June  2020, 10(2): 275-304. doi: 10.3934/mcrf.2019039

On the exact controllability and the stabilization for the Benney-Luke equation

1. 

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

2. 

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

* Corresponding author: José R. Quintero

Received  December 2018 Revised  May 2019 Published  June 2020 Early access  August 2019

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 the stabilization for the generalized Benney-Luke equation
$\begin{equation} u_{tt}-u_{xx}+a u_{xxxx}-bu_{xxtt}+ p u_t u_{x}^{p-1}u_{xx} + 2 u_x^{p}u_{xt} = f, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{equation}$
on a periodic domain
$ S $
(the unit circle on the plane) with internal control
$ f $
supported on an arbitrary sub-domain of
$ S $
. We establish that the model is exactly controllable in a Sobolev type space when the whole
$ S $
is the support of
$ f $
, without any assumption on the size of the initial and final states, and that the model is local exactly controllable when the support of
$ f $
is a proper subdomain of
$ S $
, assuming that initial and terminal states are small. Moreover, in the case that the initial data is small and
$ f $
is a special internal linear feedback, the solution of the model must have uniform exponential decay to a constant state.
Citation: José R. Quintero, Alex M. Montes. On the exact controllability and the stabilization for the Benney-Luke equation. Mathematical Control and Related Fields, 2020, 10 (2) : 275-304. doi: 10.3934/mcrf.2019039
References:
[1]

J. Ben Amara and H. Bouzidi, Exact boundary controllability for the boussinesq equation with variable coefficient, Evol. Equ. Control Theory, 7 (2018), 403-415.  doi: 10.3934/eect.2018020.

[2]

D. J. Benney and J. C. Luke, Interactions of permanent waves of finite amplitude, J. Math. Phys., 43 (1964), 309-313.  doi: 10.1002/sapm1964431309.

[3]

R. A. Capistrano-Filho and M. Cavalcante, Stabilization and control for the biharmonic schrödinger equation, preprint, arXiv: 1807.05264.

[4]

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.

[5]

E. Cerpa and I. Rivas, On the controllability of the Boussinesq equation in low regularity, Journal of Evolution Equations, 18 (2018), 1501-1519.  doi: 10.1007/s00028-018-0450-6.

[6]

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

[7]

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

[8]

C. LaurentL. Rosier and B. Y. 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.

[9]

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

[10]

S. Li, M. Chen and B.-Y. Zhang, Exact controllability and stability of the sixth order boussinesq equation, preprint, arXiv: 1811.05943.

[11]

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.

[12]

R. L. Pego and J. R. Quintero, Two-dimensional solitary waves for a Benney-Luke equation, Physica D, 132 (1999), 476-496.  doi: 10.1016/S0167-2789(99)00058-5.

[13]

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.

[14]

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.

[15]

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

[16]

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.

[17]

D. L. Russell and B. Y. 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.

[18]

T. Tao, Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equations, Amer J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.

[19]

B. Y. Zhang, Analyticity of solutions of the generalized Korteweg-de Vries equation with respect to their initial values, SIAM.I. Math. Anal., 26 (1995), 1488-1513.  doi: 10.1137/S0036141093242600.

[20]

B. Y. Zhang, Exact controllability of the generalized Boussinesq equation, Int. Series of Numerical Mathematics, Birkhüser, Basel, 126 (1998), 297–310.

show all references

References:
[1]

J. Ben Amara and H. Bouzidi, Exact boundary controllability for the boussinesq equation with variable coefficient, Evol. Equ. Control Theory, 7 (2018), 403-415.  doi: 10.3934/eect.2018020.

[2]

D. J. Benney and J. C. Luke, Interactions of permanent waves of finite amplitude, J. Math. Phys., 43 (1964), 309-313.  doi: 10.1002/sapm1964431309.

[3]

R. A. Capistrano-Filho and M. Cavalcante, Stabilization and control for the biharmonic schrödinger equation, preprint, arXiv: 1807.05264.

[4]

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.

[5]

E. Cerpa and I. Rivas, On the controllability of the Boussinesq equation in low regularity, Journal of Evolution Equations, 18 (2018), 1501-1519.  doi: 10.1007/s00028-018-0450-6.

[6]

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

[7]

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

[8]

C. LaurentL. Rosier and B. Y. 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.

[9]

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

[10]

S. Li, M. Chen and B.-Y. Zhang, Exact controllability and stability of the sixth order boussinesq equation, preprint, arXiv: 1811.05943.

[11]

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.

[12]

R. L. Pego and J. R. Quintero, Two-dimensional solitary waves for a Benney-Luke equation, Physica D, 132 (1999), 476-496.  doi: 10.1016/S0167-2789(99)00058-5.

[13]

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.

[14]

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.

[15]

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

[16]

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.

[17]

D. L. Russell and B. Y. 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.

[18]

T. Tao, Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equations, Amer J. Math., 123 (2001), 839-908.  doi: 10.1353/ajm.2001.0035.

[19]

B. Y. Zhang, Analyticity of solutions of the generalized Korteweg-de Vries equation with respect to their initial values, SIAM.I. Math. Anal., 26 (1995), 1488-1513.  doi: 10.1137/S0036141093242600.

[20]

B. Y. Zhang, Exact controllability of the generalized Boussinesq equation, Int. Series of Numerical Mathematics, Birkhüser, Basel, 126 (1998), 297–310.

[1]

Xiangqing Zhao, Bing-Yu Zhang. Global controllability and stabilizability of Kawahara equation on a periodic domain. Mathematical Control and Related Fields, 2015, 5 (2) : 335-358. doi: 10.3934/mcrf.2015.5.335

[2]

José R. Quintero. Nonlinear stability of solitary waves for a 2-d Benney--Luke equation. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 203-218. doi: 10.3934/dcds.2005.13.203

[3]

Francisco J. Vielma leal, Ademir Pastor. Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021062

[4]

Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057

[5]

Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Communications on Pure and Applied Analysis, 2010, 9 (1) : 91-102. doi: 10.3934/cpaa.2010.9.91

[6]

Giuseppe Maria Coclite, Lorenzo di Ruvo. On the solutions for a Benney-Lin type equation. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022024

[7]

Jingqun Wang, Lixin Tian, Weiwei Guo. Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2129-2148. doi: 10.3934/dcdss.2016088

[8]

Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67

[9]

Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations and Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325

[10]

Shuang Yang, Yangrong Li. Forward controllability of a random attractor for the non-autonomous stochastic sine-Gordon equation on an unbounded domain. Evolution Equations and Control Theory, 2020, 9 (3) : 581-604. doi: 10.3934/eect.2020025

[11]

Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations and Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1

[12]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations and Control Theory, 2022, 11 (2) : 373-397. doi: 10.3934/eect.2021004

[13]

Gleb G. Doronin, Nikolai A. Larkin. Kawahara equation in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 783-799. doi: 10.3934/dcdsb.2008.10.783

[14]

Melek Jellouli. On the controllability of the BBM equation. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022002

[15]

Marcel Braukhoff. Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation. Kinetic and Related Models, 2020, 13 (1) : 187-210. doi: 10.3934/krm.2020007

[16]

Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic and Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893

[17]

Jonathan Touboul. Erratum on: Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control and Related Fields, 2019, 9 (1) : 221-222. doi: 10.3934/mcrf.2019006

[18]

Jonathan Touboul. Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain. Mathematical Control and Related Fields, 2012, 2 (4) : 429-455. doi: 10.3934/mcrf.2012.2.429

[19]

Jing Cui, Guangyue Gao, Shu-Ming Sun. Controllability and stabilization of gravity-capillary surface water waves in a basin. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2035-2063. doi: 10.3934/cpaa.2021158

[20]

Morteza Fotouhi, Mohsen Yousefnezhad. Homogenization of a locally periodic time-dependent domain. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1669-1695. doi: 10.3934/cpaa.2020061

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (286)
  • HTML views (588)
  • Cited by (0)

Other articles
by authors

[Back to Top]