Multistability Analysis Of Switched Neural Networks Under State-dependent Switching

Artificial neural network is one of the most important intelligent control technolo-gies,it is a large-scale nonlinear dynamic system inspired by the human brain function.In recent years,as a new type of network system models,switched neural network has become the hot topic in international academic research due to its extensive applica-tion and theoretical research significance.According to the theory of system dynamics,stability is the prerequisite for the normal operation of the system.Therefore,the re-search on stability of switched neural network is not only the need of practice,but also the requirement of theoretical development.In this paper,on the basis of the theories switched systems,we consider the basic theory of switched neural networks with two different kinds of activation functions under state-dependent switching.The specific contents are as follows:As the introduction,in Chapter 1,a review on the development of the theory of switched neural networks is presented.Besides,the development of the theory of multistability and history of dynamics for neural networks are also briefly addressed.The motivations,methods and outlines of this work are also given in this chapter.Finally,the main contents of this paper are briefly introduced.In Chapter 2,we analyze the multistability of switched neural networks with a class of monotonically non-decreasing piecewise linear activation functions under state-dependent switching.Firstly,we establish switched neural network model under state-dependent switching,according to the locations of switching threshold and the geometric characteristics of activation functions,the state space RI is divided into multiple subsets.Then,by using contraction mapping theorem,strictly diagonally dominant matrix theory and some mathematical analysis techniques,we get the exis-tence of multiple equilibria as well as analyze the stability/instability of the equilibria.More interesting,we can find that the switching threshold plays an important,role for stable equilibria in the unsaturation regions of activation functions.Furthermore,for two-neuron switched neural networks,the precise attraction basin of each stable equi-librium point can be figured out based on the time inversion theory,and its boundary is composed of the stable manifolds of unstable equilibrium points and the switching lines.In Chapter 3,we carry out multistability analysis for switched neural networks with a class of non-monotonic activation functions-radial basis activation functions under state-dependent switching.According to the locations of switching threshold and the geometric characteristics of radial basis activation functions,we divide the state space Rn into multiple subsets.Then,by applying Brouwer's fixed point the-orem,Lyapunov function methods and some inequality techniques,it is proved that under some reasonable assumptions on the decomposition of index set and switching threshold,the number of the equilibria,their locations as well as the stability can be obtained.It is worth mentioning that we not only get the maximum number of stable equilibria,but also get sufficient conditions for almost all possible quantities of stable equilibria.Finally,based on the theoretical analysis results above,we give some examples of numerical simulations,and analyze the results of numerical simulations,which verify the correctness and effectiveness of the theoretical analysis.