Partial Differential Equations
Control, observation and polynomial decay for a coupled heat-wave system
Comptes Rendus. Mathématique, Volume 336 (2003) no. 10, pp. 823-828.

This Note is devoted to study the control, observation and polynomial decay of a linearized 1-d model for fluid–structure interaction, where a wave and a heat equation evolve in two bounded intervals, with natural transmission conditions at the point of interface. These conditions couple, in particular, the heat unknown with the velocity of the wave solution. The controllability and observability of the system through the wave component are derived from sidewise energy estimate and Carleman inequalities. As for the control and observation through the heat component, we need to develop first a careful spectral high frequency analysis for the underlying semigroup, which yields a new Ingahm-type inequality. It is shown that the controllable/observable subspace for both cases are quite different. Also, we obtain a sharp polynomial decay rate for the energy of smooth solutions.

On considère un modèle couplé ondes-chaleur 1-d. L'intervalle (−1,1) est divisé en deux parties. Dans (−1,0) l'équation des ondes a lieu pour la variable z tandis que, dans (0,1), y résout l'équation de la chaleur. Au point d'interface on impose les conditions de transmission y=zt et yx=zx. Ces sont des conditions plus naturelles dans le contexte de l'interaction fluide–structure. Dans cette Note, suivant les techniques developpées dans nos travaux précédents on donne des résultats optimaux de contrôle et d'observation depuis le bord parabolique x=1 et hyperbolique x=−1 et on montre la décroissance polynomiale des solutions régulières.

Accepted:
Published online:
DOI: 10.1016/S1631-073X(03)00204-8

Xu Zhang 1, 2; Enrique Zuazua 2

1 School of Mathematics, Sichuan University, Chengdu 610064, Sichuan Province, China
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Xu Zhang; Enrique Zuazua. Control, observation and polynomial decay for a coupled heat-wave system. Comptes Rendus. Mathématique, Volume 336 (2003) no. 10, pp. 823-828. doi : 10.1016/S1631-073X(03)00204-8. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00204-8/

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

[2] X. Zhang, E. Zuazua, Polynomial decay and control of a 1-d model for fluid–structure interaction, C. R. Acad. Sci. Paris, Ser. I, to appear

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

[4] E. Zuazua, Contrôlabilité exacte en un temps arbitrairement petit de quelques modèles de plaques, Appendix I in [1], 465–491

[5] 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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