Stability of Vector Differential Delay Equations

✍ Scribed by Michael I. Gil’ (auth.)

• Publisher: Birkhäuser Basel
• Year: 2013
• Tongue: English
• Leaves: 266
• Series: Frontiers in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-x
Preliminaries....Pages 1-24
Some Results of the Matrix Theory....Pages 25-52
General Linear Systems....Pages 53-68
Time-invariant Linear Systems with Delay....Pages 69-94
Properties of Characteristic Values....Pages 95-111
Equations Close to Autonomous and Ordinary Differential Ones....Pages 113-130
Periodic Systems....Pages 131-142
Linear Equations with Oscillating Coefficients....Pages 143-156
Linear Equations with Slowly Varying Coefficients....Pages 157-164
Nonlinear Vector Equations....Pages 165-192
Scalar Nonlinear Equations....Pages 193-210
Forced Oscillations in Vector Semi-Linear Equations....Pages 211-216
Steady States of Differential Delay Equations....Pages 217-224
Multiplicative Representations of Solutions....Pages 225-230
Back Matter....Pages 231-259

✦ Subjects

Ordinary Differential Equations; Systems Theory, Control; Applications of Mathematics