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Mean-field optimal control as Gamma-limit of finite agent controls

Published online by Cambridge University Press:  08 March 2019

M. FORNASIER ,
S. LISINI ,
C. ORRIERI  and
G. SAVARÉ Open the ORCID record for G. SAVARÉ [Opens in a new window]
M. FORNASIER
Affiliation:
Department of Mathematics, TU München, Boltzmannstr. 3, Garching bei München D-85748, Germany email: massimo.fornasier@ma.tum.de
S. LISINI
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy email: stefano.lisini@unipv.it; giuseppe.savare@unipv.it
C. ORRIERI
Affiliation:
Dipartimento di Matematica “G. Castelnuovo”, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy email: orrieri@mat.uniroma1.it
G. SAVARÉ*
Affiliation:
Dipartimento di Matematica “F. Casorati”, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy email: stefano.lisini@unipv.it; giuseppe.savare@unipv.it
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Abstract

This paper focuses on the role of a government of a large population of interacting agents as a meanfield optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a Partial Differential Equation of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a Γ -convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle.

Keywords

Finite agent optimal controlmean-field optimal controlΓ-convergencesuperposition principlePrimary: 35Q9349K20Secondary: 35Q8335Q7035Q91
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1153 - 1186
Copyright
© Cambridge University Press 2019 

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Footnotes

Massimo Fornasier acknowledges the financial support provided by the ERC-Starting Grant ‘High-Dimensional Sparse Optimal Control’ (HDSPCONTR) and the DFG-Project FO 767/7-1 ‘Identification of Energies from the Observation of Evolutions’. Giuseppe Savaré acknowledges the financial support provided by Cariplo foundation and Regione Lombardia via project ‘Variational evolution problems and optimal transport’. Carlo Orrieri acknowledges the financial support provided by PRIN 20155PAWZB ‘Large Scale Random Structures’.

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