{"title":"Exponential stability conditions for non-autonomous differential equations with unbounded commutators in a Banach space","authors":"M. Gil'","doi":"10.21136/CMJ.2023.0188-21","DOIUrl":null,"url":null,"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.","PeriodicalId":50596,"journal":{"name":"Czechoslovak Mathematical Journal","volume":"73 1","pages":"355 - 366"},"PeriodicalIF":0.4000,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Czechoslovak Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.21136/CMJ.2023.0188-21","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.