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Control Optim."],"published-print":{"date-parts":[[2026,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>This work establishes a general stochastic maximum principle for systems of partially observed optimal control of semilinear stochastic partial differential equations in a nonconvex control domain.\u00a0The state evolves in a Hilbert space driven by a cylindrical Wiener process and finitely many Brownian motions, while observations are in a Euclidean space having correlated noise.\u00a0For the convex control domain and diffusion coefficients in the state being control-independent, numerical algorithms are developed to solve the partially observed optimal control problems using a stochastic gradient descent algorithm combined with finite element approximations and the branching filtering algorithm. Numerical experiments are conducted for demonstration.<\/jats:p>","DOI":"10.1137\/25m1752912","type":"journal-article","created":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T07:14:23Z","timestamp":1780643663000},"page":"1800-1829","source":"Crossref","is-referenced-by-count":1,"title":["Optimal Control of Stochastic Partial Differential Equations with Partial Observations: Stochastic Maximum Principles and Numerical Approximation"],"prefix":"10.1137","volume":"64","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2310-3435","authenticated-orcid":true,"given":"Yanzhao","family":"Cao","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7139-6407","authenticated-orcid":true,"given":"Hongjiang","family":"Qian","sequence":"additional","affiliation":[{"name":"Corresponding author. Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"George","family":"Yin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Connecticut, Storrs, CT 06269 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2026,6,5]]},"reference":[{"key":"ref1","series-title":"Pure Appl. Math. (Amst.) 140","volume-title":"Sobolev Spaces","author":"Adams R. 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