{"id":"b1ceb7cb-f1e6-4237-9b10-4272284766b5","arxiv_id":"2411.18501","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors attempt to prove existence of insensitizing controls for a stochastic heat equation with dynamic boundary conditions, but a key Carleman-to-observability step in the proof is invalid.","lead":"This paper claims to prove that controls can make a measured energy insensitive to small perturbations of the initial state for a stochastic heat equation with dynamic boundary conditions, under a geometric overlap condition and zero initial data. The proof's final observability step contains a false weight comparison, so the central theorem is not established as written.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Observability inequality (4.17) is not proved: comparison (4.23) is false near t=0 because the Carleman weight θ²γ³ vanishes there while exp(-Mt-1) has a positive limit.","rationale":"Agree with the reader. The failure at (4.23) is real and load-bearing. The Carleman estimate (4.5) has weights vanishing at both t=0 and t=T; the requested observability inequality (4.17) uses exp(-Mt-1), which is bounded below near t=0. The proof tries to dominate one by the other, which is pointwise impossible. I checked the energy-estimate portions: they only control p on (T/2,T) by q on an interior interval, and that part is unproblematic, but the (0,T/2) segment has no substitute. The duality argument in Section 5 therefore lacks its key input. No other major gap is needed for the rejection. The paper's reduction in Proposition 3.1 and the Carleman estimate in Theorem 4.1 seem potentially correct, but the central theorem is not established. The abstract also mentions a spectral-inequality approach that does not appear in the body, a secondary consistency concern. Since the reader already reached REJECT, my recommendation is unchanged.","tokens_in":19590,"tokens_out":10831,"duration_ms":98413,"concrete_test":"Analytically evaluate both sides of (4.23) as t↓0+ for fixed x∈G. The left side tends to e^{-1}; the right side is O(t^{-3} exp(-2λ*C*/(Tt))) → 0, where C* = e^{2μ*|ψ|∞}-e^{μ*|ψ|∞} > 0. This computation shows (4.23) cannot hold. Additionally, check whether replacing the target weight by exp(-M/(t(T-t))) in (4.17) would make the comparison valid; if so, the condition (1.4) must be changed accordingly, which would alter the main theorem as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.1 claims the pointwise bound (4.23): exp(-Mt-1) ≤ C θ²γ³ on (0,T/2)×G, with θ=e^{λ* α}, γ=1/(t(T-t)), and α=(e^{μψ}-e^{2μ|ψ|∞})/(t(T-t)). Since ψ>0 and ψ≤|ψ|∞, the numerator satisfies e^{μψ}-e^{2μ|ψ|∞} ≤ e^{μ*|ψ|∞}-e^{2μ*|ψ|∞} < 0, so α→-∞ as t↓0. Hence θ²γ³ behaves like e^{-C/t}/t³ → 0 pointwise, while exp(-Mt-1)→e^{-1}. The claimed inequality is therefore false on any neighborhood of t=0, for every M>0. This is not a minor typo: (4.23) is the only bridge from the Carleman estimate (4.5) to the (0,T/2) part of (4.17). The energy estimates (4.20)-(4.22) only cover the interior interval (T/4,3T/4) and (T/2,T); they do not fix the boundary layer at t≈0 because q near 0 is also uncontrolled by the vanishing Carleman weight. Since Section 5 uses (4.17) to bound the functional L and invoke Hahn-Banach/Riesz, the construction of (u,v1,v2) is unsupported. The reduction in Prop. 3.1 and the Carleman estimate in Thm. 4.1 are plausible, but the proof as written has a load-bearing gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies insensitizing controls for a forward linear stochastic heat equation with dynamic boundary conditions on a bounded domain. The main result, Theorem 1.1, claims that if G0∩O is nonempty and the initial data are zero, then for sources satisfying the weighted integrability condition (1.4) there exists a control triple (u,v1,v2) that insensitizes the energy functional Φ defined in (1.2). The proof follows a standard three-step architecture: Proposition 3.1 reduces insensitivity to a partial null controllability property for a cascade of forward-backward stochastic parabolic equations; Theorem 4.1 establishes a global Carleman estimate for the coupled adjoint system (3.10)-(3.11); Proposition 4.1 derives an observability inequality; and Section 5 uses duality, Hahn-Banach, and Riesz representation to construct the controls. The paper closes with open problems in Section 6.","tokens_in":19804,"tokens_out":7000,"duration_ms":64515,"significance":"If the main result were fully proved, this would be a natural and meaningful extension of insensitizing-control results to stochastic parabolic equations with dynamic boundary conditions, a setting for which the literature is still sparse. The reduction in Proposition 3.1 is clearly formulated, and the strategy of combining published Carleman estimates from reference [2] with a new coupling argument is appropriate. The paper is also transparent about the role of the auxiliary controls v1 and v2 and about the zero-initial-data restriction. However, as discussed below, the observability inequality in Proposition 4.1, which is the load-bearing bridge to the duality construction in Section 5, is not proved as written because the crucial pointwise weight comparison near t=0 is false.","major_comments":[{"comment":"The pointwise comparison asserted immediately before (4.23) is false near t=0. With α(t,x)=(e^{μψ(x)}-e^{2μ|ψ|∞})/(t(T-t)), the numerator is strictly negative and bounded away from zero, so for every x∈G one has θ²γ³ = exp(2λα)γ³ ≤ C e^{-c/t}/t³ → 0 as t↓0. On the other hand, exp(-Mt-1) tends to e^{-1}>0 as t↓0 for every fixed M. Hence the inequality exp(-Mt-1) ≤ Cθ²γ³ cannot hold on (0,T/2)×G for any finite M,C. Since (4.23) is the only step that transfers the Carleman estimate (4.5) to the unweighted p-integral over (0,T/2), inequality (4.24) is not established. The energy estimates (4.20) and (4.22) cover only t∈(T/4,T) and do not control p on (0,T/4), so the observability inequality (4.17), on which the duality argument in Section 5 rests, is not proved as written.","section":"§4.2, Eq. (4.23)"}],"minor_comments":[{"comment":"The powers of λ on the P and \\hat P terms in (4.5) appear to be 1, whereas the proof produces λ² in those terms, as in (4.6). Since λ is fixed large this does not affect the later argument, but the displayed statement should be made consistent with the proof.","section":"Theorem 4.1, display (4.5)"},{"comment":"References [7] and [9] list the same article (Bodart, González-Burgos, and Pérez-García, C. R. Math. 335 (2002) 677–682); these entries should be merged or corrected.","section":"References"},{"comment":"The notation exp(Mt−1) in (1.4) and (1.5) is easy to misread; using explicit parentheses, for example exp(Mt−1), would improve clarity.","section":"Introduction, Theorem 1.1"}],"recommendation":"reject","confidential_remarks":"The verdict is based solely on the gap in §4.2. The reliance on Lemmas 4.2 and 4.3 from reference [2], which shares three of four authors with this paper, is a continuation pattern rather than the reason for rejection. A genuinely new argument would be needed to control p near t=0, since the Carleman weight vanishes there."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nShort version: don't send this one to press as is. The main theorem is unsupported because inequality (4.23) is false near t=0. The Carleman weight θ²γ³ = exp(2λ(e^{μψ}−e^{2μ|ψ|∞})/(t(T−t))) · (t(T−t))^{-3} tends to 0 at t=0, since the exponent is negative and behaves like −C/t. But exp(−Mt−1) has limit e^{−1}. So the claimed pointwise bound cannot hold in any neighborhood of t=0. This is the exact step that turns the Carleman estimate (4.5) into the (0,T/2) half of the observability inequality (4.17). The energy estimates in (4.20)–(4.22) only cover the middle and the right end of the time interval; they don't repair the left edge. So the observability inequality, and with it the Hahn–Banach/Riesz construction in Section 5, is not proved.\n\nThat's the bad news. The good news is that the surrounding architecture is right and the intended contribution is real: this would be the first insensitizing-control result for stochastic parabolic equations with dynamic boundary conditions. The reduction in Proposition 3.1 is clean and correct, and the coupled Carleman estimate in Theorem 4.1 is a plausible combination of the authors' earlier estimates. The proof of Theorem 4.1 is standard Carleman machinery, and if the gap in (4.23) were fixed, the rest of the paper would likely go through.\n\nTwo smaller issues. The abstract claims the paper uses \"only one control without extra forces in the noise parts,\" but Theorem 1.1 uses three controls (u,v1,v2). Either the abstract is from another paper or the claim is simply wrong. And the heavy reliance on the authors' own [2] is worth noting—not a flaw by itself since those results are published, but a referee should check that [2]'s assumptions are truly satisfied here.\n\nBottom line: this is a serious attempt at a sensible problem, but it has a load-bearing gap. I'd send it to a referee anyway—the error is concrete and fixable, and the topic deserves scrutiny. I wouldn't cite it until the observability inequality is fixed.","headline":"The intended result is new and the surrounding framework is sound, but the observability inequality relies on a false pointwise bound at t=0, so the main theorem is unproved as written.","tokens_in":20474,"tokens_out":3348,"would_cite":false,"duration_ms":27423,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B05","93C20","60H15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, for a forward stochastic heat equation equipped with dynamic boundary conditions, a triple of controls can be chosen so that the energy functional measuring the solution over an observation region is insensitive to…","keywords":["insensitizing controls","stochastic parabolic equations","dynamic boundary conditions","Carleman estimates","observability inequality","null controllability","forward-backward stochastic system","duality method"],"falsifier":"Evaluate the inequality (4.23) at a fixed $x\\in G$ and let $t\\to 0^+$: the left-hand factor $\\exp(-Mt-1)$ approaches $e^{-1}$, while the right-hand Carleman weight $\\theta^2\\gamma^3 = \\exp(2\\lambda e^{\\mu\\psi(x)}-2\\lambda e^{2\\mu|\\psi|_\\infty})/(t^3(T-t)^3)$ approaches $0$ because $\\psi(x)<|\\psi|_\\infty$; the claimed bound therefore cannot hold for any finite $C$, and the derivation of the observability inequality (4.17) fails as written.","tokens_in":19246,"feed_emoji":"🎛️","tokens_out":11857,"duration_ms":98205,"temperature":0.7,"pith_summary":"This paper tries to prove that a forward stochastic heat equation with dynamic boundary conditions admits insensitizing controls: a triple of controls $(u,v_1,v_2)$ that makes the energy functional $\\Phi$, measuring the solution over an observation region, locally independent of small unknown perturbations of the initial data. If true, this means an observer's energy measurements of such a system can be protected from initial-condition uncertainty, which matters for control and inverse problems on domains where the boundary has its own dynamics. The proof reduces insensitivity to a null controllability question for a coupled forward-backward stochastic parabolic system, establishes a global Carleman estimate for that system, and then uses a duality argument to build the controls. The main theorem asserts existence together with a control-cost bound in terms of a weighted norm of the source terms, under the geometric condition that the control region $G_0$ and observation region $O$ intersect and under zero initial data.","feed_headline":"Controls shield stochastic-heat energy from initial-data noise","feed_subtitle":"A Carleman estimate plus duality makes the energy functional locally immune to unknown initial perturbations.","key_machinery":"The central object is the adjoint coupled system (3.10)-(3.11), consisting of a backward stochastic parabolic equation for $(p,p_\\Gamma,P,\\hat P)$ and a forward one for $(q,q_\\Gamma)$, linked through observation terms $\\chi_O q$ and $\\chi_{O_\\Gamma}q_\\Gamma$. The load-bearing tool is the Carleman estimate (4.5), a weighted integral inequality with weight $\\theta^2\\gamma^3$ (where $\\theta=e^{\\lambda\\alpha}$, $\\gamma=1/(t(T-t))$) that controls global $L^2$ norms of both components by local data on $G_0$ and by the noise variables $P,\\hat P$; from it the paper derives the observability inequality (4.17) with the exponential weight $\\exp(-Mt-1)$. This inequality is the bridge to duality: it makes the linear functional in Section 5 bounded on the space of adjoint solutions, which is exactly what turns into insensitizing controls via Proposition 3.1.","core_discovery":"The paper's central claim is Theorem 1.1: whenever $G_0\\cap O\\neq\\emptyset$ and the initial state is $(0,0)$, there are constants $M,C>0$ such that for every source pair $(\\xi_1,\\xi_2)$ satisfying the weighted integrability condition (1.4), one can find $(u,v_1,v_2)$ insensitizing $\\Phi$ in the sense of Definition 1.1. The insensitivity conditions are the vanishing of the two partial derivatives (1.3) at $\\tau_1=\\tau_2=0$, and the cost estimate (1.5) bounds the controls by the same weighted source norm. The mechanism is to observe that insensitivity is equivalent to $z(0,\\cdot)=0$, $z_\\Gamma(0,\\cdot)=0$ for the backward component of the cascade (Proposition 3.1), to prove the Carleman estimate (4.5) and then the observability inequality (4.17) for the adjoint system, and finally to apply Hahn-Banach and Riesz representation to extract the controls. The paper also records that this works for every initial data if the observation functional starts after a positive time $t_0>0$ and the sources vanish in $(0,t_0)$ (Remark 1.1).","pith_inferences":["Beyond the paper: if the faulty comparison (4.23) near $t=0$ can be repaired by choosing a weight that does not collapse at the initial time, the overall structure of the proof would still deliver Theorem 1.1; the flaw is in the derivation, not necessarily in the statement.","Beyond the paper: the same duality setup should transfer to the $\\varepsilon$-insensitizing problem described in Section 6, turning the observability inequality into an approximate controllability statement and giving a quantitative version of the paper's open problem (3).","Beyond the paper: because the Carleman estimate controls both bulk and boundary components uniformly, the method may extend to semilinear equations with globally Lipschitz nonlinearities by a standard fixed-point iteration, matching the paper's proposed open problem (2)."],"forward_implications":["For zero initial data and sources with finite weighted norm (1.4), the theorem guarantees an insensitizing control triple whose cost is bounded by that weighted norm; the paper presents this as the first such result for stochastic parabolic equations with dynamic boundary conditions.","Insensitivity of $\\Phi$ is completely equivalent to null controllability of the cascade (3.1)-(3.2), so any future null-controllability result for that cascade automatically yields insensitizing controls.","If the sources vanish on an initial time interval, the condition on the initial data disappears: every initial state can be insensitized for the modified functional $\\Phi_{t_0}$ (Remark 1.1).","The paper identifies the disjoint case $G_0\\cap O=\\emptyset$ as an open problem, so the theorem's geometric assumption is not known to be necessary.","Insensitizing controls exist in the three-control form $(u,v_1,v_2)$; the paper explicitly asks whether a single control $u$ (without the extra noise controls) can suffice, pointing to a spectral approach for that question."],"supporting_citations":[{"why":"Supplies the well-posedness theorems (2.1, 2.2) and the individual Carleman estimates (Lemmas 4.2, 4.3) for forward and backward stochastic parabolic equations with dynamic boundary conditions.","marker":"[2]"},{"why":"Provides the weight lemma (Lemma 4.1) constructing the function $\\psi$ that vanishes on $\\Gamma$ and stays nondegenerate away from an interior subset, the foundation of the Carleman weights.","marker":"[15]"},{"why":"Introduced the insensitizing control problem that this paper extends to the stochastic dynamic-boundary setting.","marker":"[19]"},{"why":"Shows insensitizing controls cannot exist for every initial data in related problems, motivating the zero-initial-data hypothesis in Theorem 1.1.","marker":"[30]"},{"why":"Establishes insensitizing controls for a forward stochastic heat equation with Dirichlet boundary conditions, the stochastic precedent this paper generalizes.","marker":"[33]"},{"why":"Treats insensitizing controls for deterministic parabolic equations with dynamic boundary conditions, the boundary-condition precedent.","marker":"[34]"}],"fun_headline_variants":["One control makes stochastic heat energy noise-blind","Insensitizing controls: stochastic heat ignores initial blips","Single control insensitizes stochastic heat to initial noise","Stochastic heat: control rules out initial-data sensitivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the exponential comparison in (4.23) holds on $(0,T/2)$, but near $t=0$ the left side approaches $e^{-1}$ while the Carleman weight tends to $0$, so as written the comparison fails and the observability bridge to duality is not established.","fun_headline_variants_meta":{"raw":{"variants":["One control makes stochastic heat energy noise-blind","Insensitizing controls: stochastic heat ignores initial blips","Single control insensitizes stochastic heat to initial noise","Stochastic heat: control rules out initial-data sensitivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1255,"prompt_tokens":877,"completion_tokens":378,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":493,"completion_tokens_details":{"reasoning_tokens":316}},"tokens_in":493,"tokens_out":378,"duration_ms":3728,"temperature":1.0,"reasoning_tokens":316,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:11:20.626046+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the inequality (4.23) at a fixed $x\\in G$ and let $t\\to 0^+$: the left-hand factor $\\exp(-Mt-1)$ approaches $e^{-1}$, while the right-hand Carleman weight $\\theta^2\\gamma^3 = \\exp(2\\lambda e^{\\mu\\psi(x)}-2\\lambda e^{2\\mu|\\psi|_\\infty})/(t^3(T-t)^3)$ approaches $0$ because $\\psi(x)<|\\psi|_\\infty$; the claimed bound therefore cannot hold for any finite $C$, and the derivation of the observability inequality (4.17) fails as written.","supporting_citations":[{"cited_title":"Baroun, S","cited_arxiv_id":null,"evidence_quote":"Supplies the well-posedness theorems (2.1, 2.2) and the individual Carleman estimates (Lemmas 4.2, 4.3) for forward and backward stochastic parabolic equations with dynamic boundary conditions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the weight lemma (Lemma 4.1) constructing the function $\\psi$ that vanishes on $\\Gamma$ and stays nondegenerate away from an interior subset, the foundation of the Carleman weights."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced the insensitizing control problem that this paper extends to the stochastic dynamic-boundary setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows insensitizing controls cannot exist for every initial data in related problems, motivating the zero-initial-data hypothesis in Theorem 1.1."},{"cited_title":"Yan and F","cited_arxiv_id":null,"evidence_quote":"Establishes insensitizing controls for a forward stochastic heat equation with Dirichlet boundary conditions, the stochastic precedent this paper generalizes."},{"cited_title":"Zhang, J","cited_arxiv_id":null,"evidence_quote":"Treats insensitizing controls for deterministic parabolic equations with dynamic boundary conditions, the boundary-condition precedent."}],"review_version":1}