Computational Optimization and Applications

, Volume 69, Issue 2, pp 423–459 | Cite as

A Fokker–Planck approach to control collective motion

  • Souvik RoyEmail author
  • Mario Annunziato
  • Alfio Borzì
  • Christian Klingenberg


A Fokker–Planck control strategy for collective motion is investigated. This strategy is formulated as the minimisation of an expectation objective with a bilinear optimal control problem governed by the Fokker–Planck equation modelling the evolution of the probability density function of the stochastic motion. Theoretical results on existence and regularity of optimal controls are provided. The resulting optimality system is discretized using an alternate-direction implicit Chang–Cooper scheme that guarantees conservativeness, positivity, \(L^1\) stability, and second-order accuracy of the forward solution. A projected non-linear conjugate gradient scheme is used to solve the optimality system. Results of numerical experiments validate the theoretical accuracy estimates and demonstrate the efficiency of the proposed control framework.


Fokker–Planck equation Alternate direction method Chang–Cooper scheme Projected gradient method Control constrained PDE optimization 

Mathematics Subject Classification

35Q84 35Q91 35Q93 49K20 49J20 65C20 



The authors would like to gratefully acknowledge the comments by the referees which helped to improve this paper. S. Roy would like to thank A. S. Vasudeva Murthy and Praveen Chandrashekar for several fruitful discussions during the initial phases of this work. This work was supported in part by the European Union under Grant Agreement No. 304617 Marie Curie Research Training Network “Multi-ITN STRIKE—Novel Methods in Computational Finance” and the BMBF project “ROENOBIO”. S. Roy was also supported by the DAAD Passage to India Program and the AIRBUS Group Corporate Foundation Chair in Mathematics of Complex Systems established in TIFR/ICTS, Bangalore.


  1. 1.
    Annunziato, M., Borzì, A.: Optimal control of probability density functions of stochastic processes. Math. Model. Anal. 15, 393–407 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Annunziato, M., Borzì, A.: A Fokker–Planck control framework for multidimensional stochastic process. J. Comput. Appl. Math. 237, 487–507 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Annunziato, M., Borzì, A., Nobile, F., Tempone, R.: On the connection between the Hamilton–Jacobi–Bellman and the Fokker–Planck control frameworks. Appl. Math. 5, 2476–2484 (2014)CrossRefGoogle Scholar
  4. 4.
    Aronson, D.G.: Non-negative solutions of linear parabolic equations. Ann. della Scuola Normale Superiore di Pisa - Classe di Scienze 22(4), 607–694 (1968)Google Scholar
  5. 5.
    Bellomo, N., Bellouquid, A., Knopoff, D.: From the microscale to collective crowd dynamics. Multiscale Model. Simul. 11(3), 943–963 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borzì, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations. SIAM Book Series on Computational Science and Engineering 08. SIAM, Philadelphia, PA (2012)zbMATHGoogle Scholar
  7. 7.
    Chang, J.S., Cooper, G.: A practical difference scheme for Fokker–Planck equations. J. Comput. Phys. 6, 1–16 (1970)CrossRefzbMATHGoogle Scholar
  8. 8.
    Cucker, F., Mordecki, E.: Flocking in noisy environments. J. Math. Pures Appl. 89, 278–296 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dautray, R., Lions, J.-L.: Problèmes d’ évolution du premier ordre en temps, (Chap 18.) Masson, Paris (1984)Google Scholar
  10. 10.
    Deutsch, A., Theraulaz, G., Vicsek, T.: Collective motion in biological systems. Interface Focus 2(6), 689–692 (2012)CrossRefGoogle Scholar
  11. 11.
    Douglas Jr., J.: On the numerical integration of \(u_{xx}+ u_{yy}= u_t\) by implicit methods. J. Soc. Ind. Appl. Math. 3, 42–65 (1955)CrossRefGoogle Scholar
  12. 12.
    Douglas Jr., J.: Alternating direction methods for three space variables. Numer. Math. 4(1), 41–63 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Droniou, J., Vázquez, J.-L.: Noncoercive convection-diffusion elliptic problems with Neumann boundary conditions. Calc. Var. Partial Differ. Equ. 34, 413–434 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (2002)Google Scholar
  15. 15.
    Fleming, W., Soner, M.: Controlled Markov Processes and Viscosity Solutions. Springer, Berlin (2006)zbMATHGoogle Scholar
  16. 16.
    Flotron, S., Rappaz, J.: Conservation schemes for convection-diffusion equations with Robin boundary conditions. ESAIM 47, 1765–1781 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gautrais, J., Ginelli, F., Fournier, R., Blanco, S., Soria, M., Chateé, H., Theraulaz, H.G.: Deciphering interactions in moving animal groups. PLoS Comput. Biol. 8(9), e1002 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Grosskinsky, S., Klingenberg, C., Oelschläger, K.: A rigorous derivation of Smoluchowski’s equation in the moderate limit. Stoch. Anal. Appl. 22(1), 113–141 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  20. 20.
    Glowinski, R., Lions, J.-L., He, J.: Exact and approximate controllability for distributed parameter systems. Acta Numer. 3, 269–378 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Grüne, L., Pannek, J.: Nonlinear Model Predictive Control, Theory and Algorithms, Communications and Control Engineering. Springer, Berlin (2011)zbMATHGoogle Scholar
  22. 22.
    Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16(1), 170–192 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations. Springer Series in Computational Mathematics (2010)Google Scholar
  24. 24.
    Jovanović, B.S., Süli, E.: Analysis of Finite Difference Schemes For Linear Partial Differential Equations with Generalized Solutions. Springer Series in Computational Mathematics (2014)Google Scholar
  25. 25.
    Lions, J.-L.: Quelque Methodes de Résolution des Problemes aux Limites Non Linéaires. Dunod-Gauth. Vill, Paris (1969)zbMATHGoogle Scholar
  26. 26.
    Lions, J.-L.: Nonhomogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)CrossRefGoogle Scholar
  27. 27.
    Mohammadi, M., Borzì, A.: Analysis of the Chang–Cooper discretization scheme for a class of Fokker–Planck equations. J. Numer. Math. 23, 271–288 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Morale, D., Capasso, V., Oelschlaeger, K.: An interacting particle system modelling aggregation behavior: from individuals to populations. J. Math. Biol. 50(1), 49–66 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Neittaanmaki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Pure and Applied Mathematics. CRC Press, London (1994)zbMATHGoogle Scholar
  30. 30.
    Øksendal, B.K.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  31. 31.
    Peaceman, D.W., Rachford Jr., H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3(1), 28–41 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Petrelli, L., Kearsely, A.J.: Wasserstein metric convergence method for Fokker–Planck equations with point controls. Appl. Math. Lett. 22(7), 1130–1135 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)zbMATHGoogle Scholar
  34. 34.
    Roy, S., Annunziato, M., Borzì, A.: A Fokker–Planck feedback control-constrained approach for modelling crowd motion. J. Comput. Theor. Transp. 45(6), 442–458 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Sepúlveda, N., Petitjean, L., Cochet, O., Grasland-Mongrain, E., Silberzan, P., Hakim, V.: Collective cell motion in an epithelial sheet can be quantitatively described by a stochastic interacting particle model. PLoS Comput. Biol. 9(3), e1002944 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Strandburg-Peshkin, A., Farine, D.R., Couzin, I.D., Crofoot, M.C.: Shared decision-making drives collective movement in wild baboons. Science 348(6241), 1358–1361 (2015)CrossRefGoogle Scholar
  37. 37.
    Tao, T.: Nonlinear Dispersive Equations: Local and Global Analysis. American Mathematical Society, Providence, RI (2006)CrossRefzbMATHGoogle Scholar
  38. 38.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications. American Mathematical Society, Providence, RI (2010)CrossRefzbMATHGoogle Scholar
  39. 39.
    Varga, R.S.: Matrix Iterative Analysis. Springer Series in Computational Mathematics. Springer, Berlin (2000)CrossRefGoogle Scholar
  40. 40.
    Zienkiewicz, A., Barton, D.A.W., Porfiri, M., di Bernardo, M.: Data-driven stochastic modelling of zebrafish locomotion. J. Math. Biol. (2014). doi: 10.1007/s00285-014-0843-2 zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institut für MathematikUniversität WürzburgWürzburgGermany
  2. 2.Dipartimento di MatematicaUniversità degli Studi di SalernoFiscianoItaly
  3. 3.Institut für MathematikUniversität WürzburgWürzburgGermany

Personalised recommendations