Control of the surface of a fluid by a wavemaker

Lionel Rosier

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

Abstract

top
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.

How to cite

top

Rosier, Lionel. "Control of the surface of a fluid by a wavemaker." ESAIM: Control, Optimisation and Calculus of Variations 10.3 (2010): 346-380. <http://eudml.org/doc/90734>.

@article{Rosier2010,
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.; exact boundary controllability; Carleman estimate},
language = {eng},
month = {3},
number = {3},
pages = {346-380},
publisher = {EDP Sciences},
title = {Control of the surface of a fluid by a wavemaker},
url = {http://eudml.org/doc/90734},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Rosier, Lionel
TI - Control of the surface of a fluid by a wavemaker
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
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.; exact boundary controllability; Carleman estimate
UR - http://eudml.org/doc/90734
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).  
  2. P. Benilan and R. Gariepy, Strong solutions in L1 of degenerate parabolic equations. J. Differ. Equations119 (1995) 473-502.  Zbl0828.35050
  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.  Zbl1022.35044
  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. Equations28 (2003) 1391-1436.  Zbl1057.35049
  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.  Zbl0529.35069
  6. J.-M. Coron, On the controllability of the 2-D incompressible perfect fluids. J. Math. Pures Appl.75 (1996) 155-188.  Zbl0848.76013
  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: COCV8 (2002) 513-554.  Zbl1071.76012
  8. E. Crépeau, Exact boundary controllability of the Korteweg-de Vries equation around a non-trivial stationary solution. Int. J. Control74 (2001) 1096-1106.  Zbl1016.93031
  9. E. Fernández-Cara, Null controllability of the semilinear heat equation. ESAIM: COCV2 (1997) 87-103.  Zbl0897.93011
  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.  Zbl0922.93006
  11. T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equations. Stud. App. Math.8 (1983) 93-128.  
  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.  Zbl1007.93035
  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.  Zbl0958.93046
  16. S. Mottelet, Controllability and stabilization of a canal with wave generators. SIAM J. Control Optim.38 (2000) 711-735.  Zbl0966.76015
  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. Control47 (2002) 594-609.  
  19. L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV2 (1997) 33-55,  Zbl0873.93008URIhttp://www.edpsciences.org/cocv
  20. L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation – a numerical study. ESAIM Proc.4 (1998) 255-267,  Zbl0919.93039URIhttp://www.edpsciences.org/proc
  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.  Zbl0966.93055
  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.  Zbl0771.93073
  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.  Zbl0862.93035
  24. J. Simon, Compact Sets in the Space L p ( 0 , T ; B ) . Ann. Mat. Pura Appl. (IV)CXLVI (1987) 65-96.  Zbl0629.46031
  25. G.B. Whitham, Linear and nonlinear waves. A Wiley-Interscience publication, Wiley, New York (1999) reprint of the 1974 original.  
  26. E. Zeidler, Nonlinear functional analysis and its applications, Part 1. Springer-Verlag, New York (1986).  Zbl0583.47050
  27. B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation. SIAM J. Control Optim.37 (1999) 543-565.  Zbl0930.35160

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.