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On the stability of an inverse problem for waves via the Boundary Control method

Spyridon Filippas, Lauri Oksanen

TL;DR

This work connects stability in the Boundary Control method for a hyperbolic inverse problem to the cost of approximate controllability and quantitative unique continuation. By combining Blagove\v{s}tjenskii-type inner-product identities with approximate controllability and high-frequency geometric optics constructions, it derives stability bounds for the potential $q$ from boundary/source data $\Lambda_q$, valid in general geometries. The central result shows that, for $T>2\mathcal{L}(K,\omega)$ and smooth potentials, a small difference in source-to-solution maps implies a linear bound on $\|q_2-q_1\|_{L^2(K)}$, with the rate governed by the cost function $\mathcal{A}(\varepsilon)$; a complementary half-space result yields a double-log modulus. The methodology unifies stability analysis with the cost of approximate controllability, extending BC-based uniqueness results to quantitative stability in broad settings and offering explicit connections between data quality and resolvability of the unknown potential.

Abstract

We establish a link between stability estimates for a hyperbolic inverse problem via the Boundary Control method and the blowup of a constant appearing in the contexts of optimal unique continuation and cost of approximate controllability.

On the stability of an inverse problem for waves via the Boundary Control method

TL;DR

This work connects stability in the Boundary Control method for a hyperbolic inverse problem to the cost of approximate controllability and quantitative unique continuation. By combining Blagove\v{s}tjenskii-type inner-product identities with approximate controllability and high-frequency geometric optics constructions, it derives stability bounds for the potential from boundary/source data , valid in general geometries. The central result shows that, for and smooth potentials, a small difference in source-to-solution maps implies a linear bound on , with the rate governed by the cost function ; a complementary half-space result yields a double-log modulus. The methodology unifies stability analysis with the cost of approximate controllability, extending BC-based uniqueness results to quantitative stability in broad settings and offering explicit connections between data quality and resolvability of the unknown potential.

Abstract

We establish a link between stability estimates for a hyperbolic inverse problem via the Boundary Control method and the blowup of a constant appearing in the contexts of optimal unique continuation and cost of approximate controllability.

Paper Structure

This paper contains 9 sections, 15 theorems, 111 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main result and applications
  4. Finite speed of propagation and unique continuation
  5. Proof of the stability estimate
  6. Stable recovery of the inner products
  7. From inner products to domains of influence
  8. Geometric optics solutions
  9. End of the proofs

Key Result

Theorem 1.1

Assume that $T>2\mathcal{L}(K, \omega)$ and the potentials $q_j$ satisfy $\left\Vert q_j \right\Vert_{C^m(\mathbb R^n)}^{}\leq M, j \in \{1,2\}$ for some $M>0$ and $m >n$. Then given small $\varepsilon>0$ one has the following stability estimate. If where the operator norm is from $L^2((0,T) \times \omega)$ to itself, then where:

Figures (3)

  • Figure 1: A solution of the wave equation that vanishes on $(-s,s)\times \omega$ has to vanish in the set $M$. This is the largest diamond above. The domain $M(\omega,s)$ is $M\cap \{t=0\}$.
  • Figure 2: The domain $A(\eta)$ in darker color, used in the proof of Proposition \ref{['prop point values']}. As $\eta \to 0$$A(\eta)$ collapses to the point $x$. The two balls correspond to the two domains of influence under consideration and the radii are $r_1=2r+s-\eta$ and $r_2=r+s+\eta$.
  • Figure 3: Support of the geometric optics solution, localizing close to the line $\beta(t)$.

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1
  • Theorem 2.1: Finite speed of propagation for waves
  • Theorem 2.2: Unique continuation
  • Theorem 2.3: Approximate controllability
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 20 more