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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 $\mathbb{R}^3$

Mikhail I. Belishev 1, and  Aleksei F. Vakulenko 1,

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 ${\rm sing\,supp\,}h \subset \{x\in{\mathbb R}^3|\,\,|x|=\xi_0>\xi\}$.
Keywords: unobservable states., control problem, $3d$ wave equation, reachable sets.
Mathematics Subject Classification: Primary: 35Bxx, 35Lxx, 35P25, 47Ax.
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

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