Stability and Stabilization of Nonlinear Systems With Random Structures

by Kats; I. Ya

• ISBN13: 9780415272537
• ISBN10: 041527253X
• Format: Hardcover
• Copyright: 2002-08-22
• Publisher: CRC Press

Summary

Nonlinear systems with random structures arise quite frequently as mathematical models in diverse disciplines. This monograph presents a systematic treatment of stability theory and the theory of stabilization of nonlinear systems with random structure in terms of new developments in the direct Lyapunov's method. The analysis focuses on dynamic systems with random Markov parameters. This high-level research text is recommended for all those researching or studying in the fields of applied mathematics, applied engineering, and physics-particularly in the areas of stochastic differential equations, dynamical systems, stability, and control theory.

Table of Contents

Introduction to the Series	xi
Preface	xiii
Preliminary Analysis	1	(22)
Introductory Remarks	1	(1)
Random Variables and Probability Distributions	2	(3)
A multidimensional normal (Gaussian) distribution	4	(1)
A Poisson distribution with parameter λ > 0	5	(1)
Probability Processes and their Mathematical Description. Markov Processes	5	(5)
Random Differential Equations. Modelling of Systems Exposed to Random Perturbations	10	(4)
System with Random Structure. Examples	14	(8)
Comments and References	22	(1)
Stability Analysis Using Scalar Lyapunov Functions	23	(50)
Introductory Remarks	23	(1)
Stability Concepts for Stochastic Systems	24	(4)
Random Scalar Lyapunov Functions	28	(6)
Conditions of Stability in Probability	34	(12)
Converse Theorems	46	(6)
Stability in Mean Square	52	(12)
Stability in Mean Square of Linear Systems	64	(5)
Comments and References	69	(4)
Stability Analysis Using Multicomponent Lyapunov Functions	73	(62)
Introductory Remarks	73	(1)
Vector Lyapunov Functions	74	(9)
A scalar approach	75	(5)
A vector approach	80	(3)
Stochastic Matrix-Valued Lyapunov Functions	83	(10)
The structure of averaged derivatives of stochastic matrix-valued functions	87	(6)
Stability Analysis in General	93	(2)
Stability Analysis of Systems in Ito's Form	95	(9)
Decomposition of perturbed motion equations	95	(3)
The structure of averaged derivatives of hierarchical matrix-valued functions	98	(2)
Sufficient conditions of stability in probability of a stochastic Ito system	100	(4)
Stochastic Singularly Perturbed Systems	104	(13)
Statement of the problem and main definitions	104	(2)
SMVF for singularly perturbed systems	106	(3)
Theorems on stability in the whole of a singularly perturbed system	109	(3)
A stochastic linear autonomous singularly perturbed system	112	(5)
Large-Scale Singularly Perturbed Systems	117	(16)
Description of the systems	117	(2)
Independent subsystems	119	(1)
Stability conditions for stochastic singularly perturbed autonomous large-scale systems	120	(9)
Analysis of linear autonomous singularly perturbed large-scale system	129	(4)
Comments and References	133	(2)
Stability Analysis by the First-Order Approximation	135	(30)
Introductory Remarks	135	(1)
Stability Criterion by the First-Order Approximation	135	(6)
Asymptotic stability in probability in the whole and exponential stability in mean square	135	(3)
A systems with random structure and slowly-varying right-hand sides	138	(3)
Stability with Respect to the First-Order Approximation (continued)	141	(8)
The system of the first-order approximation with constant structure	141	(3)
The system of the first-order approximation with an unstable structure	144	(5)
Stability by First-Order Approximation of Systems with Random Delay	149	(7)
Convergence of Stochastic Approximation Procedures	156	(7)
The Robbins -- Monro procedure	156	(1)
The stochastic programming procedure	157	(3)
The procedure of establishing the root of an unknown function	160	(2)
Time-continuous measurements in the presence of random jumps	162	(1)
Comments and References	163	(2)
Stabilization of Controlled Systems with Random Structure	165	(36)
Introductory Remarks	165	(1)
Problems of Stabilization	166	(5)
Optimal Stabilization	171	(4)
Linear-Quadratic Optimal Stabilization	175	(5)
Sufficient Stabilization Conditions for Linear Systems	180	(6)
Optimal Solution Existence	186	(5)
The Small Parameter Method Algorithm	191	(8)
Comments and References	199	(2)
Applications	201	(18)
Introductory Remarks	201	(1)
A Stochastic Version of the Lefschetz Problem	202	(2)
Stability in Probability of Oscillating Systems	204	(4)
Stability in Probability of Regulation Systems	208	(3)
Price Stability in a Stochastic Market Model	211	(7)
A stochastic market model	212	(1)
A management problem based on stability conditions	212	(1)
Analysis of stability	213	(5)
Comments and References	218	(1)
References	219	(16)
Index	235