{"title":"Exact controllability of semilinear heat equations through a constructive approach","authors":"S. Ervedoza, Jérôme Lemoine, A. Münch","doi":"10.3934/eect.2022042","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 + f(y) = v \\, 1_{\\omega} $\\end{document}</tex-math></inline-formula> posed over multi-dimensional and bounded domains, assuming that <inline-formula><tex-math id=\"M2\">\\begin{document}$ f $\\end{document}</tex-math></inline-formula> is locally Lipschitz continuous and satisfies the growth condition <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\limsup_{| r|\\to \\infty} | f(r)| /(| r| \\ln^{3/2}| r|)\\leq \\beta $\\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\beta $\\end{document}</tex-math></inline-formula> small enough has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. Under the same assumption, by introducing a different fixed point application, we present a different and somewhat simpler proof of the exact controllability, which is not based on the cost of observability of the heat equation with respect to potentials. Then, assuming that <inline-formula><tex-math id=\"M5\">\\begin{document}$ f $\\end{document}</tex-math></inline-formula> is locally Lipschitz continuous and satisfies the growth condition <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\limsup_{| r|\\to \\infty} | f^\\prime(r)|/\\ln^{3/2}| r|\\leq \\beta $\\end{document}</tex-math></inline-formula> for some <inline-formula><tex-math id=\"M7\">\\begin{document}$ \\beta $\\end{document}</tex-math></inline-formula> small enough, we show that the above fixed point application is contracting yielding a constructive method to compute the controls for the semilinear equation. Numerical experiments illustrate the results.</p>","PeriodicalId":48833,"journal":{"name":"Evolution Equations and Control Theory","volume":"11 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Evolution Equations and Control Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.3934/eect.2022042","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
The exact distributed controllability of the semilinear heat equation \begin{document}$ \partial_{t}y-\Delta y + f(y) = v \, 1_{\omega} $\end{document} posed over multi-dimensional and bounded domains, assuming that \begin{document}$ f $\end{document} is locally Lipschitz continuous and satisfies the growth condition \begin{document}$ \limsup_{| r|\to \infty} | f(r)| /(| r| \ln^{3/2}| r|)\leq \beta $\end{document} for some \begin{document}$ \beta $\end{document} small enough has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. Under the same assumption, by introducing a different fixed point application, we present a different and somewhat simpler proof of the exact controllability, which is not based on the cost of observability of the heat equation with respect to potentials. Then, assuming that \begin{document}$ f $\end{document} is locally Lipschitz continuous and satisfies the growth condition \begin{document}$ \limsup_{| r|\to \infty} | f^\prime(r)|/\ln^{3/2}| r|\leq \beta $\end{document} for some \begin{document}$ \beta $\end{document} small enough, we show that the above fixed point application is contracting yielding a constructive method to compute the controls for the semilinear equation. Numerical experiments illustrate the results.
The exact distributed controllability of the semilinear heat equation \begin{document}$ \partial_{t}y-\Delta y + f(y) = v \, 1_{\omega} $\end{document} posed over multi-dimensional and bounded domains, assuming that \begin{document}$ f $\end{document} is locally Lipschitz continuous and satisfies the growth condition \begin{document}$ \limsup_{| r|\to \infty} | f(r)| /(| r| \ln^{3/2}| r|)\leq \beta $\end{document} for some \begin{document}$ \beta $\end{document} small enough has been obtained by Fernández-Cara and Zuazua in 2000. The proof based on a non constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized heat equation. Under the same assumption, by introducing a different fixed point application, we present a different and somewhat simpler proof of the exact controllability, which is not based on the cost of observability of the heat equation with respect to potentials. Then, assuming that \begin{document}$ f $\end{document} is locally Lipschitz continuous and satisfies the growth condition \begin{document}$ \limsup_{| r|\to \infty} | f^\prime(r)|/\ln^{3/2}| r|\leq \beta $\end{document} for some \begin{document}$ \beta $\end{document} small enough, we show that the above fixed point application is contracting yielding a constructive method to compute the controls for the semilinear equation. Numerical experiments illustrate the results.
期刊介绍:
EECT is primarily devoted to papers on analysis and control of infinite dimensional systems with emphasis on applications to PDE''s and FDEs. Topics include:
* Modeling of physical systems as infinite-dimensional processes
* Direct problems such as existence, regularity and well-posedness
* Stability, long-time behavior and associated dynamical attractors
* Indirect problems such as exact controllability, reachability theory and inverse problems
* Optimization - including shape optimization - optimal control, game theory and calculus of variations
* Well-posedness, stability and control of coupled systems with an interface. Free boundary problems and problems with moving interface(s)
* Applications of the theory to physics, chemistry, engineering, economics, medicine and biology