Skip to main content
SpringerLink
Log in

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

  • Original Article
  • Published:
Mathematics of Control, Signals, and Systems Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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):555–601

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  6. Coifman RR, Wickerhauser MV (1990) The scattering transform for the Benjamin–Ono equation. Inverse Probl 6(5):825–861

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  9. Flores C, Oh S, Smith D (2017) Stabilization of dispersion generalized Benjamin Ono. ArXiv e-prints

  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):1–10

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo Z (2008) Local well-posedness for dispersion generalized Benjamin–Ono equations in Sobolev spaces. ArXiv e-prints

  13. Herr S (2007) Well-posedness for equations of Benjamin–Ono type Illinois. J Math 51(3):951–976

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  17. Iorio RJ Jr (2003) Unique continuation principles for the Benjamin–Ono equation. Differ Integral Equ 16(11):1281–1291

    MathSciNet  MATH  Google Scholar 

  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 93–128

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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):422–443

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Linares F, Ponce G (2009) Introduction to nonlinear dispersive equations. Universitext. Springer, New York

    MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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):1531–1575

    Article  MathSciNet  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ono H (1975) Algebraic solitary waves in stratified fluids. J Phys Soc Jpn 39(4):1082–1091

    Article  Google Scholar 

  37. Ponce G (1991) On the global well-posedness of the Benjamin–Ono equation. Differ Integral Equ 4(3):527–542

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  44. Saut JC (1979) Sur quelques généralisations de l’équation de Korteweg–de Vries. J Math Pures Appl (9) 58(1):21–61

    MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  48. Young R (1980) Introduction to non harmonic Fourier series. Academic Press, New York

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. California State University Channel Islands, One University Drive, Camarillo, CA, 93012, USA

    C. Flores

Authors
  1. C. Flores
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. Flores.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Flores, C. Control and stability of the linearized dispersion-generalized Benjamin–Ono equation on a periodic domain. Math. Control Signals Syst. 30, 13 (2018). https://doi.org/10.1007/s00498-018-0219-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00498-018-0219-z

Keywords

Mathematics Subject Classification

Search

Navigation