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Exact controllability of the linear Zakharov-Kuznetsov equation
| 1. | School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, China |
| 2a. | TAG_SUP Université du Littoral Côte d'Opale, Laboratoire de Mathématiques Pures et Appliquées J. Liouville, BP 699, F-62228 Calais, France |
| 2b. | TAG_SUP CNRS FR 2956, France |
We consider the linear Zakharov-Kuznetsov equation on a rectangle with a left Dirichlet boundary control. Using the flatness approach, we prove the null controllability of that equation and provide a space of analytic reachable states.
References:
| [1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
|
| [2] |
E. Cerpa,
Control of a Korteweg-de Vries equation: A tutorial, Math. Control Relat. Fields, 4 (2014), 45-99.
doi: 10.3934/mcrf.2014.4.45. |
| [3] |
M. Chen,
Unique continuation property for the Zakharov-Kuznetsov equation, Comput. Math. Appl., 77 (2019), 1273-1281.
doi: 10.1016/j.camwa.2018.11.002. |
| [4] |
G. G. Doronin and N. A. Larkin,
Stabilization for the linear Zakharov-Kuznetsov equation without critical size restrictions, J. Math. Anal. Appl., 428 (2015), 337-355.
doi: 10.1016/j.jmaa.2015.03.010. |
| [5] |
G. G. Doronin and N. A. Larkin,
Stabilization of regular solutions for the Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip, Proc. Edinb. Math. Soc., 58 (2015), 661-682.
doi: 10.1017/S0013091514000248. |
| [6] |
A. V. Faminski,
The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equ., 31 (1995), 1002-1012.
|
| [7] |
A. V. Faminskii,
Initial-boundary value problems in a rectangle for two-dimensional Zakharov-Kuznetsov equation, J. Math. Anal. Appl., 463 (2018), 760-793.
doi: 10.1016/j.jmaa.2018.03.048. |
| [8] |
O. Glass and S. Guerrero,
Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptotic Analysis, 60 (2008), 61-100.
doi: 10.3233/ASY-2008-0900. |
| [9] |
C. Laurent and L. Rosier, Exact controllability of nonlinear heat equations in spaces of analytic functions, preprint, arXiv: 1812.06637v1. |
| [10] |
F. Linares and J.-C. Saut,
The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dynam. Syst. A, 24 (2009), 547-565.
doi: 10.3934/dcds.2009.24.547. |
| [11] |
F. Linares, A. Pastor and J.-C. Saut,
Well-posedness for the ZK equation in a cylinder and on the background of a KdV Soliton, Commun. PDEs, 35 (2010), 1674-1689.
doi: 10.1080/03605302.2010.494195. |
| [12] |
P. Martin, L. Rosier and P. Rouchon,
On the reachable states for the boundary control of the heat equation, Appl. Math. Res. Express. AMRX, 2016 (2016), 181-216.
doi: 10.1093/amrx/abv013. |
| [13] |
P. Martin, I. Rivas, L. Rosier and P. Rouchon,
Exact controllability of a linear Korteweg-de Vries equation by the flatness approach, SIAM J. Control Optim., 57 (2019), 2467-2486.
doi: 10.1137/18M1181390. |
| [14] |
G. Perla-Menzala, L. Rosier, J.-C. Saut and R. Temam, Boundary control of the Zakharov-Kuznetsov equation, in preparation. |
| [15] |
L. Rosier,
Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55.
doi: 10.1051/cocv:1997102. |
| [16] |
L. Rosier,
Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var., 10 (2004), 346-380.
doi: 10.1051/cocv:2004012. |
| [17] |
L. Rosier and B.-Y. Zhang,
Control and stabilization of the Korteweg-de Vries equation: recent progresses, J. Syst. Sci. Complex., 22 (2009), 647-682.
doi: 10.1007/s11424-009-9194-2. |
| [18] |
J.-C. Saut and R. Temam,
An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Diff. Equations, 15 (2010), 1001-1031.
|
| [19] |
J.-C. Saut, R. Temam and C. Wang, An initial and boundary-value problem for the Zakharov-Kuznestov equation in a bounded domain, Journal of Mathematical Physics, 53 (2012), 115612, 29pp.
doi: 10.1063/1.4752102. |
show all references
References:
| [1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
|
| [2] |
E. Cerpa,
Control of a Korteweg-de Vries equation: A tutorial, Math. Control Relat. Fields, 4 (2014), 45-99.
doi: 10.3934/mcrf.2014.4.45. |
| [3] |
M. Chen,
Unique continuation property for the Zakharov-Kuznetsov equation, Comput. Math. Appl., 77 (2019), 1273-1281.
doi: 10.1016/j.camwa.2018.11.002. |
| [4] |
G. G. Doronin and N. A. Larkin,
Stabilization for the linear Zakharov-Kuznetsov equation without critical size restrictions, J. Math. Anal. Appl., 428 (2015), 337-355.
doi: 10.1016/j.jmaa.2015.03.010. |
| [5] |
G. G. Doronin and N. A. Larkin,
Stabilization of regular solutions for the Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip, Proc. Edinb. Math. Soc., 58 (2015), 661-682.
doi: 10.1017/S0013091514000248. |
| [6] |
A. V. Faminski,
The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equ., 31 (1995), 1002-1012.
|
| [7] |
A. V. Faminskii,
Initial-boundary value problems in a rectangle for two-dimensional Zakharov-Kuznetsov equation, J. Math. Anal. Appl., 463 (2018), 760-793.
doi: 10.1016/j.jmaa.2018.03.048. |
| [8] |
O. Glass and S. Guerrero,
Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptotic Analysis, 60 (2008), 61-100.
doi: 10.3233/ASY-2008-0900. |
| [9] |
C. Laurent and L. Rosier, Exact controllability of nonlinear heat equations in spaces of analytic functions, preprint, arXiv: 1812.06637v1. |
| [10] |
F. Linares and J.-C. Saut,
The Cauchy problem for the 3D Zakharov-Kuznetsov equation, Discrete Contin. Dynam. Syst. A, 24 (2009), 547-565.
doi: 10.3934/dcds.2009.24.547. |
| [11] |
F. Linares, A. Pastor and J.-C. Saut,
Well-posedness for the ZK equation in a cylinder and on the background of a KdV Soliton, Commun. PDEs, 35 (2010), 1674-1689.
doi: 10.1080/03605302.2010.494195. |
| [12] |
P. Martin, L. Rosier and P. Rouchon,
On the reachable states for the boundary control of the heat equation, Appl. Math. Res. Express. AMRX, 2016 (2016), 181-216.
doi: 10.1093/amrx/abv013. |
| [13] |
P. Martin, I. Rivas, L. Rosier and P. Rouchon,
Exact controllability of a linear Korteweg-de Vries equation by the flatness approach, SIAM J. Control Optim., 57 (2019), 2467-2486.
doi: 10.1137/18M1181390. |
| [14] |
G. Perla-Menzala, L. Rosier, J.-C. Saut and R. Temam, Boundary control of the Zakharov-Kuznetsov equation, in preparation. |
| [15] |
L. Rosier,
Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var., 2 (1997), 33-55.
doi: 10.1051/cocv:1997102. |
| [16] |
L. Rosier,
Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var., 10 (2004), 346-380.
doi: 10.1051/cocv:2004012. |
| [17] |
L. Rosier and B.-Y. Zhang,
Control and stabilization of the Korteweg-de Vries equation: recent progresses, J. Syst. Sci. Complex., 22 (2009), 647-682.
doi: 10.1007/s11424-009-9194-2. |
| [18] |
J.-C. Saut and R. Temam,
An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Diff. Equations, 15 (2010), 1001-1031.
|
| [19] |
J.-C. Saut, R. Temam and C. Wang, An initial and boundary-value problem for the Zakharov-Kuznestov equation in a bounded domain, Journal of Mathematical Physics, 53 (2012), 115612, 29pp.
doi: 10.1063/1.4752102. |
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