不连续大系统的实用稳定性

本文研究具有各种不同分解形式的不连续大型动态系统的实用稳定性,而这一类问题几乎还未研究过.我们提出了不连续大系统的各种实用稳定性和不稳定性的概念,得到了判别具有各种不同分解形式的不连续大系统的实用稳定性或实用不稳定性的若干有效判别准则;末了提出了进一步可以考虑的问题.     

关于不连续动态系统的强实用稳定性

本文发展了在文献[1]中所提出的强实用稳定的概念;在此基础上,分别研究了具有内扰和外扰的两类不连续动态系统的强实用稳定性,强实用一致稳定性和非强实用稳定性;分别得到了这两类系统具有相应稳定性的判别准则;最后,还给出例子,说明这些判别准则的应用。     

带外扰的不连续系统的实用稳定性

文[1]中首先对连续系统研究并讨论了实用稳定性的概念,文[2]在[1]的基础上修改了实用稳定性的定义,并讨论了与之有关的问题。文[3]在文[2]实用稳定性定义的基础上讨论了不连续系统的实用稳定性,得到了判别实用稳定性、不稳定性的三个充分准则。但实用稳定性的研究还未受到充分的重视,也没有深入地进行研究。     

一类不连续系统的推广Matrosov稳定性定理

在一般Filippov解意义下给出利用一族满足一定条件的Lyapunov函数的更一般的Matrosov定理.     

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．     

一类不连续系统关于闭不变集的有限时间稳定性研究

主要研究右端不连续系统在Filippov解意义下关于闭不变集(未必是紧集)的有限时间稳定问题.当Liapunov函数是Lipschitz连续的正则函数情况下,给出了相关的Liapunov稳定性定理.     

一类复杂时滞系统的λ实用稳定性

主要研究了一类具有多时滞的复杂微分系统的λ实用稳定性问题,通过对Lyapunov函数方法以及比较原理的运用,建立一般形式的实用稳定性直接判据.在此基础上给出讨论这类多时滞复杂微分系统实用稳定性的新方法.     

一类时滞连续模糊系统稳定性研究

对一类状态时滞连续模糊系统模型稳定性进行了研究,基于Lyapunov稳定性理论证明其系统的稳定性,利用线性矩阵不等式(LMI)方法,得到其稳定的一个充分条件.最后用数值例子进一步说明本文所给方法的有效性.     

线性时滞系统实用稳定性的一个简明判据

利用比较的方法，获得了线性时滞系统实用稳定性的一个简单判定定理．     

不连续激活函数的时滞神经网络的多稳定性

研究了不连续激活函数的时滞神经网络的多稳定性问题,在所研究的神经网络中,激活函数并不需要是连续的和单调的．给出了判断该神经网络多个平衡点存在及局部指数稳定的充分条件．最后,给出了两个数值仿真例子来验证本文获得结果的有效性和较小的保守性．     

Connective Stability of Discontinuous Dynamic Systems

Connective stability is defined for interconnected systems under discontinuous structural perturbations. Stability conditions are established using Lipschitz and C¹-type Lyapunov functions. A considerable flexibility of the conditions is achieved by assuming the functions to be parameter dependent. For efficient testing, the obtained stability conditions are converted to convex optimization problems via the theory of M-matrices.     

On the Stability of Invariant Sets of Discontinuous Dynamical Systems

We establish sufficient conditions for the stability, asymptotic stability, and instability of invariant sets of discontinuous dynamical systems.     

Practical Stability in the pth Mean of Stochastic Differential Equations with Discontinuous Coefficients

In this paper,we give sufficient conditions to analyze the practical stability in the pth mean of stochastic differential equations with discontinuous coefficients.The Lyapunov-like function plays an important role in analysis.Some numerical computations are carried out to illustrate the theoretical results.     

Sampling of Discontinuous Dirac Systems

Homogenization and error analysis of an optimal interior control problem in the framework of Stokes’ system, on a domain with rapidly oscillating boundary, are the subject matters of this article. We consider a three dimensional domain constituted of a parallelepiped with a large number of rectangular cylinders at the top of it. An interior control is applied in a proper subdomain of the parallelepiped, away from the oscillating volume. We consider two types of functionals, namely a functional involving the L ²-norm of the state variable and another one involving its H ¹-norm. The asymptotic analysis of optimality systems for both cases, when the cross sectional area of the rectangular cylinders tends to zero, is done here. Our major contribution is to derive error estimates for the state, the co-state and the associated pressures, in appropriate functional spaces.     

初始时刻不同的奇异系统的稳定性

给出了初始时刻不同的奇异系统实用稳定性的定义,利用比较原理和Lyapunov函数方法研究了初始时刻不同的奇异系统的实用稳定问题,并得到了系统实用稳定性的判定准则.     

On the Robust Practical Global Stability of Nonlinear Time-varying Systems

The goal of this paper is twofold. The first part presents a Lyapunov statement and proof of the concept of robust global practical exponential stability (RpGES) for nonlinear time varying systems. A RpGES-Lyapunov function for the overall system is exhibited. Its proof is a consequence of some results on converse Lyapunov theorems obtained by Tsinias in 19. The second part provides sufficient conditions for the robust practical globally uniformly asymptotically stability (RpGUAS).     

Structural Stability of Discontinuous Galerkin Schemes

The goal of this work is to determine classes of traveling solitary wave solutions for a differential approximation of a discontinuous Galerkin finite difference scheme by means of an hyperbolic ansatz. It is shown that spurious solitary waves can occur in finite-difference solutions of nonlinear wave equation. The occurence of such a spurious solitary wave, which exhibits a very long life time, results in a non-vanishing numerical error for arbitrary time in unbounded numerical domain. Such a behavior is referred here to have a structural instability of the scheme, since the space of solutions spanned by the numerical scheme encompasses types of solutions (solitary waves in the present case) that are not solutions of the original continuous equations. This paper extends our previous work about classical schemes to discontinuous Galerkin schemes (David and Sagaut in Chaos Solitons Fractals 41(4):2193?C2199, 2009; Chaos Solitons Fractals 41(2):655?C660, 2009).