A second-order numerical scheme for optimal control of non-linear Fokker-Planck equations and applications in social dynamics
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918162148098048 |
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| author | Albi, Giacomo Calzola, Elisa |
| author_facet | Albi, Giacomo Calzola, Elisa |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_09848 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A second-order numerical scheme for optimal control of non-linear Fokker-Planck equations and applications in social dynamics Albi, Giacomo Calzola, Elisa Numerical Analysis Optimization and Control 35Q84, 49L12, 91D30, 65N08, 65M25 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. |
| title | A second-order numerical scheme for optimal control of non-linear Fokker-Planck equations and applications in social dynamics |
| topic | Numerical Analysis Optimization and Control 35Q84, 49L12, 91D30, 65N08, 65M25 |
| url | https://arxiv.org/abs/2503.09848 |