{"title":"Stability analysis of homogeneous cooperative positive differential systems with time-varying delays and its generalization","authors":"Le Trung Hieu , La Van Thinh , Hoang The Tuan","doi":"10.1016/j.sysconle.2024.105868","DOIUrl":null,"url":null,"abstract":"<div><p>This article is devoted to studying the asymptotic behavior of differential systems with bounded delays. We first focus on the analysis of homogeneous cooperative positive systems. Under the assumption that the vector field is homogeneous of a degree greater than 1, we show that the non-trivial solutions of the system converge to the origin at a polynomial rate. In the case when the degree of homogeneity equals 1, we prove that the solutions will decay at an exponential rate. As a generalization of these results, we consider nonlinear non-autonomous differential systems with time-varying delays that are bounded above by stable homogeneous positive systems. By some additional imposed conditions, in light of the comparison principle, we obtain the locally exponential stability and polynomial stability of the equilibrium point to these systems. Finally, specific examples and discussions are provided to illustrate the validity of the proposed theoretical results.</p></div>","PeriodicalId":49450,"journal":{"name":"Systems & Control Letters","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Systems & Control Letters","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167691124001567","RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
This article is devoted to studying the asymptotic behavior of differential systems with bounded delays. We first focus on the analysis of homogeneous cooperative positive systems. Under the assumption that the vector field is homogeneous of a degree greater than 1, we show that the non-trivial solutions of the system converge to the origin at a polynomial rate. In the case when the degree of homogeneity equals 1, we prove that the solutions will decay at an exponential rate. As a generalization of these results, we consider nonlinear non-autonomous differential systems with time-varying delays that are bounded above by stable homogeneous positive systems. By some additional imposed conditions, in light of the comparison principle, we obtain the locally exponential stability and polynomial stability of the equilibrium point to these systems. Finally, specific examples and discussions are provided to illustrate the validity of the proposed theoretical results.
期刊介绍:
Founded in 1981 by two of the pre-eminent control theorists, Roger Brockett and Jan Willems, Systems & Control Letters is one of the leading journals in the field of control theory. The aim of the journal is to allow dissemination of relatively concise but highly original contributions whose high initial quality enables a relatively rapid review process. All aspects of the fields of systems and control are covered, especially mathematically-oriented and theoretical papers that have a clear relevance to engineering, physical and biological sciences, and even economics. Application-oriented papers with sophisticated and rigorous mathematical elements are also welcome.