On Characterization of Inverse Data in the Boundary Control Method

Abstract

We deal with a dynamical system \begin{align*} & u_{tt}-Δu+qu=0 && {\rm in}\,\,\,Ω\times (0,T)\\ & u\big|_{t=0}=u_t\big|_{t=0}=0 && {\rm in}\,\,\,\overline Ω\\ & \partial_νu = f && {\rm in}\,\,\,\partialΩ\times [0,T]\,, \end{align*} where $\Omega \subset {\mathbb{R}}^n$ is a bounded domain, $q\in L_{\infty }(\Omega )$ a real-valued function, $\nu$ the outward normal to $\partial \Omega$, $u=u^f(x,t)$ a solution. The input/output correspondence is realized by a response operator $R^T:f\mapsto u^f{|}_{\partial \Omega \times [0,T]}$ and its relevant extension by hyperbolicity $R^{2T}$. Ope\-rator $R^{2T}$ is determined by $q{|}_{{\Omega }^T}$, where ${\Omega }^T:=\{x\in \Omega |\mathrm{dist}(x,\partial \Omega )<T\}$. The inverse problem is: Given $R^{2T}$ to recover $q$ in ${\Omega }^T$. We solve this problem by the boundary control method and describe the {\it ne\-ces\-sary and sufficient} conditions on $R^{2T}$, which provide its solvability.

33 pages, 1 figure