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June  2022, 11(3): 681-709. doi: 10.3934/eect.2021021

Exact controllability and stabilization for a general internal wave system of Benjamin-Ono type

1. 

Mathematics Department, Universidad del Valle, Cali, Valle del Cauca, Colombia

2. 

Mathematics Department, Universidad del Cauca, Popayán, Cauca, Colombia

* Corresponding author: Alex M. Montes

Received  July 2020 Revised  February 2021 Published  June 2022 Early access  April 2021

Fund Project: JRQ is supported by the Mathematics Department at Universidad del Valle and AMM is supported by the Mathematics Department at Universidad del Cauca

In this work we consider the exact controllability and stabilization on a periodic domain for the generalized Benjamin-Ono type system for internal waves. The exact controllability of the linearized model is proved by using the moment method and spectral analysis. In order to get the same result for the nonlinear model, we use a fixed point argument in Sobolev spaces.

Citation: José R. Quintero, Alex M. Montes. Exact controllability and stabilization for a general internal wave system of Benjamin-Ono type. Evolution Equations and Control Theory, 2022, 11 (3) : 681-709. doi: 10.3934/eect.2021021
References:
[1]

G. Arenas and J. Quintero, On the existence of solitary waves for an internal system of the Benjamin-Ono type, Diff. Eq. and Dyn. System, 3 (2020), 1-31. 

[2]

G. Arenas and J. Quintero, On the existence of internal solitary waves for a Benjamin-Ono dispersive system, work in progress.

[3]

T. BenjaminJ. Bona and D. Bose, Solitary-wave solutions of nonlinear problems, Philos. Trans. R. Soc. Lond. A Math., Phys. Eng. Sci., 331 (1990), 195-244.  doi: 10.1098/rsta.1990.0065.

[4]

J. Bona and H. Chen, Solitary waves in nonlinear dispersive systems, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 313-378.  doi: 10.3934/dcdsb.2002.2.313.

[5]

J. L. BonaM. 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.  doi: 10.1007/s00332-002-0466-4.

[6]

J. L. BonaM. Chen and J. C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: Nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[7]

E. Cerpa and E. Crépeau, On the controllability of the improved Boussinesq equation, SIAM Journal on Control and Optimization, 56 (2018), 3035-3049.  doi: 10.1137/16M108923X.

[8]

E. Crépeau, Exact controllability of the Boussinesq equation on a bounded domain, Differential Integral Equations, 16 (2003), 303-326. 

[9]

M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries, Communications in Contemporary Mathematics, 11 (2009), 495-521.  doi: 10.1142/S0219199709003454.

[10]

W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluids system, J. Fluid Mech., 313 (1996), 83-103.  doi: 10.1017/S0022112096002133.

[11]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J.Fluid Mech., 396 (1999), 1-36.  doi: 10.1017/S0022112099005820.

[12]

C. Flores, S. Oh and S. Smith, Stabilization of dispersion-generalized Benjamin-Ono, Nonlinear Dispersive Waves and Fluids, 111-136, Contemp. Math., 725, Amer. Math. Soc., Providence, RI, 2019, arXiv: 1709.10224v1. doi: 10.1090/conm/725/14548.

[13]

C. LaurentL. Rosier and B. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Communications in Partial Differential Equations, 35 (2010), 707-744.  doi: 10.1080/03605300903585336.

[14]

C. LaurentF. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation in $L^2(T)$, Arch. Rational Mech. Anal., 218 (2015), 1531-1575.  doi: 10.1007/s00205-015-0887-5.

[15]

F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain, Transactions of the American Mathematical Society, 367 (2015), 4595-4626.  doi: 10.1090/S0002-9947-2015-06086-3.

[16]

S. MicuJ. OrtegaL. Rosier and B. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst., 24 (2009), 273-313.  doi: 10.3934/dcds.2009.24.273.

[17]

J. Muñoz, Existence and numerical approximation of solutions of an improved internal wave model, Mathematical Modelling and Analysis, 19 (2014), 309-333.  doi: 10.3846/13926292.2014.924039.

[18]

J. Muñoz and J. Quintero, Solitary waves for an internal wave model, Discrete Contin. Dyn. Syst., 36 (2016), 5721-5741.  doi: 10.3934/dcds.2016051.

[19]

J. Muñoz and F. A. Pipicano, Existence of periodic travelling wave solutions for a regularized Benjamin-Ono system, J. Diff Eq., 259 (2015), 7503-7528.  doi: 10.1016/j.jde.2015.08.030.

[20]

L. Rosier and B. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Diff. Eq., 254 (2013), 141-178.  doi: 10.1016/j.jde.2012.08.014.

[21]

D. Roumégoux, A sympletic non-squeezing theorem for BBM equation, Dynamics of PDE, 7 (2010), 289-305.  doi: 10.4310/DPDE.2010.v7.n4.a1.

[22]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESIAM: COCV, 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.

[23]

D. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639-739.  doi: 10.1137/1020095.

[24]

D. Russell and B. Zhang, Controllability and stabilizability of the third order linear dispersion equation on a periodic domain, SIAM J. Control and Optim., 31 (1993), 659-676.  doi: 10.1137/0331030.

[25]

D. Russell and B. Zhang, Exact controllability and stabilizability for the Korteweg-de Vries equation, Trans. AMS, 348 (1996), 3643-3672.  doi: 10.1090/S0002-9947-96-01672-8.

[26]

B. Zhang, Exact controllability of the generalized Boussinesq equation, International Series of Numerical Mathematics, 126 (1998), 297-310. 

show all references

References:
[1]

G. Arenas and J. Quintero, On the existence of solitary waves for an internal system of the Benjamin-Ono type, Diff. Eq. and Dyn. System, 3 (2020), 1-31. 

[2]

G. Arenas and J. Quintero, On the existence of internal solitary waves for a Benjamin-Ono dispersive system, work in progress.

[3]

T. BenjaminJ. Bona and D. Bose, Solitary-wave solutions of nonlinear problems, Philos. Trans. R. Soc. Lond. A Math., Phys. Eng. Sci., 331 (1990), 195-244.  doi: 10.1098/rsta.1990.0065.

[4]

J. Bona and H. Chen, Solitary waves in nonlinear dispersive systems, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 313-378.  doi: 10.3934/dcdsb.2002.2.313.

[5]

J. L. BonaM. 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.  doi: 10.1007/s00332-002-0466-4.

[6]

J. L. BonaM. Chen and J. C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: Nonlinear theory, Nonlinearity, 17 (2004), 925-952.  doi: 10.1088/0951-7715/17/3/010.

[7]

E. Cerpa and E. Crépeau, On the controllability of the improved Boussinesq equation, SIAM Journal on Control and Optimization, 56 (2018), 3035-3049.  doi: 10.1137/16M108923X.

[8]

E. Crépeau, Exact controllability of the Boussinesq equation on a bounded domain, Differential Integral Equations, 16 (2003), 303-326. 

[9]

M. Chapouly, Global controllability of a nonlinear Korteweg-de Vries, Communications in Contemporary Mathematics, 11 (2009), 495-521.  doi: 10.1142/S0219199709003454.

[10]

W. Choi and R. Camassa, Weakly nonlinear internal waves in a two-fluids system, J. Fluid Mech., 313 (1996), 83-103.  doi: 10.1017/S0022112096002133.

[11]

W. Choi and R. Camassa, Fully nonlinear internal waves in a two-fluid system, J.Fluid Mech., 396 (1999), 1-36.  doi: 10.1017/S0022112099005820.

[12]

C. Flores, S. Oh and S. Smith, Stabilization of dispersion-generalized Benjamin-Ono, Nonlinear Dispersive Waves and Fluids, 111-136, Contemp. Math., 725, Amer. Math. Soc., Providence, RI, 2019, arXiv: 1709.10224v1. doi: 10.1090/conm/725/14548.

[13]

C. LaurentL. Rosier and B. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Communications in Partial Differential Equations, 35 (2010), 707-744.  doi: 10.1080/03605300903585336.

[14]

C. LaurentF. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation in $L^2(T)$, Arch. Rational Mech. Anal., 218 (2015), 1531-1575.  doi: 10.1007/s00205-015-0887-5.

[15]

F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain, Transactions of the American Mathematical Society, 367 (2015), 4595-4626.  doi: 10.1090/S0002-9947-2015-06086-3.

[16]

S. MicuJ. OrtegaL. Rosier and B. Zhang, Control and stabilization of a family of Boussinesq systems, Discrete Contin. Dyn. Syst., 24 (2009), 273-313.  doi: 10.3934/dcds.2009.24.273.

[17]

J. Muñoz, Existence and numerical approximation of solutions of an improved internal wave model, Mathematical Modelling and Analysis, 19 (2014), 309-333.  doi: 10.3846/13926292.2014.924039.

[18]

J. Muñoz and J. Quintero, Solitary waves for an internal wave model, Discrete Contin. Dyn. Syst., 36 (2016), 5721-5741.  doi: 10.3934/dcds.2016051.

[19]

J. Muñoz and F. A. Pipicano, Existence of periodic travelling wave solutions for a regularized Benjamin-Ono system, J. Diff Eq., 259 (2015), 7503-7528.  doi: 10.1016/j.jde.2015.08.030.

[20]

L. Rosier and B. Zhang, Unique continuation property and control for the Benjamin-Bona-Mahony equation on a periodic domain, J. Diff. Eq., 254 (2013), 141-178.  doi: 10.1016/j.jde.2012.08.014.

[21]

D. Roumégoux, A sympletic non-squeezing theorem for BBM equation, Dynamics of PDE, 7 (2010), 289-305.  doi: 10.4310/DPDE.2010.v7.n4.a1.

[22]

L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESIAM: COCV, 2 (1997), 33-55.  doi: 10.1051/cocv:1997102.

[23]

D. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 639-739.  doi: 10.1137/1020095.

[24]

D. Russell and B. Zhang, Controllability and stabilizability of the third order linear dispersion equation on a periodic domain, SIAM J. Control and Optim., 31 (1993), 659-676.  doi: 10.1137/0331030.

[25]

D. Russell and B. Zhang, Exact controllability and stabilizability for the Korteweg-de Vries equation, Trans. AMS, 348 (1996), 3643-3672.  doi: 10.1090/S0002-9947-96-01672-8.

[26]

B. Zhang, Exact controllability of the generalized Boussinesq equation, International Series of Numerical Mathematics, 126 (1998), 297-310. 

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