- Abstract: The identification of unknown dynamical systems using supervised learning enables model-based control of systems that cannot be modeled based on first principles. While most control literature focuses on the analysis of a static dataset, online learning control, where data points are added while the controller is running, has rarely been studied in depth. In this paper, we present a data-efficient approach for online learning control based on Gaussian process models. To enable real-time capability despite high computational loads with growing datasets, we propose a safe forgetting mechanism. Using an entropy criterion, data points are selected based on their utility for the future trajectory under consideration of the stability of the closed-loop system. The approach is evaluated in a simulation and in a robotic experiment to demonstrate its computational efficiency.