Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T04:55:16.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximate controllability for a linear model of fluid structure interaction

Published online by Cambridge University Press:  15 August 2002

Axel Osses  and
Jean-Pierre Puel
Axel Osses
Affiliation:
Universidad de Chile, Departamento de Ingeniería Matemática, Casilla 170/3, Correo 3, Santiago, Chile; axosses@dim.uchile.cl.
Jean-Pierre Puel
Affiliation:
Université de Versailles Saint-Quentin and Centre de Ma thé ma ti ques Appliquées, École Polytechnique, 91128 Palaiseau Cedex, France; jppuel@cmapx.polytechnique.fr.
Get access

Abstract

We consider a linear model of interaction between a viscous incompressible fluid and a thin elastic structure located on a part of the fluid domain boundary, the other part being rigid. After having given an existence and uniqueness result for the direct problem, we study the question of approximate controllability for this system when the control acts as a normal force applied to the structure. The case of an analytic boundary has been studied by Lions and Zuazua in [9] where, in particular, a counterexample is given when the fluid domain is a ball. We prove a result of approximate controllability in the 2d-case when the rigid and the elastic parts of the boundary make a rectangular corner and if the control acts on the whole elastic structure.

Keywords

Controllabilityfluid-structure interactionnonsmooth domainsunique continuation property.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 497 - 513
Copyright
© EDP Sciences, SMAI, 1999

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berenstein, C., An inverse spectral theorem and its relation to the Pompeiu problem. J. Anal. Math. 37 (1980) 128-144. CrossRef
Berenstein, C., The Pompeiu problem, what's new?, Deville R. et al. (Ed.), Complex analysis, harmonic analysis and applications. Proceedings of a conference in honour of the retirement of Roger Gay, June 7-9, 1995, Bordeaux, France. Harlow: Longman. Pitman Res. Notes Math. Ser . 347 (1996) 1-11.
Beretta, E. and Vogelius, M., An inverse problem originating from magnetohydrodynamics. III: Domains with corners of arbitrary angles. Asymptotic Anal. 11 (1995) 289-315.
H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Collection Math. Appl. Pour la Maîtrise, Masson, Paris (1983).
Brown, L., Schreiber, B.M. and Taylor, B.A., Spectral synthesis and the Pompeiu problem. Ann. Inst. Fourier 23 (1973) 125-154. CrossRef
P. Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, 24. Pitman, Boston-London-Melbourne (1985).
J.-L. Lions, Remarques sur la contrôlabilité approchée, Control of distributed systems, Span.-Fr. Days, Malaga/Spain 1990, Grupo Anal. Mat. Apl. Univ. Malaga 3 (1990) 77-87.
J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vols. I, II, III, Dunod, Paris (1968).
Lions, J.-L. and Zuazua, E., Approximate controllability of a hydro-elastic coupled system. ESAIM: Contr. Optim. Calc. Var. 1 (1995) 1-15.
A. Osses, A rotated direction multiplier technique. Applications to the controllability of waves, elasticity and tangential Stokes control, SIAM J. Cont. Optim., to appear.
A. Osses and J.-P. Puel, Approximate controllability of a linear model in solid-fluid interaction in a rectangle. to appear.
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York. Appl. Math. Sci. 44 (1983).
Serrin, J., A symmetry problem in potential theory. Arch. Rational. Mech. Anal. 43 (1971) 304-318. CrossRef
R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam (1977).
M. Vogelius, An inverse problem for the equation $\Delta u=-cu-d$ .Ann. Inst. Fourier, 44 (1994) 1181-1209.
Williams, S.A., Analyticity of the boundary for Lipschitz domains without the Pompeiu property. Indiana Univ. Math. J. 30 (1981) 357-369. CrossRef