Controllability of a simplified model of fluid-structure interaction
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 2, p. 547-575

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.

DOI : https://doi.org/10.1051/cocv/2013075
Classification:  93B05,  93B07,  93C20,  74F10
Keywords: controllability, observability, fluid-structure interaction
@article{COCV_2014__20_2_547_0,
     author = {Ervedoza, S. and Vanninathan, M.},
     title = {Controllability of a simplified model of fluid-structure interaction},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {20},
     number = {2},
     year = {2014},
     pages = {547-575},
     doi = {10.1051/cocv/2013075},
     zbl = {1285.93019},
     mrnumber = {3264216},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2014__20_2_547_0}
}
Ervedoza, S.; Vanninathan, M. Controllability of a simplified model of fluid-structure interaction. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 2, pp. 547-575. doi : 10.1051/cocv/2013075. http://www.numdam.org/item/COCV_2014__20_2_547_0/

[1] C. Bardos, G. Lebeau and J. Rauch, 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. | MR 1007364 | Zbl 0673.93037

[2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | MR 1178650 | Zbl 0786.93009

[3] N. Burq, Mesures semi-classiques et mesures de défaut. Séminaire Bourbaki, Vol. 1996/97. Astérisque, (245): Exp. No. 826 (1997) 167-195. | Numdam | MR 1627111 | Zbl 0954.35102

[4] N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 749-752. | MR 1483711 | Zbl 0906.93008

[5] N. Burq and M. Zworski, Geometric control in the presence of a black box. J. Amer. Math. Soc. 17 (2004) 443-471. | MR 2051618 | Zbl 1050.35058

[6] C. Conca, J. Planchard and M. Vanninathan, Fluids and periodic structures, vol. 38 of RAM: Res. Appl. Math. John Wiley & Sons Ltd., Chichester (1995). | MR 1652238 | Zbl 0910.76002

[7] B. Dehman and G. Lebeau, Analysis of the HUM control operator and exact controllability for semilinear waves in uniform time. SIAM J. Control Optim. 48 (2009) 521-550. | MR 2486082 | Zbl 1194.35268

[8] S. Ervedoza, Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes. Numer. Math. 113 (2009) 377-415. | MR 2534130 | Zbl 1170.93013

[9] S. Ervedoza and E. Zuazua. A systematic method for building smooth controls for smooth data. Discrete Contin. Dyn. Syst. Ser. B 14 (2010) 1375-1401. | MR 2679646 | Zbl 1219.93011

[10] 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). | MR 3220862

[11] P. Gérard, Microlocal defect measures. Commun. Partial Differ. Eqs. 16 (1991) 1761-1794. | MR 1135919 | Zbl 0770.35001

[12] 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). | MR 1065993 | Zbl 0712.35001

[13] J.-L. Lions, Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, vol. 8 RMA. Masson (1988). | MR 963060 | Zbl 0653.93002

[14] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems. SIAM Review 30 (1988) 1-68. | MR 931277 | Zbl 0644.49028

[15] R.B. Melrose and J. Sjöstrand, Singularities of boundary value problems. II. Commun. Pure Appl. Math. 35 (1982) 129-168. | MR 644020 | Zbl 0546.35083

[16] L. Miller, Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218 (2005) 425-444. | MR 2108119 | Zbl 1122.93011

[17] J.V. Ralston, Solutions of the wave equation with localized energy. Commun. Pure Appl. Math. 22 (1969) 807-823. | MR 254433 | Zbl 0209.40402

[18] J.-P. Raymond and M. Vanninathan, Exact controllability in fluid-solid structure: the Helmholtz model. ESAIM: COCV 11 (2005) 180-203. | Numdam | MR 2141885 | Zbl 1125.93007

[19] J.-P. Raymond and M. Vanninathan, Null controllability in a fluid-solid structure model. J. Differ. Eqs. 248 (2010) 1826-1865. | MR 2593609 | Zbl 1226.93030

[20] M. Tucsnak and M. Vanninathan, Locally distributed control for a model of fluid-structure interaction. Systems Control Lett. 58 (2009) 547-552. | MR 2542110 | Zbl 1166.93006

[21] M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, vol. XI of Birkäuser Advanced Texts. Springer (2009). | MR 2502023 | Zbl 1188.93002

[22] E. Zuazua, Exact controllability for semilinear wave equations in one space dimension. Ann. Inst. Henri Poincaré Anal. Non Linéaire 10 (1993) 109-129. | Numdam | MR 1212631 | Zbl 0769.93017