[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83799-en":3,"doc-seo-83799-105":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},83799,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Approximate Feedback Linearization for a Nonlinear Hyperbolic PDE Class Part I Volterra Truncation","Backstepping for nonlinear PDEs produces exact feedback linearizing laws as infinite Volterra series, but direct implementation is impractical. This paper proves that low-order finite truncations, though no longer exactly linearizing, still deliver stabilization by relying on the smallness of higher-order terms near the origin. Stability is regained by restricting the initial-condition magnitude, yielding spatial sup-norm results with finite-time practical stability and class-KL asymptotic behavior, including region-of-attraction and truncation-order dependence.","arXiv :2607 .04361v1 [ ee ss . SY] 5 Jul 2026  \nApproximate Feedback Linearization fora Nonlinear Hyperbolic PDE Class—PartI: Volterra Truncation  \nMiroslav Krstic  \nAbstract  \nBackstepping for nonlinear PDEs yields exact feedback linearizing laws in the form of infinite Volterra series—elegant in theory, but with challenges for implementation. This paper shows that even very low-order truncations of such controllers, no longer exactly linearizing, retain the stabilizing power. The key insight is that higher-order terms become negligible near the origin, so stability is recovered for any fixed truncation order by restricting the initial condition size. We establish spatial sup-norm results: finite-time practical stability and asymptotic stability characterized by a class-KL estimate. The region-of-attraction estimate grows with the truncation order and shrinks with the growth rate of the nonlinearity. The analysis overcomes the lack of pointwise kernel bounds and resolves well-posedness of the nonlinear closed loop, showing that surprisingly simple approximations already capture the essence of nonlinear PDE feedback linearization.  \n1 Introduction  \nThis is the first of two companion papers on approximate feedback linearization of a class of nonlinear hyperbolic PDEs. Part I, the present paper, replaces the exact infinite Volterra linearizer of [6] with a finite truncation and establishes closed-loop L∞ stability of the truncated feedback. Part II [5] takes the resulting finite controller and replaces it with a learned neural-operator surrogate, eliminating the plant-specific kernel solve and the N-fold nested integration that would otherwise be required at every control update.  \nPrevious results on boundary control of PDEs with Volterra nonlinearities  \nVazquez and Krstic [8,9] introduced a Volterraseries backstepping framework for boundary control of one-dimensional parabolic PDEs with analytic-type spatial nonlinearities, mapping the closed-loop plant into a stable target heat equation through an infinite Volterra transformation whose kernels are governed by parabolic PDEs on simplices of growing dimension. The result is local exponential stabilization in L2 under the infinite Volterra feedback.  \nThe first hyperbolic counterpart of [8,9] is [6], where the parabolic kernel PDEs are replaced by first-order transport PDEs on growing simplices, solvable along characteristics, and the inverse transformation is constructed by a contraction mapping in L2 ; closed-loop  \nEmail address: [mkrstic@ucsd.edu](mkrstic@ucsd.edu) (Miroslav Krstic).  \n1 Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411 .  \n2 The principal AI aid in developing the paper was Claude.  \nL2 exponential stability follows. A coefficient-level alternative is also developed there, via a noncommutativeformal-series generalization of the Volterra representations of Lesiak and Krener [7], replacing the simplex transport PDEs by scalar ODEs solvable by quadrature.  \nIn finite-dimensional nonlinear control, the idea of transforming a nonlinear system into a linear one modulo higher-order terms goes back to the approximate feedback linearization of Krener [3] and Kang [2] . The present work differs in motivation: exact feedback linearization is available here through an infinite Volterra operator, and the higher-order residual appears only when that operator is truncated for implementation.  \nResults of this paper  \nThis paper is a sequel to and extension of [6], which constructs an exact feedback-linearizing controller for the nonlinear hyperbolic Volterra PDE (1)–(4) as an infinite Volterra series in the state, and proves local exponential stabilization. The controller of [6] is itself a Volterra series, whose kernels are obtained by solving an infinite cascade of first-order transport PDEs on simplices of unbounded dimension. While [6] establishes existence and convergence of this controller, it is not mea","cbCaivjO9idA0MfE","https://ap.wps.com/l/cbCaivjO9idA0MfE","pdf",471650,1,17,"English","en",105,"# Introduction\n## Previous results on boundary control of PDEs with Volterra nonlinearities\n## Results of this paper","[{\"question\":\"What problem does Part I address in approximate feedback linearization for nonlinear hyperbolic PDEs?\",\"answer\":\"Part I studies how replacing the exact infinite Volterra feedback linearizer with a finite truncation affects closed-loop stability, and proves stability properties for the truncated controller.\"},{\"question\":\"Why do finite truncations still stabilize the system even though they are not exactly linearizing?\",\"answer\":\"Higher-order terms become negligible near the origin, so by limiting the initial condition size the residual left by truncation remains small and stabilization is recovered.\"},{\"question\":\"How are stability results quantified in this paper?\",\"answer\":\"The paper establishes L∞ (sup-norm) results including forward invariance, all-time practical stability bounds, finite-time attractivity to a residual ball, and a class-KL asymptotic estimate.\"}]",1784190486,43,{"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},"approximate-feedback-linearization-for-a-nonlinear-hyperbolic-pde-class-part-i-volterra-truncation","",{"@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/approximate-feedback-linearization-for-a-nonlinear-hyperbolic-pde-class-part-i-volterra-truncation/83799/",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},"What problem does Part I address in approximate feedback linearization for nonlinear hyperbolic PDEs?","Question",{"text":75,"@type":76},"Part I studies how replacing the exact infinite Volterra feedback linearizer with a finite truncation affects closed-loop stability, and proves stability properties for the truncated controller.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why do finite truncations still stabilize the system even though they are not exactly linearizing?",{"text":80,"@type":76},"Higher-order terms become negligible near the origin, so by limiting the initial condition size the residual left by truncation remains small and stabilization is recovered.",{"name":82,"@type":73,"acceptedAnswer":83},"How are stability results quantified in this paper?",{"text":84,"@type":76},"The paper establishes L∞ (sup-norm) results including forward invariance, all-time practical stability bounds, finite-time attractivity to a residual ball, and a class-KL asymptotic 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