Skip to main content
SpringerLink
Log in

On the Theoretical and Numerical Control of a One-Dimensional Nonlinear Parabolic Partial Differential Equation

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper deals with the analysis of the internal and boundary control of a one-dimensional parabolic partial differential equation with nonlinear diffusion. First, we prove a local null controllability result with distributed controls, locally supported in space. The proof relies on local inversion (more precisely, we use Liusternik’s Inverse Function Theorem), together with some appropriate specific estimates. We also establish a similar result with controls on one side of the boundary. Then, we consider an iterative algorithm for the computation of null controls, we prove the convergence of the iterates, and we perform some numerical experiments.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Lebeau, G., Robbiano, L.: Contrôle exact de l’équation de la chaleur. Commun. Partial Differ. Equ. 20, 335–356 (1995)

    Article  MATH  Google Scholar 

  2. Fabre, C., Puel, J.-P., Zuazua, E.: Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinb. Sect. 125A, 31–61 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fursikov, A.V., Imanuvilov, O.Y.: Controllability of Evolution Equations, Lecture Notes Series, vol. 34. Seoul National University, Research Institute of Mathematics, Seoul (1996)

    MATH  Google Scholar 

  4. Fernández-Cara, E., Zuazua, E.: Null and approximate controllability for weakly blowing up semi linear heat equations. Ann. Inst. Henri Poincaré Anal. Non Linéaire 17(5), 583–616 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alekseev, V.M., Tikhomorov, V.M., Formin, S.V.: Optimal Control. Consultants Bureau, New York (1987)

    Book  Google Scholar 

  6. Clark, H.R., Fernández-Cara, E., Limaco, J., Medeiros, L.A.: Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities. Appl. Math. Comput. 223, 483–505 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Carthel, C., Glowinski, R., Lions, J.-L.: On exact and approximate boundary controllability for the heat equation: a numerical approach. J. Optim. Theory Appl. 82(3), 429–484 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Labbé, S., Trélat, E.: Uniform controllability of semi-discrete approximations of parabolic control systems. Syst. Control Lett. 55, 597–609 (2006)

    Article  MATH  Google Scholar 

  9. Boyer, F., Hubert, F., Le Rousseau, J.: Discrete Carleman estimates for elliptic operators in arbitrary dimension and applications. SIAM J. Control Optim. 48, 5357–5397 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Boyer, F.: On the penalised HUM approach and its applications to the numerical approximation of null-controls for parabolic problems. In: CANUM 2012, Super-Besse, ESAIM Proceedings, EDP Sciences, Les Ulis (2013)

  11. Fernández-Cara, E., Limaco, J., de Menezes, S.B.: Theoretical and numerical local null controllability of a Ladyzhenskaya–Smagorinsky model of turbulence. J. Math. Fluid Mech. 17, 669–698 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Argyros, I.K.: Convergence and Applications of Newton-Type Iterations. Springer, New York (2008)

    MATH  Google Scholar 

  13. Fernández-Cara, E., Münch, A.: Strong convergence approximations of null controls for the 1D heat equation. SeMA J. 61, 49–78 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fernández-Cara, E., Münch, A.: Numerical null controllability of semi-linear 1-D heat equations: fixed point, least squares and Newton methods. Math. Control Relat. Fields 2, 217–246 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fernández-Cara, E., Münch, A.: Numerical exact controllability of the 1D heat equation: duality and Carleman weights. J. Optim. Theory Appl. 163(1), 253–285 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fernández-Cara, E., Münch, A., Souza, D.A.: On the numerical controllability of the two-dimensional heat, Stokes and Navier–Stokes equations. J. Sci. Comput. 70(2), 819–858 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author has been partially supported by DGI-MINECO (Spain), Grant MTM2016-76990-P and CAPES (Brazil).

Author information

Authors and Affiliations

  1. Dpto. E.D.A.N., Universidad de Sevilla, C/ Tarfia s/n, 41080, Sevilla, Spain

    Enrique Fernández-Cara

  2. IME, Univ. Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, s/n, Bloco H - Campus do Gragoatá, Niterói, RJ, Brazil

    Dany Nina-Huamán & Miguel R. Nuñez-Chávez

  3. Dpto. Matemática, Universidade Federal do Piauí, Campus Universitário Ministro Petrônio Portella, Bairro Ininga, Teresina, PI, Brazil

    Franciane B. Vieira

Authors
  1. Enrique Fernández-Cara
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dany Nina-Huamán
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miguel R. Nuñez-Chávez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Franciane B. Vieira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Enrique Fernández-Cara.

Additional information

Communicated by Emmanuel Trélat.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fernández-Cara, E., Nina-Huamán, D., Nuñez-Chávez, M.R. et al. On the Theoretical and Numerical Control of a One-Dimensional Nonlinear Parabolic Partial Differential Equation. J Optim Theory Appl 175, 652–682 (2017). https://doi.org/10.1007/s10957-017-1190-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1190-4

Keywords

Mathematics Subject Classification

Search

Navigation