{"title":"Riccati-based solution to the optimal control of linear evolution equations with finite memory","authors":"P. Acquistapace, F. Bucci","doi":"10.3934/eect.2023035","DOIUrl":null,"url":null,"abstract":"In this article we study the optimal control problem with quadratic functionals for a linear Volterra integro-differential equation in Hilbert spaces. With the finite history seen as an (additional) initial datum for the evolution, following the variational approach utilized in the study of the linear-quadratic problem for memoryless infinite dimensional systems, we attain a closed-loop form of the unique optimal control via certain operators that are shown to solve a coupled system of quadratic differential equations. This result provides a first extension to the partial differential equations realm of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context.","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"88 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2023-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Evolution Equations and Control Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/eect.2023035","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this article we study the optimal control problem with quadratic functionals for a linear Volterra integro-differential equation in Hilbert spaces. With the finite history seen as an (additional) initial datum for the evolution, following the variational approach utilized in the study of the linear-quadratic problem for memoryless infinite dimensional systems, we attain a closed-loop form of the unique optimal control via certain operators that are shown to solve a coupled system of quadratic differential equations. This result provides a first extension to the partial differential equations realm of the Riccati-based theory recently devised by L. Pandolfi in a finite dimensional context.
期刊介绍：
EECT is primarily devoted to papers on analysis and control of infinite dimensional systems with emphasis on applications to PDE''s and FDEs. Topics include:
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* Direct problems such as existence, regularity and well-posedness
* Stability, long-time behavior and associated dynamical attractors
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* Well-posedness, stability and control of coupled systems with an interface. Free boundary problems and problems with moving interface(s)
* Applications of the theory to physics, chemistry, engineering, economics, medicine and biology
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