Local null controllability of a two-dimensional fluid-structure interaction problem
ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 1, pp. 1-42.

In this paper, we prove a controllability result for a fluid-structure interaction problem. In dimension two, a rigid structure moves into an incompressible fluid governed by Navier-Stokes equations. The control acts on a fixed subset of the fluid domain. We prove that, for small initial data, this system is null controllable, that is, for a given T>0, the system can be driven at rest and the structure to its reference configuration at time T. To show this result, we first consider a linearized system. Thanks to an observability inequality obtained from a Carleman inequality, we prove an optimal controllability result with a regular control. Next, with the help of Kakutani’s fixed point theorem and a regularity result, we pass to the nonlinear problem.

DOI: 10.1051/cocv:2007031
Classification: 35Q30,  93C20
Keywords: controllability, fluid-solid interaction, Navier-Stokes equations, Carleman estimates
@article{COCV_2008__14_1_1_0,
     author = {Boulakia, Muriel and Osses, Axel},
     title = {Local null controllability of a two-dimensional fluid-structure interaction problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1--42},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {1},
     year = {2008},
     doi = {10.1051/cocv:2007031},
     zbl = {1149.35068},
     mrnumber = {2375750},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2007031/}
}
TY  - JOUR
AU  - Boulakia, Muriel
AU  - Osses, Axel
TI  - Local null controllability of a two-dimensional fluid-structure interaction problem
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2008
DA  - 2008///
SP  - 1
EP  - 42
VL  - 14
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2007031/
UR  - https://zbmath.org/?q=an%3A1149.35068
UR  - https://www.ams.org/mathscinet-getitem?mr=2375750
UR  - https://doi.org/10.1051/cocv:2007031
DO  - 10.1051/cocv:2007031
LA  - en
ID  - COCV_2008__14_1_1_0
ER  - 
%0 Journal Article
%A Boulakia, Muriel
%A Osses, Axel
%T Local null controllability of a two-dimensional fluid-structure interaction problem
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2008
%P 1-42
%V 14
%N 1
%I EDP-Sciences
%U https://doi.org/10.1051/cocv:2007031
%R 10.1051/cocv:2007031
%G en
%F COCV_2008__14_1_1_0
Boulakia, Muriel; Osses, Axel. Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 1, pp. 1-42. doi : 10.1051/cocv:2007031. http://www.numdam.org/articles/10.1051/cocv:2007031/

[1] S. Anita and V. Barbu, Null controllability of nonlinear convective heat equations. ESAIM: COCV 5 (2000) 157-173. | Numdam | MR | Zbl

[2] M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem. Prépublication 139, UVSQ (octobre 2005).

[3] C. Conca, J. San Martin and M. Tucsnak, Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Comm. Partial Differential Equations 25 (2000) 1019-1042. | MR | Zbl

[4] J.M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example. Asymptot. Anal. 44 (2005) 237-257. | MR | Zbl

[5] B. Desjardins and M.J. Esteban, Existence of weak solutions for the motion of rigid bodies in a viscous fluid. Arch. Ration. Mech. Anal. 146 (1999) 59-71. | MR | Zbl

[6] A. Doubova and E. Fernandez-Cara, Some control results for simplified one-dimensional models of fluid-solid interaction. Math. Models Methods Appl. Sci. 15 (2005) 783-824. | MR | Zbl

[7] C. Fabre and G. Lebeau, Prolongement unique des solutions de l'équation de Stokes. Comm. Partial Diff. Equations 21 (1996) 573-596. | MR | Zbl

[8] C. Fabre, J.-P. Puel and E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. Royal Soc. Edinburgh 125A (1995) 31-61. | MR | Zbl

[9] E. Fernandez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: the linear case. Adv. Differential Equations 5 (2000) 465-514. | MR | Zbl

[10] E. Fernandez-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system. J. Math. Pures Appl. 83 (2004) 1501-1542. | MR

[11] A.V. Fursikov and O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR | Zbl

[12] O.Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations. ESAIM: COCV 6 (2001) 39-72. | Numdam | MR | Zbl

[13] O.Yu. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Internat. Math. Res. Notices 16 (2003) 883-913. | MR | Zbl

[14] O.Yu. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system. Prépublication IECN (novembre 2005). | Zbl

[15] O. Nakoulima, Contrôlabilité à zéro avec contraintes sur le contrôle. C. R. Acad. Sci. Paris Ser. I 339 (2004) 405-410. | MR | Zbl

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

[17] J.P. Raymond and M. Vanninathan, Exact controllability in fluid-solid structure: the Helmholtz model. ESAIM: COCV 11 (2005) 180-203. | Numdam | MR | Zbl

[18] J. San Martin, V. Starovoitov and M. Tucsnak, Global weak solutions for the two dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Rational Mech. Anal. 161 (2002) 113-147. | MR | Zbl

[19] T. Takahashi, Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain, Adv. Differential Equations 8 (2003) 1499-1532. | MR | Zbl

[20] R. Temam, Behaviour at time t=0 of the solutions of semi-linear evolution equations. J. Diff. Equations 43 (1982) 73-92. | MR | Zbl

[21] J.L. Vázquez, E. Zuazua, Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations 28 (2003) 1705-1738. | MR | Zbl

[22] J.L. Vázquez and E. Zuazua, Lack of collision in a simplified 1-dimensional model for fluid-solid interaction. Math. Models Methods Apll. Sci., M3AS 16 (2006) 637-678. | MR

[23] X. Zhang and E. Zuazua, Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system. J. Differential Equations 204 (2004) 380-438. | MR | Zbl

Cited by Sources: