A Second-Order Numerical Scheme for Optimal Control of Nonlinear Fokker-Planck Equations and Applications in Social Dynamics

Abstract

In this work, we present a second-order numerical scheme to address the solution of optimal control problems constrained by the evolution of nonlinear Fokker-Planck equations arising from socio-economic dynamics. To design an appropriate numerical scheme for control realization, a coupled forward-backward system is derived based on the associated optimality conditions. The forward equation, corresponding to the Fokker-Planck dynamics, is discretized using a structure-preserving scheme able to capture steady states. On the other hand, the backward equation, modeled as a Hamilton-Jacobi-Bellman problem, is solved via a semi-Lagrangian scheme that supports large time steps while preserving stability. Coupling between the forward and backward problems is achieved through a gradient-descent optimization strategy, ensuring convergence to the optimal control. Numerical experiments demonstrate second-order accuracy, computational efficiency, and effectiveness in controlling different examples across various scenarios in social dynamics. This approach provides a reliable computational tool for the study of opinion manipulation and consensus formation in socially structured systems.

Keywords

Optimal control / Fokker-Planck equation / Hamilton-Jacobi equation / Opinion dynamics / 35Q84 / 49L12 / 91D30 / 65N08 / 65M25

References

[1] Aduamoah M, Goddard BD, Pearson JW, Roden JC. Pseudospectral methods and iterative solvers for optimization problems from multiscale particle dynamics. BIT Numer. Math., 2022, 62: 1703-1743

[2] Albi, G., Bicego, S., Kalise, D.: Control of high-dimensional collective dynamics by deep neural feedback laws and kinetic modelling. arXiv:2404.02825 (2024)

[3] Albi, G., Caliari, M., Calzola, E., Cassini, F.: Exponential integrators for a mean-field selective optimal control problem. J. Approx. Softw. 1(2) (2024). https://doi.org/10.13135/3103-1935/10973

[4] Albi G, Calzola E, Dimarco G. A data-driven kinetic model for opinion dynamics with social network contacts. Eur. J. Appl. Math., 2024, 36: 1-27

[5] Albi G, Choi Y-P, Fornasier M, Kalise D. Mean field control hierarchy. Appl. Math. Optim., 2017, 76(1): 93-135

[6] Albi G, Dimarco G, Ferrarese F, Pareschi L. Instantaneous control strategies for magnetically confined fusion plasma. J. Comput. Phys., 2025, 527 113804

[7] Albi G, Pareschi L, Zanella M. Boltzmann-type control of opinion consensus through leaders. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 2014, 372(2028): 20140138

[8] Alla A, Falcone M, Kalise D. An efficient policy iteration algorithm for dynamic programming equations. SIAM J. Sci. Comput., 2015, 371181-200

[9] Alla A, Kalise D, Simoncini V. State-dependent Riccati equation feedback stabilization for nonlinear PDEs. Adv. Comput. Math., 2023, 49(1): 9

[10] Annunziato, M., Borzì, A.: A Fokker-Planck control framework for multidimensional stochastic processes. J. Comput. Appl. Math. 237(1), 487–507 (2013)

[11] Aronna, M., Tröltzsch, F.: First and second order optimality conditions for the control of Fokker-Planck equations. ESAIM Control Optim. Calc. Var. 27, 15 (2021). https://doi.org/10.1051/cocv/2021014

[12] Barzilai J, Borwein JM. Two-point step size gradient methods. IMA J. Numer. Anal., 1988, 81141-148

[13] Bicego, S., Kalise, D., Pavliotis, G.A.: Computation and control of unstable steady states for mean field multiagent systems. arXiv:2406.11725 (2024)

[14] Bobylev, A., Nanbu, K.: Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation. Phys. Rev. E 61(4), 4576 (2000)

[15] Bonaventura L, Calzola E, Carlini E, Ferretti R. Second order fully semi-Lagrangian discretizations of advection-diffusion-reaction systems. J. Sci. Comput., 2021, 88(1): 23-29

[16] Boscarino S, Filbet F, Russo G. High order semi-implicit schemes for time dependent partial differential equations. J. Sci. Comput., 2016, 68(3): 975-1001

[17] Breiten, T., Kunisch, K.: Optimal control of a nonlinear kinetic Fokker-Planck equation. arXiv:2501.03784 (2025)

[18] Breiten, T., Kunisch, K., Pfeiffer, L.: Control strategies for the Fokker-Planck equation. ESAIM Control Optim. Calc. Var. 24(2), 741–763 (2018)

[19] Breitenbach, T., Borzì, A.: The Pontryagin maximum principle for solving Fokker-Planck optimal control problems. Comput. Optim. Appl. 76(2), 499–533 (2020)

[20] Burger M, Pinnau R, Totzeck C, Tse O. Mean-field optimal control and optimality conditions in the space of probability measures. SIAM J. Control. Optim., 2021, 59(2): 977-1006

[21] Cacace S, Ferretti R. Efficient implementation of characteristic-based schemes on unstructured triangular grids. Comput. Appl. Math., 2022, 41(1): 19-24

[22] Calzola, E., Carlini, E., Dupuis, X., Silva, F.J.: A semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with oblique derivatives boundary conditions. Numer. Math. 153(1), 49–84 (2023)

[23] Carrillo JA, Choi Y-P, Totzeck C, Tse O. An analytical framework for consensus-based global optimization method. Math. Models Methods Appl. Sci., 2018, 28(06): 1037-1066

[24] Carrillo JA, Eftimie R, Hoffmann F. Non-local kinetic and macroscopic models for self-organised animal aggregations. Kinet. Relat. Mod., 2015, 8(3): 413-441

[25] Chang, J., Cooper, G.: A practical difference scheme for Fokker-Planck equations. J. Comput. Phys. 6(1), 1–16 (1970)

[26] Degond, P., Herda, M., Mirrahimi, S.: A Fokker-Planck approach to the study of robustness in gene expression. Math. Biosci. Eng. 17(6), 6459–6486 (2020)

[27] Düring, B., Markowich, P., Pietschmann, J.-F., Wolfram, M.-T.: Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A Math. Phys. Eng. Sci. 465(2112), 3687–3708 (2009)

[28] Falcone, M., Ferretti, R.: Semi-Lagrangian schemes for Hamilton-Jacobi equations, discrete representation formulae and Godunov methods. J. Comput. Phys. 175(2), 559–575 (2002)

[29] Ferretti R. A technique for high-order treatment of diffusion terms in semi-Lagrangian schemes. Commun. Comput. Phys., 2010, 8(2): 445-470

[30] Fleig, A., Guglielmi, R.: Optimal control of the Fokker-Planck equation with space-dependent controls. J. Optim. Theory Appl. 174, 408–427 (2017)

[31] Fornasier M, Huang H, Pareschi L, Sünnen P. Consensus-based optimization on hypersurfaces: well-posedness and mean-field limit. Math. Models Methods Appl. Sci., 2020, 30(14): 2725-2751

[32] Fornasier M, Piccoli B, Rossi F. Mean-field sparse optimal control. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 2014, 372(2028): 20130400

[33] Franceschi J, Medaglia A, Zanella M. On the optimal control of kinetic epidemic models with uncertain social features. Optim. Control Appl. Methods, 2024, 452494-522

[34] Frank, T.D.: Nonlinear Fokker-Planck Equations: Fundamentals and Applications. Springer, Berlin (2005)

[35] Gugat M, Herty M, Segala C. The turnpike property for mean-field optimal control problems. Eur. J. Appl. Math., 2024, 35(6): 733-747

[36] Körner, J., Borzì, A.: Second-order analysis of Fokker-Planck ensemble optimal control problems. ESAIM Control Optim. Calc. Var. 28, 77 (2022)

[37] Lions, J.-L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin, Heidelberg (1971). (Translated from the French by S. K. Mitter)

[38] Mohammadi, M., Borzì, A.: Analysis of the Chang-Cooper discretization scheme for a class of Fokker-Planck equations. J. Numer. Math. 23(3), 271–288 (2015)

[39] Pareschi L, Toscani G. Wealth distribution and collective knowledge: a Boltzmann approach. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 2014, 372(2028): 20130396

[40] Pareschi, L., Zanella, M.: Structure preserving schemes for nonlinear Fokker-Planck equations and applications. J. Sci. Comput. 74(3), 1575–1600 (2018)

[41] Pinasco JP, Semeshenko V, Balenzuela P. Modeling opinion dynamics: theoretical analysis and continuous approximation. Chaos Solitons Fract., 2017, 98: 210-215

[42] Quarteroni A. Numerical Models for Differential Problems, 20142Milan, Springer 16

[43] Toscani G. Kinetic models of opinion formation. Commun. Math. Sci., 2006, 4(3): 481-496

Funding

Gruppo Nazionale per il Calcolo(CUP E53C23001670001)

Ministero dell’Universitá e della Ricerca(No. P2022JC95T)

RIGHTS & PERMISSIONS

The Author(s)