Dynamics Research On Several Classes Of Discontinuous Neural Network Models

Mathematical models in many research areas often need to be characterized by differential equations with discontinuous righthand sides,such as neural networks,biology and control engineering.Because the vector field of differential equations with discontinuous righthand sides is no longer continuous,thus the classical dif-ferential equation theory and research methods are often no longer applicable.At this time,we need to use new theoretical tools to research,such as Filippov differ-ential inclusion theory and so on.In this paper,under the framework of differential inclusion theory,the related dynamic properties of neural network models with dis-continuous activation functions are studied,such as the existence and uniqueness,global exponential stability and finite time stabilization about equilibrium points and almost periodic solutions.In the research process of this paper,the main theo-retical tools are differential inclusion theory,set-valued analysis,nonlinear analysis and matrix theory and so on.By constructing Lyapunov functional and using in-equality techniques,some new results are obtained,such as the robust stability of complex-valued uncertain neural networks with discontinuous activation functions and so on.The structure of this dissertation is as follows:In chapter 1,the overview about the theory of differential equations with discontinuous righthand sides is briefly introduced.Then,the research history of artificial neural network and the dynamic behaviors research of neural networks with discontinuous activation functions are summarized.Finally,the structure of this paper and the main research content,as well as the innovation of this paper are presented.In chapter 2,some important basic knowledge and theories are introduced.In chapter 3,we studied the robust stability of equilibrium point of a class of complex-valued uncertain neural networks with discontinuous activation functions.By separating the real and imaginary part,we obtain the equivalent real-valued discontinuous system of the original complex-valued system.In this paper,the dy-namic properties of the original complex-valued uncertain neural network system are studied by using the real-valued system.In this paper,it is not necessary to require that activation functions are bounded or monotonic,so that the conditions of activation functions are general.In the framework of Filippov differential inclu-sion theory,the sufficient conditions for the existence of equilibrium point of the system are obtained by using the Leray-Schauder alternative theorem in multi-valued analysis and matrix inequality techniques and so on.Then,under the condition that activation functions satisfy monotonicity,the existence,uniqueness and global exponential stability of the equilibrium point are studied by construct-ing Lyapunov functional and using nonlinear analysis and so on.We generalize and perfect some results of real-valued uncertain neural networks about the robust stability of equilibrium points.In chapter 4,we studied the dynamic properties of almost periodic solution-s for discontinuous complex-valued competitive neural networks with time delay.First,by separating the real and imaginary part,we obtain the equivalent real-valued system of the original discontinuous complex-valued system.In this paper,it is assumed that discontinuous activation functions should be monotonically non-decreasing,but don't need to be bounded.In the framework of Filippov differential inclusion theory,we have researched the boundedness of system's solutions,and obtained sufficient conditions for the existence of asymptotic almost periodic so-lutions by constructing the suitable Lyapunov functional and so on.Then,the existence of the system's almost periodic solution is discussed by using Arzela-Ascoli theorem and Lebesgue control convergence theorem.Finally,the existence,uniqueness and global exponential stability of the almost periodic solution are studied by constructing Lyapunov functional and so on.The results in this chap-ter generalize the results of almost periodic solutions of discontinuous competitive neural networks.In chapter 5,the finite time stabilization of two classes of discontinuous neural network models are studied,that is,the memristive Cohen-Grossberg neural net-works with time-varying delay and uncertain neural networks with discontinuous activations and mixed delays.With regard to the memristive Cohen-Grossberg neural networks with time-varying delay,due to the existence of memristor,the neural network system is actually differential equations with discontinuous right-hand sides from the viewpoint of mathematics.In this paper,the differential system which is easier to study is obtained from the original system by using the method of variable transformations.In the framework of Filippov differential in-clusion theory,the sufficient conditions for the existence of equilibrium point of the original neural network model are obtained by studying the transformed differential system and using the Kakutani fixed point theory of set-valued map.Then,two kinds of discontinuous state feedback controllers are designed,and the sufficient conditions for finite time stabilization of the system are obtained by constructing Lyapunov functional and using inequality techniques and so on.In addition,with regard to uncertain neural networks with discontinuous activation functions and mixed delays,it is easy to obtain that the origin is an equilibrium point of the discontinuous system.By designing two kinds of discontinuous controllers,i.e.,state feedback controller and adaptive controller,the sufficient conditions for finite time stabilization of the system are obtained by constructing Lyapunov functional and so on.In this paper,we studied the existence and stability of the equilibrium point and almost periodic solution about several classes of discontinuous neural network models under the framework of differential inclusion theory.Considering more general systems,such as complex-valued uncertain neural networks with discon-tinuous activation functions and discont,inuous complex-valued competitive neural network,we generalize and perfect some previous conclusions.