Abdulla Azamov, Gafurjan Ibragimov, Khudoyor Mamayusupov, Marks Ruziboev
{"title":"On the Stability and Null-Controllability of an Infinite System of Linear Differential Equations.","authors":"Abdulla Azamov, Gafurjan Ibragimov, Khudoyor Mamayusupov, Marks Ruziboev","doi":"10.1007/s10883-021-09587-6","DOIUrl":null,"url":null,"abstract":"<p><p>In this work, the null controllability problem for a linear system in <i>ℓ</i><sup>2</sup> is considered, where the matrix of a linear operator describing the system is an infinite matrix with <math><mi>λ</mi><mo>∈</mo><mi>ℝ</mi></math> on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if <i>λ</i> ≤- 1, which shows the fine difference between the finite and the infinite-dimensional systems. When <i>λ</i> ≤- 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered <math><msup><mrow><mi>ℓ</mi></mrow><mrow><mi>∞</mi></mrow></msup></math> is not asymptotically stable if <i>λ</i> = - 1.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10516776/pdf/","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10883-021-09587-6","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2021/12/23 0:00:00","PubModel":"Epub","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
Abstract
In this work, the null controllability problem for a linear system in ℓ2 is considered, where the matrix of a linear operator describing the system is an infinite matrix with on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤- 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤- 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered is not asymptotically stable if λ = - 1.