Linear quadratic control of nonlinear systems with Koopman operator learning and the Nyström method

Abstract

© 2025 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)In this paper, we study how the Koopman operator framework can be combined with kernel methods to effectively control nonlinear dynamical systems. While kernel methods have typically large computational requirements, we show how random subspaces (Nyström approximation) can be used to achieve huge computational savings while preserving accuracy. Our main technical contribution is deriving theoretical guarantees on the effect of the Nyström approximation. More precisely, we study the linear quadratic regulator problem, showing that the approximated Riccati operator converges at the rate m^(-1/2), and the regulator objective, for the associated solution of the optimal control problem, converges at the rate m^-1, where m is the random subspace size. Theoretical findings are complemented by numerical experiments corroborating our results.E. Caldarelli, A. Colomé and C. Torras acknowledge support from the project CLOTHILDE (“CLOTH manIpulation Learning from DEmonstrations”), funded by the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (Advanced Grant agreement No 741930). E. Caldarelli acknowledges travel support from ELISE (GA No 951847). C. Molinari is part of “GNAMPA” (INdAM) and has been supported by the projects MIUR Excellence Department awarded to DIMA UniGe CUP D33C23001110001, AFOSR FA8655-22-1-7034, MIUR-PRIN 202244A7YL and PON “Ricerca e Innovazione” 2014–2020. C. Ocampo-Martinez acknowledges the support from the project SEAMLESS (PID2023-148840OB-I00) funded by MCIN/AEI. L. Rosasco acknowledges the financial support of the European Research Council, (grant SLING 819789), the European Commission, (Horizon Europe grant ELIAS 101120237), the US Air Force Office of Scientific Research (FA8655-22-1-7034), the Ministry of Education, University and Research (FARE grant ML4IP R205T7J2KP; grant BAC FAIR PE00000013 funded by the EU - NGEU) and the Center for Brains, Minds and Machines (CBMM) .Peer ReviewedPostprint (published version