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.
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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/cocv/2011196/
[1] Representation and control of infinite-dimensional systems, Systems & Control : Foundations & Applications 1. Birkhäuser Boston Inc., Boston, MA (1992). | MR | Zbl
, , and ,[2] Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM : COCV 14 (2008) 1-42. | Numdam | MR | Zbl
and ,[3] A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185-220. | MR | Zbl
and ,[4] Some control results for simplified one-dimensional models of fluid-solid interaction. Math. Models Methods Appl. Sci. 15 (2005) 783-824. | MR | Zbl
and ,[5] Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272-292. | MR | Zbl
and ,[6] Controllability of Evolution Equations, Lecture Notes Series 34. Seoul National University Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996). | MR | Zbl
and ,[7] Regularity of the minimum time function and minimum energy problems : the linear case. SIAM J. Control Optim. 37 (1999) 1195-1221 (electronic). | MR | Zbl
and .[8] Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. (9) 87 (2007) 408-437. | MR | Zbl
and ,[9] Remarks on exact controllability for the Navier-Stokes equations. ESAIM : COCV 6 (2001) 39-72 (electronic). | Numdam | MR | Zbl
,[10] Contrôle exact de l'équation de la chaleur. Comm. Partial Differential Equations 20 (1995) 335-356. | MR | Zbl
and ,[11] 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] Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118-139. | MR | Zbl
and ,[13] On the null-controllability of diffusion equations. ESAIM : COCV 17 (2011) 1088-1100. | Numdam | MR | Zbl
and ,[14] New blow-up rates for fast controls of Schrödinger and heat equations. J. Differential Equations 243 (2007) 70-100. | MR | Zbl
and ,[15] Observation and control for operator semigroups. Birkhäuser Advanced Texts : Basler Lehrbücher [Birkhäuser Advanced Texts : Basel Textbooks], Birkhäuser Verlag, Basel (2009). | MR | Zbl
and ,[16] Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations 28 (2003) 1705-1738. | MR | Zbl
and ,[17] Lack of collision in a simplified 1D model for fluid-solid interaction. Math. Models Methods Appl. Sci. 16 (2006) 637-678. | MR
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