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

Abstract

Let $\Sigma:=[0,\infty)\times S^2$, $\mathscr F:=L_2(\Sigma)$. The {\it forward} acoustic scattering problem under consideration is to find $u=u^f(x,t)$ satisfying \begin{align} \label{Eq 01} &u_{tt}-\Delta u+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+\tau)\,\omega,s)=f(\tau,\omega), && (\tau,\omega) \in \Sigma; \end{align} for a real valued compactly supported potential $q\in L_\infty(\Bbb R^3)$ 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), \quad (\tau ,\omega ) \in \Sigma \end{align*} depends on $q$ {\it 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 {\it 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 (\ref{Eq 01})--(\ref{Eq 03}).