[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86151-en":3,"doc-seo-86151-105":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},86151,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Implicit Neural Networks as Static Controllers Certificates and Performance Separation","Implicit neural controllers (INCs) are static feedback laws evaluated via an algebraic fixed-point equation, covering neural network controllers as special cases. The work introduces an implicit representation of neural networks that yields a trainable linear interconnection closed through a known static activation map, enabling straightforward well-posedness and Lyapunov/IQC analysis. For finite-dimensional LTI plants, it develops Perron–Frobenius and norm conditions, LMI/IQC certificates for exponential stability, and LMI-based discounted infinite-horizon quadratic performance. It also proposes certification-compatible heuristic synthesis with explicit constraints and computable post-training LMIs, then proves constrained-control separation results where an INC outperforms any admissible finite-order dynamic linear controller, supported by numerical examples.","Implicit Neural Networks as Static Controllers: Certificates and Performance Separation  \nGiuseppe C. Calafiore and Laurent El Ghaoui  \narXiv :2607 . 11122v1 [ ee ss . SY] 13 Jul 2026  \nAbstract—Implicit neural controllers (INCs) are static feedback laws that are evaluated through an algebraic fixed point equation; they include as special cases neural network controllers. We propose a so-called implicit representation of neural networks as a key enabling device that exposes the controller as a trainable linear interconnection closed through a known static activation map, thereby making well-posedness and Lyapunov/IQC analysis mathematically easy to handle. For finite-dimensional LTI plants, we first develop a rigorous analysis theory for a given INC, including Perron–Frobenius and norm conditions for well posedness, LMI/IQC certificates for exponential stability, and LMIs for discounted infinite-horizon quadratic performance. We then formulate synthesis as a certification-compatible heuristic search: training is carried out under explicit well-posedness constraints, implicit-differentiation formulas provide gradients, and the resulting controller is accepted only after independent post-training LMIs or regional admissibility checks are feasible. Finally, we establish constrained-control separation results: fora specific scalar unstable plant with hard actuator bounds, an INC achieves a strictly smaller discounted infinite-horizon cost than any admissible finite-order dynamic linear controller. Additional results cover quadratic state-input costs, comparison with linear static output feedback, and computable upper/lowerbound certificates. Numerical examples illustrate the mechanism and the resulting certified performance.  \nIndex Terms—Neural network control, implicit models, stability, LTI systems, ReLU networks, constrained control, output feedback, linear matrix inequalities, integral quadratic constraints.  \nI. INTRODUCTION  \nUSING neural networks as controllers for dynamical sys  \ntems has a long history. However, so far, providing explicit stability and performance guarantees for neural control remains a challenge. A feedback law obtained by simulation or automatic differentiation is not easy to analyze mathematically; this gap is the main obstacle addressed in this paper. The difficulty is partly due to how neural networks are represented and parametrized. The standard layered notation for neural networks is ideal for implementation, but it does not easily lend itself to mathematical analysis. For feedback control synthesis, this is a serious limitation; therefore, a representation that simplifies the notation is the required mathematical entry point for certification-based neural controller synthesis.  \nA key message of the present paper is that implicit neural models, introduced in [1], provide such a mathematically convenient representation of neural networks. A control law that uses such a model will be termed an implicit neural controller (INC) . Rather than describing the neural map asa sequence of layers, an implicit model of a neural network  \ndefines a \"hidden feature vector\" as the solution of a fixedpoint equation. The model maps an input v to an output z via the following equations:  \nη = ϕ (Aη + Bv), (1a)  \nz = Cη + Dv, (1b)  \nwhere v is an external input, η is the \"hidden feature vector,\"and ϕ is a nonlinear \"activation\" vector-valued function such as a sigmoid, and matrices and vectors A, B, C, D contain the controller parameters. Note that the \"hidden\" vector η is only implicitly defined via a fixed-point equation. Perhaps surprisingly, the formalism contains standard feedforward deep networks as special cases, including convolutional, residual nets, LSTM, transformer and other architectures, see [1] . It also permits much more general network structures that have loops in the inference graph. The price of this additional modeling freedom is twofold. First, we need to guarantee that the solution to the fi","cbCaiq27fBINGupk","https://ap.wps.com/l/cbCaiq27fBINGupk","pdf",600806,4,1,13,"English","en",105,"# Introduction\n## Implicit neural controller concept\n## Key questions and contributions","[{\"question\":\"What is an implicit neural controller (INC) and how is it evaluated?\",\"answer\":\"An INC is a static feedback law expressed through an algebraic fixed-point equation. The hidden-feature vector is defined implicitly as the solution of that fixed-point problem, which then produces the controller output.\"},{\"question\":\"How does the paper make stability and performance analysis mathematically tractable?\",\"answer\":\"By representing neural networks implicitly as a linear interconnection closed through a known static activation map. This form enables well-posedness checks and Lyapunov/IQC analysis using Perron–Frobenius and norm conditions, together with LMI/IQC certificates.\"},{\"question\":\"What does the constrained-control separation result show?\",\"answer\":\"For a specific unstable scalar plant with hard actuator bounds, an INC achieves a strictly smaller discounted infinite-horizon cost than any admissible finite-order dynamic linear controller. The result is supported by computable certificates.\"}]",1784208930,33,{"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},"implicit-neural-networks-as-static-controllers-certificates-and-performance-separation","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/implicit-neural-networks-as-static-controllers-certificates-and-performance-separation/86151/",{"url":52,"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","2026-07-25","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 is an implicit neural controller (INC) and how is it evaluated?","Question",{"text":75,"@type":76},"An INC is a static feedback law expressed through an algebraic fixed-point equation. The hidden-feature vector is defined implicitly as the solution of that fixed-point problem, which then produces the controller output.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper make stability and performance analysis mathematically tractable?",{"text":80,"@type":76},"By representing neural networks implicitly as a linear interconnection closed through a known static activation map. This form enables well-posedness checks and Lyapunov/IQC analysis using Perron–Frobenius and norm conditions, together with LMI/IQC certificates.",{"name":82,"@type":73,"acceptedAnswer":83},"What does the constrained-control separation result show?",{"text":84,"@type":76},"For a specific unstable scalar plant with hard actuator bounds, an INC achieves a strictly smaller discounted infinite-horizon cost than any admissible finite-order dynamic linear controller. 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