Polynomial decay and control of a 1−<i>d</i> model for fluid–structure interaction

Xu Zhang; Enrique Zuazua

doi:10.1016/S1631-073X(03)00169-9

Partial Differential Equations/Optimal Control

Polynomial decay and control of a 1−d model for fluid–structure interaction
[Décroissance polynomiale et contrôle d'un modèle 1−d d'interaction fluide–structure]

Présenté par : Philippe G. Ciarlet

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

Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 745-750.

Résumé
Abstract

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.

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.

Reçu le : 2003-03-05
Accepté le : 2003-03-08
Publié le : 2003-04-30

DOI : 10.1016/S1631-073X(03)00169-9

Affiliations des auteurs :

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

@article{CRMATH_2003__336_9_745_0,
     author = {Xu Zhang and Enrique Zuazua},
     title = {Polynomial decay and control of a 1\ensuremath{-}\protect\emph{d} model for fluid{\textendash}structure interaction},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {745--750},
     publisher = {Elsevier},
     volume = {336},
     number = {9},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00169-9},
     language = {en},
}

TY  - JOUR
AU  - Xu Zhang
AU  - Enrique Zuazua
TI  - Polynomial decay and control of a 1−d model for fluid–structure interaction
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 745
EP  - 750
VL  - 336
IS  - 9
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00169-9
LA  - en
ID  - CRMATH_2003__336_9_745_0
ER  -

%0 Journal Article
%A Xu Zhang
%A Enrique Zuazua
%T Polynomial decay and control of a 1−d model for fluid–structure interaction
%J Comptes Rendus. Mathématique
%D 2003
%P 745-750
%V 336
%N 9
%I Elsevier
%R 10.1016/S1631-073X(03)00169-9
%G en
%F CRMATH_2003__336_9_745_0

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/

Bibliographie
Cité par

[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

Ya-Ping Guo; Jun-Min Wang; Jing Wang; Dong-Xia Zhao The null controllability of transmission wave-Schrödinger system with a boundary control, Journal of Dynamical and Control Systems, Volume 30 (2024) no. 3, p. 22 (Id/No 27) | DOI:10.1007/s10883-024-09693-1 | Zbl:1544.93051
Tiago Roux Oliveira; Miroslav Krstic Extremum seeking boundary control for PDE-PDE cascades, Systems Control Letters, Volume 155 (2021), p. 15 (Id/No 105004) | DOI:10.1016/j.sysconle.2021.105004 | Zbl:1483.93252
Boris Muha A note on optimal regularity and regularizing effects of point mass coupling for a heat-wave system, Journal of Mathematical Analysis and Applications, Volume 425 (2015) no. 2, pp. 1134-1147 | DOI:10.1016/j.jmaa.2015.01.018 | Zbl:1310.35056
Yuanting Wang; Fucheng Liao; Yonglong Liao; Zhengwei Shen Boundary control for a kind of coupled PDE-ODE system, Journal of Control Science and Engineering, Volume 2014 (2014), p. 8 (Id/No 946736) | DOI:10.1155/2014/946736 | Zbl:1298.93280
Enrique Zuazua Control and Stabilization of Waves on 1-d Networks, Modelling and Optimisation of Flows on Networks, Volume 2062 (2013), p. 463 | DOI:10.1007/978-3-642-32160-3_9
Shuxia Tang; Chengkang Xie; Zhongcheng Zhou, 2011 Chinese Control and Decision Conference (CCDC) (2011), p. 320 | DOI:10.1109/ccdc.2011.5968195
Shuxia Tang; Chengkang Xie Stabilization for a coupled PDE-ODE control system, Journal of the Franklin Institute, Volume 348 (2011) no. 8, pp. 2142-2155 | DOI:10.1016/j.jfranklin.2011.06.008 | Zbl:1231.93095
Paola Loreti; Daniela Sforza Multidimensional Controllability Problems with Memory, Modern Aspects of the Theory of Partial Differential Equations (2011), p. 261 | DOI:10.1007/978-3-0348-0069-3_15
Shuxia Tang; Chengkang Xie State and output feedback boundary control for a coupled PDE-ODE system, Systems Control Letters, Volume 60 (2011) no. 8, pp. 540-545 | DOI:10.1016/j.sysconle.2011.04.011 | Zbl:1236.93076
Shuxia Tang; Chengkang Xie, 49th IEEE Conference on Decision and Control (CDC) (2010), p. 4042 | DOI:10.1109/cdc.2010.5718141
Paola Loreti; Daniela Sforza Reachability problems for a class of integro-differential equations, Journal of Differential Equations, Volume 248 (2010) no. 7, pp. 1711-1755 | DOI:10.1016/j.jde.2009.09.016 | Zbl:1195.45034
Paola Loreti; Daniela Sforza Exact reachability for second-order integro-differential equations, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 347 (2009) no. 19-20, pp. 1153-1158 | DOI:10.1016/j.crma.2009.08.007 | Zbl:1175.93026
M. Pellicer; J. Solà-Morales Optimal decay rates and the selfadjoint property in overdamped systems, Journal of Differential Equations, Volume 246 (2009) no. 7, pp. 2813-2828 | DOI:10.1016/j.jde.2009.01.010 | Zbl:1179.35064
Leila Ouksel Observability inequality of logarithmic type and estimation of the cost function of solutions to hyperbolic equations, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 14 (2008) no. 2, pp. 318-342 | DOI:10.1051/cocv:2007052 | Zbl:1139.35016
Enrique Zuazua Controllability and Observability of Partial Differential Equations: Some Results and Open Problems, Volume 3 (2007), p. 527 | DOI:10.1016/s1874-5717(07)80010-7
Enrique Zuazua Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods, SIAM Review, Volume 47 (2005) no. 2, p. 197 | DOI:10.1137/s0036144503432862
Kangsheng Liu; Bopeng Rao Exponential stability for the wave equations with local Kelvin-Voigt damping., Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 339 (2004) no. 11, pp. 769-774 | DOI:10.1016/j.crma.2004.09.029 | Zbl:1056.35107
Xu Zhang; Enrique Zuazua Polynomial decay and control of a 1D hyperbolic-parabolic coupled system, Journal of Differential Equations, Volume 204 (2004) no. 2, pp. 380-438 | DOI:10.1016/j.jde.2004.02.004 | Zbl:1064.93008
Xu Zhang; Enrique Zuazua Control, observation and polynomial decay for a coupled heat-wave system, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 336 (2003) no. 10, pp. 823-828 | DOI:10.1016/s1631-073x(03)00204-8 | Zbl:1029.93037

Cité par 19 documents. Sources : Crossref, zbMATH

Commentaires - Politique