Functional Differential Equations

Application of i-smooth calculus

  • A. V. Kim

Part of the Mathematics and Its Applications book series (MAIA, volume 479)

Table of contents

  1. Front Matter
    Pages i-xv
  2. i—Smooth Calculus

    1. Front Matter
      Pages 1-1
    2. A. V. Kim
      Pages 3-11
  3. Functional Differential Equations

    1. Front Matter
      Pages 39-39
    2. A. V. Kim
      Pages 41-59
  4. Direct Lyapunov Method for Systems with Delays

    1. Front Matter
      Pages 69-69
    2. A. V. Kim
      Pages 71-73
    3. A. V. Kim
      Pages 74-108
    4. A. V. Kim
      Pages 109-122
    5. A. V. Kim
      Pages 123-128
  5. Dynamical Programming Method for Systems with Delays

    1. Front Matter
      Pages 129-129
    2. A. V. Kim
      Pages 131-145
    3. A. V. Kim
      Pages 146-156
  6. Back Matter
    Pages 157-168

About this book


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.


Optimal control calculus derivative differential equation functional analysis ordinary differential equation partial differential equation systems theory

Authors and affiliations

  • A. V. Kim
    • 1
  1. 1.Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesEkaterinburgRussia

Bibliographic information

  • DOI
  • Copyright Information Springer Science+Business Media B.V. 1999
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-5211-7
  • Online ISBN 978-94-017-1630-7
  • Buy this book on publisher's site