Advertisement

Control and stability of the linearized dispersion-generalized Benjamin–Ono equation on a periodic domain

  • C. FloresEmail author
Original Article
  • 119 Downloads

Abstract

We investigate the exact control problem associated to the linearized dispersion-generalized Benjamin–Ono equation which contains fractional-order spatial derivatives on a periodic domain, \(\mathbb {T}\). More specifically, we establish that a mass-preserving external force can be applied to the linear system to achieve a final state from a given initial state. The stabilization problem with a linear feedback control is also studied.

Keywords

Controllability Stabilization Linear KdV-like equations Dispersive equations 

Mathematics Subject Classification

35Q93 93D15 93B52 

Notes

Acknowledgements

The author thanks Derek Smith and Seungly Oh for fruitful conversations and Felipe Linares for helpful comments as well as the referee’s remarks which improve the presentation of this work.

References

  1. 1.
    Amick CJ, Toland JF (1991) Uniqueness of Benjamin’s solitary-wave solution of the Benjamin–Ono equation. IMA J Appl Math 46(1–2):21–28.  https://doi.org/10.1093/imamat/46.1-2.21 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benjamin T (1967) Internal waves of permanent form in fluids of great depth. J Fluid Mech 29:559–592.  https://doi.org/10.1017/S002211206700103X CrossRefzbMATHGoogle Scholar
  3. 3.
    Bona JL, Smith R (1975) The initial-value problem for the Korteweg–de Vries equation. Philos Trans Roy Soc London Ser A 278(1287):555–601MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bourgain J (1993) Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II. The KdV-equation. Geom Funct Anal 3(3):209–262.  https://doi.org/10.1007/BF01895688 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Burq N, Planchon F (2008) On well-posedness for the Benjamin–Ono equation. Math Ann 340(3):497–542.  https://doi.org/10.1007/s00208-007-0150-y MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Coifman RR, Wickerhauser MV (1990) The scattering transform for the Benjamin–Ono equation. Inverse Probl 6(5):825–861MathSciNetCrossRefGoogle Scholar
  7. 7.
    Colliander J, Keel M, Staffilani G, Takaoka H, Tao T (2003) Sharp global well-posedness for KdV and modified KdV on \(\mathbb{R}\) and \(\mathbb{T}\). J Am Math Soc 16(3):705–749.  https://doi.org/10.1090/S0894-0347-03-00421-1 (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dehman B, Gérard P, Lebeau G (2006) Stabilization and control for the nonlinear Schrödinger equation on a compact surface. Math Z 254(4):729–749.  https://doi.org/10.1007/s00209-006-0005-3 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Flores C, Oh S, Smith D (2017) Stabilization of dispersion generalized Benjamin Ono. ArXiv e-printsGoogle Scholar
  10. 10.
    Fokas AS, Ablowitz MJ (1983) The inverse scattering transform for the Benjamin–Ono equation—a pivot to multidimensional problems. Stud Appl Math 68(1):1–10MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fonseca G, Ponce G (2011) The IVP for the Benjamin–Ono equation in weighted Sobolev spaces. J Funct Anal 260(2):436–459.  https://doi.org/10.1016/j.jfa.2010.09.010 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guo Z (2008) Local well-posedness for dispersion generalized Benjamin–Ono equations in Sobolev spaces. ArXiv e-printsGoogle Scholar
  13. 13.
    Herr S (2007) Well-posedness for equations of Benjamin–Ono type Illinois. J Math 51(3):951–976MathSciNetzbMATHGoogle Scholar
  14. 14.
    Herr S, Ionescu AD, Kenig CE, Koch H (2010) A para-differential renormalization technique for nonlinear dispersive equations. Commun Partial Differ Equ 35(10):1827–1875.  https://doi.org/10.1080/03605302.2010.487232 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ionescu AD, Kenig CE (2007) Global well-posedness of the Benjamin–Ono equation in low-regularity spaces. J Am Math Soc 20(3):753–798.  https://doi.org/10.1090/S0894-0347-06-00551-0 (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iório RJ Jr (1986) On the Cauchy problem for the Benjamin–Ono equation. Commun Partial Differ Equ 11(10):1031–1081.  https://doi.org/10.1080/03605308608820456 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Iorio RJ Jr (2003) Unique continuation principles for the Benjamin–Ono equation. Differ Integral Equ 16(11):1281–1291MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kato T (1983) On the Cauchy problem for the (generalized) Korteweg–de Vries equation. In: Studies in applied mathematics. Adv Math Suppl Stud, vol 8. Academic Press, New York, pp 93–128Google Scholar
  19. 19.
    Kenig CE, Ponce G, Vega L (1991) Well-posedness of the initial value problem for the Korteweg–de Vries equation. J Am Math Soc 4(2):323–347.  https://doi.org/10.2307/2939277 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kenig CE, Koenig KD (2003) On the local well-posedness of the Benjamin–Ono and modified Benjamin–Ono equations. Math Res Lett 10(5–6):879–895.  https://doi.org/10.4310/MRL.2003.v10.n6.a13 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kenig CE, Ponce G, Vega L (1996) A bilinear estimate with applications to the KdV equation. J Am Math Soc 9(2):573–603.  https://doi.org/10.1090/S0894-0347-96-00200-7 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Koch H, Tzvetkov N (2003) On the local well-posedness of the Benjamin–Ono equation in \(H^s(\mathbb{R})\). Int Math Res Not 26:1449–1464.  https://doi.org/10.1155/S1073792803211260 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Korteweg DJ, De Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos Mag 5(39):422–443MathSciNetCrossRefGoogle Scholar
  24. 24.
    Laurent C (2010) Global controllability and stabilization for the nonlinear Schrödinger equation on an interval. ESAIM Control Optim Calc Var 16(2):356–379.  https://doi.org/10.1051/cocv/2009001 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Laurent C (2010) Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3. SIAM J Math Anal 42(2):785–832.  https://doi.org/10.1137/090749086 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Laurent C, Rosier L, Zhang BY (2010) Control and stabilization of the Korteweg–de Vries equation on a periodic domain. Comm Partial Differ Equ 35(4):707–744.  https://doi.org/10.1080/03605300903585336 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Linares F, Ortega JH (2005) On the controllability and stabilization of the linearized Benjamin–Ono equation. ESAIM Control Optim Calc Var 11(2):204–218.  https://doi.org/10.1051/cocv:2005002 (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Linares F, Ponce G (2009) Introduction to nonlinear dispersive equations. Universitext. Springer, New YorkzbMATHGoogle Scholar
  29. 29.
    Linares F, Rosier L (2015) Control and stabilization of the Benjamin–Ono equation on a periodic domain. Trans Am Math Soc 367(7):4595–4626.  https://doi.org/10.1090/S0002-9947-2015-06086-3 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Linares F, Rosier L, Laurent C (2015) Control and stabilization of the Benjamin–Ono equation in \(L^2(\mathbb{T})\). Arch Mech Anal 218(3):1531–1575MathSciNetCrossRefGoogle Scholar
  31. 31.
    M L, Pilod D (2012) The Cauchy problem for the Benjamin–Ono equation in \(L^2\) revisited. Anal PDE 5(2):365–395.  https://doi.org/10.2140/apde.2012.5.365 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Micu S, Ortega JH, Rosier L, Zhang BY (2009) Control and stabilization of a family of Boussinesq systems. Discret Contin Dyn Syst 24(2):273–313.  https://doi.org/10.3934/dcds.2009.24.273 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Micu S, Zuazua E (1997) Boundary controllability of a linear hybrid systemarising in the control of noise. SIAM J Control Optim 35(5):1614–1637.  https://doi.org/10.1137/S0363012996297972 MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Molinet L, Ribaud F (2006) On global well-posedness for a class of nonlocal dispersive wave equations. Discret Contin Dyn Syst 15(2):657–668.  https://doi.org/10.3934/dcds.2006.15.657 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Molinet L, Saut JC, Tzvetkov N (2001) Ill-posedness issues for the Benjamin–Ono and related equations. SIAM J Math Anal 33(4):982–988.  https://doi.org/10.1137/S0036141001385307 (electronic)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Ono H (1975) Algebraic solitary waves in stratified fluids. J Phys Soc Jpn 39(4):1082–1091CrossRefGoogle Scholar
  37. 37.
    Ponce G (1991) On the global well-posedness of the Benjamin–Ono equation. Differ Integral Equ 4(3):527–542MathSciNetzbMATHGoogle Scholar
  38. 38.
    Rosier L, Zhang BY (2009) Control and stabilization of the Korteweg–de Vries equation: recent progresses. J Syst Sci Complex 22(4):647–682.  https://doi.org/10.1007/s11424-009-9194-2 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Rosier L, Zhang BY (2009) Exact boundary controllability of the nonlinear Schrödinger equation. J Differ Equ 246(10):4129–4153.  https://doi.org/10.1016/j.jde.2008.11.004 CrossRefzbMATHGoogle Scholar
  40. 40.
    Rosier L, Zhang BY (2009) Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval. SIAM J Control Optim 48(2):972–992.  https://doi.org/10.1137/070709578 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Rosier L, Zhang BY (2013) Unique continuation property and control for the Benjamin–Bona–Mahony equation on a periodic domain. J Differ Equ 254(1):141–178.  https://doi.org/10.1016/j.jde.2012.08.014 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Russell DL, Zhang BY (1993) Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J Control Optim 31(3):659–676.  https://doi.org/10.1137/0331030 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Russell DL, Zhang BY (1996) Exact controllability and stabilizability of the Korteweg–de Vries equation. Trans Am Math Soc 348(9):3643–3672.  https://doi.org/10.1090/S0002-9947-96-01672-8 MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Saut JC (1979) Sur quelques généralisations de l’équation de Korteweg–de Vries. J Math Pures Appl (9) 58(1):21–61MathSciNetzbMATHGoogle Scholar
  45. 45.
    Shrira VI, Voronovich VV (1996) Nonlinear dynamics of vorticity waves in the coastal zone. J Fluid Mech 326:181–203.  https://doi.org/10.1017/S0022112096008282 MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Tao T (2004) Global well-posedness of the Benjamin–Ono equation in \(H^1({ R})\). J Hyperbolic Differ Equ 1(1):27–49.  https://doi.org/10.1142/S0219891604000032 MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Weinstein MI (1987) Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities. J Differ Equ 69(2):192–203.  https://doi.org/10.1016/0022-0396(87)90117-3 MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Young R (1980) Introduction to non harmonic Fourier series. Academic Press, New YorkGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.California State University Channel IslandsCamarilloUSA

Personalised recommendations