Time-optimal reconstruction of Riemannian manifold via boundary electromagnetic measurements

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 $\mathrm{\Omega }$ 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 $\mathrm{\partial }\mathrm{\Omega }$. 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 ${\mathrm{\Omega }}^T:=\{x\in \mathrm{\Omega }\mathrm{\mid }\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}(x,\mathrm{\partial }\mathrm{\Omega })<T\}$ of the manifold. As was shown by Belishev, Isakov, Pestov, Sharafutdinov (2000), for {\it small enough} $T$ the operator $R^{2T}$ determines ${\mathrm{\Omega }}^T$ uniquely up to isometry. Here we prove that uniqueness holds for {\it arbitrary} $T>0$ and provide a procedure that recovers ${\mathrm{\Omega }}^T$ from $R^{2T}$. Our approach is a version of the boundary control method (Belishev, 1986).