[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83094-en":3,"doc-seo-83094-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83094,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Adaptive and Neural Operator Control of Nonlinear Volterra Hyperbolic PDEs","Adaptive control learns plant behavior online while neural-operator control learns controller gains offline. The approach targets nonlinear hyperbolic PDEs whose dynamics are driven by an unknown Volterra series with arbitrarily many kernels. An observer-based passive identifier learns an online truncation, and an infinite-dimensional backstepping kernel synthesis is approximated offline by a neural operator to avoid real-time PDE cascades. Closed-loop stability and asymptotic regulation are proven on an expanding basin as neural-operator accuracy improves.","Adaptive and Neural Operator Control of Nonlinear Volterra Hyperbolic PDEs  \nMiroslav Krstic  \narXiv :2607 .06425v1 [ ee ss . SY] 7 Jul 2026  \nAbstract—Adaptive control learns the plant online; neural-operator control learns the control gains offline. We bring the two together for a class of nonlinear hyperbolic PDEs whose dynamics are governed by an unknown Volterra series of arbitrarily many kernels. An observerbased passive identifier learns a truncation of this series online. The infinite-dimensional map that synthesizes the backstepping kernels from the parameter estimates — a cascade of PDEs on simplex domains of increasing dimension, prohibitive to solve in real time — is approximated once, offline, by a neural operator. The closed loop then carries two learning processes in series: online learning of the plant feeds an offline-learned PDE solver, whose output is the online control gains. We prove closed-loop stability and asymptotic regulation of the plant state, observer state, and input, on a basin that recovers the exact-kernel basin as the neural-operator accuracy improves. With a single Lyapunov function we absorb at once the perturbations—all vanishing—of truncating an infinite Volterra series, of identifying the plant online, and of approximating the gains.  \nI. INTRODUCTION  \nIN analogy with the 1989 Sastry-Isidori classic on adaptive  \nfeedback linearization for ODEs [10], this paper solves the same problem for boundary-controlled partial differential equations (PDEs), using PDE backstepping. PDE backstepping produces the feedback gains as the solution of a kernel PDE derived from a coordinate change to a stable target system. Two obstacles separate this methodology from deployment on an imperfectly known plant. First, the kernels are functionalsof the plant coefficients and cannot be computed without a model. Second, even when the model is available, the kernel PDE must be re-solved whenever the plant estimate changes—as it does continuously under adaptation—and for many plant classes this repeated online computation is prohibitive. This paper removes both obstacles simultaneously for a class of nonlinear hyperbolic PDEs.  \nThe plant is a first-order hyperbolic PDE whose in-domain nonlinearity is a Volterra series in the state, with kernels fn defined on the n-dimensional simplices Tn. When the kernels are known, this class is feedback-linearizable: a nonlinear Volterra backstepping transformation maps it to a linear transport equation that reaches the origin in finite time [5], and truncating the series at order N yields a sup-norm stabilizer  \nM. Krstic is with the Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093 USA (e-mail: [krstic@ucsd.edu](krstic@ucsd.edu)) .  \nThe principal AI aide in developing the paper was Claude.  \nwith an explicit basin of attraction [6], in the spirit of the parabolic Volterra designs of [11] . The controller gains are the solution of a cascade of kernel PDEs, the n-th of which lives on the simplex Tn ⊂ Rn ; the domain dimension grows with the Volterra order, and the entire cascade must be resolved for every kernel tuple. The second obstacle above thus appears here in an acute form.  \nNeural operators [9] address the second obstacle by learning, once and offline, the otherwise costly map from plant data to control, so that online a single forward evaluation replaces the PDE solve. The idea was introduced for PDE backstepping in [4] and brought to the truncated Volterra controller in [7], which—for a known plant—learns the feedback operator itself and shows the approximation error to preserve closed-loop stability on a slightly reduced basin.  \nThe first obstacle—the requirement of a model—is addressed by adaptive control. For hyperbolic PDEs, identifierbased designs estimate the plant coefficients online and pass the estimates to a certainty-equivalence controller [1] . We adopt the observer-based passive identifier intro","cbCaipO8nuAninCJ","https://ap.wps.com/l/cbCaipO8nuAninCJ","pdf",1016929,2,1,16,"English","en",105,"# Introduction\n## Problem Setup and Motivation\n## Adaptive Identification and Certainty-Equivalence Control\n## Neural-Operator Approximation of Kernel Cascades\n## Main Contributions and Results","[{\"question\":\"How does the method combine adaptive control with neural-operator control?\",\"answer\":\"Adaptive control estimates the plant online using an observer-based passive identifier, while a neural operator is trained offline to approximate the PDE-based backstepping kernel synthesis. The online estimates feed the offline-learned surrogate, producing controller gains.\"},{\"question\":\"What makes the Volterra hyperbolic PDE control challenge difficult?\",\"answer\":\"The controller requires backstepping kernels that depend on unknown plant coefficients, and the kernel PDE cascade must be re-solved whenever estimates change. Additionally, the first Volterra kernel is unknown and estimated, turning stages into coupled integral equations.\"},{\"question\":\"What theoretical guarantees are provided for the closed-loop system?\",\"answer\":\"The paper proves closed-loop stability and asymptotic regulation for the plant state, observer state, and input, on a basin that expands toward the exact-kernel basin as neural-operator approximation error decreases.\"}]",1784185192,40,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"adaptive-and-neural-operator-control-of-nonlinear-volterra-hyperbolic-pdes","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/adaptive-and-neural-operator-control-of-nonlinear-volterra-hyperbolic-pdes/83094/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","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 does the method combine adaptive control with neural-operator control?","Question",{"text":75,"@type":76},"Adaptive control estimates the plant online using an observer-based passive identifier, while a neural operator is trained offline to approximate the PDE-based backstepping kernel synthesis. The online estimates feed the offline-learned surrogate, producing controller gains.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What makes the Volterra hyperbolic PDE control challenge difficult?",{"text":80,"@type":76},"The controller requires backstepping kernels that depend on unknown plant coefficients, and the kernel PDE cascade must be re-solved whenever estimates change. Additionally, the first Volterra kernel is unknown and estimated, turning stages into coupled integral equations.",{"name":82,"@type":73,"acceptedAnswer":83},"What theoretical guarantees are provided for the closed-loop system?",{"text":84,"@type":76},"The paper proves closed-loop stability and asymptotic regulation for the plant state, observer state, and input, on a basin that expands toward the exact-kernel basin as neural-operator approximation error decreases.","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":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":29,"slug":118},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]