{"id":"94e34bc2-7cdf-4bb1-a04d-12b687f46ad0","arxiv_id":"1908.03100","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"An explicit finite-dimensional sampled-data boundary feedback is shown to locally exponentially stabilize semilinear parabolic equations for arbitrarily large sampling periods.","lead":"This paper constructs an explicit boundary feedback controller that stabilizes unstable parabolic equations from state measurements taken only at discrete sampling times. It works for any sampling rate and reduces to a known continuous-time controller as the sampling period approaches zero.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Nonlinear contraction step in Theorem 3.1 fails for allowed p_i≤1: φ(η) in (3.8)/(3.15) is not small as η→0, so the proof is incomplete for part of (Hf).","rationale":"The reader's weakest assumption concerns unproved coercivity of the lifting operator and invertibility of B. Those are real gaps but likely true and can be cited from [8]; the elliptic lifting identity (2.19) and the discrete recurrences (2.28)-(2.30) check out, and the linear stabilization argument is coherent. The more serious weakness is the nonlinear contraction: the proof's smallness parameter φ(η) has the wrong behavior for exponents p_i≤1, which are explicitly allowed in (Hf). The theorem may still be true and likely repairable using the o(|y|) smallness that follows from f_y∈C(Ωbar), but the manuscript as written does not establish it. I therefore keep a conditional verdict: the authors should either restrict (Hf) to p_i>1 or replace φ(η) by a modulus-of-continuity argument, and supply the missing proofs of the coercivity/invertibility claims. No judgment is made about the authors; the concern is about the argument's internal validity.","tokens_in":11985,"tokens_out":22119,"duration_ms":230325,"concrete_test":"Apply the argument to an admissible case with p_1=1, e.g. f(y)=y+arctan y with y_e=0, so g(y)=arctan y−y satisfies (Hf) with p_1=1. Compute φ(η)=C5 (independent of η) in (3.8). Then the inequality C3(K)e^{φ(η)KT/2}φ(η)≤½e^{−μKT} required before (3.16) has a positive left-hand side independent of η for every K satisfying the preceding condition, so the instruction to take η small cannot work. Replacing φ(η) by sup_{|y|≤η}|g(y)|/|y|→0 would repair the proof, but that replacement is absent from the manuscript.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central theorem is not proved for the full range of hypothesis (Hf). In Step 2, after (3.8) the paper uses ∫‖z‖^{2p_i} ds ≤ η^{2(p_i−1)}∫‖z‖² ds on the set where ‖z(s)‖_{1/2−ε}≤η, defining φ(η)=C5Ση^{2(p_i−1)}. This is valid only when p_i≥1; for p_i<1 the inequality is reversed and the displayed bounds in (3.8)-(3.9) do not follow. In Step 3, contraction requires C3(K)e^{φ(η)KT/2}φ(η)≤½e^{−μKT}, and the text says to take η small. But if p_i=1, φ(η)=C5>0 is independent of η, so the left side cannot be made small; if p_i<1, φ(η)→∞ as η→0. Since (Hf) explicitly allows 0<p_i, the proof of Theorem 3.1 is incomplete for admissible nonlinearities. This is a gap in the main stabilization argument, not just in auxiliary lemmas.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper designs an explicit finite-dimensional boundary feedback controller of sampled-data form for semilinear parabolic equations with Dirichlet boundary conditions on part of the boundary. The controller is built from the unstable eigenfunctions of the linearized operator, uses an arbitrary sampling period T, and is shown to recover the continuous-time controller of [8,9] as T→0. The main results are a global exponential stabilization theorem for the linearized equation (Theorem 2.2) and a local exponential stabilization theorem for the semilinear equation around an equilibrium (Theorem 3.1) under a polynomial growth condition (Hf). The proof proceeds by lifting the sampled boundary data through elliptic problems Dγk, reducing the controlled linear equation to the finite-dimensional recurrence (2.30), and then treating the nonlinearity by an a priori bound and a discrete-time contraction argument.","tokens_in":12304,"tokens_out":8273,"duration_ms":84548,"significance":"If the technical gaps are repaired, the paper makes a useful contribution to constructive PDE control: it gives an explicit finite-dimensional feedback that works for any sampling period rather than only for small T, and it displays the convergence of the sampled-data law to a known continuous-time controller. The linear part of the paper is largely sound: the algebraic derivation of the discrete-time recurrence (2.30) and the spectral radius estimate (2.31) are consistent, and the high-frequency estimate in Theorem 2.2 follows along standard dissipativity lines. The paper does not provide machine-checked proofs or reproducible code, and its main nonlinear closing argument needs essential repair; nevertheless, the explicit construction and the continuous-time limit are valuable features that merit publication after the proof issues are resolved.","major_comments":[{"comment":"The proof of Theorem 3.1 does not cover all nonlinearities allowed by (Hf). In (3.8) the passage from the sum of integrals of ||z||^{2p_i} to φ(η)∫||z||^2 uses the inequality ∫||z||^{2p_i} ≤ η^{2(p_i−1)}∫||z||^2 on the set {||z||≤η}; this inequality is valid only for p_i≥1. For p_i=1, φ(η)=C5 is independent of η, and for 0<p_i<1, φ(η)→∞ as η→0. Therefore the Step 3 requirement that C3(K)e^{φ(η)KT/2}φ(η)≤(1/2)e^{−μKT} can be achieved by taking η small is not available within the stated range of (Hf). Since (Hf) explicitly allows 0<p_i<1 and p_i=1, Theorem 3.1 is not proved for all admissible nonlinearities; the argument must be repaired or the hypothesis restricts p_i.","section":"Sec. 3, Step 2 and Step 3 (Eqs. (3.8), (3.15))"},{"comment":"The feedback F in (2.12) is well defined only if the elliptic lifting problem (2.9) has a unique solution for every boundary datum v∈L2(Γ1) and if the matrix B1+⋯+BN is invertible. Both facts are asserted without proof: the coercivity estimate is introduced with 'It is not difficult to show...' before (2.9), and the invertibility is deferred with 'Following the method in [8]...' after (2.11). These are load-bearing conditions, since the matrices Λγk, B, Bk and the maps Dγk, and hence the feedback itself, do not exist if either assertion fails. The manuscript should either supply the missing proofs or give a precise theorem statement from [8] that covers these facts in the present setting.","section":"Sec. 2.2, before Eq. (2.9) and after Eq. (2.11)"}],"minor_comments":[{"comment":"The identities involving the lifted functions are dimensionally inconsistent as printed: F_j maps into L2(Γ1), so writing y=z+∑χ∑F_j(z(iT)) in (3.5) and z(0)=y0−∑F_j(y(0)) in (3.6) cannot be correct in L2(Ω); these expressions should involve h_j or Dγ_jF_j.","section":"Sec. 3, Eqs. (3.5)–(3.6)"},{"comment":"The first displayed estimate in (3.8) has a term C1||z(0)||_{1/2−ε} on the right-hand side, while the final displayed estimate uses L0||z(0)||^2_{1/2−ε}; the powers should be made consistent.","section":"Sec. 3, Eq. (3.8)"},{"comment":"The statement says 'Assume that y0∈L2(0,1)', but the domain is a general bounded open set Ω; this should read y0∈L2(Ω).","section":"Theorem 2.2 statement"},{"comment":"There are minor language issues, for example 'not necessary to be small enough' should be 'not necessarily small enough', and 'Defintion' in Definition 2.1 is a typo.","section":"Abstract and throughout"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the main concern is substantive and not merely presentational. The nonlinear closing argument in Theorem 3.1 fails for an explicitly allowed range of p_i, and the two unproved well-posedness/invertibility assertions in Section 2.2 are essential to the construction. I do not see a novelty disclosure problem, since the dependence on [8,9] is acknowledged; the issue is that the present proof must either be completed or the hypotheses narrowed. If the author supplies the missing arguments and repairs the p_i≤1 gap, the paper would be publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something real: it takes Munteanu's finite-dimensional boundary feedback and shows the same construction works as a zero-order-hold sampled-data controller for arbitrary sampling period T, with the feedback tending to the continuous-time controller as T goes to 0. The linear part of the paper is solid. The discrete-time recurrence (2.30), the semipositivity argument, and the decay estimate (2.32) all check out. This is a genuine step beyond the 1-D or small-sampling or globally Lipschitz results in [14,15,16], and the explicit form of the feedback is valuable.\n\nThe nonlinear theorem is where the trouble is. The stress-test note is correct: the step leading to (3.8) uses ∫‖z‖^{2p_i} ds ≤ η^{2(p_i−1)}∫‖z‖² ds on the set where ‖z‖≤η, and that inequality only holds for p_i≥1. Hypothesis (Hf) explicitly allows 0<p_i. For p_i=1, φ(η) is a positive constant independent of η, so the subsequent \"take η small\" does not make the contraction factor small. For p_i<1, φ(η) blows up as η→0. So the proof of Theorem 3.1 is incomplete for admissible nonlinearities. This is not a cosmetic gap; it is in the central stabilization argument. It may well be patchable—for C^1 nonlinearities one can use the fact that the remainder g(y) is o(y) as y→0 rather than a polynomial bound—but the paper does not do that.\n\nTwo smaller issues. First, the feedback F is only defined if the elliptic lifting problem (2.9) is well-posed and if the matrix B1+...+BN is invertible. The paper says these follow by \"not difficult\" estimates and \"following the method in [8]\", but gives no proof and no precise citation. This is likely fixable, but as written it is an unproved load-bearing assertion. Second, the abstract and introduction promise numerical examples in Section 4; the paper ends at the references. That is a missing advertised section, not a fatal flaw, but it should be supplied or the promise removed.\n\nWho is this for: people working on sampled-data boundary control of PDEs, especially those using Munteanu's lifting method. It deserves a serious referee—the linear result and the overall strategy are worth engaging—but I would expect major revision before acceptance. I would not cite it in its current form.","headline":"A genuine sampled-data extension of Munteanu's boundary feedback with arbitrary sampling period, but the nonlinear stabilization proof has a real gap for part of the stated hypothesis and the promised numerics are missing.","tokens_in":12772,"tokens_out":4446,"would_cite":false,"duration_ms":53578,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93C20","93D15","93C57","35K58"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sampled-data boundary feedback stabilizes parabolic equations at any sampling rate.","keywords":["parabolic equations","sampled-data control","boundary feedback stabilization","semilinear parabolic equations","finite-dimensional feedback","eigenfunction expansion","exponential stability","Dirichlet boundary control"],"falsifier":"Take an explicit multidimensional example, such as the unit square with a known unstable equilibrium, compute the normal-derivative matrix $B_0$ and the weighted matrices $B_k$, and check numerically whether $B_1+\\cdots+B_N$ is invertible and whether the spectral inequality $\\lambda_{\\max}(\\sum_k e^{-\\gamma_k T}B^{1/2}B_kB^{1/2})\\le e^{-\\gamma_1 T}$ holds. If either fails, the contraction step collapses. Alternatively, simulate the closed-loop nonlinear system with a large sampling period $T$ and a small initial deviation; the theorem predicts $\\|y(t)-y_e\\|_{1/2-\\varepsilon}\\le Ce^{-\\mu t}\\|y_0-y_e\\|_{1/2-\\varepsilon}$, so observing growth between sampling instants for arbitrarily small initial data would refute it.","tokens_in":11788,"feed_emoji":"🎛️","tokens_out":13346,"duration_ms":124679,"temperature":0.7,"pith_summary":"This paper designs an explicit finite-dimensional boundary feedback controller of sampled-data form to stabilize equilibrium solutions of semilinear parabolic equations. The controller is built from the eigenfunctions corresponding to the unstable eigenvalues of the linearized operator and works for any sampling period $T$, not necessarily small. The main result says that every initial datum sufficiently close to the equilibrium produces a solution that decays exponentially in $H^{1/2-\\varepsilon}$, at any prescribed rate $\\mu<\\rho$. The same feedback globally stabilizes the linearized equation at rate $\\rho$, and it tends to the continuous-time controller of [8,9] as $T\\to0$.","feed_headline":"Boundary feedback stabilizes parabolic equations at any sampling rate","feed_subtitle":"An explicit finite-dimensional controller works for any sampling period, not just tiny ones.","key_machinery":"The load-bearing object is the boundary feedback operator $F(w)=\\mathbf{1}_{\\Gamma_1}\\langle \\Lambda B Q_N(w), \\partial\\Phi_N/\\partial n\\rangle_N$, assembled from a weighted sum $\\Lambda=\\sum_{k=1}^N\\Lambda_{\\gamma_k}$, the invertible matrix $B=(B_1+\\cdots+B_N)^{-1}$ with $B_k=\\Lambda_{\\gamma_k}B_0\\Lambda_{\\gamma_k}$, $B_0=(\\langle\\partial\\varphi_i/\\partial n,\\partial\\varphi_j/\\partial n\\rangle_0)$, and the elliptic lifting maps $D_{\\gamma_k}:L^2(\\Gamma_1)\\to H^{1/2}(\\Omega)$ defined by (2.9). The argument turns on the discrete identity $z_N((i+1)T)=\\sum_{k=1}^N e^{-\\gamma_k T}B_kB z_N(iT)$ for the controlled low modes, which yields the contraction $|B^{1/2}z_N((i+1)T)|\\le e^{-\\gamma_1 T}|B^{1/2}z_N(iT)|$; the high-frequency modes decay at rate $\\rho$, and the nonlinear terms are controlled by a Gronwall argument over blocks of length $KT$.","core_discovery":"The central claim is Theorem 3.1: for any $0<\\mu<\\rho$, under assumption $(H_f)$, there exist constants $C>0$ and $\\delta>0$ such that for every $y_0$ with $\\|y_0-y_e\\|_{1/2-\\varepsilon}\\le\\delta$, the sampled-data feedback $u(t)=\\sum_i\\chi_{[iT,(i+1)T)}(t)F(y(iT)-y_e)+y_e$ locally exponentially stabilizes the semilinear parabolic equation, with $\\|y(t)-y_e\\|_{1/2-\\varepsilon}\\le C e^{-\\mu t}\\|y_0-y_e\\|_{1/2-\\varepsilon}$. The proof first shows that the same feedback globally stabilizes the linearized equation (Theorem 2.2) at rate $\\rho$, then treats the nonlinear remainder under the polynomial growth condition $(H_f)$ as a perturbation that can be absorbed by choosing the initial data small and the block length large. The controller is finite-dimensional, using only the $N$ eigenfunctions below the threshold $\\rho$, and the controlled low modes contract by a factor $e^{-\\gamma_1 T}$ at each sampling step, so no smallness of $T$ is required.","pith_inferences":["Not developed in the paper: the proof displays a quantitative trade-off between the sampling period and the contraction factor $e^{-\\gamma_1 T}$, so one could tune $\\gamma_1$ to keep a desired decay while increasing $T$; the paper does not optimize this choice.","A natural extension would be an output-feedback sampled-data version that estimates the $N$ low modes from boundary observations, following the same low-mode philosophy; the paper stops at state feedback.","The proof's feedback depends on $T$ through the matrices $\\Lambda_{\\gamma_k}$, but the numerical size of the stability radius $\\delta$ as a function of $T$ is left qualitative; a reader tuning the controller would need to compute it for a concrete case."],"forward_implications":["The same sampled-data feedback globally stabilizes the linearized parabolic equation at exponential rate $\\rho$, for any sampling period $T$.","For the semilinear equation under $(H_f)$, local exponential stability holds at any prescribed rate $\\mu<\\rho$, with the admissible size $\\delta$ depending on $T$ and $\\mu$.","The controller is explicit and finite-dimensional: it requires only the $N$ eigenfunctions of the linearized operator with eigenvalues below $\\rho$, together with their normal derivatives on the controlled boundary.","As $T\\to0$, the sampled-data law recovers the continuous-time boundary feedback of [8,9], so the construction is a genuine sampled-data extension rather than a separate design.","Because the gains $\\Lambda_{\\gamma_k}$ are recomputed for each $T$, the same formula applies to large sampling periods instead of requiring $T$ to be small."],"supporting_citations":[{"why":"Provides the continuous-time boundary feedback construction and the lifting method that the sampled-data design reduces to as T→0.","marker":"[8]"},{"why":"Gives the continuous-time semilinear stabilization result that this paper extends to sampled-data controls.","marker":"[9]"},{"why":"Yields the elliptic regularity estimate for the lifting maps, $\\|\\psi_k\\|_{1/2}\\le C|v|_0$, used throughout.","marker":"[18]"},{"why":"Supplies the matrix spectral radius inequality used to obtain the per-sampling-step contraction factor $e^{-\\gamma_1 T}$.","marker":"[19]"}],"fun_headline_variants":["Any sampling rate works for boundary feedback in parabolic PDEs","Finite-dimensional boundary control stabilizes parabolic equations","Sampled-data boundary feedback: exponential stability for all sample times","Parabolic stabilization without small sampling periods","Boundary feedback that works for large sampling gaps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction requires the elliptic lifting maps $D_{\\gamma_k}$ to be well posed with a coercivity estimate and the matrix $B_1+\\cdots+B_N$ to be invertible; the paper states both facts without proofs, and if either fails the feedback operator is not defined.","fun_headline_variants_meta":{"raw":{"variants":["Any sampling rate works for boundary feedback in parabolic PDEs","Finite-dimensional boundary control stabilizes parabolic equations","Sampled-data boundary feedback: exponential stability for all sample times","Parabolic stabilization without small sampling periods","Boundary feedback that works for large sampling gaps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000679,"raw_usage":{"total_tokens":3035,"prompt_tokens":847,"completion_tokens":2188,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":2114}},"tokens_in":463,"tokens_out":2188,"duration_ms":16133,"temperature":1.0,"reasoning_tokens":2114,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:25:22.798198+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an explicit multidimensional example, such as the unit square with a known unstable equilibrium, compute the normal-derivative matrix $B_0$ and the weighted matrices $B_k$, and check numerically whether $B_1+\\cdots+B_N$ is invertible and whether the spectral inequality $\\lambda_{\\max}(\\sum_k e^{-\\gamma_k T}B^{1/2}B_kB^{1/2})\\le e^{-\\gamma_1 T}$ holds. If either fails, the contraction step collapses. Alternatively, simulate the closed-loop nonlinear system with a large sampling period $T$ and a small initial deviation; the theorem predicts $\\|y(t)-y_e\\|_{1/2-\\varepsilon}\\le Ce^{-\\mu t}\\|y_0-y_e\\|_{1/2-\\varepsilon}$, so observing growth between sampling instants for arbitrarily small initial data would refute it.","supporting_citations":[{"cited_title":"Boundary stabilization of parabolic nonlinear equations","cited_arxiv_id":"1501.05737","evidence_quote":"Provides the continuous-time boundary feedback construction and the lifting method that the sampled-data design reduces to as T→0."},{"cited_title":"Liu, Hanbing, P","cited_arxiv_id":null,"evidence_quote":"Gives the continuous-time semilinear stabilization result that this paper extends to sampled-data controls."},{"cited_title":"Lasiecka , R","cited_arxiv_id":null,"evidence_quote":"Yields the elliptic regularity estimate for the lifting maps, $\\|\\psi_k\\|_{1/2}\\le C|v|_0$, used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the matrix spectral radius inequality used to obtain the per-sampling-step contraction factor $e^{-\\gamma_1 T}$."}],"review_version":1}