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Optimization of nonlocal time-delayed feedback controllers

  • Peter Nestler
  • Eckehard Schöll
  • Fredi Tröltzsch
Article

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.

Keywords

Schlögl model Nagumo equation Pyragas type feedback control Nonlocal delay Controller optimization  Numerical method 

Notes

Acknowledgments

This work was supported by DFG in the framework of the Collaborative Research Center SFB 910, Projects A1 and B6.

References

  1. 1.
    Atay, F.: Distributed delays facilitate amplitude death of coupled oscillators. Phys. Rev. Lett. 91, 094101 (2003)CrossRefGoogle Scholar
  2. 2.
    Bachmair, C., Schöll, E.: Nonlocal control of pulse propagation in excitable media. Eur. Phys. J. B. (2014). doi: 10.1140/epjb/e2014-50339-2
  3. 3.
    Borzì, A., Griesse, R.: Distributed optimal control of lambda-omega systems. J. Numer. Math. 14(1), 17–40 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brandão, A.J.V., Fernández-Cara, E., Paulo, P.M.D., Rojas-Medar, M.A.: Theoretical analysis and control results for the FitzHugh-Nagumo equation. Electron. J. Differ. Equ. 164, 1–20 (2008)MathSciNetMATHGoogle Scholar
  5. 5.
    Buchholz, R., Engel, H., Kammann, E., Tröltzsch, F.: On the optimal control of the Schlögl model. Comput. Optim. Appl. 56, 153–185 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Casas, E., Ryll, C., Tröltzsch, F.: Second order and stability analysis for optimal sparse control of the FitzHugh-Nagumo equation. SIAM J. Control Optim. 53(4), 2168–2202 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Casas, E., Ryll, C., Tröltzsch, F.: Sparse optimal control of the Schlögl and FitzHugh-Nagumo systems. Comput. Methods Appl. Math. 13, 415–442 (2014). doi: 10.1515/cmam-2013-0016 Google Scholar
  9. 9.
    Nagaiah, C., Kunisch, K., Plank, G.: Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology. Comput. Optim. Appl. 49, 149–178 (2011). doi: 10.1007/s10589-009-9280-3 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Coron, J.M.: Control and Nonlinearity. American Mathematical Society, Providence (2007)MATHGoogle Scholar
  11. 11.
    Di Benedetto, E.: On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Scuola Sup. Pisa I 13, 487–535 (1986)MathSciNetGoogle Scholar
  12. 12.
    Erneux, T.: Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 3. Springer, New York (2009)Google Scholar
  13. 13.
    Gugat, M., Tröltzsch, F.: Boundary feedback stabilization of the Schlögl system. Automatica 51, 192–199 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kunisch, K., Wang, L.: Time optimal controls of the linear Fitzhugh-Nagumo equation with pointwise control constraints. J. Math. Anal. Appl. 395(1), 114–130 (2012). doi: 10.1016/j.jmaa.2012.05.028 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kunisch, K., Wagner, M.: Optimal control of the bidomain system (iii): Existence of minimizers and first-order optimality conditions. ESAIM Math. Model. Numer. Anal. 47(4), 1077–1106 (2013)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kyrychko, Y., Blyuss, K., Schöll, E.: Synchronization of networks of oscillators with distributed delay-coupling. Chaos 24, 043117 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I: Abstract Parabolic Systems. Cambridge University Press, Cambridge (2000)CrossRefMATHGoogle Scholar
  19. 19.
    Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II. Abstract Hyperbolic-Like Systems over a Finite Time Horizon. Cambridge University Press, Cambridge (2000)CrossRefMATHGoogle Scholar
  20. 20.
    Löber, J., Coles, R., Siebert, J., Engel, H., Schöll, E.: Control of chemical wave propagation. In: Mikhailov, A., Ertl, G. (eds.) Engineering of Chemical Complexity II, pp. 185–207. World Scientific, Singapore. arXiv:1403.3363 (2014)
  21. 21.
    Pyragas, K.: Continuous control of chaos by self-controlling feedback. Phys. Rev. Lett. A 170, 421 (1992)CrossRefGoogle Scholar
  22. 22.
    Pyragas, K.: Delayed feedback control of chaos. Philos. Trans. R. Soc. A 364, 2309 (2006)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    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)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Raymond, J.P., Zidani, H.: Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39, 143–177 (1999)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Schöll, E., Schuster, H.: Handbook of Chaos Control. Wiley-VCH, Weinheim (2008)MATHGoogle Scholar
  26. 26.
    Siebert, J., Alonso, S., Bär, M., Schöll, E.: Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection. Phys. Rev. E 89, 052909 (2014). doi: 10.1103/PhysRevE.89.052909 CrossRefGoogle Scholar
  27. 27.
    Siebert, J., Schöll, E.: Front and turing patterns induced by mexican-hat-like nonlocal feedback. Europhys. Lett. 109, 40014 (2015)CrossRefGoogle Scholar
  28. 28.
    Smyshlyaev, A., Krstic, M.: Adaptive Control of Parabolic PDEs. Princeton University Press, Princeton (2010)CrossRefMATHGoogle Scholar
  29. 29.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods and Applications, vol. 112. American Mathematical Society, Providence (2010)MATHGoogle Scholar
  30. 30.
    Wille, C., Lehnert, J., Schöll, E.: Synchronization-desynchronization transitions in complex networks: an interplay of distributed time delay and inhibitory nodes. Phys. Rev. E 90, 032908 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Peter Nestler
    • 1
  • Eckehard Schöll
    • 2
  • Fredi Tröltzsch
    • 1
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Institut für Theoretische PhysikTechnische Universität BerlinBerlinGermany

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