广义泛函微分方程解的稳定性

本文首先给出广义泛函微分方程解的稳定性的概念,同时建立了有关稳定与不稳定的二个判据,然后作为例子,讨论二阶线性常系数广义微分差分程,得到了这类系统有关稳定与不稳定的若干结果,并发现了一些关于稳定性的分支或类似于分支的现象。     

The stability of the solutions of a class of integrodifferential systems with infinite delays

Various stability types, analogous to those in the theory of ordinary differential equations, are defined for some classes of integrodifferential systems of equations with infinite delays. Then, on the basis of a variation of constants formula and some results in functional analysis and Laplace transform theory, some necessary conditions and necessary and sufficient conditions are established in order that the trivial solution of a class of systems may be classified as one of the stability types considered in the definitions.     

Classification of stability-like concepts and their study using vector Lyapunov functions

In the first section, stability-like definitions for ordinary differential equations are derived from a general qualitative concept. It is shown that the classical definitions of stability in the sense of Lyapunov, and their extensions can easily be deduced from this general formulation. A classification of all the definitions which may be derived is proposed.The second section contains the main results of this paper. It deals with the “comparison method” based upon one of T. Wazewski's theorems on differential inequalities. Several authors have used this method in order to investigate stability-like properties. We display the structure of this method, in order to state and prove some general comparison principles. These apply to the class of concepts considered earlier.In the last section some new results about stability and attractivity of sets are obtained as examples for the comparison principles. A theorem on stability in tube-like domains is proved in order to emphasize the generality and the flexibility of the comparison method.     

On partial stability of a dynamic system that is integrally continuous with respect to a small parameter

For a dynamic system described by Carathéodory ordinary differential equations with a small parameter we introduce the definitions of a type of partial stability, attraction, and asymptotic stability. We state theorems giving sufficient conditions for stability in the new definitions. In particular, in terms of perturbed Lyapunov functions we obtain conditions for partial asymptotic stability that generalize known results. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 29–36.     

Local Baire classification of Lyapunov exponents

One considers two different definitions of the Baire class of a functional at a point. These definitions are in agreement with the common definition of the Baire class. The semicontinuity of a functional at a point is associated with its inclusion into the first Baire class at that point in the sense of the said definitions for Lyapunov exponents of a homogeneous nth-order system. In particular, it is shown that for the two smallest exponents, the inclusion into the first Baire class at a point is equivalent to semicontinuity in the sense of one of the two definitions and continuity in the sense of the other. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 56–70, 2007.     

Definitions of equilibrium in network formation games

We examine a variety of stability and equilibrium definitions that have been used to study the formation of social networks among a group of players. In particular we compare variations on three types of definitions: those based on a pairwise stability notion, those based on the Nash equilibria of a link formation game, and those based on equilibria of a link formation game where transfers are possible.Bloch is also affiliated with the University of Warwick.     

控制系统的广义传递函数和运动稳定性判定方法的改进

本文给出了控制系统的广义传递函数和它的连续性的定义.利用广义传递函数作为工具,建立了一组判定运动稳定性的定理,从而把对一个控制动力学系统特性的了解建立在这个系统的被观测过程之中,简化了运动稳定性问题的判定方法.     

Generalized metarationalities in the graph model for conflict resolution

A metarational tree is defined within the graph model for conflict resolution paradigm, providing a general framework within which rational behavior in models with two decision makers (DMs) can be described more comprehensively. A new definition of stability for a DM that depends on the total number, h, of moves and counter-moves allowed is proposed. Moreover, the metarational tree can be refined so that all moves must be unilateral improvements, resulting in a new set of stability definitions for each level of the tree. Relationships among stabilities at various levels of the basic and refined trees are explored, and connections are established to existing stability definitions including Nash stability, general metarationality, symmetric metarationality, sequential and limited-move stability, and policy equilibria.     

STABILITY OF SOLUTION OF TIME-VARYING SINGULAR SYSTEMS WITH DELAY

1IntroductionInmanydifferentareas)ithasbeenfoundthatmanyproblemsaremodeledbysingularsystems,suchasoptimalcontrolproblemsandconstrainedcontrolproblems,electricalcircuits,somepopulationgrowthmodelsandsingularper-turbations.Sincedelayoftenoccursintheseproble…     

关于插值的稳定性

实际计算表明,高阶多项式插值是不稳定的,分片低阶多项式插值则有良好的稳定性.在[1]中对此有简要的说明,本文将对此进行较详尽的理论分析.     

On the stability of Volterra integral equations with a lagging argument

Stability properties of numerical methods for Volterra integral equations with lagging argument are considered. Some suitable definitions for the stability of the numerical methods are included, and plots of the stability regions for particular cases are included.     

基于序列稳定性的冲突当局者行为分析模型

不同稳定性可以折射冲突当局者的不同行为类型,为了更全面且深入地了解当局者的行为类型,本文将现有四种基本稳定性拓展至六种,构建了一个能够提高冲突分析图模型理论普遍实用性的决策方法。首先,提出了强、弱序列稳定性(SEQ稳定)的逻辑概念,以此分析当局者具有不同程度的过度自信下的冲突决策过程。然后,定义了强、弱SEQ稳定的矩阵表达,以此促进相应稳定解的计算和编程算法的实现。最后,使用“制造商与零售商渠道价格冲突”案例说明强弱SEQ稳定的意义。研究发现:冲突当局者的过度自信偏差并不都会产生负面影响,有时会利于当局者的集体利益。     

Doubly-weighted pseudo almost automorphic solutions for nonlinear stochastic differential equations driven by Lévy noise

This paper introduces the definitions of Poisson doubly-weighted pseudo almost automorphy and doubly-weighted pseudo almost automorphy (DWPAA) in distribution. Based on some suitable assumptions, we establish some basic theory for these definitions, and investigate the existence, uniqueness and exponential stability of the DWPAA solution in distribution for a class of nonlinear stochastic differential equations driven by Lévy noise. Finally, an example is further given to illustrate the effectiveness of our results.     

An extension of general linear methods

General Linear Methods (GLMs) were introduced as the natural generalizations of the classical Runge–Kutta and linear multistep methods. An extension of GLMs, so-called SGLMs (GLM with second derivative), was introduced to the case in which second derivatives, as well as first derivatives, can be calculated. In this paper, we introduce the definitions of consistency, stability and convergence for an SGLM. It will be shown that in SGLMs, stability and consistency together are equivalent to convergence. Also, by introducing a subclass of SGLMs, we construct methods of this subclass up to the maximal order which possess Runge–Kutta stability property and A-stability for implicit ones.     

四元数矩阵的惯性定理与稳定性

本文给出了四元数矩阵惯性的定义,讨论了四元数体上Lyapunov矩阵方程的唯一解,推广了一般惯性定理、Lyapunov稳定性定理、Carlson-Schneider定理、Stein稳定性定理等一些重要的结果到四元数矩阵,同时得出了四元数体上稳定矩阵的一些判别条件.     

Stability of functional equations in single variable

This paper discusses Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equation, nonlinear functional equation and iterative equation. Surveying many known and related results, we clarify the relations between Hyers-Ulam stability and other senses of stability such as iterative stability, continuous dependence and robust stability, which are used for functional equations. Applying results of nonlinear functional equations we give the Hyers-Ulam stability of Böttcher's equation. We also prove a general result of Hyers-Ulam stability for iterative equations.     

Reducible Runge-Kutta methods

Reducible Runge-Kutta methods are characterized by means of special matrices. Previous definitions of reducibility are incorporated. This characterization may be useful in the study of algebraic stability and in studies of existence and uniqueness.     

On the Stability of Interpolation

Some definitions on stability of interpolating process are given and then the sufficient and necessary conditions are obtained. On this basis, we conclude that the Lagrange interpolation is unstable, whereas several types of piecewise low order polynomial interpolation are stable. For high order approximation with data on isometric nodes, we recommend the Bernstein approximation owing to its high stability. Some ideas on the relationship between stability and convergence of interpolating process are also presented.     

Stability properties of numerical methods for solving delay differential equations

Stability properties of numerical methods for delay differential equations are considered. Some suitable definitions for the stability of the numerical methods are included and Runge-Kutta type methods satisfying these properties are tested on a numerical example.     

向量值α次熵映射及熵方程的Hyers-Ulam稳定性

引入了向量值α次熵映射及熵方程的概念,定义并研究了他们的稳定性,证明了：当0〈α≠1时,向量值α次熵方程在Hyers-Ulam意义下是稳定的,给出了向量值α次熵映射的一般形式;证明了向量值α次熵映射序列是稳定的,并给出了向量值α次熵映射序列的一般形式.