doi: 10.3934/dcdsb.2022133
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A note on the control and stabilization of a higher-order water wave model

1. 

Facultad de Ingeniería, Escuela Profesional de Ingeniería Civil, Universidad Tecnológica de los Andes, Sede Abancay, Av. Perú 700, Apurímac, Peru

2. 

Institute of Mathematics, Federal University of Rio de Janeiro, UFRJ, P. O. Box 68530, CEP 21941-909, Rio de Janeiro, RJ, Brazil

*Corresponding author: Ademir Pazoto

Received  October 2021 Revised  May 2022 Early access July 2022

Fund Project: The first author was supported by Capes and Faperj (Brazil). The second author was supported by CNPq (Brazil)

The paper deals with the internal controllability and stabilizability of a family of Boussinesq systems introduced by J. L. Bona, M. Chen and J.-C. Saut to describe two-way propagation of small amplitude gravity waves on the surface of water in a canal. By applying the moment method, we first obtain the exact controllability of the linearized system in suitable Hilbert spaces, which implies its exponential stabilization by suitable feedback. Then, by using a contraction mapping principle, we establish the local exact controllability and exponential stabilization for the original nonlinear system.

Citation: George Bautista, Ademir Pazoto. A note on the control and stabilization of a higher-order water wave model. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022133
References:
[1]

G. J. BautistaS. Micu and A. F. Pazoto, On the controllability of a model system for long waves in nonlinear dispersive media, Nonlinearity, 34 (2021), 989-1013.  doi: 10.1088/1361-6544/abb453.

[2]

G. J. Bautista and A. F. Pazoto, Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain, Commun. Pure Appl. Anal., 19 (2020), 747-769.  doi: 10.3934/cpaa.2020035.

[3]

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.

[4]

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. 

[5]

J.-L. Lions, Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués, Tome 2, Masson, Paris, 1988.

[6]

K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Cont. Optim., 35 (1997), 1574-1590. 

[7]

S. MicuJ. H. OrtegaL. Rosier and B.-Y. 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.

[8]

A. F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, Systems Control Lett., 57 (2008), 595-601.  doi: 10.1016/j.sysconle.2007.12.009.

[9]

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

[10]

D. L. Russell, Controllability and stabilization theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-737.  doi: 10.1137/1020095.

[11]

M. Slemrod, A note on complete controllability and stabilizability for linear control systems in a Hilbert space, SIAM J. Cont., 12 (1974), 500-508. 

[12] R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York-London, 1980. 

show all references

References:
[1]

G. J. BautistaS. Micu and A. F. Pazoto, On the controllability of a model system for long waves in nonlinear dispersive media, Nonlinearity, 34 (2021), 989-1013.  doi: 10.1088/1361-6544/abb453.

[2]

G. J. Bautista and A. F. Pazoto, Decay of solutions for a dissipative higher-order Boussinesq system on a periodic domain, Commun. Pure Appl. Anal., 19 (2020), 747-769.  doi: 10.3934/cpaa.2020035.

[3]

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.

[4]

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. 

[5]

J.-L. Lions, Contrôlabilité Exacte Perturbations et Stabilisation de Systèmes Distribués, Tome 2, Masson, Paris, 1988.

[6]

K. Liu, Locally distributed control and damping for the conservative systems, SIAM J. Cont. Optim., 35 (1997), 1574-1590. 

[7]

S. MicuJ. H. OrtegaL. Rosier and B.-Y. 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.

[8]

A. F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV-KdV type, Systems Control Lett., 57 (2008), 595-601.  doi: 10.1016/j.sysconle.2007.12.009.

[9]

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

[10]

D. L. Russell, Controllability and stabilization theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-737.  doi: 10.1137/1020095.

[11]

M. Slemrod, A note on complete controllability and stabilizability for linear control systems in a Hilbert space, SIAM J. Cont., 12 (1974), 500-508. 

[12] R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York-London, 1980. 
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