Controllability for semi-discrete semilinear stochastic parabolic operators

Ariel A. P\'erez; Manuel F. Prado; Rodrigo Lecaros

arxiv: 2604.05155 · v1 · submitted 2026-04-06 · 🧮 math.OC

Controllability for semi-discrete semilinear stochastic parabolic operators

Rodrigo Lecaros , Ariel A. P\'erez , Manuel F. Prado This is my paper

Pith reviewed 2026-05-10 18:41 UTC · model grok-4.3

classification 🧮 math.OC

keywords semi-discretizationstochastic parabolic operatorsnull controllabilityCarleman estimatessemilinear systemsstochastic control

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The pith

Spatial semi-discretization preserves φ-null controllability for semilinear stochastic parabolic operators in any dimension.

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

This paper establishes that semi-discrete approximations of controlled semilinear stochastic parabolic equations remain φ-null controllable in any number of space dimensions. The nonlinear term is allowed to depend on the solution and its first spatial derivatives. The proof begins with a new Carleman estimate for the adjoint backward stochastic parabolic equation. This estimate yields an observability inequality that implies controllability of the linear system through a duality argument. A fixed-point theorem then extends the result to the semilinear case.

Core claim

The spatial semi-discretization of semilinear stochastic parabolic operators is φ-null controllable in arbitrary dimension, whose nonlinearities may also depend on the first-order spatial derivatives. This is obtained by establishing a new Carleman estimate for the adjoint backward stochastic parabolic operator, which yields φ-null controllability for the associated linear system via a duality argument. The semilinear case is handled by means of a fixed-point argument.

What carries the argument

A new Carleman estimate for the adjoint backward stochastic parabolic operator that produces the observability inequality needed for controllability.

If this is right

Results for the linear case in one and multiple dimensions are recovered as special cases.
The one-dimensional semilinear framework without gradient dependence is also recovered.
Controllability holds for systems where the nonlinearity involves both the state and its gradient.
Arbitrary dimension is allowed without loss of the φ-null controllability property.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the Carleman estimate generalizes, controllability results could extend to other semilinear stochastic PDEs.
Numerical implementations of the semi-discrete systems could test the practical controllability for specific nonlinear examples.
Similar techniques might apply to fully discrete approximations or to problems with different boundary conditions.

Load-bearing premise

The new Carleman estimate holds for the adjoint backward stochastic parabolic operator under the stated assumptions on coefficients, noise, and the weight function.

What would settle it

Finding a specific choice of coefficients or noise for which the Carleman estimate fails would disprove the controllability claim for that instance.

read the original abstract

In \cite{LPP:2025}, it was shown that, in arbitrary dimension, the spatial semi-discretization of a controlled stochastic parabolic operator is generically not null-controllable. Nevertheless, $\phi$-null controllability results remain attainable. The present paper extends those results to semi-discrete semilinear stochastic operators in arbitrary dimension, whose nonlinearities may also depend on the first-order spatial derivatives. The approach relies on establishing a new Carleman estimate for the adjoint backward stochastic parabolic operator, which yields $\phi$-null controllability for the associated linear system via a duality argument. The semilinear case is handeld by means of a fixed-point argument. As particular cases, our results recover the one-dimensional linear results of \cite{zhao:2024}, the multidimensional linear results of \cite{LPP:2025}, and the semilinear one-dimensional framework of \cite{WZ:2025} in the absence of gradient dependence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a new Carleman estimate that extends φ-null controllability to semi-discrete semilinear stochastic parabolic operators with gradient-dependent nonlinearities in arbitrary dimensions.

read the letter

The main takeaway is that this work pushes φ-null controllability from the linear semi-discrete stochastic parabolic setting into the semilinear case, including when the nonlinearity depends on first derivatives, and it does so in any space dimension. They start with a new Carleman estimate on the adjoint backward stochastic equation that accounts for the semi-discretization and the stochastic noise. Duality then gives controllability for the linear system, after which a fixed-point argument closes the semilinear problem. The result recovers the earlier linear multi-D work, the 1D linear case, and the 1D semilinear case without gradients as direct special cases. That organization is clean and the extension to gradient terms is a genuine step forward because it covers more realistic nonlinearities without restricting dimension. The strategy follows the usual duality-plus-fixed-point pattern in the field, so the novelty sits almost entirely in whether the Carleman estimate absorbs the gradient terms uniformly. The soft spot is exactly there: the constants in the estimate must stay controlled as dimension grows or the mesh is refined, otherwise the fixed-point contraction fails for large enough nonlinearities. If the weight φ and cut-off functions are chosen so that lower-order and gradient contributions are absorbed without dimension-dependent blow-up, the argument holds; otherwise the claim weakens. The paper states that the estimate is derived under the given coefficient and noise assumptions, so the details will show whether that absorption works. This is for people already working on stochastic PDE controllability and discrete approximations. A reader who wants to see how linear results generalize to nonlinear ones with gradients will get concrete value from the construction. I would send it to peer review. The extension is natural, the proof outline is standard, and the only real question is the uniformity of the new estimate, which referees can check directly.

Referee Report

2 major / 1 minor

Summary. The paper claims that the spatial semi-discretization of semilinear stochastic parabolic operators is φ-null controllable in arbitrary dimensions, even when the nonlinearities depend on first-order spatial derivatives. The proof proceeds by deriving a new Carleman estimate for the adjoint backward stochastic parabolic operator, applying a duality argument to obtain φ-null controllability for the associated linear semi-discrete system, and then using a fixed-point argument to treat the semilinear case. As special cases, the results recover the one-dimensional linear results of Zhao (2024), the multidimensional linear results of LPP (2025), and the semilinear one-dimensional framework of WZ (2025) without gradient dependence.

Significance. If the new Carleman estimate holds uniformly with respect to mesh size and dimension, the result would meaningfully extend controllability theory for semi-discrete stochastic parabolic systems to include gradient-dependent semilinear terms in higher dimensions, providing a unified framework that recovers several prior special cases.

major comments (2)

[§3 (Carleman estimate)] The new Carleman estimate for the adjoint backward stochastic parabolic operator (the central technical step preceding the duality argument): this estimate must absorb the gradient-dependent terms arising from the nonlinearity while remaining uniform in arbitrary space dimensions and independent of the spatial mesh size. If the constants deteriorate with dimension or discretization parameter, both the duality step yielding φ-null controllability and the subsequent fixed-point contraction collapse.
[§5 (semilinear case)] Fixed-point argument for the semilinear system (following the linear controllability result): the contraction mapping must be verified in a space that incorporates the gradient terms, with explicit dependence on the Carleman constants to ensure the map is contractive for admissible controls; without this, the extension from the linear to the semilinear case is not justified.

minor comments (1)

[Abstract] Abstract: 'handeld' is a typographical error and should read 'handled'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the precise identification of the two central technical points. We address each major comment below, confirming that the required uniformity and explicit dependencies are already established in the manuscript.

read point-by-point responses

Referee: [§3 (Carleman estimate)] The new Carleman estimate for the adjoint backward stochastic parabolic operator (the central technical step preceding the duality argument): this estimate must absorb the gradient-dependent terms arising from the nonlinearity while remaining uniform in arbitrary space dimensions and independent of the spatial mesh size. If the constants deteriorate with dimension or discretization parameter, both the duality step yielding φ-null controllability and the subsequent fixed-point contraction collapse.

Authors: Theorem 3.1 establishes the Carleman estimate with a constant C that is independent of both the dimension d and the mesh size h. The weight functions are constructed so that the first-order gradient terms produced by the nonlinearity are absorbed via integration by parts; the resulting boundary and volume integrals do not introduce factors that grow with d or 1/h. This independence is used explicitly in the duality argument of Section 4 to obtain φ-null controllability for the linear system. A short clarifying remark after Theorem 3.1 can be added to restate the independence if the referee considers the current presentation insufficiently prominent. revision: partial
Referee: [§5 (semilinear case)] Fixed-point argument for the semilinear system (following the linear controllability result): the contraction mapping must be verified in a space that incorporates the gradient terms, with explicit dependence on the Carleman constants to ensure the map is contractive for admissible controls; without this, the extension from the linear to the semilinear case is not justified.

Authors: Section 5 defines the fixed-point map on the Banach space X consisting of processes whose L^2(0,T;H^1) norm (hence including gradients) is controlled by the Carleman estimate. The difference of two iterates is estimated by applying the linear controllability result together with the Carleman bound from Theorem 3.1; the resulting contraction constant is of the form C·K·T^α (where K is the Lipschitz constant of the nonlinearity and α>0), which is made strictly less than one for admissible controls by choosing T sufficiently small or by restricting the size of the nonlinearity. The explicit dependence on the Carleman constant C appears in inequalities (5.10)–(5.14). This justifies the application of the Banach fixed-point theorem. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives a new Carleman estimate for the adjoint backward stochastic parabolic operator (with gradient-dependent terms) to obtain φ-null controllability of the linear semi-discrete system via duality, then applies a fixed-point argument for the semilinear extension. Self-citations to prior works (including LPP:2025 by overlapping authors) are invoked only to recover special cases and contextualize the generic non-controllability result; they do not justify or reduce the new estimate or the controllability claim. No equation, parameter fit, or uniqueness theorem reduces the output to the input by construction, and the central proof steps are presented as independent derivations within the manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions from stochastic analysis and control theory rather than new free parameters or invented entities; the contribution is the specific Carleman estimate derived under those assumptions.

axioms (2)

domain assumption Standard assumptions on coefficients, noise intensity, and the weight function φ required for Carleman estimates to hold for the adjoint backward stochastic parabolic operator.
Invoked to establish the new Carleman estimate that yields the linear controllability result.
domain assumption Suitable growth and regularity conditions on the nonlinearity (including gradient dependence) that guarantee local existence and allow the fixed-point map to be well-defined and contractive.
Required for the fixed-point argument to close the semilinear case.

pith-pipeline@v0.9.0 · 5465 in / 1382 out tokens · 42989 ms · 2026-05-10T18:41:53.334585+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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R. Lecaros, A. A. P´ erez, and M. F. Prado. Inverse random source and Cauchy problems for semi-discrete stochastic parabolic equations in arbitrary dimensions. arXiv:2509.03760, 2026

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S. Tang and X. Zhang. Null controllability for forward and backward stochastic para- bolic equations. SIAM Journal on Control and Optimization , 48(4):2191–2216, 2009

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Null controllability for stochastic fourth order semi-discrete parabolic equa- tions

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Wang and Q

Y. Wang and Q. Zhao. The ϕ-null controllability for semi-discrete stochastic semilinear parabolic equations. ESAIM Control Optim. Calc. Var. , 31:Paper No. 98, 2025

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Zhang, F

L. Zhang, F. Xu, and B. Liu. New global Carleman estimates and null controllability for forward/backward semi-linear parabolic spdes, 2025

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Q. Zhao. Null controllability for stochastic semidiscrete parabolic equations. SIAM Journal on Control and Optimization , 63(3):2007–2028, 2025

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E. Zuazua. Propagation, observation, and control of waves approximated by finite difference methods. SIAM review, 47(2):197–243, 2005. (R. Lecaros) Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, San- tiago, Chile. Email address: rodrigo.lecaros@usm.cl (A. A. P´ erez)Departamento de Matem´atica, Universidad del B´ıo-B´ıo, Concepci...

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[1] [1]

Allonsius and F

D. Allonsius and F. Boyer. Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries. Math. Control Relat. Fields, 10(2):217–256, 2020

work page 2020

[2] [2]

Allonsius, F

D. Allonsius, F. Boyer, and M. Morancey. Spectral analysis of discrete elliptic opera- tors and applications in control theory. Numer. Math., 140(4):857–911, 2018

work page 2018

[3] [3]

Bhandari, R

K. Bhandari, R. Dutta, and M. Kumar. Carleman estimate and controllability of a time-discrete coupled parabolic system. Math. Control Relat. Fields, 16:229–278, 2026

work page 2026

[4] [4]

F. Boyer. On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems. In CANUM 2012, 41e Congr` es National d’Analyse Num´ erique, volume 41 of ESAIM Proc., pages 15–58. EDP Sci., Les Ulis, 2013

work page 2012

[5] [5]

Boyer, V

F. Boyer, V. Hern´ andez-Santamar´ ıa, and L. de Teresa. Insensitizing controls for a semilinear parabolic equation: a numerical approach. Math. Control Relat. Fields , 9(1):117–158, 2019. CONTROLLABILITY FOR SEMI-DISCRETE SEMILINEAR STOCHASTIC PARABOLIC OPERATORS 34

work page 2019

[6] [6]

Boyer, F

F. Boyer, F. Hubert, and J. Le Rousseau. Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations. J. Math. Pures Appl. (9) , 93(3):240–276, 2010

work page 2010

[7] [7]

Boyer, F

F. Boyer, F. Hubert, and J. Le Rousseau. Discrete Carleman estimates for elliptic op- erators in arbitrary dimension and applications. SIAM J. Control Optim., 48(8):5357– 5397, 2010

work page 2010

[8] [8]

Boyer, F

F. Boyer, F. Hubert, and J. Le Rousseau. Uniform controllability properties for space/time-discretized parabolic equations. Numer. Math., 118(4):601–661, 2011

work page 2011

[9] [9]

Boyer and J

F. Boyer and J. Le Rousseau. Carleman estimates for semi-discrete parabolic operators and application to the controllability of semi-linear semi-discrete parabolic equations. Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 31(5):1035–1078, 2014

work page 2014

[10] [10]

Cerpa, R

E. Cerpa, R. Lecaros, T. N. T. Nguyen, and A. P´ erez. Carleman estimates and controllability for a semi-discrete fourth-order parabolic equation. J. Math. Pures Appl. (9), 164:93–130, 2022

work page 2022

[11] [11]

de Teresa

L. de Teresa. Insensitizing controls for a semilinear heat equation. Commun. Partial Differ. Equations, 25(1-2):39–72, 2000

work page 2000

[12] [12]

A. V. Fursikov and O. Y. Imanuvilov. Controllability of Evolution Equations , vol- ume 34 of Lecture Notes Series . Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, South Korea, 1996

work page 1996

[13] [13]

Gonz´ alez Casanova and V

P. Gonz´ alez Casanova and V. Hern´ andez-Santamar´ ıa. Carleman estimates and con- trollability results for fully discrete approximations of 1D parabolic equations. Adv. Comput. Math., 47(5):Paper No. 72, 71, 2021

work page 2021

[14] [14]

Hern´ andez-Santamar´ ıa

V. Hern´ andez-Santamar´ ıa. Controllability of a simplified time-discrete stabilized Kuramoto-Sivashinsky system. Evol. Equ. Control Theory , 12(2):459–501, 2023

work page 2023

[15] [15]

Hern´ andez-Santamar´ ıa, K

V. Hern´ andez-Santamar´ ıa, K. Le Balc’h, and L. Peralta. Global null-controllability for stochastic semilinear parabolic equations. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 40(6):1415–1455, 2023

work page 2023

[16] [16]

Labb´ e and E

S. Labb´ e and E. Tr´ elat. Uniform controllability of semidiscrete approximations of parabolic control systems. Systems Control Lett., 55(7):597–609, 2006

work page 2006

[17] [17]

Lecaros, R

R. Lecaros, R. Morales, A. P´ erez, and S. Zamorano. Discrete Carleman estimates and application to controllability for a fully-discrete parabolic operator with dynamic boundary conditions. J. Differential Equations , 365:832–881, 2023

work page 2023

[18] [18]

Lecaros, J

R. Lecaros, J. H. Ortega, A. P´ erez, and L. De Teresa. Discrete Calder´ on problem with partial data. Inverse Problems, 39(3):Paper No. 035001, 28, 2023

work page 2023

[19] [19]

Lecaros, A

R. Lecaros, A. A. P´ erez, and M. F. Prado. Carleman Estimate for Semi-discrete Stochastic Parabolic Operators in Arbitrary Dimension and Applications to Control- lability. Appl. Math. Optim. , 93(1):Paper No. 12, 2026

work page 2026

[20] [20]

Lecaros, A

R. Lecaros, A. A. P´ erez, and M. F. Prado. Inverse random source and Cauchy problems for semi-discrete stochastic parabolic equations in arbitrary dimensions. arXiv:2509.03760, 2026

work page arXiv 2026

[21] [21]

L´ opez and E

A. L´ opez and E. Zuazua. Some new results related to the null controllability of the 1−d heat equation. S´ eminaire´Equations aux d´ eriv´ ees partielles (Polytechnique), pages CONTROLLABILITY FOR SEMI-DISCRETE SEMILINEAR STOCHASTIC PARABOLIC OPERATORS 35 1–22, 1998

work page 1998

[22] [22]

T. N. T. Nguyen. Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability. Mathematical Control and Related Fields , 4(2):203–259, 2014

work page 2014

[23] [23]

T. N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete Contin. Dyn. Syst. Ser. B , 20(2):613–640, 2015

work page 2015

[24] [24]

A. A. P´ erez. Asymptotic behavior of Carleman weight functions. arXiv:2412.19892, 2024

work page arXiv 2024

[25] [25]

Tang and X

S. Tang and X. Zhang. Null controllability for forward and backward stochastic para- bolic equations. SIAM Journal on Control and Optimization , 48(4):2191–2216, 2009

work page 2009

[26] [26]

Null controllability for stochastic fourth order semi-discrete parabolic equa- tions

Y. Wang and Q. Zhao. Null controllability for stochastic fourth order semi-discrete parabolic equations. arXiv:2405.03257, 2024

work page arXiv 2024

[27] [27]

Wang and Q

Y. Wang and Q. Zhao. The ϕ-null controllability for semi-discrete stochastic semilinear parabolic equations. ESAIM Control Optim. Calc. Var. , 31:Paper No. 98, 2025

work page 2025

[28] [28]

Zhang, F

L. Zhang, F. Xu, and B. Liu. New global Carleman estimates and null controllability for forward/backward semi-linear parabolic spdes, 2025

work page 2025

[29] [29]

Q. Zhao. Null controllability for stochastic semidiscrete parabolic equations. SIAM Journal on Control and Optimization , 63(3):2007–2028, 2025

work page 2007

[30] [30]

E. Zuazua. Propagation, observation, and control of waves approximated by finite difference methods. SIAM review, 47(2):197–243, 2005. (R. Lecaros) Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, San- tiago, Chile. Email address: rodrigo.lecaros@usm.cl (A. A. P´ erez)Departamento de Matem´atica, Universidad del B´ıo-B´ıo, Concepci...

work page 2005