American Institute of Mathematical Sciences

Advanced
September  2011, 1(3): 353-389. doi: 10.3934/mcrf.2011.1.353

Global stabilization of a coupled system of two generalized Korteweg-de Vries type equations posed on a finite domain

Dugan Nina 1, , Ademir Fernando Pazoto 1, and  Lionel Rosier 2,

1. 

Instituto de Matemática, Universidade Federal do Rio de Janeiro, P.O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil, Brazil
2. 

Institut Élie Cartan, UMR 7502 UHP/CNRS/INRIA, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex

Received  November 2010 Revised  April 2011 Published  September 2011

The purpose of this work is to study the internal stabilization of a coupled system of two generalized Korteweg-de Vries equations under the effect of a localized damping term. The exponential stability, as well as, the global existence of weak solutions are investigated when the exponent in the nonlinear term ranges over the interval $[1, 4)$. To obtain the decay we use multiplier techniques combined with compactness arguments and reduce the problem to prove a unique continuation property for weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.
Keywords: Exponential Decay, Korteweg-de Vries equation, Stabilization..
Mathematics Subject Classification: Primary: 93D15, 35Q53; Secondary: 93C2.
Citation: 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 & Related Fields, 2011, 1 (3) : 353-389. doi: 10.3934/mcrf.2011.1.353
References:
[1]

M. Ablowitz, D. Kaup, A. Newell and H. Segur, Nonlinear-evolution equations of physical signifcance, Phys. Rev. Lett., 31 (1973), 125-127. doi: 10.1103/PhysRevLett.31.125.  Google Scholar

[2]

E. Alarcon, J. Angulo and J. F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035. doi: 10.1016/S0362-546X(97)00724-4.  Google Scholar

[3]

E. Bisognin, V. Bisognin and G. P. Menzala, Exponential stabilization of a coupled system of Korteweg-de Vries Equations with localized damping, Adv. Diff. Eq., 8 (2003), 443-469.  Google Scholar

[4]

J. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. doi: 10.1007/BF02099010.  Google Scholar

[5]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equation," Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[6]

J. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  Google Scholar

[7]

M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations," Ph.D thesis, Federal University of Rio de Janeiro, 1994. Google Scholar

[8]

J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves, Stud. in Appl. Math., 70 (1984), 235-258.  Google Scholar

[9]

F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431. doi: 10.3934/cpaa.2004.3.417.  Google Scholar

[10]

F. Linares and A. F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc., 135 (2007), 1515-1522. doi: 10.1090/S0002-9939-07-08810-7.  Google Scholar

[11]

C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equation as a singular limit of the Kuramoto-Sivashinsky system, Differential Integral Equations, 22 (2009), 53-68.  Google Scholar

[12]

G. P. Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quarterly of Appl. Math., 60 (2002), 111-129.  Google Scholar

[13]

G. P. Menzala, C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a class of KdV equations with localized damping,, Quarterly of Appl. Math., ().   Google Scholar

[14]

S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations, in "Mathematical and Numerical Aspects of Wave Propagation" (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, (2000), 1020-1024.  Google Scholar

[15]

S. Micu, J. H. Ortega and A. F. Pazoto, On the controllability of a coupled system of two Korteweg-de Vries equations, Commun. Contemp. Math., 11 (2009), 799-827. doi: 10.1142/S0219199709003600.  Google Scholar

[16]

A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486. doi: 10.1051/cocv:2005015.  Google Scholar

[17]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. doi: 10.1051/cocv:1997102.  Google Scholar

[18]

L. Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var., 10 (2004), 346-380. doi: 10.1051/cocv:2004012.  Google Scholar

[19]

L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line, SIAM J. Control Optim., 39 (2000), 331-351. doi: 10.1137/S0363012999353229.  Google Scholar

[20]

L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956. doi: 10.1137/050631409.  Google Scholar

[21]

J. Simon, Compact sets in the $L^p(0,T;B)$ spaces, Analli Mat. Pura Appl., 146 (1987), 65-96.  Google Scholar

[22]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Third edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[23]

O. P. Vera Villagran, "Gain of Regularity of the Solutions of a Coupled System of Equations of Korteweg-de Vries Type," Ph.D thesis, Federal University of Rio de Janeiro, 2001. Google Scholar

show all references

References:
[1]

M. Ablowitz, D. Kaup, A. Newell and H. Segur, Nonlinear-evolution equations of physical signifcance, Phys. Rev. Lett., 31 (1973), 125-127. doi: 10.1103/PhysRevLett.31.125.  Google Scholar

[2]

E. Alarcon, J. Angulo and J. F. Montenegro, Stability and instability of solitary waves for a nonlinear dispersive system, Nonlinear Anal., 36 (1999), 1015-1035. doi: 10.1016/S0362-546X(97)00724-4.  Google Scholar

[3]

E. Bisognin, V. Bisognin and G. P. Menzala, Exponential stabilization of a coupled system of Korteweg-de Vries Equations with localized damping, Adv. Diff. Eq., 8 (2003), 443-469.  Google Scholar

[4]

J. Bona, G. Ponce, J.-C. Saut and M. M. Tom, A model system for strong interaction between internal solitary waves, Comm. Math. Phys., 143 (1992), 287-313. doi: 10.1007/BF02099010.  Google Scholar

[5]

T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equation," Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[6]

J. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-value problem for the Korteweg-de Vries equation posed on a finite domain, Comm. Partial Differential Equations, 28 (2003), 1391-1436.  Google Scholar

[7]

M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations," Ph.D thesis, Federal University of Rio de Janeiro, 1994. Google Scholar

[8]

J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves, Stud. in Appl. Math., 70 (1984), 235-258.  Google Scholar

[9]

F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations, Commun. Pure Appl. Anal., 3 (2004), 417-431. doi: 10.3934/cpaa.2004.3.417.  Google Scholar

[10]

F. Linares and A. F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping, Proc. Amer. Math. Soc., 135 (2007), 1515-1522. doi: 10.1090/S0002-9939-07-08810-7.  Google Scholar

[11]

C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a nonlinear coupled system of Korteweg-de Vries equation as a singular limit of the Kuramoto-Sivashinsky system, Differential Integral Equations, 22 (2009), 53-68.  Google Scholar

[12]

G. P. Menzala, C. F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quarterly of Appl. Math., 60 (2002), 111-129.  Google Scholar

[13]

G. P. Menzala, C. P. Massarolo and A. F. Pazoto, Uniform stabilization of a class of KdV equations with localized damping,, Quarterly of Appl. Math., ().   Google Scholar

[14]

S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations, in "Mathematical and Numerical Aspects of Wave Propagation" (Santiago de Compostela, 2000), SIAM, Philadelphia, PA, (2000), 1020-1024.  Google Scholar

[15]

S. Micu, J. H. Ortega and A. F. Pazoto, On the controllability of a coupled system of two Korteweg-de Vries equations, Commun. Contemp. Math., 11 (2009), 799-827. doi: 10.1142/S0219199709003600.  Google Scholar

[16]

A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping, ESAIM Control Optim. Calc. Var., 11 (2005), 473-486. doi: 10.1051/cocv:2005015.  Google Scholar

[17]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55. doi: 10.1051/cocv:1997102.  Google Scholar

[18]

L. Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var., 10 (2004), 346-380. doi: 10.1051/cocv:2004012.  Google Scholar

[19]

L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line, SIAM J. Control Optim., 39 (2000), 331-351. doi: 10.1137/S0363012999353229.  Google Scholar

[20]

L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain, SIAM J. Control Optim., 45 (2006), 927-956. doi: 10.1137/050631409.  Google Scholar

[21]

J. Simon, Compact sets in the $L^p(0,T;B)$ spaces, Analli Mat. Pura Appl., 146 (1987), 65-96.  Google Scholar

[22]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Third edition, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, 1984.  Google Scholar

[23]

O. P. Vera Villagran, "Gain of Regularity of the Solutions of a Coupled System of Equations of Korteweg-de Vries Type," Ph.D thesis, Federal University of Rio de Janeiro, 2001. Google Scholar

[1]

Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[2]

Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & 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]

Guolian Wang, Boling Guo. Stochastic Korteweg-de Vries equation driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5255-5272. doi: 10.3934/dcds.2015.35.5255
[5]

Muhammad Usman, Bing-Yu Zhang. Forced oscillations of the Korteweg-de Vries equation on a bounded domain and their stability. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1509-1523. doi: 10.3934/dcds.2010.26.1509
[6]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[7]

Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations & Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028
[8]

Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[9]

Arnaud Debussche, Jacques Printems. Convergence of a semi-discrete scheme for the stochastic Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 761-781. doi: 10.3934/dcdsb.2006.6.761
[10]

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 & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[11]

Qifan Li. Local well-posedness for the periodic Korteweg-de Vries equation in analytic Gevrey classes. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1097-1109. doi: 10.3934/cpaa.2012.11.1097
[12]

Shou-Fu Tian. Initial-boundary value problems for the coupled modified Korteweg-de Vries equation on the interval. Communications on Pure & Applied Analysis, 2018, 17 (3) : 923-957. doi: 10.3934/cpaa.2018046
[13]

Anne de Bouard, Eric Gautier. Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[14]

John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
[15]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312
[16]

Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri. Non-standard boundary conditions for the linearized Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2625-2654. doi: 10.3934/dcdss.2021066
[17]

Julie Valein. On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021039
[18]

Ivonne Rivas, Muhammad Usman, Bing-Yu Zhang. Global well-posedness and asymptotic behavior of a class of initial-boundary-value problem of the Korteweg-De Vries equation on a finite domain. Mathematical Control & Related Fields, 2011, 1 (1) : 61-81. doi: 10.3934/mcrf.2011.1.61
[19]

Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442
[20]

Brian Pigott. Polynomial-in-time upper bounds for the orbital instability of subcritical generalized Korteweg-de Vries equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 389-418. doi: 10.3934/cpaa.2014.13.389

2020 Impact Factor: 1.284

Tools

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences