Comptes Rendus
Partial Differential Equations/Optimal Control
Polynomial decay and control of a 1−d model for fluid–structure interaction
Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 745-750.

We consider a linearized and simplified 1−d model for fluid–structure interaction. The domain where the system evolves consists in two bounded intervals in which the wave and heat equations evolve respectively, with transmission conditions at the point of interface. First, we develop a careful spectral asymptotic analysis on high frequencies. Next, according to this spectral analysis we obtain sharp polynomial decay rates for the whole energy of smooth solutions. Finally, we prove the null-controllability of the system when the control acts on the boundary of the interval where the heat equation holds. The proof is based on a new Ingham-type inequality, which follows from the spectral analysis we develop and the null controllability result in Zuazua (in: J.L. Menaldi et al. (Eds.), Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 198–210) where the control acts on the wave component.

On considère un modèle simplifié 1−d d'interaction fluide–structure. Le domaine est composé de deux sous-intervalles où l'équation des ondes et de la chaleur sont vérifiées respectivement. Au point d'interface on impose la continuité des états et des dérivées normales. Grâce à l'analyse asymptotique du spectre, on montre l'existence d'une suite de fonctions propres concentrées dans l'intervalle hyperbolique. On en déduit un taux de décroissance optimal des solutions régulières. On considère aussi le problème de contrôle à zéro moyennant un contrôle agissant sur la composante parabolique. On montre que l'espace de données contrôlables a une nature asymétrique : la composante parabolique étant L2 et la composante hyperbolique ayant des coefficients de Fourier exponentiellement petits.

Received:
Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00169-9
Xu Zhang 1, 2; Enrique Zuazua 2

1 School of Mathematics, Sichuan University, Chengdu 610064, Sichuan Province, China
2 Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
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Xu Zhang; Enrique Zuazua. Polynomial decay and control of a 1−d model for fluid–structure interaction. Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 745-750. doi : 10.1016/S1631-073X(03)00169-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00169-9/

[1] C. Bardos; G. Lebeau; J. Rauch Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., Volume 30 (1992), pp. 1024-1065

[2] E. Fernández-Cara; E. Zuazua The cost of approximate controllability for heat equations: the linear case, Adv. Differential Equations, Volume 5 (2000), pp. 465-514

[3] A.V. Fursikov; O.Yu. Imanuvilov Controllability of Evolution Equations, Lecture Notes Series, 34, Research Institute of Mathematics, Seoul National University, Seoul, Korea, 1994

[4] J.L. Lions Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués. Tome 1 : Contrôlabilité exacte, RMA, 8, Masson, Paris, 1988

[5] X. Zhang Explicit observability estimate for the wave equation with potential and its application, Proc. Roy. Soc. London Ser. A, Volume 456 (2000), pp. 1101-1115

[6] X. Zhang, E. Zuazua, Polynomial decay and control of a hyperbolic-parabolic coupled system, Preprint

[7] E. Zuazua Null control of a 1−d model of mixed hyperbolic-parabolic type (J.L. Menaldi et al., eds.), Optimal Control and Partial Differential Equations, IOS Press, 2001, pp. 198-210

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