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Controllability of a simplified model of fluid-structure interaction∗

Published online by Cambridge University Press:  28 March 2014

S. Ervedoza  and
M. Vanninathan
S. Ervedoza
Affiliation:
Institut de Mathématiques de Toulouse ; UMR5219; Université de Toulouse ; CNRS; UPS IMT, F-31062 Toulouse Cedex 9, France. ervedoza@math.univ-toulouse.fr; vanni@math.tifrbng.res.in
M. Vanninathan
Affiliation:
Institut de Mathématiques de Toulouse ; UMR5219; Université de Toulouse ; CNRS; UPS IMT, F-31062 Toulouse Cedex 9, France. ervedoza@math.univ-toulouse.fr; vanni@math.tifrbng.res.in
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Abstract

This article aims at studying the controllability of a simplified fluid structure interaction model derived and developed in [C. Conca, J. Planchard and M. Vanninathan, RAM: Res. Appl. Math. John Wiley & Sons Ltd., Chichester (1995); J.-P. Raymond and M. Vanninathan, ESAIM: COCV 11 (2005) 180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58 (2009) 547–552]. This interaction is modeled by a wave equation surrounding a harmonic oscillator. Our main result states that, in the radially symmetric case, this system can be controlled from the outer boundary. This improves previous results [J.-P. Raymond and M. Vanninathan, ESAIM: COCV 11 (2005) 180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58 (2009) 547–552]. Our proof is based on a spherical harmonic decomposition of the solution and the so-called lateral propagation of the energy for 1d waves.

Keywords

Controllabilityobservabilityfluid-structure interaction
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 547 - 575
Copyright
© EDP Sciences, SMAI, 2014

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References

Bardos, C., Lebeau, G. and Rauch, J., Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques. Nonlinear hyperbolic equations in applied sciences. Rend. Sem. Mat. Univ. Politec. Torino, (Special Issue) 1988 (1989) 11–31. Google Scholar
Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065. Google Scholar
N. Burq, Mesures semi-classiques et mesures de défaut. Séminaire Bourbaki, Vol. 1996/97. Astérisque, (245): Exp. No. 826 (1997) 167–195.
Burq, N. and Gérard, P., Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 749–752. Google Scholar
Burq, N. and Zworski, M., Geometric control in the presence of a black box. J. Amer. Math. Soc. 17 (2004) 443–471. Google Scholar
C. Conca, J. Planchard and M. Vanninathan, Fluids and periodic structures, vol. 38 of RAM: Res. Appl. Math. John Wiley & Sons Ltd., Chichester (1995).
Dehman, B. and Lebeau, G., Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time. SIAM J. Control Optim. 48 (2009) 521–550. Google Scholar
Ervedoza, S., Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numer. Math. 113 (2009) 377–415. Google Scholar
Ervedoza, S. and Zuazua, E.. A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1375–1401. Google Scholar
S. Ervedoza and E. Zuazua, The wave equation: Control and numerics. Control Partial Differ. Eqs. Lect. Notes Math., CIME Subseries. edited by P.M. Cannarsa and J.M. Coron. Springer Verlag (2011).
Gérard, P., Microlocal defect measures. Commun. Partial Differ. Eqs. 16 (1991) 1761–1794. Google Scholar
L. Hörmander, The analysis of linear partial differential operators. I, Distribution theory and Fourier analysis. Vol. 256 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2nd edn. (1990).
J.-L. Lions, Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, vol. 8 RMA. Masson (1988).
Lions, J.-L., Exact controllability, stabilization and perturbations for distributed systems. SIAM Review 30 (1988) 1–68. Google Scholar
Melrose, R.B. and Sjöstrand, J., Singularities of boundary value problems. II. Commun. Pure Appl. Math. 35 (1982) 129–168. Google Scholar
Miller, L., Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218 (2005) 425–444. Google Scholar
Ralston, J.V., Solutions of the wave equation with localized energy. Commun. Pure Appl. Math. 22 (1969) 807–823. Google Scholar
Raymond, J.-P. and Vanninathan, M., Exact controllability in fluid-solid structure: the Helmholtz model. ESAIM: COCV 11 (2005) 180–203. Google Scholar
Raymond, J.-P. and Vanninathan, M., Null controllability in a fluid-solid structure model. J. Differ. Eqs. 248 (2010) 1826–1865. Google Scholar
Tucsnak, M. and Vanninathan, M., Locally distributed control for a model of fluid-structure interaction. Systems Control Lett. 58 (2009) 547–552. Google Scholar
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, vol. XI of Birkäuser Advanced Texts. Springer (2009).
Zuazua, E., Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. Henri Poincaré Anal. Non Linéaire 10 (1993) 109–129. Google Scholar