{"title":"Optimality Conditions for Parabolic Stochastic Optimal Control Problems with Boundary Controls","authors":"Piero Visconti","doi":"10.1007/s00245-024-10204-8","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we study optimality conditions for a class of control problems driven by a cylindrical Wiener process, resulting in a stochastic maximum principle in differential form. The control acts on both the drift and volatility, potentially as unbounded operators, allowing for SPDEs with boundary control and/or noise. Through the factorization method, we establish a regularity property for the state equation, which, by duality, extends to the backward costate equation, understood in the transposition sense. Finally, we show that the cost functional is Gâteaux differentiable, with its derivative represented by the costate. The optimality condition is derived using results from set-valued analysis.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 1","pages":""},"PeriodicalIF":1.6000,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Optimization","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00245-024-10204-8","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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Abstract
In this paper, we study optimality conditions for a class of control problems driven by a cylindrical Wiener process, resulting in a stochastic maximum principle in differential form. The control acts on both the drift and volatility, potentially as unbounded operators, allowing for SPDEs with boundary control and/or noise. Through the factorization method, we establish a regularity property for the state equation, which, by duality, extends to the backward costate equation, understood in the transposition sense. Finally, we show that the cost functional is Gâteaux differentiable, with its derivative represented by the costate. The optimality condition is derived using results from set-valued analysis.
期刊介绍：
The Applied Mathematics and Optimization Journal covers a broad range of mathematical methods in particular those that bridge with optimization and have some connection with applications. Core topics include calculus of variations, partial differential equations, stochastic control, optimization of deterministic or stochastic systems in discrete or continuous time, homogenization, control theory, mean field games, dynamic games and optimal transport. Algorithmic, data analytic, machine learning and numerical methods which support the modeling and analysis of optimization problems are encouraged. Of great interest are papers which show some novel idea in either the theory or model which include some connection with potential applications in science and engineering.
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