Three-dimensional inverse acoustic scattering problem by the BC-method

Let $\mathrm{\Sigma }:=[0,\mathrm{\infty })\times S^2$, $\mathcal{F}:=L_2(\mathrm{\Sigma })$. The {\it forward} acoustic scattering problem under consideration is to find $u=u^f(x,t)$ satisfying \begin{align} \label{Eq 01} &u_{tt}-Δu+qu=0, && (x,t) \in {\mathbb R}^3 \times (-\infty,\infty); \ \label{Eq 02} &u \mid_{|x|<-t} =0 , && t<0;\ \label{Eq 03} &\lim_{s \to -\infty} s,u((-s+τ),ω,s)=f(τ,ω), && (τ,ω) \in Σ; \end{align} for a real valued compactly supported potential $q\in L_{\mathrm{\infty }}({\mathbb{R}}^3)$ and a control $f\in \mathcal{F}$. The response operator $R:\mathcal{F}\to \mathcal{F}$, \begin{align*} & (Rf)(τ,ω),:= \lim_{s \to +\infty} s, u^f((s+τ),ω,s), \quad (τ,ω) \in Σ\end{align*} depends on $q$ {\it locally}: if $\xi >0$ and $f\in {\mathcal{F}}^{\xi }:=\{f\in \mathcal{F}\mathrm{\mid }f\text{\hspace{0.17em}⁣}{\mid }_{[0,\xi )}=0\}$ holds, then the values $(Rf)\text{\hspace{0.17em}⁣}{\mid }_{\tau ⩾\xi }$ are determined by $q\text{\hspace{0.17em}⁣}{\mid }_{\mathrm{\mid }x\mathrm{\mid }⩾\xi }$ (do not depend on $q\text{\hspace{0.17em}⁣}{\mid }_{\mathrm{\mid }x\mathrm{\mid }<\xi }$). The {\it inverse problem} is: for an arbitrarily fixed $\xi >0$, to determine $q{\mid }_{\mathrm{\mid }x\mathrm{\mid }⩾\xi }$ from $X^{\xi }R\upharpoonright {\mathcal{F}}^{\xi }$, where $X^{\xi }$ is the projection in $\mathcal{F}$ onto ${\mathcal{F}}^{\xi }$. It is solved by a relevant version of the boundary control method. The key point of the approach are recent results on the controllability of the system (\ref{Eq 01})–(\ref{Eq 03}).