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arxiv: 2604.02957 · v1 · submitted 2026-04-03 · 🧮 math-ph · math.MP

On a stability of time-optimal version of the Boundary Control method

Mikhail I. Belishev This is my paper

Pith reviewed 2026-05-13 18:20 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords boundary control methodtime-optimal reconstructionstabilitytriangular factorizationresponse operatorinverse problemwave equationRiemannian manifold
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The pith

The time-optimal BC-method reconstruction from response operator R^{2T} to wave operator W^T is continuous in relevant operator topologies.

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

The paper establishes that the time-optimal version of the boundary control method reconstructs the operator W^T, which creates invisible waves in the T-neighborhood of the boundary on a Riemannian manifold, from the boundary response operator R^{2T}. This reconstruction proceeds via the intermediate operator C^T = W^{T*} W^T and its triangular factorization C^T = F^{T*} F^T with F^T = U^T W^T, where U^T is unitary. The factorization step is continuous in the relevant operator topologies, which makes the full chain R^{2T} to C^T to F^T to W^T continuous as well. As a concrete application, convergence of response operators implies convergence of the potential q in the wave equation to the limit in the H^{-2} norm over the T-neighborhood. The work leaves open the question of quantitative stability rates.

Core claim

Let Ω be a Riemannian manifold with boundary. The time-optimal version of the BC-method determines the parameters in the T-neighborhood Ω^T of ∂Ω from the boundary observations (response operator) R^{2T} on the time segment [0,2T]. It visualizes the invisible waves supported in Ω^T by reconstructing the operator W^T that creates these waves. The visualization is based on the triangular factorization of the operator C^T := W^{T*} W^T in the form C^T := F^{T*} F^T with a factor F^T = U^T W^T, where U^T is a unitary operator. The factorization C^T ↦ F^T has certain continuity properties, due to which the time-optimal reconstruction R^{2T} ↦ C^T ↦ F^T ↦ W^T turns out to be continuous (stable) in

What carries the argument

The triangular factorization C^T := F^{T*} F^T with F^T = U^T W^T (U^T unitary), which transfers continuity from the response operator R^{2T} through C^T and F^T to the reconstructed wave-creating operator W^T.

If this is right

  • Convergence of response operators R^{2T}_j to R^{2T} implies convergence of the reconstructed operators W^T_j to W^T in the relevant operator topologies.
  • For the wave equation u_tt - Δu + q u = 0, convergence of R^{2T}_j implies convergence of q_j to q in H^{-2}(Ω^T).
  • The reconstruction visualizes waves supported in Ω^T in a manner stable under small perturbations of boundary data.

Where Pith is reading between the lines

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

  • Numerical realizations of the method may tolerate measurement noise because the reconstruction map is continuous.
  • Obtaining explicit rates of convergence would turn the stability result into a practical error bound for inverse problems.
  • The same factorization-based argument could apply to other inverse problems that recover coefficients from boundary wave observations.

Load-bearing premise

The triangular factorization of C^T into F^{T*} F^T possesses continuity properties in the relevant operator topologies.

What would settle it

A sequence of response operators R^{2T}_j converging to R^{2T} such that the reconstructed potentials q_j fail to converge to q in H^{-2}(Ω^T).

read the original abstract

Let $\Omega$ be a Riemannian manifold with boundary. The time-optimal version of the BC-method determines the parameters in the $T$-neigh\-bor\-hood $\Omega^T$ of $\partial\Omega$ from the boundary observations (response operator) $R^{2T}$ on the time segment $[0,2T]$. It visualizes the invisible waves supported in $\Omega^T$, by reconstructing the operator $W^T$ that creates these waves. The visualization is based on the triangular factorization of the operator $C^T:=W^{T\,*}W^T$ in the form $C^T:=F^{T\,*}F^T$ with a factor $F^T=U^{T}W^T$, where $U^T$ is a unitary operator. The factorization $C^T\mapsto F^T$ has certain continuity properties, due to which the time-optimal reconstruction $R^{2T}\mapsto C^T\mapsto F^T\mapsto W^T$ turns out to be continuous (stable) in the sense of relevant operator topologies (convergences). As an example, determination of the potential $q$ in the wave equation $u_{tt}-\Delta u+qu=0$ from $R^{2T}$ is considered. We show that $R^{2T}_j\to R^{2T}$ implies $q_j\to q$ in $H^{-2}(\Omega^T)$. However, the question of quantitative estimates of stability (the rate of convergence) remains open.

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 / 2 minor

Summary. The manuscript establishes the stability of the time-optimal version of the Boundary Control method. It demonstrates that the reconstruction R^{2T} ↦ C^T ↦ F^T ↦ W^T is continuous in relevant operator topologies due to the continuity properties of the triangular factorization C^T = F^{T*} F^T with F^T = U^T W^T, where U^T is unitary. As an example, it shows that convergence of the response operators R^{2T}_j to R^{2T} implies convergence of the potentials q_j to q in H^{-2}(Ω^T) for the wave equation u_tt - Δu + q u = 0, although quantitative rates of convergence are left as an open question.

Significance. If the asserted continuity properties hold, this provides an important theoretical advance in the stability analysis of inverse problems for the wave equation on manifolds using boundary observations. The time-optimal aspect and the visualization of invisible waves via W^T are novel contributions. The explicit example with the potential strengthens the result by showing concrete applicability, and the honest note on open quantitative estimates is a strength.

minor comments (2)
  1. [Abstract] The term 'relevant operator topologies (convergences)' is vague; specify the exact topologies (e.g., strong operator topology) for each map in the chain R^{2T} ↦ C^T ↦ F^T ↦ W^T to improve clarity.
  2. [Introduction] Ensure that the distinction between R^{2T} and R^T is clearly explained early in the paper, as the time-optimal version uses 2T.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation for minor revision. The summary correctly identifies the core result on the continuity of the map R^{2T} to W^T through the triangular factorization of C^T and the consequent stability for the potential q in H^{-2}(Ω^T). We agree that quantitative convergence rates remain open, as already noted in the paper.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation chain R^{2T} ↦ C^T ↦ F^T ↦ W^T relies on the triangular factorization C^T = F^{T*} F^T with F^T = U^T W^T. The manuscript supplies an operator-theoretic proof that this factorization map is continuous in the relevant topologies (strong/weak/norm), which directly yields stability of the reconstruction without reducing any step to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation whose justification is internal to the present work. The explicit example (R^{2T}_j → R^{2T} implies q_j → q in H^{-2}(Ω^T)) is derived from the same continuity statement and does not presuppose the target result. The paper is therefore self-contained against external operator-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and continuity of the unitary factor U^T in the triangular factorization of C^T; these are treated as standard properties of the relevant operator algebras on the manifold.

axioms (1)
  • domain assumption The operator C^T admits a triangular factorization C^T = F^{T*} F^T with F^T = U^T W^T where U^T is unitary and the map C^T to F^T is continuous in the relevant topologies.
    Invoked directly to conclude stability of the reconstruction chain R^{2T} to W^T.

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Reference graph

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