On controllability of the acoustic scattering dynamical system in $\Bbb R^3$
M. I. Belishev,
A. F. Vakulenko St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
Abstract:
The acoustic scattering problem 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,0);\\
&u \mid_{|x|<-t} =0 , t<0\\
&\lim_{s \to \infty} s u((s+\tau) \omega,-s)=f(\tau,\omega), (\tau,\omega) \in \Sigma:=[0,\infty)\times S^2; \end{align*}
with a real valued compactly supported potential
$q\in L_\infty(\Bbb R^3)$ and a control
$f \in \mathscr F:=L_2(\Sigma)$. Let $\mathscr F^\xi:= \{f\in\mathscr F | f\big|_{0\leqslant \tau\leqslant \xi}=0\}$,
$\mathscr H:=L_2(\Bbb R^3)$, $\mathscr H^\xi:=\{y\in \mathscr H | y\big|_{|x|<\xi}=0\}$,
$\xi>0$. For the (delayed) controls
$f\in\mathscr F^\xi$, the
reachable set is $\mathscr U^\xi:=\{u^f(\cdot, 0) | f\in\mathscr F^\xi\}\subset\mathscr H^\xi$, whereas $\mathscr D^\xi:=\mathscr H^\xi\ominus\mathscr U^\xi$ is the
defect (unreachable) subspace. The paper provides a characterization of
$\mathscr D^\xi$ as follows.
We say an
$a\in\mathscr H^\xi$ to be a
$q$-polyharmonic function of the order
$n$ if
$(-\Delta +q)^n a=0$ holds for
$|x|>\xi$, and write
$a\in\mathscr A^\xi_n$. Our main result is the relation
\begin{equation*} {\mathscr D}^\xi =\overline{{\rm span }\{\mathscr A^\xi_n | n\geqslant 1\}}, \xi>0 \end{equation*}
(the closure in
$\mathscr H$). It basically concludes the study of controllability of the acoustical dynamical system governed by the locally perturbed wave equation in
$\mathbb R^3$.
Key words and phrases:
dynamical system governed by locally perturbed wave equation, scattering problem, controllability.
UDC:
517.5
Received: 13.08.2024