非线性最优控制问题中hp谱元方法的误差估计

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Nonlinear Science Pub Date : 2023-12-01 DOI:10.1007/s00332-023-09991-1

Xiuxiu Lin, Yanping Chen, Yunqing Huang

{"title":"非线性最优控制问题中hp谱元方法的误差估计","authors":"Xiuxiu Lin, Yanping Chen, Yunqing Huang","doi":"10.1007/s00332-023-09991-1","DOIUrl":null,"url":null,"abstract":"The main purpose of this paper is to discuss hp spectral element method for optimal control problem governed by a nonlinear elliptic equation with \\(L^2\\)-norm constraint for control variable. We then set up its weak formulation and hp spectral element approximation scheme. A priori error estimates of hp spectral element approximation based on some suitable projection operators are proved carefully. Using some properties of projection operators, a posteriori error estimates for both the state and the control approximation under some reasonable assumptions are established rigorously. Such estimates are useful tools, which can be used to construct reliable adaptive spectral element methods for optimal control problems.","PeriodicalId":50111,"journal":{"name":"Journal of Nonlinear Science","volume":"78 ","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Error Estimates of hp Spectral Element Methods in Nonlinear Optimal Control Problem\",\"authors\":\"Xiuxiu Lin, Yanping Chen, Yunqing Huang\",\"doi\":\"10.1007/s00332-023-09991-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The main purpose of this paper is to discuss hp spectral element method for optimal control problem governed by a nonlinear elliptic equation with \\\\(L^2\\\\)-norm constraint for control variable. We then set up its weak formulation and hp spectral element approximation scheme. A priori error estimates of hp spectral element approximation based on some suitable projection operators are proved carefully. Using some properties of projection operators, a posteriori error estimates for both the state and the control approximation under some reasonable assumptions are established rigorously. Such estimates are useful tools, which can be used to construct reliable adaptive spectral element methods for optimal control problems.\",\"PeriodicalId\":50111,\"journal\":{\"name\":\"Journal of Nonlinear Science\",\"volume\":\"78 \",\"pages\":\"\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2023-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Nonlinear Science\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00332-023-09991-1\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Nonlinear Science","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00332-023-09991-1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

本文的主要目的是讨论控制变量为\(L^2\) -范数约束的非线性椭圆方程的最优控制问题的hp谱元方法。然后建立了它的弱公式和hp谱元近似格式。仔细地证明了基于一些合适的投影算子的hp谱元逼近的先验误差估计。利用投影算子的一些性质，在一些合理的假设条件下，严格地建立了状态和控制逼近的后验误差估计。这种估计是有用的工具，可用于构建可靠的自适应谱元方法来解决最优控制问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Error Estimates of hp Spectral Element Methods in Nonlinear Optimal Control Problem

The main purpose of this paper is to discuss hp spectral element method for optimal control problem governed by a nonlinear elliptic equation with \(L^2\)-norm constraint for control variable. We then set up its weak formulation and hp spectral element approximation scheme. A priori error estimates of hp spectral element approximation based on some suitable projection operators are proved carefully. Using some properties of projection operators, a posteriori error estimates for both the state and the control approximation under some reasonable assumptions are established rigorously. Such estimates are useful tools, which can be used to construct reliable adaptive spectral element methods for optimal control problems.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal of Nonlinear Science 数学-力学

CiteScore

5.00

自引率

3.30%

发文量

审稿时长

4.5 months

期刊介绍： The mission of the Journal of Nonlinear Science is to publish papers that augment the fundamental ways we describe, model, and predict nonlinear phenomena. Papers should make an original contribution to at least one technical area and should in addition illuminate issues beyond that area''s boundaries. Even excellent papers in a narrow field of interest are not appropriate for the journal. Papers can be oriented toward theory, experimentation, algorithms, numerical simulations, or applications as long as the work is creative and sound. Excessively theoretical work in which the application to natural phenomena is not apparent (at least through similar techniques) or in which the development of fundamental methodologies is not present is probably not appropriate. In turn, papers oriented toward experimentation, numerical simulations, or applications must not simply report results without an indication of what a theoretical explanation might be. All papers should be submitted in English and must meet common standards of usage and grammar. In addition, because ours is a multidisciplinary subject, at minimum the introduction to the paper should be readable to a broad range of scientists and not only to specialists in the subject area. The scientific importance of the paper and its conclusions should be made clear in the introduction-this means that not only should the problem you study be presented, but its historical background, its relevance to science and technology, the specific phenomena it can be used to describe or investigate, and the outstanding open issues related to it should be explained. Failure to achieve this could disqualify the paper.