On equiasymptotic stability of solutions of doubly-periodic impulsive systems

We consider a system of ordinary differential equations with pulse action at fixed times that admits the trivial solution. We establish sufficient conditions for the equiasymptotic stability of the trivial solution. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1317–1325, October, 2008.     

Asymptotic stability of integral sets

We prove theorems on asymptotic, equiasymptotic, and uniform asymptotic stability of the integral sel of a nonautonomous system of ordinary differential equations.     

Exponential stability of perturbed superstable systems in Hilbert spaces

We study exponential stability of superstable systems in Hilbert spaces under perturbations. Formulas to calculate or to estimate the exponential growth bound of the perturbed systems are derived via which sufficient conditions on exponential stability are established. The obtained results are applied to a partial differential equation governing the vibration of a smart beam made of self-straining material. Several numerical simulations are given.     

含时滞的线性偏泛函微分方程解的渐近行为

本文对一类含有时滞的抛物偏泛函微分方程解的渐近行为进行了讨论。分别就两种不同的边界条件，采用Ｌｉａｐｕｎｏｖ泛函及Ｌ—ｐ—估计，获得了其解全局稳定、振动的若干简洁充分条件     

A positivity property of solutions of nonlinear diffusion equations

This paper deals with the asymptotic stability of the null solution of a semilinear partial differential equation. The La Salle Invariance Principle has been used to obtain the stability results. The first result is given under quite general hypotheses assuming only the precompactness of the orbits and the local existence. In the second part, under some restrictions, sufficient conditions for precompactness of the orbits and decay of solutions are given. An existence and uniqueness theorem is proved in the Appendix. Some examples are given.     

Exponential stability for set dynamic equations on time scales

A new theory known as set dynamic equations on time scales has been built. The criteria for the equistability, equiasymptotic stability, uniform and uniformly asymptotic stability were developed in Hong (2010) [1]. In this paper, we consider the exponential stability, exponentially asymptotic stability, uniform and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using Lyapunov-like functions.     

非Lipschitz系数下随机泛函微分方程适度解的存在唯一性及其渐近性态(英文)

本文主要运用Picard迭代和算子分数次幂方法,讨论了随机时滞偏微分方程适度解的存在性与唯一性,并对解的渐近性态进行了研究.这里方程的系数不满足Lipschitz条件,时滞r>0为有限的.最后给出了一个非Lipschitz条件的例子.     

Analytical and numerical stability of partial differential equations with piecewise constant arguments

This article is concerned with the stability analysis of the analytic and numerical solutions of a partial differential equation with piecewise constant arguments of mixed type. First, by means of the similar technique in Wiener and Debnath [Int J Math Math Sci 15 (1992), 781–788], the sufficient conditions under which the analytic solutions asymptotically stable are obtained. Then, the θ‐methods are used to solve the above‐mentioned equation, the sufficient conditions for the asymptotic stability of numerical methods are derived. Finally, some numerical experiments are given to demonstrate the conclusions.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1‐16, 2014     

可换BCH-代数

引入了可换BCH-代数的概念,给出了可换BCH-代数的两个充要条件.对偏序可换BCH-代数进行了讨论,给出了偏序BCH-代数是可换的两个充要条件.证明了偏序可换BCH-代数的每个分支是一个下半格,局部有界偏序可换BCH-代数的每个分支是一个格.     

Equiasymptotic stability of integral sets

We give the definition of the equiasymptotic stability of the integral set of a system of ordinary differential equations and prove several theorems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 180–185, February, 1995.     

ABSOLUTE STABILITY OF CONTROL SYSTEM WITH MULTI-NONLINEAR FEEDBACK TERMS

ＡＢＳＯＬＵＴＥＳＴＡＢＩＬＩＴＹＯＦＣＯＮＴＲＯＬＳＹＳＴＥＭＷＩＴＨＭＵＬＴＩ－ＮＯＮＬＩＮＥＡＲＦＥＥＤＢＡＣＫＴＥＲＭＳ￥（廖晓昕，王晓君，舒卫华，吴卫华）ＬｉａｏＸｉａｏｘｉｎ；ＷａｎｇＸｉａｏｊｕｎ（Ｄｅｐｔ．ｏｆＭａｔｈｅｍａｔｉｃｓ，...     

Discrete infinite-dimensional type-K monotone dynamical systems and time-periodic reaction-diffusion systems

The asymptotic behavior of discrete type-K monotone dynamical systems and reaction-diffusion equations is investigated. The studying content includes the index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one. A nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits. Applications are made to time-periodic Lotka-Volterra systems with diffusion, and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given. For more general time-periodic type-K monotone reaction-diffusion systems with spatial homogeneity, a simple condition is given to guarantee the convergence of all positive solutions.     

Twisted partial coactions of Hopf algebras

In this paper, the notion of a twisted partial Hopf coaction is introduced. The conditions on partial cocycles are established in order to construct partial crossed coproducts. Then the classification of partial crossed coproducts is discussed. Finally, some necessary and sufficient conditions for a class of partial crossed coproducts to be quasitriangular bialgebras are given.     

具时滞Kolmogorov系统的强区间稳定性与部分稳定性

通过得用区间动力系统的稳定性理论和构造适当的Ｌｙａｐｕｎｏｖ科学家函，我们讨论了具离散时滞Ｋｏｌｍｏｇｏｒｏｖ生态系统的渐近性，分别获得了该系统强稳定、部分稳定的充分条件。     

Absolute stability of general lurie type indirect control systems

1. IntroductionThe absolute stability problem is one of the central topics of the stability theory for itstheoretical and engineering importance. UP to date, this problem has been studied by manyauthors [1--9], and many sufficient conditions and some necessary and sufficiellt conditions inspecial cases for absolute stability are obtained. In papers [7,8], Liao gave a necessary andsufficient condition for absolute stability for partial arguments. In this paperg by using anew concept of absolut…     

On the stability of jetlike magnetohydrodynamic flows

The linear stability problem is under study for steady axisymmetric translational flows of a density-homogeneous nonviscous incompressible ideal conducting fluid with free surface and “frozen-in” poloidal magnetic field. By the direct Lyapunov method, some sufficient conditions are obtained for the stability of these flows under small long-wave perturbations with the same symmetry. These stability conditions have partial converses; and, for unstable stationary flows, an a priori exponential lower bound is constructed on the growth of small perturbations under consideration, while the increment of the appearing exponent serves as an arbitrary positive parameter. An illustrative analytical example is given of steady flows with superimposed small long-wave axisymmetric perturbations growing in time in accordance with the estimate.     

Exponential stability of mild solutions of stochastic partial differential equations with delays

A semiiinear stochastic partial differential equation with variable delays is considered. Sufficient conditions for the exponential stability in the p-th mean of mild solutions are obtained. Also, pathwise exponential stability is proved. Since the technique ofLyapunov functions is not suitable for delayed equations, the results have been proved by using the properties of the stochastic convolution. As the sufficient conditions obtained are also valid for the case without delays, one can ensure exponential stability of mild solution in some cases where the sufficient conditions in Ichikawa [11] do not give any answer. The results are illustrated with some examples     

间接控制系统部分变元的绝对稳定性

ｉ）对变量分离型间接控制系统，本文给出其平凡解关于部分变元绝对稳定的充要条件。ｉｉ）对非变量分离型间接控制系统，给出其平凡解关于部分变元绝对稳定的充分条件。     

On stability of three-dimensional periodic motions in hydrodynamics : PMM vol. 37, n6, 1973, pp. 1044–1048

We obtain exact conditions for the stability of periodic motions. We show that the conditions found in [1] are necessary and sufficient, but they are only applicable to motions not dependent on time. The conditions given in [2] are applicable in the general case but are only sufficient (necessary) conditions of instability (stability). We consider the dependence of stationary motions on parameters.     

Partial asymptotic stability via limiting equations

In this paper the limiting equations approach is applied to study the stability properties with respect to a part of the state variables for nonautonomous dynamical systems. Sufficient conditions are given for uniform asymptotic eventual strongly partial stability and for uniform asymptotic partial stability. An application of the results is given.