{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T17:54:24Z","timestamp":1767117264733,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T00:00:00Z","timestamp":1653955200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia (FCT)","doi-asserted-by":"publisher","award":["PD\/BD\/150537\/2019","UIDB\/00144\/2020","POCI-01-0145-FEDER-006933","POCI-01-0145-FEDER-032485","NORTE-01-0145-FEDER-000085"],"award-info":[{"award-number":["PD\/BD\/150537\/2019","UIDB\/00144\/2020","POCI-01-0145-FEDER-006933","POCI-01-0145-FEDER-032485","NORTE-01-0145-FEDER-000085"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"name":"ERDF | COMPETE2020 | FCT\/MEC | PT2020","award":["PD\/BD\/150537\/2019","UIDB\/00144\/2020","POCI-01-0145-FEDER-006933","POCI-01-0145-FEDER-032485","NORTE-01-0145-FEDER-000085"],"award-info":[{"award-number":["PD\/BD\/150537\/2019","UIDB\/00144\/2020","POCI-01-0145-FEDER-006933","POCI-01-0145-FEDER-032485","NORTE-01-0145-FEDER-000085"]}]},{"name":"FEDER via COMPETE 2020\u2014POCI","award":["PD\/BD\/150537\/2019","UIDB\/00144\/2020","POCI-01-0145-FEDER-006933","POCI-01-0145-FEDER-032485","NORTE-01-0145-FEDER-000085"],"award-info":[{"award-number":["PD\/BD\/150537\/2019","UIDB\/00144\/2020","POCI-01-0145-FEDER-006933","POCI-01-0145-FEDER-032485","NORTE-01-0145-FEDER-000085"]}]},{"name":"European Regional Development Fund (ERDF)","award":["PD\/BD\/150537\/2019","UIDB\/00144\/2020","POCI-01-0145-FEDER-006933","POCI-01-0145-FEDER-032485","NORTE-01-0145-FEDER-000085"],"award-info":[{"award-number":["PD\/BD\/150537\/2019","UIDB\/00144\/2020","POCI-01-0145-FEDER-006933","POCI-01-0145-FEDER-032485","NORTE-01-0145-FEDER-000085"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This work concerns the optimal control of a passive particle in viscous flows. This is relevant since, while there are many studies on optimal control in inviscid flows, there is little to no work in this context for viscous flows, and viscosity cannot always be neglected. Furthermore, in many tasks, there is a need to reduce the energy spent; thus, energy-optimal solutions to problems are important. The aim of this work is to investigate how to optimally move a passive particle advected by a Lamb\u2013Oseen (viscous) vortex between two given points in space in a given time interval while minimising the energy spent on this process. We take a control acting only on the radial component of the motion, and, by using the Pontryagin\u2019s Maximum Principle, we find an explicit time-dependent extremal. We also analyse how the energy cost changes with the viscosity of the flow.The problem is transformed into a parameter search problem with two parameters related to the radial and angular coordinates of the initial point. The energy cost of the process increases with viscosity as long as the passive particle maintains the number of full turns it makes around the vortex. However, the energy cost increases if the increase in viscosity forces the particle to make fewer full revolutions around the vortex.<\/jats:p>","DOI":"10.3390\/computation10060087","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T09:24:29Z","timestamp":1653989069000},"page":"87","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Optimal Control of a Passive Particle Advected by a Lamb\u2013Oseen (Viscous) Vortex"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4933-7136","authenticated-orcid":false,"given":"Gil","family":"Marques","sequence":"first","affiliation":[{"name":"CMUP\u2014Department of Mathematics, FCUP, University of Porto, 4169-007 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0852-8892","authenticated-orcid":false,"given":"S\u00edlvio","family":"Gama","sequence":"additional","affiliation":[{"name":"CMUP\u2014Department of Mathematics, FCUP, University of Porto, 4169-007 Porto, Portugal"}]},{"given":"Fernando Lobo","family":"Pereira","sequence":"additional","affiliation":[{"name":"SYSTEC\u2014Department of Electrical and Computer Engineering, FEUP, University of Porto, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,31]]},"reference":[{"key":"ref_1","unstructured":"Clarke, F., Ledyaev, Y., Stern, R., and Wolenski, P. (1998). Nonsmooth Analysis and Control Theory, Springer."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Lions, J. (1971). Optimal Control Of Systems Governed by Partial Differential Equations, Springer.","DOI":"10.1007\/978-3-642-65024-6"},{"key":"ref_3","unstructured":"Mahmoudian, N., Geisbert, J., and Woolsey, C. (2009). Dynamics and Control of Underwater Gliders I: Steady Motions, Virginia Center for Autonomous Systems. Technical Report."},{"key":"ref_4","unstructured":"Mahmoudian, N., and Woolsey, C. (2010). Dynamics and Control of Underwater Gliders II: Motion Planning and Control, Virginia Center for Autonomous Systems. Technical Report."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/S1672-6529(09)60184-0","article-title":"Biological Inspiration: From Carangiform Fish to Multijoint Robotic Fish","volume":"7","author":"Liu","year":"2010","journal-title":"J. Bionic Eng."},{"key":"ref_6","unstructured":"Batchelor, G.K. (1992). 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Turbulent Flows, Cambridge University Press.","DOI":"10.1017\/CBO9780511840531"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s00220-004-1254-9","article-title":"Global Stability of Vortex Solutions of the Two-Dimensional Navier-Stokes Equation","volume":"255","author":"Gallay","year":"2005","journal-title":"Commun. Math. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1007\/s00205-010-0362-2","article-title":"Interaction of Vortices in Weakly Viscous Planar Flows","volume":"200","author":"Gallay","year":"2011","journal-title":"Arch. Ration. Mech. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"455501","DOI":"10.1088\/1751-8113\/45\/45\/455501","article-title":"Interactions of Three Viscous Point Vortices","volume":"45","author":"Kim","year":"2012","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Marques, G., Grilo, T., Gama, S., and Pereira, F.L. (2021). Optimal Control of a Passive Particle Advected by a Point Vortex. Advanced Research in Technologies, Information, Innovation and Sustainability, Springer.","DOI":"10.1007\/978-3-030-90241-4_39"},{"key":"ref_15","unstructured":"Pontryagin, L., Boltyanskiy, V., Gamkrelidze, R., and Mishchenko, E. (1962). Mathematical Theory of Optimal Processes, Interscience Publish."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1088\/0951-7715\/21\/9\/R01","article-title":"Vortex dynamics models in flow control problems","volume":"21","author":"Protas","year":"2008","journal-title":"Nonlinearity"},{"key":"ref_17","unstructured":"Fehlberg, E. (1969). Low-Order Classical Runge-Kutta Formulas with Stepsize Control and Their Application to Some Heat Transfer Problems, NASA. Technical Report 315;."}],"container-title":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/6\/87\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:22:32Z","timestamp":1760138552000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/10\/6\/87"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,31]]},"references-count":17,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["computation10060087"],"URL":"https:\/\/doi.org\/10.3390\/computation10060087","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2022,5,31]]}}}