Skip to main content
SpringerLink
Log in

On the optimal control of the Schlögl-model

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

An Erratum to this article was published on 29 May 2013

Abstract

Optimal control problems for a class of 1D semilinear parabolic equations with cubic nonlinearity are considered. This class is also known as the Schlögl model. Main emphasis is laid on the control of traveling wave fronts that appear as typical solutions to the state equation.

The well-posedness of the optimal control problem and the regularity of its solution are proved. First-order necessary optimality conditions are established by standard adjoint calculus. The state equation is solved by the implicit Euler method in time and a finite element technique with respect to the spatial variable. Moreover, model reduction by Proper Orthogonal Decomposition is applied and compared with the numerical solution of the full problem. To solve the optimal control problems numerically, the performance of different versions of the nonlinear conjugate gradient method is studied. Various numerical examples demonstrate the capacities and limits of optimal control methods.

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
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Amann, H.: Abstract linear theory. In: Linear and Quasilinear Parabolic Problems, vol. I. Monographs in Mathematics, vol. 89. Birkhäuser, Boston (1995)

    Chapter  Google Scholar 

  2. Arian, E., Fahl, M., Sachs, E.W.: Trust-region proper orthogonal decomposition for flow control. Tech. rep., ICASE (2000). Technical Report 2000-25

  3. Bonnans, F., Casas, E.: Une principe de Pontryagine pour le contrôle des systèmes semilinéaires elliptiques. J. Differ. Equ. 90, 288–303 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borzì, A., Griesse, R.: Distributed optimal control of lambda-omega systems. J. Numer. Math. 14(1), 17–40 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Casas, E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35, 1297–1327 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Casas, E., De los Reyes, J.C., Tröltzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19(2), 616–643 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010). doi:10.1137/090766498

    Article  MathSciNet  MATH  Google Scholar 

  8. Di Benedetto, E.: On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 13, 487–535 (1986)

    MathSciNet  Google Scholar 

  9. Fahl, M., Sachs, E.W.: Reduced order modelling approaches to PDE-constrained optimization based on proper orthogonal decomposition. In: Biegler, L.T., Ghattas, O., Heinkenschloss, M., van Bloemen Waanders, B. (eds.) Large-Scale PDE-Constrained Optimization, vol. 30, pp. 268–280. Springer, Berlin (2003)

    Chapter  Google Scholar 

  10. Hager, W.W., Zhang, H.: CG DESCENT, a conjugate gradient method with guaranteed descent. Tech. rep., ACM TOMS (2006)

  11. Hager, W.W., Zhang, H.: A survey of nonlinear conjugate gradient methods. Pac. J. Optim. 2(1), 35–58 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Heinkenschloss, M., Sachs, E.W.: Numerical solution of a constrained control problem for a phase field model. In: Desch, W., Kappel, F., Kunisch, K. (eds.) Control and Estimation of Distributed Parameter Systems. ISNM, vol. 118, pp. 171–188. Birkhäuser, Basel (1994)

    Chapter  Google Scholar 

  13. Herzog, R., Kunisch, K.: Algorithms for PDE-constrained optimization. GAMM-Mitt. 33(2), 163–176 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: In: Optimization with PDE Constraints, vol. 23. Springer, Berlin (2009)

    Chapter  Google Scholar 

  15. Kammann, E., Tröltzsch, F., Volkwein, S.: A method of a-posteriori estimation with application to proper orthogonal decomposition. ESAIM: Math. Model. Numer. Anal. 47(2), 555–581 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90, 117–148 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kunisch, K., Wagner, M.: Optimal control of the bidomain system (I): the monodomain approximation with the Rogers-McCulloch model. Nonlinear Anal., Real World Appl. 13(4), 1525–1550 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kunisch, K., Wagner, M.: Optimal control of the bidomain system (II): Uniqueness and regularity theorems for weak solutions. Ann. Mat. Pura Appl. doi:10.1007/s10231-012-0254-1

  19. Kunisch, K., Wagner, M.: Optimal control of the bidomain system (III): Existence of minimizers and first-order optimality conditions. ESAIM: Math. Model. Numer. Anal. doi:10.1051/m2an/2012058

  20. Kunisch, K., Nagaiah, C., Wagner, M.: A parallel Newton-Krylov method for optimal control of the monodomain model in cardiac electrophysiology. Comput. Vis. Sci. 14(6), 257–269 (2011)

    Article  MathSciNet  Google Scholar 

  21. Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence. Springer Series in Synergetics, vol. 19. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  22. Leugering, G., Engell, S., Griewank, A., Hinze, M., Rannacher, R., Schulz, V., Ulbrich, M., Ulbrich, S.: In: Constrained Optimization and Optimal Control for Partial Differential Equations. ISNM, vol. 160. Birkhäuser, Basel (2012)

    Chapter  Google Scholar 

  23. Löber, J.: Nonlinear excitation waves in spatially heterogenous reaction-diffusion systems. Tech. rep, TU Berlin, Institute of Theoretical Physics (2009)

  24. Michailov, A.S.: Foundations of Synergetics, vol. 1. Springer, Berlin (1994)

    Book  Google Scholar 

  25. Müller, M.: Uniform convergence of the POD method and applications to optimal control. PhD thesis, Johannes Kepler University, Graz (2011)

  26. Murray, J.D.: Mathematical Biology. Biomathematics, vol. 19, 2nd edn. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  27. Raymond, J.P., Zidani, H.: Pontryagin’s principle for state-constrained control problems governed by parabolic equations with unbounded controls. SIAM J. Control Optim. 36, 1853–1879 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  28. Schlögl, F.: A characteristic critical quantity in nonequilibrium phase transitions. Z. Phys. B, Condens. Matter 52, 51–60 (1983)

    Article  Google Scholar 

  29. Schöll, E., Schuster, H.: Handbook of Chaos Control, 2nd edn. Wiley-VCH, Weinheim (2008)

    MATH  Google Scholar 

  30. Smoller, J.: Shock Waves and Reaction-Diffusion Equations, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258. Springer, New York (1994)

    Book  MATH  Google Scholar 

  31. Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications vol. 112. Am. Math. Soc., Providence (2010)

    MATH  Google Scholar 

  32. Volkwein, S.: Model reduction using proper orthogonal decomposition. Lecture notes, Institute of Mathematics and Scientific Computing, University of Graz (2007)

Download references

Acknowledgements

The authors are very grateful to Christopher Ryll (TU Berlin, Institute of Mathematics), who updated and partially improved the numerical results of the third author after she finished her university career.

Author information

Authors and Affiliations

  1. Physikalisches Institut, Universität Bayreuth, 95447, Bayreuth, Germany

    Rico Buchholz

  2. Institut für Theoretische Physik, Technische Universität Berlin, 10623, Berlin, Germany

    Harald Engel

  3. Institut für Mathematik, Technische Universität Berlin, 10623, Berlin, Germany

    Eileen Kammann & Fredi Tröltzsch

Authors
  1. Rico Buchholz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Harald Engel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eileen Kammann
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Fredi Tröltzsch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fredi Tröltzsch.

Additional information

This work was supported by DFG in the framework of the Collaborative Research Center Sfb 910, project B6.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Buchholz, R., Engel, H., Kammann, E. et al. On the optimal control of the Schlögl-model. Comput Optim Appl 56, 153–185 (2013). https://doi.org/10.1007/s10589-013-9550-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-013-9550-y

Keywords

Search

Navigation