no. 3
Control of the surface of a fluid by a wavemaker
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 3, pp. 346-380

The control of the surface of water in a long canal by means of a wavemaker is investigated. The fluid motion is governed by the Korteweg-de Vries equation in lagrangian coordinates. The null controllability of the elevation of the fluid surface is obtained thanks to a Carleman estimate and some weighted inequalities. The global uncontrollability is also established.

DOI : 10.1051/cocv:2004012
Classification : 35B37, 49J20, 76B15, 93B05, 93C20
Keywords: Korteweg-de Vries equation, lagrangian coordinates, exact boundary controllability, Carleman estimate 
@article{COCV_2004__10_3_346_0,
     author = {Rosier, Lionel},
     title = {Control of the surface of a fluid by a wavemaker},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {346--380},
     year = {2004},
     publisher = {EDP Sciences},
     volume = {10},
     number = {3},
     doi = {10.1051/cocv:2004012},
     mrnumber = {2084328},
     zbl = {1094.93014},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2004012/}
}
TY  - JOUR
AU  - Rosier, Lionel
TI  - Control of the surface of a fluid by a wavemaker
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2004
SP  - 346
EP  - 380
VL  - 10
IS  - 3
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv:2004012/
DO  - 10.1051/cocv:2004012
LA  - en
ID  - COCV_2004__10_3_346_0
ER  - 
%0 Journal Article
%A Rosier, Lionel
%T Control of the surface of a fluid by a wavemaker
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2004
%P 346-380
%V 10
%N 3
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/cocv:2004012/
%R 10.1051/cocv:2004012
%G en
%F COCV_2004__10_3_346_0
Rosier, Lionel. Control of the surface of a fluid by a wavemaker. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 3, pp. 346-380. doi: 10.1051/cocv:2004012

[1] S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov, Boundary values problems in mechanics of nonhomogeneous fluids. North-Holland, Amsterdam (1990). | Zbl | MR

[2] P. Benilan and R. Gariepy, Strong solutions in L 1 of degenerate parabolic equations. J. Differ. Equations 119 (1995) 473-502. | Zbl | MR

[3] J.L. Bona, M. Chen and J.-C. Saut, Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media. I: Derivation and Linear Theory. J. Nonlinear Sci. 12 (2002) 283-318. | Zbl | MR

[4] J.L. Bona, S. Sun and B.-Y. Zhang, A Non-homogeneous Boundary-Value Problem for the Korteweg-de Vries Equation Posed on a Finite Domain. Commun. Partial Differ. Equations 28 (2003) 1391-1436. | Zbl | MR

[5] J.L. Bona and R. Winther, The Korteweg-de Vries equation, posed in a quarter-plane. SIAM J. Math. Anal. 14 (1983) 1056-1106. | Zbl | MR

[6] J.-M. Coron, On the controllability of the 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155-188. | Zbl | MR

[7] J.-M. Coron, Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations, A tribute to J.L. Lions. ESAIM: COCV 8 (2002) 513-554. | Zbl | MR | Numdam

[8] E. Crépeau, Exact boundary controllability of the Korteweg-de Vries equation around a non-trivial stationary solution. Int. J. Control 74 (2001) 1096-1106. | Zbl | MR

[9] E. Fernández-Cara, Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-103. | Zbl | MR | Numdam

[10] A.V. Fursikov and O.Y. Imanuvilov, On controllability of certain systems simulating a fluid flow, in Flow Control, M.D. Gunzburger Ed., Springer-Verlag, New York, IMA Vol. Math. Appl. 68 (1995) 149-184. | Zbl | MR

[11] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Stud. App. Math. 8 (1983) 93-128. | Zbl | MR

[12] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes, Vol. 1. Dunod, Paris (1968). | Zbl

[13] G. Mathieu-Girard, Étude et contrôle des équations de la théorie “Shallow water” en dimension un. Ph.D. thesis, Université Paul Sabatier, Toulouse III (1998).

[14] S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation. SIAM J. Control Optim. 39 (2001) 1677-1696. | Zbl | MR

[15] S. Micu and J.H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations. Mathematical and numerical aspects of wave propagation (Santiago de Compostela, 2000). Philadelphia, PA SIAM (2000) 1020-1024. | Zbl | MR

[16] S. Mottelet, Controllability and stabilization of a canal with wave generators. SIAM J. Control Optim. 38 (2000) 711-735. | Zbl | MR

[17] S. Mottelet, Controllability and stabilization of liquid vibration in a container during transportation. (Preprint.)

[18] N. Petit and P. Rouchon, Dynamics and solutions to some control problems for water-tank systems. IEEE Trans. Automat. Control 47 (2002) 594-609. | MR

[19] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33-55, http://www.edpsciences.org/cocv | Zbl | MR | Numdam

[20] L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation - a numerical study. ESAIM Proc. 4 (1998) 255-267, http://www.edpsciences.org/proc | Zbl

[21] L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line. SIAM J. Control Optim. 39 (2000) 331-351. | Zbl | MR

[22] D.L. Russell and B.-Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31 (1993) 659-673. | Zbl | MR

[23] D.L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation. Trans. Amer. Math. Soc. 348 (1996) 3643-3672. | Zbl | MR

[24] J. Simon, Compact Sets in the Space L p (0,T;B). Ann. Mat. Pura Appl. (IV) CXLVI (1987) 65-96. | Zbl | MR

[25] G.B. Whitham, Linear and nonlinear waves. A Wiley-Interscience publication, Wiley, New York (1999) reprint of the 1974 original. | Zbl | MR

[26] E. Zeidler, Nonlinear functional analysis and its applications, Part 1. Springer-Verlag, New York (1986). | Zbl | MR

[27] B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation. SIAM J. Control Optim. 37 (1999) 543-565. | Zbl | MR

Cité par Sources :