doi: 10.3934/eect.2020028

Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network

1. 

Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile

2. 

Univ. Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000 Grenoble, France

3. 

Institut Elie Cartan de Lorraine, Université de Lorraine & Inria (Project-Team SPHINX), BP 70 239, F-54506 Vandœuvre-les-Nancy Cedex, France

* Corresponding author: Emmanuelle Crépeau

Received  May 2019 Revised  October 2019 Published  December 2019

Fund Project: This work has been partially supported by FONDECYT 1180528, Math-Amsud ICoPS 17-MATH-04, ANR project ISDEEC (ANR-16-CE40-0013) and Basal Project FB0008 AC3E

Controllability of coupled systems is a complex issue depending on the coupling conditions and the equations themselves. Roughly speaking, the main challenge is controlling a system with less inputs than equations. In this paper this is successfully done for a system of Korteweg-de Vries equations posed on an oriented tree shaped network. The couplings and the controls appear only on boundary conditions.

Citation: Eduardo Cerpa, Emmanuelle Crépeau, Julie Valein. Boundary controllability of the Korteweg-de Vries equation on a tree-shaped network. Evolution Equations & Control Theory, doi: 10.3934/eect.2020028
References:
[1]

F. Ammar-KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[2]

K. Ammari and E. Crépeau, Feedback stabilization and boundary controllability of the Korteweg-de Vries equation on a star-shaped network, SIAM J. Control Optim., 56 (2018), 1620-1639.  doi: 10.1137/17M113959X.  Google Scholar

[3]

S. Avdonin and M. Tucsnak, Simultaneous controllability in sharp time for two elastic strings, ESAIM Control Optim. Calc. Var., 6 (2001), 259-273.  doi: 10.1051/cocv:2001110.  Google Scholar

[4]

L. BaudouinE. Crépeau and J. Valein, Global Carleman estimate on a network for the wave equation and application to an inverse problem, Math. Control Relat. Fields, 1 (2011), 307-330.  doi: 10.3934/mcrf.2011.1.307.  Google Scholar

[5]

A. BenabdallahF. BoyerM. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680.  Google Scholar

[6]

M. Cavalcante, The Korteweg-de Vries equation on a metric star graph, Z. Angew. Math. Phys., 69 (2018), Art. 124, 22pp. doi: 10.1007/s00033-018-1018-6.  Google Scholar

[7]

C. M. CazacuL. I. Ignat and A. F. Pazoto, Null-controllability of the linear Kuramoto-Sivashinsky equation on star-shaped trees, SIAM J. Control Optim., 56 (2018), 2921-2958.  doi: 10.1137/16M1103348.  Google Scholar

[8]

E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain, SIAM J. Control Optim., 46 (2007), 877-899.  doi: 10.1137/06065369X.  Google Scholar

[9]

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

[10]

E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 457-475.  doi: 10.1016/j.anihpc.2007.11.003.  Google Scholar

[11]

E. Cerpa, E. Crépeau and C. Moreno, On the boundary controllability of the korteweg-de vries equation on a star-shaped network, IMA Journal of Math. Control and Information, 2019. doi: 10.1093/imamci/dny047.  Google Scholar

[12]

J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc. (JEMS), 6 (2004), 367-398, URL http://link.springer.de/cgi/linkref?issn=1435-9855&year=04&volume=6&page=367.  Google Scholar

[13]

J.-M. CoronO. Glass and Z. Wang, Exact boundary controllability for 1-D quasilinear hyperbolic systems with a vanishing characteristic speed, SIAM J. Control Optim., 48 (2009/10), 3105-3122.  doi: 10.1137/090749268.  Google Scholar

[14]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, vol. 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.  Google Scholar

[15]

B. Dekoninck and S. Nicaise, Control of networks of Euler-Bernoulli beams, ESAIM Control Optim. Calc. Var., 4 (1999), 57-81.  doi: 10.1051/cocv:1999103.  Google Scholar

[16]

E. Fernández-CaraM. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 259 (2010), 1720-1758.  doi: 10.1016/j.jfa.2010.06.003.  Google Scholar

[17]

O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal., 60 (2008), 61-100.   Google Scholar

[18]

G. R. Leugering and E. J. P. G. Schmidt, On exact controllability of networks of nonlinear elastic strings in 3-dimensional space, Chin. Ann. Math. Ser. B, 33 (2012), 33-60.  doi: 10.1007/s11401-011-0693-9.  Google Scholar

[19]

T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, vol. 3 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO; Higher Education Press, Beijing, 2010.  Google Scholar

[20]

D. MugnoloD. Noja and C. Seifert, Airy-type evolution equations on star graphs, Anal. PDE, 11 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625.  Google Scholar

[21]

S. Nicaise, Control and stabilization of $2\times 2$ hyperbolic systems on graphs, Math. Control Relat. Fields, 7 (2017), 53-72.  doi: 10.3934/mcrf.2017004.  Google Scholar

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[23]

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

[24]

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

[25]

E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, SIAM J. Control Optim., 30 (1992), 229-245.  doi: 10.1137/0330015.  Google Scholar

[26]

Z. A. Sobirov, H. Uecker and M. I. Akhmedov, Exact solutions of the Cauchy problem for the linearized KdV equation on metric star graphs, Uzbek. Mat. Zh., 2015,143-154.  Google Scholar

show all references

References:
[1]

F. Ammar-KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[2]

K. Ammari and E. Crépeau, Feedback stabilization and boundary controllability of the Korteweg-de Vries equation on a star-shaped network, SIAM J. Control Optim., 56 (2018), 1620-1639.  doi: 10.1137/17M113959X.  Google Scholar

[3]

S. Avdonin and M. Tucsnak, Simultaneous controllability in sharp time for two elastic strings, ESAIM Control Optim. Calc. Var., 6 (2001), 259-273.  doi: 10.1051/cocv:2001110.  Google Scholar

[4]

L. BaudouinE. Crépeau and J. Valein, Global Carleman estimate on a network for the wave equation and application to an inverse problem, Math. Control Relat. Fields, 1 (2011), 307-330.  doi: 10.3934/mcrf.2011.1.307.  Google Scholar

[5]

A. BenabdallahF. BoyerM. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680.  Google Scholar

[6]

M. Cavalcante, The Korteweg-de Vries equation on a metric star graph, Z. Angew. Math. Phys., 69 (2018), Art. 124, 22pp. doi: 10.1007/s00033-018-1018-6.  Google Scholar

[7]

C. M. CazacuL. I. Ignat and A. F. Pazoto, Null-controllability of the linear Kuramoto-Sivashinsky equation on star-shaped trees, SIAM J. Control Optim., 56 (2018), 2921-2958.  doi: 10.1137/16M1103348.  Google Scholar

[8]

E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain, SIAM J. Control Optim., 46 (2007), 877-899.  doi: 10.1137/06065369X.  Google Scholar

[9]

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

[10]

E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 457-475.  doi: 10.1016/j.anihpc.2007.11.003.  Google Scholar

[11]

E. Cerpa, E. Crépeau and C. Moreno, On the boundary controllability of the korteweg-de vries equation on a star-shaped network, IMA Journal of Math. Control and Information, 2019. doi: 10.1093/imamci/dny047.  Google Scholar

[12]

J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths, J. Eur. Math. Soc. (JEMS), 6 (2004), 367-398, URL http://link.springer.de/cgi/linkref?issn=1435-9855&year=04&volume=6&page=367.  Google Scholar

[13]

J.-M. CoronO. Glass and Z. Wang, Exact boundary controllability for 1-D quasilinear hyperbolic systems with a vanishing characteristic speed, SIAM J. Control Optim., 48 (2009/10), 3105-3122.  doi: 10.1137/090749268.  Google Scholar

[14]

R. Dáger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, vol. 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.  Google Scholar

[15]

B. Dekoninck and S. Nicaise, Control of networks of Euler-Bernoulli beams, ESAIM Control Optim. Calc. Var., 4 (1999), 57-81.  doi: 10.1051/cocv:1999103.  Google Scholar

[16]

E. Fernández-CaraM. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 259 (2010), 1720-1758.  doi: 10.1016/j.jfa.2010.06.003.  Google Scholar

[17]

O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal., 60 (2008), 61-100.   Google Scholar

[18]

G. R. Leugering and E. J. P. G. Schmidt, On exact controllability of networks of nonlinear elastic strings in 3-dimensional space, Chin. Ann. Math. Ser. B, 33 (2012), 33-60.  doi: 10.1007/s11401-011-0693-9.  Google Scholar

[19]

T. Li, Controllability and Observability for Quasilinear Hyperbolic Systems, vol. 3 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO; Higher Education Press, Beijing, 2010.  Google Scholar

[20]

D. MugnoloD. Noja and C. Seifert, Airy-type evolution equations on star graphs, Anal. PDE, 11 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625.  Google Scholar

[21]

S. Nicaise, Control and stabilization of $2\times 2$ hyperbolic systems on graphs, Math. Control Relat. Fields, 7 (2017), 53-72.  doi: 10.3934/mcrf.2017004.  Google Scholar

[22]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[23]

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

[24]

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

[25]

E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, SIAM J. Control Optim., 30 (1992), 229-245.  doi: 10.1137/0330015.  Google Scholar

[26]

Z. A. Sobirov, H. Uecker and M. I. Akhmedov, Exact solutions of the Cauchy problem for the linearized KdV equation on metric star graphs, Uzbek. Mat. Zh., 2015,143-154.  Google Scholar

Figure 1.  A tree-shaped network with $ 3 $ edges ($ N = 2 $)
[1]

Raphael Stuhlmeier. KdV theory and the Chilean tsunami of 1960. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 623-632. doi: 10.3934/dcdsb.2009.12.623

[2]

Marina Chugunova, Dmitry Pelinovsky. Two-pulse solutions in the fifth-order KdV equation: Rigorous theory and numerical approximations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 773-800. doi: 10.3934/dcdsb.2007.8.773

[3]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185

[4]

Felipe Linares, M. Panthee. On the Cauchy problem for a coupled system of KdV equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 417-431. doi: 10.3934/cpaa.2004.3.417

[5]

Enrique Fernández-Cara, Diego A. Souza. On the control of some coupled systems of the Boussinesq kind with few controls. Mathematical Control & Related Fields, 2012, 2 (2) : 121-140. doi: 10.3934/mcrf.2012.2.121

[6]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

[7]

Juan-Ming Yuan, Jiahong Wu. The complex KdV equation with or without dissipation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 489-512. doi: 10.3934/dcdsb.2005.5.489

[8]

Jiaxiang Cai, Juan Chen, Bin Yang. Fully decoupled schemes for the coupled Schrödinger-KdV system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5523-5538. doi: 10.3934/dcdsb.2019069

[9]

Pierre Lissy. Construction of gevrey functions with compact support using the bray-mandelbrojt iterative process and applications to the moment method in control theory. Mathematical Control & Related Fields, 2017, 7 (1) : 21-40. doi: 10.3934/mcrf.2017002

[10]

Jose-Luis Roca-Gonzalez. Designing dynamical systems for security and defence network knowledge management. A case of study: Airport bird control falconers organizations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1311-1329. doi: 10.3934/dcdss.2015.8.1311

[11]

Shuang Liu, Wenxue Li. Outer synchronization of delayed coupled systems on networks without strong connectedness: A hierarchical method. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 837-859. doi: 10.3934/dcdsb.2018045

[12]

Xianjin Chen, Jianxin Zhou. A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems. Conference Publications, 2009, 2009 (Special) : 151-160. doi: 10.3934/proc.2009.2009.151

[13]

Rowan Killip, Soonsik Kwon, Shuanglin Shao, Monica Visan. On the mass-critical generalized KdV equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 191-221. doi: 10.3934/dcds.2012.32.191

[14]

Annie Millet, Svetlana Roudenko. Generalized KdV equation subject to a stochastic perturbation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1177-1198. doi: 10.3934/dcdsb.2018147

[15]

S. Raynor, G. Staffilani. Low regularity stability of solitons for the KDV equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 277-296. doi: 10.3934/cpaa.2003.2.277

[16]

María Santos Bruzón, Tamara María Garrido. Symmetries and conservation laws of a KdV6 equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 631-641. doi: 10.3934/dcdss.2018038

[17]

R. Demarque, J. Límaco, L. Viana. Local null controllability of coupled degenerate systems with nonlocal terms and one control force. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020026

[18]

Shruti Agarwal, Gilles Carbou, Stéphane Labbé, Christophe Prieur. Control of a network of magnetic ellipsoidal samples. Mathematical Control & Related Fields, 2011, 1 (2) : 129-147. doi: 10.3934/mcrf.2011.1.129

[19]

Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071

[20]

Min Liu, Zhongwei Tang. Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3365-3398. doi: 10.3934/dcds.2019139

2018 Impact Factor: 1.048

Article outline

Figures and Tables

[Back to Top]