The Hopf formula for Hamilton-Jacobi Reachability analysis has been proposed for controlling stochastic, nonlinear, high-dimensional systems as a space-parallelizable method. In exchange, however, a complex, potentially non-convex optimization problem must be solved, limiting its application to linear time-varying systems. With the intent of solving Hamilton-Jacobi backwards reachable sets (BRS) and their corresponding controllers, we pair the Hopf solution with Koopman theory, which can linearize high-dimensional nonlinear systems. We find that this is a viable method for approximating the BRS and performs better than naive, local linearizations. Furthermore, we construct a Koopman-Hopf controller for robustly driving a 10-dimensional, nonlinear, stochastic, glycolysis model and find that it significantly out-competes both stochastic and game-theoretic Koopman-based model predictive controllers against stochastic disturbance. Although the demonstrations are limited, we believe the proposed method has implications for engineering efforts in diverse domains, particularly in medicine, finance and other large systems where control has remained intractable.

The video of the presentation is available here.