Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space

IF 0.4 4区 数学 Q4 MATHEMATICS Czechoslovak Mathematical Journal Pub Date : 2023-01-23 DOI:10.21136/CMJ.2023.0188-21
M. Gil'
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引用次数: 0

Abstract

We consider the equation dy(t)/dt = (A + B(t))y(t) (t ≽ 0), where A is the generator of an analytic semigroup (eAt)t≽0 on a Banach space χ, B(t) is a variable bounded operator in χ. It is assumed that the commutator K(t) = AB(t) − B(t)A has the following property: there is a linear operator S having a bounded left-inverse operator Sl−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_l^{ - 1}$$\end{document} such that ∥SeAt∥ is integrable and the operator K(t)Sl−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\left( t \right)S_l^{ - 1}$$\end{document} is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations.
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Banach空间中具有无界交换子的非自治微分方程的指数稳定性条件
我们考虑方程dy(t)/dt=(A+B(t))y(t)(t≽0),其中A是Banach空间χ上分析半群(eAt)t≴0的生成元,B(t)是χ中的可变有界算子。假设交换子K(t)=AB(t)−B(t)A具有以下性质:存在一个线性算子S,它具有一个有界左逆算子Sl−1\documentclass[12pt]{minimal}\usepackage{amsmath}\ usepackage{wasysym}\ use package{amsfonts}\usapackage{amssymb}\ usapackage{amsbsy}\usepackage{mathrsfs}\ userpackage{upgeek}\setlength{\oddsidemargin}{-69pt}\ begin{document}$S_l^{-1}$$\end{document},使得‖SeAt‖是可积的,并且运算符K(t)Sl−1\documentclass[12pt]{minimal}\usepackage{amsmath}\use package{{wasysym}\ usepackage{amsfonts}\usapackage{amssymb}\ use package{amsbsy}\usepackage{mathrsfs}\ usapackage{upgek}\setlength{\oddsedmargin}{-69pt}\begin{document}$K\left(t\right)S_l^{-1}$\end{document}是有界的。在这些条件下,导出了指数稳定性检验。作为一个例子,我们考虑一个抛物型方程的耦合系统。
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来源期刊
CiteScore
0.90
自引率
0.00%
发文量
0
审稿时长
6-12 weeks
期刊介绍: Czechoslovak Mathematical Journal publishes original research papers of high scientific quality in mathematics.
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