Razumikhin–Lyapunov functional method for the stability of impulsive switched systems with time delay

This paper investigates the uniform stability and the uniform asymptotical stability of impulsive switched systems with time delay. By employing the method of Razumikhin–Lyapunov functional, several Razumikhin-type theorems of uniform stability and uniform asymptotical stability are established, which improve some of the existing results. Several examples are also given to illustrate the results.     

关于运用Liapunov函数的滞后型系统部分稳定性的注释

Sufficient conditions for the stability with respect to part of the functional differential equation variables are given. These conditions utilize Lyapunov functions to determine the uniform stability and uniform asymptotic stability of functional differential equations. These conditions for the partial stability develop the Razumikhin theorems on uniform stability and uniform asymptotic stability of functional differential equations. An example is presented which demonstrates these results and gives insight into the new stability conditions.     

Uniform stability and ISS of discrete-time impulsive hybrid systems

This paper studies the uniform stability and ISS (input-to-state stability) properties for DIHS (discrete-time impulsive hybrid systems) via comparison approach. By employing the vector-value function, the comparison principle is established for DIHS with external inputs. Then the comparison principle is used to establish the uniform stability and ISS criteria for DIHS, respectively. Moreover, regions in which the uniform stability and ISS properties can be guaranteed are estimated. As applications, the comparison principle and the results of uniform stability and ISS are used to study the robustly globally uniformly exponential stability for uncertain DIHS and exponential ISS of DIHS. It is shown that impulses contribute to stability and ISS properties for a discrete-time system which has no such properties. Two examples with numerical simulations are worked out for illustration.     

STABILITY OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAYS

In this paper,the stability of a class of impulsive functional differential equations with infinite delays is investigated. A uniform stability theorem and a uniform asymptotic stability theorem are established.     

Banach空间中线性斜积半流指数稳定性的Holder型刻画

为了在Banach空间X的对偶空间上刻画线性斜演化半流的一致指数稳定性,借助泛函分析与算子理论得到了其一致指数稳定的一些H?lder型充要条件,所得结果推广了稳定性理论中的一些已有结论.     

Stability by Lyapunov like functions of nonlinear differential equations with non-instantaneous impulses

The stability of the solutions of a nonlinear differential equation with noninstantaneous impulses is studied using Lyapunov like functions. In these differential equation we have impulses, which start abruptly at some points and their action continue on given finite intervals. Sufficient conditions for stability, uniform stability and asymptotic uniform stability of the solutions are established. Examples are given to illustrate the results. Also, some of the results are applied to study a dynamical model in Pharmacokinetics.     

Uniform stability of switched nonlinear systems

In this paper, we investigate the stability properties of a general class of nonautonomous switched nonlinear systems. Sufficient conditions for uniform stability, uniform asymptotic stability and uniform exponential stability are derived via multiple Lyapunov functions. Our results provide stability criteria for switched systems with both stable and unstable subsystems. Particularly, our results include some existing results as special cases or improve those in the literature. Several numerical examples are worked out to illustrate our results.     

Problems of the partial stability and detectability of dynamical systems

The conditions under which uniform stability (uniform asymptotic stability) with respect to a part of the variables of the zero equilibrium position of a non-linear non-stationary system of ordinary differential equations signifies uniform stability (uniform asymptotic stability) of this equilibrium position with respect the other, larger part of the variables, which include an additional group of coordinates of the phase vector, are established. These conditions include the condition for uniform asymptotic stability of the zero equilibrium position of the “reduced” subsystem of the original system with respect to the additional group of variables. Since within the conditions obtained the stability with respect to the remaining unmeasured coordinates of the phase vector remains undetermined or is investigated additionally, partial zero-detectability of the original system occurs in this case, and the conditions obtained supplement the series of known results from partial stability theory. The application of the results obtained to problems of the partial stabilization of non-linear controlled systems, particularly to the problem of stabilizing an asymmetric rigid body relative to an assigned direction in an inertial space, is considered. The partial detectability of linear systems with constant coefficients is also investigated.     

Stable multilevel splittings of boundary edge element spaces

We establish the stability of nodal multilevel decompositions of lowest-order conforming boundary element subspaces of the trace space of on boundaries of triangulated Lipschitz polyhedra. The decompositions are based on nested triangular meshes created by uniform refinement and the stability bounds are uniform in the number of refinement levels.     

Stability criteria in terms of two measures for functional differential equations

Results involving uniform stability and uniform asymptotic stability in terms of two measures of delay differential equations by using Liapunov functional techniques are established.     

Uniform asymptotic stability for damped linear oscillators with variable parameters

In this paper, sufficient conditions for uniform asymptotic stability of the damped linear oscillators with variable coefficients are presented. The result of the present study can be applied to even the case of negative damping. The conditions presented herein are shown to be essential for uniform asymptotic stability.     

On the stability of impulsive functional differential equations with infinite delays

In this paper, the stability problem of impulsive functional differential equations with infinite delays is considered. By using Lyapunov functions and the Razumikhin technique, some new theorems on the uniform stability and uniform asymptotic stability are obtained. The obtained results are milder and more general than several recent works. Two examples are given to demonstrate the advantages of the results. Copyright © 2014 John Wiley & Sons, Ltd.     

具有可变脉冲点的脉冲微分方程的稳定性

该文考虑具有可变脉冲点的脉冲微分方程零解的稳定性。通过利用L yapunov函数以及Razumikhin技巧，可以得到关于具有可变脉冲点的脉冲微分方程零 解的一致稳定和一致渐近稳定的充分条件。     

Banach空间中线性离散时间系统的一致多项式膨胀性

给出了Banach 空间中线性离散时间系统一致多项式膨胀性的概念,并讨论了其离散特征。借助Lyapunov函数给出了线性离散时间系统满足一致多项式膨胀的充要条件。所得结论将一致指数稳定性、指数膨胀性及多项式稳定性中的若干经典结论推广到了一致多项式膨胀性的情形。     

On the stability of characterization of the normal distribution by the properties of linear forms

If the assumptions of the characterization theorem are fulfilled not exactly but only approximately, then can we state that the conclusion of this characterization is also fulfilled approximately? Theorems in which this kind of problems are considered are called stability theorems or the stability of characterizations. The problems of the stability of characterization of the normal distribution by means of identically distributed linear form and monomial are considered in a uniform metric. The moment and symmetry conditions are considerably weaker in comparison to previously published versions of the theorem. The choice of a uniform metric appears to be very natural in the framework of Zolotarev’s methodology advising to investigate stability.     

一类扩张的细胞神经网络稳定性判别法

本文研究一类含有阻尼项带有时滞细胞神经网络的全局渐进稳定性和一致稳定性的性质,通过构造适当的李雅普诺夫函数及利用分析的有关知识,给出了全局渐进稳定性和一致稳定的判别法．     

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.     

A nonlinear inequality and application to global asymptotic stability of perturbed systems

The goal of this work is to present a new nonlinear inequality which is used in a study of the Lyapunov uniform stability and uniform asymptotic stability of solutions to time‐varying perturbed differential equations. New sufficient conditions for global uniform asymptotic stability and/or practical stability in terms of Lyapunov‐like functions for nonlinear time‐varying systems is obtained. Our conditions are expressed as relation between the Lyapunov function and the existence of specific function which appear in our analysis through the solution of a scalar differential equation. Moreover, an example in dimensional two is given to illustrate the applicability of the main result. Copyright © 2014 John Wiley & Sons, Ltd.     

A population model with time-dependent delay

We present analytical and computational results concerning the linear stability and instability of the uniform steady-state solution of a system of reaction-diffusion equations where a parameter in the kinetic terms is periodic in time. Under suitable assumptions the system is equivalent to a scalar equation with a periodically varying delay. Such a varying delay can model the seasonal fluctuations to the regeneration time of a resource. We study the effect such a varying delay can have on the stability of the spatially uniform steady-state. Analytical results reveal that instability can set in if the delays are large, while computational methods of analysing the stability equations reveal the precise shape of the instability boundary. The nonlinear stability of the uniform state is also examined using ladder methods.     

Lyapunov's second method in problems of the stability of solutions of systems with impulse effect

A system of ordinary differential equations with impulse effect at fixed moments of time is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse effects are obtained under which the uniform asymptotic stability of the zero solution of the ‘unperturbed’ system implies the uniform asymptotic stability of the zero solution of the ‘perturbed’ system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)