Journal
APPLIED MATHEMATICS AND OPTIMIZATION
Volume 76, Issue 1, Pages 93-135Publisher
SPRINGER
DOI: 10.1007/s00245-017-9429-x
Keywords
PDE constrained optimization multi-agent systems; Boltzmann equations; Kinetic equations
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Funding
- ERC-Starting Grant HDSP-CONTR High-Dimensional Sparse Optimal Control
- Alexander Humboldt Foundation through the Humboldt Research Fellowship for Postdoctoral Researchers
- ERC-Advanced Grant OCLOC From Open-Loop to Closed-Loop Optimal Control of PDEs
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In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrained by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population. We show the existence of mean field optimal controls both in the stochastic and deterministic setting. We derive rigorously the first order optimality conditions useful for numerical computation of mean field optimal controls. We introduce a novel approximating hierarchy of sub-optimal controls based on a Boltzmann approach, whose computation requires a very moderate numerical complexity with respect to the one of the optimal control. We provide numerical experiments for models in opinion formation comparing the behavior of the control hierarchy.
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