Journal of Optimization Theory and Applications

, Volume 174, Issue 2, pp 408–427 | Cite as

Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls

Article

Abstract

This paper is devoted to the analysis of a bilinear optimal control problem subject to the Fokker–Planck equation. The control function depends on time and space and acts as a coefficient of the advection term. For this reason, suitable integrability properties of the control function are required to ensure well posedness of the state equation. Under these low regularity assumptions and for a general class of objective functionals, we prove the existence of optimal controls. Moreover, for common quadratic cost functionals of tracking and terminal type, we derive the system of first-order necessary optimality conditions.

Keywords

Bilinear control Fokker–Planck equation Optimal control theory Optimization in Banach spaces Probability density function Stochastic optimal control 

Mathematics Subject Classification

35Q84 35Q93 49J20 49K20 

References

  1. 1.
    Kolmogoroff, A.: Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104(1), 415–458 (1931). doi:10.1007/BF01457949 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Gardiner, C.: Stochastic Methods. A Handbook for the Natural and Social Sciences. Springer Series in Synergetics, 4th edn. Springer, Berlin (2009)MATHGoogle Scholar
  3. 3.
    Horsthemke, W., Lefever, R.: Noise-Induced Transitions. Theory and Applications in Physics, Chemistry, and Biology. Springer Series in Synergetics, vol. 15. Springer, Berlin (1984)MATHGoogle Scholar
  4. 4.
    Le Bris, C., Lions, P.L.: Existence and uniqueness of solutions to Fokker–Planck type equations with irregular coefficients. Commun. Partial Differ. Equ. 33(7–9), 1272–1317 (2008). doi:10.1080/03605300801970952 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Figalli, A.: Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254(1), 109–153 (2008). doi:10.1016/j.jfa.2007.09.020 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Porretta, A.: Weak solutions to Fokker–Planck equations and mean field games. Arch. Ration. Mech. Anal. 216(1), 1–62 (2015). doi:10.1007/s00205-014-0799-9 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brockett, R.: New issues in the mathematics of control. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited—2001 and Beyond, pp. 189–219. Springer, Berlin (2001)Google Scholar
  8. 8.
    Forbes, M.G., Forbes, J.F., Guay, M.: Regulating discrete-time stochastic systems: focusing on the probability density function. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 11(1–2), 81–100 (2004)MathSciNetMATHGoogle Scholar
  9. 9.
    Jumarie, G.: Tracking control of nonlinear stochastic systems by using path cross-entropy and Fokker–Planck equation. Int. J. Syst. Sci. 23(7), 1101–1114 (1992). doi:10.1080/00207729208949368 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kárný, M.: Towards fully probabilistic control design. Autom. J. IFAC 32(12), 1719–1722 (1996). doi:10.1016/S0005-1098(96)80009-4 MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Wang, H.: Robust control of the output probability density functions for multivariable stochastic systems with guaranteed stability. IEEE Trans. Autom. Control 44(11), 2103–2107 (1999). doi:10.1109/9.802925 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Applications of Mathematics, No. 1. Springer, Berlin, New York (1975)CrossRefMATHGoogle Scholar
  13. 13.
    Annunziato, M., Borzì, A.: Optimal control of probability density functions of stochastic processes. Math. Model. Anal. 15(4), 393–407 (2010). doi:10.3846/1392-6292.2010.15.393-407 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Annunziato, M., Borzì, A.: A Fokker–Planck control framework for multidimensional stochastic processes. J. Comput. Appl. Math. 237(1), 487–507 (2013). doi:10.1016/j.cam.2012.06.019 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Blaquière, A.: Controllability of a Fokker–Planck equation, the Schrödinger system, and a related stochastic optimal control (revised version). Dyn. Control 2(3), 235–253 (1992). doi:10.1007/BF02169515 CrossRefMATHGoogle Scholar
  16. 16.
    Porretta, A.: On the planning problem for the mean field games system. Dyn. Games Appl. 4(2), 231–256 (2014). doi:10.1007/s13235-013-0080-0 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Addou, A., Benbrik, A.: Existence and uniqueness of optimal control for a distributed-parameter bilinear system. J. Dyn. Control Syst. 8(2), 141–152 (2002). doi:10.1023/A:1015372725255 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fleig, A., Grüne, L., Guglielmi, R.: Some results on model predictive control for the Fokker–Planck equation. In: MTNS 2014: 21st International Symposium on Mathematical Theory of Networks and Systems, July 7–11, 2014, pp. 1203–1206. University of Groningen, The Netherlands (2014)Google Scholar
  19. 19.
    Aronson, D.G.: Non-negative solutions of linear parabolic equations. Ann. Sc. Norm. Super. Pisa 3(22), 607–694 (1968)MathSciNetMATHGoogle Scholar
  20. 20.
    Aronson, D.G., Serrin, J.: Local behavior of solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 25, 81–122 (1967)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Casas, E.: Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35(4), 1297–1327 (1997). doi:10.1137/S0363012995283637 MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Raymond, J.P., Zidani, H.: Hamiltonian Pontryagin’s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39(2), 143–177 (1999). doi:10.1007/s002459900102 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Primak, S., Kontorovich, V., Lyandres, V.: Stochastic methods and their applications to communications. Wiley, Hoboken (2004). doi:10.1002/0470021187 CrossRefMATHGoogle Scholar
  24. 24.
    Protter, P.E.: Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, vol. 21. Springer, Berlin (2005). doi:10.1007/978-3-662-10061-5 Google Scholar
  25. 25.
    Risken, H.: The Fokker–Planck Equation. Springer Series in Synergetics, vol. 18, 2nd edn. Springer, Berlin (1989). doi:10.1007/978-3-642-61544-3 MATHGoogle Scholar
  26. 26.
    Feller, W.: Diffusion processes in one dimension. Trans. Am. Math. Soc. 77, 1–31 (1954)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Gīhman, Ĭ.Ī., Skorohod, A.V.: Stochastic Differential Equations. Springer, New York, Heidelberg (1972). Translated from the Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72Google Scholar
  28. 28.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations, Graduate Studies in Mathematics, vol. 112. American Mathematical Society, Providence (2010). doi:10.1090/gsm/112 Google Scholar
  29. 29.
    Aubin, J.P.: Un théorème de compacité. C. R. Acad. Sci. Paris 256, 5042–5044 (1963)MathSciNetMATHGoogle Scholar
  30. 30.
    Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod; Gauthier-Villars, Paris (1969)MATHGoogle Scholar
  31. 31.
    Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. (4) 146, 65–96 (1987). doi:10.1007/BF01762360 MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987). doi:10.1017/CBO9781139171755 CrossRefMATHGoogle Scholar
  33. 33.
    Ladyzhenskaya, O., Solonnikov, V., Ural’tseva, N.: Linear and Quasilinear Equations of Parabolic Type. Izdat. ’Nauka’, Moskva (1967)MATHGoogle Scholar
  34. 34.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Translated from the French by S. K. Mitter. Die Grundlehren der mathematischen Wissenschaften, Band 170. Springer, New York, Berlin (1971)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.University of BayreuthBayreuthGermany
  2. 2.Dyrecta LabConversanoItaly

Personalised recommendations