Multistability Analysis Of Switched Neural Networks With State-Dependent Switching Rules

With the development of science and technology,neural networks have been widely applied in many fields,such as optimization,associative memory and deep learning.The theoretical study of the dynamic behavior of neural networks is the basis for their applications.Different demands are being placed on the dynamic behavior of neural networks among their various applications.For example,when considering optimization problems,a neural network is often needed to have a globally stable equilibrium point.While associative memory applications may require the neural networks to have multiple stable equilibria.The more the number of stable equilibria,the higher the storage capacity of the neural network in the associative memory application.Compared with traditional neural networks,the switching design of the switched neural network with state-dependent rules has a significant effect on the increase of the number of equilibria and stable equilibria.The study of the multistability of switched neural networks with state-dependent rules has significant research value in its theoretical expansion and practical application.In this dissertation,we study the switched neural networks with state-dependent switching rules.Given the discontinuity of the system caused by switching,we transform the models described by differential equations with discontinuous right hand sides into a differential inclusion under the framework of Filippov and define the solution and equilibrium point of the switched system.Using the theories of differential inclusion,fixed point theorem,stability,and matrix technique,we focus on the multistability of the switched neural network models with smooth activation functions,the switched T-S fuzzy neural networks with a general class of activation functions and the switched fuzzy logic neural network models with state-dependent switching rules.The details of the study are as follows:1.The multistability problem of switched neural networks with sigmoidal activation functions and state-dependent switching rules is considered.The multistability analysis with such an activation function is difficult because state-space partition is not as straightforward as that with piecewise-linear activations.For such smooth functions,this dissertation first defines the upper and lower bound functions of the system,and the functions are piece-wise continuous with respect to the switching threshold.A new state space partition is proposed by combining the zero points of the upper and lower bound functions and the designs of the switching threshold.Then,the number of equilibria and stable equilibria is available using the fixed theorem and local linearization method.Numerical simulation examples are used to verify the theoretical results under different switching threshold designs.2.The multistability problem of switched neural networks with Gaussian activation functions and state-dependent switching rules is discussed.For the connection weight parameters between neurons,we first give the definitions of four different index sets that characterize the synaptic weights of neurons,which shows that the introduction of switching allows for fuller utilization of the weight information of the neural networks.Based on the geometric properties of the Gaussian function,seven different ranges of switching are given.In this dissertation,we discuss the multistability of the switched neural network with Gaussian activation function in detail for the combination of different index sets and different switching thresholds,derive four sets of stability criteria,and determine the existence of 7p15p23p3 equilibria and 4P13P22P3 locally stable equilibria,where p1,p2,p3 are non-negative integers and satisfy 0≤p1+p2+p3≤n,n is the number of neurons.The numerical simulation examples verify the theoretical results for the cases where the system has the maximum number of stable equilibria(i.e.,4n)and the minimum number of stable equilibria(i.e.,2n),respectively.3.The multistability problem of switched T-S fuzzy neural networks with a general class of activation functions and state-dependent switching rules is analyzed.By the fixed theorem and the properties of the M-matrix,the existence and stability of equilibria are analyzed.What’s more,attraction basins of stable equilibria are estimated.In this dissertation,four sets of stability criteria are derived for switched T-S fuzzy neural networks with a general class of activation functions,and the existence of 3k12k2 exponentially stable equilibria is obtained,where k1,k2 are non-negative integers and satisfy 0<k1+k2≤n,n is the number of neurons.Finally,three numerical simulation examples are elaborated to verify the multistability results of switched T-S fuzzy neural networks with three different activation functions,and one application simulation example is elaborated to verify the validity of its theoretical results in associative memory applications.4.The multistability problem of switched fuzzy logic neural networks with ReLU activation function and state-dependent switching rules is studied.Given the popularity and versatility of unbounded activation functions and fuzzy logic operators,a switched fuzzy logic neural network model with ReLU activation function is constructed.The unboundedness of the ReLU function and the characteristics of switching of the system parameters pose difficulties for the multistability analysis of the neural networks.Based on the properties of the positively invariant set,we derive seven sets of sufficient conditions to ensure the multistability of switched fuzzy neural networks with ReLU activation function.It is shown that there are up to 3n-2n-1 exponentially stable equilibria in an n-neuron switched fuzzy neural network.Finally,We elaborate on three numerical examples to illustrate the theoretical results and a potential application in associative memories.