Article Contents
Article Contents

# Non-smooth unobservable states in control problem for the wave equation in $\mathbb{R}^3$

• The paper deals with a dynamical system \begin{align*} &u_{tt}-\Delta u=0, \qquad (x,t) \in {\mathbb R}^3 \times (-\infty,0) \\ &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*} where $u=u^f(x,t)$ is a solution ( wave), $f \in {\mathcal F}:=L_2\left([0,\infty);L_2\left(S^2\right)\right)$ is a control. For the reachable sets ${\mathcal U}^\xi:=\{u^f(\cdot,-\xi)\,|\,\, f \in {\mathcal F}\}\,\,(\xi\geq 0)$, the embedding ${\mathcal U}^\xi \subset {\mathcal H}^\xi:=\{y \in L_2({\mathbb R}^3)\,|\,\,\,y|_{|x|<\xi}=0\}$ holds, whereas the subspaces ${\mathcal D}^\xi:={\mathcal H}^\xi \ominus {\mathcal U}^\xi$ of unreachable ( unobservable) states are nonzero for $\xi> 0$. There was a conjecture motivated by some geometrical optics arguments that the elements of ${\mathcal D}^\xi$ are $C^\infty$-smooth with respect to $|x|$. We provide rather unexpected counterexamples of $h\in {\mathcal D}^\xi$ with ${\rm sing\,supp\,}h \subset \{x\in{\mathbb R}^3|\,\,|x|=\xi_0>\xi\}$.
Mathematics Subject Classification: Primary: 35Bxx, 35Lxx, 35P25, 47Axx.

 Citation:

•  [1] S. A. Avdonin, M. I. Belishev and S. I. Ivanov, Controllability in the filled domain for the wave equation with a singular boundary control, (in Russian) Zap. Nauch. Semin. POMI, 210 (1994), 7-21; English translation in J. Sov. Math., 83 (1997), 165-174.doi: 10.1007/BF02405808. [2] M. I. Belishev, Recent progress in the boundary control method, Invers Problems, 23 (2007), R1-R67.doi: 10.1088/0266-5611/23/5/R01. [3] M. I. Belishev and A. F. Vakulenko, On a control problem for the wave equation in $\mathbbR^3$, (in Russian) Zap. Nauch. Semin. POMI, 332 (2006), 19-37; English translation in J. Math. Sci., 142 (2007), 2528-2539.doi: 10.1007/s10958-007-0140-3. [4] M. I. Belishev and A. F. Vakulenko, Reachable and unreachable sets in the scattering problem for the acoustical equation in $\mathbbR^3$, SIAM J. Math. Analysis, 39 (2008), 1821-1850.doi: 10.1137/060678877. [5] M. I. Belishev and A. F. Vakulenko, $s$-points in three-dimensional acoustical scattering, SIAM J. Math. Analysis, 42 (2010), 2703-2720.doi: 10.1137/090781486. [6] S. Helgason, The Radon Transform, Birhausser, Boston, Basel, Stuttgart, 1999.doi: 10.1007/978-1-4757-1463-0. [7] L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, Heidelberg, New-York, Tokyo, 1983. [8] M. Ikawa, Hyperbolic Partial Differential Equations and Wave Fenomena, Translated from the 1997 Japanese original by Bohdan I. Kurpita, Translations of Mathematical Monographs, 189, Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2000. [9] I. Lasiecka, J-L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl, 65 (1986), 149-192. [10] P. Lax and R. Phillips, Scattering Theory, Academic Press, New-York-London, 1967. [11] D. L. Russell, Boundary value control theory of the higher-dimensional wave equation, SIAM J. Control, 9 (1971), 29-42.doi: 10.1137/0309004.