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

Let $\Sigma:=[0,\infty)\times S^2$, $\mathscr F:=L_2(\Sigma)$. The forward acoustic scattering problem under consideration is to find $u=u^f(x,t)$ satisfying
\begin{align*} &u_{tt}-\Delta u+qu=0, && (x,t) \in {\mathbb R}^3 \times (-\infty,\infty); \tag{48}\\ &u \mid_{|x|<-t} =0 , && t<0; \tag{49}\\ &\lim_{s \to -\infty} s u((-s+\tau) \omega,s)=f(\tau,\omega), && (\tau,\omega) \in \Sigma; \tag{50} \end{align*}
for a real valued compactly supported potential $q\in L_\infty(\mathbb R^3)$ and a control $f \in\mathscr F$. The response operator $R: \mathscr F\to\mathscr F$,
\begin{align*} & (Rf)(\tau ,\omega ) := \lim_{s \to +\infty} s u^f((s+\tau ) \omega ,s), (\tau ,\omega ) \in \Sigma \end{align*}
depends on $q$ locally: if $\xi>0$ and $f\in\mathscr F^\xi:=\{f\in\mathscr F | f \mid_{[0,\xi)}=0\}$ holds, then the values $(Rf) \mid_{\tau\geqslant\xi}$ are determined by $q \mid_{|x|\geqslant\xi}$ (do not depend on $q \mid_{|x|<\xi}$). The inverse problem is: for an arbitrarily fixed $\xi>0$, to determine $q\mid_{|x|\geqslant\xi}$ from $X^\xi R\upharpoonright\mathscr F^\xi$, where $X^\xi$ is the projection in $\mathscr F$ onto $\mathscr F^\xi$. 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 (48)–(50).