Global Exponential Stability of Discrete-Time Neural Networks with Time-Varying Delays

This paper presents some global stability criteria of discrete-time neural networkswith time-varying delays. Based on a discrete-type inequality, a new global stability condition for nonlinear difference equation is derived.We consider nonlinear discrete systemswith time-varying delays and independence of delay time. Numerical examples are given to illustrate the effectiveness of our theoretical results.


Introduction
In recent years, neural networks (NNs) have been investigated extensively due to their broad applications in information processing problems, associative memory, parallel computation, pattern recognition, signal processing, and optimization problems.It is well known that delays are often the sources of instability and oscillation in system.In practical studies, discrete-time systems have been used for a variety of phenomena in electrical networks, genetics, ecological systems, and so forth.Therefore, the stability analysis of discretetime neural networks (DNNs) with delays has become an important topic of theoretical studies in neural networks; for example, asymptotic stability and exponential stability of neural networks have been studied by many researchers.In [1], the authors have studied robust stability of discretetime linear-parameter-dependent (LPD) neural networks with time-varying delay.In order to derive stability criteria of discrete-time, one common approach is the use of appropriate inequalities for difference equations.Another approach is the use of Lyapunov stability theory.In [2], the authors have studied global exponential stability of impulsive discretetime neural networks with time-varying delays, based on some inequality analysis techniques.In [3], the authors have studied new discrete-type inequalities and global stability of nonlinear difference equation.In [4], the authors have studied global exponential stability of discrete-time Hopfield neural networks with variable by using the difference inequality.In [5], the authors have considered the problem of robust stability analysis of generalized neural networks with multiple discrete delays and multiple distributed delays by using the Lyapunov-Krasovskii functional method and the linear matrix inequality technique.In [6], the authors have studied delay-dependent exponential stability criteria for discretetime nonlinear system with multiple time-varying delays.In this paper, we propose to study global exponential stability of discrete-time neural networks with time-varying delays.In Section 2, we have introduced discrete-time neural networks with time-varying delays and presented some preliminaries.In Section 3, we have derived new discrete-type inequalities; global exponential stability criteria are derived by using new discrete-type inequalities.Finally, numerical examples are given to illustrate the effectiveness of our theoretical results.

Notations, Definitions, and Preliminaries Results
In this section, we give some notations definitions and preliminaries results which will be used throughout this paper.

Main Result
Throughout this section, we denote () by   .In this section, we provide global exponential stability criteria for system (5).First, we introduce new discrete-type inequalities which will be used to derive global exponential stability condition.

Theorem 3. The equilibrium point at the origin of system (5) is globally exponentially stable if
where  = max  (  ),  max = max  (  ), and max  (  ) is maximum of vector.
Proof.Let   = ‖  ‖.Then, from ( 5), the difference of system is given by where From Lemma 2, it follows assumptions of theorem that there exist  0 ∈ (0, 1) such that Thus, we obtain By Definition 1, we conclude that ( 5) is globally exponentially stable.The proof is complete.
Remark 4. In our main result, we derived global exponential stability criteria for discrete-time neural networks with multiple time-varying delays by using discrete-type inequality.
In [4], the global exponential stability criteria of discretetime Hopfield neural networks are given.Nevertheless, the stability criteria in [4] cannot be applied to discrete-time system with multiple time-varying delays.
The equilibrium point of system in Example 1 is From which it follows that Therefore, from Theorem 3, it follows that the equilibrium point at the origin of system (1) is globally exponentially stable.
Then  Therefore, from Theorem 3, it follows that the equilibrium point at the origin of system (1) is globally exponentially stable.

Conclusion
In this paper, we obtained sufficient condition for global exponential stability of discrete-time neural networks with multiple time-varying delays.The stability criteria are derived by using a discrete-type inequality.Numerical examples are given to show the effectiveness of our theoretical results.