# American Institute of Mathematical Sciences

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
 $$$u_{tt}-u_{xx}+a u_{xxxx}-bu_{xxtt}+ p u_t u_{x}^{p-1}u_{xx} + 2 u_x^{p}u_{xt} = f, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)$$$
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. Laurent, L. 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. Laurent, F. 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.

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##### 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. Laurent, L. 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. Laurent, F. 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.
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