ON GLOBAL ROBUST STABILITY FOR INTERVAL HOPFIELD NEURAL NETWORKS WITH TIME DELAY

The theorem obtained by Liao was not true (see [2]). So, this paper presents some criteria of global robust stability for interval Hopfield neural networks with time delay. The methods to judge the robust stability are practical and easily verifiable.     

ROBUST GLOBAL EXPONENTIAL STABILITY OF UNCERTAIN IMPULSIVE SYSTEMS

By using the quasi-Lyapunov function, some sufficient conditions of global exponential stability for impulsive systems are established, which is the basis for the following discussion. Then, by employing Riccati inequality and Hamilton-Jacobi inequality approach, some sufficient conditions of robust exponential stability for uncertain linear/nonlinear impulsive systems are derived, respectively. Finally, some examples are given to illustrate the applications of the theory.     

A NEW ALGORITHM FOR COMPUTING LARGEST REAL PART EIGENVALUE OF MATRICES：COLLATZ ＆ PERRON-FROBERNIUS＇ APPROACH

This paper describes a new method and algorithm for the numerical solution of eigenvalues with the largest real part of positive matrices.The method is based on a numerical implementation of Collatz’s eigenvalue inclusion theorem for non-negative irreducible matrices.Eigenvalues are analyzed for the studies of the stability of linear systems.Finally, a numerical discussion is given to derive the required number of mathematical operations of the new algorithm. Comparisons between the new algorithm and several well known ones, such as Power, and QR methods, are discussed.     

MODIFIED BERNOULLI ITERATION METHODS FOR QUADRATIC MATRIX EQUATION

We construct a modified Bernoulli iteration method for solving the quadratic matrix equation AX^2 ＋ BX ＋ C = 0, where A, B and C are square matrices. This method is motivated from the Gauss-Seidel iteration for solving linear systems and the ShermanMorrison-Woodbury formula for updating matrices. Under suitable conditions, we prove the local linear convergence of the new method. An algorithm is presented to find the solution of the quadratic matrix equation and some numerical results are given to show the feasibility and the effectiveness of the algorithm. In addition, we also describe and analyze the block version of the modified Bernoulli iteration method.     

Some Conditions for Matrices over an Incline To Be Invertible and General Linear Group on an Incline

Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn（L） = PLn （L） for any n （n 〉 2）, in which GLn（L） is the group of all n × n invertible matrices over L and PLn（L） is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.     

MAXIMUM INFORMATION AND OPTIMUM ESTIMATING FUNCTION

In order to construct estimating functions in some parametric models, this paper introduces two classes of information matrices. Some necessary and sufficient conditions for the information matrices achieving their upper bounds are given. For the problem of estimating the median, some optimum estimating functions based on the information matrices are acquired. Under some regularity conditions, an approach to carrying out the best basis function is introduced. In nonlinear regression models, an optimum estimating function based on the information matrices is obtained. Some examples are given to illustrate the results. Finally, the concept of optimum estimating function and the methods of constructing optimum estimating function are developed in more general statistical models.     

GLOBAL EXPONENTIAL STABILITY TO A CLASS OF DIFFERENTIAL SYSTEM WITH DELAY

The global exponential stability of the zero solution to a class of differential system with delay is considered.By constructing a suitable type of Lyapunov functional and using some analytical techniques,we derive some criteria to check exponential stability of this system.The results establish a relation between the delay time and the parameters of the system.Two examples are also given to illustrate the validity of the results.     

对称矩阵的β-性质及其Scaling稳定性分析

In this paper, a concept of the β-property of symmetric matrices is presented which is useful in the perturbation theory for matrices. A necessary and sufficient condition for a symmetric matrix to have the β-property, and the constant β,when it exists, are given. Further, the scaling stability of the symmetric matrix which has the β-property is investigated.     

On invertible nonnegative Hamiltonian operator matrices

Some new characterizations of nonnegative Hamiltonian operator matrices are given. Several necessary and sufficient conditions for an unbounded nonnegative Hamiltonian operator to be invertible are obtained, so that the main results in the previously published papers are corollaries of the new theorems. Most of all we want to stress the method of proof. It is based on the connections between Pauli operator matrices and nonnegative Hamiltonian matrices.     

ESTIMATES OF THE DOUBLE DETERMINANTS OF QUATERNION MATRICES

An estimate of the upper bound is given for the double determinant of the sum of two arbitrary quaternion matrices, and meanwhile the lower bound on the double determinant is established especially for the sum of two quaternion matrices which form an assortive pair. As applications, some known results are obtained as corollaries and a question in the matrix determinant theory is answered completely.     

A novel approach to exponential stability of nonlinear systems with time-varying delays

In this paper, the stability of nonlinear systems with time-varying delays is investigated by means of the concepts of generalized Dahlquist constant, generalized relative Dahlquist constant and relative minimal Lipschitz constant. In detail, two sufficient conditions are derived for the exponential stability of nonlinear systems with time-varying delays and the exponential decay of the solutions is also estimated. Compared with some existing results, our stability conditions are less conservative. Some examples are given to illustrate the effectiveness of the obtained results.     

Robust stability criteria for systems with interval time-varying delay and nonlinear perturbations

This paper considers the robust stability for a class of linear systems with interval time-varying delay and nonlinear perturbations. A Lyapunov-Krasovskii functional, which takes the range information of the time-varying delay into account, is proposed to analyze the stability. A new approach is introduced for estimating the upper bound on the time derivative of the Lyapunov-Krasovskii functional. On the basis of the estimation and by utilizing free-weighting matrices, new delay-range-dependent stability criteria are established in terms of linear matrix inequalities (LMIs). Numerical examples are given to show the effectiveness of the proposed approach.     

Stability radii of positive linear Volterra-Stieltjes equations

We study stability radii of linear Volterra-Stieltjes equations under multi-perturbations and affine perturbations. A lower and upper bound for the complex stability radius with respect to multi-perturbations are given. Furthermore, in some special cases concerning the structure matrices, the complex stability radius can precisely be computed via the associated transfer functions. Then, the class of positive linear Volterra-Stieltjes equations is studied in detail. It is shown that for this class, complex, real and positive stability radius under multi-perturbations or multi-affine perturbations coincide and can be computed by simple formulae expressed in terms of the system matrices. As direct consequences of the obtained results, we get some results on robust stability of positive linear integro-differential equations and of positive linear functional differential equations. To the best of our knowledge, most of the results of this paper are new.     

On the stability and boundedness of solutions of nonlinear vector differential equations of third order

In this paper, by using Liapunov’s second method, we establish some new results for stability and boundedness of solutions of nonlinear vector differential equations of third order. By constructing a Liapunov function, sufficient conditions for stability and boundedness of solutions of equations considered are obtained. Concerning to the subject, some explanatory examples are also given. Our results improve and include a result existing in the literature.     

Robust stability analysis for Markovian jumping interval neural networks with discrete and distributed time-varying delays

This paper investigates robust stability analysis for Markovian jumping interval neural networks with discrete and distributed time-varying delays. The parameter uncertainties are assumed to be bounded in given compact sets. The delay is assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. Based on the new Lyapunov–Krasovskii functional (LKF), some inequality techniques and stochastic stability theory, new delay-dependent stability criteria have been obtained in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are given to illustrate the less conservative and effectiveness of our theoretical results.     

On nonuniform exponential dichotomy of evolution operators in Banach spaces

The problem of nonuniform exponential dichotomy of evolution operators in Banach spaces is considered. Connections between this concept and admissibility of the pair (C ₀,C ₀) are established. Generalizations to the nonuniform case of some results of Van Min, Räbiger and Schnaubelt ([MRS]) are obtained. It is shown that an implication from the uniform case is not true for nonuniform exponential dichotomy. The results are applicable for general time-varying linear equations with unbounded coefficients in Banach spaces.     

Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay

This paper proposes an approach for the robust stability of uncertain systems with interval time-varying delay. The key features of the approach include the introduction of uncorrelated augmented matrix items into the Lyapunov functional and the use of a tighter bounding technology. Unlike existing methodologies, the proposed approach involves neither free weighting matrices nor any model transformation. It can, however, lead to much less conservative stability criteria than the existing ones for the systems under consideration. Numerical examples show that the proposed criteria improve the existing results significantly with much less computational effort.     

Improved delay-dependent robust stability criteria for a class of uncertain mixed neutral and Lur’e dynamical systems with interval time-varying delays and sector-bounded nonlinearity

The previously proposed delay-dependent robust stability criteria for a class of uncertain mixed neutral and Lur’e dynamical systems with interval time-varying delays are refined by eliminating nonlinearity and/or reducing the number of free-weighting matrices. Several robust stability criteria proposed in this paper are simpler and effective. Numerical examples show that the new method is less conservative than some previous approaches.     

A new criterion for global robust stability of interval delayed neural networks

A novel criterion for the global robust stability of Hopfield-type interval neural networks with delay is presented. An example showing the effectiveness of the present criterion is given.     

Parameter-dependent robust stability of uncertain time-delay systems

This paper presents a new delay-dependent and parameter-dependent robust stability criterion for linear continuous-time systems with polytopic parameter uncertainties and time-varying delay in the state. This criterion, expressed as a set of linear matrix inequalities, requires no matrix variable to be fixed for the entire uncertainty polytope, which produces a less conservative stability test result. Numerical examples are given to show the effectiveness of the proposed techniques.