Computational Optimization and Applications

, Volume 70, Issue 3, pp 677–707 | Cite as

Optimal control of a class of reaction–diffusion systems

  • Eduardo Casas
  • Christopher Ryll
  • Fredi TröltzschEmail author


The optimal control of a system of nonlinear reaction–diffusion equations is considered that covers several important equations of mathematical physics. In particular equations are covered that develop traveling wave fronts, spiral waves, scroll rings, or propagating spot solutions. Well-posedness of the system and differentiability of the control-to-state mapping are proved. Associated optimal control problems with pointwise constraints on the control and the state are discussed. The existence of optimal controls is proved under weaker assumptions than usually expected. Moreover, necessary first-order optimality conditions are derived. Several challenging numerical examples are presented that include in particular an application of pointwise state constraints where the latter prevent a moving localized spot from hitting the domain boundary.


Optimal control Reaction diffusion equations Pointwise control constraints Pointwise state constraints Necessary optimality conditions Propagating spot solutions 


  1. 1.
    Alonso, S., Bär, M.: Reentry near the percolation threshold in a heterogeneous discrete model for cardiac tissue. Phys. Rev. Lett. 110(15), 158,101 (2013)CrossRefGoogle Scholar
  2. 2.
    Bode, M., Liehr, A.W., Schenk, C.P., Purwins, H.G.: Interaction of dissipative solitons: particle-like behaviour of localized structures in a three-component reaction–diffusion system. Physica D 161(1), 45–66 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borzì, A., Griesse, R.: Distributed optimal control of lambda–omega systems. J. Numer. Math. 14(1), 17–40 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Buchholz, R., Engel, H., Kammann, E., Tröltzsch, F.: On the optimal control of the Schlögl model. Comput. Optim. Appl. 56, 153–185 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 31(4), 993–1006 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Casas, E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35(4), 1297–1327 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Casas, E., Kunisch, K.: Parabolic control problems in space–time measure spaces. ESAIM Control Optim. Calc. Var. 22(2), 355–370 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Casas, E., Mateos, M., Rösch, A.: Approximation of sparse parabolic control problems. Math. Control Relat. Fields 7(3), 393–417 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Casas, E., Mateos, M., Vexler, B.: New regularity results and improved error estimates for optimal control problems with state constraints. ESAIM Control Optim. Calc. Var. 20, 803–822 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Casas, E., Raymond, J., Zidani, H.: Pontryagin’s principle for local solutions of control problems with mixed control-state constraints. SIAM J. Control Optim. 39(4), 1182–1203 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Casas, E., Ryll, C., Tröltzsch, F.: Sparse optimal control of the Schlögl and FitzHugh–Nagumo systems. Comput. Methods Appl. Math. 13, 415–442 (2014)Google Scholar
  12. 12.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kuramoto, Y.: Chemical Oscillations, Waves, and Turbulence, Springer Series in Synergetics, vol. 19. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ladyzhenskaya, O., Solonnikov, V., Ural’tseva, N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence (1988)zbMATHGoogle Scholar
  16. 16.
    Löber, J.: Nonlinear excitation waves in spatially heterogenous reaction–diffusion systems. Technical report, TU Berlin, Institute of Theoretical Physics (2009)Google Scholar
  17. 17.
    Löber, J.: Optimal trajectory tracking. Ph.D. thesis, TU Berlin (2015)Google Scholar
  18. 18.
    Löber, J.: Exactly realizable desired trajectories. arXiv preprint arXiv:1603.00611 (2016)
  19. 19.
    Löber, J., Engel, H.: Controlling the position of traveling waves in reaction–diffusion systems. Phys. Rev. Lett. 112(14), 148,305 (2014)CrossRefGoogle Scholar
  20. 20.
    Mihaliuk, E., Sakurai, T., Chirila, F., Showalter, K.: Feedback stabilization of unstable propagating waves. Phys. Rev. E 65(6), 065,602 (2002)CrossRefGoogle Scholar
  21. 21.
    Mikhailov, A.S., Showalter, K.: Control of waves, patterns and turbulence in chemical systems. Phys. Rep. 425(2), 79–194 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Murray, J.D.: Mathematical Biology, Biomathematics, vol. 19, 2nd edn. Springer, Berlin (1993)CrossRefGoogle Scholar
  23. 23.
    Ryll, C.: Optimal control of patterns in some reaction–diffusion-systems. Ph.D. thesis, Technical University of Berlin (2016).
  24. 24.
    Ryll, C., Löber, J., Martens, S., Engel, H., Tröltzsch, F.: Analytical, optimal, and sparse optimal control of traveling wave solutions to reaction–diffusion systems. In: Schöll, E., Klapp, S., Hövel, P. (eds.) Understanding Complex Systems, Control of Self-organizing Nonlinear Systems, pp. 189–210. Springer, Berlin (2016)CrossRefGoogle Scholar
  25. 25.
    Sakurai, T., Mihaliuk, E., Chirila, F., Showalter, K.: Design and control of wave propagation patterns in excitable media. Science 296(5575), 2009–2012 (2002)CrossRefGoogle Scholar
  26. 26.
    Schenk, C.P., Or-Guil, M., Bode, M., Purwins, H.G.: Interacting pulses in three-component reaction–diffusion systems on two-dimensional domains. Phys. Rev. Lett. 78(19), 3781 (1997)CrossRefGoogle Scholar
  27. 27.
    Schlesner, J., Zykov, V.S., Brandtstädter, H., Gerdes, I., Engel, H.: Efficient control of spiral wave location in an excitable medium with localized heterogeneities. New J. Phys. 10(1), 015,003 (2008)CrossRefGoogle Scholar
  28. 28.
    Schlögl, F.: A characteristic critical quantity in nonequilibrium phase transitions. Z. Phys. B Condens. Matter. 52, 51–60 (1983)CrossRefGoogle Scholar
  29. 29.
    Schöll, E., Schuster, H.: Handbook of Chaos Control. Wiley-VCH, Weinheim (2008)zbMATHGoogle Scholar
  30. 30.
    Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs, vol. 49. American Mathematical Society, Providence, RI (1997)zbMATHGoogle Scholar
  31. 31.
    Smoller, J.: Shock Waves and Reaction–Diffusion Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, 3rd edn. Springer, New York (1994)Google Scholar
  32. 32.
    Zykov, V.S., Bordiougov, G., Brandtstädter, H., Gerdes, I., Engel, H.: Global control of spiral wave dynamics in an excitable domain of circular and elliptical shape. Phys. Rev. Lett. 92(1), 018,304 (2004)CrossRefGoogle Scholar
  33. 33.
    Zykov, V.S., Engel, H.: Feedback-mediated control of spiral waves. Physica D 199(1–2), 243–263 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de TelecomunicaciónUniversidad de CantabriaSantanderSpain
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany

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