Three-dimensional inverse acoustic scattering problem by the BC-method
M. I. Belishev,
A. F. Vakulenko St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Abstract:
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).
Key words and phrases:
three-dimensional dynamical system governed by the locally perturbed wave equation, determination of potential from inverse scattering data, boundary control method.
UDC:
517 Received: 29.08.2024