半线性一维热方程的构造精确控制

IF 1 4区 数学 Q1 MATHEMATICS
Jérôme Lemoine, A. Munch
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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.
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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.

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来源期刊
Mathematical Control and Related Fields
Mathematical Control and Related Fields MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
2.50
自引率
8.30%
发文量
67
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.
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