Koopman-Hopf Hamilton-Jacobi Reachability and Control

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Publication date: 21 March 2023

Abstract: The Hopf formula for Hamilton-Jacobi Reachability analysis has been proposed for solving viscosity solutions of high-dimensional differential games as a space-parallelizeable 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 online 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 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.

Has companion code repository: https://github.com/ucsd-saslab/hopfreachability

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