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Computational Optimization and Applications

, Volume 56, Issue 1, pp 153–185 | Cite as

On the optimal control of the Schlögl-model

  • Rico Buchholz
  • Harald Engel
  • Eileen Kammann
  • Fredi TröltzschEmail author
Article

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.

Keywords

Schlögl model Semilinear parabolic equation Travelling wave front Optimal control Model reduction 

Notes

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.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Rico Buchholz
    • 1
  • Harald Engel
    • 2
  • Eileen Kammann
    • 3
  • Fredi Tröltzsch
    • 3
    Email author
  1. 1.Physikalisches InstitutUniversität BayreuthBayreuthGermany
  2. 2.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany
  3. 3.Institut für MathematikTechnische Universität BerlinBerlinGermany

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