Stability of Vector Differential Delay Equations

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.

List of contents

Preface.- 1 Preliminaries.- 2 Some Results of the Matrix Theory.- 3 General Linear Systems.- 4 Time-invariant Linear Systems with Delay.- 5 Properties of Characteristic Values.- 6 Equations Close to Autonomous and Ordinary Differential Ones.- 7 Periodic Systems.- 8 Linear Equations with Oscillating Coefficients.- 9 Linear Equations with Slowly Varying Coefficients.- 10 Nonlinear Vector Equations.- 11 Scalar Nonlinear Equations.- 12 Forced Oscillations in Vector Semi-Linear Equations.- 13 Steady States of Differential Delay Equations.- 14 Multiplicative Representations of Solutions.- Appendix A. The General Form of Causal Operators.- Appendix B. Infinite Block Matrices.- Bibliography.- Index.

About the author

Michael I. Gil' is professor of mathematics at Ben Gurion University of the Negev in Beer Sheva, Israel.

Summary

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.

Additional text

From the reviews:
“This research monograph presents and studies the stability of linear and nonlinear delay systems. … Every chapter begins with a brief description of its contents and is completed by ‘Comments’ with some remarks and references. … the book contains many interesting results on stability analysis and can be recommended to specialists in linear stability theory and researchers interested by analysis of delay systems.” (Nataliya Olegovna Sedova, Mathematical Reviews, November, 2013)

Report

From the reviews:
"This research monograph presents and studies the stability of linear and nonlinear delay systems. ... Every chapter begins with a brief description of its contents and is completed by 'Comments' with some remarks and references. ... the book contains many interesting results on stability analysis and can be recommended to specialists in linear stability theory and researchers interested by analysis of delay systems." (Nataliya Olegovna Sedova, Mathematical Reviews, November, 2013)

Product details

• Authors: Michael I Gil, Michael I. Gil, Michael I Gil', Michael I. Gil', Michael I. Gil¿, Michael I. Gil’
• Publisher: Springer, Basel
• Languages: English
• Product format: Paperback / Softback
• Released: 01.03.2013
• EAN: 9783034805766
• ISBN: 978-3-0-3480576-6
• No. of pages: 259
• Dimensions: 170 mm x 16 mm x 239 mm
• Weight: 452 g
• Illustrations: X, 259 p.
• Series: Frontiers in Mathematics; Frontiers in Mathematics
• Subject: Natural sciences, medicine, IT, technology > Mathematics > Analysis