Convergence Rates of Time Discretization in Extended Mean Field Control
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914015255461888 |
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| author | Reisinger, Christoph Stockinger, Wolfgang Tsianni, Maria Olympia Zhang, Yufei |
| author_facet | Reisinger, Christoph Stockinger, Wolfgang Tsianni, Maria Olympia Zhang, Yufei |
| contents | Piecewise constant control approximation provides a practical framework for designing numerical schemes of continuous-time control problems. We analyze the accuracy of such approximations for extended mean field control (MFC) problems, where the dynamics and costs depend on the joint distribution of states and controls. For linear-convex extended MFC problems, we show that the optimal control is $1/2$-Hölder continuous in time. Using this regularity, we prove that the optimal cost of the continuous-time problem can be approximated by piecewise constant controls with order $1/2$, while the optimal control itself can be approximated with order $1/4$. For general extended MFC problems, we further show that, under sufficient regularity of the value functions, the value functions converge with an improved first-order rate, matching the best-known rate for classical control problems without mean field interaction, and consistent with the numerical observations for MFC of Cucker-Smale models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00904 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence Rates of Time Discretization in Extended Mean Field Control Reisinger, Christoph Stockinger, Wolfgang Tsianni, Maria Olympia Zhang, Yufei Optimization and Control Numerical Analysis 49N80, 49N60, 60H35, 65L70 Piecewise constant control approximation provides a practical framework for designing numerical schemes of continuous-time control problems. We analyze the accuracy of such approximations for extended mean field control (MFC) problems, where the dynamics and costs depend on the joint distribution of states and controls. For linear-convex extended MFC problems, we show that the optimal control is $1/2$-Hölder continuous in time. Using this regularity, we prove that the optimal cost of the continuous-time problem can be approximated by piecewise constant controls with order $1/2$, while the optimal control itself can be approximated with order $1/4$. For general extended MFC problems, we further show that, under sufficient regularity of the value functions, the value functions converge with an improved first-order rate, matching the best-known rate for classical control problems without mean field interaction, and consistent with the numerical observations for MFC of Cucker-Smale models. |
| title | Convergence Rates of Time Discretization in Extended Mean Field Control |
| topic | Optimization and Control Numerical Analysis 49N80, 49N60, 60H35, 65L70 |
| url | https://arxiv.org/abs/2509.00904 |