{"id":"5e6b3ee4-262d-41b9-9e05-369e016e7eea","arxiv_id":"2411.13234","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper presents boundary-control designs and local stability theorems for extremum and Nash equilibrium seeking when player actions propagate through delays or PDEs, with convergence to small neighborhoods of the optimum or Nash equilibrium.","lead":"This paper extends model-free extremum seeking and Nash equilibrium seeking to systems where player actions pass through time delays or partial differential equations such as heat and transport equations. It derives local stability and convergence guarantees using boundary control, small-gain analysis, and averaging, and illustrates the methods on traffic, drilling, cable, and other applications.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central stability theorems hinge on an unverified small-gain verification; in the parabolic case the listed parameter mapping does not transparently match the actual boundary condition, so exponential stability of the average system is not established.","rationale":"The reader's weakest assumption is exactly the point on which the central claim rests: the small-gain verification for the averaged closed loop is asserted rather than proved. My review found a concrete reason to take that gap seriously: in Appendix A the parameter list for the parabolic loop assigns φ0=b1u(0) with b1<0, whereas the actual boundary condition is ∂x u(0)=0. That inconsistency means the claimed ISS gains of order O(ε) cannot be accepted without a fresh derivation. Appendix B and C repeat the same pattern, using phrases like 'can be readily verified' and 'not difficult to see' at the precise step where the small-gain hypotheses must be checked. The subsequent averaging step and the residual bounds O(|a|+1/ω) all depend on this exponential stability, so if the small-gain verification fails the central theorems fail with it. The authors' own simulation uses ε=1, outside the stated 0<ε<1 theoretical regime, and the numerical experiment is not reproducible from the paper alone, so it does not compensate for the missing proof. I therefore agree with the reader's conditional verdict: the paper is plausible and the framework is valuable, but the load-bearing proof step needs to be completed or corrected before the theorem claims can be taken as established.","tokens_in":54430,"tokens_out":9997,"duration_ms":115858,"concrete_test":"Take the single-player heat-PDE loop (278)–(281) with the actual boundary conditions ∂x u_i(0,t)=0 and u_i(1,t)=k_i ¯G_i + ε k_i φ_i(1), re-derive the ISS estimate required by [34, Thm. 8.2] directly from the variation-of-constants formula, and compute the gain product γ0γ1 appearing in the small-gain condition. If the product cannot be shown to be <1 for all sufficiently small ε, the exponential stability of the average system, and hence Theorems 1 and 3, is not established; if it can, the corrected parameter mapping must be stated explicitly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorems 1–3 all reduce to exponential stability of an averaged PDE–ODE loop, and the only proof offered is an appeal to the Karafyllis–Krstic small-gain theorems with parameter lists that are asserted rather than derived. Appendix A claims the loop (278)–(281) satisfies (A1)–(A7) with φ0(0,u_i,¯G_i)=b1u_i(0,t), b1<0, and with gains γ0 and b3 of order O(ε). But the actual boundary condition at x=0 is the homogeneous Neumann condition ∂x u_i(0,t)=0 (Eq. (280)), not a Robin-type condition of the form used in the listed φ0. Since the ISS gains and the small-gain condition (8.3.24) depend on the true boundary operator, this is not a harmless notational slip: the claimed O(ε) gain scaling has not been demonstrated for the actual plant. The same pattern appears in Appendices B and C, where the hyperbolic and mixed hyperbolic–parabolic loops are said to satisfy (H1)–(H2) or (A1)–(A7) with the verification left to phrases such as 'can be readily verified' and 'it is not difficult to see.' Because local exponential stability of the average system is the load-bearing premise for the subsequent averaging argument and for the residual bounds O(|a|+1/ω), the central claim is not yet supported by the written proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents boundary-control extremum seeking (ES) and Nash equilibrium seeking (NES) algorithms for quadratic noncooperative games in which players' actions are filtered through delays, heat PDEs, or a heterogeneous transport-heat PDE pair. The main theoretical results, Theorems 1-3, assert that with sufficiently large filter gains and frequencies and sufficiently small coupling ε, the closed-loop system has a locally exponentially stable periodic solution and converges to an O(|a|+1/ω) neighborhood of the Nash equilibrium (with additional exponentially growing amplitude factors in the input-side bounds). The paper also reviews applications to traffic, drilling, source seeking, additive manufacturing, bioreactors, and neuromuscular stimulation, and includes a duopoly simulation.","tokens_in":54770,"tokens_out":8582,"duration_ms":89771,"significance":"If fully proven, the results would be a significant extension of model-free ES/NES to infinite-dimensional actuator dynamics, providing constructive boundary controllers and explicit residual bounds. The paper is well organized, connects to a large body of prior work, and the simulation illustrates the proposed laws. The main limitations are in the proof apparatus: the central stability theorems are reduced to small-gain verifications that are asserted rather than demonstrated, and at least one parameter mapping appears inconsistent with the plant boundary condition. The survey and application sections are informative, but the technical contribution is not yet fully supported.","major_comments":[{"comment":"The verification of assumptions (A1)-(A7) of [34, Theorem 8.2] is not carried out. In particular, the plant's boundary at x=0 is the homogeneous Neumann condition ∂x u_i^av(0,t)=0 (Eq. (280)), while the parameter list assigns φ0(0,u_i^av,¯G_i^av)=b1 u_i^av(0,t) with b1<0. Since the small-gain condition (8.3.24) and the asserted O(ε) scaling of γ0 and b3 depend on the boundary operator at x=0, the current text does not establish the exponential stability of the averaged system (70)-(73). Please provide the correct parameter mapping for a Neumann boundary or show directly that the resulting ISS gains scale as claimed.","section":"Appendix A (Theorem 1), Eqs. (278)-(281)"},{"comment":"The proofs of Theorems 2 and 3 rely on the same pattern: the average closed-loop systems (286)-(288) and (292)-(296) are claimed to satisfy the hypotheses (H1)-(H2) or (A1)-(A7) of [34, Theorems 8.1 and 8.2], and the small-gain condition is asserted to hold for sufficiently small ε. No explicit verification of the gains (b3, γ0, γ1) or of the small-gain inequality is given; phrases such as 'can be readily verified' (after Eq. (297)) and 'it is not difficult to see' (Eqs. (66) and (134)) are not sufficient for a journal publication. Because local exponential stability of the average system is the load-bearing premise for the subsequent averaging argument and residual bounds, these steps must be supplied or replaced by precise pointers to the corresponding theorems in [60].","section":"Appendices B and C (Theorems 2 and 3)"},{"comment":"The residual bounds for the input-side estimates contain factors e^{max(D_i)√ω/2} and e^{D2√ω/2}, which grow with ω. The theorems state existence of sufficiently large ω while simultaneously using bounds that deteriorate as ω grows. Without an explicit scaling of the dither amplitudes a_i with ω (e.g., a_i = o(e^{-D_i√ω/2})), the assertion that θ(t) converges to a small neighborhood of θ* is not justified for the claimed parameter regime. Please state the admissible parameter scaling or reformulate the residual bound as a trade-off.","section":"Theorems 1 and 3, Eqs. (76) and (148)"}],"minor_comments":[{"comment":"The word 'dealts' in the opening sentence should be 'deals'.","section":"Section I"},{"comment":"The symbol D is overloaded: it denotes the delay matrix, the vector of domain lengths, the diffusion operator, and the spatial domain length. This causes confusion in equations such as (33) and (62); please use distinct notation or explicitly disambiguate.","section":"Section IV"},{"comment":"In the sentence defining the low-pass filter, 'steads for convolution' should be 'stands for convolution'.","section":"Section X"},{"comment":"The heading 'THE BASIC IDEA OF NASH EQULIBRIUM SEEKING' contains a typo; it should be 'EQUILIBRIUM'.","section":"Appendix H"},{"comment":"The simulation uses ε=1, which is outside the theorem's stated 0<ε<1. The text acknowledges that this suggests the analysis may be conservative, but the simulation should not be read as validation of the theorem's assumptions; this limitation should be stated more prominently.","section":"Section VII"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is submitted to a magazine special issue, so one might argue that proofs can be sketches with pointers to the authors' book [60]. However, Theorems 1-3 are presented as formal results with appendices, and the omissions are load-bearing. The heavy self-citation pattern is understandable given the topic, but the paper would be strengthened by stating exactly which results in [60] contain the missing verifications. The editor may wish to decide whether full proofs are required for a magazine paper or whether a theorem-number pointer to the book is acceptable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is worth a read if you want the state of the art on extremum and Nash equilibrium seeking when player actions are filtered through delays or PDEs. What is genuinely new here is the heterogeneous duopoly case—one player through a transport PDE, the other through a heat PDE—and the demonstration that a unified small-gain/averaging framework covers both. The authors also do a good job of collecting the design tools: predictor feedback for delays, boundary control for heat/transport equations, and the trajectory-generation dither signals that make the probing work. The applications section is a useful broad map of where these ideas can go.\n\nThe soft spot is exactly where the reader put it: the proofs of Theorems 1–3 are sketches, and the load-bearing step—verifying that the averaged PDE–ODE loops satisfy the small-gain assumptions with the claimed O(ε) gain scalings—is asserted, not shown. The stress-test note sharpens this to a specific mapping problem in Appendix A: the actual x=0 boundary condition is homogeneous Neumann, but the listed φ0 and coefficients correspond to a different boundary operator. That is not a harmless notational slip; the gain estimates used in the small-gain condition depend on the true boundary condition. Unless the missing verification can be supplied, the convergence claims are not fully supported by the written proof. The simulation's ε=1 is a separate but real credibility gap, since the theory explicitly requires 0<ε<1; the authors' suggestion that it shows conservatism is fine, but it doesn't fix the proof.\n\nThe paper is honest about its reliance on prior conference and journal papers, and self-citation is not a flaw when those are the actual sources of the results. There is no sign of parameter fitting or circularity. The missing verification is the main issue, and it is fixable in principle: either supply the full check or reframe the paper as a design/review contribution with the theorems deferred to the cited works.\n\nWho should read this: control engineers and researchers in ES/PDE control who want a single entry point to this line of work. For a magazine like IEEE Control Systems Magazine, it deserves serious peer review, but the referee should insist on the small-gain verification before the theoretical claims are accepted.","headline":"A useful consolidation of the authors' ES/NES results for PDE-constrained games, but the central stability claims rest on asserted small-gain verifications that the appendices do not actually carry out.","tokens_in":55233,"tokens_out":4998,"would_cite":true,"duration_ms":49886,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C20","91A10","93C23"],"pacs":[],"model":"deepseek-v4-flash","headline":"Boundary-control extremum seeking drives games whose players act through delays or heat PDEs to a small neighborhood of the Nash equilibrium.","keywords":["extremum seeking","Nash equilibrium seeking","partial differential equations","time delays","boundary control","small-gain analysis","averaging theory","noncooperative games"],"falsifier":"Compute the input-to-state gains $\\gamma_0$ and $b_3$ for the averaged two-player heat-PDE loop (70)-(73) from the variation-of-constants formula and the integral bounds in Appendix A, for example with $D_1=1$, $D_2=2$, and diagonal-dominant Hessian entries, and test the small-gain inequalities (283) along a sequence $\\epsilon\\to 0$. If the gain product fails to stay below one, or if a direct simulation of the averaged loop for those parameters shows no exponential decay, the asserted verification is wrong and Theorem 1's conclusion is unsupported.","tokens_in":54260,"feed_emoji":"🎯","tokens_out":12668,"duration_ms":125678,"temperature":0.7,"pith_summary":"Extremum seeking is a model-free way to optimize an unknown map in real time using only output measurements; Nash equilibrium seeking extends it to games where competing players each maximize their own payoff. This paper claims that the method keeps working when a player's action reaches the payoff function through infinite-dimensional dynamics: a time delay, a heat (diffusion) equation, or a mixed pair of one transport and one heat equation. For quadratic noncooperative games with weak coupling and a diagonally dominant Hessian, the proposed decentralized boundary-control laws make the closed-loop system locally exponentially converge to a small neighborhood of the Nash equilibrium, with propagated actions within $O(|a|+1/\\omega)$ of the optimum. The significance is that real-time, model-free optimization can be applied to physical and networked systems where actuation is blurred by transport or diffusion, such as traffic bottlenecks, thermal manufacturing, and source seeking.","feed_headline":"PDE-delayed players can still reach Nash equilibrium","feed_subtitle":"Boundary-control extremum seeking converges locally exponentially to within O(|a|+1/ω) of the equilibrium.","key_machinery":"The load-bearing object is the averaged infinite-dimensional closed loop: a finite-dimensional ODE for the estimated gradient $\\hat{G}$ coupled through its boundary condition to a transport or heat PDE that represents the compensated actuator error. A reduction-like transformation, the finite-spectrum assignment in equations (65) and (132)-(133), rewrites the loop so that the ODE sees the boundary control signal directly, and the boundary laws (58), (27), and (122) use only the player's own Hessian diagonal estimate plus an integral of the PDE state. Exponential stability of the average system is established by checking an input-to-state small-gain condition for the PDE-ODE interconnection, where the game coupling $\\epsilon$ appears in the interconnection gains, and an averaging theorem for infinite-dimensional systems then converts that stability into convergence of the original periodic closed loop.","core_discovery":"On the paper's own terms, the central discovery is a design-and-stability statement: in an $N$-player quadratic game where each player measures only its own payoff and knows only its own delay or diffusion domain, a boundary controller that compensates that player's actuation PDE lets all players collectively find the Nash equilibrium. Theorem 1 covers arbitrary distinct heat PDEs, Theorem 2 covers arbitrary distinct delays, and Theorem 3 covers a duopoly with one transport player and one heat player. In each case, the closed-loop error system has a unique locally exponentially stable periodic solution, and the propagated action vector satisfies $\\limsup_{t\\to\\infty}|\\Theta(t)-\\Theta^*|=O(|a|+1/\\omega)$, with the finer bounds $|\\theta_1(t)-\\theta_1^*|=O(a_1+1/\\omega)$ and $|\\theta_2(t)-\\theta_2^*|=O(a_2e^{D_2\\sqrt{\\omega}/2}+1/\\omega)$ in the heterogeneous duopoly. The paper also shows that scalar extremum seeking with PDE actuation is a corollary of the Nash results, and it catalogues a broad set of PDE types for which the same template, trajectory-generated probing signals plus boundary control, applies.","pith_inferences":["The numerical example uses $\\epsilon=1$, outside the theorem's stated $0<\\epsilon<1$ range, and still converges; I take this as evidence that the small-coupling assumption is conservative, though the paper does not promote this observation to a theorem.","The residual bound for a heat-PDE player contains $e^{D\\sqrt{\\omega}/2}$, so a long diffusion domain forces a practical trade-off: to keep the Nash neighborhood tight one must shrink the probing amplitude, which in turn weakens the gradient estimate. I expect this trade-off to be the practical bottleneck in thermal and diffusion applications.","The small-gain framework used for two PDE classes is generic enough that the same proof structure should cover games with more than two heterogeneous players, or mixtures involving wave and reaction-advection-diffusion dynamics, once the missing gain verifications are supplied.","Since the payoffs are quadratic, a standard local-quadratic approximation argument suggests the theorems extend to smooth strictly concave payoffs in a neighborhood of the equilibrium; checking this explicitly would be a natural follow-up."],"forward_implications":["Theorem 1: in the heat-PDE game, $\\limsup_{t\\to\\infty}|\\Theta(t)-\\Theta^*|=O(|a|+1/\\omega)$ while the boundary inputs satisfy $\\limsup_{t\\to\\infty}|\\theta(t)-\\theta^*|=O(|a|e^{\\max_i D_i\\sqrt{\\omega}/2}+1/\\omega)$, so the exponential probing amplification is confined to the input side.","Theorem 2: with arbitrary distinct input delays, the predictor feedback (27) gives $\\limsup_{t\\to\\infty}|\\theta(t)-\\theta^*|=O(|a|+1/\\omega)$, the same residual order as a delay-free game.","Theorem 3: in the transport-heat duopoly, the transport player converges to $O(a_1+1/\\omega)$ and the heat player to $O(a_2e^{D_2\\sqrt{\\omega}/2}+1/\\omega)$ of their Nash actions, despite each player compensating only its own PDE.","Scalar extremum seeking with PDE actuation follows as a corollary (Remark 1), so every single-agent result in the paper's PDE catalogue inherits the same stability template.","The boundary-control-plus-probing template is listed for reaction-advection-diffusion, wave, variable-delay, and distributed-delay actuation (Table I), indicating the design is not tied to heat or transport equations specifically."],"supporting_citations":[{"why":"Supplies the original Nash equilibrium seeking algorithm for noncooperative games, whose assumptions and algorithm structure this paper extends to delay and PDE actuation.","marker":"[21]"},{"why":"Provides extremum seeking for static maps with diffusion PDE actuation, including the heat-PDE probing signal and boundary-control structure adapted here.","marker":"[18]"},{"why":"Provides extremum seeking for static maps with delays, including the dither design and predictor-feedback structure used for transport-PDE players.","marker":"[61]"},{"why":"Establishes Nash equilibrium seeking with delayed player actions, the direct predecessor for the delay-compensation design in Theorem 2.","marker":"[64]"},{"why":"Supplies the small-gain theorems for ODE-PDE loops that form the stability engine for Theorems 1, 2, and 3.","marker":"[34]"},{"why":"Supplies the averaging theorems for infinite-dimensional and functional-differential systems that convert exponential stability of the average system into the periodic-solution estimates of Theorems 1-3.","marker":"[25]"},{"why":"Supplies the dither frequency nonresonance conditions and Hessian estimation signals used in the Nash seeking algorithms.","marker":"[23]"},{"why":"Provides the reduction (finite-spectrum assignment) transformation that makes the delayed or diffused control appear directly in the averaged ODE.","marker":"[6]"},{"why":"Supplies the backstepping and trajectory-generation tools used for boundary control design and for constructing the probing signals through PDE dynamics.","marker":"[44]"}],"fun_headline_variants":["Boundary control finds Nash equilibrium in PDE games","Extremum seeking tames delays and PDEs for Nash","Delayed and PDE agents converge to Nash equilibrium","Local exponential Nash seeking with actuation PDEs","Heterogeneous PDE duopolies reach Nash via ES"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole theorem rests on the claim, stated but not fully derived in the appendices, that the averaged PDE-ODE loop satisfies the small-gain condition needed for exponential stability for sufficiently small coupling $\\epsilon$; if that verification fails, the convergence result collapses.","fun_headline_variants_meta":{"raw":{"variants":["Boundary control finds Nash equilibrium in PDE games","Extremum seeking tames delays and PDEs for Nash","Delayed and PDE agents converge to Nash equilibrium","Local exponential Nash seeking with actuation PDEs","Heterogeneous PDE duopolies reach Nash via ES"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000205,"raw_usage":{"total_tokens":1417,"prompt_tokens":996,"completion_tokens":421,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":612,"completion_tokens_details":{"reasoning_tokens":345}},"tokens_in":612,"tokens_out":421,"duration_ms":4367,"temperature":1.0,"reasoning_tokens":345,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:40:30.349251+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the input-to-state gains $\\gamma_0$ and $b_3$ for the averaged two-player heat-PDE loop (70)-(73) from the variation-of-constants formula and the integral bounds in Appendix A, for example with $D_1=1$, $D_2=2$, and diagonal-dominant Hessian entries, and test the small-gain inequalities (283) along a sequence $\\epsilon\\to 0$. If the gain product fails to stay below one, or if a direct simulation of the averaged loop for those parameters shows no exponential decay, the asserted verification is wrong and Theorem 1's conclusion is unsupported.","supporting_citations":[{"cited_title":"Frihauf, M","cited_arxiv_id":null,"evidence_quote":"Supplies the original Nash equilibrium seeking algorithm for noncooperative games, whose assumptions and algorithm structure this paper extends to delay and PDE actuation."},{"cited_title":"Feiling, S","cited_arxiv_id":null,"evidence_quote":"Provides extremum seeking for static maps with diffusion PDE actuation, including the heat-PDE probing signal and boundary-control structure adapted here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides extremum seeking for static maps with delays, including the dither design and predictor-feedback structure used for transport-PDE players."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes Nash equilibrium seeking with delayed player actions, the direct predecessor for the delay-compensation design in Theorem 2."},{"cited_title":"Karafyllis and M","cited_arxiv_id":null,"evidence_quote":"Supplies the small-gain theorems for ODE-PDE loops that form the stability engine for Theorems 1, 2, and 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the averaging theorems for infinite-dimensional and functional-differential systems that convert exponential stability of the average system into the periodic-solution estimates of Theorems 1-3."},{"cited_title":"Ghaffari, M","cited_arxiv_id":null,"evidence_quote":"Supplies the dither frequency nonresonance conditions and Hessian estimation signals used in the Nash seeking algorithms."},{"cited_title":"Artstein","cited_arxiv_id":null,"evidence_quote":"Provides the reduction (finite-spectrum assignment) transformation that makes the delayed or diffused control appear directly in the averaged ODE."},{"cited_title":"Krsti ´c and A","cited_arxiv_id":null,"evidence_quote":"Supplies the backstepping and trajectory-generation tools used for boundary control design and for constructing the probing signals through PDE dynamics."}],"review_version":1}