几何波方程的半全局可控性

IF 0.5 4区数学 Q3 MATHEMATICS

Pure and Applied Mathematics Quarterly Pub Date : 2024-07-18 DOI:10.4310/pamq.2024.v20.n4.a9

Joachim Krieger, Shengquan Xiang

{"title":"几何波方程的半全局可控性","authors":"Joachim Krieger, Shengquan Xiang","doi":"10.4310/pamq.2024.v20.n4.a9","DOIUrl":null,"url":null,"abstract":"We prove the semi-global controllability and stabilization of the $(1 + 1)$-dimensional wave maps equation with spatial domain $\\mathbb{S}^1$ and target $\\mathbb{S}^k$. First, we show that damping stabilizes the system when the energy is strictly below the threshold $2\\pi$, where harmonic maps appear as obstruction for global stabilization. Then, we adapt an iterative control procedure to get low-energy exact controllability of the wave maps equation. This result is optimal in the case $k = 1$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Semi-global controllability of a geometric wave equation\",\"authors\":\"Joachim Krieger, Shengquan Xiang\",\"doi\":\"10.4310/pamq.2024.v20.n4.a9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove the semi-global controllability and stabilization of the $(1 + 1)$-dimensional wave maps equation with spatial domain $\\\\mathbb{S}^1$ and target $\\\\mathbb{S}^k$. First, we show that damping stabilizes the system when the energy is strictly below the threshold $2\\\\pi$, where harmonic maps appear as obstruction for global stabilization. Then, we adapt an iterative control procedure to get low-energy exact controllability of the wave maps equation. This result is optimal in the case $k = 1$.\",\"PeriodicalId\":54526,\"journal\":{\"name\":\"Pure and Applied Mathematics Quarterly\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-07-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Pure and Applied Mathematics Quarterly\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/pamq.2024.v20.n4.a9\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Pure and Applied Mathematics Quarterly","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/pamq.2024.v20.n4.a9","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

我们证明了空间域为$\mathbb{S}^1$、目标为$\mathbb{S}^k$的$(1 + 1)$维波图方程的半全局可控性和稳定性。首先，我们证明当能量严格低于阈值 $2\pi$ 时，阻尼会使系统趋于稳定，此时谐波图会成为全局稳定的障碍。然后，我们采用迭代控制程序来获得波映射方程的低能量精确可控性。这一结果在 $k = 1$ 的情况下是最优的。

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

Semi-global controllability of a geometric wave equation

We prove the semi-global controllability and stabilization of the $(1 + 1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$. First, we show that damping stabilizes the system when the energy is strictly below the threshold $2\pi$, where harmonic maps appear as obstruction for global stabilization. Then, we adapt an iterative control procedure to get low-energy exact controllability of the wave maps equation. This result is optimal in the case $k = 1$.

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

Pure and Applied Mathematics Quarterly 数学-数学

CiteScore

0.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Publishes high-quality, original papers on all fields of mathematics. To facilitate fruitful interchanges between mathematicians from different regions and specialties, and to effectively disseminate new breakthroughs in mathematics, the journal welcomes well-written submissions from all significant areas of mathematics. The editors are committed to promoting the highest quality of mathematical scholarship.