{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T09:32:33Z","timestamp":1767864753361,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,31]],"date-time":"2021-10-31T00:00:00Z","timestamp":1635638400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We study an optimal control problem for the stationary Stokes equations with variable density and viscosity in a 2D bounded domain under mixed boundary conditions. On in-flow and out-flow parts of the boundary, nonhomogeneous Dirichlet boundary conditions are used, while on the solid walls of the flow domain, the impermeability condition and the Navier slip condition are provided. We control the system by the external forces (distributed control) as well as the velocity boundary control acting on a fixed part of the boundary. We prove the existence of weak solutions of the state equations, by firstly expressing the fluid density in terms of the stream function (Frolov formulation). Then, we analyze the control problem and prove the existence of global optimal solutions. Using a Lagrange multipliers theorem in Banach spaces, we derive an optimality system. We also establish a second-order sufficient optimality condition and show that the marginal function of this control system is lower semi-continuous.<\/jats:p>","DOI":"10.3390\/sym13112050","type":"journal-article","created":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T22:17:23Z","timestamp":1635891443000},"page":"2050","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Control Problem Related to 2D Stokes Equations with Variable Density and Viscosity"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1514-4475","authenticated-orcid":false,"given":"Evgenii S.","family":"Baranovskii","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Informatics and Mechanics, Voronezh State University, 394018 Voronezh, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3880-1279","authenticated-orcid":false,"given":"Eber","family":"Lenes","sequence":"additional","affiliation":[{"name":"\u00c1rea de Ciencias B\u00e1sicas Exactas, Grupo de Investigaci\u00f3n Deartica, Universidad del Sin\u00fa, Cartagena 130001, Colombia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7726-2362","authenticated-orcid":false,"given":"Exequiel","family":"Mallea-Zepeda","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidad de Tarapac\u00e1, Av. 18 de Septiembre 2222, Arica 1000000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7657-2986","authenticated-orcid":false,"given":"Jonnathan","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Ciencias B\u00e1sicas, Universidad de Antofagasta, Antofagasta 1240000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7104-2895","authenticated-orcid":false,"given":"Lautaro","family":"V\u00e1squez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidad de Tarapac\u00e1, Av. 18 de Septiembre 2222, Arica 1000000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,31]]},"reference":[{"key":"ref_1","first-page":"389","article-title":"Sur le lois du mouvement des fluides","volume":"6","author":"Navier","year":"1823","journal-title":"Mem. 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