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

TL;DR

This paper demonstrates the stability of the time-optimal Boundary Control method for reconstructing wave parameters on Riemannian manifolds from boundary data, with a focus on the continuity of operator factorizations.

Contribution

It establishes the stability of the boundary control method's reconstruction process via operator factorizations, ensuring continuous dependence on boundary observations.

Findings

Reconstruction of wave-supporting operators is stable under boundary data convergence.

The method determines the potential in the wave equation from boundary measurements with proven stability.

Quantitative stability estimates, such as convergence rates, remain an open problem.

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.