Mean-field control of non exchangeable systems

We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal probability laws. We prove the well-posedness of such systems a...

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Bibliographic Details
Published inarXiv.org
Main Authors De Crescenzo, Anna, Fuhrman, Marco, Kharroubi, Idris, Pham, Huyên
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 26.07.2024
Subjects
Control systems
Dynamic programming
Optimal control
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Summary:We study the optimal control of mean-field systems with heterogeneous and asymmetric interactions. This leads to considering a family of controlled Brownian diffusion processes with dynamics depending on the whole collection of marginal probability laws. We prove the well-posedness of such systems and define the control problem together with its related value function. We next prove a law invariance property for the value function which allows us to work on the set of collections of probability laws. We show that the value function satisfies a dynamic programming principle (DPP) on the flow of collections of probability measures. We also derive a chain rule for a class of regular functions along the flows of collections of marginal laws of diffusion processes. Combining the DPP and the chain rule, we prove that the value function is a viscosity solution of a Bellman dynamic programming equation in a \(L^2\)-set of Wasserstein space-valued functions.
ISSN:2331-8422