  Reachability analysis is important for studying optimal control problems and
differential games, which are powerful theoretical tools for analyzing and
modeling many practical problems in robotics, aircraft control, among other
application areas. In reachability analysis, one is interested in computing the
reachable set, defined as the set of states from which there exists a control,
despite the worst disturbance, that can drive the system into a set of target
states. The target states can be used to model either unsafe or desirable
configurations, depending on the application. Many Hamilton-Jacobi formulations
allow the computation of reachable sets; however, due to the exponential
complexity scaling in computation time and space, problems involving
approximately 5 dimensions become intractable. A number of methods that compute
an approximate solution exist in the literature, but these methods trade off
complexity for optimality. In this paper, we eliminate complexity-optimality
trade-offs for time-invariant decoupled systems using a decoupled
Hamilton-Jacobi formulation that enables the exact reconstruction of high
dimensional solutions via low dimensional solutions of the decoupled
subsystems. Our formulation is compatible with existing numerical tools, and we
show the accuracy, computation benefits, and an application of our novel
approach using two numerical examples.
