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

Previous Article
Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks
MCRF Home
This Issue
Next Article
Coefficient identification and fault detection in linear elastic systems; one dimensional problems

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

Abstract
Related Papers

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. doi: 10.1103/PhysRevLett.31.125.
[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. doi: 10.1016/S0362-546X(97)00724-4.
[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.
[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. doi: 10.1007/BF02099010.
[5]	T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equation,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998).
[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.
[7]	M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations,", Ph.D thesis, (1994).
[8]	J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves,, Stud. in Appl. Math., 70 (1984), 235.
[9]	F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations,, Commun. Pure Appl. Anal., 3 (2004), 417. doi: 10.3934/cpaa.2004.3.417.
[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. doi: 10.1090/S0002-9939-07-08810-7.
[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.
[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.
[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., ().
[14]	S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations,, in, (2000), 1020.
[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. doi: 10.1142/S0219199709003600.
[16]	A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping,, ESAIM Control Optim. Calc. Var., 11 (2005), 473. doi: 10.1051/cocv:2005015.
[17]	L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain,, ESAIM Control Optim. Calc. Var., 2 (1997), 33. doi: 10.1051/cocv:1997102.
[18]	L. Rosier, Control of the surface of a fluid by a wavemaker,, ESAIM Control Optim. Calc. Var., 10 (2004), 346. doi: 10.1051/cocv:2004012.
[19]	L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line,, SIAM J. Control Optim., 39 (2000), 331. doi: 10.1137/S0363012999353229.
[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. doi: 10.1137/050631409.
[21]	J. Simon, Compact sets in the $L^p(0,T;B)$ spaces,, Analli Mat. Pura Appl., 146 (1987), 65.
[22]	R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Third edition, 2 (1984).
[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, (2001).

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. doi: 10.1103/PhysRevLett.31.125.
[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. doi: 10.1016/S0362-546X(97)00724-4.
[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.
[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. doi: 10.1007/BF02099010.
[5]	T. Cazenave and A. Haraux, "An Introduction to Semilinear Evolution Equation,", Oxford Lecture Series in Mathematics and its Applications, 13 (1998).
[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.
[7]	M. Davila, "On the Unique Continuation Property for a Coupled System of Korteweg-de Vries Equations,", Ph.D thesis, (1994).
[8]	J. A. Gear and R. Grimshaw, Weak and strong interaction between internal solitary waves,, Stud. in Appl. Math., 70 (1984), 235.
[9]	F. Linares and M. Panthee, On the Cauchy problem for a coupled system of KdV equations,, Commun. Pure Appl. Anal., 3 (2004), 417. doi: 10.3934/cpaa.2004.3.417.
[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. doi: 10.1090/S0002-9939-07-08810-7.
[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.
[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.
[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., ().
[14]	S. Micu and J. H. Ortega, On the controllability of a linear coupled system of Korteweg-de Vries equations,, in, (2000), 1020.
[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. doi: 10.1142/S0219199709003600.
[16]	A. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping,, ESAIM Control Optim. Calc. Var., 11 (2005), 473. doi: 10.1051/cocv:2005015.
[17]	L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain,, ESAIM Control Optim. Calc. Var., 2 (1997), 33. doi: 10.1051/cocv:1997102.
[18]	L. Rosier, Control of the surface of a fluid by a wavemaker,, ESAIM Control Optim. Calc. Var., 10 (2004), 346. doi: 10.1051/cocv:2004012.
[19]	L. Rosier, Exact boundary controllability for the linear Korteweg-de Vries equation on the half-line,, SIAM J. Control Optim., 39 (2000), 331. doi: 10.1137/S0363012999353229.
[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. doi: 10.1137/050631409.
[21]	J. Simon, Compact sets in the $L^p(0,T;B)$ spaces,, Analli Mat. Pura Appl., 146 (1987), 65.
[22]	R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Third edition, 2 (1984).
[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, (2001).

[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 - A, 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 - A, 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]	Ludovick Gagnon. Qualitative description of the particle trajectories for the N-solitons solution of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1489-1507. doi: 10.3934/dcds.2017061
[8]	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
[9]	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
[10]	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 - A, 2010, 26 (3) : 857-871. doi: 10.3934/dcds.2010.26.857
[11]	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
[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 & Related Fields, 2018, 8 (3&4) : 583-605. doi: 10.3934/mcrf.2018024
[13]	John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
[14]	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
[15]	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
[16]	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
[17]	Olivier Goubet. Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 625-644. doi: 10.3934/dcds.2000.6.625
[18]	Zhaosheng Feng, Yu Huang. Approximate solution of the Burgers-Korteweg-de Vries equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 429-440. doi: 10.3934/cpaa.2007.6.429
[19]	Terence Tao. Two remarks on the generalised Korteweg de-Vries equation. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 1-14. doi: 10.3934/dcds.2007.18.1
[20]	Massimiliano Gubinelli. Rough solutions for the periodic Korteweg--de~Vries equation. Communications on Pure & Applied Analysis, 2012, 11 (2) : 709-733. doi: 10.3934/cpaa.2012.11.709

2018 Impact Factor: 1.292

American Institute of Mathematical Sciences