{"id":"beceb488-89f2-444b-8d16-c29c310c5263","arxiv_id":"2501.02995","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper extends a known controllability method to impulsive systems, but the main theorem's proof is incomplete.","lead":"This paper attempts to prove finite-approximate controllability for impulsive evolution equations in Hilbert spaces by adapting a known resolvent-operator technique. The proof of the main semilinear result contains invalid convergence steps, so the central claim is not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.2's proof collapses before the DCT step: the fixed-point radius r(α) in Theorem 4.1 blows up as α→0, so {z_α} is never shown to be uniformly bounded and the weak-compactness argument is unjustified.","rationale":"I re-read Theorem 4.2's proof. The reader's identified flaw — weak convergence in L2 does not imply the strong convergence used after (4.12) — is a genuine invalid step. However, I find an earlier, more load-bearing gap: the proof never establishes that the family of fixed points {z_α} is uniformly bounded. Theorem 4.1's invariant-ball argument produces a radius r(α) that depends on α. From (4.4)–(4.5) and the control bound (4.3), any r that works must be at least M3(α) ~ C/α; hence r(α)→∞. In Theorem 4.2 the notation z_α∈B_r is used as though r were a single constant, but it is not. Consequently the assertion that {z_α(t):α>0} is bounded in H for each t is unsupported. This invalidates the Banach–Alaoglu step, Lemma 4.1, and hence the entire subsequence argument that leads to (4.12). Even if the DCT step were corrected by a compactness argument for the integral operators, the proof would still need a uniform bound to have a subsequence at all. There is also a smaller assumption mismatch: Theorem 4.2 lists (A3)–(A6) and not (A2), but Theorem 4.1's continuity proof uses (A2); replacement by the boundedness (A5) alone does not verify continuity. This reinforces that the advertised theorem is not proved as written.","tokens_in":19614,"tokens_out":16052,"duration_ms":134231,"concrete_test":"Take a concrete instance of the heat-equation setting of Section 5 with a bounded nonzero μ (e.g., μ(t,ξ)=sin(ξ)), and compute the minimal radius r(α) satisfying G_α(B_{r(α)})⊂B_{r(α)} using the estimates of Theorem 4.1. If r(α) grows like C/α as α→0, the family {z_α} is not bounded uniformly in α, so the weak-compactness extraction in Theorem 4.2 fails. Equivalently, attempt to derive a bound ||z_α(t)||≤C independent of α directly from the fixed-point equation; if the control estimate (4.3) forces C/α, the proof breaks.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central semilinear FA-controllability claim rests on Theorem 4.2, whose proof applies Theorem 4.1 for every α>0 and then extracts a weakly convergent subsequence from {z_α}. The extraction is not justified because the closed ball B_r in Theorem 4.1 has radius r = r(α) that depends on α. Indeed, Step 1 of Theorem 4.1 shows by contradiction that any admissible radius must exceed M3(α) + M4(α)Λ_μ(r). Under assumption (A5), μ is bounded, so Λ_μ(r) ≤ N; hence any r > M3(α)+M4(α)N works, and the minimal radius is at least M3(α). The term M3(α) contains contributions of order 1/α, originating from the control estimate (4.3), which has the factor 1/(α(1−δ)). Since (A6) gives δ→0 as α→0, M3(α)→∞. Thus r(α)→∞. Therefore the statement 'The condition z_α ∈ B_r implies that the set {z_α(t): α>0} is bounded in H' in the proof of Theorem 4.2 is false: the bound depends on α. Without a uniform bound, Banach–Alaoglu cannot be applied to produce a weakly convergent subsequence, and Lemma 4.1's L2-boundedness of {μ(·,z_α(·))} is likewise unsupported (its proof uses the same uniform boundedness). The later DCT error noted by the reader is real but secondary: it concerns convergence of a subsequence that may not exist. A repair would require an a priori bound on the closed-loop trajectories independent of α, which the paper does not provide.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies finite-approximate controllability (FA-controllability) of impulsive evolution systems in Hilbert spaces. Section 3 establishes, via a resolvent-like operator and the Gramian Γ^b_{t_p}+tildeΓ^b_{t_p}+Θ^{t_p}_0+tildeΘ^{t_p}_0, an equivalence between approximate controllability, strict positivity of the Gramian, strong convergence of the resolvent, and FA-controllability for the linear impulsive system (3.1). Section 4 considers the semilinear system (1.2) with a bounded nonlinear perturbation; using Schauder's fixed-point theorem the authors prove existence of mild solutions for each α>0 (Theorem 4.1) and then claim FA-controllability as α→0 (Theorem 4.2). Section 5 sketches an application to a heat equation with impulses. The linear part appears to follow standard arguments from [18,21]; the semilinear FA-controllability proof has a critical gap.","tokens_in":19994,"tokens_out":8681,"duration_ms":77053,"significance":"The potential contribution is an extension of FA-controllability to impulsive semilinear evolution equations with compact semigroup and bounded nonlinearity, building on the authors' prior work. The linear equivalence result and the explicit construction of the finite-approximating control are useful and likely correct. However, Theorem 4.2, which is the main new claim for semilinear systems, rests on an unjustified uniform-boundedness assertion; without it, the weak-compactness argument collapses. Since the paper's advertised semilinear result is not established, the contribution is currently not significant enough for publication in its present form.","major_comments":[{"comment":"The assertion 'The condition z_α ∈ B_r implies that the set {z_α(t): α>0} is bounded in H' is unjustified because the radius r produced by Theorem 4.1 depends on α. In Step 1 of Theorem 4.1, the admissible radius must dominate M3(α) + M4(α)Λ_μ(r), and M3(α) contains terms proportional to 1/(α(1−δ)), which diverges as α→0+ under (A6). Consequently, no α-independent bound for {z_α} is established, so the Banach–Alaoglu extraction of a weakly convergent subsequence and the L²-boundedness in Lemma 4.1 are unsupported. This is the central step of the semilinear FA-controllability claim.","section":"Section 4, proof of Theorem 4.2"},{"comment":"The convergence '→0 as α_i→0+' in (4.12) is inferred from the weak convergence (4.11) by invoking the Dominated Convergence Theorem. Weak convergence in L² does not imply convergence of the H-norms of the integrals in (4.12). A repair would require a compactness argument for the operator f ↦ ∫ S(·−s)f(s)ds under assumption (A1), which is absent. The subsequence appearing in (4.12) is also in doubt because of the preceding uniform-boundedness gap.","section":"Section 4, equations (4.11)–(4.12)"},{"comment":"The proof begins 'By applying Theorem 4.1', but the hypotheses do not match. Theorem 4.1 requires (A1)–(A4) plus condition (4.2), whereas Theorem 4.2 assumes (A1), (A3)–(A6), thereby omitting (A2) and adding (A5). Since (A5) does not imply the continuity or strong measurability of μ(t,·) used in Step 3 of Theorem 4.1, the existence of the fixed points z_α is not guaranteed by the quoted theorem. This should be repaired either by adding (A2) to the assumptions of Theorem 4.2 or by proving an analogue of Theorem 4.1 under (A5).","section":"Theorem 4.2, assumptions versus Theorem 4.1"}],"minor_comments":[{"comment":"The typo 'non-neqative' should read 'nonnegative'.","section":"Lemma 2.1"},{"comment":"In the proof of (iii)⇔(iv), the expression 'lim_{α→0−}' should be 'lim_{α→0+}'.","section":"Theorem 3.1 proof"},{"comment":"The target state is called h in the definition but z_b in the preceding paragraph; the notation should be made consistent.","section":"Section 2, definition of FA-controllability"},{"comment":"The interval [r1,r2] is never defined in the statement or proof of Lemma 4.1; it should be specified, presumably as [0,b] or an appropriate subinterval.","section":"Lemma 4.1"},{"comment":"The text says 'let μ satisfies Assumptions (A2) and (A3)', but Theorem 4.2 uses (A5); the application should either verify (A5) or state matching assumptions.","section":"Section 5, application"}],"recommendation":"reject","confidential_remarks":"If the uniform-boundedness gap can be closed, the linear part and the existence theorem might form the basis of a revised submission; in its current form the semilinear FA-controllability claim is unsupported. The incremental relationship to [18,19,21] should also be clarified."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a routine extension of Mahmudov's resolvent-like operator technique to first-order impulsive semilinear systems. The linear part (Theorem 3.1) is a competent adaptation—explicit controls for the impulsive case are written out, and the equivalence with approximate controllability follows known lines. The existence theorem (Theorem 4.1) is a standard Schauder fixed-point argument, and the heat equation application is a reasonable sanity check. The citation pattern is honest; the authors build on their own prior work, which is fine for an incremental contribution.\n\nThe problem is the advertised semilinear FA-controllability result, Theorem 4.2. The stress-test note is correct and more fundamental than the reader's report suggests. Theorem 4.1 guarantees a fixed point in a ball B_r, but the radius r depends on α, and the minimal admissible r grows like 1/α as α→0 (the M3 term). So the set {z_α} is not uniformly bounded in H. Without that uniform bound, the Banach–Alaoglu step fails, and Lemma 4.1's L2-boundedness claim also has no support. The DCT step in (4.12) is a further error—weak L2 convergence does not give pointwise convergence, so the integral convergence is not justified. (The compactness of the semigroup could potentially repair this, but no such argument appears.)\n\nOne small correction to the reader's report: condition (4.2) is actually automatically satisfied under (A5) because d=0, so the failure to \"verify\" it is not the real issue. The real issue is the α-dependent radius.\n\nCould the paper be fixed? Possibly—an a priori bound on the trajectories independent of α plus a compactness argument to upgrade weak L2 convergence to strong convergence of the integrals might salvage the main theorem. But the paper as written does none of this, and the gap is load-bearing. The linear section might be useful to someone working specifically on impulsive controllability, but the semilinear claim should not be cited as established.\n\nI would not send this to peer review in its current form. If the authors supply a genuine uniform boundedness argument and correct the convergence step, it might be worth a re-submission. As it stands, I'd reject or, at best, invite a revision with these concrete demands.","headline":"Routine extension of the authors' own resolvent-like operator work to impulsive systems, but the main semilinear FA-controllability theorem has a load-bearing gap: the fixed-point radius depends on α and blows up, so the weak-compactness argument is unjustified.","tokens_in":20522,"tokens_out":4729,"would_cite":false,"duration_ms":83314,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B05","93C25","47D06","34A37"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves finite-approximate controllability for semilinear impulsive evolution systems in Hilbert spaces, using a resolvent-like operator and Schauder's fixed-point theorem.","keywords":["finite-approximate controllability","impulsive evolution systems","resolvent-like operator","Hilbert space","Schauder fixed-point theorem","compact semigroup","approximate controllability","heat equation"],"falsifier":"Compute the remainder in (4.12) for a standard weak-not-strong example: in H=L2(0,1), take z_α(t)=sin(αt) (constant in space) and choose μ(t,z)=$z^{2}$ so that {μ(·,z_α(·))} converges weakly to 1/2 in L2 but its L2-distance to the weak limit does not vanish. If such a sequence satisfies the paper's bounds, the estimate in (4.12) fails and the endpoint-error argument collapses.","tokens_in":19413,"feed_emoji":"🎛️","tokens_out":5054,"duration_ms":49364,"temperature":0.7,"pith_summary":"The paper aims to establish that impulsive semilinear evolution systems in Hilbert spaces are finite-approximate controllable: one can steer the state arbitrarily close to a target while matching the target exactly on any finite-dimensional subspace. A resolvent-like operator is built from the controllability Gramian and used to define explicit approximating controls, and Schauder's fixed-point theorem supplies existence of mild solutions for the nonlinear system. The authors first prove the linear case, give an equivalence theorem linking approximate and finite-approximate controllability, and then extend the result to an impulsive heat equation. The interest is that exact controllability typically fails for infinite-dimensional systems such as the heat equation, while finite-approximate controllability is the natural workable notion for such systems, particularly in the presence of state jumps.","feed_headline":"Impulsive evolution systems proven finite-approximate controllable","feed_subtitle":"Explicit controls steer the state near target while matching finite-dimensional projections exactly.","key_machinery":"The central object is the resolvent-like operator (α(I−π_D)+$Γ_b^{{t_p}}$+Γ̃$_b^{{t_p}}$+$Θ_0^{{t_p}}$+Θ̃$_0^{{t_p}}$)^{-1} formed from the controllability Gramian $Γ_b^{{t_p}}$+Γ̃$_b^{{t_p}}$+$Θ_0^{{t_p}}$+Θ̃$_0^{{t_p}}$. This operator arises as the minimizer of a regularized quadratic functional and yields the explicit control formulas (3.2)–(3.3) and (4.1). The paper combines this with Schauder's fixed-point theorem to obtain existence of mild solutions and with compactness of the semigroup S(t) to ensure the image of the fixed-point operator is relatively compact. A named identity used is Lemma 2.1, the invertibility bound ||(α(I−π_D)+Γ)^{-1}h|| ≤ ||h||/min(α,δ), which is load-bearing throughout the estimates.","core_discovery":"The central claim is Theorem 4.2: if the semigroup is compact, the nonlinear term satisfies growth and continuity conditions (A3)–(A5), and the associated linear impulsive system is approximately controllable (A6), then the semilinear impulsive system (1.2) is finite-approximate controllable on [0,b]. The proof constructs, for each α>0, a mild solution z_α driven by the explicit control (4.9), shows the family is bounded in PC([0,b],H), extracts weakly convergent subsequences for both the state and the nonlinear term, and then uses the resolvent-like operator estimate to show the endpoint error tends to zero while the projection onto the finite-dimensional subspace D matches exactly. The linear predecessor, Theorem 3.1, asserts the equivalence of approximate controllability, strict positivity of the Gramian, strong convergence of the α-family, and finite-approximate controllability for the linear impulsive system.","pith_inferences":["A repair of the strong-convergence gap would likely require an additional compactness or monotonicity assumption on the Nemitskii operator z ↦ μ(·,z(·)) in L2, since the weak convergence in (4.11) alone does not justify the Dominated Convergence step in (4.12).","If the proof gap is closed, the same scheme should extend to other parabolic systems with analytic semigroups, where compactness of the semigroup often gives the needed strong L2 compactness of the nonlinear term.","The finite-dimensional projection equality π_D z_α(b)=π_D h appears to hold for every α>0 in the construction, suggesting that exact reachability of the projection may persist in the limit even as the full-state error tends to zero; this is an inference, not a claim the paper states.","A testable extension would be to check whether the same resolvent-like construction remains effective when the impulse operators are only Lipschitz rather than linear, which is a direction the paper does not pursue."],"forward_implications":["If Theorem 4.2 is correct, finite-approximate controllability holds for impulsive heat equations with controls acting through a smoothing operator, where exact controllability is known to fail.","The explicit controls (3.2)–(3.3) and (4.9) provide a constructive steering law: from any initial state, one can compute controls achieving the finite-approximate target to any desired tolerance.","The equivalence theorem for linear systems transfers approximate-controllability criteria directly to finite-approximate controllability, so existing results on A-controllability of linear impulsive systems automatically yield FA-controllability.","The resolvent-like operator method, previously used for non-impulsive evolution equations, extends to systems with state jumps, showing that the method does not rely on continuity of the trajectory.","The application to the heat equation illustrates that the abstract Hilbert-space framework is flexible enough to cover parabolic systems with impulses on an interval."],"supporting_citations":[{"why":"Supplies the resolvent operator invertibility lemma and the strong-convergence equivalence that underpin the linear controllability results.","marker":"[18]"},{"why":"Establishes approximate controllability of the linear impulsive system, which is the baseline Assumption (A6) for the semilinear theorem.","marker":"[19]"},{"why":"Introduces the Schauder fixed-point method for finite-approximate controllability of evolution equations, which the semilinear proof follows.","marker":"[20]"},{"why":"Provides the resolvent-like operator construction for finite-approximate controllability, which the paper adapts to impulsive systems.","marker":"[21]"},{"why":"Supplies the generalized Arzelà–Ascoli theorem used to prove compactness of the fixed-point operator G_α.","marker":"[29]"},{"why":"Defines the concepts of controllability and the resolvent operator condition for linear systems that are extended in the present paper.","marker":"[5]"}],"fun_headline_variants":["Impulsive systems: finite-approximate controllability proven","Resolvent-like operator proves impulsive finite-approximate control","Impulsive systems: finite-mode exact, infinite-mode approximate","Semilinear impulsive systems: finite-approximate control"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof that the nonlinear term converges strongly enough in L2 to force the endpoint error to zero relies on applying the Dominated Convergence Theorem to a weakly convergent subsequence, which requires strong convergence that the argument does not establish.","fun_headline_variants_meta":{"raw":{"variants":["Impulsive systems: finite-approximate controllability proven","Resolvent-like operator proves impulsive finite-approximate control","Impulsive systems: finite-mode exact, infinite-mode approximate","Semilinear impulsive systems: finite-approximate control"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.002313,"raw_usage":{"total_tokens":8847,"prompt_tokens":796,"completion_tokens":8051,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":412,"completion_tokens_details":{"reasoning_tokens":7996}},"tokens_in":412,"tokens_out":8051,"duration_ms":61729,"temperature":1.0,"reasoning_tokens":7996,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:58:50.462944+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the remainder in (4.12) for a standard weak-not-strong example: in H=L2(0,1), take z_α(t)=sin(αt) (constant in space) and choose μ(t,z)=$z^{2}$ so that {μ(·,z_α(·))} converges weakly to 1/2 in L2 but its L2-distance to the weak limit does not vanish. If such a sequence satisfies the paper's bounds, the estimate in (4.12) fails and the endpoint-error argument collapses.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the resolvent operator invertibility lemma and the strong-convergence equivalence that underpin the linear controllability results."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes approximate controllability of the linear impulsive system, which is the baseline Assumption (A6) for the semilinear theorem."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the Schauder fixed-point method for finite-approximate controllability of evolution equations, which the semilinear proof follows."},{"cited_title":"Finite-approximate controllability of evolution systems via resolvent-like operators","cited_arxiv_id":"1806.06930","evidence_quote":"Provides the resolvent-like operator construction for finite-approximate controllability, which the paper adapts to impulsive systems."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the generalized Arzelà–Ascoli theorem used to prove compactness of the fixed-point operator G_α."},{"cited_title":"E., & Mahmudov, N","cited_arxiv_id":null,"evidence_quote":"Defines the concepts of controllability and the resolvent operator condition for linear systems that are extended in the present paper."}],"review_version":1}