{"title":"半线性一维热方程的构造精确控制","authors":"Jérôme Lemoine, A. Munch","doi":"10.3934/mcrf.2022001","DOIUrl":null,"url":null,"abstract":"<p style='text-indent:20px;'>The exact distributed controllability of the semilinear heat equation <inline-formula><tex-math id=\"M1\">\\begin{document}$ \\partial_{t}y-\\Delta y + g(y) = f \\,1_{\\omega} $\\end{document}</tex-math></inline-formula> posed over multi-dimensional and bounded domains, assuming that <inline-formula><tex-math id=\"M2\">\\begin{document}$ g\\in C^1(\\mathbb{R}) $\\end{document}</tex-math></inline-formula> satisfies the growth condition <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\limsup_{r\\to \\infty} g(r)/ (\\vert r\\vert \\ln^{3/2}\\vert r\\vert) = 0 $\\end{document}</tex-math></inline-formula> has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that <inline-formula><tex-math id=\"M4\">\\begin{document}$ g^\\prime $\\end{document}</tex-math></inline-formula> does not grow faster than <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\beta \\ln^{3/2}\\vert r\\vert $\\end{document}</tex-math></inline-formula> at infinity for <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\beta>0 $\\end{document}</tex-math></inline-formula> small enough and that <inline-formula><tex-math id=\"M7\">\\begin{document}$ g^\\prime $\\end{document}</tex-math></inline-formula> is uniformly Hölder continuous on <inline-formula><tex-math id=\"M8\">\\begin{document}$ \\mathbb{R} $\\end{document}</tex-math></inline-formula> with exponent <inline-formula><tex-math id=\"M9\">\\begin{document}$ p\\in [0,1] $\\end{document}</tex-math></inline-formula>, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order <inline-formula><tex-math id=\"M10\">\\begin{document}$ 1+p $\\end{document}</tex-math></inline-formula> after a finite number of iterations.</p>","PeriodicalId":48889,"journal":{"name":"Mathematical Control and Related Fields","volume":"43 3 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Constructive exact control of semilinear 1D heat equations\",\"authors\":\"Jérôme Lemoine, A. Munch\",\"doi\":\"10.3934/mcrf.2022001\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p style='text-indent:20px;'>The exact distributed controllability of the semilinear heat equation <inline-formula><tex-math id=\\\"M1\\\">\\\\begin{document}$ \\\\partial_{t}y-\\\\Delta y + g(y) = f \\\\,1_{\\\\omega} $\\\\end{document}</tex-math></inline-formula> posed over multi-dimensional and bounded domains, assuming that <inline-formula><tex-math id=\\\"M2\\\">\\\\begin{document}$ g\\\\in C^1(\\\\mathbb{R}) $\\\\end{document}</tex-math></inline-formula> satisfies the growth condition <inline-formula><tex-math id=\\\"M3\\\">\\\\begin{document}$ \\\\limsup_{r\\\\to \\\\infty} g(r)/ (\\\\vert r\\\\vert \\\\ln^{3/2}\\\\vert r\\\\vert) = 0 $\\\\end{document}</tex-math></inline-formula> has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that <inline-formula><tex-math id=\\\"M4\\\">\\\\begin{document}$ g^\\\\prime $\\\\end{document}</tex-math></inline-formula> does not grow faster than <inline-formula><tex-math id=\\\"M5\\\">\\\\begin{document}$ \\\\beta \\\\ln^{3/2}\\\\vert r\\\\vert $\\\\end{document}</tex-math></inline-formula> at infinity for <inline-formula><tex-math id=\\\"M6\\\">\\\\begin{document}$ \\\\beta>0 $\\\\end{document}</tex-math></inline-formula> small enough and that <inline-formula><tex-math id=\\\"M7\\\">\\\\begin{document}$ g^\\\\prime $\\\\end{document}</tex-math></inline-formula> is uniformly Hölder continuous on <inline-formula><tex-math id=\\\"M8\\\">\\\\begin{document}$ \\\\mathbb{R} $\\\\end{document}</tex-math></inline-formula> with exponent <inline-formula><tex-math id=\\\"M9\\\">\\\\begin{document}$ p\\\\in [0,1] $\\\\end{document}</tex-math></inline-formula>, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order <inline-formula><tex-math id=\\\"M10\\\">\\\\begin{document}$ 1+p $\\\\end{document}</tex-math></inline-formula> after a finite number of iterations.</p>\",\"PeriodicalId\":48889,\"journal\":{\"name\":\"Mathematical Control and Related Fields\",\"volume\":\"43 3 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2021-03-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Control and Related Fields\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.3934/mcrf.2022001\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Control and Related Fields","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/mcrf.2022001","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 6
摘要
The exact distributed controllability of the semilinear heat equation \begin{document}$ \partial_{t}y-\Delta y + g(y) = f \,1_{\omega} $\end{document} posed over multi-dimensional and bounded domains, assuming that \begin{document}$ g\in C^1(\mathbb{R}) $\end{document} satisfies the growth condition \begin{document}$ \limsup_{r\to \infty} g(r)/ (\vert r\vert \ln^{3/2}\vert r\vert) = 0 $\end{document} has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that \begin{document}$ g^\prime $\end{document} does not grow faster than \begin{document}$ \beta \ln^{3/2}\vert r\vert $\end{document} at infinity for \begin{document}$ \beta>0 $\end{document} small enough and that \begin{document}$ g^\prime $\end{document} is uniformly Hölder continuous on \begin{document}$ \mathbb{R} $\end{document} with exponent \begin{document}$ p\in [0,1] $\end{document}, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order \begin{document}$ 1+p $\end{document} after a finite number of iterations.
Constructive exact control of semilinear 1D heat equations
The exact distributed controllability of the semilinear heat equation \begin{document}$ \partial_{t}y-\Delta y + g(y) = f \,1_{\omega} $\end{document} posed over multi-dimensional and bounded domains, assuming that \begin{document}$ g\in C^1(\mathbb{R}) $\end{document} satisfies the growth condition \begin{document}$ \limsup_{r\to \infty} g(r)/ (\vert r\vert \ln^{3/2}\vert r\vert) = 0 $\end{document} has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. In the one dimensional setting, assuming that \begin{document}$ g^\prime $\end{document} does not grow faster than \begin{document}$ \beta \ln^{3/2}\vert r\vert $\end{document} at infinity for \begin{document}$ \beta>0 $\end{document} small enough and that \begin{document}$ g^\prime $\end{document} is uniformly Hölder continuous on \begin{document}$ \mathbb{R} $\end{document} with exponent \begin{document}$ p\in [0,1] $\end{document}, we design a constructive proof yielding an explicit sequence converging to a controlled solution for the semilinear equation, at least with order \begin{document}$ 1+p $\end{document} after a finite number of iterations.
期刊介绍:
MCRF aims to publish original research as well as expository papers on mathematical control theory and related fields. The goal is to provide a complete and reliable source of mathematical methods and results in this field. The journal will also accept papers from some related fields such as differential equations, functional analysis, probability theory and stochastic analysis, inverse problems, optimization, numerical computation, mathematical finance, information theory, game theory, system theory, etc., provided that they have some intrinsic connections with control theory.