This preprint was accepted December 25, 2006
ABSTRACT:
The scattering problem is to find $u=u^f(x,t)$
satisfying
\begin{align*}
&u_{tt}-\Delta u+qu=0, \qquad (x,t) \in {\Bbb R}^3 \times (-\infty,\infty) \\
&u \mid_{|x|<-t} =0 , \qquad t<0\\
&\lim_{s \to \infty} su((s+\tau)\omega,-s)=f(\tau,\omega),
\qquad (\tau,\omega) \in [0,\infty)\times S^2
\end{align*}
for a real smooth compactly supported potential $q=q(x)$ and a
control $f \in {\cal F} =L_2([0,\infty);L_2(S^2))$. The
corresponding control problem is: given $y \in {\cal H} =L_2({\Bbb
R}^3)$ find $f \in {\cal F}$ providing $u^f(\cdot,0)=y$; the
reachable set is $ {\cal U}=\left \{ u^f(\cdot,0) \mid f \in {\cal
F} \right \}$; the subspace of unreachable states is ${\cal
D}={\cal H} \ominus {\cal U}$. The main subject of the paper is
the structure of ${\cal U}$ and ${\cal D}$. We present an example
of the finite energy solution $u^f$ satisfying $u^f|_{|x|<|t|}=0$,
i.e., vanishing simultaneously in the past and future cones
(reversing wave) and introduce the set of points at which such a
``revers effect'' occurs. The existence of the reversing waves turns
out to be equivalent to the lack of controllability $ {\cal D}
\neq \{0\}$. Cauchy data of such waves belong to the classes
$D_\mp$ of the incoming and outgoing data simultaneously,
providing $D_- \cap D_+ \not= \{0\}$. We also describe the simple
conditions on $f$ ensuring $\|u^f(\cdot,t)\|_{\cal H} \leq c
\|f\|_{\cal F}$ for all $t \in (-\infty, \infty)$.
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