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Hamilton-Jacobi (HJ) reachability is a method that provides rigorous analysesof the safety properties of dynamical systems. This method has beensuccessfully applied to many low-dimensional dynamical system models such ascoarse models of aircraft and quadrotors in order to provide safety guaranteesin potentially dangerous scenarios. These guarantees can be provided by thecomputation of a backward reachable set (BRS), which represents the set ofstates from which the system may be driven into violating safety propertiesdespite the system's best effort to remain safe. Unfortunately, HJ reachabilityis not practical for high-dimensional systems because the complexity of the BRScomputation scales exponentially with the number of state dimensions. Althoughnumerous approximation techniques are able to tractably provide conservativeestimates of the BRS, they often require restrictive assumptions about systemdynamics without providing an exact solution. In this paper we propose ageneral method for decomposing dynamical systems. Even when the resultingsubsystems are coupled, relatively high-dimensional BRSs that were previouslyintractable or expensive to compute can now be quickly and exactly computed inlower-dimensional subspaces. As a result, the curse of dimensionality isalleviated to a large degree without sacrificing optimality. We demonstrate ourtheoretical results through two numerical examples: a 3D Dubins Car model and a6D Acrobatic Quadrotor model.

Exact and Efficient Hamilton-Jacobi-based Guaranteed Safety Analysis via System Decomposition