Time-optimal reconstruction of Riemannian manifold via boundary electromagnetic measurements

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).