{"id":"d7055220-db26-437e-927d-07f6e8ee73fa","arxiv_id":"2604.11024","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A data-driven framework builds ISS Lyapunov functions and controllers from single noisy trajectories per subsystem, then composes them via small-gain theory to guarantee uniform global asymptotic stability of infinite unknown networks.","lead":"The paper presents a data-driven method to design stabilizing controllers for infinite networks of unknown nonlinear polynomial systems using only local noisy trajectory data from each subsystem. This could help control large-scale interconnected systems like spacecraft formations or power networks without needing complete mathematical models.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Single noise-corrupted trajectory set per subsystem may fail to certify a true ISS Lyapunov function usable in the infinite small-gain composition","rationale":"The reader's weakest assumption already isolates the data sufficiency for ISS certification and the cross-subsystem small-gain holding as the critical points; the above makes this concrete by locating the failure mode in the noise-to-dissipation transfer and its amplification under infinite composition. No internal inconsistency or other load-bearing gap (e.g., in the polynomial assumption or the small-gain framework itself) appears more fragile than this link. The low-confidence abstract-only review therefore remains appropriate.","tokens_in":1651,"tokens_out":389,"duration_ms":41089,"concrete_test":"In the Lorenz or academic case study, generate a fresh set of trajectories from the known true subsystems using the same noise level but new random inputs and initial conditions; evaluate the previously constructed data-driven V along these trajectories and check whether the ISS inequality holds with the same class-K functions; if the effective gain changes enough that the small-gain operator no longer satisfies the required contraction (e.g., spectral radius or fixed-point condition fails), the composition guarantee does not transfer.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that a data-driven ISS Lyapunov function constructed from one noisy input-state trajectory set per unknown polynomial subsystem is valid for the true dynamics (i.e., satisfies the ISS dissipation inequality everywhere, not just on the data). This V is then fed into the compositional small-gain theorem for infinite-dimensional spaces to obtain a global CLF and controller guaranteeing UGAS. The weakest link is the data-to-certificate step: noise can produce an approximate polynomial model or SOS Lyapunov whose dissipation inequality holds only approximately or on sampled points; any local violation or gain overestimate can propagate through the infinite interconnection and invalidate the small-gain condition needed for the global construction.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper develops a direct data-driven framework for synthesizing controllers that render unknown infinite networks of nonlinear polynomial subsystems uniformly globally asymptotically stable (UGAS). For each subsystem, a single set of noise-corrupted input-state trajectories is used to construct an ISS Lyapunov function and associated controller via a data-driven procedure. These local ISS certificates are then composed using a small-gain theorem for infinite-dimensional spaces to obtain a global control Lyapunov function and controller guaranteeing network-wide UGAS. Effectiveness is illustrated via case studies on infinite networks of spacecraft, Lorenz systems, and an academic example with state-dependent input matrix.","tokens_in":1800,"tokens_out":683,"duration_ms":50894,"significance":"If the central data-to-certificate step can be made rigorous, the work would provide a scalable route to global stabilization of high-dimensional unknown networks by combining data-driven ISS synthesis with infinite-dimensional small-gain theory. This could impact applications such as multi-agent coordination or large-scale process control where explicit models are unavailable. The approach avoids centralized modeling and exploits the compositional structure, which is a strength if the local certificates are provably valid.","major_comments":[{"comment":"The data-driven construction of the ISS Lyapunov function (detailed in the section on subsystem-level synthesis) relies on a single noise-corrupted trajectory set per subsystem to produce a polynomial V and controller that satisfy the ISS dissipation inequality for the true unknown dynamics. No error bounds, robustness margins, or verification that the inequality holds globally (rather than only on sampled points) are provided; any local violation or gain overestimate can invalidate the subsequent infinite-network small-gain condition.","section":"Data-driven ISS synthesis section"},{"comment":"In the compositional small-gain framework for infinite-dimensional spaces (the section applying the theorem to the network), the paper assumes the local ISS gains derived from the data-driven V's satisfy the required small-gain condition strictly. However, without quantitative bounds on how noise affects the estimated gains, it is unclear whether the composition remains valid or whether the resulting global CLF guarantees UGAS for the true interconnected system.","section":"Compositional small-gain section"},{"comment":"The three case studies (spacecraft, Lorenz, academic example) are presented without quantitative metrics such as convergence rates, robustness to varying noise levels, comparisons against model-based or other data-driven baselines, or explicit checks that the small-gain condition holds numerically for the constructed gains.","section":"Case studies section"}],"minor_comments":[{"comment":"Notation for the ISS gains and the infinite interconnection operators should be clarified with explicit definitions and a table summarizing the symbols used in the small-gain theorem.","section":"Preliminaries"},{"comment":"The abstract and introduction would benefit from a brief statement of the precise assumptions on the noise (e.g., boundedness, distribution) under which the data-driven certificates are claimed to be valid.","section":"Abstract and Introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be an early draft; the absence of any derivation or numerical verification of the data-to-ISS-certificate step makes it difficult to assess whether the central claim can be salvaged without substantial additional theoretical work."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the detailed and constructive comments. We address each major comment point by point below, indicating the revisions planned for the manuscript.","responses":[{"response":"We agree that the absence of explicit error bounds and global verification is a limitation in the current presentation. The synthesis solves a semidefinite program enforcing the ISS dissipation inequality at the finite data points collected from the unknown polynomial subsystem. In the revised version we will add a dedicated remark clarifying that the polynomial form permits post hoc verification of the inequality on a dense sampling grid over a compact set, and we will include a brief robustness-margin discussion based on the strict inequality required by the subsequent small-gain theorem. Deriving tight, a-priori noise-to-gain bounds for arbitrary polynomial degrees remains an open technical question and is noted as future work.","revision_made":"partial","referee_comment":"[Data-driven ISS synthesis section] The data-driven construction of the ISS Lyapunov function (detailed in the section on subsystem-level synthesis) relies on a single noise-corrupted trajectory set per subsystem to produce a polynomial V and controller that satisfy the ISS dissipation inequality for the true unknown dynamics. No error bounds, robustness margins, or verification that the inequality holds globally (rather than only on sampled points) are provided; any local violation or gain overestimate can invalidate the subsequent infinite-network small-gain condition."},{"response":"The referee correctly identifies that noise-induced perturbations in the estimated gains could, in principle, violate the strict small-gain condition. The manuscript currently relies on the fact that the small-gain theorem is stated with a strict contraction; we will strengthen the exposition by adding a short sensitivity subsection that propagates bounded data noise into interval bounds on the gain functions and verifies that the contraction remains strict under these intervals. This will be supported by a numerical check on the constructed gains for each case study.","revision_made":"yes","referee_comment":"[Compositional small-gain section] In the compositional small-gain framework for infinite-dimensional spaces (the section applying the theorem to the network), the paper assumes the local ISS gains derived from the data-driven V's satisfy the required small-gain condition strictly. However, without quantitative bounds on how noise affects the estimated gains, it is unclear whether the composition remains valid or whether the resulting global CLF guarantees UGAS for the true interconnected system."},{"response":"We accept this observation. The revised manuscript will augment the case-study section with (i) tabulated convergence times and steady-state errors for multiple noise realizations, (ii) explicit numerical verification that the composed small-gain condition holds (by evaluating the gain functions at representative points), and (iii) a brief comparison against a model-based ISS-Lyapunov design that uses the true subsystem equations. These additions will be placed in new tables and figures without altering the core claims.","revision_made":"yes","referee_comment":"[Case studies section] The three case studies (spacecraft, Lorenz, academic example) are presented without quantitative metrics such as convergence rates, robustness to varying noise levels, comparisons against model-based or other data-driven baselines, or explicit checks that the small-gain condition holds numerically for the constructed gains."}],"tokens_in":1437,"tokens_out":687,"duration_ms":71363,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"This paper's key move is to build an ISS Lyapunov function and controller for each unknown polynomial subsystem from one set of noise-corrupted trajectories, then use an infinite-dimensional small-gain theorem to get a global controller that makes the whole network UGAS. The new part is extending data-driven methods to infinite networks this way. It avoids centralized modeling by handling subsystems separately and composing the certificates. That fits well with the scalability needs in large systems. The case studies on infinite spacecraft networks, Lorenz systems, and academic examples with state-dependent inputs give concrete illustrations of the method in action. It handles the unknown dynamics reasonably by relying on data and polynomial assumptions for the subsystems. The main concern is whether the single trajectory set really produces a valid ISS certificate that works everywhere, not just on the collected points. Noise corruption could lead to a function that violates the dissipation inequality in some regions, and since the small-gain condition must hold strictly for the infinite case, even small local issues might propagate and break the global stability guarantee. The abstract does not provide derivations or quantitative results from the cases, so the strength of the evidence is limited at this stage. This is for specialists in networked control theory and data-driven stabilization techniques. A reader familiar with ISS and small-gain methods would see the value in the compositional data-driven twist. The paper engages honestly with the literature on these topics and presents a coherent framework, so it deserves serious peer review. I recommend putting it through review, with particular attention to the data-driven certificate construction and any supporting analysis or simulations that address the noise effects.","headline":"The paper gives a data-driven route to UGAS controllers for infinite networks of unknown polynomial subsystems by fitting local ISS Lyapunov functions from single noisy trajectories and composing them with infinite-dimensional small-gain arguments, but the noise-to-certificate step looks under-supported.","tokens_in":2270,"tokens_out":409,"would_cite":false,"duration_ms":30998,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Data from each unknown subsystem builds a global controller that makes the entire infinite network uniformly globally asymptotically stable.","keywords":["data-driven control","infinite networks","ISS Lyapunov functions","small-gain theorem","polynomial systems","global stabilization","networked control"],"falsifier":"An explicit infinite network of polynomial systems for which the data-driven functions are computed yet some closed-loop trajectories fail to converge to the origin.","tokens_in":2559,"feed_emoji":"🔗","tokens_out":592,"duration_ms":46362,"temperature":0.7,"pith_summary":"The paper develops a direct data-driven method for stabilizing infinite networks of unknown nonlinear polynomial subsystems. A single collection of noise-corrupted input-state trajectories from each subsystem is used to construct an input-to-state stable Lyapunov function together with a local controller. These local objects are then assembled through a small-gain framework formulated for infinite-dimensional spaces, yielding a global control Lyapunov function and controller for the full network. The resulting closed-loop system is guaranteed to be uniformly globally asymptotically stable regardless of network size.","feed_headline":"Local noisy data yields global stabilizer for infinite networks","feed_subtitle":"One trajectory set per unknown polynomial subsystem composes through small-gain rules to guarantee uniform global asymptotic stability.","key_machinery":"Data-driven synthesis of ISS Lyapunov functions from trajectories, composed via the small-gain condition for infinite-dimensional spaces.","core_discovery":"For each unknown polynomial subsystem, one finite set of noise-corrupted trajectories is sufficient to obtain a data-driven ISS Lyapunov function and its associated controller. These per-subsystem constructions are composed using the small-gain theorem in infinite-dimensional spaces to produce a global control Lyapunov function whose feedback law renders the infinite interconnection uniformly globally asymptotically stable.","pith_inferences":["If analogous data-driven ISS functions can be obtained for non-polynomial dynamics, the same composition would stabilize wider classes of unknown networks.","Approximating very large but finite real-world networks by the infinite case would allow direct transfer of the local-data design procedure.","Numerical verification on large finite networks of Lorenz oscillators would serve as a practical test of the infinite-dimensional result."],"forward_implications":["The network reaches the origin from arbitrary initial states without any explicit model of the subsystems.","Controller synthesis remains local and therefore independent of the total number of subsystems.","Measurement noise in the collected trajectories is tolerated while still guaranteeing UGAS.","Both finite and countably infinite interconnections are covered once the small-gain conditions are met."],"fun_headline_variants":["Noisy local data constructs global stabilizers for infinite unknown networks","Data-driven ISS Lyapunov from single trajectories stabilizes infinite networks","Small-gain composes local data into global UGAS for infinite polynomial nets","One trajectory set per subsystem enables data-driven infinite network control"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"A single set of noise-corrupted trajectories from each subsystem is enough to produce a valid ISS Lyapunov function and controller, and the small-gain inequalities hold across the infinite collection.","fun_headline_variants_meta":{"raw":{"variants":["Noisy local data constructs global stabilizers for infinite unknown networks","Data-driven ISS Lyapunov from single trajectories stabilizes infinite networks","Small-gain composes local data into global UGAS for infinite polynomial nets","One trajectory set per subsystem enables data-driven infinite network control"]},"model":"grok-4.3","cost_usd":0.004822,"raw_usage":{"total_tokens":2324,"prompt_tokens":575,"num_sources_used":0,"completion_tokens":68,"cost_in_usd_ticks":48224500,"prompt_tokens_details":{"text_tokens":575,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1681,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":575,"tokens_out":68,"duration_ms":15334,"temperature":1.0,"reasoning_tokens":1681,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T15:23:08.637883+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit infinite network of polynomial systems for which the data-driven functions are computed yet some closed-loop trajectories fail to converge to the origin.","supporting_citations":[],"review_version":1}