  Learning-based methods have been successful in solving complex control tasks
without significant prior knowledge about the system. However, these methods
typically do not provide any safety guarantees, which prevents their use in
safety-critical, real-world applications. In this paper, we present a
learning-based model predictive control scheme that can provide provable
high-probability safety guarantees. To this end, we exploit regularity
assumptions on the dynamics in terms of a Gaussian process prior to construct
provably accurate confidence intervals on predicted trajectories. Unlike
previous approaches, we do not assume that model uncertainties are independent.
Based on these predictions, we guarantee that trajectories satisfy safety
constraints. Moreover, we use a terminal set constraint to recursively
guarantee the existence of safe control actions at every iteration. In our
experiments, we show that the resulting algorithm can be used to safely and
efficiently explore and learn about dynamic systems.
