Exact controllability in fluid-solid structure : the Helmholtz model
ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 2, pp. 180-203.

A model representing the vibrations of a fluid-solid coupled structure is considered. Following Hilbert Uniqueness Method (HUM) introduced by Lions, we establish exact controllability results for this model with an internal control in the fluid part and there is no control in the solid part. Novel features which arise because of the coupling are pointed out. It is a source of difficulty in the proof of observability inequalities, definition of weak solutions and the proof of controllability results.

DOI : 10.1051/cocv:2005006
Classification : 35B37, 93C20
Mots-clés : fluid - solid structure, exact controllability 
@article{COCV_2005__11_2_180_0,
     author = {Raymond, Jean-Pierre and Vanninathan, Muthusamy},
     title = {Exact controllability in fluid-solid structure : the {Helmholtz} model},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {180--203},
     publisher = {EDP-Sciences},
     volume = {11},
     number = {2},
     year = {2005},
     doi = {10.1051/cocv:2005006},
     zbl = {1125.93007},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2005006/}
}
TY  - JOUR
AU  - Raymond, Jean-Pierre
AU  - Vanninathan, Muthusamy
TI  - Exact controllability in fluid-solid structure : the Helmholtz model
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2005
SP  - 180
EP  - 203
VL  - 11
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2005006/
DO  - 10.1051/cocv:2005006
LA  - en
ID  - COCV_2005__11_2_180_0
ER  - 
%0 Journal Article
%A Raymond, Jean-Pierre
%A Vanninathan, Muthusamy
%T Exact controllability in fluid-solid structure : the Helmholtz model
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2005
%P 180-203
%V 11
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2005006/
%R 10.1051/cocv:2005006
%G en
%F COCV_2005__11_2_180_0
Raymond, Jean-Pierre; Vanninathan, Muthusamy. Exact controllability in fluid-solid structure : the Helmholtz model. ESAIM: Control, Optimisation and Calculus of Variations, Tome 11 (2005) no. 2, pp. 180-203. doi : 10.1051/cocv:2005006. http://archive.numdam.org/articles/10.1051/cocv:2005006/

[1] G. Avalos, I. Lasiecka, Exact controllability of structural acoustic interactions. J. Math. Pures Appl. 82 (2003) 1047-1073. | Zbl

[2] V. Barbu, T. Precupanu, Convexity and Optimization in Banach Spaces, 2nd ed., D. Reidel, Dordrecht (1986). | MR | Zbl

[3] A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems. Birkhäuser, Boston 1 (1992). | MR | Zbl

[4] C. Conca, J. Planchard, B. Thomas and M. Vanninathan, Problèmes mathématiques en couplage fluide-structure. Eyrolles, Paris (1994).

[5] C. Conca, J. Planchard and M. Vanninathan, Fluids and periodic structures. Masson and J. Wiley, Paris (1995). | MR | Zbl

[6] L. Cot, J.-P. Raymond and J. Vancostenoble, Exact controllability of an aeroacoustic model. In preparation.

[7] R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Scientifique. Masson, Paris (1987). | MR

[8] P. Destuynder and E. Gout D'Henin, Existence and uniqueness of a solution to an aeroacoustic model. Chin. Ann. Math. 23B (2002) 11-24. | Zbl

[9] E. Gout D'Henin, Ondes de Stoneley en interaction fluide-structure. Ph.D. Thesis, Université de Poitiers (2002).

[10] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Masson, Paris (1988). | Zbl

[11] S. Micu and E. Zuazua, Boundary controllability of a linear hybrid system arising in the control of noise. SIAM J. Control Optim. 35 (1997) 531-555. | Zbl

[12] J.J. Moreau, Bounded variation in time, in Topics in Nonsmooth Mechanics, J.J. Moreau, P.D. Panagiotopoulos, G. Strang Eds. Birkhäuser, Boston (1988) 1-74. | Zbl

Cité par Sources :