Stability Analysis Of Mix-valued Logical Networks

In recent years,with the development,of research on neural networks and genetic systems,logical networks have played increasingly important roles both in theory and application.Motivated by the semi-tensor product of matrices,the algebraic expression of logical networks is given for easier research.As the generalization of Boolean networks,mix-valued logical networks are present-ed which are more generic and can describe more practical problems.In the research field of mix-valued logical systems,the stability analysis is an impor-tant research orientation.Therefore,some stability issues of mix-valued logical networks are studied by using the semi-tensor product of matrices in this pa-per.Two main components are included in this research:the set stability analysis of mix-valued logical networks,and the global stochastic stability and robust stability analyses of Markov jump mix-valued logical networks.The main contents of this thesis are listed as follows:In Chapter 1,the background of logical networks and the semi-tensor product of matrices is introduced.Also,results concerning stability issues in logical networks are provided.At last,the main research results and organi-zations of this paper are given.In Chapter 2,the definition and operation methods of the semi-tensor product of matrices are firstly presented.Based on the permutation matrix,some useful and important properties of the semi-tensor product of matrices are introduced.Then,the description of mix-valued logical networks is proposed.By using the semi-tensor product of matrices,the algebraic form of mix-valued logical networks is given for easier research.Chapter 3 investigates the set stability of mix-valued logical networks.First,the fixed point of mix-valued logical networks is introduced,based on which,the global stability of mix-valued logical networks is studied.Then,the concept of invariant,subset is introduced,and a method of searching for the largest invariant subset is proposed.Furthermore,a necessary and sufficient condition for the setstability of the concerned networks is developed.At last,the stability of partial states is proposed,and is converted to the study of the set stability for a certain subset.One numerical example is shown to illustrate the effectiveness of the obtained results.In Chapter 4,the Lyapunov function of mix-valued logical networks is pro-posed firstly,based on which,a Lyapunov stability theorem is derived.Then,switched mix-valued logical networks are presented,the stability of which is studied based on the common Lyapunov function.Next,Markov jump mix-valued logical networks are investigated.A necessary and sufficient condi-tion for the global stochastic stability of the concerned networks is developed.Meanwhile,a Lyapunov theorem is derived.In addition,the robust stabili-ty of the concerned networks with uncertain transition probability matrices is concerned,and an equivalent condition for the robuststability is provided.At last,numerical examples are shown to illustrate the effectiveness of the obtained results.In Chapter 5,main results of this article are reviewed and summarized.