Quantitative stability rates for time‑optimal reconstruction (TOR) via the Boundary Control method

Develop quantitative stability estimates (rates of convergence) for the time‑optimal reconstruction mapping R^{2T} \mapsto C^T \mapsto F^T \mapsto W^T that determines the potential q on the T‑neighborhood \Omega^T of the boundary from the response operator R^{2T} measured on [0,2T], for example by deriving bounds that control the distance between reconstructed parameters (such as \|q_j-q\| on \Omega^T) in terms of the discrepancy of the data \|R^{2T}_j-R^{2T}\|.

Background

The paper proves a qualitative stability result for the time‑optimal version of the Boundary Control method: convergence of response operators R^{2T}_j to R^{2T} implies weak operator convergence of the associated visualization and control operators and, consequently, convergence of the recovered potentials q_j to q in H^{-2}(\Omega^T).

However, unlike other variants of the BC‑method that work with observations on longer time intervals [0,2T'] with T'>T and admit quantitative estimates, the time‑optimal reconstruction (using exactly [0,2T]) currently lacks rates of convergence. The authors emphasize that obtaining such quantitative estimates is difficult and important for applications.