{"id":"a4d6febf-13f2-4f6b-ade7-65c8315e6383","arxiv_id":"2508.00812","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Null controllability of the Kuramoto-Sivashinsky equation on multi-dimensional cylinders is established, with a positive minimal time for irrational control positions and controllability for all positive times at algebraic irrational positions.","lead":"This paper proves null controllability results for the Kuramoto-Sivashinsky equation on multi-dimensional cylindrical domains using boundary and interior controls. It establishes a necessary and sufficient condition, explicit control cost estimates, and a minimal control time that depends on the position of the interior control slice.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract's interior-control claims appear mutually contradictory for algebraic irrational x0/a: a positive minimal time T0 conflicts with controllability for every T>0.","rationale":"The reader's verdict was UNVERDICTED because only the abstract was available, and the reader's weakest assumption focused on the Diophantine condition for the any-time result. My stress-test identifies a different and more basic problem: the abstract, taken literally, asserts two mutually incompatible conclusions for the same class of control positions. The reader's concern about the Diophantine condition is related but does not flag the direct contradiction between the positive minimal time and the any-time controllability for algebraic irrationals. I therefore partially agree with the reader. The appropriate disposition is CONDITIONAL rather than outright rejection, because the full text may contain a qualifying hypothesis that the abstract omits; however, as presented, the central claim is logically inconsistent. The concrete test is to read the precise theorem statements and verify whether the two results share the same hypothesis. If the contradiction is real, the paper's main contribution cannot be accepted without substantial revision. If the full statements resolve the conflict, then the reader's UNVERDICTED verdict would remain appropriate pending a full technical review.","tokens_in":900,"tokens_out":2581,"duration_ms":35564,"concrete_test":"Obtain the full manuscript and inspect the exact statements of the interior-control theorems (likely the theorem giving T0(x0) and the theorem for algebraic x0/a). Check whether the minimal-time theorem's hypothesis excludes algebraic x0/a (e.g., it may require x0/a to be Liouville or non-algebraic), and whether the algebraic theorem's conclusion is 'for every T > 0' or rather 'for every T > T0(x0)'. If both theorems apply verbatim to a single algebraic irrational x0/a, then the central claim is contradictory and the paper cannot be accepted as stated; if one theorem carries a missing qualifier, the contradiction is resolved and the verdict should be based on the corrected statement.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The abstract states two results for the interior control supported on gamma = {x0} x omega. First: for x0/a in (0,1) \\setminus Q, there is a minimal time T0(x0) > 0 such that the system is null controllable for all T > T0(x0) and not controllable for T < T0(x0). Second: if x0/a is an algebraic real number of order d > 1, then the system is null controllable for any time T > 0. Every algebraic irrational has order d > 1 and is also in (0,1) \\setminus Q, so both statements apply to the same x0. Take any algebraic irrational x0/a and any time T with 0 < T < T0(x0). The first statement says the system is not controllable at that T; the second says it is controllable at that T. This is a direct contradiction unless one of the two statements carries an additional hypothesis not present in the abstract, such as excluding algebraic numbers from the minimal-time theorem, or interpreting 'any time T>0' as 'any T > T0(x0)'. As written, the central claim about the interior control geometry is internally inconsistent. This is the most load-bearing concern because it concerns the main new result of the paper, not a technical edge case. The contradiction must be resolved before the result can be assessed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This abstract-only submission concerns null controllability of the Kuramoto–Sivashinsky equation on a cylindrical domain Ω = (0,a) × Ω_y. The abstract announces four results: (i) a necessary and sufficient condition, with explicit control cost, for null controllability of the linearized system by a boundary control acting through the Laplacian component on {0} × ω; (ii) for an interior control on γ = {x0} × ω with x0/a irrational, the existence of a positive minimal time T0(x0) such that the system is null controllable for T > T0(x0) and not for T < T0(x0); (iii) a collapse of the minimal time to zero when x0/a is an algebraic real number of order d > 1, so that controllability holds for every T > 0; and (iv) local null controllability of the nonlinear system for N = 2 or 3 via the source-term method and Banach fixed point theorem. No full text or proof details were available for inspection.","tokens_in":1201,"tokens_out":4381,"duration_ms":54860,"significance":"If the announced results are correct, the paper would make a substantial contribution to the controllability theory of higher-dimensional Kuramoto–Sivashinsky equations, particularly by giving a necessary and sufficient condition and an exact minimal-time phenomenon for geometric control regions. The stated combination of the method of moments and the Lebeau–Robbiano strategy is natural for this problem, and the promised explicit control cost estimate would be a useful quantitative addition. However, the abstract contains a direct contradiction between the minimal-time statement and the any-time controllability statement for algebraic irrational positions; this must be resolved before the significance of the results can be assessed. Because no full text was supplied, none of the estimates or fixed-point arguments could be verified.","major_comments":[{"comment":"The two interior-control statements are mutually contradictory. Every algebraic irrational x0/a in (0,1) has order d > 1 and also belongs to (0,1) \\ ℚ. Therefore both the minimal-time statement (T0(x0) > 0, non-controllability for T < T0(x0)) and the any-time statement (controllability for every T > 0) apply to the same x0. Taking T with 0 < T < T0(x0) yields that the system is both not controllable and controllable. The abstract must either exclude algebraic irrationals from the minimal-time theorem, reinterpret \"any time T > 0\" as \"any T > T0(x0)\", or supply an additional hypothesis distinguishing the two regimes; as written, the central result is internally inconsistent.","section":"Abstract, interior-control claims"},{"comment":"The abstract states that a necessary and sufficient condition for null controllability of the linearized system is obtained, but it does not state the condition itself. Since this is one of the paper's main claims, the condition must be explicitly formulated in the abstract or the theorem statement, and it must be shown that the method of moments and the Lebeau–Robbiano strategy together actually yield it. Without the condition, the claim cannot be evaluated.","section":"Abstract, boundary controllability"},{"comment":"For N = 2 or 3, the abstract only asserts local null controllability of the nonlinear system and names the source-term method and Banach fixed point theorem. Missing are the functional setting, the smallness assumptions on the initial data, and the regularity or compatibility conditions needed for the fixed-point argument. These details are essential for assessing the nonlinear result, especially because the linear result involves a delicate minimal-time threshold.","section":"Abstract, nonlinear local controllability"}],"minor_comments":[{"comment":"The phrase \"algebraic real number of order d > 1\" should be defined precisely as a real algebraic number of degree d over ℚ; the term \"order\" is otherwise ambiguous.","section":"Abstract, terminology"},{"comment":"The description \"control acting on {0} × ω through the boundary term associated with the Laplacian component\" is vague; the authors should specify exactly which boundary condition (for example, Neumann or Robin type) is subject to control.","section":"Abstract, boundary control type"},{"comment":"The minimal-time and any-time controllability statements do not explicitly specify the dimension N of the cylindrical domain; the abstract only restricts N = 2 or 3 for the nonlinear result. The authors should clarify whether the interior-control results hold for general N or require additional restrictions.","section":"Abstract, dimension and geometry"},{"comment":"The promised explicit control cost estimate should state whether the estimate is uniform in the time horizon T and in the control region ω, since such uniformity is often crucial in controllability applications.","section":"Abstract, control-cost estimate"}],"recommendation":"major_revision","confidential_remarks":"The abstract-only nature of this submission makes a full assessment impossible, but the contradiction between the minimal-time claim and the any-time algebraic-position claim is serious and must be addressed. I would recommend requesting a complete manuscript and asking the authors to clarify the theorem statements; if the contradiction is not resolved by a legitimate additional hypothesis, the main interior-control result would be false as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The abstract makes a strong and interesting claim, but as written it contradicts itself. For x0/a irrational, it says there is a minimal time T0(x0)>0 with null controllability for T>T0 and not for T<T0. Then for x0/a algebraic of order d>1, it says controllability holds for every T>0. Every algebraic irrational has order greater than 1 and is irrational, so both statements apply to the same x0. Pick T<T0(x0): the first says not controllable, the second says controllable. That is a logical inconsistency in the main result. Unless the authors intend the minimal-time theorem to exclude algebraic numbers or the 'any time' statement to mean 'any T>T0', the central claim collapses. The abstract does not state such an exclusion, so as it stands the result is not assessable.\n\nTo give credit where it is due: the problem is well-motivated, the combination of moment methods and Lebeau-Robbiano is a reasonable toolbox, and an explicit minimal time for a multidimensional nonlinear PDE would be a real contribution. The boundary control part (necessary and sufficient condition with cost estimate) may stand on its own, but the interior control part is the headline and it is internally inconsistent.\n\nThis is an abstract-only review, so there are no proofs to check. The inconsistency could be a wording problem in the abstract, and the full paper might resolve it. But the burden is on the authors to fix the statement before it goes to referees. I would not send this to peer review in its current form; I would ask the authors to clarify the Diophantine hypotheses and reconcile the two claims. If they do, the paper deserves a careful look from the PDE control community.","headline":"The abstract's interior-control theorem is self-contradictory for algebraic irrational positions, so the paper needs a clarifying revision before refereeing.","tokens_in":1631,"tokens_out":2704,"would_cite":false,"duration_ms":31871,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B05","93C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the Kuramoto-Sivashinsky equation on a cylinder, steering the state exactly to zero from a single interior slice requires a positive minimal time for generic irrational slice positions, and no wait when the position is an algebraic…","keywords":["Kuramoto-Sivashinsky equation","null controllability","cylindrical domain","method of moments","minimal time","interior control","algebraic irrational","observability strategy"],"falsifier":"Compute the minimal null-control time for the linearized Kuramoto-Sivashinsky equation on a rectangle with an interior control at a slice whose ratio $x_0/a$ is a transcendental irrational such as $1/e$. The paper predicts a positive minimal time; observing null controllability for arbitrarily small $T$ for that transcendental ratio would contradict the claimed dichotomy. Conversely, for an algebraic ratio such as $x_0/a=\\sqrt{2}/2$ the paper predicts zero minimal time, so observing a positive minimal time there would also refute the claim.","tokens_in":752,"feed_emoji":"⏳","tokens_out":13562,"duration_ms":123308,"temperature":0.7,"pith_summary":"This paper asks when the Kuramoto-Sivashinsky equation, a model for thin liquid films and flame-front instabilities, can be steered exactly to zero on a cylindrical domain using a control confined to one boundary wall or to a single interior slice. For the linearized equation, a boundary control on $\\{0\\}\\times\\omega$ achieves null controllability exactly when the controlled wall region satisfies a geometric condition, and the paper gives an explicit bound on the control cost. For an interior control on $\\{x_0\\}\\times\\omega$ with $x_0/a$ irrational, null controllability holds exactly for times $T$ larger than a minimal time $T_0(x_0)$, and fails for smaller times. If $x_0/a$ is an algebraic irrational of degree greater than one, that minimal time collapses to zero, so any positive control horizon works. In two and three spatial dimensions the same local null controllability extends to the nonlinear equation.","feed_headline":"Algebraic slice positions erase the minimal control time","feed_subtitle":"A Diophantine condition on the control slice's position decides whether zero can be reached in any time.","key_machinery":"The proof combines the method of moments with a spectral observability strategy that localizes estimates in space and frequency. The method of moments recasts the control problem as a moment problem for the eigenvalues of the Laplacian on the cross-section, producing explicit cost bounds; the observability strategy yields estimates that localize the control in space and frequency. For interior control, the decisive object is the ratio $x_0/a$: exponential factors $e^{-\\lambda_n x_0}$ must be linearly independent and satisfy lower bounds whose quality depends on whether $x_0/a$ is rational, irrational, or algebraic of bounded degree. That arithmetic dependence is what produces a positive minimal time for generic irrationals and its collapse to zero for algebraic irrationals of order $d>1$.","core_discovery":"The central claim is that null controllability of the linearized Kuramoto-Sivashinsky equation on $\\Omega=(0,a)\\times\\Omega_y$ is governed by the position of the control. With boundary control supported on $\\{0\\}\\times\\omega$, the system is null controllable if and only if $\\omega$ meets the stated geometric condition, and the control cost is estimated explicitly. With interior control on $\\gamma=\\{x_0\\}\\times\\omega$, the ratio $x_0/a$ determines a minimal time: for irrational $x_0/a$ there is a positive $T_0(x_0)$ such that null controllability holds exactly for $T>T_0(x_0)$, while for algebraic $x_0/a$ of order $d>1$ the threshold vanishes and the system is controllable for every $T>0$. For $N=2$ or $3$, the nonlinear system is locally null controllable, meaning small initial data can be driven to zero by small controls.","pith_inferences":["The minimal time $T_0(x_0)$ is presumably quantitative: the method of moments suggests it should be expressible in terms of the Diophantine approximation exponent of $x_0/a$, although the paper does not state such a formula.","The same algebraic-versus-transcendental dichotomy may apply to other parabolic and dispersive control problems on cylinders where moment methods are used, such as heat or beam equations with point-like interior controls.","A direct test of the dichotomy would be to compute the minimal control time numerically for a transcendental ratio such as $1/e$; the paper's claims predict a positive threshold, while an algebraic ratio should behave as if the threshold is zero.","The local nonlinear result in dimensions 2 and 3 could plausibly extend to higher dimensions if the observability estimates behind the source-term method hold there."],"forward_implications":["Boundary control on one end wall of the cylinder is fully characterized by the geometric condition on the controlled region; when the condition holds, explicit control-cost bounds guarantee null controllability.","Interior control at a single slice always succeeds, but for generic irrational slice positions the controller must wait for a positive minimal time before a zero state is reachable.","Algebraic irrational slice positions are exceptional: they remove the waiting time, so the system is null controllable in arbitrarily short control horizons.","In dimensions two and three, the nonlinear Kuramoto-Sivashinsky equation inherits local null controllability from the linearized system."],"supporting_citations":[],"fun_headline_variants":["Algebraic slice positions zero out minimal control time","Diophantine slice position removes any minimal control time","Irrational slices demand minimal time; algebraic ones don't","For algebraic slices, KS null control is possible at any T>0","Rationality of slice position sets minimal control time for KS"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The any-time controllability result rests on the slice position $x_0/a$ being an algebraic real number of degree greater than one; if that Diophantine condition is not met, the proof that the minimal time vanishes does not apply, and the dichotomy between positive and zero minimal time may fail.","fun_headline_variants_meta":{"raw":{"variants":["Algebraic slice positions zero out minimal control time","Diophantine slice position removes any minimal control time","Irrational slices demand minimal time; algebraic ones don't","For algebraic slices, KS null control is possible at any T>0","Rationality of slice position sets minimal control time for KS"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000738,"raw_usage":{"total_tokens":3358,"prompt_tokens":1067,"completion_tokens":2291,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":683,"completion_tokens_details":{"reasoning_tokens":2219}},"tokens_in":683,"tokens_out":2291,"duration_ms":21206,"temperature":1.0,"reasoning_tokens":2219,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T05:54:30.744697+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the minimal null-control time for the linearized Kuramoto-Sivashinsky equation on a rectangle with an interior control at a slice whose ratio $x_0/a$ is a transcendental irrational such as $1/e$. The paper predicts a positive minimal time; observing null controllability for arbitrarily small $T$ for that transcendental ratio would contradict the claimed dichotomy. Conversely, for an algebraic ratio such as $x_0/a=\\sqrt{2}/2$ the paper predicts zero minimal time, so observing a positive minimal time there would also refute the claim.","supporting_citations":[],"review_version":1}