Control, observation and polynomial decay for a coupled heat-wave system

Xu Zhang; Enrique Zuazua

doi:10.1016/S1631-073X(03)00204-8

Partial Differential Equations

Control, observation and polynomial decay for a coupled heat-wave system
[Contrôle, observation et décroissance polynomiale pour un système couplé ondes-chaleur]

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. 10, pp. 823-828.

Résumé
Abstract

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=z_t et y_x=z_x. 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.

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.

Reçu le : 2003-03-18
Accepté le : 2003-03-25
Publié le : 2003-05-15

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

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_10_823_0,
     author = {Xu Zhang and Enrique Zuazua},
     title = {Control, observation and polynomial decay for a coupled heat-wave system},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {823--828},
     publisher = {Elsevier},
     volume = {336},
     number = {10},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00204-8},
     language = {en},
}

TY  - JOUR
AU  - Xu Zhang
AU  - Enrique Zuazua
TI  - Control, observation and polynomial decay for a coupled heat-wave system
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 823
EP  - 828
VL  - 336
IS  - 10
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00204-8
LA  - en
ID  - CRMATH_2003__336_10_823_0
ER  -

%0 Journal Article
%A Xu Zhang
%A Enrique Zuazua
%T Control, observation and polynomial decay for a coupled heat-wave system
%J Comptes Rendus. Mathématique
%D 2003
%P 823-828
%V 336
%N 10
%I Elsevier
%R 10.1016/S1631-073X(03)00204-8
%G en
%F CRMATH_2003__336_10_823_0

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/

Bibliographie
Cité par

[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

Yan-Fang Li; Yang Wang Sharp polynomial decay rate of multi-link hyperbolic-parabolic systems with different network configurations, Evolution Equations and Control Theory, Volume 13 (2024) no. 1, pp. 12-25 | DOI:10.3934/eect.2023034 | Zbl:1532.35067
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
Manish Kumar; Subrata Majumdar On the controllability of a system coupling Kuramoto-Sivashinsky-Korteweg-de Vries and transport equations, MCSS. Mathematics of Control, Signals, and Systems, Volume 36 (2024) no. 4, pp. 875-926 | DOI:10.1007/s00498-024-00390-9 | Zbl:7927397
Zhong-Jie Han; Han-Qi Song; Kai Yu Sharp decay rates of degenerate hyperbolic-parabolic coupled system: rectangular domain vs one-dimensional domain, Journal of Differential Equations, Volume 349 (2023), pp. 53-82 | DOI:10.1016/j.jde.2022.12.005 | Zbl:1507.35039
Louis Tebou Stability and Gevrey regularity for some transmission systems involving a degenerate parabolic component, Journal of Evolution Equations, Volume 23 (2023) no. 2, p. 31 (Id/No 26) | DOI:10.1007/s00028-023-00876-0 | Zbl:1529.74015
Fu Zheng; Sijia Zhang; Huakun Wang; Bao-Zhu Guo The exponential stabilization of a heat-wave coupled system and its approximation, Journal of Mathematical Analysis and Applications, Volume 521 (2023) no. 1, p. 20 (Id/No 126927) | DOI:10.1016/j.jmaa.2022.126927 | Zbl:1506.65134
Zhao-Zhong Su; Ya-Ping Guo Exact controllability of the transmission string-beam equations with a single boundary control, Mathematical Methods in the Applied Sciences, Volume 46 (2023) no. 14, pp. 15352-15366 | DOI:10.1002/mma.9382 | Zbl:1536.93092
Fengyan Yang Boundary controllability of a coupled inhomogeneous plates system, The Journal of Geometric Analysis, Volume 33 (2023) no. 2, p. 23 (Id/No 55) | DOI:10.1007/s12220-022-01115-7 | Zbl:1504.35597
Louis Tebou Sharp decay estimates for semigroups associated with some one-dimensional fluid-structure interactions involving degeneracy, SIAM Journal on Control and Optimization, Volume 60 (2022) no. 5, pp. 2787-2810 | DOI:10.1137/21m141854x | Zbl:1498.93663
Kaïs Ammari; Fathi Hassine Energy Decay Estimates of Elastic Transmission Wave/Beam Systems with a Local Kelvin–Voigt Damping, Stabilization of Kelvin-Voigt Damped Systems, Volume 47 (2022), p. 85 | DOI:10.1007/978-3-031-12519-5_5
Genqi Xu; Min Li Stability of wave networks on elastic and viscoelastic media, Acta Applicandae Mathematicae, Volume 175 (2021), p. 40 (Id/No 11) | DOI:10.1007/s10440-021-00437-y | Zbl:1476.35055
Bopeng Rao; Xu Zhang Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system, Communications on Pure and Applied Analysis, Volume 20 (2021) no. 7-8, pp. 2789-2809 | DOI:10.3934/cpaa.2021119 | Zbl:1500.35048
Zahra Andleeb; Sohail Malik; Hassan Abbas Khawaja; Anders Samuelsen Nordli; Ståle Antonsen; Ghulam Hussain; Mojtaba Moatamedi Thermoelastic Investigation of Carbon-Fiber-Reinforced Composites Using a Drop-Weight Impact Test, Applied Sciences, Volume 11 (2020) no. 1, p. 207 | DOI:10.3390/app11010207
Zhong-Jie Han; Gengsheng Wang; Jing Wang Explicit decay rate for a degenerate hyperbolic-parabolic coupled system, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 26 (2020), p. 20 (Id/No 116) | DOI:10.1051/cocv/2020040 | Zbl:1459.35041
Caihong Zhang; Yinuo Huang; Licheng Wang; Chongxiong Duan; Tiezhu Zhang; Kai Wang Stability of two weakly coupled elastic beams with partially local damping, Mathematical Problems in Engineering, Volume 2020 (2020), p. 9 (Id/No 7169526) | DOI:10.1155/2020/7169526 | Zbl:1544.35172
Amritam Das; Sachin Shivakumar; Siep Weiland; Matthew M. Peet, 2019 IEEE 58th Conference on Decision and Control (CDC) (2019), p. 262 | DOI:10.1109/cdc40024.2019.9029595
Xue-Lian Jin; Yan Li; Fu Zheng Spectrum and stability of a $1 - d$ heat-wave coupled system with dynamical boundary control, Mathematical Problems in Engineering, Volume 2019 (2019), p. 9 (Id/No 5716729) | DOI:10.1155/2019/5716729 | Zbl:1435.35196
Fathi Hassine Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping, International Journal of Control, Volume 89 (2016) no. 10, pp. 1933-1950 | DOI:10.1080/00207179.2015.1135509 | Zbl:1364.35043
Zhong-Jie Han; Enrique Zuazua Decay rates for $1 - d$ heat-wave planar networks, Networks and Heterogeneous Media, Volume 11 (2016) no. 4, pp. 655-692 | DOI:10.3934/nhm.2016013 | Zbl:1355.35023
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
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
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
Weijiu Liu Higher-dimensional Wave Equation, Elementary Feedback Stabilization of the Linear Reaction-Convection-Diffusion Equation and the Wave Equation, Volume 66 (2010), p. 233 | DOI:10.1007/978-3-642-04613-1_6
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
Xu Zhang; Enrique Zuazua Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Archive for Rational Mechanics and Analysis, Volume 184 (2007) no. 1, pp. 49-120 | DOI:10.1007/s00205-006-0020-x | Zbl:1178.74075
Xu Zhang; Enrique Zuazua Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction, Free Boundary Problems, Volume 154 (2006), p. 445 | DOI:10.1007/978-3-7643-7719-9_43
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
Michael Renardy Are viscoelastic flows under control or out of control?, Systems Control Letters, Volume 54 (2005) no. 12, pp. 1183-1193 | DOI:10.1016/j.sysconle.2005.04.006 | Zbl:1129.93325
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

Cité par 35 documents. Sources : Crossref, zbMATH

Commentaires - Politique