{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:42:33Z","timestamp":1760143353608,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2024,1,31]],"date-time":"2024-01-31T00:00:00Z","timestamp":1706659200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CeDRI","award":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"],"award-info":[{"award-number":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"]}]},{"name":"SusTEC","award":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"],"award-info":[{"award-number":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"]}]},{"name":"Ministerio de Ciencia e Innovaci\u00f3n, Agencia Estatal de Investigaci\u00f3n","award":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"],"award-info":[{"award-number":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"]}]},{"name":"Conseller\u00eda de Educaci\u00f3n, Universidade e Formaci\u00f3n Profesional","award":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"],"award-info":[{"award-number":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"]}]},{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"],"award-info":[{"award-number":["UIDB\/05757\/2020","UIDP\/05757\/2020","LA\/P\/0007\/2020","PID2020-115155GB-I00","ED431C 2023\/31","UIDB\/00144\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This work focuses on optimizing the displacement of a passive particle interacting with vortices located on the surface of a sphere. The goal is to minimize the energy expended during the displacement within a fixed time. The modeling of particle dynamics, whether in Cartesian or spherical coordinates, gives rise to alternative formulations of the identical problem. Thanks to these two versions of the same problem, we can assert that the algorithm, employed to transform the optimal control problem into an optimization problem, is effective, as evidenced by the obtained controls. The numerical resolution of these formulations through a direct approach consistently produces optimal solutions, regardless of the selected coordinate system.<\/jats:p>","DOI":"10.3390\/computation12020023","type":"journal-article","created":{"date-parts":[[2024,1,31]],"date-time":"2024-01-31T10:44:24Z","timestamp":1706697864000},"page":"23","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Exploring Controlled Passive Particle Motion Driven by Point Vortices on a Sphere"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2431-8665","authenticated-orcid":false,"given":"Carlos","family":"Balsa","sequence":"first","affiliation":[{"name":"Research Centre in Digitalization and Intelligent Robotics (CeDRI), Laborat\u00f3rio para a Sustentabilidade e Tecnologia em Regi\u00f5es de Montanha (SusTEC), Instituto Polit\u00e9cnico de Bragan\u00e7a, 5300-253 Bragan\u00e7a, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0201-0523","authenticated-orcid":false,"given":"M. Victoria","family":"Otero-Espinar","sequence":"additional","affiliation":[{"name":"Departamento de Estat\u00edstica, An\u00e1lise Matem\u00e1tica e Optimizaci\u00f3n, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0852-8892","authenticated-orcid":false,"given":"S\u00edlvio","family":"Gama","sequence":"additional","affiliation":[{"name":"Mathematics Center of the Porto University (CMUP), Mathematics Department, Faculty of Sciences, University of Porto, R. Campo Alegre s\/n, 4169-007 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,31]]},"reference":[{"key":"ref_1","first-page":"25","article-title":"\u00dcber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen","volume":"55","author":"Helmholtz","year":"1858","journal-title":"J. F\u00fcr Die Reine Angew. 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