Koopman-Hopf Hamilton-Jacobi Reachability and Control

TL;DR

This paper introduces a novel method combining Koopman theory with the Hopf formula to efficiently approximate Hamilton-Jacobi reachability for high-dimensional nonlinear systems, enabling better control under disturbances.

Contribution

It extends the Hopf formula to nonlinear systems via Koopman linearization, providing a scalable approach for reachability analysis and control in high-dimensional settings.

Findings

Koopman-Hopf method accurately approximates reachable sets for nonlinear systems.

The approach outperforms traditional controllers in a 10D nonlinear glycolysis model.

Open-source Julia toolbox facilitates application of the method.

Abstract

The Hopf formula for Hamilton-Jacobi reachability (HJR) analysis has been proposed to solve high-dimensional differential games, producing the set of initial states and corresponding controller required to reach (or avoid) a target despite bounded disturbances. As a space-parallelizable method, the Hopf formula avoids the curse of dimensionality that afflicts standard dynamic-programming HJR, but is restricted to linear time-varying systems. To compute reachable sets for high-dimensional nonlinear systems, we pair the Hopf solution with Koopman theory for global linearization. By first lifting a nonlinear system to a linear space and then solving the Hopf formula, approximate reachable sets can be efficiently computed that are much more accurate than local linearizations. Furthermore, we construct a Koopman-Hopf disturbance-rejecting controller, and test its ability to drive a 10-dimensional nonlinear glycolysis model. We find that it significantly out-competes expectation-minimizing and game-theoretic model predictive controllers with the same Koopman linearization in the presence of bounded stochastic disturbance. In summary, we demonstrate a dimension-robust method to approximately solve HJR, allowing novel application to analyze and control high-dimensional, nonlinear systems with disturbance. An open-source toolbox in Julia is introduced for both Hopf and Koopman-Hopf reachability and control.