# Approximate controllability for a linear model of fluid structure interaction

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 4, page 497-513
- ISSN: 1292-8119

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topOsses, Axel, and Puel, Jean-Pierre. "Approximate controllability for a linear model of fluid structure interaction." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 497-513. <http://eudml.org/doc/197380>.

@article{Osses2010,

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.
},

author = {Osses, Axel, Puel, Jean-Pierre},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Controllability; fluid-structure interaction; nonsmooth domains;
unique continuation property.; unique continuation property; eigenvalue problem},

language = {eng},

month = {3},

pages = {497-513},

publisher = {EDP Sciences},

title = {Approximate controllability for a linear model of fluid structure interaction},

url = {http://eudml.org/doc/197380},

volume = {4},

year = {2010},

}

TY - JOUR

AU - Osses, Axel

AU - Puel, Jean-Pierre

TI - Approximate controllability for a linear model of fluid structure interaction

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 4

SP - 497

EP - 513

AB -
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.

LA - eng

KW - Controllability; fluid-structure interaction; nonsmooth domains;
unique continuation property.; unique continuation property; eigenvalue problem

UR - http://eudml.org/doc/197380

ER -

## References

top- C. Berenstein, An inverse spectral theorem and its relation to the Pompeiu problem. J. Anal. Math.37 (1980) 128-144. Zbl0449.35024
- C. Berenstein, 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.
- E. Beretta and M. Vogelius, An inverse problem originating from magnetohydrodynamics. III: Domains with corners of arbitrary angles. Asymptotic Anal.11 (1995) 289-315. Zbl0853.76093
- H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Collection Math. Appl. Pour la Maîtrise, Masson, Paris (1983).
- L. Brown, B.M. Schreiber and B.A. Taylor, Spectral synthesis and the Pompeiu problem. Ann. Inst. Fourier23 (1973) 125-154. Zbl0265.46044
- P. Grisvard, Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, 24. Pitman, Boston-London-Melbourne (1985). Zbl0695.35060
- 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). Zbl0165.10801
- J.-L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system. ESAIM: Contr. Optim. Calc. Var.1 (1995) 1-15. Zbl0878.93034
- A. Osses, A rotated direction multiplier technique. Applications to the controllability of waves, elasticity and tangential Stokes control, SIAM J. Cont. Optim., to appear. Zbl0997.93013
- A. Osses and J.-P. Puel, Approximate controllability of a linear model in solid-fluid interaction in a rectangle. to appear. Zbl0919.35019
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York. Appl. Math. Sci.44 (1983). Zbl0516.47023
- J. Serrin, A symmetry problem in potential theory. Arch. Rational. Mech. Anal.43 (1971) 304-318. Zbl0222.31007
- 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. Zbl0813.35136
- S.A. Williams, Analyticity of the boundary for Lipschitz domains without the Pompeiu property. Indiana Univ. Math. J.30 (1981) 357-369. Zbl0439.35046

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