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}).
Categories: math.AP math-ph math.MP