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October  2020, 25(10): 3889-3916. doi: 10.3934/dcdsb.2020080

Exact controllability of the linear Zakharov-Kuznetsov equation

Mo Chen 1, and  Lionel Rosier 2a,2b,,

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

* Corresponding author

Received  August 2019 Published  January 2020

Fund Project: The first author (MC) is supported by NSFC Grant (11701078) and China Scholarship Council. The second author (LR) is supported by the ANR project Finite4SoS (ANR-15-CE23-0007)

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.

Keywords: Zakharov-Kuznetsov equation, null controllability, reachable states, exact controllability, flatness approach, Gevrey functions.
Mathematics Subject Classification: 37L50, 93B05.
Citation: Mo Chen, Lionel Rosier. Exact controllability of the linear Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3889-3916. doi: 10.3934/dcdsb.2020080
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. V. Faminski, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equ., 31 (1995), 1002-1012.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Laurent and L. Rosier, Exact controllability of nonlinear heat equations in spaces of analytic functions, preprint, arXiv: 1812.06637v1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. Perla-Menzala, L. Rosier, J.-C. Saut and R. Temam, Boundary control of the Zakharov-Kuznetsov equation, in preparation. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Diff. Equations, 15 (2010), 1001-1031.   Google Scholar

[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.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

A. V. Faminski, The Cauchy problem for the Zakharov-Kuznetsov equation, Differential Equ., 31 (1995), 1002-1012.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Laurent and L. Rosier, Exact controllability of nonlinear heat equations in spaces of analytic functions, preprint, arXiv: 1812.06637v1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

G. Perla-Menzala, L. Rosier, J.-C. Saut and R. Temam, Boundary control of the Zakharov-Kuznetsov equation, in preparation. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J.-C. Saut and R. Temam, An initial boundary-value problem for the Zakharov-Kuznetsov equation, Adv. Diff. Equations, 15 (2010), 1001-1031.   Google Scholar

[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.  Google Scholar

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