[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84699-en":3,"doc-seo-84699-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},84699,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Learning Robust Control Lyapunov Functions through Lipschitz Neural Networks","This work introduces a learning framework for robust control Lyapunov functions and stabilizing state-feedback controllers for nonlinear dynamical systems with additive disturbances bounded by a state-dependent function. Lipschitz neural networks are used to learn both the Lyapunov function and the controller simultaneously. The method derives explicit Hessian and third-order derivative bounds in spectral norm and proposes a GPU-friendly branch-and-bound verification algorithm that accelerates Lyapunov condition checking. Extensive simulations on six dynamical systems validate the approach.","IEEE TRANSACTIONS AND JOURNALS TEMPLATE 1  \nLearning Robust Control Lyapunov Functions through Lipschitz Neural Networks  \nShiqing Wei, Graduate Student Member, IEEE, Prashanth Krishnamurthy, Member, IEEE,  \nand Farshad Khorrami, Fellow, IEEE  \narXiv :2607 .03713v1 [ ee ss . SY] 4 Jul 2026  \nAbstract—This work presents a novel framework for learning robust control Lyapunov functions and stabilizing controllers for nonlinear dynamical systems subject to additive disturbances upper bounded by a state-dependent function. We leverage recent advances in Lipschitz neural networks to jointly learn both the Lyapunov functions and state-feedback controllers. We establish explicit boundson the Hessian and third-order derivatives of these neural networks in the spectral norm, and introduce a GPUfriendly branch-and-bound algorithm that utilizes higherorder bounds to significantly accelerate the verification of the Lyapunov conditions. Finally, we validate the proposed approach through extensive simulations on six different dynamical systems.  \nIndex Terms—Stability of nonlinear systems, neural networks, robust control, computer-aided control design  \nI. INTRODUCTION  \nS  \nTABILITY controlling  \nanalysis of dynamical systems is crucial for nonlinear systems where linear approxima-  \ntions are insufficient. In many scenarios, model mismatch or external disturbances may exist and necessitate robust control strategies, e.g., a robotic manipulator operating in an environment with unpredictable loads and friction variations.  \nLyapunov functions are a foundational tool in the stability analysis of dynamical systems [1] . They offer a sufficient condition for the stability of a nonlinear system by designing a positive definite scalar function that decreases along solution trajectories, which enables characterization of the forward invariant region (or region of attraction) and robustification of control designs. While converse theorems guarantee the existence of a suitable Lyapunov function given certain stability properties of the system [2], these results are typically non-constructive in nature. Consequently, extensive research has focused on constructing Lyapunov functions using both analytical [3] and computational methods [4] .  \nNumerous computational methods have been investigated since the 1950s. Zubov introduced a specific Lyapunov function as the solution to a first-order partial differential equa  \nThis work was supported in part by ARO grants W911NF-21-1-0155 and W911NF-22-1-0028 and by the New York University Abu Dhabi (NYUAD) Center for Artificial Intelligence and Robotics (CAIR), funded by Tamkeen under the NYUAD Research Institute Award CG010 .  \nShiqing Wei, Prashanth Krishnamurthy, and Farshad Khorrami are with Control/Robotics Research Laboratory, Electrical and Computer Engineering Department, Tandon School of Engineering, New York University, Brooklyn, NY 11201, USA. E-mail: {shiqing.wei, prashanth.krishnamurthy, [khorrami](khorrami}@nyu.edu)[}](khorrami}@nyu.edu)[@nyu.edu](khorrami}@nyu.edu).  \ntion (now known as Zubov’s equation) whose solutions are computed using series expansions and other techniques [5],[6] . Later, Linear Programming (LP) methods were used to compute Lyapunov functions, first for switched linear systems [7] and subsequently for nonlinear systems [8] . Since the 1990s, significant effort has been dedicated to Linear Matrix Inequality (LMI) methods [9] for nonlinear systems, which paved the way for a variety of techniques based on sums-ofsquares (SOS) approaches [10]–[12] . However, the automatic construction of Lyapunov functions for general nonlinear systems remains an open question [4] .  \nAdvances in computational hardware and deep learning frameworks have led many researchers to construct (control) Lyapunov functions for nonlinear systems using neural networks—a line of inquiry that also allows the joint learning of a state-feedback controller for a control system. Unlike earlier computational meth","cbCaicUkYHgo5ZJq","https://ap.wps.com/l/cbCaicUkYHgo5ZJq","pdf",2302824,1,17,"English","en",105,"# Introduction\n## Stability and Lyapunov foundations\n## Classical computational approaches\n## Neural-network-based Lyapunov learning and verification\n## Robustness gap and related work","[{\"question\":\"What problem does the document address in robust control?\",\"answer\":\"It addresses learning robust control Lyapunov functions and stabilizing controllers for nonlinear systems with additive disturbances that are bounded by a state-dependent function.\"},{\"question\":\"How does the proposed method use Lipschitz neural networks?\",\"answer\":\"It leverages Lipschitz neural networks to jointly learn the Lyapunov functions and the state-feedback controllers while enabling formal verification of Lyapunov conditions.\"},{\"question\":\"How is verification accelerated in the framework?\",\"answer\":\"The work derives explicit bounds on higher-order derivatives and uses a GPU-friendly branch-and-bound algorithm that exploits these bounds to significantly speed up checking the Lyapunov conditions.\"}]",1784197725,43,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"learning-robust-control-lyapunov-functions-through-lipschitz-neural-networks","",{"@graph":35,"@context":84},[36,53,67],{"@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-robust-control-lyapunov-functions-through-lipschitz-neural-networks/84699/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the document address in robust control?","Question",{"text":74,"@type":75},"It addresses learning robust control Lyapunov functions and stabilizing controllers for nonlinear systems with additive disturbances that are bounded by a state-dependent function.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the proposed method use Lipschitz neural networks?",{"text":79,"@type":75},"It leverages Lipschitz neural networks to jointly learn the Lyapunov functions and the state-feedback controllers while enabling formal verification of Lyapunov conditions.",{"name":81,"@type":72,"acceptedAnswer":82},"How is verification accelerated in the framework?",{"text":83,"@type":75},"The work derives explicit bounds on higher-order derivatives and uses a GPU-friendly branch-and-bound algorithm that exploits these bounds to significantly speed up checking the Lyapunov 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