On Characterization of Inverse Data in the Boundary Control Method

Abstract

We deal with a dynamical system \begin{align*} & u_{tt}-\Delta u+qu=0 && {\rm in}\,\,\,\Omega \times (0,T)\\ & u\big|_{t=0}=u_t\big|_{t=0}=0 && {\rm in}\,\,\,\overline \Omega\\ & \partial_\nu u = f && {\rm in}\,\,\,\partial\Omega \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 $R^T: f \mapsto u^f\big|_{\partial\Omega \times [0,T]}$ and its relevant extension by hyperbolicity $R^{2T}$. Ope\-rator $R^{2T}$ is determined by $q\big|_{\Omega^T}$, where $\Omega^T:=\{x \in \Omega\,|\,\,{\rm dist\,}(x,\partial \Omega)<T\}$. 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.