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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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract
we recall very briefly some basic notions of the theory of Banach and Hilbert spaces. More details can be found in any textbook on Banach and Hilbert spaces (e.g., [2] and [16]).
Michael I. Gil’

Chapter 2. Some Results of the Matrix Theory

Abstract
This chapter is devoted to norm estimates for matrix-valued functions, in particular, for resolvents. These estimates will be applied in the rest of the book chapters.
Michael I. Gil’

Chapter 3. General Linear Systems

Abstract
This chapter is devoted to general linear systems including the Bohl–Perron principle.
Michael I. Gil’

Chapter 4. Time-invariant Linear Systems with Delay

Abstract
This chapter is devoted to time-invariant (autonomous) linear systems with delay. In the terms of the characteristic matrix-valued function we derive estimates for the \(L^p\)- and C-norms of fundamental solutions. By these estimates below we obtain stability conditions for linear time-variant and nonlinear systems. Moreover, these estimates enable us to establish bounds for the region of attraction of the stationary solutions of nonlinear systems.
Michael I. Gil’

Chapter 5. Properties of Characteristic Values

Abstract
In this chapter we investigate properties of the characteristic values of autonomous systems. In particular some identities for characteristic values and perturbations results are derived.
Michael I. Gil’

Chapter 6. Equations Close to Autonomous and Ordinary Differential Ones

Abstract
In this chapter we establish explicit stability conditions for linear time variant systems with delay “close” to ordinary differential systems and for systems with small delays. We also investigate perturbations of autonomous equations.
Michael I. Gil’

Chapter 7. Periodic Systems

Abstract
This chapter deals with a class of periodic systems. Explicit stability conditions are derived. The main tool is the invertibility conditions for infinite block matrices. In the case of scalar equations we apply regularized determinants.
Michael I. Gil’

Chapter 8. Linear Equations with Oscillating Coefficients

Abstract
In the present chapter we investigate vector and scalar linear equations with “quickly” oscillating coefficients.
Michael I. Gil’

Chapter 9. Linear Equations with Slowly Varying Coefficients

Abstract
This chapter deals with vector differential-delay equations having slowly varying coefficients. The main tool in this chapter is the “freezing” method.
Michael I. Gil’

Chapter 10. Nonlinear Vector Equations

Abstract
In the present chapter we investigate nonlinear systems with causal mappings. The main tool is that of norm estimates for fundamental solutions. The generalized norm is also applied. It enables us to use information about a system more completely than the usual (number) norm.
Michael I. Gil’

Chapter 11. Scalar Nonlinear Equations

Abstract
In this chapter, nonlinear scalar first- and higher-order equations with differentialdelay linear parts and nonlinear causal mappings are considered. Explicit stability conditions are derived.
Michael I. Gil’

Chapter 12. Forced Oscillations in Vector Semi-Linear Equations

Abstract
This chapter deals with forced periodic oscillations of coupled systems of semilinear functional differential equations. Explicit conditions for the existence and uniqueness of periodic solutions are derived. These conditions are formulated in terms of the roots of characteristic matrix functions.
Michael I. Gil’

Chapter 13. Steady States of Differential Delay Equations

Abstract
In this chapter we investigate steady states of differential delay equations. Steady states of many differential delay equations are described by equations of the type \(F_0(x)=0\), where \(F_0 \; : \;\mathbb{C}^n\rightarrow\mathbb{C}^n \) is a function satisfying various conditions.
Michael I. Gil’

Chapter 14. Multiplicative Representations of Solutions

Abstract
In this chapter we suggest a representation for solutions of differential delay equations via multiplicative operator integrals.
Michael I. Gil’

Backmatter

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