Stability of Vector Differential Delay Equations

  • Michael I. Gil’

Part of the Frontiers in Mathematics book series (FM)

Table of contents

  1. Front Matter
    Pages i-x
  2. Michael I. Gil’
    Pages 1-24
  3. Michael I. Gil’
    Pages 25-52
  4. Michael I. Gil’
    Pages 53-68
  5. Michael I. Gil’
    Pages 69-94
  6. Michael I. Gil’
    Pages 95-111
  7. Michael I. Gil’
    Pages 131-142
  8. Michael I. Gil’
    Pages 143-156
  9. Michael I. Gil’
    Pages 157-164
  10. Michael I. Gil’
    Pages 165-192
  11. Michael I. Gil’
    Pages 193-210
  12. Michael I. Gil’
    Pages 211-216
  13. Michael I. Gil’
    Pages 217-224
  14. Michael I. Gil’
    Pages 225-230
  15. Back Matter
    Pages 231-259

About this book

Introduction

Differential equations with delay naturally arise in various applications, such as control systems, viscoelasticity, mechanics, nuclear reactors, distributed networks, heat flows, neural networks, combustion, interaction of species, microbiology, learning models, epidemiology, physiology, and many others. This book systematically investigates the stability of linear as well as nonlinear vector differential equations with delay and equations with causal mappings. It presents explicit conditions for exponential, absolute and input-to-state stabilities. These stability conditions are mainly formulated in terms of the determinants and eigenvalues of auxiliary matrices dependent on a parameter; the suggested approach allows us to apply the well-known results of the theory of matrices. In addition, solution estimates for the considered equations are established which provide the bounds for regions of attraction of steady states.  

 

The main methodology presented in the book is based on a combined usage of the recent norm estimates for matrix-valued functions and the following methods and results: the generalized Bohl-Perron principle and the integral version of the generalized Bohl-Perron principle; the freezing method; the positivity of fundamental solutions. A significant part of the book is devoted to  the Aizerman-Myshkis problem and  generalized Hill theory of periodic systems.  

 

The book is intended not only for specialists in the theory of functional differential equations and control theory, but also for anyone with a sound mathematical background interested in their various applications.

Keywords

Aizerman - Myshkis problem differential delay equations freezing method matrix methods positive fundamental solutions stability analysis

Authors and affiliations

  • Michael I. Gil’
    • 1
  1. 1., Department of MathematicsBen Gurion University of the NegevBeer ShevaIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-0348-0577-3
  • Copyright Information Springer Basel 2013
  • Publisher Name Birkhäuser, Basel
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-0348-0576-6
  • Online ISBN 978-3-0348-0577-3
  • Series Print ISSN 1660-8046
  • Series Online ISSN 1660-8054
  • About this book
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