Weak Solutions to the Riccati Equation and the Application in the Closed-Loop Control

IF 2.6 2区 数学 Q1 MATHEMATICS, APPLIED
Deqin Su, Xiaoying Wang, Yong Li
{"title":"Weak Solutions to the Riccati Equation and the Application in the Closed-Loop Control","authors":"Deqin Su,&nbsp;Xiaoying Wang,&nbsp;Yong Li","doi":"10.1111/sapm.70078","DOIUrl":null,"url":null,"abstract":"<div>\n \n <p>In this paper, we establish the existence of a weak solution to the Riccati equation via the canonical Hamiltonian formulation and Ekeland variational principle and present an application in the closed-loop control. It is well-known that the general Riccati equation admits no classical solution due to its blow-up behavior. Nevertheless, by introducing a residual <span></span><math>\n <semantics>\n <mi>μ</mi>\n <annotation>$\\mu$</annotation>\n </semantics></math> through the combined application of the canonical Hamiltonian formulation and the Ekeland variational principle, we observe that under appropriate conditions, the weak solution to the Riccati equation exists. Initially, we derive the Hamilton–Jacobi equation from the Riccati equation incorporating the residual <span></span><math>\n <semantics>\n <mi>μ</mi>\n <annotation>$\\mu$</annotation>\n </semantics></math>, utilizing the canonical Hamiltonian formalism. Subsequently, we elucidate the relationship between the viscosity solution <span></span><math>\n <semantics>\n <msup>\n <mi>S</mi>\n <mo>∗</mo>\n </msup>\n <annotation>$\\mathcal {S}^*$</annotation>\n </semantics></math> of the Hamilton–Jacobi equation and the residual <span></span><math>\n <semantics>\n <mi>μ</mi>\n <annotation>$\\mu$</annotation>\n </semantics></math>, thereby justifying the introduction of <span></span><math>\n <semantics>\n <mi>μ</mi>\n <annotation>$\\mu$</annotation>\n </semantics></math> and establishing the existence of the weak solution. Finally, we present the application of the Riccati equation with the residual <span></span><math>\n <semantics>\n <mi>μ</mi>\n <annotation>$\\mu$</annotation>\n </semantics></math> in a closed-loop control setting, thereby further substantiating the existence of the weak solution.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Studies in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/sapm.70078","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

Abstract

In this paper, we establish the existence of a weak solution to the Riccati equation via the canonical Hamiltonian formulation and Ekeland variational principle and present an application in the closed-loop control. It is well-known that the general Riccati equation admits no classical solution due to its blow-up behavior. Nevertheless, by introducing a residual μ $\mu$ through the combined application of the canonical Hamiltonian formulation and the Ekeland variational principle, we observe that under appropriate conditions, the weak solution to the Riccati equation exists. Initially, we derive the Hamilton–Jacobi equation from the Riccati equation incorporating the residual μ $\mu$ , utilizing the canonical Hamiltonian formalism. Subsequently, we elucidate the relationship between the viscosity solution S $\mathcal {S}^*$ of the Hamilton–Jacobi equation and the residual μ $\mu$ , thereby justifying the introduction of μ $\mu$ and establishing the existence of the weak solution. Finally, we present the application of the Riccati equation with the residual μ $\mu$ in a closed-loop control setting, thereby further substantiating the existence of the weak solution.

Riccati方程的弱解及其在闭环控制中的应用
本文利用正则哈密顿公式和Ekeland变分原理,建立了Riccati方程弱解的存在性,并给出了它在闭环控制中的应用。众所周知,一般里卡第方程由于其爆破性质,不存在经典解。然而,通过联合应用正则哈密顿公式和Ekeland变分原理引入残差μ $\mu$,我们观察到在适当条件下Riccati方程存在弱解。首先,我们利用正则哈密顿形式,从包含残差μ $\mu$的Riccati方程推导出哈密顿-雅可比方程。随后,我们阐明了Hamilton-Jacobi方程的黏度解S∗$\mathcal {S}^*$与残差μ $\mu$之间的关系,从而证明了μ $\mu$的引入,并建立了弱解的存在性。最后,我们给出了带有残差μ $\mu$的Riccati方程在闭环控制设置中的应用,从而进一步证明了弱解的存在性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Studies in Applied Mathematics
Studies in Applied Mathematics 数学-应用数学
CiteScore
4.30
自引率
3.70%
发文量
66
审稿时长
>12 weeks
期刊介绍: Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信