    June  2014, 3(2): 247-256. doi: 10.3934/eect.2014.3.247

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

 1 Saint-Petersburg Department of the Steklov Mathematical Institute, Saint-Petersburg State University, Russian Federation, Russian Federation

Received  November 2013 Revised  April 2014 Published  May 2014

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 ${\rm sing\,supp\,}h \subset \{x\in{\mathbb R}^3|\,\,|x|=\xi_0>\xi\}$.
Citation: Mikhail I. Belishev, Aleksei F. Vakulenko. Non-smooth unobservable states in control problem for the wave equation in $\mathbb{R}^3$. Evolution Equations and Control Theory, 2014, 3 (2) : 247-256. doi: 10.3934/eect.2014.3.247
##### References:
  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.   M. I. Belishev, Recent progress in the boundary control method, Invers Problems, 23 (2007), R1-R67. doi: 10.1088/0266-5611/23/5/R01.   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.   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.   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.   S. Helgason, The Radon Transform, Birhausser, Boston, Basel, Stuttgart, 1999. doi: 10.1007/978-1-4757-1463-0.   L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, Heidelberg, New-York, Tokyo, 1983. 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.  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.  P. Lax and R. Phillips, Scattering Theory, Academic Press, New-York-London, 1967.  D. L. Russell, Boundary value control theory of the higher-dimensional wave equation, SIAM J. Control, 9 (1971), 29-42. doi: 10.1137/0309004.   show all references

##### References:
  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.   M. I. Belishev, Recent progress in the boundary control method, Invers Problems, 23 (2007), R1-R67. doi: 10.1088/0266-5611/23/5/R01.   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.   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.   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.   S. Helgason, The Radon Transform, Birhausser, Boston, Basel, Stuttgart, 1999. doi: 10.1007/978-1-4757-1463-0.   L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin, Heidelberg, New-York, Tokyo, 1983. 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.  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.  P. Lax and R. Phillips, Scattering Theory, Academic Press, New-York-London, 1967.  D. L. Russell, Boundary value control theory of the higher-dimensional wave equation, SIAM J. Control, 9 (1971), 29-42. doi: 10.1137/0309004.   Tomoyuki Tanaka, Kyouhei Wakasa. On the critical decay for the wave equation with a cubic convolution in 3D. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4545-4566. doi: 10.3934/dcds.2021048  Felipe Linares, Jean-Claude Saut. The Cauchy problem for the 3D Zakharov-Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 547-565. doi: 10.3934/dcds.2009.24.547  Indranil SenGupta, Weisheng Jiang, Bo Sun, Maria Christina Mariani. Superradiance problem in a 3D annular domain. Conference Publications, 2011, 2011 (Special) : 1309-1318. doi: 10.3934/proc.2011.2011.1309  Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519  Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507  Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407  Roberta Fabbri, Sylvia Novo, Carmen Núñez, Rafael Obaya. Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042  Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete and Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361  Alp Eden, Varga K. Kalantarov. 3D convective Cahn--Hilliard equation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1075-1086. doi: 10.3934/cpaa.2007.6.1075  Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251  Wenjing Liu, Rong Yang, Xin-Guang Yang. Dynamics of a 3D Brinkman-Forchheimer equation with infinite delay. Communications on Pure and Applied Analysis, 2021, 20 (5) : 1907-1930. doi: 10.3934/cpaa.2021052  Kunio Hidano, Dongbing Zha. Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1735-1767. doi: 10.3934/cpaa.2019082  Masaru Yamaguchi. 3D wave equations in sphere-symmetric domain with periodically oscillating boundaries. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 385-396. doi: 10.3934/dcds.2001.7.385  A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289  Yang Liu, Xin Zhong. On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5219-5238. doi: 10.3934/cpaa.2020234  Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems and Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991  Michael V. Klibanov, Dinh-Liem Nguyen, Loc H. Nguyen, Hui Liu. A globally convergent numerical method for a 3D coefficient inverse problem with a single measurement of multi-frequency data. Inverse Problems and Imaging, 2018, 12 (2) : 493-523. doi: 10.3934/ipi.2018021  Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5621-5632. doi: 10.3934/dcdsb.2019075  Xuecheng Wang. Global solution for the $3D$ quadratic Schrödinger equation of $Q(u, \bar{u}$) type. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5037-5048. doi: 10.3934/dcds.2017217  Chuntian Wang. The existence of strong solutions to the $3D$ Zakharov-Kuznestov equation in a bounded domain. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4897-4910. doi: 10.3934/dcds.2014.34.4897

2020 Impact Factor: 1.081