Qualitative Methods in Nonlinear Dynamics

by Martynyuk; A.A.

• ISBN13: 9780824707354
• ISBN10: 0824707354
• Format: Hardcover
• Copyright: 2001-11-05
• Publisher: CRC Press

Summary

Presents new approaches to qualitative analysis of continuous, discrete-time, and impulsive nonlinear systems via Liapunov matrix valued functions.

Author Biography

A. A. Martynyuk is Chief, Stability of Processes Department, Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev.

Table of Contents

Preface	iii
Preliminaries	1	(48)
Introduction	1	(1)
Nonlinear Continuous Systems	2	(3)
General equations of nonlinear dynamics	2	(1)
Perturbed motion equations	3	(2)
Definitions of Stability	5	(3)
Scalar, Vector and Matrix-Valued Liapunov Functions	8	(9)
Auxiliary scalar functions	8	(1)
Comparison functions	9	(5)
Vector Liapunov functions	14	(1)
Matrix-valued metafunction	15	(2)
Comparison Principle	17	(6)
Liapunov-Like Theorems	23	(10)
Matrix-valued function and its properties	23	(3)
A version of the original theorems of Liapunov	26	(7)
Advantages of Cone-Valued Liapunov Functions	33	(8)
Stability with respect to two measures	34	(2)
Stability analysis of large scale systems	36	(5)
Liapunov's Theorems for Large Scale Systems in General	41	(6)
Why are matrix-valued Liapunov functions needed?	41	(1)
Stability and instability of large scale systems	42	(5)
Notes	47	(2)
Qualitative Analysis of Continuous Systems	49	(90)
Introduction	49	(1)
Nonlinear Systems with Mixed Hierarchy of Subsystems	50	(18)
Mixed hierarchical structures	50	(2)
Hierarchical matrix function structure	52	(4)
Structure of hierarchical matrix function derivative	56	(3)
Stability and instability conditions	59	(1)
Linear autonomous system	60	(3)
Examples of third order systems	63	(5)
Dynamics of the Systems with Regular Hierarchy Subsystems	68	(22)
Ikeda-Siljak hierarchical decomposition	68	(1)
Hierarchical Liapunov's matrix-valued functions	69	(5)
Stability and instability conditions	74	(5)
Linear nonautonomous systems	79	(11)
Stability Analysis of Large Scale Systems	90	(10)
A class of large scale systems	90	(1)
Construction of nondiagonal elements of matrix-valued function	91	(3)
Test for stability analysis	94	(1)
Linear large scale system	94	(3)
Discussion and numerical example	97	(3)
Overlapping Decomposition and Matrix-Valued Function Construction	100	(8)
Dynamical system extension	100	(5)
Liapunov matrix-valued function construction	105	(1)
Test for stability of system (2.5.1)	105	(1)
Numerical example	106	(2)
Exponential Polystability Analysis of Separable Motions	108	(19)
Statement of the Problem	108	(2)
A method for the solution of the problem	110	(8)
Autonomous system	118	(4)
Polystability by the first order approximations	122	(5)
Integral and Lipschitz Stability	127	(8)
Definitions	127	(1)
Sufficient conditions for integral and asymptotic integral stability	128	(5)
Uniform Lipschitz stability	133	(2)
Notes	135	(4)
Qualitative Analysis of Discrete-Time Systems	139	(44)
Introduction	139	(1)
Systems Described by Difference Equations	140	(3)
Matrix-Valued Liapunov Functions Method	143	(6)
Auxiliary results	143	(1)
Comparison principle application	144	(3)
General theorems on stability	147	(2)
Large Scale System Decomposition	149	(2)
Stability and Instability of Large Scale Systems	151	(8)
Auxiliary estimates	151	(6)
Stability and instability conditions	157	(2)
Autonomous Large Scale Systems	159	(7)
Hierarchical Analysis of Stability	166	(13)
Hierarchical decomposition and stability conditions	166	(6)
Novel tests for connective stability	172	(7)
Controlled Systems	179	(2)
Notes	181	(2)
Nonlinear Dynamics of Impulsive Systems	183	(56)
Introduction	183	(1)
Large Scale Impulsive Systems in General	184	(17)
Notations and definitions	184	(2)
Auxiliary results	186	(9)
Sufficient stability conditions	195	(2)
Instability conditions	197	(4)
Hierarchical Impulsive Systems	201	(3)
Analytical Construction of Liapunov Function	204	(11)
Structure of hierarchical matrix-valued Liapunov function	204	(3)
Structure of the total derivative of hierarchical matrix-valued function	207	(8)
Uniqueness and Continuability of Solutions	215	(7)
On Boundedness of the Solutions	222	(6)
Novel Methodology for Stability	228	(10)
Stability conditions	228	(10)
Notes	238	(1)
Applications	239	(38)
Introduction	239	(1)
Estimations of Asymptotic Stability Domains in General	239	(24)
A fundamental Zubov's result	239	(2)
Some estimates for quadratic matrix-valued functions	241	(4)
Algorithm of constructing a point network covering boundary of domain E	245	(5)
Numerical realization and discussion of the algorithm	250	(4)
Illustrative examples	254	(9)
Construction of Estimate for the Domain E of Power System	263	(4)
Oscillations and Stability of Some Mechanical Systems	267	(3)
Three-mass systems	267	(2)
Nonautonomous oscillator	269	(1)
Absolute Stability of Discrete Systems	270	(4)
Notes	274	(3)
References	277	(18)
Subject Index	295