Single input controllability of a simplified fluid-structure interaction model
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 20-42.

In this paper we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. The control variable is the velocity of the fluid at one end. One of the novelties brought in with respect to the existing literature consists in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem. This methodology is based on an abstract argument for the null controllability of parabolic equations in the presence of source terms and it avoids tackling linearized problems with time dependent coefficients.

DOI : 10.1051/cocv/2011196
Classification : 35L10, 65M60, 93B05, 93B40, 93D15
Mots-clés : null-controllability, fluid-structure interaction, viscous Burgers equation 
@article{COCV_2013__19_1_20_0,
     author = {Liu, Yuning and Takahashi, Tak\'eo and Tucsnak, Marius},
     title = {Single input controllability of a simplified fluid-structure interaction model},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {20--42},
     publisher = {EDP-Sciences},
     volume = {19},
     number = {1},
     year = {2013},
     doi = {10.1051/cocv/2011196},
     mrnumber = {3023058},
     zbl = {1270.35259},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2011196/}
}
TY  - JOUR
AU  - Liu, Yuning
AU  - Takahashi, Takéo
AU  - Tucsnak, Marius
TI  - Single input controllability of a simplified fluid-structure interaction model
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2013
SP  - 20
EP  - 42
VL  - 19
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2011196/
DO  - 10.1051/cocv/2011196
LA  - en
ID  - COCV_2013__19_1_20_0
ER  - 
%0 Journal Article
%A Liu, Yuning
%A Takahashi, Takéo
%A Tucsnak, Marius
%T Single input controllability of a simplified fluid-structure interaction model
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2013
%P 20-42
%V 19
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2011196/
%R 10.1051/cocv/2011196
%G en
%F COCV_2013__19_1_20_0
Liu, Yuning; Takahashi, Takéo; Tucsnak, Marius. Single input controllability of a simplified fluid-structure interaction model. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 1, pp. 20-42. doi : 10.1051/cocv/2011196. http://archive.numdam.org/articles/10.1051/cocv/2011196/

[1] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite-dimensional systems, Systems & Control : Foundations & Applications 1. Birkhäuser Boston Inc., Boston, MA (1992). | MR | Zbl

[2] M. Boulakia and A. Osses, Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM : COCV 14 (2008) 1-42. | Numdam | MR | Zbl

[3] S. Dolecki and D.L. Russell, A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185-220. | MR | Zbl

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

[5] H.O. Fattorini and D.L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272-292. | MR | Zbl

[6] 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

[7] F. Gozzi and P. Loreti. Regularity of the minimum time function and minimum energy problems : the linear case. SIAM J. Control Optim. 37 (1999) 1195-1221 (electronic). | MR | Zbl

[8] O. Imanuvilov and T. Takahashi, Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. (9) 87 (2007) 408-437. | MR | Zbl

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

[10] G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur. Comm. Partial Differential Equations 20 (1995) 335-356. | MR | Zbl

[11] L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete Contin. Dyn. Syst., Ser. B 14 (2010) 1465-1485. | MR | Zbl

[12] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118-139. | MR | Zbl

[13] G. Tenenbaum and M. Tucsnak, On the null-controllability of diffusion equations. ESAIM : COCV 17 (2011) 1088-1100. | Numdam | MR | Zbl

[14] G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrödinger and heat equations. J. Differential Equations 243 (2007) 70-100. | MR | Zbl

[15] M. Tucsnak and G. Weiss, Observation and control for operator semigroups. Birkhäuser Advanced Texts : Basler Lehrbücher [Birkhäuser Advanced Texts : Basel Textbooks], Birkhäuser Verlag, Basel (2009). | MR | Zbl

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

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

Cité par Sources :