Skip to main content
SpringerLink
Log in

Boundary control of the generalized Korteweg–de Vries–Burgers equation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper considers the boundary control problem of the generalized Korteweg–de Vries–Burgers (GKdVB) equation on the interval [0, 1]. We derive a control law of the form \(u(0,t)$ $=$ $u_{x}(1,t)$ $=$ $u_{xx}(1,t)$ $+$ $\frac{1}{\mu(\alpha+2)}u^{\alpha+1}$ $(1,t) = 0$, where $\mu > 0\) and α is a positive integer, and prove that it guarantees L 2-global exponential stability, H 1-global asymptotic stability, and H 1-semiglobal exponential stability. Numerical results supporting the analytical ones for both the controlled and uncontrolled equations are presented using a finite element method.

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. Armaou, A., Christofides, P.D.: Wave suppression by nonlinear finite-dimensional control. Chem. Eng. Sci. 55, 2627–2640 (2000)

    Article  Google Scholar 

  2. Atwell, J.A., King, B.B.: Stabilized finite element methods and feedback control for Burgers' equation. In: Proceedings of the American Control Conference, Chicago, IL (2000)

  3. Baker, J.A., Armaou, A., Christofides, P.D.: Nonlinear control of incompressible fluid flows: application to Burgers equation and 2D chaneel flow. J. Math. Anal. Appl. 252, 230–255 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Balogh, A., Krstić, M.: Global boundary stabilization and regularization of Burgers' equation. In: Proceedings of the American Control Conference, San Diego, CA, pp. 1712–1716 (1999)

  5. Balogh, A., Krstić, M.: Boundary control of the Korteweg–de Vries–Burgers equation: further results on stabilization and well-posedness, with numerical demonstration. IEEE Trans. Autom. Control 45(9), 1739–1745 (2000)

    Article  MATH  Google Scholar 

  6. Bona, J.L., Luo, L.: Decay of solutions to nonlinear dispersive wave equations. Differ. Int. Equations 6, 961–980 (1993)

    MATH  MathSciNet  Google Scholar 

  7. Bona, J.L., Schonbek, M.E.: Traveling-wave solutions to the Korteweg–de Vries–Burgers equation. Proc. R. Soc. Edinburgh 101A, 207–226 (1985)

    MathSciNet  Google Scholar 

  8. Bona, J.L., Dougalis, V.A., Karakashian, O.A., McKinney, W.R.: Computations of blow-up and decay for periodic solutions of the generalized Korteweg–de Vries–Burgers equation. Appl. Numer. Math. 10, 335–355 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bracken, P.: Some methods for generating solutions to the Korteweg–de Vries equation. Physica A 335, 70–78 (2004)

    Article  MathSciNet  Google Scholar 

  10. Christofides, P.D., Armaou, A.: Global stabilization of the Kuramoto–Sivashinsky equation via distributed output feedback control. Syst. Control Lett. 39, 283–294 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Èdel'man, I.Ya.: Propagation of nonlinear waves on a porous medium with two-phase saturation by a liquid and a gas. Fluid Dyn. 31(4), 552–559 (1996)

    Article  MATH  Google Scholar 

  12. Gao, P., Zhao, Y.: Boundary stabilization for the general Korteweg–de Vries–Burgers equation. Acta Anal. Funct. Appl. 5(2), 110–118 (2003)

    MATH  MathSciNet  Google Scholar 

  13. Karakashian, O., McKinney, W.: On the approximation of solutions of the generalized Korteweg–de Vries–Burgers equation. Math. Comput. Simul. 37(4/5), 405–416 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Karch, G.: Self-similar lage time behavior of solutions to Korteweg–de Vries–Burgers equation. Nonlinear Anal. 35, 199–219 (1999)

    Article  MathSciNet  Google Scholar 

  15. Komornik, V., Russell, D.L., Zhang, B.Y.: Stabilization de l'équation de Korteweg–de Vries. C. R. Acad. Sci. Paris Ser. I Math. 312(11), 841–843 (1991)

    MATH  MathSciNet  Google Scholar 

  16. Liu, W.-J., Krstic, M.: Controlling nonlinear water waves: boundary stabilization of the Korteweg–de Vries–Burgers equation. In: Proceedings of the American Control Conference, San Diego, CA, pp. 1637–1641 (1999)

  17. Lü, S., Zhang, F.: The spectral method for long time behavior of generalized KdV–Burgers equations. Math. Numer. Sin. 21(2), 129–138 (1999)

    MATH  MathSciNet  Google Scholar 

  18. Naumkin, P.I., Shishmarev, I.A.: A problem on the decay of step-like data for the Korteweg–de Vries–Burgers equation (Russian). Funct. Anal. Appl. 25(1), 16–25 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  19. Nishihara, K., Rajopadhye, S.V.: Asymptotic behavior of solutions to the Korteweg–de Vries–Burgers equation. Differ. Integral Equations 11(1), 85–93 (1998)

    MATH  MathSciNet  Google Scholar 

  20. Rosier, L.: Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain. ESAIM Conrol Optim. Cal. Var. 2, 33–55 (electronic) (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rosier, L.: Exact boundary controllability for the linear Korteweg–de Vries equation – a numerical study. ESAIM Proc. 4, 255–267 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Russell, D.L., Zhang, B.Y.: Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain. SIAM J. Control Optim. 31(3), 659–676 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  23. Russell, D.L., Zhang, B.Y.: Smoothing and decay properties of solutions of the Korteweg–de Vries equation on a periodic domain with point dissipation. J. Math. Anal. Appl. 190(2), 449–488 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  24. Russell, D.L., Zhang, B.Y.: Exact controllability and stabilizability of the Korteweg–de Vries equation. Trans. Am. Math. Soc. 348(9), 3643–3672 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  25. Smaoui, N.: Nonlinear boundary control of the generalized Burgers equation. Nonlinear Dyn. J. 37(1), 75–86 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Smaoui, N.: Boundary and distributed control of the viscous Burgers equation. J. Comput. Appl. Math. 182, 91–104 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  27. Smaoui, N.: Analyzing the dynamics of the forced Burgers equation. J. Appl. Math. Stochastic Anal. 13(3), 269–285 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  28. Taha, T.R.: A parallel algorithm for solving higher KdV equation on hypercube. In: Proceedings of the Fifth IEEE Distributed Memory Computing Conference, Charleston, SC, pp. 564–567 (1990)

  29. Xia, Z., Huang, F.: Fourier spectral methods for a class of generalized KdV–Burgers equations with variable coefficients. In: Proceedings of the First World Congress, 4 volumes, Tampa, FL, August 19–26 (1992)

  30. Zhang, B.Y.: Boundary stabilization of the Korteweg–de Vries equation. In: Control Estimation of Distributed Parameter Systems: Nonlinear Phenomena (Vorau, Austria, 1993). Int. Ser. Numer. Math. 118, 371–389 (1994)

    Google Scholar 

  31. Zhang, L.H.: Decay of solutions of a higher order multi-dimensional nonlinear Korteweg–de Vries–Burgers system. Proc. R. Soc. Edinburgh Sect. A 124(2), 263–271 (1994)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Kuwait University, P.O. Box 5969, Safat, 13060, Kuwait

    Nejib Smaoui & Rasha H. Al-Jamal

Authors
  1. Nejib Smaoui
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rasha H. Al-Jamal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nejib Smaoui.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Smaoui, N., Al-Jamal, R.H. Boundary control of the generalized Korteweg–de Vries–Burgers equation. Nonlinear Dyn 51, 439–446 (2008). https://doi.org/10.1007/s11071-007-9222-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-007-9222-5

Keywords

Search

Navigation