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1999 | Buch

Functional Differential Equations

Application of i-smooth calculus

verfasst von: A. V. Kim

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

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Über dieses Buch

Beginning with the works of N.N.Krasovskii [81, 82, 83], which clari­ fied the functional nature of systems with delays, the functional approach provides a foundation for a complete theory of differential equations with delays. Based on the functional approach, different aspects of time-delay system theory have been developed with almost the same completeness as the corresponding field of ODE (ordinary differential equations) the­ ory. The term functional differential equations (FDE) is used as a syn­ onym for systems with delays 1. The systematic presentation of these re­ sults and further references can be found in a number of excellent books [2, 15, 22, 32, 34, 38, 41, 45, 50, 52, 77, 78, 81, 93, 102, 128]. In this monograph we present basic facts of i-smooth calculus ~ a new differential calculus of nonlinear functionals, based on the notion of the invariant derivative, and some of its applications to the qualitative theory of functional differential equations. Utilization of the new calculus is the main distinction of this book from other books devoted to FDE theory. Two other distinguishing features of the volume are the following: - the central concept that we use is the separation of finite dimensional and infinite dimensional components in the structures of FDE and functionals; - we use the conditional representation of functional differential equa­ tions, which is convenient for application of methods and constructions of i~smooth calculus to FDE theory.

Inhaltsverzeichnis

Frontmatter

i—Smooth Calculus

Chapter 1. Structure of Functionals
Abstract
The conception of a functional originates in V.Volterra’s works as “functions of line”. The term “functional”, introduced by J.Hadamard, at present is understood as a mapping V of an arbitrary set X into the set of real R (or complex C) numbers.
A. V. Kim
Chapter 2. Properties of Functionals. Invariant Derivative
A. V. Kim
Chapter 3. Generalized Derivatives of Nonlinear Functionals
Abstract
In the present chapter we develop the theory of generalized derivatives of nonlinear functionals on D(R). 13 Though the generalized derivative is closely related to the invariant derivative, the material of this chapter is not directly connected with other chapters of the book. As the application of this new derivative, we consider an approach to solution of the multiplication problem of distribution theory and, also, the concept of generalized solutions of nonlinear functionals.
A. V. Kim

Functional Differential Equations

Frontmatter
Chapter 4. Functional Differential Equations
A. V. Kim
Chapter 5. Neutral Functional Differential Equations
Abstract
The present chapter contains some remarks and ideas concerning application of i—smooth calculus to functional differential equations of neutral type. Taking into account essential features of neutral functional differential equations (NFDE) subsequent elaboration of these aspects requires additional investigating properties of invariant differentiable functionals and NFDE.
A. V. Kim

Direct Lyapunov Method for Systems with Delays

Frontmatter
Chapter 6. The Problem Statement
Abstract
The method of the Lyapunov function 28 is one of the most effective methods for investigation of ODE dynamics. Efficiency of the Lyapunov function method for ODE is based on the fact that application of Lyapunov’s function allows us to investigate stability of solutions without solving corresponding ODE, and existence of the converse theorems shows its universal nature.
A. V. Kim
Chapter 7. The Lyapunov Functional Method
A. V. Kim
Chapter 8. The Lyapunov Function Method
Abstract
In Chapter 7 we discussed the first approach to realization of the Lyapunov direct method for FDE based on application of the Lyapunov functionals. In the second approach finite dimensional Lyapunov’s functions v(t, x): R × R n R are used.
A. V. Kim
Chapter 9. Instability
Abstract
Systems with delays are only mathematical models of some real processes with aftereffect, so, generally speaking, we should consider initial functions to be some functions that occur in practice. However for different real processes there can be different admissible initial functions, and it would be reasonable to consider initial value problem only for admissible initial conditions. But this would be a formal theory, because we do not know admissible initial functions a priori and besides a set of initial functions can have a complicated structure.
A. V. Kim

Dynamical Programming Method for Systems with Delays

Frontmatter
Chapter 10. Systems with State Delays
Abstract
Part IV is concerned with some aspects of the dynamic programming method and the invariant differentiability of the Bellman functionals in optimal control problems for systems with state and control delays. Investigations in this direction were initiated by N.N.Krasovskii [84, 85] and then were developed in [1, 63, 66, 75, 76, 26, 27, 80, 90, 127].
A. V. Kim
Chapter 11. Systems with Control Delays
A. V. Kim
Backmatter
Metadaten
Titel
Functional Differential Equations
verfasst von
A. V. Kim
Copyright-Jahr
1999
Verlag
Springer Netherlands
Electronic ISBN
978-94-017-1630-7
Print ISBN
978-90-481-5211-7
DOI
https://doi.org/10.1007/978-94-017-1630-7