Certifiable Reachability Learning Using a New Lipschitz Continuous Value Function

We propose a new reachability learning framework for high-dimensional nonlinear systems, focusing on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reach-avoid problems</i>. These problems require computing the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reach-avoid set</i>, which ensures that all its elements can safely reach a target set despite disturbances within pre-specified bounds. Our framework has two main parts: offline learning of a newly designed reach-avoid value function, and post-learning certification. Compared to prior work, our new value function is Lipschitz continuous and its associated Bellman operator is a contraction mapping, both of which improve the learning performance. To ensure deterministic guarantees of our learned reach-avoid set, we introduce two efficient post-learning certification methods. Both methods can be used online for real-time local certification or offline for comprehensive certification. We validate our framework in a 12-dimensional crazyflie drone racing hardware experiment and a simulated 10-dimensional highway take-over example.