Geodesic
On controllability of the acoustic scattering dynamical system in $\Bbb R^3$
Zapiski Nauchnykh Seminarov POMI, Investigations on applied mathematics and informatics. Part III, Tome 539 (2024), pp. 31-43 
M. I. Belishev; A. F. Vakulenko. On controllability of the acoustic scattering dynamical system in $\Bbb R^3$. Zapiski Nauchnykh Seminarov POMI, Investigations on applied mathematics and informatics. Part III, Tome 539 (2024), pp. 31-43. http://geodesic.mathdoc.fr/item/ZNSL_2024_539_a1/
@article{ZNSL_2024_539_a1,
     author = {M. I. Belishev and A. F. Vakulenko},
     title = {On controllability of the acoustic scattering dynamical system in $\Bbb R^3$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {31--43},
     year = {2024},
     volume = {539},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_539_a1/}
}
TY  - JOUR
AU  - M. I. Belishev
AU  - A. F. Vakulenko
TI  - On controllability of the acoustic scattering dynamical system in $\Bbb R^3$
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2024
SP  - 31
EP  - 43
VL  - 539
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2024_539_a1/
LA  - ru
ID  - ZNSL_2024_539_a1
ER  - 
%0 Journal Article
%A M. I. Belishev
%A A. F. Vakulenko
%T On controllability of the acoustic scattering dynamical system in $\Bbb R^3$
%J Zapiski Nauchnykh Seminarov POMI
%D 2024
%P 31-43
%V 539
%U http://geodesic.mathdoc.fr/item/ZNSL_2024_539_a1/
%G ru
%F ZNSL_2024_539_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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 $\mathbb R^3$.

[1] M. I. Belishev, “vBoundary control in reconstruction of manifolds and metrics (the BC method)”, Inverse Problems, 13:5 (1997), R1–R45 | DOI | MR | Zbl

[2] M. I. Belishev, A. F. Vakulenko, “Ob odnoi zadache upravleniya dlya volnovogo uravneniya v $R^3$”, Zap. nauchn. semin. POMI, 332, 2006, 19–73

[3] M. I. Belishev, A. F. Vakulenko, “Reachable and unreachable sets in the scattering problem for acoustical equation in $\mathbb R^3$”, SIAM J. Math. Anal., 39:6 (2008), 1821–1850 | DOI | MR | Zbl

[4] M. I. Belishev, A. F. Vakulenko, “Inverse scattering problem for the wave equation with locally perturbed centrifugal potential”, J. Inverse Ill-Posed Probl., 17:2 (2009), 127–157 | DOI | MR | Zbl

[5] M. I. Belishev, A. F. Vakulenko, “$s$-points in three-dimensional acoustical scattering”, SIAM J. Math. Anal., 42:6 (2010), 2703–2720 | DOI | MR | Zbl

[6] V. I. Smirnov, Kurs vysshei matematiki, Chast 2, v. 4, Nauka, M., 1981 | MR

[7] I. Lasiecka, R. Triggiani, “Recent advances in regularity of second-order hyperbolic mixed problems, and applications”, Dynamics reported. Expositions in dynamical systems, v. 3, eds. Ch. K. R. T. Jones et al., Springer-Verlag, Berlin, 1994, 104–162 | Zbl

[8] P. Lax, R. Phillips, Scattering Theory, Academic Press, New-York–London, 1967 | MR | Zbl

[9] D. Tataru, “Unique continuation for solutions to PDE's: between Hormander's and Holmgren's theorem”, Comm. PDE, 20 (1995), 855–884 | DOI | MR | Zbl