Efficient Computation of State-Constrained Reachability Problems Using Hopf–Lax Formulae

This article considers two state-constrained reachability problems: computing 1) control-invariant and 2) reach–avoid sets, both under state constraints. Prior research has developed Hamilton–Jacobi (HJ) partial differential equations (PDEs) that characterize the optimal cost functions of these two problems. Unfortunately, solving the HJ PDEs by grid-based methods, such as level-set methods, suffers from exponentially growing computational complexity in the system state dimension. In order to alleviate this computational issue, this article proposes a Hopf–Lax formula for each reachability problem's HJ PDE. The advantage of the Hopf–Lax formulae is that they have more favorable convexity conditions than the corresponding problems. Thus, direct methods may be used to solve Hopf–Lax formulae and thus efficiently compute the optimal solution of the reachability problems under specified conditions. This article provides an example for each reachability problem and demonstrates the performance of the proposed Hopf–Lax method.