{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:48:05Z","timestamp":1777448885255,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"1-2","license":[{"start":{"date-parts":[[2011,1,1]],"date-time":"2011-01-01T00:00:00Z","timestamp":1293840000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>We are interested in the optimal control problem of a parabolic equation with no state constraints, where the quadratic cost functional involves a final observation and the control variable is a Dirichlet boundary condition. Practical considerations lead us to use Dirichlet controls with no more regularity than square integrability, which arise some technical difficulties in the mathematical analysis. After setting the state equation and the related adjoint equation we fit the two coupled equations into a mixed space\u2013time variational framework. The resulting saddle-point problem turns out to be well posed. We then use a Robin penalization on the Dirichlet control which enables us to re-transcript the mixed problem in a form better suited to numerical computations. We analyze and establish the convergence when the penalty parameter tends to zero, first without additional smoothness assumptions on the optimal control and then for smooth controls.<\/jats:p>","DOI":"10.3233\/asy-2010-1015","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:15:03Z","timestamp":1575054903000},"page":"101-121","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":7,"title":["Dirichlet boundary control for a parabolic equation with a final observation I: A space\u2013time mixed formulation and penalization"],"prefix":"10.1177","volume":"71","author":[{"given":"Faker","family":"Ben Belgacem","sequence":"first","affiliation":[{"name":"Universit\u00e9 de Technologie de Compi\u00e8gne, BP 20529, 60205 Compi\u00e8gne Cedex, France. E-mail: faker.ben-belgacem@utc.fr"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christine","family":"Bernardi","sequence":"additional","affiliation":[{"name":"Laboratoire Jacques-Louis Lions, CNRS and Universit\u00e9 Pierre et Marie Curie, BC 187, 4 place Jussieu, 75252 Paris Cedex 05, France. E-mail: bernardi@ann.jussieu.fr"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Henda","family":"El Fekih","sequence":"additional","affiliation":[{"name":"LAMSIN, \u00c9cole Nationale d'Ing\u00e9nieurs de Tunis, BP 37, 1002 Tunis-Belv\u00e9d\u00e8re, Tunisie. E-mail: henda.elfekih@enit.rnu.tn"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2011,1,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2010-1015","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2010-1015","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:39:16Z","timestamp":1777379956000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-2010-1015"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":0,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.3233\/ASY-2010-1015"],"URL":"https:\/\/doi.org\/10.3233\/asy-2010-1015","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}