On the controllability and stabilization of the linearized Benjamin-Ono equation

Felipe Linares; Jaime H. Ortega

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 2, page 204-218
  • ISSN: 1292-8119

Abstract

top
In this work we are interested in the study of controllability and stabilization of the linearized Benjamin-Ono equation with periodic boundary conditions, which is a generic model for the study of weakly nonlinear waves with nonlocal dispersion. It is well known that the Benjamin-Ono equation has infinite number of conserved quantities, thus we consider only controls acting in the equation such that the volume of the solution is conserved. We study also the stabilization with a feedback law which gives us an exponential decay of the solutions.

How to cite

top

Linares, Felipe, and Ortega, Jaime H.. "On the controllability and stabilization of the linearized Benjamin-Ono equation." ESAIM: Control, Optimisation and Calculus of Variations 11.2 (2005): 204-218. <http://eudml.org/doc/245969>.

@article{Linares2005,
abstract = {In this work we are interested in the study of controllability and stabilization of the linearized Benjamin-Ono equation with periodic boundary conditions, which is a generic model for the study of weakly nonlinear waves with nonlocal dispersion. It is well known that the Benjamin-Ono equation has infinite number of conserved quantities, thus we consider only controls acting in the equation such that the volume of the solution is conserved. We study also the stabilization with a feedback law which gives us an exponential decay of the solutions.},
author = {Linares, Felipe, Ortega, Jaime H.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {exact controllability; stabilization; Benjamin-Ono equation; dispersive equation},
language = {eng},
number = {2},
pages = {204-218},
publisher = {EDP-Sciences},
title = {On the controllability and stabilization of the linearized Benjamin-Ono equation},
url = {http://eudml.org/doc/245969},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Linares, Felipe
AU - Ortega, Jaime H.
TI - On the controllability and stabilization of the linearized Benjamin-Ono equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 2
SP - 204
EP - 218
AB - In this work we are interested in the study of controllability and stabilization of the linearized Benjamin-Ono equation with periodic boundary conditions, which is a generic model for the study of weakly nonlinear waves with nonlocal dispersion. It is well known that the Benjamin-Ono equation has infinite number of conserved quantities, thus we consider only controls acting in the equation such that the volume of the solution is conserved. We study also the stabilization with a feedback law which gives us an exponential decay of the solutions.
LA - eng
KW - exact controllability; stabilization; Benjamin-Ono equation; dispersive equation
UR - http://eudml.org/doc/245969
ER -

References

top
  1. [1] M. Abdelouhab, J.L. Bona, M. Felland and J.-C. Saut, Nonlocal models for nonlinear, dispersive waves. Physica D 40 (1989) 360–392. Zbl0699.35227
  2. [2] M.J. Ablowitz and A.S. Fokas, The inverse scattering transform for the Benjamin-Ono equation-a pivot to multidimensional problems. Stud. Appl. Math. 68 (1983) 1–10. Zbl0505.76031
  3. [3] T.B. Benjamin, Internal waves of permanent form in fluids of great depth. J. Fluid Mech. 29 (1967) 559–592. Zbl0147.46502
  4. [4] J. Bona and R. Winther, The Korteweg-de Vries equation, posed in a quarter-plane. SIAM J. Math. Anal. 14 (1983) 1056–1106. Zbl0529.35069
  5. [5] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geom. Funct. Anal. 3 (1993) 107–156, 209–262. Zbl0787.35098
  6. [6] K.M. Case, Benjamin-Ono related equations and their solutions. Proc. Nat. Acad. Sci. USA 76 (1979) 1–3. Zbl0395.76020
  7. [7] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equation. Oxford Sci. Publ. (1998). Zbl0926.35049MR1691574
  8. [8] J. Colliander and C.E. Kenig, The generalized Korteweg-de Vries equation on the half line. Comm. Partial Differential Equations 27 (2002) 2187–2266. Zbl1041.35064
  9. [9] K.D. Danov and M.S. Ruderman, Nonlinear waves on shallow water in the presence of a horizontal magnetic field. Fluid Dynamics 18 (1983) 751–756. Zbl0557.76029
  10. [10] A.E. Ingham, A further note on trigonometrical inequalities. Proc. Cambridge Philos. Soc. 46 (1950) 535–537. Zbl0037.32901
  11. [11] R. Iorio, On the Cauchy problem for the Benjamin-Ono equation. Comm. Partial Differentiel Equations 11 (1986) 1031–1081. Zbl0608.35030
  12. [12] Y. Ishimori, Solitons in a one-dimensional Lennard/Mhy Jones lattice. Progr. Theoret. Phys. 68 (1982) 402–410. Zbl1074.82512
  13. [13] C.E. Kenig and K. Koenig, On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations. Math. Res. Lett. 10 (2003) 879–895. Zbl1044.35072
  14. [14] C.E. Kenig, G. Ponce and L. Vega, A bilinear estimate with application to the KdV equation. J. Amer. Math Soc. 9 (1996) 573–603. Zbl0848.35114
  15. [15] H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in H s ( ) . Int. Math. Res. Not. 26 (2003) 1449–1464. Zbl1039.35106
  16. [16] Y. Matsuno and D.J. Kaup, Initial value problem of the linearized Benjamin-Ono equation and its applications. J. Math. Phys. 38 (1997) 5198–5224. Zbl0891.35141
  17. [17] S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation. SIAM J. Control Optim. 39 (2001) 1677–1696. Zbl1007.93035
  18. [18] H. Ono, Algebraic solitary waves in stratified fluids. J. Phys. Soc. Japan 39 (1975) 1082–1091. Zbl1334.76027
  19. [19] A. Pazy. Semigroups of linear operators and applications to partial differential equations. Springer-Verlag, New York, Appl. Math. Sci. 44 (1983). Zbl0516.47023MR710486
  20. [20] G. Perla-Menzala, F. Vasconcellos and E. Zuazua. Stabilization of the Korteweg-de Vries equation with localized damping. Quart. Appl. Math. 60 (2002) 111–129. Zbl1039.35107
  21. [21] G. Ponce, On the global well-posedness of the Benjamin-Ono equation. Diff. Integral Equations 4 (1991) 527–542. Zbl0732.35038
  22. [22] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33–55. Zbl0873.93008
  23. [23] D.L. Russell and B.-Y. Zhang, Controllability and stabilizability of the third order linear dispersion equation on a periodic domain. SIAM J. Cont. Optim. 31 (1993) 659–676. Zbl0771.93073
  24. [24] 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
  25. [25] T. Tao, Global well-posedness of the Benjamin-Ono equation in H 1 ( ) , preprint (2003). Zbl1055.35104MR2052470

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.