Volume 7 Issue 7 - February 20, 2009
Stability analysis of neural networks with interval time-varying delays
Yi-You Hou1, Teh-Lu Liao1,*, Chang-Hua Lien2, Jun-Juh Yan3

1Department of Engineering Science, National Cheng Kung University, Tainan, 701 Taiwan
2Department of Marine Engineering, National Kaohsiung Marine University, Kaohsiung 811, Taiwan
3Department of Computer and Communication, Shu-Te University, Kaohsiung 824, Taiwan
tlliao@mail.ncku.edu.tw

CHAOS 17 (3): Art. No. 033120 SEP 2007

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The stability analysis of a class of neural networks, such as Hopfield neural networks (HNNs), cellular neural networks (CNNs), and Cohen-Grossberg neural networks (CGNNs) has been extensively investigated; these particular networks have been applied to image processing and signal processing problems. However, in this class of neural networks, the interactions between neurons are generally asynchronous which inevitably result in time-delay. The existence of time-delay may cause oscillations and instability in neural networks. Therefore, the stability analysis of DNNs has emerged as a new and attractive research field in neural networks, such as delayed cellular neural networks (DCNNs), delayed Hopfield neural networks (DHNNs), and delayed Cohen-Grossberg neural networks (DCGNNs). The criteria for time-delay systems can be classified into two categories, namely delay-independent criteria and delay-dependent criteria. Generally speaking, the latter group is less conservative than its counterpart when the time-delay values are small. In both of these analysis problems, the time-delay is known to reside within an interval including zero and the upper bound of the time derivative of time-varying delay is known and smaller than one.  This paper proposes a less conservative delay-dependent stability formula to guarantee the global exponential stability of neural networks with interval delays, which may exclude zero. The obtained results are derived based on the Lyapunov theory, the linear matrix inequaility (LMI) technique, and the Leibniz-Newton formula. Furthermore, the supplementary requirement that the time-derivative of time-varying delays must be smaller than one is not necessary in this study.

The dynamics of the neural networks with interval time-varying delays are described by the following differential equation:
                     (1)
where and , is a positive diagonal matrix, while and are known constant matrices, A is referred to as the feedback matrix, B represents the delayed feedback matrix with the time-varying delay and J  is the external bias vector.  In (1), it is assumed that and , for all and .  The initial condition of (1) is given by .  Each function , is monotonically nondecreasing, bounded and satisfies the Lipschitz condition with a Lipschitz constant .

Assume that an equilibrium point of (1) exists and is given as . Shifting the equilibrium point to the origin and defining, where , then (1) can be transformed into the following form:
                                (2a)
                                                                    (2b)
where ,
,
, .
Clearly, if the origin of the system (2) is globally exponentially stable, then the equilibrium point of system (1) is also globally exponentially stable.  Let , then system (2) can be transformed into the following form:

        (3a)
        (3b)
where , , , and .

Now, a delay-dependent criterion for the globally exponential stability of the system (1) with the same time-varying delays, i.e. , is derived as follows:

The equilibrium point of system (1) associated with is globally exponentially stable with convergence rate , if there are some symmetric positive definite matrices , , , , matrices , , and some positive constants and such that the following LMI condition holds
        (4)
where
, ,, , , ,, , , , , , , , , , , , .

If the upper bound of the time derivative of time-varying delay () is unknown, is obtained the following result.

The equilibrium point of system (1) associated with , is globally exponentially stable with convergence rate , if there exist some symmetric positive definite matrices , , , matrices , , and some positive constants and such that the following LMI condition holds
        (5)
where
, other , are defined in (4).

The delay-dependent criterion of the exponential stability of system (1) associated with different time delays, i.e. if is derived as follows:

The equilibrium point of system (1) associated with is globally exponentially stable with convergence rate , if there exist some symmetric positive definite matrices , , some positive diagonal matrices , , matrices , , and some positive constants and such that the LMI (4) is satisfied.

The proposed several sufficient conditions will guarantee the globally exponential stability for neural networks with interval time-varying delays. The results can be easily derived by the Lyapunov theory, the linear matrix inequaility (LMI) technique, and the Leibniz-Newton formula.
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