Weakly Connected Nonlinear Systems

Weakly Connected Nonlinear Systems: Boundedness and Stability of Motion

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Features

  • Presents the mathematical foundations of integral and differential inequalities, comparison techniques, the direct Lyapunov method, and stability definitions for systems with a small parameter
  • Covers recently developed approaches based on the direct Lyapunov method and the comparison technique for the investigation of the boundedness of motion of weakly connected nonlinear systems with two different measures
  • Examines the stability conditions of solutions of the weakly connected systems of differential equations based on the direct Lyapunov method and the comparison technique
  • Applies a general approach to the study of the stability of solutions for nonlinear systems with small perturbing forces
  • Describes fundamental results on the boundedness and stability of systems in Banach spaces with weakly connected subsystems

Summary

Weakly Connected Nonlinear Systems: Boundedness and Stability of Motion provides a systematic study on the boundedness and stability of weakly connected nonlinear systems, covering theory and applications previously unavailable in book form. It contains many essential results needed for carrying out research on nonlinear systems of weakly connected equations.

After supplying the necessary mathematical foundation, the book illustrates recent approaches to studying the boundedness of motion of weakly connected nonlinear systems. The authors consider conditions for asymptotic and uniform stability using the auxiliary vector Lyapunov functions and explore the polystability of the motion of a nonlinear system with a small parameter. Using the generalization of the direct Lyapunov method with the asymptotic method of nonlinear mechanics, they then study the stability of solutions for nonlinear systems with small perturbing forces. They also present fundamental results on the boundedness and stability of systems in Banach spaces with weakly connected subsystems through the generalization of the direct Lyapunov method, using both vector and matrix-valued auxiliary functions.

Designed for researchers and graduate students working on systems with a small parameter, this book will help readers get up to date on the knowledge required to start research in this area.

Table of Contents

Preliminaries
Introductory Remarks
Fundamental Inequalities
Stability in the Sense of Lyapunov
Comparison Principle
Stability of Systems with a Small Parameter

Analysis of the Boundedness of Motion
Introductory Remarks
Statement of the Problem
μ-Boundedness with Respect to Two Measures
Boundedness and the Comparison Technique
Boundedness with Respect to a Part of Variables
Algebraic Conditions of μ-Boundedness
Applications

Analysis of the Stability of Motion
Introductory Remarks
Statement of the Problem
Stability with Respect to Two Measures
Equistability via Scalar Comparison Equations
Dynamic Behavior of an Individual Subsystem
Asymptotic Behavior
Polystability of Motion
Applications

Stability of Weakly Perturbed Systems
Introductory Remarks
Averaging and Stability
Stability on a Finite Time Interval
Methods of Application of Auxiliary Systems
Systems with Nonasymptotically Stable Subsystems
Stability with Respect to a Part of Variables
Applications

Stability of Systems in Banach Spaces
Introductory Remarks
Preliminary Results
Statement of the Problem
Generalized Direct Lyapunov Method
μ-Stability of Motion of Weakly Connected Systems
Stability Analysis of a Two-Component System

Bibliography

Index

Comments and References appear at the end of each chapter.

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