# American Institute of Mathematical Sciences

September  2020, 9(3): 673-692. 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  September 2020 Early access  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 and Control Theory, 2020, 9 (3) : 673-692. doi: 10.3934/eect.2020028
##### References:
 [1] F. Ammar-Khodja, A. Benabdallah, M. 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. [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. [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. [4] L. Baudouin, E. 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. [5] A. Benabdallah, F. Boyer, M. 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. [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. [7] C. M. Cazacu, L. 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. [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. [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. [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. [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. [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. [13] J.-M. Coron, O. 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. [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. [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. [16] E. Fernández-Cara, M. 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. [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. [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. [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. [20] D. Mugnolo, D. Noja and C. Seifert, Airy-type evolution equations on star graphs, Anal. PDE, 11 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625. [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. [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. [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. [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. [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. [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.

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##### References:
 [1] F. Ammar-Khodja, A. Benabdallah, M. 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. [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. [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. [4] L. Baudouin, E. 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. [5] A. Benabdallah, F. Boyer, M. 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. [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. [7] C. M. Cazacu, L. 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. [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. [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. [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. [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. [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. [13] J.-M. Coron, O. 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. [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. [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. [16] E. Fernández-Cara, M. 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. [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. [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. [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. [20] D. Mugnolo, D. Noja and C. Seifert, Airy-type evolution equations on star graphs, Anal. PDE, 11 (2018), 1625-1652.  doi: 10.2140/apde.2018.11.1625. [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. [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. [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. [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. [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. [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.
A tree-shaped network with $3$ edges ($N = 2$)
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