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Optimization of nonlocal time-delayed feedback controllers

Author

Listed:
  • Peter Nestler

    (Technische Universität Berlin)

  • Eckehard Schöll

    (Technische Universität Berlin)

  • Fredi Tröltzsch

    (Technische Universität Berlin)

Abstract

A class of Pyragas type nonlocal feedback controllers with time-delay is investigated for the Schlögl model. The main goal is to find an optimal kernel in the controller such that the associated solution of the controlled equation is as close as possible to a desired spatio-temporal pattern. An optimal kernel is the solution to a nonlinear optimal control problem with the kernel taken as control function. The well-posedness of the optimal control problem and necessary optimality conditions are discussed for different types of kernels. Special emphasis is laid on time-periodic functions as desired patterns. Here, the cross correlation between the state and the desired pattern is invoked to set up an associated objective functional that is to be minimized. Numerical examples are presented for the 1D Schlögl model and a class of simple step functions for the kernel.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:64:y:2016:i:1:d:10.1007_s10589-015-9809-6
    DOI: 10.1007/s10589-015-9809-6
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    References listed on IDEAS

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    1. Chamakuri Nagaiah & Karl Kunisch & Gernot Plank, 2011. "Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology," Computational Optimization and Applications, Springer, vol. 49(1), pages 149-178, May.
    2. 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.
    3. Rico Buchholz & Harald Engel & Eileen Kammann & Fredi Tröltzsch, 2013. "Erratum to: On the optimal control of the Schlögl-model," Computational Optimization and Applications, Springer, vol. 56(1), pages 187-188, September.
    4. Clemens Bachmair & Eckehard Schöll, 2014. "Nonlocal control of pulse propagation in excitable media," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(11), pages 1-10, November.
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