{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,16]],"date-time":"2026-02-16T09:03:05Z","timestamp":1771232585718,"version":"3.50.1"},"reference-count":29,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["CPAA"],"published-print":{"date-parts":[[2022]]},"abstract":"<jats:p xml:lang=\"fr\">&lt;p style='text-indent:20px;'&gt;This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton\u2013Jacobi equation coupled with a Fokker\u2013Planck equation. Applying Poincar\u00e9's Lemma to the Fokker\u2013Planck equation, we deduce the existence of a potential. Rewriting the Hamilton\u2013Jacobi equation in terms of the potential, we obtain a system of Euler\u2013Lagrange 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.&lt;\/p&gt;&lt;p style='text-indent:20px;'&gt;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.&lt;\/p&gt;<\/jats:p>","DOI":"10.3934\/cpaa.2022054","type":"journal-article","created":{"date-parts":[[2022,3,21]],"date-time":"2022-03-21T09:39:42Z","timestamp":1647855582000},"page":"2147","source":"Crossref","is-referenced-by-count":4,"title":["A potential approach for planning mean-field games in one dimension"],"prefix":"10.3934","volume":"21","author":[{"given":"Tigran","family":"Bakaryan","sequence":"first","affiliation":[{"name":"King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia"}]},{"given":"Rita","family":"Ferreira","sequence":"additional","affiliation":[{"name":"King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia"}]},{"given":"Diogo","family":"Gomes","sequence":"additional","affiliation":[{"name":"King Abdullah University of Science and Technology (KAUST), CEMSE Division, Thuwal 23955-6900, Saudi Arabia"}]}],"member":"2321","reference":[{"key":"key-10.3934\/cpaa.2022054-1","doi-asserted-by":"publisher","unstructured":"Y. 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