Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T10:31:14.573Z Has data issue: false hasContentIssue false
  • English
  • Français

Single input controllability of a simplified fluid-structure interaction model

Published online by Cambridge University Press:  23 February 2012

Yuning Liu ,
Takéo Takahashi  and
Marius Tucsnak
Yuning Liu
Affiliation:
Institut Elie Cartan, Nancy Université/CNRS/INRIA, BP 70239, 54506 Vandoeuvre-lès-Nancy, France. [email protected]; [email protected]; [email protected]
Takéo Takahashi
Affiliation:
Institut Elie Cartan, Nancy Université/CNRS/INRIA, BP 70239, 54506 Vandoeuvre-lès-Nancy, France. [email protected]; [email protected]; [email protected]
Marius Tucsnak
Affiliation:
Institut Elie Cartan, Nancy Université/CNRS/INRIA, BP 70239, 54506 Vandoeuvre-lès-Nancy, France. [email protected]; [email protected]; [email protected]
Get access

Abstract

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.

Keywords

Null-controllabilityfluid-structure interactionviscous Burgers equation
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 20 - 42
Copyright
© EDP Sciences, SMAI, 2012

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

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).
Boulakia, M. and Osses, A., Local null controllability of a two-dimensional fluid-structure interaction problem. ESAIM : COCV 14 (2008) 1–42. Google Scholar
Dolecki, S. and Russell, D.L., A general theory of observation and control. SIAM J. Control Optim. 15 (1977) 185–220. Google Scholar
Doubova, A. and Fernández-Cara, E., Some control results for simplified one-dimensional models of fluid-solid interaction. Math. Models Methods Appl. Sci. 15 (2005) 783–824. Google Scholar
Fattorini, H.O. and Russell, D.L., Exact controllability theorems for linear parabolic equations in one space dimension. Arch. Rational Mech. Anal. 43 (1971) 272–292. Google Scholar
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).
Gozzi, F. and Loreti, P.. Regularity of the minimum time function and minimum energy problems : the linear case. SIAM J. Control Optim. 37 (1999) 1195–1221 (electronic). Google Scholar
Imanuvilov, O. and Takahashi, T., Exact controllability of a fluid-rigid body system. J. Math. Pures Appl. (9) 87 (2007) 408–437. Google Scholar
Imanuvilov, O.Y., Remarks on exact controllability for the Navier-Stokes equations. ESAIM : COCV 6 (2001) 39–72 (electronic). Google Scholar
Lebeau, G. and Robbiano, L., Contrôle exact de l’équation de la chaleur. Comm. Partial Differential Equations 20 (1995) 335–356. Google Scholar
Miller, L., A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups. Discrete Contin. Dyn. Syst., Ser. B 14 (2010) 1465–1485. Google Scholar
Saut, J.-C. and Scheurer, B., Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118–139. Google Scholar
Tenenbaum, G. and Tucsnak, M., On the null-controllability of diffusion equations. ESAIM : COCV 17 (2011) 1088–1100. Google Scholar
Tenenbaum, G. and Tucsnak, M., New blow-up rates for fast controls of Schrödinger and heat equations. J. Differential Equations 243 (2007) 70–100. Google Scholar
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).
Vázquez, J.L. and Zuazua, E., Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations 28 (2003) 1705–1738. Google Scholar
Vázquez, J.L. and Zuazua, E., Lack of collision in a simplified 1D model for fluid-solid interaction. Math. Models Methods Appl. Sci. 16 (2006) 637–678. Google Scholar