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On the optimal control of the Schlögl-model

Author

Listed:
  • Rico Buchholz
  • Harald Engel
  • Eileen Kammann
  • Fredi Tröltzsch

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. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Rico Buchholz & Harald Engel & Eileen Kammann & Fredi Tröltzsch, 2013. "On the optimal control of the Schlögl-model," Computational Optimization and Applications, Springer, vol. 56(1), pages 153-185, September.
    • Handle: RePEc:spr:coopap:v:56:y:2013:i:1:p:153-185
    DOI: 10.1007/s10589-013-9550-y
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    Citations

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    Cited by:

    1. Eduardo Casas & Christopher Ryll & Fredi Tröltzsch, 2018. "Optimal control of a class of reaction–diffusion systems," Computational Optimization and Applications, Springer, vol. 70(3), pages 677-707, July.
    2. Peter Nestler & Eckehard Schöll & Fredi Tröltzsch, 2016. "Optimization of nonlocal time-delayed feedback controllers," Computational Optimization and Applications, Springer, vol. 64(1), pages 265-294, May.
    3. Daraghmeh, Adnan & Hartmann, Carsten & Qatanani, Naji, 2019. "Balanced model reduction of linear systems with nonzero initial conditions: Singular perturbation approximation," Applied Mathematics and Computation, Elsevier, vol. 353(C), pages 295-307.
    4. Uzunca, M. & Karasözen, B. & Küçükseyhan, T., 2017. "Moving mesh discontinuous Galerkin methods for PDEs with traveling waves," Applied Mathematics and Computation, Elsevier, vol. 292(C), pages 9-18.

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