Ergodic control of McKean-Vlasov systems on the Wasserstein space
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914169016549376 |
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| author | Fuhrman, Marco Rudà, Silvia |
| author_facet | Fuhrman, Marco Rudà, Silvia |
| contents | We consider an optimal control problem with ergodic (long term average) reward for a McKean-Vlasov dynamics, where the coefficients of a controlled stochastic differential equation depend on the marginal law of the solution. Starting from the associated infinite time horizon expected discounted reward, we construct both the value $λ$ of the ergodic problem and an associated function $ϕ$, which provide a viscosity solution to an ergodic Hamilton-Jacobi-Bellman (HJB) equation of elliptic type. In contrast to previous results, we consider the function $ϕ$ and the HJB equation on the Wasserstein space, using concepts of derivatives with respect to probability measures. The pair $(λ,ϕ)$ also provides information on limit behavior of related optimization problems, for instance, results of Abelian-Tauberian type or limits of value functions of control problems for finite time horizon when the latter tends to infinity. Many arguments are simplified by the use of a functional relation for $ϕ$ in the form of a suitable dynamic programming principle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17958 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ergodic control of McKean-Vlasov systems on the Wasserstein space Fuhrman, Marco Rudà, Silvia Optimization and Control 60H10, 60H30, 93E20 We consider an optimal control problem with ergodic (long term average) reward for a McKean-Vlasov dynamics, where the coefficients of a controlled stochastic differential equation depend on the marginal law of the solution. Starting from the associated infinite time horizon expected discounted reward, we construct both the value $λ$ of the ergodic problem and an associated function $ϕ$, which provide a viscosity solution to an ergodic Hamilton-Jacobi-Bellman (HJB) equation of elliptic type. In contrast to previous results, we consider the function $ϕ$ and the HJB equation on the Wasserstein space, using concepts of derivatives with respect to probability measures. The pair $(λ,ϕ)$ also provides information on limit behavior of related optimization problems, for instance, results of Abelian-Tauberian type or limits of value functions of control problems for finite time horizon when the latter tends to infinity. Many arguments are simplified by the use of a functional relation for $ϕ$ in the form of a suitable dynamic programming principle. |
| title | Ergodic control of McKean-Vlasov systems on the Wasserstein space |
| topic | Optimization and Control 60H10, 60H30, 93E20 |
| url | https://arxiv.org/abs/2504.17958 |