Time-optimal reconstruction of Riemannian manifold via boundary electromagnetic measurements

TL;DR

This paper proves that the boundary control method can uniquely reconstruct a part of a Riemannian manifold from electromagnetic boundary measurements for any time T, extending previous results limited to small T.

Contribution

It extends the uniqueness result for reconstructing manifold parts from boundary data to arbitrary time T, and provides a reconstruction procedure.

Findings

Proves uniqueness of reconstruction for any T>0.

Develops a boundary control method-based reconstruction procedure.

Extends previous small T results to all T>0.

Abstract

A dynamical Maxwell system is \begin{align*} & e_t={\rm curl\,} h, \quad h_t=-{\rm curl\,} e &&{\rm in}\,\,\Omega \times (0,T) & e|_{t=0}=0,\,\,\,\,h|_{t=0}=0 &&{\rm in}\,\,\Omega & e_\theta =f &&{\rm in}\,\,\, \partial\Omega \times [0,T] \end{align*} where $\Omega$ is a smooth compact oriented $3$-dimensional Riemannian manifold with boundary, $(\,\cdot\,)_\theta$ is a tangent component of a vector at the boundary, $e=e^f(x,t)$ and $h=h^f(x,t)$ are the electric and magnetic components of the solution. With the system one associates a response operator $R^T: f \mapsto -\nu \wedge h^f|_{\partial\Omega \times (0,T)}$, where $\nu$ is an outward normal to $\partial\Omega$. The time-optimal setup of the inverse problem, which is relevant to the finiteness of the wave speed propagation, is: given $R^{2T}$ to recover the part $\Omega^T:=\{x\in \Omega\,|\,{\rm dist\,}(x,\partial \Omega)<T\}$ of the manifold. As was shown by Belishev, Isakov, Pestov, Sharafutdinov (2000), for {\it small enough} $T$ the operator $R^{2T}$ determines $\Omega^T$ uniquely up to isometry. Here we prove that uniqueness holds for {\it arbitrary} $T>0$ and provide a procedure that recovers ${\Omega^T}$ from $R^{2T}$. Our approach is a version of the boundary control method (Belishev, 1986).