Safe Over- and Under-Approximation of Reachable Sets for Autonomous Dynamical Systems

We present a method based on the Hamilton-Jacobi framework that is able to compute over- and under-approximations of reachable sets for autonomous dynamical systems beyond polynomial dynamics. The method does not resort to user-supplied candidate polynomials, but rather relies on an expansion of the evolution function whose convergence in compact state space is guaranteed. Over- and under-approximations of the reachable state space up to any designated precision can consequently be obtained based on truncations of that expansion. As the truncations used in computing over- and under-approximations as well as their associated error bounds agree, double-sided enclosures of the true reach-set can be computed in a single sweep. We demonstrate the precision of the enclosures thus obtained by comparison of benchmark results to related simulations.