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  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 & Related Fields, 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[1]

Xiangqing Zhao, Bing-Yu Zhang. Global controllability and stabilizability of Kawahara equation on a periodic domain. Mathematical Control & 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 & Continuous Dynamical Systems - A, 2005, 13 (1) : 203-218. doi: 10.3934/dcds.2005.13.203

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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 & Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Andrei Fursikov. Stabilization of the simplest normal parabolic equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1815-1854. doi: 10.3934/cpaa.2014.13.1815

[14]

Marcel Braukhoff. Semiconductor Boltzmann-Dirac-Benney equation with a BGK-type collision operator: Existence of solutions vs. ill-posedness. Kinetic & Related Models, 2019, 12 (2) : 445-482. doi: 10.3934/krm.2019019

[15]

Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021

[16]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305

[17]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143

[18]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020

[19]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

[20]

Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (32)
  • HTML views (194)
  • Cited by (0)

Other articles
by authors

[Back to Top]