Hamilton-Jacobi Reachability Analysis

Solving a partial differential equation to compute the safe region — the set of states from which danger is guaranteed avoidable.

A formal safety-verification method long championed by Claire Tomlin and colleagues, with a 2017 survey by Bansal, Chen, Herbert, and Tomlin. Given the system's dynamics, bounded disturbances, and a ‘failure set’ (states already counted as a collision, say), solving a Hamilton-Jacobi partial differential equation yields a value function V(x) whose sign carves out the backward reachable set: states from which, under worst-case disturbance, no control strategy can avoid entering the failure set; every other state forms the safe set, and the boundary's optimal safe control also falls out of the solution. It supports nonlinear dynamics and gives strict guarantees, but computation grows exponentially with state dimension (the curse of dimensionality), limiting it to low-dimensional models. It's commonly used as a safety filter: a learned policy controls the robot normally, and control switches to the HJ-derived safe action only near the boundary of the safe set. Compared to a control barrier function, it computes the safe set directly, while a CBF usually requires a human to supply a candidate function first.

ExampleTwo drones flying toward each other: computing the backward reachable set from their relative position and heading, the original controller is left alone while the relative state stays outside that set, and the moment it touches the boundary, the HJ-derived avoidance maneuver takes over immediately.

Also called

HJ Reachability

Related

Control Barrier Function · Safety Filter · Safe Reinforcement Learning · Optimal Control · Value Function · Embodied Safety

Sources

Bansal, Chen, Herbert, Tomlin, Hamilton-Jacobi Reachability: A Brief Overview and Recent Advances (arXiv 1709.07523)