The existence and controllability of nonautonomous system influenced by impulses on both state and control

Garima Gupta; Jaydev Dabas

REVIEW 3 major objections 4 minor 1 cited by

The existence and controllability of nonautonomous system influenced by impulses on both state and control

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2412.01355 v1 pith:L6CESFQ4 submitted 2024-12-02 math.OC

Garima Gupta , Jaydev Dabas This is my paper · ORCID

classification math.OC MSC 35A1237L0593C2793B0593C10

keywords nonautonomousintegro-differentialequationsimpulsivesystemsapproximatecontrollabilitymildsolutionsevolutionfamilyresolventoperatorsHilbertspacesemilinearcontrol

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The 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.

What'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.

The 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.

The 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.

Recommendation: 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.

Best.

Referee Report

3 major / 4 minor

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.

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 (3)

[Section 3, Theorem 3.1, condition (3.10)] 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 3, Theorem 3.1, Step 2] 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 4, Application] 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.

minor comments (4)

[Section 2.2, adjoint operator M*] 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 3, Theorem 3.1 definitions] 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 3, Step 1 estimate of u(s)] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the existence and controllability proofs rest on explicit hypotheses, explicit control construction, and external fixed-point and resolvent results.

full rationale

The paper's central claims are not obtained by assuming their own conclusions. Theorem 3.1 is proved by Krasnoselskii's fixed point theorem with explicit estimates for F1 and F2; the contraction and compactness steps do not presuppose the controllability conclusion. Theorem 3.2 constructs a control explicitly via formula (3.13) and then derives xλ(b) - h = λ(λI + Θ + Γ + Θ̃ + Γ̃)^{-1} g(xλ(.)). Convergence is obtained from assumption (A1), which is a standard resolvent condition for the associated linear system and is stated as an explicit hypothesis, not as a consequence of the semilinear theorem. The equivalences in Theorem 2.1 invoke external results by Pazy and Mahmudov, and the only self-citation, reference [1], appears in the introduction as background and is not used in any proof. The serious concern that Theorem 3.1's condition max{M,L}<1 cannot be satisfied because Definition 2.1 and Lemma 2.1 force M ≥ 1 (since U(s,s)=I), and that N is positive, is a correctness and feasibility problem about empty hypotheses, not a circularity in the derivation chain.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard fixed point theorems and on the a priori assumption that the linearized system is approximately controllable. No genuinely new entities are introduced. The most fragile assumptions are the unsatisfiable contraction condition M<1 and the unverified condition (A1) in the example.

assumptions (6)

domain assumption Assumptions (R1)-(R4) on the family {A(t)}: closed operators with common dense domain, resolvent bound ‖R(λ,A(t))‖ ≤ K/(|λ|+1) for Reλ≤0, Hölder continuity of A(t)A^{-1}(τ), and compact resolvent.
Used to construct the evolution family U(t,s) via Lemma 2.1 and to obtain compactness via Lemma 2.2.
domain assumption Assumption (A1): λ(λI + Θ + Γ)^{-1} converges strongly to the zero operator as λ→0+.
This is the statement that the linear system is approximately controllable. It is assumed for the semilinear theorem and never verified in the example.
domain assumption Assumptions (A2)-(A3): f and ξ are continuous, Lipschitz in the state, and uniformly bounded.
These ensure the nonlinear terms behave well enough for the fixed point argument.
standard math Krasnoselskii's fixed point theorem.
Invoked in Lemma 2.3 and used in Theorem 3.1 as the main existence tool.
standard math Equivalence (a)⇔(c) of Theorem 2.1 is taken from [15] (Mahmudov).
The paper says this part is 'clear from [15]' and does not reprove it.
domain assumption The example's operator A(t) satisfies (R1)-(R4) and the linear system satisfies (A1).
Referred to as 'the application section of [17]' and asserted without verification.

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Pith. "Pith review of The existence and controllability of nonautonomous system influenced by impulses on both state and control." pith.science (2026). https://pith.science/paper/L6CESFQ4

@misc{pith2026241201355,
  author       = {Pith},
  title        = {Pith review of: The existence and controllability of nonautonomous system influenced by impulses on both state and control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L6CESFQ4}},
  note         = {Machine review of arXiv:2412.01355}
}

read the original abstract

This paper examines impulsive controls related to nonautonomous impulsive integro-differential equations in Hilbert space, highlighting their significance. We establish the existence of the mild solution by using fixed point approach and present conditions for approximate controllability using impulsive resolvent operators and the adjoint problem, supported by an illustrative example.

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Forward citations

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Existence and ap proximate control- lability of non-autonomous functional impulsive evolution inclusions in b anach spaces

Sumit Arora, Manil T Mohan, and Jaydev Dabas. Existence and ap proximate control- lability of non-autonomous functional impulsive evolution inclusions in b anach spaces. Journal of Diﬀerential Equations , 307:83–113, 2022

work page 2022

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Javad A Asadzade and Nazim I Mahmudov. Approximate controllab ility of impulsive semilinear evolution equations in hilbert spaces. arXiv preprint arXiv:2411.02766 , 2024

arXiv 2024

[3] [3]

Solvability and Optimal Controls of Impulsive Stochastic Evolution Equations in Hilbert Spaces

Javad A Asadzade and Nazim I Mahmudov. Solvability and optimal controls of impulsive stochastic evolution equations in hilbert spaces. arXiv preprint arXiv:2407.13496 , 2024

work page Pith review arXiv 2024

[4] [4]

Analysis and control of nonlinear inﬁnite dimensional systems

Viorel Barbu. Analysis and control of nonlinear inﬁnite dimensional systems. (No Title), 1993

work page 1993

[5] [5]

Controllability and stabilization of parabolic equations

Viorel Barbu. Controllability and stabilization of parabolic equations . Springer, 2018

work page 2018

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TA Burton. A ﬁxed-point theorem of krasnoselskii. Applied Mathematics Letters, 11(1): 85–88, 1998

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[7] [7]

Ergodicity for inﬁnite dimensional systems , volume 229

Giuseppe Da Prato and Jerzy Zabczyk. Ergodicity for inﬁnite dimensional systems , volume 229. Cambridge university press, 1996

work page 1996

[8] [8]

Semilinear functional diﬀerential equations in bana ch space

WE Fitzgibbon. Semilinear functional diﬀerential equations in bana ch space. Journal of Diﬀerential Equations , 29(1):1–14, 1978

work page 1978