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Journal of Dynamics and Differential Equations

, Volume 31, Issue 4, pp 2095–2108 | Cite as

Exponential Stability Tests for Linear Delayed Differential Systems Depending on All Delays

  • Leonid Berezansky
  • Josef DiblíkEmail author
  • Zdeněk Svoboda
  • Zdeněk Šmarda
Article
  • 148 Downloads

Abstract

Linear delayed differential systems
$$\begin{aligned} \dot{x}_i(t)=\sum _{j=1}^m \sum _{k=1}^{r_{ij}}a_{ij}^{k}(t)x_j\left( h_{ij}^{k}(t)\right) ,\quad i=1,\dots ,m \end{aligned}$$
are considered on a half-infinity interval \(t\ge 0\). It is assumed that m and \(r_{ij}\), \(i,j=1,\dots ,m\) are natural numbers and the coefficients \(a_{ij}^{k}:[0,\infty )\rightarrow \mathbb {R}\) and delays \(h_{ij}^{k}:[0,\infty )\rightarrow {\mathbb {R}}\) are Lebesgue measurable functions. New explicit results on uniform exponential stability, depending on all delays, are derived. The conditions obtained do not require the dominance of diagonal terms over the off-diagonal terms as most of the existing stability tests for non-autonomous delay differential systems do.

Keywords

Linear delayed differential system Exponential stability Conditions depending on all delays Matrix measure 

Mathematics Subject Classification

34K20 34K06 34K25 

Notes

Acknowledgements

The authors greatly appreciate the work of the anonymous referee, whose comments and suggestions have helped to improve the paper in many aspects. The second and fourth authors have been supported by the Czech Science Foundation under the Project 16-08549S. The third author has been supported by the project CEITEC 2020 (LQ1601) with financial support from the Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme II. This work was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBen-Gurion University of the NegevBeer-ShevaIsrael
  2. 2.CEITEC - Central European Institute of TechnologyBrno University of TechnologyBrnoCzech Republic

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