Uniform exponential stability of linear delayed integro-differential vector equations

Leonid Berezansky, Josef Diblík, Zdeněk Svoboda, Zdeněk Šmarda

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

Uniform exponential stability of a linear delayed integro-differential vector equation x˙(t)=∑k=1mAk(t)x(hk(t))+∑k=1l∫gk(t)tPk(t,s)x(s)ds,t∈[0,∞), where x=(x1,…,xn)T is an unknown vector-function, is considered. It is assumed that m, l are positive integers, matrices Ak, Pk and delays hk, gk are Lebesgue measurable. The main result is of an explicit type, depending on all delays, and its proof is based on an a priori estimation of solutions, a Bohl-Perron type result, and utilization of the matrix measure. As particular cases, it includes (2m+l−1) mutually different sufficient conditions. Some of them are formulated separately as corollaries. Advantages of derived explicit results over the existing ones are demonstrated on examples and open problems are proposed as well.

Original languageEnglish
Pages (from-to)573-595
Number of pages23
JournalJournal of Differential Equations
Volume270
DOIs
StatePublished - 5 Jan 2021

Keywords

  • A priori estimation
  • Bohl-Perron type result
  • Delay
  • Exponential stability
  • Integro-differential systems
  • Linear systems

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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