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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.16721 |
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| _version_ | 1866910962778374144 |
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| author | La Scala, Giuseppe |
| author_facet | La Scala, Giuseppe |
| contents | We consider a generic, suitable class of optimal control problems under a constraint given by a finite-dimensional SDE-ODE system, describing a system of two interacting species of particles: the herd, described by SDEs, and the herders, described by ODEs with the addition of a control function.
In particular, we firstly show that for a low number of herders and for the limit of large number of herd individuals, the SDE-ODE system can be approximated by an infinite-dimensional system given by a McKean-Vlasov single SDE coupled with ODEs. Then, thanks to this we show the $Γ-$convergence of the optimal control problem for the finite-dimensional system to a certain optimal control problem for the mean-field system.
Differently from Ascione-Castorina-Solombrino [9] (SIAM J. Math. Anal., Vol. 55, No. 6, pp. 6965-6990 (2023)), we do not consider an additive noise for the herd, but a more general class, given by idiosyncratic noises (due to a single herd individual) together with common noise (due to how the environment affects the whole herd), and they are independent one from another. As well as this, we consider a more general class of control functions in the ODEs for herders, where the control is applied not only on the herd dynamics, but also on the herd one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16721 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal control of mean-field limit of multiagent systems with and without common noise La Scala, Giuseppe Optimization and Control We consider a generic, suitable class of optimal control problems under a constraint given by a finite-dimensional SDE-ODE system, describing a system of two interacting species of particles: the herd, described by SDEs, and the herders, described by ODEs with the addition of a control function. In particular, we firstly show that for a low number of herders and for the limit of large number of herd individuals, the SDE-ODE system can be approximated by an infinite-dimensional system given by a McKean-Vlasov single SDE coupled with ODEs. Then, thanks to this we show the $Γ-$convergence of the optimal control problem for the finite-dimensional system to a certain optimal control problem for the mean-field system. Differently from Ascione-Castorina-Solombrino [9] (SIAM J. Math. Anal., Vol. 55, No. 6, pp. 6965-6990 (2023)), we do not consider an additive noise for the herd, but a more general class, given by idiosyncratic noises (due to a single herd individual) together with common noise (due to how the environment affects the whole herd), and they are independent one from another. As well as this, we consider a more general class of control functions in the ODEs for herders, where the control is applied not only on the herd dynamics, but also on the herd one. |
| title | Optimal control of mean-field limit of multiagent systems with and without common noise |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2505.16721 |