A discounted reach-cost Bellman operator is contractive, has a unique fixed point identical to the HJ reachability value function, and can be approximated by sample-based RL.
Robust control barrier–value functions for safety- critical control
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Reachability-Preserving Bellman Operator for the Discounted Reach-Cost Value Function: Uniting Hamilton-Jacobi Reachability and Reinforcement Learning
A discounted reach-cost Bellman operator is contractive, has a unique fixed point identical to the HJ reachability value function, and can be approximated by sample-based RL.