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Control of the surface of a fluid by a wavemaker

Published online by Cambridge University Press:  15 June 2004

Lionel Rosier
Lionel Rosier*
Affiliation:
Institut Elie Cartan, Université Henri Poincaré Nancy 1, BP 239, 54506 Vandœuvre-lès-Nancy Cedex, France; rosier@iecn.u-nancy.fr.
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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.

Keywords

Korteweg-de Vries equationLagrangian coordinatesexact boundary controllabilityCarleman estimate.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 3 , July 2004 , pp. 346 - 380
Copyright
© EDP Sciences, SMAI, 2004

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References

S.N. Antontsev, A.V. Kazhikov and V.N. Monakhov, Boundary values problems in mechanics of nonhomogeneous fluids. North-Holland, Amsterdam (1990).
Benilan, P. and Gariepy, R., Strong solutions in L 1 of degenerate parabolic equations. J. Differ. Equations 119 (1995) 473-502. CrossRef
Bona, J.L., Chen, M. and Saut, J.-C., 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. CrossRef
Bona, J.L., Sun, S. and Zhang, B.-Y., Non-homogeneous Boundary-Value Problem, A for the Korteweg-de Vries Equation Posed on a Finite Domain. Commun. Partial Differ. Equations 28 (2003) 1391-1436. CrossRef
Bona, J.L. and Winther, R., The Korteweg-de Vries equation, posed in a quarter-plane. SIAM J. Math. Anal. 14 (1983) 1056-1106. CrossRef
Coron, J.-M., On the controllability of the 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155-188.
Coron, J.-M., 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. CrossRef
Crépeau, E., Exact boundary controllability of the Korteweg-de Vries equation around a non-trivial stationary solution. Int. J. Control 74 (2001) 1096-1106. CrossRef
Fernández-Cara, E., Null controllability of the semilinear heat equation. ESAIM: COCV 2 (1997) 87-103. CrossRef
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.
Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Stud. App. Math. 8 (1983) 93-128.
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes, Vol. 1. Dunod, Paris (1968).
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).
Micu, S., On the controllability of the linearized Benjamin-Bona-Mahony equation. SIAM J. Control Optim. 39 (2001) 1677-1696. CrossRef
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.
Mottelet, S., Controllability and stabilization of a canal with wave generators. SIAM J. Control Optim. 38 (2000) 711-735. CrossRef
S. Mottelet, Controllability and stabilization of liquid vibration in a container during transportation. (Preprint.)
Petit, N. and Rouchon, P., Dynamics and solutions to some control problems for water-tank systems. IEEE Trans. Automat. Control 47 (2002) 594-609. CrossRef
Rosier, L., Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33-55, http://www.edpsciences.org/cocv CrossRef
Rosier, L., Exact boundary controllability for the linear Korteweg-de Vries equation – a numerical study. ESAIM Proc. 4 (1998) 255-267, http://www.edpsciences.org/proc CrossRef
Rosier, L., Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line. SIAM J. Control Optim. 39 (2000) 331-351. CrossRef
Russell, D.L. and Zhang, B.-Y., Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31 (1993) 659-673. CrossRef
Russell, D.L. and Zhang, B.-Y., Exact controllability and stabilizability of the Korteweg-de Vries equation. Trans. Amer. Math. Soc. 348 (1996) 3643-3672. CrossRef
J. Simon, Compact Sets in the Space $L^p(0,T;B)$ . Ann. Mat. Pura Appl. (IV) CXLVI (1987) 65-96.
G.B. Whitham, Linear and nonlinear waves. A Wiley-Interscience publication, Wiley, New York (1999) reprint of the 1974 original.
E. Zeidler, Nonlinear functional analysis and its applications, Part 1. Springer-Verlag, New York (1986).
Zhang, B.-Y., Exact boundary controllability of the Korteweg-de Vries equation. SIAM J. Control Optim. 37 (1999) 543-565. CrossRef