Study on Control Problem of a Impulsive Neutral Integro-Differential Equations with Fading Memory

Garima Gupta; Jaydev Dabas

REVIEW 3 major objections 6 minor 19 references

Study on Control Problem of a Impulsive Neutral Integro-Differential Equations with Fading Memory

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper sets out to prove that a class of impulsive neutral integro-differential equations with fading memory in a Banach space is approximately controllable: for every target state and tolerance, some control drives the solution…

desk verdict A plausible abstract controllability theorem undermined by an example that does not satisfy its own key hypothesis. read the letter →

arxiv 2507.16560 v1 pith:MDNCCXIN submitted 2025-07-22 math.OC math.DS

Garima Gupta , Jaydev Dabas This is my paper · ORCID

classification math.OCmath.DS MSC 93C2393B0534K3034G2047H1045J05

keywords approximatecontrollabilityimpulsivesystemsneutralintegro-differentialequationsfadingmemoryresolventfamilyBanachspacedualitymappingfeedbackcontrol

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

Semilinear impulsive neutral integro-differential equations with fading memory model systems whose current rate of change depends on the whole past history and which undergo sudden jumps at fixed times. The paper's goal is to show such systems are approximately controllable: for any target state and any tolerance, a control input exists that steers the solution within the tolerance of the target. It achieves this by constructing a resolvent family for the linearized equation without memory, assembling the contributions of continuous controls and impulse controls into a single nonnegative operator, and then using a fixed-point argument with a feedback control built from the duality map. The core result, Theorem 3.2, states approximate controllability of the semilinear system under hypotheses (H1)--(H3), and the paper applies it to a heat-conduction model with fading memory.

What carries the argument

The load-bearing object is the resolvent family $R(t)$, a strongly continuous family of bounded operators with $R(0)=I$ solving the linear neutral integro-differential equation (2.1); it plays the role that the semigroup plays for ordinary evolution equations. Around it the paper builds the impulsive controllability Gramian $MM^*=\Theta_0^{t_m}+\Gamma_{t_m}^b+\widetilde{\Theta}_0^{t_m}+\widetilde{\Gamma}_{t_m}^b$, which records how distributed controls and impulsive controls affect the final state; strict positivity of this operator is shown to be equivalent to approximate controllability of the linear system. The semilinear argument is carried by the fixed-point map $G_\alpha$, defined on a ball of piecewise-continuous functions, together with the duality map $J:X\to X^*$ used to convert terminal residuals into the feedback control $u_\alpha$. Compactness of $G_\alpha$ comes from compactness of $R(t)$ for $t>0$, and its ball-invariance comes from the uniform bound on the nonlinearity.

What would settle it

Fix a concrete system satisfying (H1)--(H3), set $f\equiv 0$, and compute $MM^*$; if there is a nonzero $\varphi\in X^*$ with $M^*\varphi=0$ (equivalently, with $B^*R^*(b-t)\varphi=0$ for $t\in(t_m,b]$ and the analogous impulse-time conditions), then the linear system is not approximately controllable and the semilinear controllability conclusion cannot hold for that datum.

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Extended reading notes

Core claim

The central claim is Theorem 3.2: under hypotheses (H1)--(H3), system (1.3) is approximately controllable on $J=[0,b]$. The proof represents mild solutions through the resolvent family $R(t)$ of the associated linear integro-differential equation, with impulse effects encoded by products of the form $(I+D_j)R(t_j-t_{j-1})$, and collects every control channel into the operator $MM^*=\Theta_0^{t_m}+\Gamma_{t_m}^b+\widetilde{\Theta}_0^{t_m}+\widetilde{\Gamma}_{t_m}^b$. For each $\alpha>0$, the feedback control $u_\alpha$ is designed so that the terminal state satisfies $x_\alpha(b)=h-\alpha(\alpha I+MM^*J)^{-1}\sigma(x(\cdot))$; a Schauder fixed-point argument supplies a mild solution for every $\alpha$, and hypothesis (H1) makes the residual term vanish as $\alpha\to0^+$. The paper first proves, for the linear system, that approximate controllability is equivalent to strict positivity of this Gramian operator, then extends that conclusion to the semilinear system.

Load-bearing premise

Hypothesis (H3)(c) -- that the nonlinear memory term $f(t,\phi)$ is bounded in norm by an integrable function $\gamma(t)$ independent of the history $\phi$ -- is the load-bearing premise: it gives the fixed-point operator $G_\alpha$ its invariant ball, and without it the existence and controllability argument stops at Step 1.

Editorial extensions

If this is right

Theorem 2.1 reduces approximate controllability of the linear impulsive memory system to a positivity check on the single operator $MM^*$, so the property can be verified without solving the equation.
For each $\alpha>0$ the controlled semilinear system has at least one mild solution, even though no uniqueness is asserted; existence holds for every initial history in the phase space.
If (H1) holds, Theorem 3.2 turns the fixed-point family into a steering procedure: the controls $u_\alpha$ and jump inputs $v_k$ drive the terminal state within $\epsilon$ of any target $h$ as $\alpha\to0^+$.
The paper concludes that the heat-conduction-with-fading-memory model (4.21), with impulses, falls under the theorem once the linearized system satisfies the injectivity condition $B^*R^*(b-t)w^*=0$ forces $w^*=0$.

Reading between the lines

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

The invariant-ball argument relies on the uniform bound (H3)(c), so the theorem's method covers nonlinearities whose size does not grow with the history; extending it to Lipschitz or locally bounded nonlinearities would require a different fixed-point device.
The Gramian structure suggests a computational design rule: evaluating $x_\alpha=\alpha(\alpha I+MM^*J)^{-1}\sigma$ for small $\alpha$ gives both the steering control and a residual that estimates how close the system can actually get to the target.
The linear equivalence in Theorem 2.1 indicates a route to finite-approximate controllability: if only finitely many output functionals must match the target, strict positivity on that finite-dimensional subspace should suffice, though the paper does not prove this version.

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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

Let me give you the short version. The paper combines known resolvent-family techniques to prove a conditional approximate controllability result for semilinear impulsive neutral integro-differential equations with fading memory in reflexive Banach spaces. The abstract theorem (3.2) is standard in structure: assume linear approximate controllability (H1) and a uniform bound on the nonlinearity (H3)(c), then get existence of a fixed point by Schauder and pass to the limit by the usual resolvent arguments. The combination of impulses, neutral memory, and the Banach-space duality mapping is genuinely new, and the linear controllability characterization via the Gramian operator follows the expected pattern. There is real work here, and the main proof seems plausible.

The problem is the application. The Section 4 example defines f(t,φ)(ζ)=∫_{-∞}^0 H(-s)φ(s,ζ)ds, a nonzero linear map in the history φ. Hypothesis (H3)(c) requires ||f(t,φ)||_X ≤ γ(t) for all φ in the phase space B. Since B contains histories of arbitrary size, a nonzero linear map can only satisfy a bound proportional to ||φ||_B, not a uniform one. The paper claims the bound ||f||_{L(B;X)} ≤ K_f, but that is an operator norm, not a pointwise uniform bound. So the example does not fall under Theorems 3.1 and 3.2. If H≡0 the example satisfies H3(c) but loses the memory coupling. This is a real flaw in the only illustration of the abstract results.

A second, smaller issue: the paper leans heavily on conditions (Cd1)-(Cd4) and several lemmas from [6] and [18] without stating them. Lemma 2.1 and Lemma 2.2 are proved only by reference. That makes independent verification needlessly hard. There are also numerous typos.

The abstract theory should be correct under the stated hypotheses; I don't see a hidden circularity or a fundamental gap in the fixed-point argument. But the example is wrong, and that has to be fixed before publication. The authors could either replace the nonlinearity with a bounded one (e.g., saturate the linear map) or restructure the hypothesis and proof. The paper will likely be of interest to researchers who work on resolvent operators and impulsive controllability in Banach spaces. I would not cite it in my own work, but I would send it to a referee rather than desk-reject, because the core theorem is plausible and the flaw is localized to the example.

Referee Report

3 major / 6 minor

Summary. The paper studies approximate controllability of a semilinear impulsive neutral integro-differential equation with infinite memory/fading memory in a separable reflexive Banach space. It constructs an impulsive resolvent-like operator, proves equivalence statements for linear approximate controllability, and then, under hypotheses (H1)-(H3), proves existence of mild solutions via Schauder's fixed point theorem and approximate controllability of the semilinear system. Section 4 presents a heat-conduction example and claims that Theorem 3.2 applies to it.

Significance. If the abstract theorems are correct, the paper is a modest but legitimate extension of the resolvent-like operator technique of Mahmudov to impulsive neutral integro-differential equations with fading memory in Banach spaces. The proof of Theorem 3.2 is conditional on the linear controllability hypothesis (H1) and does not assume the semilinear conclusion, so the reduction is not circular. However, the only application, Section 4, does not satisfy the uniform boundedness hypothesis (H3)(c), because the nonlinearity is a nonzero linear map in the history variable. The claimed illustration is therefore false as written, and the paper's applied contribution is not established.

major comments (3)

[Section 4, Eq. (4.21)] The function f(t, phi)(zeta) = integral_{-infinity}^0 H(-s) phi(s,zeta) ds is linear in phi. Unless H is identically zero, sup_{phi in B} ||f(t,phi)||_X = infinity, so hypothesis (H3)(c) cannot hold for any finite gamma in L^1. The cited bound ||f||_{L(B;X)} <= K_f only gives ||f(t,phi)|| <= K_f ||phi||_B, which is not a uniform-in-phi bound. Consequently the heat-conduction example does not satisfy the hypotheses of Theorems 3.1 and 3.2, and the asserted application is not valid as written.
[Section 2, after Definition 2.1] The manuscript assumes 'conditions (Cd1)-(Cd4) in [6]' without stating them. These conditions are load-bearing: they guarantee existence, strong continuity, exponential boundedness, and compactness of the resolvent family R(·), and the mild solution formula (2.4) and the control estimates (3.10)-(3.12) rely on the norm bound ||R(t)|| <= M_R. The reader cannot verify the hypotheses for the Section 4 model without consulting [6]. Please state the conditions explicitly or formulate resolvent existence and regularity as standing assumptions.
[Section 4, linear controllability verification] The proof of approximate controllability of the linearized system uses the chain B*R*(T-t)*w* = 0 => R*(T-t)*w* = 0 => w* = 0. The first implication requires B* to be injective, which is not implied by the asserted injectivity of B; one needs density of the range of B. The second implication requires R*(t) to be injective, which is not justified anywhere in the paper. Thus hypothesis (H1) is not actually verified for the example.

minor comments (6)

[Section 2] Definition 2.1 is used twice: once for the resolvent operator and again for approximate controllability (labeled 'Definition 2.1 ([18])'). The second should be renumbered, for example as Definition 2.7.
[Theorem 3.2] In the first line of the proof, 'h ∈ H' should be 'h ∈ X'; the space H is not defined in the paper.
[Theorem 3.1, estimate (3.11)] The constant M_X appears in the second inequality and then M_R in the third; M_X is never defined and should be removed or replaced by M_R.
[Theorem 3.1, Step 2] The index of summation is written as 'p' and 'n' inconsistently; the number of impulses is m throughout, so the same symbol should be used everywhere.
[Theorem 3.1 and Eq. (3.9)] The history variable x_tilde_s is used without being defined; state explicitly how x in PC(J;X) is extended to an element of the phase space B before evaluating f(s, x_tilde_s).
[Title and abstract] The title contains typographical spacing artifacts ('A Impulsive', 'F ADING'); these should be corrected to 'An Impulsive' and 'Fading'.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the semilinear controllability theorem is a conditional reduction to the explicit linear controllability hypothesis (H1) and to external resolvent and compactness results; the Section 4 example has a hypothesis-compliance defect, but that is a correctness issue, not a circular one.

full rationale

The derivation chain is not circular. Theorem 3.2 proves semilinear approximate controllability by first assuming (H1), which is exactly the strong convergence of the resolvent-family elements x_alpha(y) to zero, and by Remark 2.3 this is equivalent to approximate controllability of the linear system (2.5). This is a standard and honest reduction: the semilinear theorem does not pretend to prove the linear controllability condition, it takes it as an explicit hypothesis. The fixed-point existence argument (Theorem 3.1) uses (H2)-(H3) and borrowed compactness/equicontinuity verifications from Arora-Nandakumaran [6] and Dos Santos et al. [5], which are external works. The control construction follows the resolvent-like operator method of Mahmudov [18], and the final passage uses Theorem 2.5 of [18] together with the assumed (H1). None of these steps defines the target conclusion into the hypotheses. The paper does contain self-citations: [9], [10], and [14] list earlier work by the same research group, but these appear only in the literature review and are not load-bearing for any proof. A genuine mathematical defect exists in Section 4: the example defines f(t,phi)(zeta)=int_{-infinity}^0 H(-s)phi(s,zeta)ds and states only ||f||_{L(B;X)} <= K_f, whereas (H3)(c) requires a uniform bound ||f(t,phi)|| <= gamma(t) independent of phi. For nonzero H this linear-in-phi map cannot satisfy such a uniform bound, so the application does not meet the theorem's hypotheses. However, this is a failure of the example to satisfy assumptions, not a circularity in the theorem; the proof does not assume its conclusion. The minor self-citations are non-load-bearing, so the circularity score is 1 rather than 0, but the central claim is not circular.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper introduces no fitted constants. It relies on standard functional-analysis theorems and on domain assumptions: the unstated conditions (Cd1)-(Cd4) of [6] that guarantee a resolvent family with the needed properties, the reflexivity and uniform-convexity structure that makes the duality mapping well behaved, and hypotheses H1-H3, one of which (H3(c)) is so strong that the paper's example violates it.

assumptions (8)

domain assumption Conditions (Cd1)-(Cd4) from reference [6] hold.
These conditions guarantee existence, strong continuity, exponential boundedness, and other properties of the resolvent family R(t). They are assumed but not stated in the paper, yet they underlie all estimates involving R(t).
domain assumption X is a separable reflexive Banach space whose dual X* is uniformly convex.
This ensures the duality mapping J is single-valued, bijective, and demicontinuous (Remark 2.2), which is used in the resolvent-like operator and in the proofs of Theorems 2.1 and 2.2.
domain assumption (H1) the linear system is approximately controllable, i.e., x_alpha(y) converges strongly to zero.
This is the standing controllability assumption for the linearized system, carried over to the semilinear theorem.
domain assumption (H2) the resolvent operator R(alpha_0, A) is compact for some alpha_0 in rho(A).
Used to conclude compactness of R(t) and R(t)*, which is essential for the compactness of the fixed-point operator G_alpha and for the weak-to-strong convergence in Theorem 3.2.
domain assumption (H3)(a-b) f is continuous in its second argument and strongly measurable in its first.
Standard Caratheodory-type conditions for the nonlinearity, used in the continuity argument for the fixed-point operator.
domain assumption (H3)(c) f is uniformly bounded by an integrable gamma(t) independent of phi in B.
This strong condition makes the ball B_r invariant in Step 1 of Theorem 3.1, but it is not satisfied by the paper's own example, which defines a nonzero linear f.
standard math Hale-Kato phase space axioms (A1)-(A3).
The functional framework for infinite delay, adapted for impulsive systems, is taken as background.
standard math Standard results from functional analysis: Schauder fixed point theorem, Banach-Alaoglu theorem, Dunford-Pettis theorem, generalized Arzela-Ascoli theorem.
These are the classical tools used in the fixed-point and convergence arguments; the paper relies on them without proof.

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Cite this review

Pith. "Pith review of Study on Control Problem of a Impulsive Neutral Integro-Differential Equations with Fading Memory." pith.science (2026). https://pith.science/paper/MDNCCXIN

@misc{pith2026250716560,
  author       = {Pith},
  title        = {Pith review of: Study on Control Problem of a Impulsive Neutral Integro-Differential Equations with Fading Memory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MDNCCXIN}},
  note         = {Machine review of arXiv:2507.16560}
}

read the original abstract

This article addresses control problems for semilinear impulsive neutral integro-differential equations with memory in a Banach space. It investigates the approximate controllability of linear and semilinear systems and proves the establishment of mild solutions in the semilinear setting. The approach involves constructing a resolvent family for the corresponding integro-differential equation of linear type without memory. The results for the linear system are established first, then extended to the semilinear scenario, followed by a detailed example to illustrate the theoretical findings.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[6]

Arora, A

S. Arora, A. Nandakumaran, Controllability problems of a neutral integro-differential equation with memory, Nonlinear Analysis: Real World Applications 84 (2025) 104317

work page 2025
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[1]

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work page 1990
[4]

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J. Wu, Theory and applications of partial functional differential equations, Vol. 119, Springer Science & Business Media, 2012

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[5]

J. P. C. Dos Santos, H. Henr ´ ıquez, E. Hern´ andez, Existence results for neutral integro-differential equations with unbounded delay, The Journal of Integral Equations and Applications (2011) 289–330

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[7]

Lakshmikantham, P

V. Lakshmikantham, P. S. Simeonov, et al., Theory of impulsive differential equations, Vol. 6, World scientific, 1989

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[8]

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S. Arora, M. T. Mohan, J. Dabas, Approximate controllability of non-instantaneous impulsive fractional evolution equations of order α ∈ (1, 2) with state-dependent delay in banach spaces, arXiv preprint arXiv:2106.15122. 23

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [6]

Arora, A

S. Arora, A. Nandakumaran, Controllability problems of a neutral integro-differential equation with memory, Nonlinear Analysis: Real World Applications 84 (2025) 104317

work page 2025

[2] [18]

N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equa- tions in abstract spaces, SIAM journal on control and optimization 42 (5) (2003) 1604–1622

work page 2003

[3] [1]

M. E. Gurtin, A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Archive for Rational Mechanics and Analysis 31 (1968) 113–126

work page 1968

[4] [2]

J. W. Nunziato, On heat conduction in materials with memory, Quarterly of Applied Mathematics 29 (2) (1971) 187–204

work page 1971

[5] [3]

Lunardi, On the linear heat equation with fading memory, SIAM Journal on Mathematical analysis 21 (5) (1990) 1213–1224

A. Lunardi, On the linear heat equation with fading memory, SIAM Journal on Mathematical analysis 21 (5) (1990) 1213–1224

work page 1990

[6] [4]

Wu, Theory and applications of partial functional differential equations, Vol

J. Wu, Theory and applications of partial functional differential equations, Vol. 119, Springer Science & Business Media, 2012

work page 2012

[7] [5]

J. P. C. Dos Santos, H. Henr ´ ıquez, E. Hern´ andez, Existence results for neutral integro-differential equations with unbounded delay, The Journal of Integral Equations and Applications (2011) 289–330

work page 2011

[8] [7]

Lakshmikantham, P

V. Lakshmikantham, P. S. Simeonov, et al., Theory of impulsive differential equations, Vol. 6, World scientific, 1989

work page 1989