{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,16]],"date-time":"2026-04-16T18:42:53Z","timestamp":1776364973078,"version":"3.51.2"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,5,27]],"date-time":"2023-05-27T00:00:00Z","timestamp":1685145600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,27]],"date-time":"2023-05-27T00:00:00Z","timestamp":1685145600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003407","name":"Ministero dell\u2019Istruzione, dell\u2019Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","award":["2017FKHBA8 001"],"award-info":[{"award-number":["2017FKHBA8 001"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we study a first extension of the theory of mild solutions for Hamilton\u2013Jacobi\u2013Bellman (HJB) equations in Hilbert spaces to the case where the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear HJB equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, which are continuously differentiable in the space variable. We also provide an application of our results to an exit-time optimal control problem, and we show that the corresponding value function is the unique solution to a semilinear HJB equation, possessing sufficient regularity to express the optimal control in feedback form. Finally, we give an illustrative example.<\/jats:p>","DOI":"10.1007\/s10957-023-02208-1","type":"journal-article","created":{"date-parts":[[2023,5,27]],"date-time":"2023-05-27T10:01:47Z","timestamp":1685181707000},"page":"710-744","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["HJB Equations and Stochastic Control on Half-Spaces of Hilbert Spaces"],"prefix":"10.1007","volume":"198","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4448-6877","authenticated-orcid":false,"given":"Alessandro","family":"Calvia","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gianluca","family":"Cappa","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fausto","family":"Gozzi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrico","family":"Priola","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,5,27]]},"reference":[{"key":"2208_CR1","doi-asserted-by":"crossref","unstructured":"Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton\u2013Jacobi\u2013Bellman Equations. Foundations and Applications. Birkh\u00e4user, Boston, Systems and Control (1997)","DOI":"10.1007\/978-0-8176-4755-1"},{"key":"2208_CR2","unstructured":"Biagini, S., Biffis, E., Gozzi, F., Zanella, M.: Wage rigidity and retirement in optimal portfolio choice. Preprint, arXiv:2101.09732 (2021)"},{"issue":"3","key":"2208_CR3","doi-asserted-by":"publisher","first-page":"1004","DOI":"10.1137\/21M1429722","volume":"13","author":"S Biagini","year":"2022","unstructured":"Biagini, S., Gozzi, F., Zanella, M.: Robust portfolio choice with sticky wages. SIAM J. Financ. Math. 13(3), 1004\u20131039 (2022)","journal-title":"SIAM J. Financ. Math."},{"issue":"4","key":"2208_CR4","doi-asserted-by":"publisher","first-page":"1906","DOI":"10.1137\/19M1259687","volume":"58","author":"E Biffis","year":"2020","unstructured":"Biffis, E., Gozzi, F., Prosdocimi, C.: Optimal portfolio choice with path dependent labor income: the infinite horizon case. SIAM J. Control Optim. 58(4), 1906\u20131938 (2020)","journal-title":"SIAM J. Control Optim."},{"issue":"6","key":"2208_CR5","doi-asserted-by":"publisher","first-page":"2719","DOI":"10.1016\/j.jet.2013.09.013","volume":"148","author":"R Boucekkine","year":"2013","unstructured":"Boucekkine, R., Camacho, C., Fabbri, G.: Spatial dynamics and convergence: the spatial AK model. J. Econ. Theory 148(6), 2719\u20132736 (2013). https:\/\/doi.org\/10.1016\/j.jet.2013.09.013","journal-title":"J. Econ. Theory"},{"issue":"6","key":"2208_CR6","doi-asserted-by":"publisher","first-page":"1287","DOI":"10.1093\/jeg\/lby041","volume":"19","author":"R Boucekkine","year":"2018","unstructured":"Boucekkine, R., Fabbri, G., Federico, S., Gozzi, F.: Growth and agglomeration in the heterogeneous space: a generalized AK approach. J. Econ. Geogr. 19(6), 1287\u20131318 (2018). https:\/\/doi.org\/10.1093\/jeg\/lby041","journal-title":"J. Econ. Geogr."},{"issue":"6","key":"2208_CR7","doi-asserted-by":"publisher","first-page":"4481","DOI":"10.1137\/20M1376959","volume":"59","author":"A Calvia","year":"2021","unstructured":"Calvia, A., Federico, S., Gozzi, F.: State constrained control problems in Banach lattices and applications. SIAM J. Control Optim. 59(6), 4481\u20134510 (2021)","journal-title":"SIAM J. Control Optim."},{"key":"2208_CR8","volume-title":"Semiconcave functions, Hamilton\u2013Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and Their Applications","author":"P Cannarsa","year":"2004","unstructured":"Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton\u2013Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and Their Applications. Birkh\u00e4user, Boston (2004)"},{"issue":"11\u201312","key":"2208_CR9","doi-asserted-by":"publisher","first-page":"1997","DOI":"10.1080\/03605300008821576","volume":"25","author":"MG Crandall","year":"2000","unstructured":"Crandall, M.G., Kocan, M., Swiech, A.: L-P-Theory for fully nonlinear uniformly parabolic equations. Commun. Partial Differ. Equ. 25(11\u201312), 1997\u20132053 (2000)","journal-title":"Commun. Partial Differ. Equ."},{"issue":"3","key":"2208_CR10","first-page":"183","volume":"8","author":"G Da Prato","year":"1997","unstructured":"Da Prato, G., Zabczyk, J.: Differentiability of the Feynman-Kac semigroup and a control application. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 8(3), 183\u2013188 (1997)","journal-title":"Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl."},{"key":"2208_CR11","volume-title":"Second order partial differential equations in Hilbert spaces. London Mathematical Society Lecture Note Series","author":"G Da Prato","year":"2002","unstructured":"Da Prato, G., Zabczyk, J.: Second order partial differential equations in Hilbert spaces. London Mathematical Society Lecture Note Series, vol. 293. Cambridge University Press, Cambridge (2002)"},{"key":"2208_CR12","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107295513","volume-title":"Stochastic equations in infinite dimensions, volume 152 of Encyclopedia of Mathematics and its Applications","author":"G Da Prato","year":"2014","unstructured":"Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions, volume 152 of Encyclopedia of Mathematics and its Applications, 2nd edn. Cambridge University Press, Cambridge (2014)","edition":"2"},{"issue":"4","key":"2208_CR13","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1016\/S0764-4442(97)85631-5","volume":"325","author":"G Da Prato","year":"1997","unstructured":"Da Prato, G., Goldys, B., Zabczyk, J.: Ornstein\u2013Uhlenbeck semigroups in open sets of Hilbert spaces. C. R. Acad. Sci. Paris S\u00e9r. I Math. 325(4), 433\u2013438 (1997)","journal-title":"C. R. Acad. Sci. Paris S\u00e9r. I Math."},{"key":"2208_CR14","doi-asserted-by":"publisher","first-page":"48","DOI":"10.1016\/j.spa.2021.11.010","volume":"145","author":"B Djehiche","year":"2022","unstructured":"Djehiche, B., Gozzi, F., Zanco, G., Zanella, M.: Optimal portfolio choice with path dependent benchmarked labor income: a mean field model. Stochastic Process. Appl. 145, 48\u201385 (2022)","journal-title":"Stochastic Process. Appl."},{"key":"2208_CR15","series-title":"Textbooks in Mathematics","doi-asserted-by":"publisher","DOI":"10.1201\/b18333","volume-title":"Measure Theory and Fine Properties of Functions","author":"LC Evans","year":"2015","unstructured":"Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Textbooks in Mathematics, revised CRC Press, Boca Raton, FL (2015)","edition":"revised"},{"key":"2208_CR16","unstructured":"Fabbri, G., Gozzi, F., Swiech, A.: Stochastic optimal control in infinite dimension, volume\u00a082 of Probability Theory and Stochastic Modelling. Springer, Cham (2017). Dynamic programming and HJB equations, With a contribution by Marco Fuhrman and Gianmario Tessitore"},{"issue":"2","key":"2208_CR17","doi-asserted-by":"publisher","first-page":"620","DOI":"10.1137\/21M1414206","volume":"60","author":"F Gozzi","year":"2022","unstructured":"Gozzi, F., Leocata, M.: A stochastic model of economic growth in time-space. SIAM J. Control Optim. 60(2), 620\u2013651 (2022)","journal-title":"SIAM J. Control Optim."},{"key":"2208_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4260-4","volume-title":"Optimal Control Theory for Infinite-Dimensional Systems. Systems & Control: Foundations & Applications","author":"XJ Li","year":"1995","unstructured":"Li, X.J., Yong, J.M.: Optimal Control Theory for Infinite-Dimensional Systems. Systems & Control: Foundations & Applications. Birkh\u00e4user Boston Inc, Boston, MA (1995). https:\/\/doi.org\/10.1007\/978-1-4612-4260-4"},{"key":"2208_CR19","unstructured":"Priola, E.: Partial differential equations with infinitely many variables. PhD thesis, University of Milan, (1999). http:\/\/hdl.handle.net\/2318\/1559581"},{"key":"2208_CR20","unstructured":"Priola, E.: The fundamental solution for a degenerate parabolic Dirichlet problem. In: Partial Differential Equations (Torino, 2000), pp. 137\u2013148. Univ. Torino, Turin (2000)"},{"issue":"5, Ser. A: Theo","key":"2208_CR21","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1016\/S0362-546X(99)00300-4","volume":"44","author":"E Priola","year":"2001","unstructured":"Priola, E.: Schauder estimates for a homogeneous Dirichlet problem in a half-space of a Hilbert space. Nonlinear Anal. 44(5, Ser. A: Theory Methods), 679\u2013702 (2001)","journal-title":"Nonlinear Anal."},{"issue":"2","key":"2208_CR22","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1142\/S0219025702000729","volume":"5","author":"E Priola","year":"2002","unstructured":"Priola, E.: Dirichlet problems in a half-space of a Hilbert space. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5(2), 257\u2013291 (2002)","journal-title":"Infin. Dimens. Anal. Quantum Probab. Relat. Top."},{"issue":"3","key":"2208_CR23","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1023\/A:1020933325029","volume":"18","author":"E Priola","year":"2003","unstructured":"Priola, E.: On a Dirichlet problem involving an Ornstein\u2013Uhlenbeck operator. Potential Anal. 18(3), 251\u2013287 (2003)","journal-title":"Potential Anal."},{"issue":"3","key":"2208_CR24","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1137\/0324032","volume":"24","author":"HM Soner","year":"1986","unstructured":"Soner, H.M.: Optimal control with state-space constraint. I. SIAM J. Control Optim. 24(3), 552\u2013561 (1986)","journal-title":"SIAM J. Control Optim."},{"issue":"6","key":"2208_CR25","doi-asserted-by":"publisher","first-page":"1110","DOI":"10.1137\/0324067","volume":"24","author":"HM Soner","year":"1986","unstructured":"Soner, H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim. 24(6), 1110\u20131122 (1986)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"2208_CR26","doi-asserted-by":"publisher","first-page":"109","DOI":"10.4064\/sm-141-2-109-142","volume":"141","author":"A Talarczyk","year":"2000","unstructured":"Talarczyk, A.: Dirichlet problem for parabolic equations on Hilbert spaces. Studia Math. 141(2), 109\u2013142 (2000)","journal-title":"Studia Math."},{"issue":"1A","key":"2208_CR27","doi-asserted-by":"publisher","first-page":"424","DOI":"10.1214\/aop\/1078415841","volume":"32","author":"F-Y Wang","year":"2004","unstructured":"Wang, F.-Y.: Gradient estimates of Dirichlet heat semigroups and application to isoperimetric inequalities. Ann. Probab. 32(1A), 424\u2013440 (2004)","journal-title":"Ann. Probab."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-023-02208-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-023-02208-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-023-02208-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,30]],"date-time":"2023-08-30T18:08:45Z","timestamp":1693418925000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-023-02208-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,27]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,8]]}},"alternative-id":["2208"],"URL":"https:\/\/doi.org\/10.1007\/s10957-023-02208-1","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,27]]},"assertion":[{"value":"29 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 March 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}