Remarks on finite-approximate controllability of impulsive evolution systems via resolvent-like operator in Hilbert spaces

Javad A. Asadzade; Nazim I. Mahmudov

REVIEW 3 major objections 5 minor 1 cited by

Remarks on finite-approximate controllability of impulsive evolution systems via resolvent-like operator in Hilbert spaces

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2501.02995 v2 pith:5SDNSMNI submitted 2025-01-06 math.OC

Javad A. Asadzade , Nazim I. Mahmudov This is my paper · ORCID

classification math.OC MSC 93B0593C2547D0634A37

keywords finite-approximatecontrollabilityimpulsiveevolutionsystemsresolvent-likeoperatorHilbertspaceSchauderfixed-pointtheoremcompactsemigroupapproximateheatequation

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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

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.

The 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.)

One 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.

Could 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.

I 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[Section 4, proof of Theorem 4.2] 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 4, equations (4.11)–(4.12)] 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.
[Theorem 4.2, assumptions versus Theorem 4.1] 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).

minor comments (5)

[Lemma 2.1] The typo 'non-neqative' should read 'nonnegative'.
[Theorem 3.1 proof] In the proof of (iii)⇔(iv), the expression 'lim_{α→0−}' should be 'lim_{α→0+}'.
[Section 2, definition of FA-controllability] The target state is called h in the definition but z_b in the preceding paragraph; the notation should be made consistent.
[Lemma 4.1] 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 5, application] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the semilinear FA-controllability claim is derived by an explicit resolvent-based control construction and standard fixed-point arguments, with prior-work citations used as tools rather than as the conclusion.

full rationale

No circular step is established. Theorem 4.2 constructs, for each alpha>0, a control u_alpha via the resolvent-like operator (alpha(I-pi_D)+Gamma+tildeGamma+Theta+tildeTheta)^-1 and then uses the identity z_alpha(b)=h-alpha(I-pi_D)(...)^-1 sigma_alpha, which is computed from the mild-solution formula rather than assumed. The linear FA-controllability result in Section 3 relies on the equivalence (i)-(iii), cited to [18], and Lemma 3.1(a)-(b) is likewise taken from [18]; these are external prior results used as lemmas, not the paper's own conclusion, so the self-citations do not reduce the central claim to its inputs. Assumption (A6) is an explicit hypothesis representing approximate controllability of the linear part, not a consequence derived from the semilinear theorem. The proof does contain a genuine correctness gap: Theorem 4.1 only gives a fixed point in B_{r(alpha)} with r(alpha) depending on alpha, so the uniform boundedness of {z_alpha(t): alpha>0} used in Lemma 4.1 and in the Banach-Alaoglu extraction is not justified; additionally, the convergence of sigma_{alpha_i} to eta at equation (4.12) is asserted via the Dominated Convergence Theorem from weak L2 convergence, which is not valid without strong convergence. These are mathematical flaws in the proof rather than circularity, because the desired FA-controllability is not assumed or hidden in the hypotheses.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new postulated entities. Its proof rests on standard functional analysis tools and on domain assumptions A1-A6 about the semigroup, nonlinearity, and linear controllability. The key issue is the misuse of dominated convergence in Theorem 4.2, not an extra axiom.

assumptions (9)

domain assumption A1: S(t) is compact for t > 0.
Used in Theorem 4.1 for compactness of the operator G_alpha via Arzela-Ascoli, and in the application to the heat equation.
domain assumption A2: mu(t, .) is continuous and mu(., z) is strongly measurable.
Assumed for the continuity of G_alpha and for the Dominated Convergence Theorem step in the semilinear proofs.
domain assumption A3: Growth condition on mu with liminf_{r to infinity} Lambda(r)/r = d < infinity.
Used to ensure the fixed-point set is invariant and to derive condition (4.2).
domain assumption A4: Omega is linear and continuous.
Standard boundedness assumption for the control operator.
domain assumption A5: Bounded nonlinearity, alternative to A3.
Remark 4.1 notes that if mu is bounded, then d=0 and condition (4.2) holds automatically.
domain assumption A6: The linear system is approximately controllable.
Equivalent to the strong convergence of alpha(alpha I + Gramian)^{-1} to zero, which is used in the final FA-controllability step.
standard math Schauder fixed-point theorem
Used in Theorem 4.1 to prove existence of a mild solution for each alpha.
standard math Banach-Alaoglu theorem
Used in Theorem 4.2 to extract weakly convergent subsequences from bounded sets.
standard math Generalized Arzela-Ascoli theorem
Used in Theorem 4.1 Step 2 to show compactness of G_alpha.

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Pith. "Pith review of Remarks on finite-approximate controllability of impulsive evolution systems via resolvent-like operator in Hilbert spaces." pith.science (2026). https://pith.science/paper/5SDNSMNI

@misc{pith2026250102995,
  author       = {Pith},
  title        = {Pith review of: Remarks on finite-approximate controllability of impulsive evolution systems via resolvent-like operator in Hilbert spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5SDNSMNI}},
  note         = {Machine review of arXiv:2501.02995}
}

read the original abstract

In this manuscript, we examine impulsive evolution systems in Hilbert spaces. Using a resolvent-like operator, we first establish the finite-approximate controllability for linear systems. Subsequently, by applying the Schauder fixed-point theorem (SFPT), we prove the existence of a solution and demonstrate the finite-approximate controllability of semilinear impulsive systems in Hilbert spaces. Finally, we extend these results to a broader application, specifically to the heat equation.

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Forward citations

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Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

T., & Dabas, J

Arora, S., Mohan, M. T., & Dabas, J. (2023). Finite-Approximate Controllability of Impulsive Fractional Functional Evolution Equations of Order 1 < α < 2. Journal of Optimization Theory and Applications, 197(3), 855-890

work page 2023

[2] [2]

A., & Mahmudov, N

Asadzade, J. A., & Mahmudov, N. I. (2024). Approximate contr ollability of Linear Fractional Impulsive Evolution Equations in Hilbert Spaces. arXiv preprint arXiv:2406.1511 4

work page arXiv 2024

[3] [3]

A., & Mahmudov, N

Asadzade, J. A., & Mahmudov, N. I. (2024). Solvability and Optima l Controls of Impulsive Stochastic Evolution Equations in Hilbert Spaces. arXiv preprint arXiv:2407.1349 6. 18

work page arXiv 2024

[4] [4]

A., & Mahmudov, N

Asadzade, J. A., & Mahmudov, N. I. (2024). Approximate contr ollability of impulsive semilinear evolu- tion equations in Hilbert spaces. arXiv preprint arXiv:2411.02766

arXiv 2024

[5] [5]

E., & Mahmudov, N

Bashirov, A. E., & Mahmudov, N. I. (1999). On concepts of cont rollability for deterministic and stochas- tic systems. SIAM Journal on Control and Optimization, 37(6), 18 08-1821

work page 1999

[6] [6]

(1993, June)

Benzaid, Z., & Sznaier, M. (1993, June). Constrained controllab ility of linear impulse diﬀerential sys- tems. In 1993 American Control Conference (pp. 216-220). IEE E

work page 1993

[7] [7]

G., & O’Regan, D

Wang, J., Ibrahim, A. G., & O’Regan, D. (2020). Finite approximate controllability of Hilfer fractional semilinear diﬀerential equations. Miskolc Mathematical Notes, 21(1 ), 489-507

work page 2020

[8] [8]

K., Nandakumaran, A

George, R. K., Nandakumaran, A. K., & Arapostathis, A. (2000) . A note on controllability of impulsive systems. Journal of Mathematical Analysis and Applications, 241( 2), 276-283

work page 2000