  Hamilton-Jacobi (HJ) reachability is a method that provides rigorous analyses
of the safety properties of dynamical systems. This method has been
successfully applied to many low-dimensional dynamical system models such as
coarse models of aircraft and quadrotors in order to provide safety guarantees
in potentially dangerous scenarios. These guarantees can be provided by the
computation of a backward reachable set (BRS), which represents the set of
states from which the system may be driven into violating safety properties
despite the system's best effort to remain safe. Unfortunately, HJ reachability
is not practical for high-dimensional systems because the complexity of the BRS
computation scales exponentially with the number of state dimensions. Although
numerous approximation techniques are able to tractably provide conservative
estimates of the BRS, they often require restrictive assumptions about system
dynamics without providing an exact solution. In this paper we propose a
general method for decomposing dynamical systems. Even when the resulting
subsystems are coupled, relatively high-dimensional BRSs that were previously
intractable or expensive to compute can now be quickly and exactly computed in
lower-dimensional subspaces. As a result, the curse of dimensionality is
alleviated to a large degree without sacrificing optimality. We demonstrate our
theoretical results through two numerical examples: a 3D Dubins Car model and a
6D Acrobatic Quadrotor model.
