{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:32:59Z","timestamp":1787319179476,"version":"build-2736575974"},"reference-count":42,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"7","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>In this paper we study the optimal control problem of the heat equation with a cubic nonlinearity by a distributed control over a subset of the domain, in the presence of a state constraint. The latter is integral over the space and has to be satisfied each time. Using for the first time the technique of alternative optimality systems in the context of optimal control of partial differential equations, we show that both the control and multiplier are continuous in time. Under some natural geometric hypotheses, we can prove that extended polyhedricity holds, allowing us to obtain no-gap second-order optimality conditions, that characterize quadratic growth. An expansion of the value function and of approximate solutions can be computed for a directional perturbation of the right-hand side of the state equation.<\/jats:p>","DOI":"10.1137\/080744608","type":"journal-article","created":{"date-parts":[[2010,8,10]],"date-time":"2010-08-10T18:07:59Z","timestamp":1281463679000},"page":"4550-4571","source":"Crossref","is-referenced-by-count":22,"title":["Optimal Control of a Parabolic Equation with Time-Dependent State Constraints"],"prefix":"10.1137","volume":"48","author":[{"given":"J. Fr\u00e9d\u00e9ric","family":"Bonnans","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pascal","family":"Jaisson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,8,10]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/0327001"},{"key":"R2","unstructured":"R. Adams,\n                      Sobolev Spaces\n                      , Academic Press, New York, 1975."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998341411"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S036301299733035X"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012999357926"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-001-0036-4"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0140040"},{"key":"R8","unstructured":"J. Bonnans,\n                      Contr\u00f4le optimal de syst\u00e8mes instables gouvern\u00e9s par des \u00e9quations aux d\u00e9riv\u00e9es partielles non lin\u00e9aires\n                      , Ph.D. thesis, Universit\u00e9 Paris IX Dauphine, Paris, France, 1982."},{"key":"R9","first-page":"249","volume":"6","author":"Bonnans J. F.","year":"1984","journal-title":"Large Scale Systems","ISSN":"https:\/\/id.crossref.org\/issn\/0165-0777","issn-type":"print"},{"key":"R10","unstructured":"J. F. Bonnans and E. Casas,\n                      Contr\u00f4le de syst\u00e8mes elliptiques semilin\u00e9aires comportant des contraintes sur l'\u00e9tat\n                      , in Nonlinear Partial Differential Equations and Their Applications, Coll\u00e8ge de France seminar Vol. VII, H. Br\u00e9zis and J. Lions, eds., Pitman Res. Notes Math. 166, Longman Scientific & Technical, Harlow, UK, 1988, pp. 69\u201386."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0327023"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"J. F. Bonnans and E. Casas,\n                      On the choice of the function spaces for some state-constrained control problems\n                      , Numer. Funct. Anal. Optim., 7 (1984\/85), pp. 333\u2013348.","DOI":"10.1080\/01630568508816197"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012994267273"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012994267285"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012994276883"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/06065756X"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1051\/cocv:2008016"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-007-0167-8"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1016\/j.anihpc.2007.12.002"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"J. F. Bonnans and A. Shapiro,\n                      Perturbation Analysis of Optimization Problems\n                      , Non Lin\u00e9aire Springer Ser. Oper. Res. Springer-Verlag, New York, 2000.","DOI":"10.1007\/978-1-4612-1394-9"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012995283637"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/07068240X"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012998345627"},{"key":"R24","first-page":"1631","volume":"321","author":"Cominetti R.","year":"1995","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-4442","issn-type":"print"},{"key":"R25","unstructured":"N. Dunford and J. Schwartz,\n                      Linear Operators, Vol.\n                      I, John Wiley & Sons, New York, 1958."},{"key":"R26","unstructured":"N. Dunford and J. Schwartz,\n                      Linear Operators, Vol.\n                      II John Wiley & Sons, New York, 1963."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1137\/0331045"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/0317026"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(71)90219-8"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1007\/BF01580754"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1007\/BF01580771"},{"key":"R32","doi-asserted-by":"crossref","unstructured":"H. Kawasaki,\n                      Second order necessary optimality conditions for minimizing a sup-type function\n                      , Math. Programming, 49 (1990\/91), pp. 213\u2013229.","DOI":"10.1007\/BF01588788"},{"key":"R33","unstructured":"J.L. Lions,\n                      Contr\u00f4le des syst\u00e8mes distribu\u00e9s singuliers\n                      , Dunod, Paris, 1983."},{"key":"R34","unstructured":"J.L. Lions,\n                      Quelques m\u00e9thodes de r\u00e9solution des probl\u00e8mes aux limites non lin\u00e9aires\n                      , Dunod Gauthier-villars, Paris, 1969."},{"key":"R35","unstructured":"J.L. Lions and E. Magenes,\n                      Probl\u00e8mes aux limites non homog\u00e8nes et applications.\n                      Vol. 1, Travaux et Recherches Math\u00e9matiques 17, Dunod, Paris, 1968."},{"key":"R36","unstructured":"H. Maurer,\n                      On the Minimum Principle for Optimal Control Problems with State Constraints\n                      , Schriftenreihe des Rechenzentrum 41, Universit\u00e4t M\u00fcnster, M\u00fcnster, Germany, 1979."},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2000.6.431"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012996302470"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1137\/0130053"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1137\/0713043"},{"key":"R41","unstructured":"L. Tartar,\n                      Topics in Nonlinear Analysis\n                      , Publ. Math. 'Orsay 78 13, Universit\u00e9 de Paris-Sud D\u00e9partement de Math\u00e9matique, Orsay, France, 1978."},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1007\/BF01442543"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080744608","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:46:01Z","timestamp":1787316361000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080744608"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":42,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/080744608"],"URL":"https:\/\/doi.org\/10.1137\/080744608","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}