  The control of complex systems faces a trade-off between high performance and
safety guarantees, which in particular restricts the application of
learning-based methods to safety-critical systems. A recently proposed
framework to address this issue is the use of a safety controller, which
guarantees to keep the system within a safe region of the state space. This
paper introduces efficient techniques for the synthesis of a safe set and
control law, which offer improved scalability properties by relying on
approximations based on convex optimization problems. The first proposed method
requires only an approximate linear system model and Lipschitz continuity of
the unknown nonlinear dynamics. The second method extends the results by
showing how a Gaussian process prior on the unknown system dynamics can be used
in order to reduce conservatism of the resulting safe set. We demonstrate the
results with numerical examples, including an autonomous convoy of vehicles.
