Formulas of Liapunov functions for systems of linear partial differential equations

Formulas of explicit quadratic Liapunov functions for showing asymptotic stability of the system of linear partial differential equations on (0,∞)×Ω, are constructed, where A is an n×n real matrix, u=T(u₁,u₂,…,un), Ω is a bounded domain in Rk with smooth boundary ∂Ω, and Δ denotes the Laplacian operator on Rk with Δu=T(Δu₁,Δu₂,…,Δun). These formulas are also modified and applied to a number of nonautonomous linear and nonlinear systems and models in structural stability, traveling wave, and Navier-Stokes equations.     

基于向量Liapunov函数不连续系统的稳定性研究

基于局部Lipschitz连续且正则(Clarke意义下)的向量Liapunov函数,讨论不连续自治系统的稳定性(Filippov解意义下)．通过定义一类新的向量Liapunov函数的“集值导数”,给出了关于不连续系统的广义比较原理．基于Lipschitz连续且正则的向量Liapunov函数,进一步的给出不连续自治系统的Liapunov稳定性定理．     

差分系统基于两种度量的极端稳定性

首先引入了差分系统基于两种度量的极端稳定性概念 ,然后建立了一些关于差分系统 ( h0 ,h)极端稳定性 (极端一致稳定性 ,极端渐近稳定性 ,极端一致渐近稳定性 )的判定准则 .在所得到的定理中 ,对△ V的限制较弱 ,特别地 ,△V甚至可以恒为正 ,从而便于实际应用 .     

二阶线性矩阵差分方程的解及渐近稳定性

讨论了二阶线性矩阵差分方程AXn+2+BXn+1+CXn=0的解及其渐近稳定性.首先,给出了它的特征方程有解的一个充要条件,然后利用特征方程两个相异的解刻划出该矩阵差分方程的通解,并分析其解的渐近稳定性,最后运用一实例验证了相关结果.     

On the Construction of Globally Asymptotically Stable Liapunov Functions of a Type of Third－order Systems with Nonseparable Variables

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Stability Criteria for Solutions of Systems of Linear Deterministic or Stochastic Delay Difference Equations with Continuous Time

We give spectral and algebraic coefficient criteria (necessary and sufficient conditions) as well as sufficient algebraic coefficient conditions for the Lyapunov asymptotic stability of solutions to systems of linear deterministic or stochastic delay difference equations with continuous time under white noise coefficient perturbations for the case in which all delay ratios are rational. For stochastic systems, mean-square asymptotic stability is studied. The Lyapunov function method is used. Our criteria on algebraic coefficients and our sufficient conditions are stated in terms of matrix Lyapunov equations (for deterministic systems) and matrix Sylvester equations (for stochastic systems).     

一类差分系统周期解的存在性

This paper studies the nonautonomous nonlinear system of difference equations △x(n) = A(n)x(n) f(n, x(n)), n ∈ Z, (*)where x(n) ∈ RN,A(n) = (aij(n))N×N is an N × N matrix, with aij∈ C(R,R) for i,j =ω, z) = f(t, z) for any t ∈ R, (t, z) ∈ R × RN and ω is a positive integer. Sufficient conditions for the existence of ω-periodic solutions to equations (*) are obtained.     

A Mathod to Solve Systems of Fuzzy Linear Equations

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线性方程组的异步松弛迭代法*

本文考虑解线性方程组经典迭代法的异步形式,对系数矩阵为H矩阵,给出了异步迭代过程收敛性的充分条件,这不仅降低了文献[3]对系数矩阵的要求,而且收敛区域比文献[3]的大.     

多时滞微分方程数值稳定性

考虑了时滞微分方程的初值问题,分析了用线性多步法求解一类滞后型微分系统数值解的稳定性,在一定的Lagrange插值条件下,给出并证明了求解滞后型微分系统的线性多步法数值稳定的充分必要条件.     

有限时滞差分方程的稳定性

在本文中,作者建立了一些定理可以确保系统：x(n 1)=f(n,xn）（1）为一致稳定、渐近稳定或一致渐近稳定。在所获得的定理中,不要求△V为常负,这里△V（n,xn)≡V(n 1,x(n 1)-V(n,x(n))=V(n 1,f(n,xn)))-V(n,x(n)),特别地,在定理1中,△V甚至可以恒为正,这大大地改进了已有结果并且更加便于应用。     

正线性方程组的半正解

通过引入正线性方程组Ax=b的可去变量,得到了没有可去变量的正线性方程组有无穷多个半正解情形的充分必要条件以及有惟一半正解情形的充分必要条件.     

Stability of Linear Integrodifferential Systems with Time-varied Confficients with Large Scale

51.IntroductionConsideralinearintegrodifferentialsystemswithtime-variedconfficientswherex6R",A(t)isanXnrealmatrixfunctioncontinuousforoO.ManyscholarshavecontributedtheirpaPers[1-4]todiscussstabilityofzerosolutionofsystem(1.l).Inthispaper,weassumethatsystem(1.1)canbedecomPOsedintotheform=n,A,j(t)isaniXnjrealmatrixfunctioncontinuousforo相似文献   

On the Liapunov Second Method for Difference Equations

We study various kinds of stability of a constant solution of an autonomous difference equation by using auxiliary functions like in the Liapunov Second Method. We establish new general principles and we improve several classical results.     

Asymptotic Stability of Solutions to Perturbed Linear Difference Equations with Periodic Coefficients

We consider perturbed linear systems of difference equations with periodic coefficients. The zero solution of a nonperturbed system is assumed asymptotically stable, i.e., all eigenvalues of the monodromy matrix belong to the unit disk {||<1}. We obtain conditions on the perturbation of this system under which the zero solution of the system is asymptotically stable and also establish continuous dependence of one class of numeric characteristics of asymptotic stability of solutions on the coefficients of the system.     

时滞差分系统基于两种测度的极端稳定性

首先引入了时滞差分系统基于两种测度的极端稳定性概念,然后建立了一些关于时滞差分系统（犺０,犺）极端稳定性（极端一致稳定性,极端渐近稳定性,极端一致渐近稳定性）的判定准则．在所得到的定理中,对Δ犞的限制较弱,特别地,Δ犞甚至可以恒为正,从而便于实际应用．     

常系数齐次线性差分方程组的一个新解法

常系数齐次线性差分方程组的求解方法,已有作者讨论过,但都没有给出一个比较简便的计算方法.本文将给出一个十分简明而有效的常系数齐次线性差分方程组的新求解方法.     

Contours of Liapunov functions

As is well known, the stability of a dynamical system in two dimensions may be demonstrated in a very intuitive fashion from the existence of a suitable positive-definite Liapunov function, providing the contours of this function in a neighborhood of the stable point are Jordan curves. It is shown that the Liapunov function will certainly have this property if the stable point is an isolated stationary point in the sense of the Clarke calculus, but a counterexample is given if this assumption is weakened to the stable point being an isolated local extremum.This work was carried out with the support of the Natural Sciences and Engineering Council of Canada, which is gratefully acknowledged.