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Lyapunov Functionals and Stability of Stochastic Functional Differential Equations

  • Book
  • © 2013

Overview

Authors:
  1. Leonid Shaikhet

    1. Department of Higher Mathematics, Donetsk State University of Management, Donetsk, Ukraine

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  • Detailed description of Lyapunov functional construction will allow researchers to analyse stability results for hereditary systems more easily
  • Profuse analytical and numerical examples help to explain the methods used
  • Demonstrates a method that can be usefully applied in economic, mechanical, biological and ecological systems
  • Includes supplementary material: sn.pub/extras

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Table of contents (12 chapters)

  1. Front Matter

    Pages I-XII
    Download chapter PDF

  2. Short Introduction to Stability Theory of Deterministic Functional Differential Equations

    • Leonid Shaikhet
    Pages 1-28

  3. Stochastic Functional Differential Equations and Procedure of Constructing Lyapunov Functionals

    • Leonid Shaikhet
    Pages 29-52

  4. Stability of Linear Scalar Equations

    • Leonid Shaikhet
    Pages 53-96

  5. Stability of Linear Systems of Two Equations

    • Leonid Shaikhet
    Pages 97-111

  6. Stability of Systems with Nonlinearities

    • Leonid Shaikhet
    Pages 113-152

  7. Matrix Riccati Equations in Stability of Linear Stochastic Differential Equations with Delays

    • Leonid Shaikhet
    Pages 153-185

  8. Stochastic Systems with Markovian Switching

    • Leonid Shaikhet
    Pages 187-208

  9. Stabilization of the Controlled Inverted Pendulum by a Control with Delay

    • Leonid Shaikhet
    Pages 209-250

  10. Stability of Equilibrium Points of Nicholson’s Blowflies Equation with Stochastic Perturbations

    • Leonid Shaikhet
    Pages 251-256

  11. Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator–Prey with Aftereffect and Stochastic Perturbations

    • Leonid Shaikhet
    Pages 257-282

  12. Stability of SIR Epidemic Model Equilibrium Points

    • Leonid Shaikhet
    Pages 283-296

  13. Stability of Some Social Mathematical Models with Delay Under Stochastic Perturbations

    • Leonid Shaikhet
    Pages 297-323

  14. Back Matter

    Pages 325-342
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Keywords

About this book

Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for difference equations with discrete and continuous time. The text begins with both a description and a delineation of the peculiarities of deterministic and stochastic functional differential equations. There follows basic definitions for stability theory of stochastic hereditary systems, and the formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of different mathematical models such as: • inverted controlled pendulum; • Nicholson's blowflies equation; • predator-prey relationships; • epidemic development; and • mathematical models that describe human behaviours related to addictions and obesity. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations is primarily addressed to experts in stability theory but will also be of interest to professionals and students in pure and computational mathematics, physics, engineering, medicine, and biology.

Reviews

From the reviews:

 “This is a book entirely devoted to the stability of stochastic functional differential equations, including various stochastic delay differential equations. This book is well written by a true expert in the field. In addition to analysis, it contains many simulation results. This book should be beneficial to researchers both in mathematics and control areas and in various applied areas who need to use stability.” (Fuke Wu, Mathematical Reviews, January, 2014)

Authors and Affiliations

  • Department of Higher Mathematics, Donetsk State University of Management, Donetsk, Ukraine

    Leonid Shaikhet

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