A second-order numerical scheme for optimal control of non-linear 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. In order 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
Cite
@article{arxiv.2503.09848,
title = {A second-order numerical scheme for optimal control of non-linear Fokker-Planck equations and applications in social dynamics},
author = {Giacomo Albi and Elisa Calzola},
journal= {arXiv preprint arXiv:2503.09848},
year = {2025}
}