New Sufficient Condition of Discrete-Time Systems of Neural Networks

: Problem statement: In this study, we derive a new sufficient condition for asymptotic stability of the zero solution of delay-differential system of neural networks in terms of certain matrix inequalities by using a discrete analog of the new Lyapunov second method. Conclusion/Recommendations: The problem is solved by applying a novel Lyapunov functional and an improved delay-dependent stability criterion is obtained in terms of a linear matrix inequality.


INTRODUCTION
In recent decades, neural networks have been extensively studied in many aspects and successfully applied to many fields such as pattern identifying, voice recognizing, system controlling, signal processing systems, static image treatment and solving nonlinear algebraic system (Alfaris et al., 2008;Bay and Phat, 2002;Bezzarga and Bucur, 2005;El-Said and EL-Sherbeny, 2005;Lekhmissi, 2006;Sen, 2004;2005a;2005b;Sen et al., 2005;Sharif and Saad, 2005;Waziri et al., 2005). Such applications are based on the existence of equilibrium points and qualitative properties of systems. In electronic implementation, time delays occur due to some reasons such as circuit integration, switching delays of the amplifiers and communication delays, etc. Therefore, the study of the asymptotic stability of neural networks with delays is of particular importance to manufacturing high quality microelectronic neural networks.
In this study, we consider delay-differential system of neural networks of the form where n u(k) ∈ Ω ⊆ R is the neuron state vector, h 0, The asymptotic stability of the zero solution of the delay-differential system of neural networks has been developed during the past several years. Much less is known regarding the asymptotic stability of the zero solution of the delay-differential system of neural networks. Therefore, the purpose of this study is to establish new sufficient condition for the asymptotic stability of the zero solution of (1) in terms of certain matrix inequalities.
Preliminaries: Lemma 1 (Agarwal, 1992) the zero solution of difference system is asymptotic stability if there exists a positive definite function n V(x(k)) : along the solution of the system. In the case the above condition holds for all x (k)∈V δ , we say that the zero solution is locally asymptotically stable. We present the following technical lemmas, which will be used in the proof of our main result.

Improved stability criterion:
In this section, we consider the new sufficient condition for asymptotic stability of the zero solution u * of (1) in terms of certain matrix inequalities. Without loss of generality, we can assume that * By condition (2)

MATERIALS AND METHODS
The zero solution of difference system is asymptotic stability if there exists a positive definite function n V(x(k)) : along the solution of the system. In the case the above condition holds for all x (k)∈V δ , we say that the zero solution is locally asymptotically stable.

RESULTS AND DISCUSSION
Theorem 3.1 The zero solution of the delaydifferential system (1)  [ ] y(k) x(k), x(k h(k)) .

= −
Then difference of V(y(k)) along trajectory of solution of (1) is given by: where (3)  hence, the asymptotic stability of the system immediately follows from Lemma 1. This completes the proof.
Remark 1: Theorem 1 gives a sufficient condition for the asymptotic stability of delay-difference system (1) via matrix inequalities. These conditions are described in terms of certain symmetric matrix inequalities But Wahab and Mohamed (2008) these conditions are described in terms of certain nonsymmetric matrix inequalities. CONCLUSION In this study, an improved delay-dependent stability condition for discrete-time linear systems with interval-like time-varying delays has been presented in terms of an LMI.