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

Let $\mathrm{\Omega }$ be a Riemannian manifold with boundary. The time-optimal version of the BC-method determines the parameters in the $T$-neigh-bor-hood ${\mathrm{\Omega }}^T$ of $\mathrm{\partial }\mathrm{\Omega }$ from the boundary observations (response operator) $R^{2T}$ on the time segment $[0,2T]$. It visualizes the invisible waves supported in ${\mathrm{\Omega }}^T$, by reconstructing the operator $W^T$ that creates these waves. The visualization is based on the triangular factorization of the operator $C^T:=W^{T\ast }{W}^T$ in the form $C^T:=F^{T\ast }{F}^T$ 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 $u_{tt}-\mathrm{\Delta }u+qu=0$ from $R^{2T}$ is considered. We show that $R_j^{2T}\to R^{2T}$ implies $q_j\to q$ in $H^{-2}({\mathrm{\Omega }}^T)$. However, the question of quantitative estimates of stability (the rate of convergence) remains open.