GLOBAL EXPONENTIAL STABILITY FOR A CLASS OF IMPULSIVE INTEGRO-DIFFERENTIAL EQUATION

2008 ◽  
Vol 18 (03) ◽  
pp. 735-743 ◽  
Author(s):  
QINGHUA ZHOU
Keyword(s):  
Differential Equation ◽  
Exponential Stability ◽  
Spectral Radius ◽  
Differential Inequality ◽  
Global Exponential Stability ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Integro Differential Equation ◽  
Dynamical Behaviors ◽  
Theoretical Results

The convergent dynamical behaviors of a class of impulsive integro-differential equation are discussed. By establishing an integro-differential inequality with impulsive initial conditions and using the properties of M-cone and eigenspace of the spectral radius of non-negative matrices, some new sufficient conditions to guarantee the global exponential stability are obtained. The results extend and improve the earlier publications. An example is given to illustrate the theoretical results.

scholarly journals Exponential Stability of Stochastic Differential Equation with Mixed Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Wenli Zhu ◽  
Jiexiang Huang ◽  
Xinfeng Ruan ◽  
Zhao Zhao
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Stochastic Differential Equation ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Martingale Inequality ◽  
Mixed Delay ◽  
Exponentially Stable ◽  
Theoretical Results

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

scholarly journals Existence and Global Exponential Stability of Almost Periodic Solutions for SICNNs with Nonlinear Behaved Functions and Mixed Delays

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Xinsong Yang ◽  
Jinde Cao ◽  
Chuangxia Huang ◽  
Yao Long
Keyword(s):  
Exponential Stability ◽  
Periodic Solutions ◽  
Differential Inequality ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Mixed Delays ◽  
Time Varying ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Inequality Techniques

By using the Leray-Schauder fixed point theorem and differential inequality techniques, several new sufficient conditions are obtained for the existence and global exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with discrete and distributed delays. The model in this paper possesses two characters: nonlinear behaved functions and all coefficients are time varying. Hence, our model is general and applicable to many known models. Moreover, our main results are also general and can be easily deduced to many simple cases, including some existing results. An example and its simulation are employed to illustrate our feasible results.

scholarly journals Existence and Global Exponential Stability of Periodic Solutions for General Neural Networks with Time-Varying Delays

2008 ◽  
Vol 2008 ◽  
pp. 1-14 ◽  
Author(s):  
Xinsong Yang
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Periodic Solutions ◽  
Differential Inequality ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Time Varying ◽  
Inequality Techniques ◽  
Coincidence Degree Theorem

By using the coincidence degree theorem and differential inequality techniques, sufficient conditions are obtained for the existence and global exponential stability of periodic solutions for general neural networks with time-varying (including bounded and unbounded) delays. Some known results are improved and some new results are obtained. An example is employed to illustrate our feasible results.

scholarly journals Existence and global exponential stability of periodic solution of high-order Cohen-Grossberg neural network with impulses

2009 ◽  
Vol 42 (2) ◽  
Author(s):  
Jing Liu
Keyword(s):  
Neural Network ◽  
Periodic Solution ◽  
Exponential Stability ◽  
Differential Inequality ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
High Order ◽  
Continuation Theorem ◽  
Mawhin’S Continuation Theorem

AbstractSufficient conditions are obtained for the existence and global exponential stability of periodic solution of high-order Cohen-Grossberg neural network with impulses by using Mawhin’s continuation theorem of coincidence degree and by means of a method based differential inequality.

GLOBAL EXPONENTIAL STABILITY CONDITIONS FOR DELAYED PARABOLIC NEURAL NETWORKS WITH VARIABLE COEFFICIENTS

2007 ◽  
Vol 17 (12) ◽  
pp. 4409-4415
Author(s):  
XUYANG LOU ◽  
BAOTONG CUI
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Differential Inequality ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Lyapunov Functionals ◽  
The Stability ◽  
Delay Differential ◽  
Delay Differential Inequality

In this paper, we present a class of delayed parabolic neural networks (DPNN) with variable coefficients. Some sufficient conditions for the global exponential stability of the DPNN with variable coefficients are derived by a method based on delay differential inequality. The method, which does not make use of Lyapunov functionals, is simple and effective for the stability analysis of DPNN with variable coefficients.

scholarly journals Global Exponential Stability for DCNNs with Impulses on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yongkun Li ◽  
Yuanhong Zhi
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Time Scales ◽  
Continuous Time ◽  
Global Exponential Stability ◽  
Sufficient Conditions ◽  
Degree Theory ◽  
Topological Degree Theory ◽  
Dynamical Behaviors ◽  
Time Scale Calculus

A class of delayed cellular neural networks (DCNNs) with impulses on time scales is considered. By using the topological degree theory, and the time scale calculus theory some sufficient conditions are derived to ensure the existence, uniqueness, and global exponential stability of equilibria for this class of neural networks. Finally, a numerical example illustrates the feasibility of our results and also shows that the continuous-time neural network and the discrete-time analogue have the same dynamical behaviors. The results of this paper are completely new and complementary to the previously known results.

scholarly journals Exponential Stability of BAM Fuzzy Cellular Neural Networks with Time-Varying Delays in Leakage Terms and Impulses

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Yongkun Li ◽  
Youqin Li
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Differential Inequality ◽  
Sufficient Conditions ◽  
Cellular Neural Networks ◽  
Time Varying ◽  
Fuzzy Cellular Neural Networks ◽  
Inequality Techniques ◽  
Theoretical Results

BAM fuzzy cellular neural networks with time-varying delays in leakage terms and impulses are considered. Some sufficient conditions for the exponential stability of the networks are established by using differential inequality techniques. The results of this paper are completely new and complementary to the previously known results. Finally, an example is given to demonstrate the effectiveness and conservativeness of our theoretical results.

scholarly journals Existence and exponential stability of a periodic solution for fuzzy cellular neural networks with time-varying delays

2011 ◽  
Vol 21 (4) ◽  
pp. 649-658 ◽  
Author(s):  
Qianhong Zhang ◽  
Lihui Yang ◽  
Daixi Liao
Keyword(s):  
Neural Networks ◽  
Periodic Solution ◽  
Exponential Stability ◽  
Differential Inequality ◽  
Sufficient Conditions ◽  
Coincidence Degree ◽  
Cellular Neural Networks ◽  
Time Varying ◽  
Fuzzy Cellular Neural Networks ◽  
Inequality Technique

Existence and exponential stability of a periodic solution for fuzzy cellular neural networks with time-varying delays Fuzzy cellular neural networks with time-varying delays are considered. Some sufficient conditions for the existence and exponential stability of periodic solutions are obtained by using the continuation theorem based on the coincidence degree and the differential inequality technique. The sufficient conditions are easy to use in pattern recognition and automatic control. Finally, an example is given to show the feasibility and effectiveness of our methods.

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