New Exponential Stability Conditions of Switched BAM Neural Networks with Delays

The exponential stability problem is considered in this paper for discrete-time switched BAM neural networks with time delay. The average dwell time method is introduced to deal with the exponential stability analysis of the systems for the first time. By constructing a new switching-dependent Lyapunov-Krasovskii functional, somenewdelay-dependent criteria are developed,which guarantee the exponential stability. A numerical example is provided to demonstrate the potential and effectiveness of the proposed algorithms.


Introduction
It is well known that bidirectional associative memory (BAM) neural networks have been proposed by Kosko [1,2], which include two layers: the -layer and the -layer.The neurons in one layer are fully interconnected to the neurons in another layer.Recently, the dynamics analysis for BAM neural networks has received much attention due to their extensive applications in pattern recognition, solving optimization, automatic control engineering, and so forth.It is known that time delay, which will inevitably occur in the communication owing to the unavoidable finite switching speed of amplifiers, is the main cause of instability and poor performance of neural networks.Hence, it is of great importance to study the stability of BAM neural networks with time delay.Many asymptotic or exponential stability conditions for BAM neural networks with time delay were developed, see, for example [3][4][5][6][7][8][9][10] and the references therein.
On the other hand, switched systems are an important class of hybrid dynamical systems which consist of a family of continuous-time or discrete-time subsystems and a rule that orchestrates the switching among them.Switched systems provide a natural and convenient unified framework for mathematical modeling of many physical phenomena and practical applications such as autonomous transmission systems, computer disc driver, room temperature control, power electronics, and chaos generators, to name a few.Lots of valuable results concerning the stability analysis and stabilization for linear or nonlinear hybrid and switched systems were established, see, for example [11][12][13][14] and the references cited therein.
Recently, the switched neural networks, whose individual subsystems are a set of neural networks, have found their applications in the field of high-speed signal processing and artificial intelligence.Many researchers have been devoted to studying the stability issues for switched neural networks; see, for example, [15][16][17].In [15], by using switched Lyapunov function method and a generalized Halanay inequality technique, the authors illustrated the asymptotic and exponential stability conditions for hybrid impulsive and switching Hopfield neural networks.While the switched Hopfield neural networks with time-varying delay were considered in [16], a robust stability condition was proposed based on the Lyapunov-Krasovskii functional approach.By combining Cohen-Grossberg neural networks with an arbitrary switching rule, the model of the switched Cohen-Grossberg neural networks with mixed time-varying delays was established in [17], and the robust stability criteria were established for these systems.However, all these results are related to the continuous-time switched neural networks.To the best of the authors' knowledge, stability issues of the discrete-time switched neural networks have not been fully investigated to date.Particularly for the exponential stability analysis of the discrete-time switched BAM neural networks under some constrained switching, few results have been available in the literature so far, which motivates us to carry out the present study.
In this paper, the exponential stability analysis of discretetime switched BAM neural networks with time delay is considered.To begin with, the mathematical model of the discrete-time switched BAM neural networks with time delay is established.Then by constructing a new switchingdependent Lyapunov-Krasovskii functional, some sufficient criteria are developed to guarantee the discrete-time switched BAM neural networks to be exponentially stable based on the average dwell time approach and finite sum inequality technology.Finally, A numerical example is provided to demonstrate the potential and effectiveness of the proposed algorithms.

Problem Formulation and Preliminaries
In this section, firstly, we will establish the model of discretetime switched BAM neural networks.Consider the following discrete-time BAM neural networks with time delay (Σ 1 ): where x (), ỹ () are states of the th neuron from the neural field   and the th neuron from the neural field   at time , respectively.  ,   ∈ (0, 1) describe the stability of internal neuron processes on the -layer and the -layer, respectively.  , V  are constants and denote the synaptic connection weights.f (⋅) and g (⋅) denote the activation functions of the th neuron from the neural field   and the th neuron from the neural field   , respectively.  and   are the external constant inputs from outside of the network acting on the th neuron from the neural field   and the th neuron from the neural field   , respectively. and  are constant delays.
(G 3 ) There exist constants ℓ (1)   > 0 and ℓ (2)   > 0 such that        ()      ≤ ℓ (1) With the rapid development of intelligent control, hybrid systems have been investigated due to their extensive applications.In recent years, considerable efforts have been focused on analysis and design of switched systems.The discretetime switched system can be characterized by the following difference equation (Σ 4 ): where () is a switching signal which takes its values in the finite set N = {1, 2, . . ., }.Γ () = Γ  , when () = , are the functions of the switching signals.
Combining the theories of switched systems and discretetime BAM neural networks, the discrete-time switched BAM neural networks can be formulated as the following system (Σ): where () is a switching signal which takes its values in the finite set N = {1, 2, . . ., }.
For the discrete-time switched BAM neural networks (Σ), we have the following assumptions.
When  ∈ [  ,  +1 ), the   th subsystem is activated and the states of system (Σ) do not jump when switch occurs.
Remark 1.By combining the switched systems theory and the discrete-time BAM neural networks model, the mathematical model of discrete-time switched BAM neural networks is introduced as above.A set of discrete-time BAM neural networks with time delay are used as the subsystems, and an arbitrary switching rule is assumed to coordinate the switching between these neural networks.
To present the main results of this paper more precisely, the following definitions and lemmas are introduced, which will be essential for the later development.
Remark 4. Without loss of generality, in this paper, we assume  0 = 0 for simplicity as commonly used in the literature.
Remark 5. Based on the definition of exponential stability for BAM neural networks in [5] and the definition of exponential stability for switched systems in [13], we give the above definition of exponential stability for discrete-time switched BAM neural networks.

An Illustrative Example
Consider the discrete-time switched BAM neural networks (Σ) combining two subsystems with the following parameters: Let  = 1 and  = 1.Solving LMI ( 17), (28), and (29), it is found that the LMIs are feasible for all  ≤ 1.47.The calculated values of the delay upper bound  and decay rate  for different values of  and  are given in Table 1 when   = 1.From Table 1, we can see that the delay is related to the decay rate.For a given , a smaller decay rate  allows a larger delay  max .Moreover, for every , the delay  max decreases when the delay  increases.Letting  = 1.4, = 3, and  = 1.2, we obtain that  *  = ceil[0.5419].Based on (30),   = 1 is satisfied.Then we can calculate that the decay rate  =  (− ln /  ln )+1 = 1.1667.Therefore, the discrete-time switched BAM neural networks with time delay are exponentially stable with the decay rate  = 1.1667 if the delay  is not larger than 13 based on Table 1.

Mathematical Problems in Engineering
For  = 1.

Conclusions
In this paper, the exponential stability problem for the discrete-time switched BAM neural networks with time delay has been proposed.At first, the mathematical model of the discrete-time switched BAM neural networks with time delay has been established.And then by constructing a new switching-dependent Lyapunov-Krasovskii functional, some sufficient criteria have been developed to guarantee the discrete-time switched BAM neural networks to be exponentially stable based on the average dwell time approach and finite sum inequality technology.Finally, a numerical example has been provided to demonstrate the potential and effectiveness of the proposed algorithms.

5 Figure 1 :
Figure 1: State response of the given system.
10mark10.In (30), the function ceil() is used, which represents rounding real number  to the nearest integer greater than or equal to .The reason that we introduce the function ceil is that the dwell time length of the currently active subsystem is the number of sampling periods between the two consecutive switching times.

Table 1 :
The maximum delay bound ,  and decay rate .