New Exponential Stability Conditions of Switched BAM Neural Networks with Delays

Abstract

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, some new delay-dependent criteria are developed, which guarantee the exponential stability. A numerical example is provided to demonstrate the potential and effectiveness of the proposed algorithms.

1. Introduction

It is well known that bidirectional associative memory (BAM) neural networks have been proposed by Kosko [1, 2], which include two layers: the 𝑋-layer and the 𝑌-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 [3–10] 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 [11–14] 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, [15–17]. 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 discrete-time 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 switching-dependent 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.

Notations. In this paper, we use 𝐴>0(𝐴<0) to denote a positive- (negative-) definite matrix 𝐴; 𝐴^𝑇 represents the transpose of matrix 𝐴; 𝜆_𝑀(⋅) (resp., 𝜆_𝑚(⋅)) means the maximum (resp., minimum) eigenvalue of (⋅). Let ℝ denote the set of real numbers; ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space; ℝ^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices; ℝ⁺ denotes the set of {0,1,2,…}. 𝒩={1,2,…,𝑁} means a set of positive integers; ℕ={1,2,…,𝑛}. The notation diag(⋅) denotes a diagonal matrix. For given 𝜏>0 and 𝜃∈[−𝜏,0], ‖𝑥(𝑡)‖ denotes vector norm defined by ‖𝑥(𝑡)‖=sup_{−𝜏≤𝜃≤0}‖𝑥(𝑡+𝜃)‖. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Problem Formulation and Preliminaries

In this section, firstly, we will establish the model of discrete-time switched BAM neural networks. Consider the following discrete-time BAM neural networks with time delay (Σ₁): 𝑥_𝑝(𝑘+1)=𝑎_𝑝𝑥_𝑝(𝑘)+^𝑛_𝑞=1𝑤_𝑞𝑝𝑓_𝑞𝑦_𝑞(𝑘−𝑑)+𝐼_𝑝𝑦,𝑝∈ℕ,_𝑞(𝑘+1)=𝑏_𝑞𝑦_𝑞(𝑘)+^𝑛_𝑝=1𝑣_𝑝𝑞𝑔_𝑝𝑥_𝑝(𝑘−𝜏)+𝐽_𝑞,𝑞∈ℕ, (1) where 𝑥_𝑝(𝑘),  𝑦_𝑞(𝑘) are states of the 𝑝th neuron from the neural field 𝐹_𝑋 and the 𝑞th neuron from the neural field 𝐹_𝑌 at time 𝑘, respectively. 𝑎_𝑝,  𝑏_𝑞∈(0,1) describe the stability of internal neuron processes on the 𝑋-layer and the 𝑌-layer, respectively. 𝑤_𝑞𝑝,  𝑣_𝑝𝑞 are constants and denote the synaptic connection weights. 𝑓_𝑞(⋅) and 𝑔_𝑝(⋅) denote the activation functions of the 𝑞th neuron from the neural field 𝐹_𝑌 and the 𝑝th neuron from the neural field 𝐹_𝑋, respectively. 𝐼_𝑝 and 𝐽_𝑞 are the external constant inputs from outside of the network acting on the 𝑝th neuron from the neural field 𝐹_𝑋 and the 𝑞th neuron from the neural field 𝐹_𝑌, respectively. 𝑑 and 𝜏 are constant delays.

The system (Σ₁) can be rewritten as the vector form (Σ₂): 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝑊^𝑇𝑓𝑦(𝑦()𝑦(𝑘)𝑘−𝑑)+𝐼,𝑘+1=𝐵+𝑉^𝑇𝑔(𝑥(𝑘−𝜏))+𝐽, (2) where 𝑥(𝑘)=𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)^𝑇,𝑦(𝑘)=𝑦₁(𝑘),𝑦₂(𝑘),…,𝑦_𝑛(𝑘)^𝑇,𝐴=diag𝑎₁,𝑎₂,…,𝑎_𝑛,𝐵=diag𝑏₁,𝑏₂,…,𝑏_𝑛,𝑊=𝑤_{𝑞𝑝𝑛×𝑛},𝑉=𝑣_{𝑝𝑞𝑛×𝑛},𝑓𝑦(𝑘)=𝑓₁𝑦₁(𝑘),𝑓₂𝑦₂(𝑘),…,𝑓_𝑛𝑦_𝑛(𝑘)^𝑇,𝑔(𝑥(𝑘=𝑔))₁𝑥₁(𝑘),𝑔₂𝑥₂(𝑘),…,𝑔_𝑛𝑥_𝑛(𝑘)^𝑇,𝐼=𝐼₁,𝐼₂,…,𝐼_𝑛,𝐽=𝐽₁,𝐽₂,…,𝐽_𝑛. (3) Throughout this paper, we always assume the following. (G₁)The neurons activation functions 𝑓_𝑞(⋅) and 𝑔_𝑝(⋅)  (𝑝,𝑞∈ℕ) are bounded on ℝ. (G₂)There exist constants ℓ_𝑞⁽¹⁾>0 and ℓ_𝑝⁽²⁾>0 such that 𝑓_𝑞𝜉₁−𝑓_𝑞𝜉₂≤ℓ_𝑞⁽¹⁾𝜉₁−𝜉₂,𝑔_𝑝𝜉₁−𝑔_𝑝𝜉₂≤ℓ_𝑝⁽²⁾𝜉₁−𝜉₂,∀𝜉₁,𝜉₂∈ℝ,𝑝,𝑞∈ℕ. (4)Then, under the assumptions (G₁) and (G₂), system (Σ₂) has at least one equilibrium.

Now, we shift equilibrium point 𝑥^∗=[𝑥^∗₁𝑥^∗₂…𝑥^∗_𝑛],   𝑦^∗=[𝑦^∗₁𝑦^∗₂⋯𝑦^∗_𝑛] of system (Σ₂) to the origin. Let 𝑥(𝑘)=𝑥(𝑘)−𝑥^∗,   𝑦(𝑘)=𝑦(𝑘)−𝑦^∗; then the system (Σ₂) can be transformed to the following system (Σ₃): 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝑊^𝑇𝑓𝑦(,𝑦()(𝑘)𝑘−𝑑)𝑘+1=𝐵𝑦+𝑉^𝑇𝑔(𝑥(𝑘−𝜏)), (5) where 𝑥(𝑘)=𝑥₁(𝑘),𝑥₂(𝑘),…,𝑥_𝑛(𝑘)^𝑇,𝑦(𝑘)=𝑦₁(𝑘),𝑦₂(𝑘),…,𝑦_𝑛(𝑘)^𝑇,𝑓𝑦(𝑘)=𝑓₁𝑦₁(𝑘),𝑓₂𝑦₂(𝑘),…,𝑓_𝑛𝑦_𝑛(𝑘)^𝑇,𝑔(𝑥(𝑘))=𝑔₁𝑥₁(𝑘),𝑔₂𝑥₂(𝑘),…,𝑔_𝑛𝑥_𝑛(𝑘)^𝑇,𝑓_𝑞𝑦_𝑞(𝑘)=𝑓_𝑞𝑦_𝑞(𝑘)−𝑓_𝑞𝑦^∗_𝑞𝑔,𝑞∈ℕ,_𝑝𝑥_𝑝(𝑘)=𝑔_𝑝𝑥_𝑝(𝑘)−𝑔_𝑝𝑔^∗_𝑝,𝑝∈ℕ. (6) Obviously, the activation functions 𝑓_𝑞(⋅) and 𝑔_𝑝(⋅) satisfy the following conditions. (G₃) There exist constants ℓ_𝑞⁽¹⁾>0 and ℓ_𝑝⁽²⁾>0 such that 𝑓_𝑞(𝜉)≤ℓ_𝑞⁽¹⁾𝜉,𝑔_𝑝(𝜉)≤ℓ_𝑝⁽²⁾𝜉,∀𝜉∈ℝ,𝑝,𝑞∈ℕ. (7)

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 discrete-time switched system can be characterized by the following difference equation (Σ₄): 𝑥(𝑘+1)=Γ_𝜎(𝑘)𝑥(𝑘), (8) where 𝜎(𝑘) is a switching signal which takes its values in the finite set 𝒩={1,2,…,𝑁}. Γ_𝜎(𝑘)=Γ_𝑖, when 𝜎(𝑘)=𝑖, are the functions of the switching signals.

Combining the theories of switched systems and discrete-time BAM neural networks, the discrete-time switched BAM neural networks can be formulated as the following system (Σ): 𝑥(𝑘+1)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝑊^𝑇_𝜎(𝑘)𝑓𝑦(,𝑦()𝑘−𝑑)𝑘+1=𝐵_𝜎(𝑘)𝑦(𝑘)+𝑉^𝑇_𝜎(𝑘)𝑔(𝑥(𝑘−𝜏)), (9) where 𝜎(𝑘) is a switching signal which takes its values in the finite set 𝒩={1,2,…,𝑁}.

For the discrete-time switched BAM neural networks (Σ), we have the following assumptions. (H₁) The initial value is 𝑥(𝑠)=𝜙(𝑠),   𝑦(𝑠)=𝜓(𝑠),𝑠∈[−ℎ,0], where ℎ=max{𝑑,𝜏}.(H₂) There exist matrices 𝐿₁>0 and 𝐿₂>0 such that 𝑓(𝜉)≤𝐿₁𝜉,𝑔(𝜉)≤𝐿₂𝜉,∀𝜉∈ℝ, (10) where 𝐿₁=diag(ℓ₁⁽¹⁾,ℓ₂⁽¹⁾,…,ℓ_𝑛⁽¹⁾) and 𝐿₂=diag(ℓ₁⁽²⁾,ℓ₂⁽²⁾,…,ℓ_𝑛⁽²⁾). (H₃) Switching sequence is defined as  𝜁={[𝑥_𝑘0𝑦_𝑘0]^𝑇;(𝑖₀,𝑘₀),(𝑖₁,𝑘₁),…,(𝑖_𝑚,𝑘_𝑚),…,|𝑖_𝑚∈𝒩,𝑚∈ℝ⁺}. When 𝑘∈[𝑘_𝑚,𝑘_𝑚+1), the 𝑘_𝑚th 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.

Definition 2 (see [12]). For any 𝑘≥𝑘₀ and any switched signal 𝜎(𝜍),  𝑘₀≤𝜍<𝑘, let 𝑁_𝜎 denote the switching numbers of 𝜎(𝜍) during the interval [𝑘₀,𝑘]. If there exist 𝑁₀≥0 and 𝑇_𝑎>0 such that 𝑁_𝜎(𝑘₀,𝑘)≤𝑁₀+(𝑘−𝑘₀)/𝑇_𝑎, then 𝑇_𝑎 and 𝑁₀ are called average dwell time and the chatter bound, respectively.

Definition 3. The discrete-time switched BAM neural network (Σ) is said to be exponentially stable if its solution satisfies ‖𝑥(𝑘)‖²+𝑦(𝑘)²≤𝐾𝜙²_𝐿+𝜓²_𝐿𝜆^{−(𝑘−𝑘0)},∀𝑘≥𝑘₀, (11) for any initial condition (𝑘₀,𝜙)∈ℝ⁺×𝐶^𝑛 and (𝑘₀,𝜓)∈ℝ⁺×𝐶^𝑛⋅‖𝜙‖_𝐿=sup_{𝑘0−ℎ≤ℓ≤𝑘0}‖𝜙(ℓ)‖, and ‖𝜓‖_𝐿=sup_{𝑘0−ℎ≤ℓ≤𝑘0}‖𝜓(ℓ)‖,   ℎ=max{𝑑,𝜏}.    𝐾>0 is the decay coefficient, and 𝜆>1 is the decay rate.

Remark 4. Without loss of generality, in this paper, we assume 𝑁₀=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.

Lemma 6 (the Schur complement [18]). For any symmetric matrix 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22}<0, the following conditions are equivalent:(i)𝑆₁₁<0,  and 𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0,(ii)𝑆₂₂<0,  and 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

Lemma 7 (finite sum inequality [13]). For any constant matrix 𝑌=[𝑀₁𝑀₂]∈ℝ^𝑛×2𝑛,   𝑅>0,   ℎ≥0, the following inequality holds: −^𝑘−1_{𝑗=𝑘−ℎ}ℓ^𝑇𝑗𝑅ℓ𝑗≤𝜉^𝑇(𝑘)𝑀^𝑇₁+𝑀₁−𝑀^𝑇₁+𝑀₂∗−𝑀^𝑇₂−𝑀₂×𝜉(𝑘)+ℎ𝜉^𝑇(𝑘)𝑌^𝑇𝑅⁻¹𝑌𝜉(𝑘), (12) where ℓ(𝑘)=𝑥(𝑘+1)−𝑥(𝑘) and 𝜉(𝑘)=[𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−ℎ)]^𝑇.

3. Main Result

In this section, the exponential stability condition for the discrete-time switched BAM neural networks (Σ) will be presented using the average dwell time method.

When 𝜎(𝑘)=𝑖, we have the following subsystem (Σ^𝑖): 𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝑊^𝑇_𝑖𝑓𝑦(,𝑦()𝑘−𝑑)𝑘+1=𝐵_𝑖𝑦(𝑘)+𝑉^𝑇_𝑖𝑔(𝑥(𝑘−𝜏)). (13) Choose the Lyapunov-Krasovskii functional candidate for the subsystem (Σ^𝑖) as 𝑉_𝑖(𝑘)=𝑉_1𝑖(𝑘)+𝑉_2𝑖(𝑘)+𝑉_3𝑖(𝑘), (14) where 𝑉_1𝑖(𝑘)=𝑥^𝑇(𝑘)𝑃_1𝑖𝑥(𝑘)+𝑦^𝑇(𝑘)𝑃_2𝑖𝑦(𝑘),𝑉_2𝑖(𝑘)=^𝑘−1_{𝜃=𝑘−𝜏}𝑟^{𝜃−𝑘+1}𝑥^𝑇(𝜃)𝑄_1𝑖𝑥(𝜃)+^𝑘−1_{𝜃=𝑘−𝑑}𝑟^{𝜃−𝑘+1}𝑦^𝑇(𝜃)𝑄_2𝑖𝑦(𝜃),𝑉_3𝑖(𝑘)=⁻¹_𝑠=−𝜏^𝑘−1_{𝜃=𝑘+𝑠}𝑟^{𝜃−𝑘+1}ℓ^𝑇₁(𝜃)𝑅_1𝑖ℓ₁(𝜃)+⁻¹_𝑠=−𝑑^𝑘−1_{𝜃=𝑘+𝑠}𝑟^{𝜃−𝑘+1}ℓ^𝑇₂(𝜃)𝑅_2𝑖ℓ₂(𝜃),ℓ₁(𝑘)()(𝑘),ℓ=𝑥𝑘+1−𝑥₂(𝑘)=𝑦(𝑘+1)−𝑦(𝑘). (15)

Now we give the following theorem, which plays an important role in the derivation of the exponential stability condition for the discrete-time switched BAM neural networks (Σ).

Theorem 8. Under the assumptions (𝐻₁)–(𝐻₃), for given scalar 𝑟>1, the decay estimation 𝑉_𝑖(𝑘)≤𝑟^{−(𝑘−𝑘0)}𝑉_𝑖𝑘₀ (16) is satisfied along any trajectory of system (Σ^𝑖) if there exist matrices 𝑃_1𝑖>0,  𝑃_2𝑖>0,  𝑄_1𝑖>0,  𝑄_2𝑖>0,  𝑅_1𝑖>0,  𝑅_2𝑖>0,  𝑁_1𝑖,𝑁_2𝑖,𝑀_1𝑖,𝑀_2𝑖,𝑇_1𝑖>0, and 𝑇_2𝑖>0,  𝑖∈𝒩, such that the following linear matrix inequality holds: Ω_𝑖Γ^𝑇_1𝑖Γ^𝑇_2𝑖∗−𝜏𝑟^𝜏𝑅_1𝑖0∗∗−𝑑𝑟^𝑑𝑅_2𝑖<0, (17) where Ω_𝑖=Ω₁₁Ω₁₂000Ω₁₆∗Ω₂₂0000∗∗Ω₃₃Ω₃₄00∗∗∗Ω₄₄Ω₄₅0∗∗∗∗Ω₅₅0∗∗∗∗∗Ω₆₆,Γ_1𝑖=𝜏𝑁_1𝑖𝜏𝑁_2𝑖,Γ0000_2𝑖=000𝑑𝑀_1𝑖𝑑𝑀_2𝑖0,Ω₁₁=𝐴_𝑖𝑃_1𝑖𝐴_𝑖−𝑟⁻¹𝑃_1𝑖+𝑄_1𝑖+𝐴_𝑖−𝐼𝜏𝑅_1𝑖𝐴_𝑖−𝐼+𝑟^−𝜏𝑁^𝑇_1𝑖+𝑁_1𝑖,Ω₁₂=𝑟^−𝜏−𝑁^𝑇_1𝑖+𝑁_2𝑖,Ω₂₂=−𝑟^−𝜏𝑄_1𝑖+𝑁^𝑇_2𝑖+𝑁_2𝑖+𝑇_1𝑖,Ω₃₃=𝑉_𝑖𝑃_2𝑖+𝑑𝑅_2𝑖𝑉^𝑇_𝑖−𝐿⁻¹_1𝑖𝑇_1𝑖𝐿⁻¹_1𝑖,Ω₃₄=𝑉_𝑖𝑃_2𝑖𝐵_𝑖+𝑉_𝑖𝑑𝑅_2𝑖𝐵_𝑖,Ω−𝐼₄₄=𝐵_𝑖𝑃_2𝑖𝐵_𝑖−𝑟⁻¹𝑃_2𝑖+𝑄_2𝑖+𝐵_𝑖−𝐼𝑑𝑅_2𝑖𝐵_𝑖−𝐼+𝑟^−𝑑𝑀^𝑇_1𝑖+𝑀_1𝑖,Ω₄₅=𝑟^−𝑑−𝑀^𝑇_1𝑖+𝑀_2𝑖,Ω₅₅=−𝑟^−𝑑𝑄_2𝑖+𝑀^𝑇_2𝑖+𝑀_2𝑖+𝑇_2𝑖,Ω₁₆=𝐴_𝑖𝑃_1𝑖𝑊^𝑇_𝑖+𝐴_𝑖−𝐼𝜏𝑅_1𝑖𝑊^𝑇_𝑖,Ω₆₆=𝑊_𝑖𝑃_1𝑖+𝜏𝑅_1𝑖𝑊^𝑇_𝑖−𝐿⁻¹_2𝑖𝑇_2𝑖𝐿⁻¹_2𝑖. (18)

Proof. Calculating the differential of 𝑉_𝑖(𝑘) along the trajectory of system (Σ^𝑖), we obtain Δ𝑉_1𝑖(𝑘)=𝑥^𝑇(𝑘+1)𝑃_1𝑖𝑥(𝑘+1)+𝑦^𝑇(𝑘+1)𝑃_2𝑖𝑦(𝑘+1)−𝑥^𝑇(𝑘)𝑃_1𝑖𝑥(𝑘)−𝑦^𝑇(𝑘)𝑃_2𝑖𝑦(𝑘)=𝑥^𝑇(𝑘+1)𝑃_1𝑖𝑥(𝑘+1)−𝑟⁻¹𝑥^𝑇(𝑘)𝑃_1𝑖𝑥(𝑘)+𝑦^𝑇(𝑘+1)𝑃_2𝑖𝑦(𝑘+1)−𝑟⁻¹𝑦^𝑇(𝑘)𝑃_2𝑖𝑦(𝑘)+𝑟⁻¹−1𝑉_1𝑖(𝑘),Δ𝑉_2𝑖(𝑘)=^𝑘_{𝜃=𝑘+1−𝜏}𝑟^𝜃−𝑘𝑥^𝑇(𝜃)𝑄_1𝑖𝑥(𝜃)+^𝑘_{𝜃=𝑘+1−𝑑}𝑟^𝜃−𝑘𝑦^𝑇(𝜃)𝑄_2𝑖𝑦(𝜃)−^𝑘−1_{𝜃=𝑘−𝜏}𝑟^{𝜃−𝑘+1}𝑥^𝑇(𝜃)𝑄_1𝑖𝑥(𝜃)−^𝑘−1_{𝜃=𝑘−𝑑}𝑟^{𝜃−𝑘+1}𝑦^𝑇(𝜃)𝑄_2𝑖𝑦(𝜃)=𝑥^𝑇(𝑘)𝑄_1𝑖𝑥(𝑘)−𝑟^−𝜏𝑥^𝑇(𝑘−𝜏)𝑄_1𝑖𝑥(𝑘−𝜏)+𝑦^𝑇(𝑘)𝑄_2𝑖𝑦(𝑘)−𝑟^−𝑑𝑦^𝑇(𝑘−𝑑)𝑄_2𝑖𝑦(𝑘−𝑑)+𝑟⁻¹−1𝑉_2𝑖(𝑘),Δ𝑉_3𝑖(𝑘)=^−1𝑘_{𝑠=−𝜏𝜃=𝑘+1+𝑠}𝑟^𝜃−𝑘ℓ^𝑇₁(𝜃)𝑅_1𝑖ℓ₁(𝜃)+^−1𝑘_{𝑠=−𝑑𝜃=𝑘+1+𝑠}𝑟^𝜃−𝑘ℓ^𝑇₂(𝜃)𝑅_2𝑖ℓ₂(𝜃)−⁻¹_𝑠=−𝜏^𝑘−1_{𝜃=𝑘+𝑠}𝑟^{𝜃−𝑘+1}ℓ^𝑇₁(𝜃)𝑅_1𝑖ℓ₁(𝜃)−⁻¹_𝑠=−𝑑^𝑘−1_{𝜃=𝑘+𝑠}𝑟^{𝜃−𝑘+1}ℓ^𝑇₂(𝜃)𝑅_2𝑖ℓ₂(𝜃)=⁻¹_𝑠=−𝜏ℓ^𝑇₁(𝑘)𝑅_1𝑖ℓ₁(𝑘)+^𝑘−1_{𝜃=𝑘+𝑠}𝑟^𝜃−𝑘ℓ^𝑇₁(𝜃)𝑅_1𝑖ℓ₁(𝜃)−𝑟^𝑠ℓ^𝑇₁(𝑘+𝑠)𝑅_1𝑖ℓ₁(+𝑘+𝑠)_{−1𝑠=−𝑑}ℓ^𝑇₂(𝑘)𝑅_2𝑖ℓ₂(𝑘)+^𝑘−1_{𝜃=𝑘+𝑠}𝑟^𝜃−𝑘ℓ^𝑇₂(𝜃)𝑅_2𝑖ℓ₂(𝜃)−𝑟^𝑠ℓ^𝑇₂(𝑘+𝑠)𝑅_2𝑖ℓ₂(𝑘+𝑠)−𝑉_3𝑖(𝑘)=𝜏ℓ^𝑇₁(𝑘)𝑅_1𝑖ℓ₁(𝑘)−^𝑘−1_{𝑠=𝑘−𝜏}𝑟^𝑠−𝑘ℓ^𝑇₁(𝑠)𝑅_1𝑖ℓ₁(𝑠)+𝑑ℓ^𝑇₂(𝑘)𝑅_2𝑖ℓ₂(𝑘)−^𝑘−1_{𝑠=𝑘−𝑑}𝑟^𝑠−𝑘ℓ^𝑇₂(𝑠)𝑅_2𝑖ℓ₂(𝑠)+𝑟⁻¹−1𝑉_3𝑖(𝑘)≤𝜏ℓ^𝑇₁(𝑘)𝑅_1𝑖ℓ₁(𝑘)+𝑑ℓ^𝑇₂(𝑘)𝑅_2𝑖ℓ₂(𝑘)−^𝑘−1_{𝑠=𝑘−𝜏}𝑟^−𝜏ℓ^𝑇₁(𝑠)𝑅_1𝑖ℓ₁(𝑠)−^𝑘−1_{𝑠=𝑘−𝑑}𝑟^−𝑑ℓ^𝑇₂(𝑠)𝑅_2𝑖ℓ₂(𝑠)+𝑟⁻¹−1𝑉_3𝑖(𝑘). (19)(20)
Note that −𝑟^{−𝜏𝑘−1}_{𝑠=𝑘−𝜏}ℓ^𝑇₁(𝑠)𝑅_1𝑖ℓ₁(𝑠)≤𝑟^−𝜏𝜂^𝑇₁(𝑘)𝑁^𝑇_1𝑖+𝑁_1𝑖−𝑁^𝑇_1𝑖+𝑁_2𝑖∗−𝑁^𝑇_2𝑖−𝑁_2𝑖𝜂₁(𝑘)+𝑟^−𝜏𝜂^𝑇₁(𝑘)𝑁^𝑇_1𝑖𝑁^𝑇_2𝑖𝜏𝑅⁻¹_1𝑖𝑁_1𝑖𝑁_2𝑖𝜂₁(𝑘),−𝑟^{−𝑑𝑘−1}_{𝑠=𝑘−𝑑}ℓ^𝑇₂(𝑠)𝑅_2𝑖ℓ₂(𝑠)≤𝑟^−𝑑𝜂^𝑇₂(𝑘)𝑀^𝑇_1𝑖+𝑀_1𝑖−𝑀^𝑇_1𝑖+𝑀_2𝑖∗−𝑀^𝑇_2𝑖−𝑀_2𝑖𝜂₂(𝑘)+𝑟^−𝑑𝜂^𝑇₂(𝑘)𝑀^𝑇_1𝑖𝑀^𝑇_2𝑖𝑑𝑅⁻¹_2𝑖𝑀_1𝑖𝑀_2𝑖𝜂₂(𝑘), (21)
where 𝜂₁(𝑘)=𝑥(𝑘)𝑥(𝑘−𝜏),𝜂₂(𝑘)=𝑦(𝑘)𝑦()𝑥𝑘−𝑑_𝑇,(𝑘−𝜏)𝑇_1𝑖𝑥(𝑘−𝜏)−𝑔^𝑇(𝑥(𝑘−𝜏))𝐿⁻¹_1𝑖𝑇_1𝑖𝐿⁻¹_1𝑖𝑔(𝑥(𝑦𝑘−𝜏))≥0,_𝑇(𝑘−𝑑)𝑇_2𝑖𝑥(𝑘−𝑑)−𝑓^𝑇𝑦(𝑘−𝑑)𝐿⁻¹_2𝑖𝑇_2𝑖𝐿⁻¹_2𝑖𝑓𝑦(𝑘−𝑑)≥0. (22)(23)(24) From (20) to (24), the following inequality is satisfied: Δ𝑉_𝑖(𝑘)≤𝑥(𝑘)𝑓(𝑦(𝑘−𝑑))^𝑇𝐴_𝑖𝑊_𝑖𝑃_1𝑖𝐴_𝑖𝑊^𝑇_𝑖𝑥(𝑘)𝑓𝑦(𝑘−𝑑)−𝑥^𝑇(𝑘)𝑟⁻¹𝑃_1𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝑄_1𝑖𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏)𝑟^−𝜏𝑄_1𝑖𝑥()+𝑘−𝜏𝑦(𝑘)𝑔(𝑥(𝑘−𝜏))^𝑇𝐵_𝑖𝑉_𝑖𝑃_2𝑖𝐵_𝑖𝑉^𝑇_𝑖𝑦(𝑘)𝑔(𝑥(𝑘−𝜏))−𝑦^𝑇(𝑘)𝑟⁻¹𝑃_2𝑖𝑦(𝑘)+𝑦^𝑇(𝑘)𝑄_2𝑖𝑦(𝑘)−𝑦^𝑇(𝑘−𝑑)𝑟^−𝑑𝑄_2𝑖𝑦()+𝑘−𝑑𝑥(𝑘)𝑓(𝑦(𝑘−𝑑))^𝑇𝐴_𝑖𝑊−𝐼_𝑖×𝜏𝑅_1𝑖𝐴_𝑖−𝐼𝑊^𝑇_𝑖𝑥(𝑘)𝑓𝑦()+𝑘−𝑑𝑦(𝑘)𝑔(𝑥(𝑘−𝜏))^𝑇𝐵_𝑖𝑉−𝐼_𝑖×𝑑𝑅_2𝑖𝐵_𝑖−𝐼𝑉^𝑇_𝑖𝑦(𝑘)𝑔(𝑥(𝑘−𝜏))+𝑟^−𝜏𝜂^𝑇₁(𝑘)𝑁^𝑇_1𝑖+𝑁_1𝑖−𝑁^𝑇_1𝑖+𝑁_2𝑖∗−𝑁^𝑇_2𝑖−𝑁_2𝑖𝜂₁(𝑘)+𝑟^−𝜏𝜂^𝑇₁(𝑘)𝑁^𝑇_1𝑖𝑁^𝑇_2𝑖𝜏𝑅⁻¹_1𝑖𝑁_1𝑖𝑁_2𝑖𝜂₁(𝑘)+𝑟^−𝑑𝜂^𝑇₂(𝑘)𝑀^𝑇_1𝑖+𝑀_1𝑖−𝑀^𝑇_1𝑖+𝑀_2𝑖∗−𝑀^𝑇_2𝑖−𝑀_2𝑖𝜂₂(𝑘)+𝑟^−𝑑𝜂^𝑇₂(𝑘)𝑀^𝑇_1𝑖𝑀^𝑇_2𝑖𝑑𝑅⁻¹_2𝑖𝑀_1𝑖𝑀_2𝑖𝜂₂(𝑘)+𝑥^𝑇(𝑘−𝜏)𝑇_1𝑖𝑥(𝑘−𝜏)−𝑔^𝑇(𝑥(𝑘−𝜏))𝐿⁻¹_1𝑖𝑇_1𝑖𝐿⁻¹_1𝑖𝑔(𝑥(𝑘−𝜏))+𝑦^𝑇(𝑘−𝑑)𝑇_2𝑖𝑥(𝑘−𝑑)−𝑓^𝑇𝑦(𝑘−𝑑)𝐿⁻¹_2𝑖𝑇_2𝑖𝐿⁻¹_2𝑖𝑓𝑦()+𝑟𝑘−𝑑₋₁−1𝑉_𝑖(𝑘)=𝜁^𝑇(𝑘)Ω_𝑖𝜁(𝑘)+𝑟^−𝜏𝜂^𝑇₁(𝑘)𝑁^𝑇_1𝑖𝑁^𝑇_2𝑖×𝜏𝑅⁻¹_1𝑖𝑁_1𝑖𝑁_2𝑖𝜂₁(𝑘)+𝑟^−𝑑𝜂^𝑇₂(𝑘)𝑀^𝑇_1𝑖𝑀^𝑇_2𝑖𝑑𝑅⁻¹_2𝑖𝑀_1𝑖𝑀_2𝑖𝜂₂(𝑘)+𝑟⁻¹−1𝑉_𝑖(𝑘), (25) where𝜁(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝜏)𝑔^𝑇(𝑥(𝑘−𝜏))𝑦^𝑇(𝑘)𝑦^𝑇(𝑘−𝑑)𝑓^𝑇𝑦(𝑘−𝑑)^𝑇. (26) Therefore, from (17), we haveΔ𝑉_𝑖(𝑘)≤𝑟⁻¹−1𝑉_𝑖(𝑘), (27) which implies (16) is true. This completes the proof of Theorem 8.

In what follows, we are in a position to derive the delay-dependent exponential stability condition for the discrete-time switched BAM neural networks (Σ), and the results are given in the following theorem.

Theorem 9. Under the assumptions (𝐻₁)–(𝐻₃), for given scalars 𝑟>1,   𝜇≥1, the system (Σ) is exponentially stable and ensures a decay rate 𝜆, where 𝜆=𝑟^{(−ln𝜇/(𝑇𝑎ln𝑟))+1}, if there exist matrices 𝑃_1𝑖>0,𝑃_2𝑖>0,𝑄_1𝑖>0,𝑄_2𝑖>0,𝑅_1𝑖>0,𝑅_2𝑖>0,𝑁_1𝑖,𝑁_2𝑖,𝑀_1𝑖,𝑀_2𝑖, and 𝑇_1𝑖>0,  𝑇_2𝑖>0,  𝑖∈𝒩, such that (17) and the following inequalities hold: 𝑃_1𝛼≤𝜇𝑃_1𝛽,𝑃_2𝛼≤𝜇𝑃_2𝛽,𝑄_1𝛼≤𝜇𝑄_1𝛽,𝑄_2𝛼≤𝜇𝑄_2𝛽,𝑅_1𝛼≤𝜇𝑅_1𝛽,𝑅_2𝛼≤𝜇𝑅_2𝛽,𝑇_𝑎≥𝑇^∗_𝑎=ceilln𝜇ln𝑟,𝛼,𝛽∈𝒩. (28)(29)(30)

Proof. Choose the Lyapunov-Krasovskii functional candidate of system (Σ) as 𝑉_𝜎(𝑘)(𝑘)=𝑉_1𝜎(𝑘)(𝑘)+𝑉_2𝜎(𝑘)(𝑘)+𝑉_3𝜎(𝑘)(𝑘), (31) where 𝑉_1𝜎(𝑘)(𝑘)=𝑥^𝑇(𝑘)𝑃_1𝜎(𝑘)𝑥(𝑘)+𝑦^𝑇(𝑘)𝑃_2𝜎(𝑘)𝑦(𝑘),𝑉_2𝜎(𝑘)(𝑘)=^𝑘−1_{𝜃=𝑘−𝜏}𝑟^{𝜃−𝑘+1}𝑥^𝑇(𝜃)𝑄_1𝜎(𝑘)𝑥(𝜃)+^𝑘−1_{𝜃=𝑘−𝑑}𝑟^{𝜃−𝑘+1}𝑦^𝑇(𝜃)𝑄_2𝜎(𝑘)𝑦(𝜃),𝑉_3𝜎(𝑘)(𝑘)=⁻¹_𝑠=−𝜏^𝑘−1_{𝜃=𝑘+𝑠}𝑟^{𝜃−𝑘+1}ℓ^𝑇₁(𝜃)𝑅_1𝜎(𝑘)ℓ₁(𝜃)+⁻¹_𝑠=−𝑑^𝑘−1_{𝜃=𝑘+𝑠}𝑟^{𝜃−𝑘+1}ℓ^𝑇₂(𝜃)𝑅_2𝜎(𝑘)ℓ₂(𝜃). (32) From (16), (28), and (29), when 𝑘∈[𝑘_𝑚,𝑘_𝑚+1), there holds 𝑉_𝜎(𝑘)(𝑘)≤𝑟^{−(𝑘−𝑘𝑚)}𝑉_{𝜎(𝑘𝑚)}𝑘_𝑚≤𝜇𝑟^{−(𝑘−𝑘𝑚)}𝑉_{𝜎(𝑘𝑚−1)}𝑘_𝑚≤𝜇𝑟^{−(𝑘−𝑘𝑚)}𝑟^{−(𝑘𝑚−𝑘𝑚−1)}𝑉_{𝜎(𝑘𝑚−1)}𝑘_𝑚−1=𝜇𝑟^{−(𝑘−𝑘𝑚−1)}𝑉_{𝜎(𝑘𝑚−1)}𝑘_𝑚−1≤⋯≤𝜇^𝑁𝜎𝑟^{−(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀. (33) Observe that 𝑁_𝜎≤𝑘−𝑘₀𝑇_𝑎,𝜇=𝑟^{ln𝜇/ln𝑟}. (34) This together with (30) and (33) yields 𝜇^𝑁𝜎𝑟^{−(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀≤𝜆^{−(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀. (35) This further implies 𝑉_𝜎(𝑘)(𝑘)≤𝜆^{−(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀. (36) Let 𝛽₁=minmin_𝑖∈𝒩𝜆_𝑚𝑃_1𝑖,min_𝑖∈𝒩𝜆_𝑚𝑃_2𝑖,𝛽₂=max𝛽₂₁,𝛽₂₂, (37) where 𝛽₂₁=max_𝑖∈𝒩𝜆_𝑀𝑃_1𝑖+𝑟1−𝑟^−𝜏𝑟−1max_𝑖∈𝒩𝜆_𝑀𝑄_1𝑖()+2𝑟𝜏𝑟−1−𝑟1−𝑟^−𝜏(𝑟−1)²max_𝑖∈𝒩𝜆_𝑀𝑅_1𝑖,𝛽₂₂=max_𝑖∈𝒩𝜆_𝑀𝑃_2𝑖+𝑟1−𝑟^−𝑑𝑟−1max_𝑖∈𝒩𝜆_𝑀𝑄_2𝑖()+2𝑟𝑑𝑟−1−𝑟1−𝑟^−𝑑(𝑟−1)²max_𝑖∈𝒩𝜆_𝑀𝑅_2𝑖. (38)(39) It can be verified from (31) that 𝑉_𝜎(𝑘)(𝑘)≥𝛽₁‖𝑥(𝑘)‖²+𝑦(𝑘)²,𝑉_𝜎(𝑘0)𝑘₀≤𝛽₂₁𝜙(𝑘)²_𝐿+𝛽₂₂𝜓(𝑘)²_𝐿, (40)(41) which gives rise to 𝛽₁‖𝑥(𝑘)‖²+𝑦(𝑘)²≤𝑉_𝜎(𝑘)(𝑘)≤𝜆^{−(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑘₀≤𝜆^{−(𝑘−𝑘0)}𝛽₂𝜙²_𝐿+𝜓²_𝐿. (42) Therefore, we have ‖𝑥(𝑘)‖²+𝑦(𝑘)²≤𝛽₂𝛽₁𝜙²_𝐿+𝜓²_𝐿𝜆^{−(𝑘−𝑘0)}, (43) which implies that the discrete-time switched BAM neural networks (Σ) are exponentially stable. This completes the proof of Theorem 9.

Remark 10. In (30), the function ceil(𝑡) is used, which represents rounding real number 𝑡 to the nearest integer greater than or equal to 𝑡. 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.

Remark 11. In [3–6], the asymptotic or exponential stability problem is considered for continuous-time BAM neural networks with time delay 𝑥̇(𝑡)(𝑡)𝑦(),𝑦̇(𝑡)(𝑡)(𝑥(.=−𝐴𝑥+𝑊𝑓𝑡−𝑑=−𝐵𝑥+𝑉𝑔𝑡−𝜏)) (44)(45) However, the dynamics of discrete-time neural networks may be quite different from those of continuous-time ones, and the stability criteria established for continuous-time BAM neural networks model are not necessarily applicable to discrete-time systems. Considering the importance in both theory and practice, it is necessary to study the dynamics of the discrete-time BAM neural networks.

Remark 12. There are few references concerning exponential stability analysis for discrete-time switched BAM neural networks. In this paper, some delay-dependent sufficient conditions checking the exponential stability of discrete-time switched BAM neural networks using average dwell time approach are presented. These conditions are proposed in the form of linear matrix inequalities, which can be easily solved by using the recently developed interior algorithms.

4. An Illustrative Example

Consider the discrete-time switched BAM neural networks (Σ) combining two subsystems with the following parameters: 𝐴₁=,𝑊0.5000.5₁=−0.0483−0.01−0.03−0.04,𝐵₁=,𝑉0.3000.2₁=,𝐴−0.05−0.01−0.05−0.06₂=0.7000.4,𝐵₂=,𝑊0.6000.7₂=−0.032−0.06−0.01−0.05,𝑉₂=−0.05−0.01−0.05−0.06. (46)(47)(48) The activation functions are taken as 𝑓𝑦=12−,𝑔(𝑥)=1𝑦+1𝑦−12(|𝑥+1|−|𝑥−1|). (49)

Let 𝑑=1 and 𝜏=1. Solving LMI (17), (28), and (29), it is found that the LMIs are feasible for all 𝑟≤1.47. The calculated values of the delay upper bound 𝜏 and decay rate 𝜆 for different values of 𝑑 and 𝑟 are given in Table 1 when 𝑇_𝑎=1.

Table 1 

The maximum delay bound 𝜏,𝑑 and decay rate 𝜆.

From Table 1, we can see that the delay is related to the decay rate. For a given 𝑑, a smaller decay rate 𝜆 allows a larger delay 𝜏_max. Moreover, for every 𝑟, the delay 𝜏_max decreases when the delay 𝑑 increases.

Letting 𝑟=1.4,   𝑑=3, and 𝜇=1.2, we obtain that 𝑇^∗_𝑎=ceil[0.5419]. Based on (30), 𝑇_𝑎=1 is satisfied. Then we can calculate that the decay rate 𝜆=𝑟^{(−ln𝜇/𝑇𝑎ln𝑟)+1}=1.1667. Therefore, the discrete-time switched BAM neural networks with time delay are exponentially stable with the decay rate 𝜆=1.1667 if the delay 𝜏 is not larger than 13 based on Table 1.

For 𝑟=1.4,   𝑑=3, and  𝜏=2, based on Definition 3, the discrete-time switched BAM neural networks ensure the following exponential decay estimation: ‖𝑥(𝑘)‖²+𝑦(𝑘)²≤2.2866×1.1667^{−(𝑘−𝑘0)}𝜙²_𝐿+𝜓²_𝐿,∀𝑘≥𝑘₀. (50) Let 𝑘₀=0. Suppose the switching sequence is: 121212…. It can be seen from switched sequence that 𝑇_𝑎=1. Choose initial value as 𝜙(𝑠)=[0.5−0.6]^𝑇 and 𝜓(𝑠)=[0.8−0.7]^𝑇; then we obtain Figure 1, which depicts the trajectories of the system state.

Figure 1 

State response of the given system.

5. 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.

Acknowledgments

This work is supported by National Nature Science Foundation of China under Grant 61104132 and Director of the Foundation under Grant SJ201002.