Contributed articleOn stability of nonlinear continuous-time neural networks with delays
Introduction
Continuous-time analog neural networks with symmetric connections will always converge to fixed points when the inter-neuronal transmission delays are ignored (Chua & Yang, 1988, Hopfield, 1984), but may become unstable when time delays are considered. So it is significant to investigate stability conditions of neural networks with delays.
For neural networks without delays, some conditions imposed on connection weight matrices have been derived by many researchers to ensure the stability of the networks (see, for example, Hirch, 1989, Kelly, 1990, Matsuoka, 1992). These conditions are mostly given by matrix norms or matrix measures. For neural networks with delays, however, it is more difficult to analyze their stability properties due to introduction of delays. There are usually two ways to do this. One is to linearize the system near an equilibrium, the original system has the same stability properties as the linearized system near the equilibrium considered, conditions obtained by this way concern the local stability around an equilibrium. Another way is to construct a suitable Lyapunov function for the system and then to derive sufficient conditions ensuring stability, this usually involves global stability. However, construction of a suitable Lyapunov function is usually not an easy task. For detailed theory of stability analysis for systems with delays, one can consult some tutorial books, e.g. Hale, 1977, Qin et al., 1989. Marcus and Westervelt (1989) studied stability of analog neural networks with delay by linearizing the systems. Civalleri et al., 1993, Gilli, 1994 have established some sufficient conditions for delay-dependent stability of cellular neural networks (CNNs) with delay using the Lyapunov functional approach. Gopalsamy & He, 1994a, Gopalsamy & He, 1994b have obtained some sufficient conditions for globally asymptotic stability of delayed bidirectional associative memory networks (BAM) and Hopfield networks, respectively. In this paper we will construct different Lyapunov functions to obtain some sufficient conditions for delay-independent globally asymptotic stability of a class of continuous-time continuous-state neural networks which includes delayed Hopfield networks, BAM networks and CNNs as its special cases. Our conditions relaxed some of the restrictions on the weight matrices and so improved some results developed by other authors. Our condition is without requirement of symmetry of the weight matrices.
Section snippets
Model description
In this paper, we deal with delayed continuous-time neural networks described by the following differential equations with delayswhere wij0,wijτ,Ii,τj are constant real numbers, τj represents delays which are nonnegative. Suppose further the following assumptions are satisfied
(A1) gi:R→R is differentiable and strictly monotone increasing, i.e. i=1,2,…,n, where g′i(x) represents the derivative of gi(x).
Sufficient conditions for global stability
In this section we will derive some sufficient conditions for globally asymptotic stability of the neural network model (1), globally asymptotic stability means global stability independent of delays. Theorem 1 For neural network model (1), if the spectral radius of the matrix M−1(|W0|+|Wτ|)K is less than 1, i.e. ρ(M−1(|W0|+|Wτ|)K)<1, then there exists at most one equilibrium for Eq. (1) and when Eq. (1) does have an equilibrium, the equilibrium is globally asymptotically stable. Proof Note that the equilibrium
Conclusion
Some sufficient conditions for globally asymptotic stability independent of delays for a kind of continuous-time continuous-state nonlinear neural networks with delays have been obtained. The network model considered here is general and includes delayed Hopfield-type networks, BAM networks and CNNs as its special cases. Some of our results are improvements on previous works established by other researchers.
Acknowledgements
The author wishes to thank the anonymous references and editors for their valuable suggestions. This work is supported by the National Natural Science Foundation of China and Center for Software and Theory of Universities in Shanghai.
References (17)
- et al.
Stability analysis of delayed cellular neural networks
Neural Networks
(1998) - et al.
Stability in asymmetric Hopfield nets with transmission delays
Physica D
(1994) Stability conditions for nonlinear continuous neural networks with asymmetric connection weights
Neural Networks
(1992)Matrices with positive principal minors
Linear algebra and its applications
(1977)- et al.
- et al.
Cellular neural networks: theory
IEEE Transactions on Circuits and Systems
(1988) - et al.
On stability of cellular neural networks with delay
IEEE Transactions on Circuits and Systems
(1993) Stability of cellular neural networks and delayed cellular neural networks with nonpositive templates and nonmonotonic output functions
IEEE Transactions on Circuits and Systems
(1994)
Cited by (128)
Existence and global exponential stability of equilibrium of competitive neural networks with different time scales and multiple delays
2010, Journal of the Franklin InstituteDynamics of solution for a class of delayed diffusive neural networks with mixed boundary conditions
2010, NeurocomputingCitation Excerpt :□ By comparison with the earlier works on neural networks with or without diffusion, we will find that they discussed constant equilibrium point and its asymptotic behavior [4–22]. In this paper, we first investigate the nonconstant equilibrium solution and its properties.
Stability analysis of Cohen-Grossberg neural networks with discontinuous neuron activations
2010, Applied Mathematical ModellingCoexistence of 2<sup>N</sup> domains of attraction of nonautonomous neural networks with time-varying delays
2010, Applied Mathematical ModellingLMI conditions for global robust stability of delayed neural networks with discontinuous neuron activations
2009, Applied Mathematics and Computation