Boundary bilinear control of semilinear parabolic PDEs: quadratic convergence of the SQP method

Eduardo Casas; Mariano Mateos

arxiv: 2505.24237 · v2 · submitted 2025-05-30 · 🧮 math.OC · math.AP

Boundary bilinear control of semilinear parabolic PDEs: quadratic convergence of the SQP method

Eduardo Casas , Mariano Mateos This is my paper

Pith reviewed 2026-05-19 13:51 UTC · model grok-4.3

classification 🧮 math.OC math.AP

keywords bilinear boundary controlsemilinear parabolic equationsequential quadratic programmingquadratic convergenceRobin coefficientsecond-order sufficient conditionsstrict complementarityPDE-constrained optimization

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The pith

The SQP algorithm converges quadratically to local solutions of a boundary bilinear control problem for semilinear parabolic PDEs when started in an L2 neighborhood and the solution satisfies a no-gap second-order condition plus strict,com,

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

This paper studies the problem of controlling a semilinear parabolic equation through the Robin boundary coefficient. The authors derive first-order necessary and second-order sufficient optimality conditions for the resulting optimization problem. They introduce a sequential quadratic programming method and prove that it is stable with quadratic convergence in Lp for finite p and in L∞, provided the target local solution obeys a no-gap second-order sufficient optimality condition and a strict complementarity condition.

Core claim

For the bilinear boundary control of a semilinear parabolic PDE, the sequential quadratic programming algorithm, when initialized sufficiently close in the L2 norm to a local solution that satisfies a no-gap second-order sufficient optimality condition together with strict complementarity, is stable and converges quadratically in the Lp norm for all finite p and in the L∞ norm.

What carries the argument

The sequential quadratic programming (SQP) algorithm that replaces the original nonlinear control problem by a sequence of quadratic subproblems built from the current iterate and the linearized state equation.

If this is right

The algorithm remains stable and converges quadratically in Lp and L∞ when the initialization lies inside a sufficiently small L2 ball around the local solution.
The same quadratic rate holds simultaneously in every Lp space with finite p and in L∞ under the stated second-order conditions.
The proof supplies an explicit radius for the L2 neighborhood from which quadratic convergence is guaranteed.
The result applies directly to the Robin-coefficient control setting without requiring additional regularization of the control variable.

Where Pith is reading between the lines

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

The quadratic convergence guarantee may allow practitioners to stop the iteration after a small number of steps once the residual falls below a chosen tolerance.
The same SQP framework could be tested on related boundary-control problems whose nonlinearity satisfies analogous second-order conditions.
If strict complementarity can be verified a posteriori from the computed solution, the theoretical quadratic rate becomes a practical stopping criterion.

Load-bearing premise

The target local solution must satisfy both a no-gap second-order sufficient optimality condition and a strict complementarity condition.

What would settle it

Run the SQP iteration starting from an L2-nearby point to a local solution that violates strict complementarity and observe whether the convergence rate in L∞ remains quadratic or drops to linear.

read the original abstract

We analyze a bilinear control problem governed by a semilinear parabolic equation. The control variable is the Robin coefficient on the boundary. First-order necessary and second-order sufficient optimality conditions are derived. A sequential quadratic programming algorithm is then proposed to compute local solutions. Starting the iterations from an initial point in an $L^2$-neighborhood of the local solution we prove stability and quadratic convergence of the algorithm in $L^p$ ($p < \infty$) and $L^\infty$ assuming that the local solution satisfies a no-gap second-order sufficient optimality condition and a strict complementarity condition.

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

This paper proves local quadratic convergence for SQP on a semilinear parabolic PDE with bilinear Robin boundary control, under no-gap SOSOC plus strict complementarity.

read the letter

This paper proves local quadratic convergence for SQP on a semilinear parabolic PDE with bilinear Robin boundary control, under no-gap SOSOC plus strict complementarity. The authors first derive the first-order necessary and second-order sufficient optimality conditions for the problem. They then analyze an SQP iteration started from a small L2 ball around a local solution and show stability with quadratic rate in Lp for p finite and in L∞ when the no-gap condition and strict complementarity hold at that solution. The extension to boundary bilinear control is the main new piece. Prior SQP quadratic convergence results exist for distributed controls or other structures, and this work adapts the arguments to the trace operators and the bilinear boundary term while still recovering the stronger-norm convergence. That step requires some care with the function spaces and is handled cleanly. The assumptions are the standard ones for these local analyses, but they are strong. Verifying a no-gap second-order sufficient condition and strict complementarity is rarely straightforward in a concrete problem, and the result gives no information on the size of the basin or behavior from distant starts. The paper stays entirely at the continuous level and does not discuss discretization or numerical realization of the quadratic subproblems. This is a paper for specialists in PDE-constrained optimization who work on convergence theory for iterative methods. Readers already familiar with SQP analysis in infinite dimensions will see the value in the boundary-control case. It deserves a serious referee. The claims are stated precisely, the hypotheses match what is needed to close the estimates, and the setting fills a specific gap without obvious internal contradictions.

Referee Report

1 major / 2 minor

Summary. The manuscript analyzes a bilinear optimal control problem for a semilinear parabolic PDE with the control entering as the Robin boundary coefficient. First-order necessary and second-order sufficient optimality conditions are derived. A sequential quadratic programming (SQP) algorithm is proposed, and local stability together with quadratic convergence of the iterates is proved in L^p (p < ∞) and L^∞, provided the initial guess lies in a sufficiently small L²-neighborhood of a local solution that satisfies a no-gap second-order sufficient optimality condition and a strict complementarity condition.

Significance. If the proofs hold, the result supplies a rigorous local convergence theory for SQP applied to an infinite-dimensional bilinear boundary-control problem. Establishing quadratic rates in both L^p and L^∞ under standard second-order conditions is a technically useful contribution to the numerical analysis of PDE-constrained optimization, especially for applications where boundary Robin control appears.

major comments (1)

[§4, Theorem 4.5] §4, Theorem 4.5: The quadratic convergence statement is conditional on the no-gap SOSC and strict complementarity; while these are the standard hypotheses needed to close the estimate, the manuscript provides no concrete verification or sufficient conditions under which they hold for the specific semilinear parabolic operator with bilinear Robin control, leaving the practical scope of the theorem unclear.

minor comments (2)

[§2] Notation for the state and adjoint equations in §2 could be made more uniform; the bilinear term is introduced in (2.3) but its precise mapping properties are only referenced later.
[Algorithm 3.1] The statement of the SQP subproblem in Algorithm 3.1 would benefit from an explicit display of the quadratic objective and the linearized PDE constraint.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

Referee: [§4, Theorem 4.5] §4, Theorem 4.5: The quadratic convergence statement is conditional on the no-gap SOSC and strict complementarity; while these are the standard hypotheses needed to close the estimate, the manuscript provides no concrete verification or sufficient conditions under which they hold for the specific semilinear parabolic operator with bilinear Robin control, leaving the practical scope of the theorem unclear.

Authors: We agree that the no-gap SOSC and strict complementarity are the key hypotheses under which Theorem 4.5 establishes local stability and quadratic convergence of the SQP iterates. These conditions are standard in the literature on SQP methods for PDE-constrained problems precisely because they close the gap between first- and second-order conditions and permit the quadratic rate in the chosen function spaces. Section 3 of the manuscript already derives the first-order necessary conditions and a second-order sufficient condition for the specific bilinear Robin boundary-control problem; the convergence analysis in §4 then proceeds under the additional no-gap and strict-complementarity assumptions. We do not supply explicit, easily verifiable sufficient conditions guaranteeing that these two assumptions hold for arbitrary data, because such conditions would typically involve smallness restrictions on the nonlinearity or on the target state and would constitute an independent technical development. To clarify the practical scope, we will add a short remark immediately after Theorem 4.5 (and a corresponding sentence in the introduction) that recalls the standard character of the assumptions, points to analogous verifications in related parabolic control papers, and notes that the hypotheses are expected to hold when the semilinear term is locally Lipschitz and the control is sufficiently small in L^∞. This revision addresses the referee’s concern without changing the main theoretical result. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper first derives first-order necessary and second-order sufficient optimality conditions for the boundary bilinear control problem governed by a semilinear parabolic PDE. It then introduces an SQP algorithm and establishes its local stability and quadratic convergence in L^p and L^∞ spaces, conditional on the local solution satisfying a no-gap second-order sufficient optimality condition together with strict complementarity. These hypotheses are standard requirements for closing quadratic convergence estimates in infinite-dimensional optimization and are not shown to reduce to the paper's own fitted quantities or self-citations by construction. No load-bearing step equates a prediction to its input via definition or renaming; the central claims remain independent of the target convergence result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard well-posedness assumptions for semilinear parabolic PDEs and the validity of the derived optimality conditions; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption The semilinear parabolic PDE admits unique solutions for admissible controls in appropriate function spaces
Required to define the control-to-state map and to formulate the optimality conditions.

pith-pipeline@v0.9.0 · 5619 in / 1344 out tokens · 154964 ms · 2026-05-19T13:51:17.055737+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We analyze a bilinear control problem governed by a semilinear parabolic equation... prove stability and quadratic convergence... assuming... no-gap second-order sufficient optimality condition and a strict complementarity condition.
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unclear
Relation between the paper passage and the cited Recognition theorem.

J(u) := ∫_Q L(x,t,y_u) dx dt + ∫_Ω l(x,y_u(T)) dx + (κ/2)∫_Σ u² dx dt

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unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Works this paper leans on

15 extracted references · 15 canonical work pages · 1 internal anchor

[1]

H. Amann. Linear parabolic problems involving measures.Rev. R. Acad. Cien. Serie A. Mat., 95(1):85–119, 2001

work page 2001
[2]

Casas, K

E. Casas, K. Chrysafinos, and M. Mateos. Semismooth Newton method for boundary bilinear control.IEEE Control Syst. Lett., 7:3549–3554, 2023.doi:10.1109/lcsys. 2023.3337747

work page doi:10.1109/lcsys 2023
[3]

Casas, K

E. Casas, K. Chrysafinos, and M. Mateos. Bilinear control of semilinear elliptic PDEs: Convergence of a semismooth Newton method.Numerische Mathematik, 2024.doi: 10.1007/s00211-024-01448-1

work page doi:10.1007/s00211-024-01448-1 2024
[4]

Casas and K

E. Casas and K. Kunisch. Space-timeL ∞-estimates for solutions of infinite horizon semilinear parabolic equations.Commun. Pure Appl. Annal., 24(4):482–506, 2025. doi:10.3934/cpaa.2025002

work page doi:10.3934/cpaa.2025002 2025
[5]

Casas and M

E. Casas and M. Mateos. Quadratic convergence of an SQP method for some optimiza- tion problems with applications to control theory, 2025. To appear in SIAM Journal on Control and Optimization.arXiv:2505.22750

work page internal anchor Pith review arXiv 2025
[6]

Casas, M

E. Casas, M. Mateos, and A. R¨ osch. Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity.Comput. Optim. Appl., 70(1):239–266, 2018.doi:10.1007/s10589-018-9979-0

work page doi:10.1007/s10589-018-9979-0 2018
[7]

Casas and F

E. Casas and F. Tr¨ oltzsch. Second order analysis for optimal control problems: Im- proving results expected from abstract theory.SIAM J. Optim., 22(1):261–279, 2012. doi:10.1137/110840406

work page doi:10.1137/110840406 2012
[8]

P. G. Ciarlet.Introduction ` a l’analyse num´ erique matricielle et ` a l’optimisation. Mas- son, Paris, 1982. Collection Math´ ematiques Appliqu´ ees pour la Maˆ ıtrise

work page 1982
[9]

Dautray and J

R. Dautray and J. Lions.Mathematical Analysis and Numerical Methods for Science and Technology, volume 5. Springer-Verlag, Berlin-Heidelberg-New York, 2000

work page 2000
[10]

Disser, A

K. Disser, A. F. M. ter Elst, and J. Rehberg. H¨ older estimates for parabolic operators on domains with rough boundary.Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 17(1):65–79, 2017.doi:10.2422/2036-2145.201503_013

work page doi:10.2422/2036-2145.201503_013 2017
[11]

Goldberg and F

H. Goldberg and F. Tr¨ oltzsch. On a Lagrange-Newton method for a nonlinear parabolic boundary control problem.Optim. Methods Softw., 8(3-4):225–247, 1998.doi:10. 1080/10556789808805678

work page 1998
[12]

Hoppe and I

F. Hoppe and I. Neitzel. Convergence of the SQP method for quasilinear parabolic optimal control problems.Optim. Eng., 22(4):2039–2085, 2021.doi:10.1007/ s11081-020-09547-2

work page 2039
[13]

Triebel.Interpolation Theory, Function Spaces, Differential Operators

H. Triebel.Interpolation Theory, Function Spaces, Differential Operators. North- Holland, Berlin, 1978

work page 1978
[14]

Tr¨ oltzsch

F. Tr¨ oltzsch. On the Lagrange–Newton–SQP method for the optimal control of semi- linear parabolic equations.SIAM Journal on Control and Optimization, 38(1):294–312, 1999.doi:10.1137/S0363012998341423. Bilinear control problems: convergence of the SQP method39

work page doi:10.1137/s0363012998341423 1999
[15]

Wachsmuth

D. Wachsmuth. Analysis of the SQP-method for optimal control problems governed by the nonstationary Navier–Stokes equations based onL p-theory.SIAM Journal on Control and Optimization, 46(3):1133–1153, 2007.doi:10.1137/S0363012904443506

work page doi:10.1137/s0363012904443506 2007

[1] [1]

H. Amann. Linear parabolic problems involving measures.Rev. R. Acad. Cien. Serie A. Mat., 95(1):85–119, 2001

work page 2001

[2] [2]

Casas, K

E. Casas, K. Chrysafinos, and M. Mateos. Semismooth Newton method for boundary bilinear control.IEEE Control Syst. Lett., 7:3549–3554, 2023.doi:10.1109/lcsys. 2023.3337747

work page doi:10.1109/lcsys 2023

[3] [3]

Casas, K

E. Casas, K. Chrysafinos, and M. Mateos. Bilinear control of semilinear elliptic PDEs: Convergence of a semismooth Newton method.Numerische Mathematik, 2024.doi: 10.1007/s00211-024-01448-1

work page doi:10.1007/s00211-024-01448-1 2024

[4] [4]

Casas and K

E. Casas and K. Kunisch. Space-timeL ∞-estimates for solutions of infinite horizon semilinear parabolic equations.Commun. Pure Appl. Annal., 24(4):482–506, 2025. doi:10.3934/cpaa.2025002

work page doi:10.3934/cpaa.2025002 2025

[5] [5]

Casas and M

E. Casas and M. Mateos. Quadratic convergence of an SQP method for some optimiza- tion problems with applications to control theory, 2025. To appear in SIAM Journal on Control and Optimization.arXiv:2505.22750

work page internal anchor Pith review arXiv 2025

[6] [6]

Casas, M

E. Casas, M. Mateos, and A. R¨ osch. Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity.Comput. Optim. Appl., 70(1):239–266, 2018.doi:10.1007/s10589-018-9979-0

work page doi:10.1007/s10589-018-9979-0 2018

[7] [7]

Casas and F

E. Casas and F. Tr¨ oltzsch. Second order analysis for optimal control problems: Im- proving results expected from abstract theory.SIAM J. Optim., 22(1):261–279, 2012. doi:10.1137/110840406

work page doi:10.1137/110840406 2012

[8] [8]

P. G. Ciarlet.Introduction ` a l’analyse num´ erique matricielle et ` a l’optimisation. Mas- son, Paris, 1982. Collection Math´ ematiques Appliqu´ ees pour la Maˆ ıtrise

work page 1982

[9] [9]

Dautray and J

R. Dautray and J. Lions.Mathematical Analysis and Numerical Methods for Science and Technology, volume 5. Springer-Verlag, Berlin-Heidelberg-New York, 2000

work page 2000

[10] [10]

Disser, A

K. Disser, A. F. M. ter Elst, and J. Rehberg. H¨ older estimates for parabolic operators on domains with rough boundary.Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 17(1):65–79, 2017.doi:10.2422/2036-2145.201503_013

work page doi:10.2422/2036-2145.201503_013 2017

[11] [11]

Goldberg and F

H. Goldberg and F. Tr¨ oltzsch. On a Lagrange-Newton method for a nonlinear parabolic boundary control problem.Optim. Methods Softw., 8(3-4):225–247, 1998.doi:10. 1080/10556789808805678

work page 1998

[12] [12]

Hoppe and I

F. Hoppe and I. Neitzel. Convergence of the SQP method for quasilinear parabolic optimal control problems.Optim. Eng., 22(4):2039–2085, 2021.doi:10.1007/ s11081-020-09547-2

work page 2039

[13] [13]

Triebel.Interpolation Theory, Function Spaces, Differential Operators

H. Triebel.Interpolation Theory, Function Spaces, Differential Operators. North- Holland, Berlin, 1978

work page 1978

[14] [14]

Tr¨ oltzsch

F. Tr¨ oltzsch. On the Lagrange–Newton–SQP method for the optimal control of semi- linear parabolic equations.SIAM Journal on Control and Optimization, 38(1):294–312, 1999.doi:10.1137/S0363012998341423. Bilinear control problems: convergence of the SQP method39

work page doi:10.1137/s0363012998341423 1999

[15] [15]

Wachsmuth

D. Wachsmuth. Analysis of the SQP-method for optimal control problems governed by the nonstationary Navier–Stokes equations based onL p-theory.SIAM Journal on Control and Optimization, 46(3):1133–1153, 2007.doi:10.1137/S0363012904443506

work page doi:10.1137/s0363012904443506 2007