Certifying Hamilton-Jacobi Reachability Learned via Reinforcement Learning

We present a framework to \emph{certify} Hamilton–Jacobi (HJ) reachability learned by reinforcement learning (RL). Building on a discounted initial time \emph{travel-cost} formulation that makes small-step RL value iteration provably equivalent to a forward Hamilton–Jacobi (HJ) equation with damping, we convert certified learning errors into calibrated inner/outer enclosures of strict backward reachable tube. The core device is an additive-offset identity: if $W_{\lambda }$ solves the discounted travel-cost Hamilton–Jacobi–Bellman (HJB) equation, then $W_{\epsilon }:=W_{\lambda }+\epsilon$ solves the same PDE with a constant offset $\lambda \epsilon$. This means that a uniform value error is \emph{exactly} equal to a constant HJB offset. We establish this uniform value error via two routes: (A) a Bellman operator-residual bound, and (B) a HJB PDE-slack bound. Our framework preserves HJ-level safety semantics and is compatible with deep RL. We demonstrate the approach on a double-integrator system by formally certifying, via satisfiability modulo theories (SMT), a value function learned through reinforcement learning to induce provably correct inner and outer backward-reachable set enclosures over a compact region of interest.