{"id":"d7dff8ba-8bd5-42f5-b5e8-94d2b6880fe8","arxiv_id":"2412.01355","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A fixed point and adjoint resolvent argument is used to claim existence and approximate controllability for nonautonomous impulsive integro-differential systems, with an illustrative heat equation example.","lead":"This paper proves existence and approximate controllability theorems for a class of nonautonomous impulsive integro-differential equations in Hilbert spaces. A heat equation example is presented, but the proofs contain gaps that undermine the central claims.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1's contraction condition max{M,L}<1 cannot hold because U(s,s)=I forces the evolution bound M≥1; the existence and controllability claims rest on empty hypotheses.","rationale":"The reader's weakest_assumption is exactly the load-bearing problem: the evolution bound M cannot be strictly less than 1 because U(s,s)=I forces M≥1. I agree with that identification, and the stress-test confirms it is not a minor typo. The condition enters at two independent points: the stated hypotheses (3.9)-(3.10) and the Step 1 choice of r0, which requires 1−N and 1−K1 in denominators. Since M≥1, N>1 and max{M,L}<1 is impossible, so Theorem 3.1 is vacuous and Theorem 3.2, which relies on Theorem 3.1, has no active hypotheses. The example then cannot 'satisfy all conditions' as claimed. I also note the Step 2 contraction proof would need to control the x-dependent control terms inside F1, and it does not, compounding the issue, but the unsatisfiable M<1 condition is sufficient on its own. For these reasons the reader's REJECT verdict should stand unchanged.","tokens_in":18232,"tokens_out":7653,"duration_ms":71593,"concrete_test":"Take any evolution family satisfying Definition 2.1, in particular the explicit family U(t,s)=Σ e^{-n²∫_s^t a(τ)dτ}⟨·,w_n⟩w_n from Section 4, and compute M=sup_{0≤s≤t≤1} ‖U(t,s)‖. Since U(s,s)=I, this supremum is at least 1; substituting M≥1 into (3.10) and into the Step-1 denominator 1−N shows the hypotheses of Theorem 3.1 have no instances. This single check decides whether the existence and controllability theorems have any non-vacuous content.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence theorem is unsupported by any satisfiable hypotheses. Lemma 2.1 defines M by ‖U(t,s)‖≤M for 0≤s≤t≤b, but Definition 2.1 requires U(s,s)=I, so ‖U(s,s)‖=1 and hence M=sup‖U(t,s)‖≥1. Therefore condition (3.10), max{M,L}<1, cannot be met; moreover, the stated condition (3.9) also includes N=M+M^3 M_B^2 b/λ, which is >1 for every λ>0. This is not cosmetic: in the proof of Theorem 3.1 the choice of r0 in Step 1 divides by 1−N and 1−K1, and Step 2 concludes contraction from M<1. Independently, the Step 2 contraction estimate omits the x-dependent control terms u(s) and v_k appearing inside F1 through φ̂λ=(λI+Θ+Γ+Θ̃+Γ̃)^{-1}g(x); those terms must be included in any Lipschitz estimate for F1. Since Theorem 3.2 assumes the conditions of Theorem 3.1, the approximate-controllability conclusion inherits the empty hypothesis. The Section 4 example claims all conditions are satisfied but never computes M or verifies (A1), and its explicit evolution family U(t,s)=Σ e^{-n²∫_s^t a(τ)dτ}⟨g,w_n⟩w_n has U(s,s)=I. Thus the central claims are not established by the presented argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies a nonautonomous impulsive integro-differential equation in Hilbert space, with impulses affecting both the state and the control. The authors formulate a linear nonautonomous impulsive system, introduce adjoint resolvent operators, and prove an approximate controllability criterion for the linear system (Theorem 2.1). They then attempt to prove existence of mild solutions for the semilinear system (Theorem 3.1) via Krasnoselskii's fixed point theorem, and approximate controllability (Theorem 3.2) under an additional hypothesis (A1). The paper closes with a heat-equation example. The central claims are Theorem 3.1 and Theorem 3.2, which depend on numerical hypotheses (3.9) and (3.10) involving the evolution family bound M.","tokens_in":18524,"tokens_out":5362,"duration_ms":46065,"significance":"If correct, the paper would extend approximate controllability results to nonautonomous impulsive integro-differential equations with impulses on both state and control, a meaningful and currently active direction. The linear-system part (Theorem 2.1) follows a standard adjoint/resolvent approach and is coherent. However, the main existence theorem rests on hypotheses that are unsatisfiable in general, specifically the requirement max{M,L}<1 with M the evolution family bound, which must satisfy M>=1 because U(s,s)=I. Consequently Theorem 3.1 has no admissible case, and Theorem 3.2, which explicitly assumes the conditions of Theorem 3.1, inherits this failure. The paper therefore does not establish its central existence or controllability results.","major_comments":[{"comment":"The condition max{M,L}<1 is impossible for any evolution family. Lemma 2.1(1) states that ||U(t,s)|| <= M for 0 <= s <= t <= b, and Definition 2.1 requires U(s,s)=I, so ||U(s,s)||=1 and hence M>=1. The proof of Theorem 3.1 uses exactly this M in Step 2 to conclude that F1 is a contraction by requiring \"M<1\". Thus the hypotheses of Theorem 3.1 are empty, and the existence claim is not established. Additionally, the condition (3.9), with N defined as M + M^3 M_B^2 b/lambda, is also always at least 1 for every lambda>0 because M>=1, so the denominators 1-N and 1-K1 in the choice of r0 in Step 1 are nonpositive or zero. The theorem therefore has no satisfiable hypotheses.","section":"Section 3, Theorem 3.1, condition (3.10)"},{"comment":"The contraction estimate for F1 omits all control-dependent terms. The operator F1 as defined in Step 1 contains terms involving u(s) and v_k, and these controls are defined through (3.8) with phi_hat_lambda = (lambda I + Theta + Gamma + Theta_tilde + Gamma_tilde)^{-1} g(x(.)), so they depend on x. The displayed bound for ||F1x - F1y|| estimates only the x0 term and the term containing f and xi, while the terms U(t,t_k) sum_i ... integral U(t_i,s) B u(s) ds and U(t,t_k) sum_i ... E_{i-1} v_{i-1} + U(t,t_k) E_k v_k are dropped without justification. The resulting Lipschitz constant L is therefore not the Lipschitz constant of F1, and the conclusion that F1 is a contraction on B_r is unsupported. A correct estimate would have to include lambda^{-1} factors and resolvent norms, which would not be bounded by 1 under any stated condition.","section":"Section 3, Theorem 3.1, Step 2"},{"comment":"The application claims that \"all the conditions are satisfied\" for system (4.15), but it never verifies the numerical hypotheses (3.9) and (3.10). In particular, no bound M is computed, the Lipschitz constants Lf and Lxi are stated but the composite constant L is not evaluated, and the condition (A1) is not checked. For the explicit evolution family U(t,s)g = sum_n e^{-n^2 int_s^t a(tau) dtau} <g,w_n> w_n presented in Section 4, one necessarily has U(t,t)=I, so any valid global bound M satisfies M>=1. Since (3.10) requires max{M,L}<1, the example cannot satisfy the theorem's hypotheses. Thus the illustrative example does not substantiate the results.","section":"Section 4, Application"}],"minor_comments":[{"comment":"The displayed formula for M* has index inconsistencies: the term B*U*(tk,t)(I+D*_k) prod_{i=k+1}^m U*(ti,t_{i-1})(I+D*_i)U*(b,t_m) mixes k and i subscripts, and the product limits are written inconsistently. The formula should be checked and written with uniform indices.","section":"Section 2.2, adjoint operator M*"},{"comment":"The symbol N is used both for the quantity N = M + M^3 M_B^2 b/lambda in condition (3.9) and for N = sum_{i=1}^k C_i in the definition of L and in the proof. This collision makes the hypotheses and the proof ambiguous and should be resolved by using distinct symbols.","section":"Section 3, Theorem 3.1 definitions"},{"comment":"In the estimate of ||u(s)|| for t0 < t <= t1, the resolvent (lambda I + Gamma^{t1}_0)^{-1} appears, whereas controls for the whole system are defined through (lambda I + Theta + Gamma + Theta_tilde + Gamma_tilde)^{-1}. The notation is inconsistent and the displayed formula is not used later; the authors should align the notation or remove the redundant estimate.","section":"Section 3, Step 1 estimate of u(s)"},{"comment":"There are numerous typographical errors and awkward phrasings, for example \"Impusive\" in the abstract, \"esists\" for \"exists\", \"Supose\" for \"Suppose\", and inconsistent use of T(t_j - t_{j-1}) versus U(t_j,t_{j-1}) in the proof of Theorem 2.1. A careful editorial revision is needed.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The central existence theorem has unsatisfiable hypotheses (M>=1 contradicts max{M,L}<1), and the contraction argument omits essential control-dependent terms. Because Theorem 3.2 inherits the hypotheses of Theorem 3.1, the paper's main results are not established. The linear-system analysis is standard and the problem choice is reasonable, but the current manuscript would require a fundamentally revised existence proof and example to be reconsidered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe short version: the linear controllability machinery here is a genuine, if routine, extension of Mahmudov's impulsive resolvent approach, but the semilinear existence theorem is not supported. The condition max{M,L}<1 in Theorem 3.1 cannot be satisfied: the evolution family satisfies U(s,s)=I, so any uniform bound M must be at least 1. The contraction argument for F1 also drops the control terms that depend on x. The central claims rest on empty hypotheses.\n\nWhat's actually new: combining nonautonomous A(t), an integral term, and impulses on both state and control is a real combination and is not in the cited literature. The decomposition of the operator MM* into Θ, Γ, Θ̃, Γ̃ and the equivalence in Theorem 2.1 are a faithful extension of Mahmudov's results for the linear impulsive system. The example, a heat equation with time-dependent diffusion coefficient and explicit evolution family, is appropriate in spirit.\n\nThe soft spots are serious. First, (3.10) asks for max{M,L}<1. Since U(s,s)=I, ‖U(s,s)‖=1, so M≥1. This is not a typo: Step 2 uses M<1 to get contraction, and Step 1 divides by 1-N and 1-K1, where N≥M>1. Second, the contraction estimate for F1 on the intervals after the first neglects the terms Bu(s) and E_k v_k, which depend on x through φ̂λ. Those terms belong in the Lipschitz estimate. Third, the application section never computes M or checks (A1); it simply asserts all conditions hold, which is not verified.\n\nThe linear section might be useful to someone working on impulsive controllability, but as written the existence theorem has no satisfiable hypotheses and the controllability theorem inherits that. This is not a matter of a missing technical assumption; the main argument is broken.\n\nRecommendation: I would not cite it, and I would not send it to a serious referee. The authors should fix the existence proof before this is publishable.\n\nBest.","headline":"The semilinear existence theorem is built on an impossible contraction condition (M<1 contradicts U(s,s)=I), so despite a legitimate linear extension, the main claims are unsupported.","tokens_in":19031,"tokens_out":3809,"would_cite":false,"duration_ms":32473,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A12","37L05","93C27","93B05","93C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a semilinear nonautonomous integro-differential system in Hilbert space, with impulses acting on both the state and the control, has at least one mild solution and, under an extra resolvent condition, is…","keywords":["nonautonomous integro-differential equations","impulsive systems","approximate controllability","mild solutions","evolution family","impulsive resolvent operators","Hilbert space","semilinear control systems"],"falsifier":"Compute sup_{0≤s≤t≤b} ‖U(t,s)‖ for any evolution family satisfying Definition 2.1, for instance the heat kernel in Section 4: the initial-condition property U(t,t)=I forces this supremum to be at least 1, so the theorem's condition max{M,L}<1—and, since N contains M, also max{N,K1}<1—cannot be met, and this single calculation shows the hypotheses of Theorem 3.1 have no concrete instance.","tokens_in":18015,"feed_emoji":"🎯","tokens_out":14381,"duration_ms":121898,"temperature":0.7,"pith_summary":"This paper studies a time-dependent integro-differential equation in Hilbert space whose state jumps at prescribed instants by a rule that mixes the pre-jump state with a separate impulsive control. Its aim is to prove two things: that for every regularizing parameter λ>0 there is a mild solution—a solution of the associated integral equation rather than a classical differentiable one—and that the system is approximately controllable, meaning every target state can be approached arbitrarily closely by choosing the continuous control together with the impulse controls. The proof splits the solution operator into a contraction part and a compact part, then uses the adjoint of the input-to-state map to build controls whose terminal error vanishes as λ tends to zero. If correct, this extends approximate-controllability criteria from linear impulsive equations to semilinear integro-differential equations with time-varying generators.","feed_headline":"Impulsive controls make nonautonomous systems approximately controllable","feed_subtitle":"State jumps at fixed times add extra control inputs; the paper proves mild solutions exist and targets are nearly reachable.","key_machinery":"The carrying mechanism is the evolution family U(t,s), the two-parameter propagator of the time-dependent linear equation, whose global norm bound M appears in every estimate. Impulses are encoded as products ∏(I+D_j), which concatenate the state jumps, plus the summands E_k v_k, which place control inputs at the jump instants. The controllability criterion is built around the nonnegative operator Θ+Γ—a Gramian-type object formed from B, E_k, the evolution family, and the jump products—and the control is constructed from its adjoint through the resolvent (λI+Θ+Γ)^{-1}. Existence is handled by a fixed-point theorem that splits the solution equation into a contraction F1 and a compact continuous map F2; the contraction property of F1 is exactly the step that requires the numerical bounds involving M and the Lipschitz constants of the nonlinearities to be less than 1.","core_discovery":"On its own terms, the central claim is Theorem 3.1 and Theorem 3.2. Under conditions (R1)–(R4), (A2)–(A3), and the numerical inequalities max{N,K1}<1 and max{M,L}<1, the semilinear system (1.2) has at least one mild solution in PC([0,b];H)—the space of piecewise-continuous H-valued functions with jumps at the t_k—for each λ>0 and each target h; if, in addition, the strong-resolvent assumption (A1) holds, the system is approximately controllable. The mild solution is written in closed form by the evolution family U(t,s), with the jump rule at each t_k entered through products of (I+D_j) and the impulse controls E_k v_k entering as separate summands. Controllability is formulated through the operator M of the linear system: approximate controllability is equivalent to the strict positivity of the nonnegative operator Θ+Γ and to the strong convergence of λ(λI+Θ+Γ)^{-1} to zero as λ→0+. The proof then shows that the control built from M* makes the terminal error x_λ(b)−h equal to that resolvent applied to a remainder, and compactness of U(t,s) forces the remainder to vanish in the limit.","pith_inferences":["The resolvent identity that produces the error estimate is independent of the contraction step used to prove existence; a different fixed-point or topological existence argument could in principle supply the mild solution while keeping the controllability conclusion unchanged.","A numerical discretization of the heat example would let one estimate the practical rate at which ‖x_λ(b)−h‖ tends to zero as λ→0, giving a quantitative sense of how close approximate controllability is for finite time horizons.","Because the impulse controls enter the Gramian through additional nonnegative terms, adding more impulse times or larger E_k should monotonically enlarge the effective Gramian; comparing reachable sets with and without E_k would test directly what the discrete controls contribute."],"forward_implications":["Approximate controllability would hold for semilinear nonautonomous integro-differential systems with the impulse law ∆x(t_k)=D_k x(t_k)+E_k v_k, not merely for the linear problem on which the resolvent operator was first studied.","The control law is explicit: choose λ, solve the adjoint-type expression for the target, feed the resulting continuous control and impulse controls into the system, and the terminal miss is controlled by λ times a resolvent operator acting on the nonlinear remainder.","Controllability becomes a checkable operator condition: strict positivity of Θ+Γ, or strong convergence of λ(λI+Θ+Γ)^{-1} to zero, is equivalent to approximate controllability of the linearized system and is the hypothesis used for the semilinear one.","The method is stated to extend to second-order systems, so the same impulse-on-control structure is expected to carry over to hyperbolic or beam-type models."],"supporting_citations":[{"why":"Supplies the definition and existence theory of the evolution family U(t,s), including the global bound M used throughout the estimates.","marker":"[16]"},{"why":"Provides the fixed-point theorem used to split the solution operator into a contraction plus a compact map.","marker":"[6]"},{"why":"Gives the compactness result for the evolution family that is used to prove the relative compactness of the integral operator F2.","marker":"[8]"},{"why":"Establishes the impulsive resolvent operator and the equivalence criteria for approximate controllability of the linear impulsive system that the nonlinear proof extends.","marker":"[15]"},{"why":"Supplies the weak-convergence corollary used to show that the nonlinear remainder g(xλ(.)) converges to the limiting vector ω.","marker":"[13]"},{"why":"Provides the compactness of the operator Q acting on L2 that lets the nonlinear integral terms vanish in the controllability estimate.","marker":"[17]"}],"fun_headline_variants":["State and control impulses enable approximate controllability","Impulsive jumps on both state and control lead to near-reachability","Nonautonomous systems with dual impulses: existence and controllability","Fixed point proof: mild solutions and approximate control under impulses"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the numerical inequality M<1, where M is the global norm bound of the evolution family; every evolution family has U(t,t)=I, so any such bound is at least 1, and the contraction step of the existence proof needs exactly this inequality to close.","fun_headline_variants_meta":{"raw":{"variants":["State and control impulses enable approximate controllability","Impulsive jumps on both state and control lead to near-reachability","Nonautonomous systems with dual impulses: existence and controllability","Fixed point proof: mild solutions and approximate control under impulses"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000246,"raw_usage":{"total_tokens":1493,"prompt_tokens":856,"completion_tokens":637,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":569}},"tokens_in":472,"tokens_out":637,"duration_ms":5895,"temperature":1.0,"reasoning_tokens":569,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:25:44.797286+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute sup_{0≤s≤t≤b} ‖U(t,s)‖ for any evolution family satisfying Definition 2.1, for instance the heat kernel in Section 4: the initial-condition property U(t,t)=I forces this supremum to be at least 1, so the theorem's condition max{M,L}<1—and, since N contains M, also max{N,K1}<1—cannot be met, and this single calculation shows the hypotheses of Theorem 3.1 have no concrete instance.","supporting_citations":[{"cited_title":"Semigroups of linear operators and applications to partial differential equations, volume 44","cited_arxiv_id":null,"evidence_quote":"Supplies the definition and existence theory of the evolution family U(t,s), including the global bound M used throughout the estimates."},{"cited_title":"A fixed-point theorem of krasnoselskii","cited_arxiv_id":null,"evidence_quote":"Provides the fixed-point theorem used to split the solution operator into a contraction plus a compact map."},{"cited_title":"Semilinear functional differential equations in bana ch space","cited_arxiv_id":null,"evidence_quote":"Gives the compactness result for the evolution family that is used to prove the relative compactness of the integral operator F2."},{"cited_title":"A study on approximate controllability of linear impulsive equa- tions in hilbert spaces","cited_arxiv_id":null,"evidence_quote":"Establishes the impulsive resolvent operator and the equivalence criteria for approximate controllability of the linear impulsive system that the nonlinear proof extends."},{"cited_title":"Optimal control theory for infinite dimensional systems","cited_arxiv_id":null,"evidence_quote":"Supplies the weak-convergence corollary used to show that the nonlinear remainder g(xλ(.)) converges to the limiting vector ω."},{"cited_title":"Approximate controllability of a non-autonomous evolution equation in Banach spaces","cited_arxiv_id":"2004.10460","evidence_quote":"Provides the compactness of the operator Q acting on L2 that lets the nonlinear integral terms vanish in the controllability estimate."}],"review_version":1}