A potential approach for planning mean-field games in one dimension

Tigran Bakaryan; Rita Ferreira; Diogo Gomes

doi:10.3934/cpaa.2022054

Article Contents

2022, Volume 21, Issue 6: 2147-2187. Doi: 10.3934/cpaa.2022054

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A potential approach for planning mean-field games in one dimension

King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia

^*Corresponding author
^*Corresponding author

Received: March 31, 2021

Revised: October 31, 2021

Published: June 2022

T. Bakaryan, R. Ferreira, and D. Gomes were partially supported by King Abdullah University of Science and Technology (KAUST) baseline funds and KAUST OSR-CRG2021-4674

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Abstract

This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. Applying Poincaré's Lemma to the Fokker–Planck equation, we deduce the existence of a potential. Rewriting the Hamilton–Jacobi equation in terms of the potential, we obtain a system of Euler–Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. The variational approach has the advantage of eliminating the continuity equation.

We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities. We end the paper by presenting an application to the one-dimensional Hughes' model.

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Mathematics Subject Classification: 35J47, 35A01, 35J50.

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