  Control theory can provide useful insights into the properties of controlled,
dynamic systems. One important property of nonlinear systems is the region of
attraction (ROA), a safe subset of the state space in which a given controller
renders an equilibrium point asymptotically stable. The ROA is typically
estimated based on a model of the system. However, since models are only an
approximation of the real world, the resulting estimated safe region can
contain states outside the ROA of the real system. This is not acceptable in
safety-critical applications. In this paper, we consider an approach that
learns the ROA from experiments on a real system, without ever leaving the true
ROA and, thus, without risking safety-critical failures. Based on regularity
assumptions on the model errors in terms of a Gaussian process prior, we use an
underlying Lyapunov function in order to determine a region in which an
equilibrium point is asymptotically stable with high probability. Moreover, we
provide an algorithm to actively and safely explore the state space in order to
expand the ROA estimate. We demonstrate the effectiveness of this method in
simulation.
