American Institute of Mathematical Sciences

Advanced
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  October 2020 Early access  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

[1]

Felipe Linares, Mahendra Panthee, Tristan Robert, Nikolay Tzvetkov. On the periodic Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3521-3533. doi: 10.3934/dcds.2019145
[2]

Nathan Glatt-Holtz, Roger Temam, Chuntian Wang. Martingale and pathwise solutions to the stochastic Zakharov-Kuznetsov equation with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1047-1085. doi: 10.3934/dcdsb.2014.19.1047
[3]

Raphaël Côte, Frédéric Valet. Polynomial growth of high sobolev norms of solutions to the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1039-1058. doi: 10.3934/cpaa.2021005
[4]

Felipe Linares, Gustavo Ponce. On special regularity properties of solutions of the Zakharov-Kuznetsov equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1561-1572. doi: 10.3934/cpaa.2018074
[5]

Axel Grünrock, Sebastian Herr. The Fourier restriction norm method for the Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2061-2068. doi: 10.3934/dcds.2014.34.2061
[6]

Felipe Linares, Jean-Claude Saut. The Cauchy problem for the 3D Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 547-565. doi: 10.3934/dcds.2009.24.547
[7]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075
[8]

Mohamad Darwich. Local and global well-posedness in the energy space for the dissipative Zakharov-Kuznetsov equation in 3D. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3715-3724. doi: 10.3934/dcdsb.2020087
[9]

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
[10]

José R. Quintero, Alex M. Montes. On the exact controllability and the stabilization for the Benney-Luke equation. Mathematical Control & Related Fields, 2020, 10 (2) : 275-304. doi: 10.3934/mcrf.2019039
[11]

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
[12]

Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953
[13]

Abdelaziz Khoutaibi, Lahcen Maniar. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evolution Equations & Control Theory, 2020, 9 (2) : 535-559. doi: 10.3934/eect.2020023
[14]

Umberto Biccari, Mahamadi Warma. Null-controllability properties of a fractional wave equation with a memory term. Evolution Equations & Control Theory, 2020, 9 (2) : 399-430. doi: 10.3934/eect.2020011
[15]

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
[16]

André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021024
[17]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743
[18]

Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639
[19]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1
[20]

Patrick Martinez, Judith Vancostenoble. Exact controllability in "arbitrarily short time" of the semilinear wave equation. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 901-924. doi: 10.3934/dcds.2003.9.901

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (151)
  • HTML views (333)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences