The acoustic scattering problem is to find \(u=u^f(x,t)\) satisfying \(\begin{aligned}&u_{tt}-\Delta u+qu=0, \qquad (x,t) \in {\mathbb {R}}^3 \times (-\infty ,0); \\&u \mid _{|x|<-t} =0 , \qquad t<0\;\\&\lim _{s \rightarrow \infty } s\,u((s+\tau )\,\omega ,-s)=f(\tau ,\omega ), \qquad (\tau ,\omega ) \in \Sigma :=[0,\infty )\times S^2; \end{aligned}\) with a real valued compactly supported potential \(q\in L_\infty (\mathbb {R}^3)\) and a control \(f \in \mathscr {F}:=L_2(\Sigma )\) . Let \(\mathscr {F}^\xi := \{f\in \mathscr {F}\,|\,\,f\big |_{0\leqslant \tau \leqslant \xi }=0\}\) , \(\mathscr {H}:=L_2(\mathbb {R}^3)\) , \(\mathscr {H}^\xi :=\{y\in \mathscr {H}\,|\,\,y\big |_{|x|<\xi }=0\}\) , \(\xi >0\) . 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 the set \(\mathscr {D}^\xi \) . We say that \(a\in \mathscr {H}^\xi \) is a q-polyharmonic function of the order n if the relation \((-\Delta +q)^n\, a=0\) holds for \(|x|>\xi \) , and write \(a\in \mathscr {A}^\xi _n\) . Our main result is the relation \(\begin{aligned}{\mathscr {D}}^\xi \,=\overline{\mathrm{span\,}\{\mathscr {A}^\xi _n\,|\,\,n\ge 1\}},\qquad \xi >0,\end{aligned}\) where bar denotes the closure in \(\mathscr {H}\) . It basically concludes the study ofcontrollability of the acoustical dynamical system governed by the locally perturbed wave equation in \(\mathbb {R}^3\) . Bibliography: 9 titles.
