doi: 10.3934/eect.2021021
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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 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 & Control Theory, 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.   Google Scholar

[2]

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

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

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.   Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

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