{"title":"Optimal control problem governed by a kind of Kirchhoff-type equation","authors":"","doi":"10.1016/j.chaos.2024.115422","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, we consider an optimal control problem governed by a kind of Kirchhoff-type equation, which plays an important role in the phenomenon of beam vibration. Firstly, the existence of solution to the state equation is proved by the variational method. Secondly, for the given cost functional, we get that there exists at least an optimal state-control pair via the Sobolev’s embedding theorem under the constraint of the state equation. Next, the necessary optimality condition for the optimal solution is derived by using the cone method. Finally, we give the pointwise variational inequality, minimum principles and an equivalent necessary condition for the optimal control problem according to the discussion of the variational inequality.</p></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":null,"pages":null},"PeriodicalIF":5.3000,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Chaos Solitons & Fractals","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0960077924009743","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we consider an optimal control problem governed by a kind of Kirchhoff-type equation, which plays an important role in the phenomenon of beam vibration. Firstly, the existence of solution to the state equation is proved by the variational method. Secondly, for the given cost functional, we get that there exists at least an optimal state-control pair via the Sobolev’s embedding theorem under the constraint of the state equation. Next, the necessary optimality condition for the optimal solution is derived by using the cone method. Finally, we give the pointwise variational inequality, minimum principles and an equivalent necessary condition for the optimal control problem according to the discussion of the variational inequality.
期刊介绍：
Chaos, Solitons & Fractals strives to establish itself as a premier journal in the interdisciplinary realm of Nonlinear Science, Non-equilibrium, and Complex Phenomena. It welcomes submissions covering a broad spectrum of topics within this field, including dynamics, non-equilibrium processes in physics, chemistry, and geophysics, complex matter and networks, mathematical models, computational biology, applications to quantum and mesoscopic phenomena, fluctuations and random processes, self-organization, and social phenomena.
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