Mean Field Control Hierarchy

Giacomo Albi, Young Pil Choi, Massimo Fornasier, Dante Kalise

Research output: Contribution to journalArticlepeer-review

56 Scopus citations


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.

Original languageEnglish
Pages (from-to)93-135
Number of pages43
JournalApplied Mathematics & Optimization
Issue number1
StatePublished - 1 Aug 2017


  • Boltzmann equations
  • Kinetic equations
  • PDE constrained optimization multi-agent systems


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