Synopses & Reviews
This monograph presents the basics of i-smooth calculus, a new differential calculus of nonlinear functionals based on the notion of invariant derivative, and its application to some problems of the qualitative theory of functional differential equations.
This book is unique in its separation of finite and infinite dimensional components in the structures of functional differential equations and functionals, as well as in its use of conditional representation of FDEs, which is expedient for the application of methods and constructions of i-smooth calculus.
Part I contains a foundation of i-smooth calculus. Part II is an introduction to FDEs based on i-smooth calculus. Part III describes the direct Lyapunov method for systems with delays in terms of i- smooth functionals. Part IV considers an approach to the development of a dynamical programming method for systems with delays in terms of i-smooth Bellman's functionals.
Audience: This volume will be of interest to students and researchers in mathematics, applied mathematicians, and engineers whose work involves ordinary differential equations, functional analysis, partial differential equations, optimal control and mathematics systems theory.
Review
`The book is warmly recommended for students and researchers in mathematics, applied mathematicians, and engineers whose work involves ordinary differential equations, functional analysis, partial equations, optimal control and mathematics system theory.' Acta Scientiarum Mathematicarum
Review
`The book is warmly recommended for students and researchers in mathematics, applied mathematicians, and engineers whose work involves ordinary differential equations, functional analysis, partial equations, optimal control and mathematics system theory.'
Acta Scientiarum Mathematicarum
Synopsis
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.
Synopsis
This monograph presents the basics of i-smooth calculus, a new differential calculus of nonlinear functionals based on the notion of invariant derivative, and its application to some problems of the qualitative theory of functional differential equations. This book is unique in its separation of finite and infinite dimensional components in the structures of functional differential equations and functionals, as well as in its use of conditional representation of FDEs, which is expedient for the application of methods and constructions of i-smooth calculus. Part I contains a foundation of i-smooth calculus. Part II is an introduction to FDEs based on i-smooth calculus. Part III describes the direct Lyapunov method for systems with delays in terms of i- smooth functionals. Part IV considers an approach to the development of a dynamical programming method for systems with delays in terms of i-smooth Bellman's functionals. Audience: This volume will be of interest to students and researchers in mathematics, applied mathematicians, and engineers whose work involves ordinary differential equations, functional analysis, partial differential equations, optimal control and mathematics systems theory.
Description
Includes bibliographical references (p. [157]-162) and index.
Table of Contents
Preface.
Part I: i-Smooth Calculus. 1. Structure of Functionals.
2. Properties of Functionals. Invariant Derivative.
3. Generalized Derivatives of Nonlinear Functionals.
Part II: Functional Differential Equations. 4. Functional Differential Equations.
5. Neutral Functional Differential Equations.
Part III: Direct Lyapunov Method for Systems with Delays. 6. The Problem Statement.
7. The Lyapunov Functional Method.
8. The Lyapunov Function Method.
9. Instability.
Part IV: Dynamical Programming Method for Systems with Delays. 10. Systems with State Delays.
11. Systems with Control Delays. References. Index.