American Institute of Mathematical Sciences

Advanced
May  2009, 11(3): 655-668. doi: 10.3934/dcdsb.2009.11.655

Rapid exponential stabilization for a linear Korteweg-de Vries equation

Eduardo Cerpa 1, and  Emmanuelle Crépeau 2,

1. 

Laboratoire de Mathématiques, Université Paris-Sud 11, Bât. 425, 91405 Orsay Cedex, France
2. 

INRIA Rocquencourt, Domaine de Voluceau, 78150 Le Chesnay, France

Received  February 2008 Revised  November 2008 Published  March 2009

We consider a control system for a Korteweg-de Vries equation with homogeneous Dirichlet boundary conditions and Neumann boundary control. We address the rapid exponential stabilization problem. More precisely, we build some feedback laws forcing the solutions of the closed-loop system to decay exponentially to zero with arbitrarily prescribed decay rates. We also perform some numerical computations in order to illustrate this theoretical result.
Keywords: controllability, feedback., stabilization, Korteweg-de Vries equation.
Mathematics Subject Classification: Primary: 93D15, 35Q53; Secondary: 93B0.
Citation: Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[1]

Ahmat Mahamat Taboye, Mohamed Laabissi. Exponential stabilization of a linear Korteweg-de Vries equation with input saturation. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021052
[2]

Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control and Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
[3]

M. Agrotis, S. Lafortune, P.G. Kevrekidis. On a discrete version of the Korteweg-De Vries equation. Conference Publications, 2005, 2005 (Special) : 22-29. doi: 10.3934/proc.2005.2005.22
[4]

Julie Valein. On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021039
[5]

Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations and Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028
[6]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[7]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[8]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[9]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain. Mathematical Control and Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353
[10]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[11]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[12]

Roberto A. Capistrano-Filho, Shuming Sun, Bing-Yu Zhang. General boundary value problems of the Korteweg-de Vries equation on a bounded domain. Mathematical Control and Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[13]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097
[14]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure and Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[15]

Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[16]

John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
[17]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[18]

Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri. Non-standard boundary conditions for the linearized Korteweg-de Vries equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2625-2654. doi: 10.3934/dcdss.2021066
[19]

Rusuo Ye, Yi Zhang. Initial-boundary value problems for the two-component complex modified Korteweg-de Vries equation on the interval. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022111
[20]

Ryan McConnell. Global attractor for the periodic generalized Korteweg-De Vries equation through smoothing. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022115

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (107)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy