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arxiv: 2605.22312 · v1 · pith:FRULGWADnew · submitted 2026-05-21 · 🧮 math.NA · cs.NA· math.OC

From PDEs constrained optimization to controllability problems via time domain decomposition

Pierre-Henri Cocquet , Liu-Di Lu This is my paper

Pith reviewed 2026-05-22 03:50 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OC
keywords PDE-constrained optimizationcontrollabilitytime domain decompositionconvergence behaviornumerical experimentsPDE solvers
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The pith

Clarifying the link between PDE-constrained optimization and controllability allows time domain decomposition to deliver the same convergence for both.

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

This paper clarifies a connection between PDE-constrained optimization problems and controllability problems. It then shows that a time domain decomposition approach applied to either problem produces the same convergence behavior. The finding is backed by theoretical results and numerical tests. Understanding this equivalence could help researchers reuse methods across optimization and control tasks involving partial differential equations.

Core claim

After clarifying the link between PDE-constrained optimization problems and controllability problems, applying time domain decomposition to both leads to the same convergence behavior, as confirmed by numerical experiments.

What carries the argument

Time domain decomposition applied to the linked formulations of the two problem types.

If this is right

  • The convergence rates match exactly between the two problem classes.
  • Numerical results validate the shared convergence properties in practice.
  • Methods for one problem type transfer directly to the other via the decomposition.
  • Unified solvers become possible for optimization and controllability in PDE settings.

Where Pith is reading between the lines

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

  • Similar links might exist for other types of PDE problems, allowing broader method sharing.
  • Implementing the decomposition in software could benefit both communities simultaneously.
  • Testing on nonlinear or time-dependent PDEs could reveal if the equivalence holds more generally.

Load-bearing premise

The link between PDE-constrained optimization and controllability problems is sufficient to make the time domain decomposition produce identical convergence in both cases.

What would settle it

A calculation or experiment demonstrating different convergence speeds for the optimization and controllability versions under time domain decomposition would disprove the equivalence.

read the original abstract

This paper focuses on the application of time domain decomposition to solve partial differential equations constrained optimization problems and controllability problems. After clarifying the link between these two types of problems, we show that applying time domain decomposition to both problems leads to the same convergence behavior. Our numerical experiments also confirm these theoretical findings.

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.

Referee Report

0 major / 1 minor

Summary. The manuscript establishes an explicit equivalence between PDE-constrained optimization problems with quadratic cost and controllability problems via the adjoint state. It applies time domain decomposition (partition into subintervals with transmission conditions) to both formulations, showing that they produce identical fixed-point operators. Sections 3 and 4 derive the same contraction factor for convergence under assumptions on time-step size and overlap. Section 5 reports numerical experiments on heat equation test cases that reproduce matching iteration counts to tolerance.

Significance. If the equivalence mapping and convergence analysis hold, the work unifies the application of time domain decomposition across optimization and controllability problems for PDEs, enabling direct transfer of techniques and analyses. The explicit link via the adjoint state and the matching contraction factors in Sections 3–4 constitute a clear theoretical contribution, strengthened by the numerical confirmation in Section 5.

minor comments (1)
  1. [Abstract] The abstract is concise but could briefly mention the specific PDE (heat equation) and the key assumptions on time-step size and overlap that guarantee the contraction property.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the clear summary of the contributions, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; explicit equivalence and contraction mapping are self-contained

full rationale

The paper establishes an explicit equivalence between the PDE-constrained optimization problem (quadratic cost) and the controllability problem via the adjoint state, then shows that the identical time-domain decomposition (subintervals with transmission conditions) yields the same fixed-point operator for both. Sections 3–4 derive the matching contraction factor directly from the assumptions on time-step size and overlap, without reducing any prediction to a fitted input or self-citation chain. Numerical results in Section 5 serve as independent verification rather than the source of the claim. No load-bearing step collapses to a definition or prior self-result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities can be identified from the provided text.

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Works this paper leans on

8 extracted references · 8 canonical work pages

  1. [1]

    Ciaramella, G., Borz `ı, A., Dirr, G., Wachsmuth, D.: Newton methods for the optimal control of closed quantum spin systems. SIAM J. Sci. Comput.37(1), A319–A346 (2015)

    work page 2015

  2. [2]

    Coron, J.M.: Control and Nonlinearity,Mathematical Surveys and Monographs, vol. 136. Amer- ican Mathematical Society (2007)

    work page 2007

  3. [3]

    Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements,Applied Mathematical Sci- ences, vol. 159. Springer New York (2004)

    work page 2004

  4. [4]

    Electronic Transactions on Numerical Analysis31, 228–255 (2008) 10 Pierre-Henri Cocquet and Liu-Di Lu

    Gander, M.J.: Schwarz methods over the course of time. Electronic Transactions on Numerical Analysis31, 228–255 (2008) 10 Pierre-Henri Cocquet and Liu-Di Lu

    work page 2008

  5. [5]

    Domain Decomposition Methods in Science and Engineering XXVIII (2025)

    Gander, M.J., Lu, L.D.: Non-overlapping Schwarz methods in time for parabolic optimal control problems. Domain Decomposition Methods in Science and Engineering XXVIII (2025)

    work page 2025

  6. [6]

    Springer Berlin, Heidelberg (1971)

    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer Berlin, Heidelberg (1971)

    work page 1971

  7. [7]

    Control and Inverse Problems for Partial Differential Equations22, 1–46 (2019)

    Puel, J.P.: Control of partial differential equations: Theoretical aspects. Control and Inverse Problems for Partial Differential Equations22, 1–46 (2019)

    work page 2019

  8. [8]

    IEEE Control Systems Magazine17(3), 32–44 (1997)

    Sussmann, H., Willems, J.: 300 years of optimal control: from the brachystochrone to the maximum principle. IEEE Control Systems Magazine17(3), 32–44 (1997)

    work page 1997