[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85316-en":3,"doc-seo-85316-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85316,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Learning to Control Switching Nonlinear Systems with Koopman Operator Regression","Identification and control of nonlinear systems with finite action spaces are addressed by estimating unknown dynamics from finite samples using Koopman operator regression in a reproducing kernel Hilbert space. The approach yields a linear switching predictive model whose switches depend on the control variable value. Closed-loop control is performed through an infinite-horizon time-varying stage-cost optimal control problem solved via model predictive control, with learning-rate and sub-optimality guarantees. Numerical experiments on the Duffing oscillator validate the theory.","arXiv :2607 . 11344v1 [math .OC] 13 Jul 2026  \nLearning to Control Switching Nonlinear Systems with  \nKoopman Operator Regression ∗  \nEdoardo Caldarelli†1, Oleksii Kachaiev†2, Cesare Molinari2 , and Lorenzo Rosasco 1, 3  \n1 Istituto Italiano di Tecnologia, Genoa, Italy  \n2 MaLGa center, DIMA, Universit`a degli Studi di Genova, Genoa, Italy  \n3 MaLGa center, DIBRIS, Universit`a degli Studi di Genova, Genoa, Italy  \nCorrespondence to: edoardo . caldarelli@iit . it  \nAbstract  \nIn this work, we consider the identification and control of nonlinear systems with finite action spaces. The unknown dynamics are estimated from finite samples with Koopman operator regression in a reproducing kernel Hilbert space, yielding a linear switching predictive model, the switches governed by the value of the control variable. In order to perform control in closed-loop, the learned dynamics are employed in an infinite-horizon optimal control problem with time-varying stage cost, which is solved by means of model predictive control. In a theoretical analysis, we derive learning rates for the Koopman dynamics approximation. We further quantify, under suitable assumptions, the sub-optimality of the model predictive control strategy, both in the case of exact Koopman dynamics, and in the case of learned ones. Numerical simulations on the Duffing oscillator complement our theoretical findings.  \nKeywords Koopman operator; kernel methods; switching controllers; model predictive control; databased control.  \n1 Introduction and related works  \nOptimal control is a well established, powerful tool to control dynamical systems in closed-loop [1] . Given a state space X, a control space U, a flow map f : X × U → X, and a dynamical system described, e.g. , by a difference equation  \nxt+1 = f(xt , ut) , (1)  \noptimal control entails solving an optimization problem of the form  \n∞  \num0,ui1n, .. X g(t, xt , ut) , subject to xt+1 = f(xt , ut), x0 ∈ X . (2)  \nt=0  \n∗ Preprint. Under review.†Equal contribution.  \nfor some stage cost g : N0 × X × U → R≥0 .  \nWhile optimal control for linear systems enjoys a long-lasting history of results and well-developed techniques [2,3], when the system’s dynamics are nonlinear, the solution of problem (2) is challenging. The Koopman operator formalism [4–7] has emerged as an effective paradigm to obtain a global linearization of the dynamics of interest, by lifting the state to a possibly infinite-dimensional space of functions H, named observable space. Having introduced a lifting map x →7 ψx , we can define the lifted state as  \nzt = ψxt . (3)  \nWe can therefore describe how such a lifted state evolves over time, in terms of the Koopman operator.  \nWhen considering controlled dynamical systems, the classical formulation of the Koopman operator needs to be adapted to account for the control variable u [6] . State-of-the-art approaches consider additional linearity assumptions on the impact of the control on the lifted dynamics [8–10] . Other concurrent works, such as [11–13], restrict the dynamics (1) to be bi-linear, which transfers to a similar structure in the Koopman model. In this work, we assume the set of available controls to be finite (a common feature, e.g., in power electronics applications [14]) . In this way, the nonlinear dynamics translate into a family of systems, one for each value of the control variable. Consequently, we are able to define a corresponding family of Koopman operators Ku : H → H, that evolve the lifted state z over time according to the rule  \nzt+1 = Kutzt. (4)  \nThe linear switching system in (4) was firstly introduced in the seminal work by Peitz & Klus [15], and is analogous to (1) under suitable assumptions. Furthermore, it can be related to state-of-the-art models in reinforcement learning with operator world models, see the work by Novelli et al. [16] .  \nSuch lifted dynamics can be used to define an optimal control in the lifted state space, i.e. ,  \n∞  \num0,ui1n, .. X ℓ(t, zt , ut) , s","cbCaiiFdWGYd1ETS","https://ap.wps.com/l/cbCaiiFdWGYd1ETS","pdf",3800220,1,35,"English","en",105,"# Abstract\n# Introduction and related works\n## Optimal control for nonlinear systems\n## Koopman operator and controlled lifted dynamics\n## Finite control sets and time-varying costs\n## Learning Koopman operators via RKHS regression","[{\"question\":\"How are the nonlinear dynamics identified in this work?\",\"answer\":\"Unknown dynamics are learned from finite samples via Koopman operator regression formulated in a reproducing kernel Hilbert space.\"},{\"question\":\"What does the resulting model look like for control?\",\"answer\":\"The method produces a linear switching predictive model in a lifted observable space, where switching is governed by the control variable value.\"},{\"question\":\"How is closed-loop control computed?\",\"answer\":\"Learned dynamics are used inside an infinite-horizon optimal control problem with a time-varying stage cost, solved using model predictive control.\"}]",1784202434,88,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"learning-to-control-switching-nonlinear-systems-with-koopman-operator-regression","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/learning-to-control-switching-nonlinear-systems-with-koopman-operator-regression/85316/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How are the nonlinear dynamics identified in this work?","Question",{"text":75,"@type":76},"Unknown dynamics are learned from finite samples via Koopman operator regression formulated in a reproducing kernel Hilbert space.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the resulting model look like for control?",{"text":80,"@type":76},"The method produces a linear switching predictive model in a lifted observable space, where switching is governed by the control variable value.",{"name":82,"@type":73,"acceptedAnswer":83},"How is closed-loop control computed?",{"text":84,"@type":76},"Learned dynamics are used inside an infinite-horizon optimal control problem with a time-varying stage cost, solved using model predictive control.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]