A second-order numerical scheme for optimal control of non-linear Fokker-Planck equations and applications in social dynamics

TL;DR

The paper develops a coupled, second‑order numerical framework for optimal control of nonlinear, nonlocal Fokker–Planck equations arising in socio‑economic and opinion‑dynamics contexts. It combines a structure‑preserving forward discretization (Chang–Cooper) for the FP equation with a second‑order semi‑Lagrangian scheme for the Hamilton–Jacobi–Bellman backward equation, solved within a reduced‑gradient loop augmented by Barzilai–Borwein step updates. Numerical experiments in 1D and 2D settings validate second‑order convergence, mass conservation, and efficiency, and demonstrate the method’s capability to steer opinion distributions and network‑based dynamics toward prescribed targets. The approach offers a practical computational tool for analyzing and shaping collective behavior in socially structured systems, with potential extensions to uncertainty and multi‑scale coupling.

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.

Figures (9)

• Figure 4.1: Time evolution of the numerical approximation of $f(v,t)$ (left) and convergence to the stationary state (right). The direction of the horizontal axis is inverted in the second picture.
• Figure 4.2: Time evolution of the numerical approximation of $\psi(v,t)$ (left) and convergence to the reference solution (right). The direction of the horizontal axis is inverted in the second picture.
• Figure 4.3: Uncontrolled density $f(\mathbf{v},t)$ at time $\textcolor{black}{t}=0$ (left), $t=1.5$ (center), and $t=3$ (right).
• Figure 4.4: Case with $s_1(v_1,v_2)$ as in \ref{['eq:noinflcontr']}. Contour plots of the density $f(\mathbf{v},t)$ (upper row), and the control $\mathbf{u}(\mathbf{v},t)$ (bottom row) for time $t=0$ (left), $t=1.5$ (center) and $t=3$ (right).
• Figure 4.5: Case with $s_1(v_1,v_2)$ as in \ref{['eq:inflcontr']}. Contour plots of the density $f(\mathbf{v},t)$ (upper row), and the control $\mathbf{u}(\mathbf{v},t)$ (bottom row) at time $t=0$ (left), $t=1.5$ (center) and $t=3$ (right).