Open Access
Issue
ESAIM: COCV
Volume 29, 2023
Article Number 7
Number of page(s) 47
DOI https://doi.org/10.1051/cocv/2022085
Published online 11 January 2023
  1. J.J. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci. 12 (2002) 283–318. [CrossRef] [MathSciNet] [Google Scholar]
  2. J.L. Bona, J. Cohen and G. Wang, Global well-posedness of a system of KdV-type with coupled quadratic nonlinearities. Nagoya Math. J. 215 (2014) 67–149. [CrossRef] [MathSciNet] [Google Scholar]
  3. J.L. Bona, S.-M. Sun and B.-Y. Zhang, The initial-boundary value problem for the Korteweg-de Vries equation in a quarter plane. Trans. Am. Math. Soc. 354 (2001) 427–490. [CrossRef] [Google Scholar]
  4. J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, part II: The KdV equation. Geom. Funct. Anal. 3 (1993) 209–262. [CrossRef] [MathSciNet] [Google Scholar]
  5. J.M. Boussinesq, Thórie de l’intumescence liquide, applelée onde solitaire ou de, translation, se propageant dans un canal rectangulaire. C.R. Acad. Sci. Paris 72 (1871) 755–759. [Google Scholar]
  6. J.M. Boussinesq, Theorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal. C.R. Acad. Sci. Paris 72 (1871) 755–759. [Google Scholar]
  7. R.A. Capistrano-Filho, Weak damping for the Korteweg-de Vries equation. Electr. J. Qual. Theory Differ. Equ. 43 (2021) 1–25. [Google Scholar]
  8. R.A. Capistrano-Filho and M. Cavalcante, Stabilization and control for the biharmonic Schrödinger equation. Appl. Math. Optim. 84 (2021) 103–144. [CrossRef] [MathSciNet] [Google Scholar]
  9. R.A. Capistrano-Filho, A.F. Pazoto and L. Rosier, Internal controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 21 (2015) 1076–1107. [CrossRef] [EDP Sciences] [Google Scholar]
  10. R.A. de Capistrano-Filho, V. Komornik and A. Pazoto, Pointwise control of the linearized Gear-Grimshaw system. Evolut. Equ. Control Theory 9 (2020) 693–719. [CrossRef] [Google Scholar]
  11. R.A. Capistrano-Filho, S.-M. Sun and B.-Y. Zhang, Initial boundary value problem for Korteweg-de Vries equation: a review and open problems. Sao Paulo J. Math. Sci. 13 (2019) 402–417. [CrossRef] [MathSciNet] [Google Scholar]
  12. E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain. SIAM J. Control Optim. 43 (2007) 877–899. [CrossRef] [MathSciNet] [Google Scholar]
  13. J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T. Comput. Phys. Commun. 184 (2013) 812–823. [CrossRef] [MathSciNet] [Google Scholar]
  14. B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface. Math. Z. 254 (2006) 729–749. [CrossRef] [MathSciNet] [Google Scholar]
  15. J.A. Gear and R. Grimshaw, Weak and strong interactions between internal solitary waves. Stud. Appl. Math. 70 (1984) 235–258. [CrossRef] [MathSciNet] [Google Scholar]
  16. A. Gruunrock, New applications of the Fourier restriction norm method to wellposedness problems for nonlinear evolution equations. Doctoral Thesis. Bergischen University (2002). [Google Scholar]
  17. R. Hirota and J. Satsuma, Soliton solutions of a coupled Korteweg-de Vries equation. Phys. Lett. A. 85 (1981) 407–408. [CrossRef] [MathSciNet] [Google Scholar]
  18. T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equations, Advances in Mathematics Supplementary Studies. Academic Press, New York (1983), pp. 93–128. [Google Scholar]
  19. C.E. Kenig, G. Ponce and L. Vega, A bilinear estimate with applications to the KdV equation. J. Am. Math. Soc. 9 (1996) 573–603. [CrossRef] [Google Scholar]
  20. V. Komornik and P. Loreti, Fourier Series in Control Theory. Springer Verlag (2005). [Google Scholar]
  21. D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39 (1895) 422–443. [CrossRef] [MathSciNet] [Google Scholar]
  22. C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval. ESAIM: COCV 16 (2010) 356–379. [CrossRef] [EDP Sciences] [Google Scholar]
  23. C. Laurent, F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation in L2(T). Arch Ratl. Mech. Anal. 218 (2015) 1531–1575. [CrossRef] [Google Scholar]
  24. C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain. Commun. Partial Differ. Equ. 35 (2010) 707–744. [CrossRef] [Google Scholar]
  25. F. Linares and J. Ortega, On the controllability and stabilization of the linearized Benjamin-Ono equation. ESAIM: COCV 11 (2005) 204–218. [CrossRef] [EDP Sciences] [Google Scholar]
  26. F. Linares and L. Rosier, Control and stabilization of the Benjamin-Ono equation on a periodic domain. Trans. Am. Math. Soc. 7367 (2015) 4595–4626. [CrossRef] [Google Scholar]
  27. A.J. Majda and J.A. Biello, The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves. J. Atmosp. Sci. 60 (2003) 1809-1821. [CrossRef] [Google Scholar]
  28. S. Micu, J.H. Ortega, L. Rosier and B.-Y. Zhang, Control and stabilization of a family of Boussinesq systems. Discr. Contin. Dyn. Syst. 24 (2009) 273–313. [CrossRef] [Google Scholar]
  29. R.M. Miura, The Korteweg-de Vries equation: a survey of results. SIAM Rev. 18 (1976) 412–459. [CrossRef] [MathSciNet] [Google Scholar]
  30. T. Oh, Diophantine conditions in well-posedness theory of coupled KdV-type systems: local theory. Int. Math. Res. Not. IMRN 18 (2009) 3516–3556. [Google Scholar]
  31. M. Panthee and F. Vielma Leal, On the controllability and stabilization of the Benjamin equation on a periodic domain. To apper Annales de l'Institut Henri Poincaré C, Analyse non linéaire (2020). [Google Scholar]
  32. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Vol. 44 of Applied mathematical sciences. Springer (1983). [CrossRef] [Google Scholar]
  33. J.R. Quintero and A.M. Montes, On the exact controllability and the stabilization for the Benney-Luke equation. Math. Control Related Fields 10 (2020) 275–304. [MathSciNet] [Google Scholar]
  34. L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33–55. [CrossRef] [EDP Sciences] [Google Scholar]
  35. D. Russell, Controllability and stabilizabilization theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639–739. [CrossRef] [MathSciNet] [Google Scholar]
  36. D. Russell and B.-Y. Zhang, Controllability and stabilizability of the third order linear dispersion equation on a periodic domain. SIAM J. Cont. Optim. 31 (1993) 659–676. [CrossRef] [Google Scholar]
  37. D. Russell and B.-Y. Zhang, Exact contollability and stabilizability of the Korteweg-De Vries equation. Trans. Am. Math. Soc. 348 (1996) 3643–3672. [CrossRef] [Google Scholar]
  38. M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space. SIAM J. Control. 12 (1974) 500–508. [CrossRef] [MathSciNet] [Google Scholar]
  39. T. Tao, Nonlinear Dispersive Equations, Local and Global Analysis. CBMS Regional Conference Series in Mathematics, 106. Providence, RI: American Mathematical Society (2006). [CrossRef] [Google Scholar]
  40. X. Yang and B.-Y. Zhang, Well-posedness and critical index set of the cauchy problem for the coupled KdV-KdV systems on 𝕋, arXiv:1907.05580v1 [math.AP]. [Google Scholar]
  41. X. Zhao and B.-Y. Zhang, Global controllability and stabilizability of Kawahara equation on a periodic domain. Math. Control Related Fields 5 (2015) 335–358. [CrossRef] [MathSciNet] [Google Scholar]
  42. B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation. SIAM J. Cont. Optim. 37 (1999) 543–565. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.