Efficient and Guaranteed Hamilton - Jacobi Reachability via Self-Contained Subsystem Decomposition and Admissible Control Sets

Hamilton-Jacobi reachability analysis is a useful tool for generating reachable sets and corresponding optimal control policies, but its use in high-dimensional systems is hindered by the "curse of dimensionality." Self-contained subsystem decomposition is a proposed solution, but it can produce conservative or incorrect results due to the "leaking corner issue." This issue arises from the inexact decomposition of the target set and inconsistencies across the computed control policies for each coupled subsystem. In this letter, we define and resolve this issue by introducing the notion of an admissible control set that enforces consistent control actions across the coupled subsystems. Our method efficiently computes exact reachable sets and the corresponding optimal control policy for self-contained subsystems with a decomposable goal (or failure) set. We also provide conservative under-approximations for goal (or failure) sets with inexact decomposition. In this conservative case, a local update method in the full dimensional space can be applied to recover exact results. We validate our approach on a 3D system and demonstrate its scalability on a 6D system.