A Minimum Discounted Reward Hamilton-Jacobi Formulation for Computing Reachable Sets
This work addresses the challenge of computing reachable sets more efficiently for control systems, with potential applications in reinforcement learning for systems with unknown dynamics, though it appears incremental in improving existing formulations.
The authors tackled the problem of approximating reachable sets by proposing a novel formulation based on a minimum discounted reward optimal control problem, which yields a continuous solution via a Hamilton-Jacobi equation and enables more efficient solution methods through a contraction mapping, as demonstrated on benchmark examples like double integrator and pursuit-evasion games.
We propose a novel formulation for approximating reachable sets through a minimum discounted reward optimal control problem. The formulation yields a continuous solution that can be obtained by solving a Hamilton-Jacobi equation. Furthermore, the numerical approximation to this solution can be obtained as the unique fixed-point to a contraction mapping. This allows for more efficient solution methods that could not be applied under traditional formulations for solving reachable sets. In addition, this formulation provides a link between reinforcement learning and learning reachable sets for systems with unknown dynamics, allowing algorithms from the former to be applied to the latter. We use two benchmark examples, double integrator, and pursuit-evasion games, to show the correctness of the formulation as well as its strengths in comparison to previous work.