Skip to main content
Log in

Finite element approximation to global stabilization of the Burgers’ equation by Neumann boundary feedback control law

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this article, we discuss global stabilization results for the Burgers’ equation using nonlinear Neumann boundary feedback control law. As a result of the nonlinear feedback control, a typical nonlinear problem is derived. Then, based on C 0-conforming finite element method, global stabilization results for the semidiscrete solution are analyzed. Further, introducing an auxiliary projection, optimal error estimates in \(L^{\infty }(L^{2})\), \(L^{\infty }(H^{1})\) and \(L^{\infty }(L^{\infty })\)-norms for the state variable are obtained. Moreover, superconvergence results are established for the first time for the feedback control laws, which preserve exponential stabilization property. Finally, some numerical experiments are conducted to confirm our theoretical findings.

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

Access this article

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. Allen, E.J., Burns, J.A., Gilliam, D.S., Hill, J., Shubov, V.: The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations. Math. Comput. Model. 35, 1165–1195 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Allen, E.J., Burns, J.A., Gilliam, D.S.: Numerical approximations of the dynamical system generated by Burgers equation with Neumann—Dirichlet boundary conditions. ESAIM: Math. Modell. Numer. Anal. 47, 1465–1492 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balogh, A., Krstic, M.: Burgers’ equation with nonlinear boundary feedback: H 1 stability well-posedness and simulation. Math. Problems Engg. 6, 189–200 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burns, J.A., Kang, S.: A stabilization problem for Burgers’ equation with unbounded control and observation. In: Proceedings of an International Conference on Control and Estimation of Distributed Parameter Systems. Vorau (1990)

  5. Burns, J.A., Kang, S.: A control problem for Burgers’ equation with bounded input/output. Nonlinear Dyn. 2, 235–262 (1991)

    Article  Google Scholar 

  6. Burns, J.A., Marrekchi, H.: Optimal fixed-finite-dimensional compensator for Burgers’ equation with unbounded input/output operators. Comput. Control III(2), 83–104 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Byrnes, C.I., Gilliam, D.S., Shubov, V.I.: On the global dynamics of a controlled viscous Burgers’ equation. J. Dyn. Control Syst. 4, 457–519 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Doss, L.J.T., Pani, A.K., Padhy, S.: Galerkin method for a Stefan-type problem in one space dimension. Numer. Methods Partial Diff. Equa. 13, 393–416 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ito, K., Kang, S.: A dissipative feedback control for systems arising in fluid dynamics. SIAM J. Control Optim. 32, 831–854 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ito, K., Yan, Y.: Viscous scalar conservation laws with nonlinear flux feedback and global attractors. J. Math. Anal. Appl. 227, 271–299 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Krstic, M.: On global stabilization of Burgers’ equation by boundary control. Syst. Control Lett. 37, 123–141 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear equations of parabolic type. In: Translations of AMS, vol. 23 (1968)

  13. Liu, W.J., Krstic, M.: Adaptive control of Burgers equation with unknown viscosity. Int. J. Adapt. Control Signal Process. 15, 745–766 (2001)

    Article  MATH  Google Scholar 

  14. Ly, H.V., Mease, K.D., Titi, E.S.: Distributed and boundary control of the viscous Burgers’ equation. Numer. Funct. Anal. Optim. 18, 143–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Marrekchi, H.: Dynamic compensators for a nonlinear conservation law, PhD. Thesis Department of Mathematics. Virginia Polytechnic Institute and State University (1993)

  16. Pani, A.K.: A finite element method for a diffusion equation with constrained energy and nonlinear boundary conditions. J. Austral. Math. Soc. Ser. B 35, 87–102 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (1997)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The second author acknowledges the support provided by the National Programme on Differential Equations: Theory, Computation and Applications (NPDE-TCA) vide the DST project No.SR/S4/MS:639/90. The first author acknowledges the financial support from UGC, Govt. India. Both authors thank the referees for their valuable suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Amiya Kumar Pani.

Additional information

Communicated by: Enrique Zuazua

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kundu, S., Pani, A.K. Finite element approximation to global stabilization of the Burgers’ equation by Neumann boundary feedback control law. Adv Comput Math 44, 541–570 (2018). https://doi.org/10.1007/s10444-017-9553-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-017-9553-9

Keywords

Mathematics Subject Classification (2010)

Navigation