The Stability Of Uncertain Dynamical Systems And Applications

A system is called certain system if its history and future are determined by its state at one moment. Differential equation, difference equation or iterative equation with fixed parameters are important examples of certain system. A system is called uncertain system if its history and future "depend not only on its state at one moment but also on some uncertain factors. Interval differential (difference) equation, fuzzy differential equation or stochastic differential equation are important examples of uncertain system. The purpose of uncertain dynamical systems theory is to study rules of change in state which depends oe time. There are many important applications of uncertain dynamical system on control theory, artificial intelligence, language theory, finance, quantum theory and network theory.In introduction, the uncertain dynamical system is defined. Three kinds of uncertain factors that will be studied in this paper are pointed out. That is,1. the uncertainty that is caused by varying of uncertain parameters of system;2. the vagueness that is caused by imprecise knowledge of notions in system; 3. the probability that is caused by numerus stochastic factors which influence on va/ying of system. Finally, many known results on Robust stability of interval dynamical systems, fuzzy differential equation, stochastic differential equation and stochastic neural network are summarized.In chapter 1, the distribution of polynomial zeros in complex plane is studied. A algebraic relationship of the distribution of polynomial zeros and its coefficients is obtained. Two classical theorems of polynomial zeros by Cauchy andPellet are improved by means of the relationship. Furthermore, the result in and H. [52] is improved to fit complex coefficients. New tests of stability for linear systems is obtained. As applications of main theorem, D stability and estimation of Robustness are studied. On the other hand, by means of the method in D.Xu [70], a special D stability, r stability, is discussed. As a application of main theorem, Schur stability of a state feedback closed-loop system is studied.In chapter 2, by means of the properties of a differential and integral calculus for fuzzy-set-valued mappings and completeness of metric space of fuzzy numbers, the existence, uniqueness and continuous dependence of a solution to a fuzzy retarded functional differential equation with nonlocal condition, to a fuzzy retarded integro-differential equation and to a general neutral functional equation are derived. To the best knowledge of author, there are the known results of continuous dependence only for fuzzy differential equation. The results of continuous dependence for fuzzy functional differential equation are obtained firstly.In chapter 3, by means of the properties of a differentir! and integral calculus for martingale, local martingale and semi-martingale, Ito's formula as well Young's inequality, Holder inequality and Burkholder-Davis-Gundy inequality, the almost sure exponential stability for stochastic delays Recurrent neural networks is studied. The moment exponential stability of stochastic/Cohen-Grossberg neural network with time-varying delays and of stochastic delays Hopfield neural network with reaction-diffusion terms is discussed. To the best knowledge of author, there are not known results of exponential stability for stochastic delays Hopfield neural network with reaction-diffusion terms.