Stability analysis of numerical methods for systems of neutral delay-differential equations

Stability analysis of some representative numerical methods for systems of neutral delay-differential equations (NDDEs) is considered. After the establishment of a sufficient condition of asymptotic stability for linear NDDEs, the stability regions of linear multistep, explicit Runge-Kutta and implicitA-stable Runge-Kutta methods are discussed when they are applied to asymptotically stable linear NDDEs. Some mentioning about the extension of the results for the multiple delay case is given.  

农村产业结构布局优化的数学模型及其稳定性分析

本文主要介绍作者多年来在农业生产结构优化研究中构造的数学模型，并从理论上证明了该模型的稳定性  

高阶广义KDV方程和KP方程的数值解法

采用一种线性隐格式来解3-阶和5-阶的广义Korteweg-De Vries(KDV)方程,对这种方法做一推广,就能应用到它的二维形式广义Kadomtsew-Petviashvili(KP)方程．这种方法是无条件稳定的,且是无损耗的．数值实验描述了一个线性孤波运动的情形以及两个孤波交互的情形,从结果来看,它们满足孤立子解的两个守恒-动量守恒和能量守恒．  

Computing Stability of Differential Equations with Bounded Distributed Delays

This paper deals with the stability analysis of scalar delay integro-differential equations (DIDEs). We propose a numerical scheme for computing the stability determining characteristic roots of DIDEs which involves a linear multistep method as time integration scheme and a quadrature method based on Lagrange interpolation and a Gauss–Legendre quadrature rule. We investigate to which extent the proposed scheme preserves the stability properties of the original equation. We derive and prove a sufficient condition for (asymptotic) stability of a DIDE (with a constant kernel) which we call RHP-stability. Conditions are obtained under which the proposed scheme preserves RHP-stability. We compare the obtained results with corresponding ones using Newton–Cotes formulas. Results of numerical experiments on computing the stability of DIDEs with constant and nonconstant kernel functions are presented.  

A Stable Numerical Approach for Implicit Non-Linear Neutral Delay Differential Equations

In this paper we consider implicit non-linear neutral delay differential equations to derive efficient numerical schemes with good stability properties. The basic idea is to reformulate the original problem eliminating the dependence on the derivative of the solution in the past values. Our hypothesis on the original equation allow us to study the boundedness and asymptotic stability of the true and numerical solutions by the theory of stability with respect to the forcing term.  

On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization

We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values are evaluated approximately. We show how these approximations should be controlled in order to satisfy the desired optimality tolerance. For example, this is relevant in the context of Lagrangian relaxation, where obtaining exact information about the function and subgradient values involves solving exactly a certain optimization problem, which can be relatively costly (and as we show, in any case unnecessary). We show that approximation with some finite precision is sufficient in this setting and give an explicit characterization of this precision. Alternatively, our result can be viewed as a stability analysis of standard proximal bundle methods, as it answers the following question: for a given approximation error, what kind of approximate solution can be obtained and how does it depend on the magnitude of the perturbation?  

非线性系统稳定分析的特征函数法及其应用

本文引入一个特征函数，用于定量刻画非线性常微分方程的指数稳定性。与常用的Lyapunov方法相比，该方法简单、易用、而且易获得对一族范数（所有单调范数）皆成立的稳定性条件。所获结果推广了稳定理论中的一些著名结论，并应用于非线性连续神经网络的指数稳定性分析，推广和深化了[1-3]所获得的基本结论。  

一类不确定双线性系统的状态反馈Robust控制

本文针对一类含不确定性的双线性系统设计了一种线性状态反馈控制．在一定的条件下，利用改进的李雅普诺夫第二方法关于稳定性的理论，证明了系统的稳定性．并给出例子说明．  

New results for global stability of a class of neutral-type neural systems with time delays

This paper studies the global convergence properties of a class of neutral-type neural networks with discrete time delays. This class of neutral systems includes Cohen–Grossberg neural networks, Hopfield neural networks and cellular neural networks. Based on the Lyapunov stability theorems, some delay independent sufficient conditions for the global asymptotic stability of the equilibrium point for this class of neutral-type systems are derived. It is shown that the results presented in this paper for neutral-type delayed neural networks are the generalization of a recently reported stability result. A numerical example is also given to demonstrate the applicability of our proposed stability criteria.  

Dynamic system methods for solving mixed linear matrix inequalities and linear vector inequalities and equalities

A novel idea is proposed for solving a system of mixed linear matrix inequalities and linear vector inequalities and equalities. First, the problem is converted into an unconstrained minimization problem with a continuously differentiable convex objective function. Then, a continuous-time dynamic system and a discrete-time dynamic system are proposed for solving it. Under some mild conditions, the proposed dynamic systems are shown to be globally convergent to a solution of the problem. The merits of the methods refer to their simple numerical implementations and capability for handling nonstrict LMIs easily. In addition, the methods are promising in neural circuits realization, and therefore have potential applications in many online control problems. Several numerical examples are presented to illustrate the performance of the methods and substantiate the theoretical results.