Optimal control problems driven by nonlinear degenerate Fokker-Planck equations
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916463354314752 |
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| author | Anceschi, Francesca Ascione, Giacomo Castorina, Daniele Solombrino, Francesco |
| author_facet | Anceschi, Francesca Ascione, Giacomo Castorina, Daniele Solombrino, Francesco |
| contents | The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as mean-field limits of Stochastic Differential models for multipopulation dynamics, where a large number of agents (followers) is steered through parsimonious intervention on a selected class of leaders. The proposed approach combines stability estimates for measure solutions of nonlinear degenerate Fokker-Planck equations with a general framework of assumptions on the cost functional, ensuring compactness and lower semicontinuity properties. The Lie structure of the state equations allows one for considering non-Lipschitz nonlinearities, provided some suitable dissipativity assumptions are considered in addition to non-Euclidean Hölder and sublinearity conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_24000 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal control problems driven by nonlinear degenerate Fokker-Planck equations Anceschi, Francesca Ascione, Giacomo Castorina, Daniele Solombrino, Francesco Optimization and Control Analysis of PDEs Probability The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as mean-field limits of Stochastic Differential models for multipopulation dynamics, where a large number of agents (followers) is steered through parsimonious intervention on a selected class of leaders. The proposed approach combines stability estimates for measure solutions of nonlinear degenerate Fokker-Planck equations with a general framework of assumptions on the cost functional, ensuring compactness and lower semicontinuity properties. The Lie structure of the state equations allows one for considering non-Lipschitz nonlinearities, provided some suitable dissipativity assumptions are considered in addition to non-Euclidean Hölder and sublinearity conditions. |
| title | Optimal control problems driven by nonlinear degenerate Fokker-Planck equations |
| topic | Optimization and Control Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2410.24000 |