Journal of Optimization Theory and Applications

, Volume 180, Issue 1, pp 290–302 | Cite as

Control Problems with an Integral Condition for Korteweg–de Vries Equation on Unbounded Domains

  • Andrei V. FaminskiiEmail author


The initial and initial-boundary value problems, posed on infinite domains for Korteweg–de Vries equation, are considered. The right-hand side of the equation contains an unknown function, regarded as a control, which must be chosen such that the corresponding solution should satisfy certain additional integral condition. Results on existence and uniqueness are established in the cases of small input data or small time interval.


Control Initial-boundary value problem Integral condition Korteweg–de Vries equation 

Mathematics Subject Classification

49N45 35Q53 35Q93 



The publication was prepared with the support of the “RUDN University Program 5-100”, Ministry of Education and Science of the Russian Federation (Project 1.962.2017/PCh) and RFBR Grants 17-01-00849, 17-51-52022, 18-01-00590.


  1. 1.
    Korteweg, D.G., de Vries, G.: 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, 422–443 (1895)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and application to non-linear evolution equations, part II: the KdV equation. Geom. Funct. Anal. 3, 209–262 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Kenig, C.E., Ponce, G., Vega, L.: A bilinear estimate with application to the KdV equation. J. Am. Math. Soc. 9, 573–603 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Colliander, J.E., Kenig, C.E.: The generalized Korteweg–de Vries equation on the half line. Commun. Partial Differ. Equ. 27, 2187–2266 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Faminskii, A.V.: An initial boundary-value problem in a half-strip for the Korteweg–de Vries equation in fractional-order Sobolev spaces. Commun. Partial Differ. Equ. 29, 1653–1695 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bona, J.L., Sun, S., Zhzng, B.-Y.: Boundary smoothing properties of the Korteweg–de Vries equation in a quarter plane and applications. Dyn. Partial Differ. Equ. 3, 1–70 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Faminskii, A.V.: Global well-posedness of two initial-boundary-value problems for the Korteweg–de Vries equation. Differ. Integral Equ. 20, 601–642 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Russel, D.L., Zhang, B.-Y.: Exact controllability and stabilizability of the Korteweg–de Vries equation. Trans. Am. Math. Soc. 348, 3643–3672 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Rosier, L.: Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain. ESAIM Control Optim. Calc. Var. 2, 33–55 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Zhang, B.-Y.: Exact boundary controllability of the Korteweg–de Vries equation. SIAM J. Control Optim. 37, 543–565 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Coron, J.-M., Crépeau, E.: Exact boundary controllability of a nonlinear KdV equation with critical lengths. J. Eur. Math. Soc. 6, 367–398 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rosier, L.: Control of the surface of a fluid by a wavemaker. ESAIM Control Optim. Cal. Var. 10, 346–380 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cerpa, E.: Exact controllability of a nonlinear Korteweg–de Vries equation on a critical spatial domain. SIAM J. Control Optim. 46, 877–899 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Glass, O., Guerrero, S.: Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit. Asymptot. Anal. 60, 61–100 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cerpa, E., Crépeau, E.: Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain. Ann. Inst. H. Poincaré (C) Non Linear Anal 26, 457–475 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Laurent, C., Rosier, L., Zhang, B.-Y.: Control and stabilization of the Korteweg–de Vries equation on a periodic domain. Commun. Partial Differ. Equ. 35, 707–744 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Glass, O., Guerrero, S.: Controllability of the Korteweg–de Vries equation from the right Dirichlet boundary condition. Syst. Control Lett. 59, 390–395 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cerpa, E., Rivas, I., Zhang, B.-Y.: Boundary controllability of the Korteweg–de vries equation on a bounded domain. SIAM J. Control Optim. 51, 2976–3010 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Capistrano-Filho, R.A., Pazoto, A.F., Rosier, L.: Internal controllability of the Korteweg–de Vries equation on a bounded domain. ESAIM Control Optim. Calc. Var. 21, 1076–1107 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rosier, L., Zhang, B.-Y.: Control and stabilization of the Korteweg–de Vries equation: recent progresses. J. Syst. Sci. Complex. 22, 647–682 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Prilepko, A.I., Orlovsky, D.G., Vasin, I.A.: Methods for Solving Inverse Problems in Mathematical Physics. Marcel Dekker Inc., New York-Basel (1999)zbMATHGoogle Scholar
  22. 22.
    Trebel, H.: Interpolation Theory. Differential Operators, Function Spaces. Veb Deutscher Verlag Der Wissenschaften, Berlin (1978)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Peoples’ Friendship University of Russia (RUDN University)MoscowRussian Federation

Personalised recommendations