  Data-driven models are subject to model errors due to limited and noisy
training data. Key to the application of such models in safety-critical domains
is the quantification of their model error. Gaussian processes provide such a
measure and uniform error bounds have been derived, which allow safe control
based on these models. However, existing error bounds require restrictive
assumptions. In this paper, we employ the Gaussian process distribution and
continuity arguments to derive a novel uniform error bound under weaker
assumptions. Furthermore, we demonstrate how this distribution can be used to
derive probabilistic Lipschitz constants and analyze the asymptotic behavior of
our bound. Finally, we derive safety conditions for the control of unknown
dynamical systems based on Gaussian process models and evaluate them in
simulations of a robotic manipulator.
