  Safely controlling unknown dynamical systems is one of the biggest challenges
in the field of control. Oftentimes, an approximate model of a system's
dynamics exists which provides beneficial information for the selection of
controls. However, differences between the approximate and true systems present
challenges as well as safety concerns. We propose an algorithm called
SAFE-SLOPE to safely evaluate points from a Gaussian process model of a
function when its Lipschitz constant is unknown. We establish theoretical
guarantees for the performance of SAFE-SLOPE and quantify how multi-fidelity
modeling improves the algorithm's performance. Finally, we demonstrate how
SAFE-SLOPE achieves lower cumulative regret than a naive sampling method by
applying it to find the control gains of a linear time-invariant system.
