Certifying Hamilton-Jacobi Reachability Learned via Reinforcement Learning

We present a framework to certify Hamilton--Jacobi (HJ) reachability learned by reinforcement learning (RL). Building on a discounted initial time 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 Wλ solves the discounted travel-cost Hamilton--Jacobi--Bellman (HJB) equation, then W_varepsilon:=Wλ + varepsilon solves the same PDE with a constant offset λvarepsilon. This means that a uniform value error is 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.