{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/koopman-hopf-hamilton-jacobi-reachability-and","title":"Koopman-Hopf Hamilton-Jacobi Reachability and Control","arxiv_id":"2303.11590","date":"2023-03-21","proceeding":null,"authors":["Will Sharpless","Nikhil Shinde","Matthew Kim","Yat Tin Chow","Sylvia Herbert"],"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.","url_abs":"https://arxiv.org/abs/2303.11590v7","url_pdf":"https://arxiv.org/pdf/2303.11590v7.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"koopman-hopf-hamilton-jacobi-reachability-and","repo_url":"https://github.com/ucsd-saslab/hopfreachability","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"jax","reach":{"status":"ok"}},{"paper_slug":"koopman-hopf-hamilton-jacobi-reachability-and","repo_url":"https://github.com/UCSD-SASLab/HopfReachability/tree/main/Koopman-Hopf","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"jax","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}