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

Osses, Axel; Puel, Jean-Pierre

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 279-294.

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 $0<{\theta }_0<2\pi$, ${\theta }_0\ne \pi$ and ${\theta }_0\ne 3\pi /2$, 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 Izvestiya 34 (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.

Classification : 35B60, 35B37
Mots clés : continuation of solutions of PDE, fluid-structure control, domains with corners

[1] T. Chatelain and A. Henrot, Some results about Schiffer's conjectures. Inverse Problems 15 (1999) 647-658. | MR | Zbl

[2] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman Advanced Publishing Program, Boston-London-Melbourne (1985). | MR | Zbl

[3] V.A. Kozlov, V.A. Kondratiev and V.G. Mazya, On sign variation and the absence of strong zeros of solutions of elliptic equations. Math. USSR Izvestiya 34 (1990) 337-353. | MR | Zbl

[4] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and applications. Springer-Verlag, Berlin (1972). | Zbl

[5] J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system. ESAIM: COCV 1 (1995) 1-15. | Numdam | MR | Zbl

[6] V.A. Kozlov, V.G. Mazya and J. Rossmann, Elliptic boundary value problems in domains with point singularities, Mathematical Surveys and Monographs 52. AMS, Providence (1997). | MR | Zbl

[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. | MR | Zbl

[8] A. Osses and J.-P. Puel, Approximate controllability of a linear model in solid-fluid interaction. ESAIM: COCV 4 (1999) 497-513. | Numdam | MR | Zbl

[9] S. Williams, A partial solution of the Pompeiu problem. Math. Anal. 223 (1976) 183-190. | MR | Zbl

[10] S. Williams, Analyticity of the boundary of Lipschitz domains without the Pompeiu property. Indiana Univ. Math. J. 30 (1981) 357-369. | MR | Zbl