New delay-dependent exponential stability criteria of BAM neural networks with time delays

doi:10.1016/j.matcom.2008.08.014

Mathematics and Computers in Simulation

Volume 79, Issue 5, January 2009, Pages 1679-1697

https://doi.org/10.1016/j.matcom.2008.08.014 Get rights and content

Abstract

In this paper, the global exponential stability is investigated for the bi-directional associative memory networks with time delays. Several new sufficient conditions are presented to ensure global exponential stability of delayed bi-directional associative memory neural networks based on the Lyapunov functional method as well as linear matrix inequality technique. To the best of our knowledge, few reports about such “linearization” approach to exponential stability analysis for delayed neural network models have been presented in literature. The method, called parameterized first-order model transformation, is used to transform neural networks. The obtained conditions show to be less conservative and restrictive than that reported in the literature. Two numerical simulations are also given to illustrate the efficiency of our result.

Introduction

Kosko has proposed a series of neural networks related to bi-directional associative memory (BAM) neural networks with the ability of information memory and information association [11], [12], [13]. However, dynamics of neural networks often have time delays due to the finite switching speed of amplifiers in electronic neural networks, or to the finite signal propagation time in biological networks. Time delay will affect the stability of a network by creating instability, oscillation and chaos phenomena. In view of this, the stability issue of delayed neural networks is a topic of great practical importance [1], [7], [14], [16], [26], which has gained increasing interest in the potential applications in signal processing, optimization problem, image processing and other fields [8], [20].

Recently, some sufficient conditions for the stability of BAM networks with time delays have been derived (see [2], [4], [6], [9], [10], [15], [17], [19], [22], [23], [24], [25], [27], [28]). In [4], the global asymptotic stability and the existence of equilibrium are considered as continuous bi-directional associative memory (BAM) neural networks with axonal signal transmission delay by using Lyapunov method and new sufficient conditions are derived to ascertain the global asymptotic stability of BAM. In [28], the authors present a new sufficient condition for the existence, uniqueness and global exponential stability of the equilibrium point for bi-directional associative memory (BAM) neural networks with constant delays. Huang et al. [10] proposed a set of criteria for the exponential stability of BAM neural networks with constant or time-varying delays, based on the Lyapunov–Krasovskii functional in combination with linear matrix inequality (LMI) approach. These criteria manifest explicitly the influence of time delay on exponential convergence rate and show the differences between the excitatory and inhibitory effect.

In this paper, inspired by [1], [7], [14], [16], [26], we present some criteria on the exponential stability and estimate the exponential convergence rate for BAM neural networks with time delays. Some new conditions for global exponential stability of BAM with constant delay are given in terms of LMIs by constructing a suitable Lyapunov functional. Firstly, we shifted the nonlinear neural network model to the linear one by employing a simple transformation. Secondly, a process, which is called a parameterized first-order model transformation, is used to transform the linear system. Then, we establish novel sufficient conditions to ensure the existence, uniqueness, and global exponential stability of equilibrium point for BAM neural networks with time delays. The obtained result in this paper is different from those in the earlier literatures [5], [10], [15], [27]. Compared with the earlier results in the literature, those results are less restrictive and conservative.

The rest of this paper is organized as follows. In Section 2, the problem to be studied is stated and some needed preliminaries and lemmas are given. In Section 3, some global exponential stability criteria are presented for delayed BAM neural networks by means of the Lyapunov-type stability theorem and linear matrix inequality (LMI). In Section 4, two illustrative examples are given to show the effectiveness of our results. Finally, some conclusions are drawn in Section 5.

Section snippets

Neural network model and preliminaries

In this paper, we consider the following BAM neural network model with time delays: $\begin{array}{c} {\dot{u}}_{i} (t) = - a_{i} u_{i} (t) + \sum_{j = 1}^{m} w_{i j} f_{j} (h_{j} (t - τ)) + I_{i}, i = 1,2, \dots, n, \\ {\dot{h}}_{j} (t) = - b_{j} h_{j} (t) + \sum_{i = 1}^{n} v_{j i} g_{i} (u_{i} (t - σ)) + J_{j}, j = 1,2, \dots, m, \end{array}$ or equivalently $\begin{array}{c} \dot{u} (t) = - A u (t) + W f (h (t - τ)) + I, \\ \dot{h} (t) = - B h (t) + V g (u (t - σ)) + J, \end{array}$ where a_i and b_i denote the neuron charging time constants and passive decay rates, respectively, $w_{i j}$ and $v_{j i}$ are the synaptic connection strengths, f_j and g_i represent the activation functions of the neurons and the propagational signal functions,

Global exponential stability analysis for delayed BAM neural networks

In this section, we present sufficient conditions for the uniqueness and global exponential stability of the equilibrium point for the delayed BAM neural networks.

In Ref. [18], the equality $x (t - τ) = x (t) - \int_{- τ}^{0} \dot{x} (t + ξ) d ξ = x (t) - \int_{- τ}^{0} [A x (t + ξ) + A_{d} x (t + ξ - τ)] ξ$ was used to transform the system $\dot{x} (t) = A x (t) + A_{d} x (t - τ)$ into a distributed delay system $\dot{x} (t) = (A + C) x (t) + (A_{d} - C) x (t - τ) - C \int_{- τ}^{0} [A x (t + θ) + A_{d} x (t + θ - τ)] d θ,$ where C is a parameter matrix which makes the stability result less restrictive to some degree. Such process is

Examples

In this section, we give two examples to illustrate our results.

Example 1

Consider the following system [10]:

\begin{array}{l} {\dot{x}}_{1} (t) = - x_{1} (t) + 0.05 f (y_{1} (t - τ)) + 0.25 f (y_{2} (t - τ)) + 0.05 f (y_{3} (t - τ)), \\ {\dot{x}}_{2} (t) = - x_{2} (t) + 0.1 f (y_{1} (t - τ)) + 0.05 f (y_{2} (t - τ)) + 0.15 f (y_{3} (t - τ)), \\ {\dot{x}}_{3} (t) = - x_{3} (t) + 0.15 f (y_{1} (t - τ)) + 0.15 f (y_{2} (t - τ)) + 0.05 f (y_{3} (t - τ)), \\ {\dot{y}}_{1} (t) = - 4 y_{1} (t) + 0.75 f (x_{1} (t - σ)) + 0.75 f (x_{2} (t - σ)) + 0.95 f (x_{3} (t - σ)), \\ {\dot{y}}_{2} (t) = - 4 y_{2} (t) + 0.5 f (x_{2} (t - σ)) + 0.75 f (x_{3} (t - σ)), \\ {\dot{y}}_{3} (t) = - 4 y_{3} (t) + 0.15 f (x_{1} (t - σ)) + 0.95 f (x_{2} (t - σ)) + 0.95 f (x_{3} (t - σ)), \end{array}

where the activation function is described by f(x) = 1/2(|x + 1| −

Conclusions

We have investigated the global exponential stability problem via a novel approach. By employing a simple transformation, we first shifted the nonlinear neural network model to the linear one. Then, via an approach combining the Lyapunov stability theorem and linear matrix inequality (LMI) technique, we have derived some new sufficient conditions for the global exponential stability of a general class of delayed BAM neural networks. In comparison with some recent results reported in the

Acknowledgements

The authors are greatly indebted to anonymous referees for their constructive comments. The work described in this paper was partially supported by the National Natural Science Foundation of China (Grant No. 60573047, 60574024) and Natural Science Foundation Project of CQ CSTC (Grant No. 2008BB2366, 2007BB2231), Program for New Century Excellent Talents in University, the Doctoral Foundation Project of Chongqing Normal University (Grant No. 08XLB003) and the Applying Basic Research Program of

References (28)

S. Arik et al.
Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays
Neurocomputing
(2005)
J. Cao
Global asymptotic stability of delayed bi-directional associative memory neural networks
Appl. Math. Comp.
(2003)
J. Cao et al.
Exponential stability of delayed bidirectional associative memory networks
Appl. Math. Comp.
(2003)
K. Gopalsamy et al.
Stability in asymmetric BAM networks with transmission delays
Physica D
(1994)
X. Huang et al.
LMI-based approach for delay-dependent exponential stability analysis of BAM neural networks, Chaos
Sol. Frac.
(2005)
C. Li et al.
Delay-dependent exponential stability analysis of bi-directional associative memory neural networks with time delay: an LMI approach, Chaos
Sol. Frac.
(2005)
S. Mohamad
Global exponential stability in continuous-time and discrete-time delayed bidirectional neural networks
Physica D
(2001)
Q.K. Song et al.
Global exponential stability and existence of periodic-solutions in BAM networks with delays and reaction diffusion terms
Chaos Sol. Frac.
(2005)
Q.K. Song et al.
Global exponential stability of BAM neural networks with distributed delays and reaction-diffusion terms
Phys. Lett. A
(2005)
H. Wang et al.
Existence and exponential stability of periodic solution of BAM neural networks with impulse and time-varying delay, Chaos
Sol. Frac.
(2007)

Q. Zhang et al.

An improved result for complete stability of delayed cellular neural networks

Automatica

(2005)

W. Zhao et al.

An analysis of global exponential stability of bidirectional associative memory neural networks with constant time delays

Neurocomputing

(2007)

H. Zhao

Exponential stability and periodic oscillatory of bi-directional associative memory neural network involving delays

Neurocomputing

(2006)

S. Arik

Stability analysis of delayed neural networks

IEEE Trans. Circuits Syst. I

(2000)

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New delay-dependent exponential stability criteria of BAM neural networks with time delays

Abstract

Introduction

Section snippets

Neural network model and preliminaries

Global exponential stability analysis for delayed BAM neural networks

Examples

Conclusions

Acknowledgements

Neurocomputing

Appl. Math. Comp.

Appl. Math. Comp.

Physica D

Sol. Frac.

Sol. Frac.

Physica D

Chaos Sol. Frac.

Phys. Lett. A

Sol. Frac.

Automatica

Neurocomputing

Neurocomputing

Stability analysis of delayed neural networks

IEEE Trans. Circuits Syst. I