Uncertain Dynamical Systems: Stability and Motion Control

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Features

    • Details all proofs of stability conditions for five classes of uncertain systems
    • Clearly defines all used notions of stability and control theory
    • Contains an extensive bibliography, facilitating quick access to specific subject areas in each chapter

    Summary

    This self-contained book provides systematic instructive analysis of uncertain systems of the following types: ordinary differential equations, impulsive equations, equations on time scales, singularly perturbed differential equations, and set differential equations. Each chapter contains new conditions of stability of unperturbed motion of the above-mentioned type of equations, along with some applications. Without assuming specific knowledge of uncertain dynamical systems, the book includes many fundamental facts about dynamical behaviour of its solutions. Giving a concise review of current research developments, Uncertain Dynamical Systems: Stability and Motion Control

    • Details all proofs of stability conditions for five classes of uncertain systems
    • Clearly defines all used notions of stability and control theory
    • Contains an extensive bibliography, facilitating quick access to specific subject areas in each chapter

    Requiring only a fundamental knowledge of general theory of differential equations and calculus, this book serves as an excellent text for pure and applied mathematicians, applied physicists, industrial engineers, operations researchers, and upper-level undergraduate and graduate students studying ordinary differential equations, impulse equations, dynamic equations on time scales, and set differential equations.

    Table of Contents

    Introduction
    Parametric Stability
    Stability with Respect to Moving Invariant Sets

    Lyapunov’s Direct Method for Uncertain Systems
    Problem Setting and Auxiliary Results
    Classes of Lyapunov Functions
    Theorems on Stability and Uniform Stability
    Exponential Convergence of Motions to a Moving Invariant Set
    Instability of Solutions with Respect to a Given Moving Set
    Stability with Respect to a Conditionally Invariant Moving Set

    Stability of Uncertain Controlled Systems
    Problem Setting
    Synthesis of Controls
    Convergence of Controlled Motions to a Moving Set
    Stabilization of Rotary Motions of a Rigid Body in an Environment with Indefinite Resistance
    Stability of an Uncertain Linear System with Neuron Control
    Conditions for Parametric Quadratic Stabilizability

    Stability of Quasilinear Uncertain Systems
    Uncertain Quasilinear System and Its Transformation
    Application of the Canonical Matrix-Valued Function
    Isolated Quasilinear Systems
    Quasilinear Systems with Nonautonomous Uncertainties
    Synchronizing of Motions in Uncertain Quasilinear Systems

    Stability of Large-Scale Uncertain Systems
    Description of a Large-Scale System
    Stability of Solutions with Respect to a Moving Set
    Application of the Hierarchical Lyapunov Function
    Stability of a Class of Time Invariant Uncertain Systems

    Interval and Parametric Stability of Uncertain Systems
    Conditions for the Stability of a Quasilinear System (Continued)
    Interval Stability of a Linear Mechanical System
    Parametric Stability of an Uncertain Time Invariant System

    Stability of Solutions of Uncertain Impulsive Systems
    Problem Setting
    Principle of Comparison with a Block-Diagonal Matrix Function
    Conditions for Strict Stability
    Application of the Vector Approach
    Robust Stability of Impulsive Systems
    Concluding Remarks

    Stability of Solutions of Uncertain Dynamic Equations on a Time Scale
    Elements of the Analysis on a Time Scale
    Theorems of the Direct Lyapunov Method
    Applications and the Discussion of the Results

    Singularly Perturbed Systems with Uncertain Structure
    Structural Uncertainties in Singularly Perturbed Systems
    Tests for Stability Analysis
    Tests for Instability Analysis
    Linear Systems under Structural Perturbations

    Qualitative Analysis of Solutions of Set Differential Equations
    Some Results of the General Theory of Metric Spaces
    Existence of Solutions of Set Differential Equations
    The Matrix-Valued Lyapunov Function and Its Application
    Stability of a Set Stationary Solution
    Theorems on Stability
    The Application of the Strengthened Lyapunov Function
    Boundedness Theorems

    Set Differential Equations with a Robust Causal Operator
    Preliminary Results
    Comparison Principle
    Estimates of Funnel for Solutions
    Test for Stability

    Stability of a Set of Impulsive Equations
    Auxiliary Results
    Heterogeneous Lyapunov Function
    Sufficient Stability Conditions
    Impulsive Equations with Delay under Small Perturbations

    Comments and References

    Appendix
    Bibliography
    Index

    Editorial Reviews

    "This book contains very interesting and advanced material on the application of various forms of Lyapunov functions for tackling a myriad of problems in differential equations. … it is very rich in both theory and applications and is recommended to all major libraries. It is also recommended to researchers and teachers of differential equations … ."
    —Yawvi A. Fiagbedzi, Mathematical Reviews, May 2013