Unique continuation property near a corner and its fluid-structure controllability consequences

Axel Osses; Jean-Pierre Puel

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 15, Issue: 2, page 279-294
  • ISSN: 1292-8119

Abstract

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We study a non standard unique continuation property for the biharmonic spectral problem Δ 2 w = - λ Δ w in a 2D corner with homogeneous Dirichlet boundary conditions and a supplementary third order boundary condition on one side of the corner. We prove that if the corner has an angle 0 < θ 0 < 2 π , θ 0 ≠ π and θ 0 ≠ 3 π / 2 , a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing a Stokes fluid. The proof of the main result is based in a power series expansion of the eigenfunctions near the corner, the resolution of a coupled infinite set of finite dimensional linear systems, and a result of Kozlov, Kondratiev and Mazya, concerning the absence of strong zeros for the biharmonic operator [Math. USSR Izvestiya34 (1990) 337–353]. We also show how the same methodology used here can be adapted to exclude domains with corners to have a local version of the Schiffer property for the Laplace operator.

How to cite

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Osses, Axel, and Puel, Jean-Pierre. "Unique continuation property near a corner and its fluid-structure controllability consequences." ESAIM: Control, Optimisation and Calculus of Variations 15.2 (2008): 279-294. <http://eudml.org/doc/90914>.

@article{Osses2008,
abstract = { We study a non standard unique continuation property for the biharmonic spectral problem $\Delta^2 w=-\lambda\Delta w$ in a 2D corner with homogeneous Dirichlet boundary conditions and a supplementary third order boundary condition on one side of the corner. We prove that if the corner has an angle $0<\theta_0<2\pi$, $\theta_0\not=\pi$ and $\theta_0\not=3\pi/2$, a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing a Stokes fluid. The proof of the main result is based in a power series expansion of the eigenfunctions near the corner, the resolution of a coupled infinite set of finite dimensional linear systems, and a result of Kozlov, Kondratiev and Mazya, concerning the absence of strong zeros for the biharmonic operator [Math. USSR Izvestiya34 (1990) 337–353]. We also show how the same methodology used here can be adapted to exclude domains with corners to have a local version of the Schiffer property for the Laplace operator. },
author = {Osses, Axel, Puel, Jean-Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Continuation of solutions of PDE; fluid-structure control; domains with corners; approximate controllability; biharmonic spectral problem; homogeneous Dirichlet boundary conditions; supplementary third order boundary condition},
language = {eng},
month = {3},
number = {2},
pages = {279-294},
publisher = {EDP Sciences},
title = {Unique continuation property near a corner and its fluid-structure controllability consequences},
url = {http://eudml.org/doc/90914},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Osses, Axel
AU - Puel, Jean-Pierre
TI - Unique continuation property near a corner and its fluid-structure controllability consequences
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/3//
PB - EDP Sciences
VL - 15
IS - 2
SP - 279
EP - 294
AB - We study a non standard unique continuation property for the biharmonic spectral problem $\Delta^2 w=-\lambda\Delta w$ in a 2D corner with homogeneous Dirichlet boundary conditions and a supplementary third order boundary condition on one side of the corner. We prove that if the corner has an angle $0<\theta_0<2\pi$, $\theta_0\not=\pi$ and $\theta_0\not=3\pi/2$, a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing a Stokes fluid. The proof of the main result is based in a power series expansion of the eigenfunctions near the corner, the resolution of a coupled infinite set of finite dimensional linear systems, and a result of Kozlov, Kondratiev and Mazya, concerning the absence of strong zeros for the biharmonic operator [Math. USSR Izvestiya34 (1990) 337–353]. We also show how the same methodology used here can be adapted to exclude domains with corners to have a local version of the Schiffer property for the Laplace operator.
LA - eng
KW - Continuation of solutions of PDE; fluid-structure control; domains with corners; approximate controllability; biharmonic spectral problem; homogeneous Dirichlet boundary conditions; supplementary third order boundary condition
UR - http://eudml.org/doc/90914
ER -

References

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  6. V.A. Kozlov, V.G. Mazya and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs52. AMS, Providence (1997).  
  7. A. Osses and J.-P. Puel, Approximate controllability for a hydro-elastic model in a rectangular domain, in Optimal Control of partial Differential Equations (Chemnitz, 1998), Internat. Ser. Numer. Math.133, Birkhäuser, Basel (1999) 231–243.  
  8. A. Osses and J.-P. Puel, Approximate controllability of a linear model in solid-fluid interaction. ESAIM: COCV4 (1999) 497–513.  
  9. S. Williams, A partial solution of the Pompeiu problem. Math. Anal.223 (1976) 183–190.  
  10. S. Williams, Analyticity of the boundary of Lipschitz domains without the Pompeiu property. Indiana Univ. Math. J.30 (1981) 357–369.  

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