Eigenvalues Of Tensors And Its Applications In Stability Analysis Of Nonlinear Systems

Tensors are widely used in the areas of signal processing,Big data science,higher-order Markov chain,machine learning,quantum computing and so on.Recently,the problem on eigenvalues of tensors was proposed and attract much attention,which has been found ap-plications in many fields,such as positive definiteness of tensors,spectral of hypergraphs,partition of hypergraphs,automatic control,image processing,higher-order Markov chain and polynomial optimization etc.The nonlinear system is an important dynamic system.The stability is a basic property of a system,and is an important research topic in system theo-ry.Studying the stability of nonlinear systems by tensor eigenvalues is a new research topic,which is significant for the study on the system theoryThe product of tensors is a fundamental operation of tensors,which is used in many prob-lems of tensors.We first study the product of tensors by maps,give the the more gener-al products of tensors and the properties.Since the eigenvalues of tensors are the roots of characteristic polynomials of tensors,and which is defined by the resultant of the system of high-order equations.There is no good method to calculate the eigenvalues of tensors.In this thesis,we study the theory of tensors eigenvalues and its applications in stability of nonlinear systems.We give some inclusion sets of tensor eigenvalues and some bounds on the spectral radius,further give some criteria of the positive definiteness of tensors.It is important to study the application of tensor eigenvalues in the stability of nonlinear systems.The main work of this thesis is as follows:By the composition of linear maps,we propose and study the more general products of tensors.We naturally obtain two multiplications of tensors,which generalize general product of tensors,Einstein product and n-mode product etc.And the properties of the two multi-plications are presented as well,such as the associate law.We give the properties of tensor representations of quivers,by using it,the properties of eigenvalues of tensors are gotWe give the Brauer-type eigenvalue inclusion sets of stochastic tensors and irreducible tensors,respectively.Furthermore,the Gersgorin-type,Brauer-type and Brualdi-type eigen-value inclusion sets of the general product of tensors are given.The positive definiteness of polynomials is important to the study of the stability of nonlinear systems.The positive defi-niteness of homogeneous polynomials is exactly the positive definiteness of tensors.By using the Brauer-type eigenvalue inclusion sets of tensors,some sufficient conditions of positive(semi-)definiteness of tensors are obtainedWe extend the the Mine-type bound on spectral radius of nonnegative matrices to tensors,give the Mine-type bound on spectral radius of nonnegative tensors.And we further give a criterion of the positive definiteness of tensors via the Mine-type bound.Furthermore,the generalized Minimax Theorem of nonnegative weakly irreducible tensors is presented.More-over,we give some bounds on spectral radius of the general product of tensors,which are characterized by the eigenvalues of tensors A and B.In this thesis,we represent the nonlinear polynomial systems by tensors.And using the eigenvalues of tensors,the positive(semi-)definiteness of tensors and Lyapunov stability the-orem,the sufficient conditions of stable in sense of Lyapunov,asymptotically stable and in-stability of some high-order nonlinear systems are obtainedA tensor representation method for a nonlinear polynomial system is presented,and give the Lyapunov function by the tensor.Further the analysis on the stability of the nonlinear system is presented in terms of the tensor eigenvalues and Lyapunov stability theorem.And the sufficient conditions of stable in sense of Lyapunov,asymptotically stable and instability of some high-order nonlinear systems are obtained.