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We present a framework that uses control Lyapunov functions (CLFs) to implement provably stable path-following controllers for autonomous mobile platforms. Our approach is based on learning a guaranteed CLF for path following by using recent approaches — combining machine learning with automated theorem proving — to train a neural network feedback law along with a CLF that guarantees stabilization for driving along low-curvature reference paths. We discuss how key properties of the CLF can be exploited to extend the range of the curvatures for which the stability guarantees remain valid. We then demonstrate that our approach yields a controller that obeys theoretical guarantees in simulation, but also performs well in practice. We show our method is both a verified method of control and better than a common MPC implementation in computation time. Additionally, we implement the controller on-board on a 18 -scale autonomous vehicle testing platform and present results for various robust path following scenarios.