Reachability Theory

Reachability theory, a branch of optimal control theory, describes how objects within a set of initial states may possibly reach a new volume of state space after a known period of time. Most often reachability is used in the context of safety, wherein it is rigorously shown that certain dangerous states are not achievable in fixed time. Unfortunately, solving reachability problems involves finding a solution to the Hamilton Jacobi Bellman equation, which is well known to require a computation time that is exponential with the dimension of the state space under consideration. The extreme computational cost of the method effectively limits its application to state spaces of 3-4 dimensions.