# Control of the surface of a fluid by a wavemaker

ESAIM: Control, Optimisation and Calculus of Variations (2004)

- Volume: 10, Issue: 3, page 346-380
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topRosier, Lionel. "Control of the surface of a fluid by a wavemaker." ESAIM: Control, Optimisation and Calculus of Variations 10.3 (2004): 346-380. <http://eudml.org/doc/245305>.

@article{Rosier2004,

abstract = {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.},

author = {Rosier, Lionel},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Korteweg-de Vries equation; lagrangian coordinates; exact boundary controllability; Carleman estimate; Lagrangian coordinates},

language = {eng},

number = {3},

pages = {346-380},

publisher = {EDP-Sciences},

title = {Control of the surface of a fluid by a wavemaker},

url = {http://eudml.org/doc/245305},

volume = {10},

year = {2004},

}

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

PB - EDP-Sciences

VL - 10

IS - 3

SP - 346

EP - 380

AB - 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.

LA - eng

KW - Korteweg-de Vries equation; lagrangian coordinates; exact boundary controllability; Carleman estimate; Lagrangian coordinates

UR - http://eudml.org/doc/245305

ER -

## References

top- [1] S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov, Boundary values problems in mechanics of nonhomogeneous fluids. North-Holland, Amsterdam (1990). Zbl0696.76001MR1035212
- [2] P. Benilan and R. Gariepy, Strong solutions in ${L}^{1}$ of degenerate parabolic equations. J. Differ. Equations 119 (1995) 473-502. Zbl0828.35050MR1340548
- [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. Zbl1022.35044MR1915939
- [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. Zbl1057.35049MR1998942
- [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. Zbl0529.35069MR718811
- [6] J.-M. Coron, On the controllability of the 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155-188. Zbl0848.76013MR1380673
- [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. Zbl1071.76012MR1932962
- [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. Zbl1016.93031MR1848887
- [9] E. Fernández-Cara, Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-103. Zbl0897.93011MR1445385
- [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. Zbl0922.93006MR1348646
- [11] T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Stud. App. Math. 8 (1983) 93-128. Zbl0549.34001MR759907
- [12] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes, Vol. 1. Dunod, Paris (1968). Zbl0165.10801
- [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. Zbl1007.93035MR1825859
- [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. Zbl0958.93046MR1786022
- [16] S. Mottelet, Controllability and stabilization of a canal with wave generators. SIAM J. Control Optim. 38 (2000) 711-735. Zbl0966.76015MR1741435
- [17] S. Mottelet, Controllability and stabilization of liquid vibration in a container during transportation. (Preprint.) Zbl0966.76015
- [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. MR1893517
- [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 Zbl0873.93008MR1440078
- [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 Zbl0919.93039
- [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. Zbl0966.93055MR1788062
- [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. Zbl0771.93073MR1214759
- [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. Zbl0862.93035MR1360229
- [24] J. Simon, Compact Sets in the Space ${L}^{p}(0,T;B)$. Ann. Mat. Pura Appl. (IV) CXLVI (1987) 65-96. Zbl0629.46031MR916688
- [25] G.B. Whitham, Linear and nonlinear waves. A Wiley-Interscience publication, Wiley, New York (1999) reprint of the 1974 original. Zbl0940.76002MR1699025
- [26] E. Zeidler, Nonlinear functional analysis and its applications, Part 1. Springer-Verlag, New York (1986). Zbl0583.47050MR816732
- [27] B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation. SIAM J. Control Optim. 37 (1999) 543-565. Zbl0930.35160MR1670653

## Citations in EuDML Documents

top- Eduardo Cerpa, Emmanuelle Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain
- O. Glass, S. Guerrero, On the controllability of the fifth-order Korteweg-de Vries equation
- Olivier Glass, Problèmes de contrôle pour des équations dispersives unidimensionnelles

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.