Accepted Paper

Inserted: 14 feb 2024
Last Updated: 4 mar 2025

Journal: ESAIM: COCV
Year: 2025
Doi: 10.1051/cocv/2025025

Abstract:

We derive a first order optimality condition for a class of agent-based systems, as well as for their mean-field counterpart. A relevant difficulty of our analysis is that the state equation is formulated on possibly infinite-dimensional convex subsets of Banach spaces. This is a typical feature of many problems in multi-population dynamics, where a convex set of probability measures may account for the population, the degree of influence or the strategy attached to each agent. Due to the lack of a linear structure and of local compactness, the usual tools of needle variations and linearisation procedures used to derive Pontryagin type conditions have to be generalised to the setting at hand. This is done by considering suitable notions of differentials and by a careful inspection of the underlying functional structures.

Keywords: Pontryagin Maximum Principle, mean field optimal control,, agent-based systems, differential equations on convex state spaces

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