{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:04:35Z","timestamp":1787321075272,"version":"build-2736575974"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>In this paper we study the compatibility (or viability) of a given state constraint K with respect to a controlled stochastic evolution equation in a real Hilbert space H. We allow the noise to be a cylindrical Wiener process and admit an unbounded linear operator in the state equation. Our assumptions cover, for instance, controlled heat equations with space-time white noise. Our main result is to prove that if K is $\\varepsilon$-viable, then the square of the distance from K: $d_K^2(x):= \\inf_{y\\in K}|x-y|^2$ is a viscosity supersolution of a suitable class of fully nonlinear Hamilton\u2013Jacobi\u2013Bellman equations in H. This extends already obtained results into the finite dimensional case. We use the definition of viscosity supersolutions for \u201cunbounded\u201d elliptic equations in infinite variables that have been recently introduced by \u015awi\u0119ch and Kelome. We discuss several cases where the above necessary condition is also sufficient.<\/jats:p>","DOI":"10.1137\/060674284","type":"journal-article","created":{"date-parts":[[2008,1,22]],"date-time":"2008-01-22T15:11:16Z","timestamp":1201014676000},"page":"218-250","source":"Crossref","is-referenced-by-count":6,"title":["Controlled Stochastic Differential Equations under Constraints in Infinite Dimensional Spaces"],"prefix":"10.1137","volume":"47","author":[{"given":"Rainer","family":"Buckdahn","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marc","family":"Quincampoix","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gianmario","family":"Tessitore","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,1,22]]},"reference":[{"key":"R1","unstructured":"J.P. Aubin,\n                      Viability Theory.\n                      Birkh\u00e4user Boston, Boston, 1992."},{"key":"R2","unstructured":"J.P.Aubin and G. Da Prato,\n                      Stochastic viability and invariance\n                      , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 17 (1990) pp. 595\u2013613."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"J.P. Aubin and G. Da Prato,\n                      Stochastic Nagumo's viability theorem\n                      , Stochastic Anal. Appl., 13 (1995) pp. 1\u201311.","DOI":"10.1080\/07362999508809379"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"J.P. Aubin and G. Da Prato,\n                      The viability theorem for stochastic differential inclusion\n                      , Stochastic Anal. Appl., 16 (1998) 1\u201315.","DOI":"10.1080\/07362999808809512"},{"key":"R5","unstructured":"J.P. Aubin and H. Frankowska,\n                      Set-Valued Analysis\n                      , Systems Control Found. Appl. 2, Birkh\u00e4user Boston, Boston, 1990."},{"key":"R6","unstructured":"D. P. Bertsekas and S. E. Shreve,\n                      Stochastic Optimal Control: The Discrete Time Case\n                      , Academic Press, New York, 1978."},{"key":"R7","doi-asserted-by":"crossref","unstructured":"R. Buckdahn, S. Peng, M. Quincampoix, and C. Rainer,\n                      Existence of stochastic control under state constraints\n                      , C. R. Acad. Sci. Paris S\u00e9r. I Math. 327, (1998), pp. 17\u201322.","DOI":"10.1016\/S0764-4442(98)80096-7"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141000380334"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"R. Buckdahn, M. Quincampoix, C. Rainer, and R. Rascanu,\n                      Viability of moving sets for stochastic differential equation\n                      , Adv. Differential Equations, 7 (2002),pp. 1045\u20131072.","DOI":"10.57262\/ade\/1367241459"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/s00245-003-0789-z"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(90)90084-X"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"G. Da Prato and J. Zabczyk,\n                      Stochastic Equations in Infinite Dimensions\n                      , Cambridge University Press, Cambridge, UK, 1992.","DOI":"10.1017\/CBO9780511666223"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"G. Da Prato and J. Zabczyk,\n                      Ergodicity for Infinite-Dimensional Systems\n                      , London Math. Soc. Lecture Note Ser. 229, Cambridge University Press, Cambridge, UK, 1996.","DOI":"10.1017\/CBO9780511662829"},{"key":"R14","unstructured":"E. B. Davies,\n                      One-Parameter Semigroups\n                      , Academic Press, London, 1980."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1080\/17442508708833443"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1029867132"},{"key":"R17","doi-asserted-by":"crossref","first-page":"1395","DOI":"10.57262\/die\/1370019765","volume":"6","author":"Gautier S.","year":"1993","journal-title":"Differential Integral Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0893-4983","issn-type":"print"},{"key":"R18","unstructured":"D. Gatarek,\n                      Existence of optimal controls for stochastic evolution systems\n                      , in Control of Partial Differential Equations, Lecture Notes in Pure and Appl. Math. 165, G. Da Prato et al., eds., Marcel Dekker, New York, 1994, pp. 81\u201386."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012992226260"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1080\/03605309308820943"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392299"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"P. L. Lions,\n                      Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions,\n                      II\n                      . Optimal control of Zakai's equation\n                      , in Stochastic Partial Differential Equations and Applications, II, Lecture Notes in Math. 1390, G. Da Prato and L. Tubaro eds., Springer, Berlin, 1989, pp. 147\u2013170.","DOI":"10.1007\/BFb0083943"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/0022-1236(89)90062-1"},{"key":"R24","unstructured":"D. A. Kelome,\n                      Viscosity Solutions of Second-Order Equations in a Separable Hilbert Space and Applications to Stochastic Optimal Control\n                      , Ph.D. thesis, Georgia Institute of Technology, Atlanta, GA, 2002."},{"key":"R25","doi-asserted-by":"crossref","unstructured":"D. A. Kelome and A. \u015awi\u0119ch,\n                      Viscosity solutions of an infinite-dimensional Black-Scholes-Barenblatt equation\n                      , Appl. Math. Optim., 47 (2003), 253\u2013278.","DOI":"10.1007\/s00245-003-0764-8"},{"key":"R26","first-page":"551","volume":"24","author":"Nagumo M.","year":"1942","journal-title":"Proc. Phys.-Math. Soc. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0370-1239","issn-type":"print"},{"key":"R27","first-page":"313","volume":"11","author":"Nakayama T.","year":"2004","journal-title":"J. Math. Sci. Univ. 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