Conservative Linear Envelopes for High-Dimensional, Hamilton-Jacobi Reachability for Nonlinear Systems via the Hopf Formula

Hamilton-Jacobi reachability (HJR) analysis provides a value function that encodes (1) the set of states from which a nonlinear system with bounded control inputs can reach a goal (or avoid a failure set) despite any bounded disturbance, and (2) the corresponding optimal control policy to reach (or avoid). Though powerful, traditional methods for HJR rely on dynamic programming and suffer from exponential computation growth with respect to state dimension. The recently favored Hopf formula mitigates this ''curse of dimensionality'' by providing an efficient and space-parallelizable approach for solving the reachability problem. However, the Hopf formula can only be applied to linear time-varying systems. To overcome this limitation, we show that the error between a nonlinear system and a linear model can be transformed into an adversarial bounded artificial disturbance, making an envelope of the true value. One may then solve the dimension-robust Hopf formula for a linear game with this ''antagonistic error" to perform guaranteed conservative reachability analysis and control synthesis of nonlinear systems; this can be done for problem formulations in which no other HJR method is both computationally feasible and guaranteed. In addition, we offer several technical methods for reducing conservativeness in the analysis. We demonstrate the theory by solving the safe linear envelope in the controlled Van der Pol system, where the true reachable set may be observed, and by solving a 5 agent (15D) pursuit-evasion game with Dubins cars.