Stability by the linear approximation for discrete equations

This paper is concerned with exponential stability of solutions of perturbed discrete equations. For a given m>1 we will provide necessary and sufficient conditions for exponential stability of all perturbed systems with perturbation of order m under the assumption that the unperturbed linear system is exponentially stable. Basing on this result we obtained necessary and sufficient conditions for exponential stability of the perturbed system for all perturbations of order m>1 for regular systems. Our results are expressed in terms of regular coefficients of the unperturbed system.     

Exponential Stability of Singularly Perturbed Systems with Time Delay

In this article, we study the exponential stability of singularly perturbed systems with time delay. By using vector delay inequalities and Lyapunov functions, exponential stability criteria are derived for both linear and some classes of nonlinear singularly perturbed systems with time delay. Examples are given to verify the stability criteria.     

有噪及无噪的时滞细胞神经网络稳定性分析

分析了一类时滞细胞神经网络(DCNN)系统在无噪声和有噪声干扰情况下的稳定性．首先针对确定性系统给出了一种简单且容易验证的全局指数稳定性条件,然后讨论了噪声干扰下系统的稳定性．当DCNN被外部噪声扰动时,系统是全局稳定的．重要的是,当系统被内在噪声扰动时,只要噪声总强度控制在一定范围内,系统是全局指数稳定的．鉴于随机共振现象在越来越多的非线性生物系统中被发现,这种稳定性具有重要意义．     

Exponential stability of singularly perturbed switched systems with time delay

This paper investigates exponential stability of singularly perturbed switched systems with time delay. The multiple Lyapunov functions technique and dwell time approach are used to establish stability criteria for a switched system consisting of both stable and unstable subsystems. Examples are presented to illustrate the criteria.     

Well-posedness and long time behavior of a hyperbolic Caginalp phase-field system

The aim of this paper is to prove the continuity of exponential attractors for a hyperbolic perturbed Caginalp system to an exponential attractor for the limit parabolichyperbolic Caginalp system. The symmetric distance between the perturbed and unperturbed exponential attractors in terms of the perturbation parameter is obtained.     

一类扰动系统的指数稳定性问题

主要讨论了一类扰动系统的指数稳定性问题 .若扰动项的控制函数满足无穷可积、 L2 可积或者 Lp可积时 ,x =0是常系统的指数稳定点 ,则也是扰动系统的指数稳定点 .推广和丰富了 Khalil[1] 的结果     

具有时滞的奇异扰动随机微分方程的均方指数稳定性

本文利用常数变易公式,随机过程数学期望的性质,矩阵范数,测度的相关理论以及不等式技巧,对一类具有时滞的奇异扰动随机微分方程的均方指数稳定性进行了讨论,得到了该类方程均方指数稳定的充分条件的代数判据.     

Stability of singularly perturbed nonlinear stochastic hybrid systems

The problem of exponential mean-square stability of nonlinear singularly perturbed, stochastic hybrid systems is studied in this article. Two groups of nonlinear systems are considered separately. To obtain the sufficient conditions of stability, two basic approaches of stability analysis for hybrid systems with a given Markovian switching rule and any Markovian switching rule and singularly perturbed non–hybrid systems were combined. The Lyapunov techniques were used in both approaches. The obtained results are illustrated by examples.     

Weak Exponential Stability for Time-Periodic Differential Inclusions via First Approximation Averaging

In this work we propose a method to study a weak exponential stability for time-varying differential inclusions applying an averaging procedure to a first approximation. Namely, we show that a weak exponential stability of the averaged first approximation to the differential inclusion implies the weak exponential stability of the original time-varying inclusion. The result is illustrated by an example.     

The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag

This paper is concerned with the exponential stability of singularly perturbed delay differential equations with a bounded (state-independent) lag. A generalized Halanay inequality is derived in Section 2, and in Section 3 a sufficient condition will be provided to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is exponentially stable uniformly for sufficiently small ε>0. This type of exponential asymptotic stability can obviously be applied to general delay differential equations with a bounded lag.     

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.     

On stability and bifurcation of periodic solutions of delay differential equations

The purpose of this paper is to study a class of delay differential equations with two delays. first, we consider the existence of periodic solutions for some delay differential equations. Second, we investigate the local stability of the zero solution of the equation by analyzing the correlocal stability of the zero solution of the equation by analyzing the corresponding characteristic equation of the linearized equation. The exponential stability of a perturbed delay differential system with a bounded lag is studied. Finally, by choosing one of the delays as a bifurcation parameter, we show that the equation exhibits Hopf and saddle-node bifurcations.     

Stability of impulsive systems depending on a parameter

In this paper, we present a practical exponential stability result for impulsive dynamic systems depending on a parameter. Stability theorem and converse stability theorem are established by employing the second Lyapunov method. These theorems are used to analyze the practical exponential stability of the solution of perturbed impulsive systems and cascaded impulsive systems, depending on a parameter. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability of stochastic systems in the diffusion-approximation scheme

By using a solution of a singular perturbation problem, we obtain sufficient conditions for the stability of a dynamical system with rapid Markov switchings under the condition of exponential stability of the averaged diffusion process. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 36–47, January, 1998.     

On the Robust Stability of Solutions to Periodic Systems of Neutral Type

Under consideration is some class of linear systems of neutral type with periodic coefficients. We obtain the conditions on perturbations of the coefficients which preserve the exponential stability of the zero solution. Using a special Lyapunov–Krasovskii functional, we establish some estimates that characterize the rate of exponential decay at infinity of the solutions of the perturbed systems.     

Characterization of the optimal trajectories for the averaged dynamics associated to singularly perturbed control systems

The aim of this paper is to study singularly perturbed control systems. Firstly, we provide linearized formulation version for the calculus of the value function associated with the averaged dynamics. Secondly, we obtain necessary and sufficient conditions in order to identify the optimal trajectory of the averaged system.     

Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions

We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt-Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions.     

Well-posedness and asymptotic stability to a laminated beam in thermoelasticity of type III

This paper is concerned with the well-posedness and asymptotic behaviour of solutions to a laminated beam in thermoelasticity of type III. We first obtain the well-posedness of the system by using semigroup method. We then investigate the asymptotic behaviour of the system through the perturbed energy method. We prove that the energy of system decays exponentially in the case of equal wave speeds and decays polynomially in the case of nonequal wave speeds. Under the case of nonequal wave speeds, we also investigate the lack of exponential stability of the system.     

Exponential stability of singularly perturbed systems with mixed impulses

This paper investigates the exponential stability problem for a class of singularly perturbed impulsive systems in which the flow dynamics is unstable and is affected at discrete time instants by impulses that have both destabilizing and stabilizing effects. More precisely the impulses have stabilizing effects on the slow variables but destabilizing effects on the fast ones. Thus, a first contribution of our work is related to stability analysis of singularly perturbed impulsive systems in the case when neither the flow dynamics nor the impulsive one is stable. In order to take full advantage of the jump matrix structure and its stabilizing effects on the slow dynamics, we introduce a new impulse-dependent vector Lyapunov function. This function allows us to better describe the behavior between two consecutive impulses as well as the jumps at impulse instants. Several numerically tractable criteria for stability of singularly perturbed impulsive systems are established based on vector comparison principle. Additionally, upper bounds on the singular perturbation parameter are derived. Finally, the validity of our results is verified by two numerical examples.     

Input-to-state stability and sliding mode control of the nonlinear singularly perturbed systems via trajectory-based small-gain theorem

This paper presents the trajectory-based input-to-state stability (ISS) and input-to-output stability (IOS) small-gain theorem, and the finite-time ISS (FTISS) and finite-time IOS (FTIOS) of nonlinear singularly perturbed systems. The contribution of this paper is threefold. Firstly, a novel idea is proposed to analyze the stability of the nonlinear singularly perturbed system, which is regarded as an interconnected system by using two-time-scale decomposition. Secondly, the trajectory-based approach is applied to establish ISS and IOS small-gain theorem for singularly perturbed systems and the FTISS and FTIOS properties are proposed. Thirdly, a novel sliding mode controller is developed for a class of nonlinear singularly perturbed systems. Finally, the effectiveness of proposed method is illustrated by using a numerical example, a DC motor simulation and a multi-agent singularly perturbed system.