{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:41:56Z","timestamp":1787319716368,"version":"build-2736575974"},"reference-count":13,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>An optimal control problem for semilinear elliptic partial differential equations is considered. The equation is in divergence form with the leading term containing the control. Necessary conditions for optimal controls are established by the method of homogenized spike variation. The key to such a method is to modify the usual spike variational technique by taking into account the homogenization techniques for elliptic equations, together with an idea from the theory of relaxed controls. Problems with state constraints are also discussed by further adding some well-known penalty arguments involving the application of Ekeland's variational principle and finite codimensionality of certain sets.<\/jats:p>","DOI":"10.1137\/080740301","type":"journal-article","created":{"date-parts":[[2009,7,18]],"date-time":"2009-07-18T04:19:08Z","timestamp":1247890748000},"page":"2366-2387","source":"Crossref","is-referenced-by-count":7,"title":["Optimality Conditions for Semilinear Elliptic Equations with Leading Term Containing Controls"],"prefix":"10.1137","volume":"48","author":[{"given":"Hongwei","family":"Lou","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jiongmin","family":"Yong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,7,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0523084"},{"key":"R2","unstructured":"A. Bensoussan, J. L. Lions, and G. Papanicolaou,\n                      Asymptotic Analysis for Periodic Structures\n                      , North\u2013Holland, Amsterdam, 1978."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012902411714"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/070685890"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01218394"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2004029"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1017\/S0334270000011814"},{"key":"R8","doi-asserted-by":"crossref","unstructured":"X. Li and J. Yong,\n                      Optimal Control Theory for Infinite-Dimensional Systems\n                      , Birkh\u00e4user, Boston, 1995.","DOI":"10.1007\/978-1-4612-4260-4"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"F. Murat and L. Tartar,\n                      Calculus of variations and homogenization\n                      , in Collection d'tudes d'electricit de France, rewritten in Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl. 31, L. Cherkaev and R. V. Kohn, eds., Birkh\u00e4user, Boston, 1998, pp. 139\u2013174.","DOI":"10.1007\/978-3-319-97184-1_6"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1080\/02331930500050632"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1515\/156939403770888255"},{"key":"R12","first-page":"247","volume":"24","author":"Tagiyev R. K.","year":"2004","journal-title":"Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci."},{"key":"R13","unstructured":"A. Zygmund,\n                      Trigonometric Series\n                      , 3rd ed., Cambridge University Press, Cambridge, UK, 2002."}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080740301","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:51:34Z","timestamp":1787316694000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080740301"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/080740301"],"URL":"https:\/\/doi.org\/10.1137\/080740301","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}