# American Institute of Mathematical Sciences

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 & Control Theory, 2014, 3 (2) : 247-256. doi: 10.3934/eect.2014.3.247
##### References:
 [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.  doi: 10.1007/BF02405808.  Google Scholar [2] M. I. Belishev, Recent progress in the boundary control method,, Invers Problems, 23 (2007).  doi: 10.1088/0266-5611/23/5/R01.  Google Scholar [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.  doi: 10.1007/s10958-007-0140-3.  Google Scholar [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.  doi: 10.1137/060678877.  Google Scholar [5] M. I. Belishev and A. F. Vakulenko, $s$-points in three-dimensional acoustical scattering,, SIAM J. Math. Analysis, 42 (2010), 2703.  doi: 10.1137/090781486.  Google Scholar [6] S. Helgason, The Radon Transform,, Birhausser, (1999).  doi: 10.1007/978-1-4757-1463-0.  Google Scholar [7] L. Hörmander, The Analysis of Linear Partial Differential Operators I,, Springer-Verlag, (1983).   Google Scholar [8] M. Ikawa, Hyperbolic Partial Differential Equations and Wave Fenomena,, Translated from the 1997 Japanese original by Bohdan I. Kurpita, (1997).   Google Scholar [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.   Google Scholar [10] P. Lax and R. Phillips, Scattering Theory,, Academic Press, (1967).   Google Scholar [11] D. L. Russell, Boundary value control theory of the higher-dimensional wave equation,, SIAM J. Control, 9 (1971), 29.  doi: 10.1137/0309004.  Google Scholar

show all references

##### References:
 [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.  doi: 10.1007/BF02405808.  Google Scholar [2] M. I. Belishev, Recent progress in the boundary control method,, Invers Problems, 23 (2007).  doi: 10.1088/0266-5611/23/5/R01.  Google Scholar [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.  doi: 10.1007/s10958-007-0140-3.  Google Scholar [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.  doi: 10.1137/060678877.  Google Scholar [5] M. I. Belishev and A. F. Vakulenko, $s$-points in three-dimensional acoustical scattering,, SIAM J. Math. Analysis, 42 (2010), 2703.  doi: 10.1137/090781486.  Google Scholar [6] S. Helgason, The Radon Transform,, Birhausser, (1999).  doi: 10.1007/978-1-4757-1463-0.  Google Scholar [7] L. Hörmander, The Analysis of Linear Partial Differential Operators I,, Springer-Verlag, (1983).   Google Scholar [8] M. Ikawa, Hyperbolic Partial Differential Equations and Wave Fenomena,, Translated from the 1997 Japanese original by Bohdan I. Kurpita, (1997).   Google Scholar [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.   Google Scholar [10] P. Lax and R. Phillips, Scattering Theory,, Academic Press, (1967).   Google Scholar [11] D. L. Russell, Boundary value control theory of the higher-dimensional wave equation,, SIAM J. Control, 9 (1971), 29.  doi: 10.1137/0309004.  Google Scholar
 [1] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [2] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [3] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [4] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [5] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264 [6] Héctor Barge. Čech cohomology, homoclinic trajectories and robustness of non-saddle sets. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020381 [7] Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297 [8] Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019 [9] Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011 [10] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [11] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273 [12] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [13] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [14] H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433 [15] Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020 [16] Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215 [17] Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 [18] Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466 [19] Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046 [20] Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

2019 Impact Factor: 0.953