{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:43:01Z","timestamp":1787323381807,"version":"3.56.0"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We consider optimal transport problems where the cost is optimized over controlled dynamics and the end time is free. Unlike the classical setting, the search for optimal transport plans also requires the identification of optimal \u201cstopping plans,\" and the corresponding Monge--Kantorovich duality involves the resolution of a Hamilton--Jacobi--Bellman quasi-variational inequality. We discuss both Lagrangian and Eulerian formulations of the problem and its natural connection to Pontryagin's maximum principle. We also exhibit a purely dynamic situation, where the optimal stopping plan is a hitting time of a barrier given by the free boundary problem associated to the dual variational inequality. This problem was motivated by its stochastic counterpart, which will be studied in a companion paper.<\/jats:p>","DOI":"10.1137\/18m1178037","type":"journal-article","created":{"date-parts":[[2018,9,13]],"date-time":"2018-09-13T11:28:07Z","timestamp":1536838087000},"page":"3239-3259","source":"Crossref","is-referenced-by-count":10,"title":["Optimal Transport with Controlled Dynamics and Free End Times"],"prefix":"10.1137","volume":"56","author":[{"given":"Nassif","family":"Ghoussoub","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6920-603X","authenticated-orcid":true,"given":"Young-Heon","family":"Kim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2731-6895","authenticated-orcid":true,"given":"Aaron Zeff","family":"Palmer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,9,13]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-09-04813-2"},{"key":"atypb2","unstructured":"M. Bardi and I. Capuzzo-Dolcetta,\n                      Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations\n                      , Mod. Birkha\u0308user Class., Birkha\u0308user Boston, 2008."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-016-0692-2"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050002"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"P. Bernard and B. Buffoni,\n                      Optimal mass transportation and Mather theory\n                      , J. Eur. Math. Soc. (JEMS), 9 (2007), pp. 85-121.","DOI":"10.4171\/jems\/74"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160440402"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-01-00376-9"},{"key":"atypb8","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1002\/cpa.20315","volume":"63","author":"Cannarsa P.","year":"2010","journal-title":"Comm. Pure Appl. Math."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1214\/12-AAP857"},{"key":"atypb10","first-page":"653","volume":"137","author":"Evans L. C.","year":"1999","journal-title":"Mem. Amer. Math. Soc."},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/PL00004236"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/S0925-7535(00)00021-7"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392620"},{"key":"atypb14","unstructured":"N. Ghoussoub, Y.H. Kim, and A. Z. Palmer,\n                      PDE Methods for Stochastic Transportation Problems\n                      , manuscript, 2018."},{"key":"atypb15","unstructured":"N. Ghoussoub, Y.H. Kim, and A. Z. Palmer,\n                      PDE Methods for Optimal Skorokhod Embeddings\n                      ,https:\/\/arxiv.org\/abs\/1807.10347, 2018; J. Eur. Math. Soc. (JEMS), submitted."},{"key":"atypb16","unstructured":"G. Monge,\n                      Me\u0301moire sur la the\u0301orie des de\u0301blais et des remblais\n                      , Histoire de l'Acade\u0301mie Royale des Sciences de Paris, (1781), pp. 666-704."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-017-1183-3"},{"key":"atypb18","unstructured":"L. S. Pontryagin,\n                      Mathematical Theory of Optimal Processes\n                      , CRC Press, 1987."},{"key":"atypb19","first-page":"3","volume":"141","author":"Sudakov V. N.","year":"1979","journal-title":"Trudy Mat. Inst. Steklov."},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"C. Villani,\n                      Topics in Optimal Transportation\n                      , Grad. Stud. Math. 58, American Mathematical Society, 2003.","DOI":"10.1090\/gsm\/058"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"C. Villani,\n                      Optimal Transport: Old and New\n                      , Grundlehren Math. Wiss. 338, Springer, 2008.","DOI":"10.1007\/978-3-540-71050-9"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1178037","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:59:09Z","timestamp":1787320749000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1178037"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":21,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/18M1178037"],"URL":"https:\/\/doi.org\/10.1137\/18m1178037","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}