This technical note studies the adaptive control for linear systems with set-valued observations to track a given periodic target. Based on the system parameters, accessorial parameters with the same order as that of the tracking targets are introduced and estimated. Considering the system parameters are unknown and set-valued observations can supply only limited information each time, a two-scale adaptive control algorithm is designed. Each control input is designed at the large time scale and lasts for a holding time (small scale), during which the parameter estimation algorithm is constructed. From the estimate of accessorial parameters, the control signal is updated at the large time scale by the certainty equivalence principle. As the holding time goes to infinity, the algorithm can be proved to be asymptotically efficient in a certain sense. Meanwhile, the adaptive tracking algorithm is shown to be asymptotically optimal. A numerical example is given to demonstrate the effectiveness of the algorithms and the main results obtained.