Archive for Rational Mechanics and Analysis

, Volume 233, Issue 3, pp 1383–1440 | Cite as

On Stochastic Optimal Control in Ferromagnetism

  • Thomas Dunst
  • Ananta K. Majee
  • Andreas ProhlEmail author
  • Guy Vallet


A model is proposed to, e.g., control the domain wall motion in ferromagnets in the presence of thermal fluctuations, and the existence of an optimal stochastic control process is proved. The convergence of a finite element approximation of the problem is shown in space-dimension one, which then allows one to apply Pontryagin’s maximum principle for this finite dimensional setting. The resulting coupled system of forward-backward stochastic differential equations is numerically solved by means of the stochastic gradient method to enable practical simulations.


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  1. 1.
    Agarwal, S., Carbou, G., Labbé, S., Prieur, C.: Control of a network of magnetic ellipsoidal samples. Math. Control Relat. Fields 1(2), 129–147, 2011MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alouges, F., Beauchard, K.: Magnetization switching on small ferromagnetic ellipsoidal samples. ESAIM Control Optim. Calc. Var. 15(3), 676–711, 2009MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balder, E.J.: Lectures on Young measure theory and its applications in economics. Rend. Istit. Mat. Univ. Trieste 31(Suppl. 1), 1–69, 2000MathSciNetzbMATHGoogle Scholar
  4. 4.
    Baňas, Ľ.; Brzeźniak, Z.; Neklyudov, M.; Prohl, A.: Stochastic Ferromagnetism. Analysis and Numerics. De Gruyter Studies in Mathematics, vol. 58. De Gruyter, Berlin, 2014Google Scholar
  5. 5.
    Bertotti, G., Mayergoyz, I.D., Serpico, C.: Nonlinear magnetization dynamics in nanosystems. Elsevier series in electromagnetism, Amsterdam, 2009Google Scholar
  6. 6.
    Blackwell, D., Dubins, L.E.: An extension of Skorokhod’s almost sure representation theorem. Proc. Am. Math. Soc. 89(4), 691–692, 1983zbMATHGoogle Scholar
  7. 7.
    Bouchard, B., Touzi, N.: Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations. Stoch. Proc. Appl. 111, 175–206, 2004MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin 1994CrossRefzbMATHGoogle Scholar
  9. 9.
    Brzeźniak, Z., Goldys, B., Jegaraj, T.: Weak solutions of a stochastic Landau–Lifshitz–Gilbert equation. Appl. Math. Res. Express. AMRX 1, 1–33, 2013MathSciNetzbMATHGoogle Scholar
  10. 10.
    Brzeźniak, Z., Goldys, B., Jegaraj, T.: Large deviations and transitions between equilibria for stochastic Landau-Lifshitz equation. Arch. Ration. Mech. Anal. 226(2), 497–558, 2017MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brzeźniak, Z., Serrano, R.: Optimal relaxed control of dissipative stochastic partial differential equations in Banach spaces. SIAM J. Control Optim. 51(3), 2664–2703, 2013MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Castaing, C.; Raynaud de Fitte, P.; Valadier, M., Young Measures on Topological Spaces. With Applications in Control Theory and Probability Theory. Mathematics and Its Applications, vol. 571. Kluwer Academic Publishers, Dordrecht, 2004Google Scholar
  13. 13.
    Dunst, T.: Convergence with rates for a time-discretization of the stochastic Landau–Lifschitz–Gilbert equation. IMA J. Numer. Anal. 35(3), 1342–1380, 2015MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dunst, T., Klein, M., Prohl, A., Schäfer, A.: Optimal control in evolutionary micromagnetism. IMA J. Numer. Anal. 35(3), 1342–1380, 2015MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dunst, T., Prohl, A.: The forward-backward stochastic heat equation: numerical analysis and simulation. SIAM J. Sci. Comput. 38(5), A2725–A2755, 2016MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Dunst, T., Prohl, A.: Stochastic optimal control of finite ensembles of nanomagnets. Sci. Compt. 74(2), 872–894, 2018Google Scholar
  17. 17.
    El Karoui N., Huang, S.-J.: A general result of existence and uniqueness of backward stochastic differential equations. Backward Stochastic Differential Equations (Ed. N. El Karoui et al.), Pitman Research Notes in Math. Series, 1997Google Scholar
  18. 18.
    Fernique, X.: Un modèle presque sûr pour la convergence en loi. (French) [An almost sure model for weak convergence]. C. R. Acad. Sci. Paris Sér. I Math. 306(7), 335–338, 1988MathSciNetzbMATHGoogle Scholar
  19. 19.
    Flandoli, F., Gatarek, D.: Martingale and stationary solutions for stochastic Navier–Stokes equations. Prob. Theory Relat. Fields 102(3), 367–391, 1995MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fleming, W.H.: Measure-valued processes in the control of partially-observable stochastic systems. Appl. Math. Optim. 6, 271–285, 1980MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Florescu, L.C., Godet-Thobie, C.: Young Measures and Compactness in Measure Spaces. De Gruyter, Berlin 2012CrossRefzbMATHGoogle Scholar
  22. 22.
    Fuhrman, M., Hu, Y., Tessitore, G.: Stochastic Maximum Principle for Optimal Control of SPDEs. Appl. Math. Optim. 68(2), 181–217, 2013MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fuhrman, M., Orrier, C.: Stochastic maximum principle for optimal control of a class of nonlinear SPDEs with dissipative drift. SIAM J. Control Optim. 54(1), 341–371, 2016MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gobet, E., Lemor, J., Warin, X.: A regression-based Monte Carlo Method to solve backward stochastic differential equations. Ann. Appl. Probab. 15(3), 2172–2202, 2005MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gobet, E., Lopez-Salas, J.G., Turkedjiew, P., Vazquez, C.: Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs. hal-01186000, 2015Google Scholar
  26. 26.
    Gyöngy, I., Krylov, N.: Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Relat. Fields 105, 143–158, 1996CrossRefzbMATHGoogle Scholar
  27. 27.
    Hofmanová, M.: Degenerate parabolic stochastic partial differential equations. Stoch. Process. Appl. 123(12), 4294–4336, 2013MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jakubowski, A.: The almost sure Skorokhod representation for subsequences in nonmetric spaces. Theory Probab. Appl. 42(1), 164–174, 1998MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kružík, M., Prohl, A.: Recent developments in the modeling, analysis, and numerics of ferromagnetism. SIAM Rev. 48(3), 439–483, 2006MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Nagase, N., Nisio, M.: Optimal controls for stochastic partial differential equations. SIAM. Control Optim. 28(1), 186–213, 1990MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38(158), 437–445, 1982MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Simon, J.: Compact sets in the space \(L^p(0, T, B)\). Ann. Mat. Pura Appl. 4(146), 65–96, 1987zbMATHGoogle Scholar
  33. 33.
    Yong, J., Zhou, X.Y.: Stochastic controls. Hamiltonian systems and HJB equations. Applications of Mathematics, vol. 43. Springer, New York 1999Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany
  2. 2.Department of MathematicsIndian Institute of Technology DelhiNew DelhiIndia
  3. 3.Laboratoire de Mathématiques et de leurs Applications de Pau IPRA, UMR5142IFCAM UMI CNRS 3494, CNRS/Univ Pau & Pays AdourPauFrance

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