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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 |
$\begin{equation} u_{tt}-u_{xx}+a u_{xxxx}-bu_{xxtt}+ p u_t u_{x}^{p-1}u_{xx} + 2 u_x^{p}u_{xt} = f, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{equation}$ |
$ S $ |
$ f $ |
$ S $ |
$ S $ |
$ f $ |
$ f $ |
$ S $ |
$ f $ |
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. |
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. 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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