discounted objective functions. We demonstrate that global discounting potentially leads to unstable controllers and propose a partially discounted objective function instead, which we show to have a unique, globally asymptotically stabilizing solution in the linear-quadratic case. Based on this result, we present a model-free adaptive tracking control architecture for linear continuous-time systems. Once trained, the controller can be used to track flexible reference trajectories. We demonstrate the functionality of our approach with an example.