Global Robust Stability of Switched Interval Neural Networks with Discrete and Distributed Time-Varying Delays of Neural Type

Abstract

By combing the theories of the switched systems and the interval neural networks, the mathematics model of the switched interval neural networks with discrete and distributed time-varying delays of neural type is presented. A set of the interval parameter uncertainty neural networks with discrete and distributed time-varying delays of neural type are used as the individual subsystem, and an arbitrary switching rule is assumed to coordinate the switching between these networks. By applying the augmented Lyapunov-Krasovskii functional approach and linear matrix inequality (LMI) techniques, a delay-dependent criterion is achieved to ensure to such switched interval neural networks to be globally asymptotically robustly stable in terms of LMIs. The unknown gain matrix is determined by solving this delay-dependent LMIs. Finally, an illustrative example is given to demonstrate the validity of the theoretical results.

1. Introduction

In the past few decades, neural networks have been a subject of intense research activities due to their wide applications in different areas such as image processing, pattern recognition, associative memory, and combinational optimization. A fundamental problem is the stability which is the prerequisite to ensure that the developed neural network can work [1–40]. In hardware implementation of the neural networks, time delay is inevitably encountered and is usually time varying due to the finite switching speed of amplifiers. It is known that time delay is often the main cause for instability and poor performance of neural networks. Moreover, due to unavoidable factors, such as modeling error, external perturbation, and parameter fluctuation, the neural networks model certainly involves uncertainties such as perturbations and component variations, which will change the stability of neural networks. To analyze uncertainty of neural networks, one reasonable method is to assume parameters in certain intervals. Therefore, it is of great importance to study the global robust stability of interval neural networks with time-varying delay. Recently, some sufficient conditions for the global robust stability of interval neural networks with time-varying delays and parametric uncertainties have been obtained in terms of LMIs [1–9].

Since neural networks usually have a spatial extent, there is a distribution of propagation delays over a period of time. In these circumstances, the signal propagation is not instantaneous and cannot be modeled with discrete delays, and a more appropriate way is to incorporate continuously distributed delays in neural network model. On the other hand, in certain physical systems, mathematical models have been described by some functional differential equations of neutral type, which depend on the delays of state and state derivative. In practice, neutral type phenomenon always appear in studies of automatic control, chemical reactors, population ecology, heat exchanges, microwave oscillators, and so on. Hence, the stability for neutral type neural networks with time-varying delay has been also considered in the recent years [10–18].

A class of hybrid systems has attracted significant attention because it can model several practical control problems that involve the integration of supervisory logic-based control schemes and feedback control algorithms. As a special class of hybrid systems, switched systems are regarded as nonlinear systems, which are composed of a family of continuous-time or discrete-time subsystems and a rule that orchestrates the switching between the subsystems. Recently, switched neural networks, whose individual subsystems are a set of neural networks, have found applications in fields of high-speed signal processing, artificial intelligence, and gene selection in a DNA microarray analysis [19–21]. Therefore, some researchers have studied the stability issues for switched neural networks [22–27]. In [22], based on the Lyapunov-Krasovskii method and LMI approach, some sufficient conditions were derived for global robust exponential stability of a class of switched Hopfield neural networks with time-varying delay under uncertainty. In [23], by combining Cohen-Grossberg neural networks with an arbitrary switching rule, the mathematical model of a class of switched Cohen-Grossberg neural networks with mixed time-varying delays were established, and the robust stability for such switched Cohen-Grossberg neural networks was analyzed. In [24], by employing nonlinear measure and LMI techniques, some new sufficient conditions were obtained to ensure global robust asymptotical stability and global robust stability of the unique equilibrium for a class of switched recurrent neural networks with time-varying delay. In [25], authors investigated a large class of switched recurrent neural networks with time-varying structured uncertainties and time-varying delay; some delay-dependent robust periodicity criteria guaranteeing the existence, uniqueness, and global asymptotic stability of periodic solution for all admissible parametric uncertainties were devised by taking free weighting matrices and LMIs. In [26], based on multiple Lyapunov functions method and LMI techniques, the authors presented some sufficient conditions in terms of LMIs which guarantee the robust exponential stability for uncertain switched Cohen-Grossberg neural networks with interval time-varying delay and distributed time-varying delay under the switching rule with the average dwell time property.

It should be noted that, in the above literature, almost all results treated of the robust stability for switched neural networks with norm-bounded uncertainty. However, as it well known that there are two forms of parametric uncertainties, namely the interval uncertainty and the norm-bounded uncertainty. To the best of our knowledge, up to now, there are few researchers to deal with the global robust stability for switched neural networks with the interval uncertainty, despite its potential and practical importance.

Motivated by the preceding discussion, the aim of this paper is to present a new class of the switched interval neural networks with discrete and distributed time-varying delays of neural type under interval parameter uncertainties by integrating the theory of switched systems with neural networks. By constructing a suitable Lyapunov-Krasovskii functional and employing Jensens inequality, a delay-dependent criterion will be derived such that the proposed switched interval neural networks are globally robustly asymptotically stable. The proposed criterion is represented in terms of LMIs, which can be solved efficiently by using recently developed convex optimization algorithms [28].

The rest of this paper is organized as follows. In Section 2, the model formulation and some preliminaries are given. The main result are stated in Section 3. In Section 4, a numerical example is presented to demonstrate the validity of the proposed results. Some conclusions are made in Section 5.

Notations. Throughout this paper, and 𝑅 denotes the set of real numbers, 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space, 𝑅^𝑚×𝑛 denotes the set of all 𝑚×𝑛 real matrices. For any matrix 𝐴, 𝐴^𝑇 denotes the transpose of 𝐴. 𝐴>0(𝐴<0) means that 𝐴 is a positive definite (negative definite). Given the column vectors 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇,𝑦=(𝑦₁,…,𝑦_𝑛)^𝑇∈𝑅^𝑛,and𝑥^𝑇𝑦=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖. 𝑥̇(𝑡) denotes the derivative of 𝑥(𝑡), and ∗ represents the symmetric form of matrix.

2. Neural Network Model and Preliminaries

Consider the interval neural network model with discrete and distributed time-varying delays of neutral type described by the system of differential equations in the form𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝑊₁𝑔(𝑥(𝑡))+𝑊₂𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑊₃^𝑡_{(𝑡)𝑡−𝜏}𝑔(𝑥(𝑠))𝑑𝑠+𝑊₄𝑥̇𝑡−𝜇(𝑡)+𝑢,𝐴∈𝐴_𝑙,𝑊_𝑘∈𝑊_𝑙^(𝑘),𝑘=1,2,3,4, (2.1) where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑅^𝑛 denotes the state vector associated with 𝑛 neurons; 𝑔(𝑥)=(𝑔₁(𝑥₁),…,𝑔_𝑛(𝑥_𝑛))^𝑇∶𝑅^𝑛→𝑅^𝑛 is a vector-valued neuron activation function; 𝑢=(𝑢₁,…,𝑢_𝑛)^𝑇 is a constant external input vector; 𝜏(𝑡) denotes the discrete and distributed time-varying delays 𝜇(𝑡) represents neutral time-varying delays; 𝐴=diag(𝑎₁,…,𝑎_𝑛) is 𝑛×𝑛 constant diagonal matrices; 𝑎_𝑖>0,𝑖=1,…,𝑛, are the neural self-inhibitions; 𝑊_𝑘=[𝑤^(𝑘)_𝑖𝑗]∈𝑅^𝑛×𝑛,𝑘=1,2,3,4, are the connection weight matrices; 𝐴_𝑙=[𝐴,𝐴]={𝐴=diag(𝑎_𝑖)∶0<𝑎_𝑖≤𝑎_𝑖≤𝑎_𝑖,𝑖=1,2,…,𝑛},𝑊_𝑙^(𝑘)=[𝑊_𝑘,𝑊_𝑘]={𝑊_𝑘=[𝑤^(𝑘)_𝑖𝑗𝑤]∶_{(𝑘)𝑖𝑗}≤𝑤^(𝑘)_𝑖𝑗≤𝑤^(𝑘)_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛} with 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛),𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛),𝑊_𝑘=[𝑤^(𝑘)_𝑖𝑗]_𝑛×𝑛,𝑊_𝑘=[𝑤^(𝑘)_𝑖𝑗]_𝑛×𝑛.

Throughout this paper, the following assumptions are made on the activation functions 𝑔_𝑗,𝑗=1,2,…,𝑛, the discrete and distributed time-varying delay 𝜏(𝑡), and the neutral time-varying delay 𝜇(𝑡),(ℋ₁):|𝑔_𝑗(𝑥₁)−𝑔_𝑗(𝑥₂)|≤𝑘_𝑗|𝑥₁−𝑥₂|,𝑥₁,𝑥₂∈𝑅,𝑗=1,2,…,𝑛,(ℋ₂):0≤𝜏(𝑡)≤𝜏_𝑁,𝜏̇(𝑡)≤𝜏<1,0≤𝜇(𝑡)≤𝜇_𝑁,𝜇̇(𝑡)≤𝜇<1,

where 𝜏_𝑁,𝜇_𝑁 are constants. The initial value associated with (2.1) is assumed to be 𝑥(𝑠)=𝜓(𝑠), and 𝜓(𝑠) is a continuous function on [−ℎ,0], ℎ=max{𝜏_𝑁,𝜇_𝑁}.

With loss of generality, it is assumed that the above neural networks have only one equilibrium point and are denoted by 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇. For the purpose of simplicity, the equilibrium 𝑥^∗ will be always shifted to the origin by letting 𝑦(𝑡)=𝑥(𝑡)−𝑥^∗, and the neural network system (2.1) can be represented as follows:𝑦̇(𝑡)=−𝐴𝑦(𝑡)+𝑊₁𝑓𝑦(𝑡)+𝑊₂𝑓𝑦(𝑡−𝜏(𝑡))+𝑊₃^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝑑𝑠+𝑊₄𝑦̇𝑡−𝜇(𝑡), (2.2) where 𝑓_𝑗(𝑦_𝑗(𝑡))=𝑔_𝑗(𝑦_𝑗(𝑡)+𝑥^∗_𝑗)−𝑔_𝑗(𝑥^∗_𝑗), and 𝑓_𝑗(0)=0,𝑗=1,2,…,𝑛.

The initial condition associated with (2.2) is given in the form 𝑦(𝑠)=𝑥(𝑠)−𝑥^∗=𝜑(𝑠)=𝜓(𝑠)−𝑥^∗,𝑠∈[−ℎ,0]. From the assumption (ℋ₁), it follows that 𝑓(𝑦(𝑡))≤𝐾𝑦(𝑡),𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑛). Based on some transformations, the system (2.2) can be written as an equivalent form𝑦̇(𝑡)=−𝐴₀+𝐸_𝐴Σ_𝐴𝐹_𝐴𝑦(𝑡)+𝑊₁₀+𝐸₁Σ₁𝐹₁𝑓𝑦(𝑡)+𝑊₂₀+𝐸₂Σ₂𝐹₂𝑓𝑦((𝑡+𝑊𝑡−𝜏))₃₀+𝐸₃Σ₃𝐹₃^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝑑𝑠+𝑊₄₀+𝐸₄Σ₄𝐹₄𝑦̇𝑡−𝜇(𝑡), (2.3) where Σ_𝐴∈Σ,Σ_𝑘∈Σ,𝑘=1,2,3,4Σ=diag𝛿₁₁,…,𝛿_1𝑛,…,𝛿_𝑛1,…,𝛿_𝑛𝑛∈𝑅^𝑛2×𝑛2∶𝛿_𝑖𝑗,𝐴≤1,𝑖,𝑗=1,2,…,𝑛₀=𝐴+𝐴2,𝐻_𝐴=𝛼_{𝑖𝑗𝑛×𝑛}=𝐴−𝐴2,𝑊_𝑘0=𝑊_𝑘+𝑊_𝑘2,𝐻_𝑊^(𝑘)=𝛽_{𝑖𝑗𝑛×𝑛}=𝑊_𝑘−𝑊_𝑘2,𝐸_𝐴=√𝛼₁₁𝑒₁,…,√𝛼_1𝑛𝑒₁,…,√𝛼_𝑛1𝑒_𝑛,…,√𝛼_𝑛𝑛𝑒_{𝑛𝑛×𝑛2},𝐹_𝐴=√𝛼₁₁𝑒₁,…,√𝛼_1𝑛𝑒_𝑛,…,√𝛼_𝑛1𝑒₁,…,√𝛼_𝑛𝑛𝑒_𝑛^𝑇_𝑛2×𝑛,𝐸_𝑘=𝛽^(𝑘)₁₁𝑒₁,…,𝛽^(𝑘)_1𝑛𝑒₁,…,𝛽^(𝑘)_𝑛1𝑒_𝑛,…,𝛽^(𝑘)_𝑛𝑛𝑒_{𝑛𝑛×𝑛2},𝐹_𝑘=𝛽^(𝑘)₁₁𝑒₁,…,𝛽^(𝑘)_1𝑛𝑒_𝑛,…,𝛽^(𝑘)_𝑛1𝑒₁,…,𝛽^(𝑘)_𝑛𝑛𝑒_𝑛^𝑇_𝑛2×𝑛, (2.4) where 𝑒_𝑖∈𝑅^𝑛 denotes the column vector with 𝑖th element to be 1 and others to be 0.

The switched interval neural networks with discrete and distributed time-varying delays of neural type consists of a set of interval neural network with discrete and distributed time-varying delays of neural type and a switching rule. Each of the interval neural networks is regarded as an individual subsystem. The operation mode of the switched neural networks is determined by the switching rule. According to (2.2), the switched interval neural network with discrete and distributed delays of neural type can be described as follows:𝑦̇(𝑡)=−𝐴_𝜎(𝑡)𝑦(𝑡)+𝑊_1𝜎(𝑡)𝑓𝑦(𝑡)+𝑊_2𝜎(𝑡)𝑓𝑦(𝑡−𝜏(𝑡))+𝑊_3𝜎(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝑑𝑠+𝑊_4𝜎(𝑡)𝑦̇𝑡−𝜇(𝑡),𝐴_𝜎(𝑡)∈𝐴_{𝑙𝜎(𝑡)},𝑊_{𝑘𝜎(𝑡)}∈𝑊_𝑙^(𝑘)_𝜎(𝑡),𝑘=1,2,3,4, (2.5) where 𝐴_{𝑙𝜎(𝑡)}=[𝐴_𝜎(𝑡),𝐴_𝜎(𝑡)]={𝐴_𝜎(𝑡)=diag(𝑎_{𝑖𝜎(𝑡)})∶0<𝑎_{𝑖𝜎(𝑡)}≤𝑎_{𝑖𝜎(𝑡)}≤𝑎_{𝑖𝜎(𝑡)},𝑖=1,2,…,𝑛}, 𝑊_𝑙^(𝑘)_𝜎(𝑡)=[𝑊_{𝑘𝜎(𝑡)},𝑊_{𝑘𝜎(𝑡)}]={𝑊_{𝑘𝜎(𝑡)}=[𝑤^(𝑘)_{𝑖𝑗𝜎(𝑡)}]∶0<𝑤^(𝑘)_{𝑖𝑗𝜎(𝑡)}≤𝑤^(𝑘)_{𝑖𝑗𝜎(𝑡)}≤𝑤^(𝑘)_{𝑖𝑗𝜎(𝑡)},𝑖,𝑗=1,2,…,𝑛} with 𝐴_𝜎(𝑡)=diag(𝑎_1𝜎(𝑡),𝑎_2𝜎(𝑡),…,𝑎_{𝑛𝜎(𝑡)}), 𝐴_𝜎(𝑡)=diag(𝑎_1𝜎(𝑡),𝑎_2𝜎(𝑡),…,𝑎_{𝑛𝜎(𝑡)}),𝑊_{𝑘𝜎(𝑡)}=[𝑤^(𝑘)_{𝑖𝑗𝜎(𝑡)}]_𝑛×𝑛,𝑊_{𝑘𝜎(𝑡)}=[𝑤^(𝑘)_{𝑖𝑗𝜎(𝑡)}]_𝑛×𝑛𝐴_0𝜎(𝑡)=𝐴_𝜎(𝑡)+𝐴_𝜎(𝑡)2,𝐻_{𝐴𝜎(𝑡)}=𝛼_{𝑖𝑗𝜎(𝑡)𝑛×𝑛}=𝐴_𝜎(𝑡)−𝐴_𝜎(𝑡)2,𝑊_{𝑘0𝜎(𝑡)}=𝑊_{𝑘𝜎(𝑡)}+𝑊_{𝑘𝜎(𝑡)}2,𝐻_𝑊^(𝑘)_𝜎(𝑡)=𝛽_{𝑖𝑗𝜎(𝑡)𝑛×𝑛}=𝑊_{𝑘𝜎(𝑡)}−𝑊_{𝑘𝜎(𝑡)}2,𝐸_{𝐴𝜎(𝑡)}=𝛼_11𝜎(𝑡)𝑒₁,…,𝛼_{1𝑛𝜎(𝑡)}𝑒₁,…,𝛼_{𝑛1𝜎(𝑡)}𝑒_𝑛,…,𝛼_{𝑛𝑛𝜎(𝑡)}𝑒_{𝑛𝑛×𝑛2},𝐹_{𝐴𝜎(𝑡)}=𝛼_11𝜎(𝑡)𝑒₁,…,𝛼_{1𝑛𝜎(𝑡)}𝑒_𝑛,…,𝛼_{𝑛1𝜎(𝑡)}𝑒₁,…,𝛼_{𝑛𝑛𝜎(𝑡)}𝑒_𝑛^𝑇_𝑛2×𝑛,𝐸_{𝑘𝜎(𝑡)}=𝛽^(𝑘)_11𝜎(𝑡)𝑒₁,…,𝛽^(𝑘)_{1𝑛𝜎(𝑡)}𝑒₁,…,𝛽^(𝑘)_{𝑛1𝜎(𝑡)}𝑒_𝑛,…,𝛽^(𝑘)_{𝑛𝑛𝜎(𝑡)}𝑒_{𝑛𝑛×𝑛2},𝐹_{𝑘𝜎(𝑡)}=𝛽^(𝑘)_11𝜎(𝑡)𝑒₁,…,𝛽^(𝑘)_{1𝑛𝜎(𝑡)}𝑒_𝑛,…,𝛽^(𝑘)_{𝑛1𝜎(𝑡)}𝑒₁,…,𝛽^(𝑘)_{𝑛𝑛𝜎(𝑡)}𝑒_𝑛^𝑇_𝑛2×𝑛, (2.6)

𝜎(𝑡)∶[0,+∞)→Γ={1,2,…,𝑁} is the switching signal, which is a piecewise constant function of time. For any 𝑖∈{1,2,…,𝑙}, 𝐴_𝑖=𝐴_0𝑖+𝐸_𝐴𝑖Σ_𝐴𝑖𝐹_𝐴𝑖, 𝑊_𝑘𝑖=𝑊_𝑘0𝑖+𝐸_𝑘𝑖Σ_𝑘𝑖𝐹_𝑘𝑖, and Σ_𝐴𝑖∈Σ,Σ_𝑘𝑖∈Σ,𝑘=1,2,3,4. This means that the matrices (𝐴_𝜎(𝑡),𝑊_1𝜎(𝑡),𝑊_2𝜎(𝑡),𝑊_3𝜎(𝑡),𝑊_4𝜎(𝑡)) are allowed to take values, at an arbitrary time, in the finite set {(𝐴₁,𝑊₁₁,𝑊₂₁,𝑊₃₁,𝑊₄₁),(𝐴₂,𝑊₁₂,𝑊₂₂,𝑊₃₂,𝑊₄₂),…,(𝐴_𝑁,𝑊_1𝑁,𝑊_2𝑁,𝑊_3𝑁,𝑊_4𝑁)}. Throughout this paper, it is assumed that the switching rule 𝜎 is not known a priori, and its instantaneous value is available in real time. The initial condition associated with the switching system (2.5) is 𝑦(𝑠)=𝜑(𝑠),𝑠∈[−ℎ,0].

By (2.3), the system (2.5) can be written as𝑦̇(𝑡)=−𝐴_0𝜎(𝑡)𝑦(𝑡)+𝑊_10𝜎(𝑡)𝑓𝑦(𝑡)+𝑊_20𝜎(𝑡)𝑓𝑦(𝑡−𝜏(𝑡))+𝑊_30𝜎(𝑡)^𝑡_{𝑡−𝜏(𝑡)}𝑓𝑦(𝑠)𝑑𝑠+𝑊_40𝜎(𝑡)𝑦̇𝑡−𝜇(𝑡)+𝐸_𝜎(𝑡)Δ_𝜎(𝑡)(𝑡),𝐴_𝜎(𝑡)∈𝐴_{𝑙𝜎(𝑡)},𝑊_{𝑘𝜎(𝑡)}∈𝑊_𝑙^(𝑘)_𝜎(𝑡),𝑘=1,2,3,4, (2.7) where 𝐸_𝜎(𝑡)=[𝐸_{𝐴𝜎(𝑡)},𝐸_1𝜎(𝑡),𝐸_2𝜎(𝑡),𝐸_3𝜎(𝑡),𝐸_4𝜎(𝑡)]Δ_𝜎(𝑡)(𝑡)=−Σ_{𝐴𝜎(𝑡)}𝐹_{𝐴𝜎(𝑡)}𝑦(𝑡)Σ_1𝜎(𝑡)𝐹_1𝜎(𝑡)𝑓𝑦(𝑡)Σ_2𝜎(𝑡)𝐹_2𝜎(𝑡)𝑓𝑦((𝑡Σ𝑡−𝜏))_3𝜎(𝑡)𝐹_3𝜎(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠Σ)𝑑𝑠_4𝜎(𝑡)𝐹_4𝜎(𝑡)(𝑡)Σ𝑦̇𝑡−𝜇=diag_{𝐴𝜎(𝑡)},Σ_1𝜎(𝑡),Σ_2𝜎(𝑡),Σ_3𝜎(𝑡),Σ_4𝜎(𝑡)−𝐹_{𝐴𝜎(𝑡)}𝑦(𝑡)𝐹_1𝜎(𝑡)𝑓𝑦(𝑡)𝐹_2𝜎(𝑡)𝑓𝑦((𝑡𝐹𝑡−𝜏))_{3𝜎(𝑡)𝑡(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝐹𝑑𝑠_4𝜎(𝑡)(𝑡),𝑦̇𝑡−𝜇 (2.8) and Δ_𝜎(t)(𝑡) satisfies the following quadratic inequality:Δ^𝑇_𝜎(𝑡)(𝑡)Δ_𝜎(𝑡)(𝑡)≤𝑦(𝑡)𝑓𝑦(𝑡)𝑓𝑦(𝑡−𝜏(𝑡))_{𝑡(𝑡)𝑡−𝜏}𝑓𝑦(𝑠(𝑡))𝑑𝑠𝑦̇𝑡−𝜇^𝑇𝐹^𝑇_{𝐴𝜎(𝑡)}𝐹^𝑇_1𝜎(𝑡)𝐹^𝑇_2𝜎(𝑡)𝐹^𝑇_3𝜎(𝑡)𝐹^𝑇_4𝜎(𝑡)𝐹^𝑇_{𝐴𝜎(𝑡)}𝐹^𝑇_1𝜎(𝑡)𝐹^𝑇_2𝜎(𝑡)𝐹^𝑇_3𝜎(𝑡)𝐹^𝑇_4𝜎(𝑡)^𝑇𝑦(𝑡)𝑓𝑦(𝑡)𝑓𝑦(𝑡−𝜏(𝑡))_{𝑡(𝑡)𝑡−𝜏}𝑓𝑦(𝑠(𝑡).)𝑑𝑠𝑦̇𝑡−𝜇 (2.9)

Define the indicator function 𝜉(𝑡)=[𝜉₁(𝑡),𝜉₂(𝑡),…,𝜉_𝑁(𝑡)]^𝑇, where 𝜉_𝑖(𝑡)=𝐴1,whentheswitchedsystemisdescribedbythe𝑖thmode,_0𝑖,𝑊_𝑘0𝑖,𝑘=1,2,3,4,𝐸_𝑖,0,otherwise (2.10) where 𝑖=1,2,…,𝑁. Therefore, the system model (2.7) can also be written as𝑦̇(𝑡)=^𝑁_𝑖=1𝜉_𝑖(𝑡)−𝐴_0𝑖𝑦(𝑡)+𝑊_10𝑖𝑓𝑦(𝑡)+𝑊_20𝑖𝑓𝑦(𝑡−𝜏(𝑡))+𝑊_30𝑖^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝑑𝑠+𝑊_40𝑖𝑦̇𝑡−𝜇(𝑡)+𝐸_𝑖Δ_𝑖(𝑡), (2.11) where ∑^𝑁_𝑖=1𝜉_𝑖(𝑡)=1 is satisfied under any switching rules.

To derive the main result in the next section, the following definitions and lemmas will be need.

Definition 2.1. The switched interval neural network model (2.5) is said to be globally robustly asymptotically stable if the neural network model (2.5) is globally asymptotically stable for any 𝐴_𝜎(𝑡)∈𝐴_{𝑙𝜎(𝑡)},𝑊_{𝑘𝜎(𝑡)}∈𝑊_𝑙^(𝑘)_𝜎(𝑡),𝑘=1,2,3,4.

Lemma 2.2 (see [6]). Let Γ₀(𝑥) and Γ₁(𝑥) be two arbitrary quadratic forms over 𝑅^𝑛, then Γ₀(𝑥)<0 for all 𝑥∈𝑅^𝑛−{0} satisfying Γ₁(𝑥)≤0 if and only if there exists 𝜀≥0 such that Γ₀(𝑥)−𝜀Γ₁(𝑥)<0,∀𝑥∈𝑅^𝑛−{0}. (2.12)

Lemma 2.3 (Jensen's inequality, see [23]). For any constant matrix Ω∈𝑅^𝑛×𝑛,Ω=Ω^𝑇>0,𝑠𝑐𝑎𝑙𝑎𝑟0<𝛾(𝑡)<𝛾, vector function 𝜔∶[𝑡−𝛾,𝑡]→𝑅^𝑛,𝑡≥0 such that the integrations concerned are well defined, then ₀^𝛾(𝑡)𝜔(𝑠)𝑑𝑠^𝑇Ω₀^𝛾(𝑡)𝜔(𝑠)𝑑𝑠≤𝛾(𝑡)₀^𝛾(𝑡)𝜔(𝑠)^𝑇Ω𝜔(𝑠)𝑑𝑠. (2.13)

3. Main Results

In this section, the global robust asymptotic stability of the proposed model (2.5) will be discussed. By constructing a suitable Lyapunov functional, a robust delay-dependent criterion for the global asymptotic stability of the neural network system (2.5) is derived in terms of LMIs.

Theorem 3.1. Under the assumptions (ℋ₁) and (ℋ₂), if there exist matrices 𝑃>0,𝑄₁>0,𝑄₂>0,𝑄₃>0,𝑄₄>0,𝑁_𝑖(𝑖=1,2,…,7) such that the following LMIs hold: Π_𝑖=^{𝑖11𝑖12𝑖13𝑖14𝑖15𝑖16𝑖17}∗−𝑄₁𝑁₃𝑊_20𝑖𝑁₃𝑊_30𝑖𝑁₃𝑊_40𝑖𝑁₃𝐸_𝑖−𝑁₃∗∗_{𝑖33𝑖34𝑖35𝑖36}𝑁₂𝑊_20𝑖−𝑁₄∗∗∗_𝑖44𝑖45𝑁₅𝐸_𝑖+𝑁₇𝑊_30𝑖𝑁₂𝑊_30𝑖−𝑁₅∗∗∗∗_𝑖55𝑁₆𝐸_𝑖+𝑁₇𝑊_40𝑖𝑁₂𝑊_40𝑖−𝑁₆∗∗∗∗∗𝑁₇𝐸_𝑖−𝐼𝑁₂𝐸_𝑖−𝑁₇∗∗∗∗∗∗𝑄₄−𝑁₂−𝑁^𝑇₂<0, (3.1)

where_𝑖11=−𝑃𝐴_0𝑖−𝐴^𝑇_0𝑖𝑃+𝑃𝑊_10𝑖𝑊^𝑇_10𝑖𝑃+𝐾^𝑇𝐾+𝐾𝑄₂𝐾+𝜏_𝑁𝐾𝑄₃𝐾−𝑁₁𝐴_0𝑖−𝐴^𝑇_0𝑖𝑁^𝑇₁+𝑁₁𝑊_10𝑖+𝑊^𝑇_10𝑖𝑁^𝑇₁+𝐹^𝑇_𝐴𝑖𝐹_𝐴𝑖+𝐹^𝑇_𝐴𝑖𝐹_1𝑖+𝐹^𝑇_1𝑖𝐾𝐹_1𝑖,_𝑖12=−𝑁₃𝐴_0𝑖+𝑁₃𝑊_10𝑖𝐾,_𝑖13=−𝑃𝑊_20𝑖+𝑁₁𝑊_20𝑖−𝑁₄𝐴_0𝑖+𝑁₄𝑊_10𝑖𝐾+𝐹^𝑇_𝐴𝑖𝐹_2𝑖+𝐹^𝑇_1𝑖𝐾𝐹_2𝑖,_𝑖14=−𝑃𝑊_30𝑖+𝑁₁𝑊_30𝑖−𝑁₅𝐴_0𝑖+𝑁₅𝑊_10𝑖𝐾+𝐹^𝑇_𝐴𝑖𝐹_3𝑖+𝐹^𝑇_1𝑖𝐾𝐹_3𝑖,_𝑖15=−𝑃𝑊_40𝑖+𝑁₁𝑊_40𝑖−𝑁₆𝐴_0𝑖+𝑁₆𝑊_10𝑖𝐾+𝐹^𝑇_𝐴𝑖𝐹_4𝑖+𝐹^𝑇_1𝑖𝐾𝐹_4𝑖,_𝑖16=−𝑃𝐸_𝑖+𝑁₁𝐸_𝑖−𝑁₇𝐴_0𝑖+𝑁₇𝑊_10𝑖𝐾,_𝑖17=−𝑁₂𝐴_0𝑖+𝑁₂𝑊_10𝑖𝐾−𝑁₁,_𝑖33=−(1−𝜏)𝑄₂+𝑁₄𝑊_20𝑖+𝐹^𝑇_2𝑖𝐹_2𝑖,_𝑖34=𝑁₄𝑊_30𝑖+𝑁₅𝑊_20𝑖+𝐹^𝑇_2𝑖𝐹_3𝑖,_𝑖35=𝑁₄𝑊_40𝑖+𝑁₆𝑊_20𝑖+𝐹^𝑇_2𝑖𝐹_4𝑖,_𝑖36=𝑁₄𝐸_𝑖+𝑁₇𝑊_20𝑖,_𝑖44=−(1−𝜏)1/𝜏_𝑁𝑄₂+𝑁₅𝑊_30𝑖+𝐹^𝑇_3𝑖𝐹_3𝑖,_𝑖45=𝑁₅𝑊_40𝑖+𝑁₆𝑊_30𝑖+𝐹^𝑇_3𝑖𝐹_4𝑖,_𝑖55=−1−𝜇𝑄₄+𝑁₆𝑊_40𝑖+𝐹^𝑇_4𝑖𝐹_4𝑖, (3.2)

then the switched interval neural network model (2.5) is globally robustly asymptotically stable under any switching rules.

Proof. Consider the following Lyapunov-Krasovskii functional: 𝑉(𝑡)=𝑦^𝑇(𝑡)𝑃𝑦(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑦(𝑠)^𝑇𝑄₁𝑦(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)^𝑇𝑄₂𝑓𝑦(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−𝜏}(𝑠−𝑡+𝜏(𝑡))𝑓𝑦(𝑠)^𝑇𝑄₃𝑓𝑦(𝑠)𝑑𝑠+^𝑡_{(𝑡)𝑡−𝜇}𝑦(̇𝑠)^𝑇𝑄₄𝑦̇(𝑠)𝑑𝑠. (3.3) Calculating the time derivative of 𝑉(𝑡) along the trajectory of (2.11), it can follow that 𝑉̇(𝑡)=2𝑦^𝑇(𝑡)𝑃𝑦̇(𝑡)+𝑦^𝑇(𝑡)𝑄₁𝑦(𝑡)−𝑦(𝑡−𝜏(𝑡))^𝑇𝑄₁𝑦(𝑡−𝜏(𝑡))+𝑓𝑦(𝑡)^𝑇𝑄₂𝑓𝑦(𝑡−(𝜏̇(𝑡𝑓)1−))^𝑇𝑦(𝑡−𝜏(𝑡))𝑄₂𝑓𝑦(𝑡−𝜏(𝑡))+𝜏(𝑡)𝑓^𝑇𝑦(𝑡)𝑄₃𝑓𝑦(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}(−1+𝜏̇(𝑡))𝑓𝑦(𝑠)^𝑇𝑄₃𝑓𝑦(𝑠)𝑑𝑠+𝑦^𝑇̇(𝑡)𝑄₄𝑦̇(𝑡)−𝑦^𝑇̇𝑡−𝜇(𝑡)𝑄₄𝑦̇𝑡−𝜇(𝑡)1−𝜇̇(𝑡)=2𝑦^𝑇(𝑡)𝑃^𝑁_𝑖=1𝜉_𝑖(𝑡)−𝐴_0𝑖𝑦(𝑡)+𝑊_10𝑖𝑓𝑦(𝑡)+𝑊_20𝑖𝑓𝑦(𝑡−𝜏(𝑡))+𝑊_30𝑖^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝑑𝑠+𝑊_40𝑖𝑦̇𝑡−𝜇(𝑡)+𝐸_𝑖Δ_𝑖(𝑡)+𝑦^𝑇(𝑡)𝑄₁𝑦(𝑡)−𝑦(𝑡−𝜏(𝑡))^𝑇𝑄₁𝑦(𝑡−𝜏(𝑡))+𝑓𝑦(𝑡)^𝑇𝑄₂𝑓𝑦(𝑡)−(1−𝜏̇(𝑡))𝑓^𝑇𝑦(𝑡−𝜏(𝑡))𝑄₂𝑓𝑦(𝑡−𝜏(𝑡))+𝜏(𝑡)𝑓^𝑇𝑦(𝑡)𝑄₃𝑓𝑦(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}(−1+𝜏̇(𝑡))𝑓𝑦(𝑠)^𝑇𝑄₃𝑓𝑦(𝑠)𝑑𝑠+𝑦^𝑇̇(𝑡)𝑄₄𝑦̇(𝑡)−𝑦^𝑇̇𝑡−𝜇(𝑡)𝑄₄(𝑡(𝑡=𝑦̇𝑡−𝜇)1−𝜇̇)_𝑁𝑖=1𝜉_𝑖(𝑡)𝑦^𝑇(𝑡)−𝑃𝐴_0𝑖−𝐴^𝑇_0𝑖𝑃𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃𝑊_10𝑖𝑓𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃𝑊_20𝑖𝑓𝑦(𝑡−𝜏(𝑡))+2𝑦^𝑇(𝑡)𝑃𝑊_30𝑖^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝑑𝑠+2𝑦^𝑇(𝑡)𝑃𝑊_40𝑖𝑦̇𝑡−𝜇(𝑡)+2𝑦^𝑇(𝑡)𝑃𝐸_𝑖Δ_𝑖(𝑡)+𝑦^𝑇(𝑡)𝑄₁𝑦(𝑡)−𝑦(𝑡−𝜏(𝑡))^𝑇𝑄₁𝑦(𝑡−𝜏(𝑡))+𝑓𝑦(𝑡)^𝑇𝑄₂𝑓𝑦(𝑡)−(1−𝜏̇(𝑡))𝑓^𝑇𝑦(𝑡−𝜏(𝑡))𝑄₂𝑓𝑦(𝑡−𝜏(𝑡))+𝜏(𝑡)𝑓^𝑇𝑦(𝑡)𝑄₃𝑓𝑦(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}(−1+𝜏̇(𝑡))𝑓𝑦(𝑠)^𝑇𝑄₃𝑓𝑦(𝑠)𝑑𝑠+𝑦^𝑇̇(𝑡)𝑄₄𝑦̇(𝑡)−𝑦^𝑇̇𝑡−𝜇(𝑡)𝑄₄𝑦̇𝑡−𝜇(𝑡)1−𝜇̇(𝑡). (3.4) By the assumption (ℋ₁) and Lemma 2.3, 2𝑦^𝑇(𝑡)𝑃𝑊_10𝑖𝑓𝑦(𝑡)≤𝑦^𝑇(𝑡)𝑃𝑊_10𝑖𝑊^𝑇_10𝑖𝑃+𝐾^𝑇𝐾𝑦(𝑡),𝑓𝑦(𝑡)^𝑇𝑄₂𝑓𝑦(𝑡)≤𝑦^𝑇(𝑡)𝐾𝑄₂𝐾𝑦_{𝑡(𝑡)𝑡−𝜏}(𝑡),(−1+𝜏̇(𝑡))𝑓𝑦(𝑠)^𝑇𝑄₃𝑓𝑦(𝑠)()1𝑑𝑠≤−1−𝜏𝜏_𝑁^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝑑𝑠^𝑇𝑄₃^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝑑𝑠. (3.5)(3.6)(3.7) Noting the following zero equation with free weighting matrices 𝑁_𝑖(𝑖=1,2,…,7), which indicate the relationship between the terms in the state equation (2.11) and can easily be determined by solving the corresponding LMIs, it follows that 2𝜁(𝑡)^𝑇𝑁×^𝑙_𝑖=1𝜉_𝑖(𝑡)−𝑦̇(𝑡)−𝐴_0𝑖𝑦(𝑡)+𝑊_10𝑖𝑓𝑦(𝑡)+𝑊_20𝑖𝑓𝑦(𝑡−𝜏(𝑡))+𝑊_30𝑖^𝑡_{(𝑡)𝑡−𝜏}𝑓𝑦(𝑠)𝑑𝑠+𝑊_40𝑖𝑦̇𝑡−𝜇(𝑡)+𝐸_𝑖Δ_𝑖(𝑡)=0, (3.8)𝜁(𝑡)=[𝑦^𝑇(𝑡)𝑦^𝑇̇(𝑡)𝑦^𝑇(𝑡−𝜏(𝑡))𝑓^𝑇(𝑦(𝑡−𝜏(𝑡)))(∫^𝑡_{𝑡−𝜏(𝑡)}𝑓(𝑦(𝑠))𝑑𝑠)^𝑇𝑦^𝑇̇(𝑡−𝜇(𝑡))Δ^𝑇_𝑖(𝑡)]^𝑇.𝑁=[𝑁^𝑇₁𝑁^𝑇₂𝑁^𝑇₃𝑁^𝑇₄𝑁^𝑇₅𝑁^𝑇₆𝑁^𝑇₇]^𝑇. By substituting (2.9) and (3.5)–(3.8) into (3.4), it follows that 𝜀𝑉̇(𝑡)−𝜀^𝑁_𝑖=1𝜉_𝑖(𝑡)Δ^𝑇_𝑖(𝑡)Δ_𝑖(𝑡)−𝑓𝑦(𝑡)𝑓𝑦((𝑡𝑦(𝑡)𝑡−𝜏))_{𝑡(𝑡)𝑡−𝜏}𝑓𝑦(𝑠(𝑡))𝑑𝑠𝑦̇𝑡−𝜇^𝑇𝐹^𝑇_𝐴𝑖𝐹^𝑇_1𝑖𝐹^𝑇_2𝑖𝐹^𝑇_3𝑖𝐹^𝑇_4𝑖𝐹^𝑇_𝐴𝑖𝐹^𝑇_1𝑖𝐹^𝑇_2𝑖𝐹^𝑇_3𝑖𝐹^𝑇_4𝑖^𝑇𝑦(𝑡)𝑓𝑦(𝑡)𝑓𝑦(𝑡−𝜏(𝑡))_{𝑡(𝑡)𝑡−𝜏}𝑓𝑦(𝑠(𝑡))𝑑𝑠𝑦̇𝑡−𝜇≤𝜀^𝑁_𝑖=1𝜉_𝑖(𝑡)𝜙^𝑇(𝑡)Π_𝑖𝜙(𝑡), (3.9) where 𝜙(𝑡)=[𝑦^𝑇(𝑡)𝑦^𝑇(𝑡−𝜏(𝑡))𝑓^𝑇(𝑦(𝑡−𝜏(𝑡)))(∫^𝑡_{𝑡−𝜏(𝑡)}𝑓(𝑦(𝑠))𝑑𝑠)^𝑇𝑦^𝑇̇(𝑡−𝜇(𝑡))Δ^𝑇_𝑖(𝑡)𝑦^𝑇̇(𝑡)]^𝑇 and 𝜀>0. From Lemma 2.2 with the conditions (3.1) and (3.9), this implies that 𝑉̇(𝑡)<0 for all 𝜙(𝑡)≠0. Hence, the neural network system (2.7) is globally asymptotically stable for 𝐴_𝜎(𝑡)∈𝐴_{𝑙𝜎(𝑡)},𝑊_{𝑘𝜎(𝑡)}∈𝑊_𝑙^(𝑘)_𝜎(𝑡),𝑘=1,2,3,4, that is, the switched neural network model (2.5) is globally robustly asymptotically stable. The proof is completed.

When the time-varying delay 𝜏(𝑡) and 𝜇(𝑡) in (2.5) becomes into 𝜏(𝑡)=𝜏=const, 𝜇(𝑡)=𝜇=const, according to Theorem 3.1, it is easy to obtain the following corollary.

Corollary 3.2. Under the assumption (ℋ₁) and (ℋ₂), if there exist matrices 𝑃>0,𝑄₁>0,𝑄₂>0,𝑄₃>0,𝑄₄>0,𝑁_𝑖(𝑖=1,2,…,7) such that the following LMIs hold: ∗−𝑄₁𝑁₃𝑊_20𝑖𝑁₃𝑊_30𝑖𝑁₃𝑊_40𝑖𝑁₃𝐸_𝑖−𝑁^{𝑖11𝑖12𝑖13𝑖14𝑖15𝑖16𝑖17}₃∗∗_{𝑖33𝑖34𝑖35𝑖36}𝑁₂𝑊_20𝑖−𝑁₄∗∗∗_𝑖44𝑖45𝑁₅𝐸_𝑖+𝑁₇𝑊_30𝑖𝑁₂𝑊_30𝑖−𝑁₅∗∗∗∗_𝑖55𝑁₆𝐸_𝑖+𝑁₇𝑊_40𝑖𝑁₂𝑊_40𝑖−𝑁₆∗∗∗∗∗𝑁₇𝐸_𝑖−𝐼𝑁₂𝐸_𝑖−𝑁₇∗∗∗∗∗∗𝑄₄−𝑁₂−𝑁^𝑇₂<0, (3.10) where _𝑖11=−𝑃𝐴_0𝑖−𝐴^𝑇_0𝑖𝑃+𝑃𝑊_10𝑖𝑊^𝑇_10𝑖𝑃+𝐾^𝑇𝐾+𝐾𝑄₂𝐾+𝜏_𝑁𝐾𝑄₃K−𝑁₁𝐴_0𝑖−𝐴^𝑇_0𝑖𝑁^𝑇₁+𝑁₁𝑊_10𝑖+𝑊^𝑇_10𝑖𝑁^𝑇₁+𝐹^𝑇_𝐴𝑖𝐹_𝐴𝑖+𝐹^𝑇_𝐴𝑖𝐹_1𝑖+𝐹^𝑇_1𝑖𝐾𝐹_1𝑖,_𝑖12=−𝑁₃𝐴_0𝑖+𝑁₃𝑊_10𝑖𝐾,_𝑖13=−𝑃𝑊_20𝑖+𝑁₁𝑊_20𝑖−𝑁₄𝐴_0𝑖+𝑁₄𝑊_10𝑖𝐾+𝐹^𝑇_𝐴𝑖𝐹_2𝑖+𝐹^𝑇_1𝑖𝐾𝐹_2𝑖,_𝑖14=−𝑃𝑊_30𝑖+𝑁₁𝑊_30𝑖−𝑁₅𝐴_0𝑖+N₅𝑊_10𝑖𝐾+𝐹^𝑇_𝐴𝑖𝐹_3𝑖+𝐹^𝑇_1𝑖𝐾𝐹_3𝑖,_𝑖15=−𝑃𝑊_40𝑖+𝑁₁𝑊_40𝑖−𝑁₆𝐴_0𝑖+𝑁₆𝑊_10𝑖𝐾+𝐹^𝑇_𝐴𝑖𝐹_4𝑖+𝐹^𝑇_1𝑖𝐾𝐹_4𝑖,_𝑖16=−𝑃𝐸_𝑖+𝑁₁𝐸_𝑖−𝑁₇𝐴_0𝑖+𝑁₇𝑊_10𝑖𝐾,_𝑖17=−𝑁₂𝐴_0𝑖+𝑁₂𝑊_10𝑖𝐾−𝑁₁,_𝑖33=−𝑄₂+𝑁₄𝑊_20𝑖+𝐹^𝑇_2𝑖𝐹_2𝑖,_𝑖34=𝑁₄𝑊_30𝑖+𝑁₅𝑊_20𝑖+𝐹^𝑇_2𝑖𝐹_3𝑖,_𝑖35=𝑁₄𝑊_40𝑖+𝑁₆𝑊_20𝑖+𝐹^𝑇_2𝑖𝐹_4𝑖,_𝑖36=𝑁₄𝐸_𝑖+𝑁₇𝑊_20𝑖,_𝑖44=−1/𝜏𝑄₂+𝑁₅𝑊_30𝑖+𝐹^𝑇_3𝑖𝐹_3𝑖,_𝑖45=𝑁₅𝑊_40𝑖+𝑁₆𝑊_30𝑖+𝐹^𝑇_3𝑖𝐹_4𝑖,_𝑖55=−𝑄₄+𝑁₆𝑊_40𝑖+𝐹^𝑇_4𝑖F_4𝑖, (3.11) then the switched interval neural network model (2.5) is global robust asymptotically stable.

Remark 3.3. When 𝑁=1, the switched system model (2.7) degenerated into the interval neural network model (2.1) with discrete and distributed time-varying delays of neutral type which contain neural network models studied in [1–3, 5, 7]. Moreover, note that when 𝑁=1,𝑊₁=0, and 𝜏(𝑡)=𝜇(𝑡)=const, Theorem 3.1 in this paper coincides with Theorem 1 in [12]. Without considering interval parameter uncertainty, the neural network models presented in [11, 13–15] are the special case. Hence, the results obtained in this paper extend and improve the stability results available in the existing literature [1–3, 5, 7, 11, 13–15].

Remark 3.4. In this paper, augmented Lyapunov functional is used to analyze the stability of the interval neural network model (2.1) with discrete and distributed time-varying delays of neutral type. In the Lyapunov functional, both state and activation function are considered in the same term. Hence, the novel Lyapunov function contains structures more general than the traditional ones, and the negative matrices Π_𝑖 in Theorem 3.1 contain more elements. This shows that it is easy to find more appropriate elements in Π_𝑖 to ensure that the LMIs (3.1) hold. Thus, sufficient conditions given in this paper are less conservative than the existing results.

Remark 3.5. In this paper, the activation function is Lipschitz continuous, which is first introduced in [29] and used also in [30, 31] is more general than the usual sigmoid functions. Therefore, the stability results obtained in this paper are less conservative than those in [10–13, 27].

Remark 3.6. In [32, 33], the mixed time-delay problems have been considered for the stochastic system with Markovian jump parameters and discrete-time stochastic complex networks with randomly occurred nonlinearities. By applying the Lyapunov-Krasovskii functional approach and linear matrix inequality techniques, the conditions of exponential stabilization and global synchronization were presented for the stochastic system and discrete-time stochastic complex networks. In [34], Bounded H-infinity synchronization and state estimation were considered for discrete time-varying stochastic complex networks over a finite- horizon via linear matrix inequality. [35] The authors studied the robust H-infinity fuzzy output-feedback control; a controller design is given for the system with multiple probabilistic delays and multiple missing measurements. In the future, based on [32, 33], the model of the switched interval stochastic system and the switched interval discrete-time stochastic complex networks will be expected to be established, and the stability strategy proposed in this paper will be utilized to investigate the stability problems.

4. An Illustrative Example

In this section, an example will be given to illustrate the validity and effectiveness of the proposed stability criterion for the switched interval neural network with discrete and distributed delays of neural type when 𝑁=2.

Example 4.1. Consider the following second-order switched interval neural networks with discrete and distributed delays of neural type: 𝑦_𝑖̇(𝑡)=−𝑎_{𝑖𝜎(𝑡)}𝑦_𝑖(𝑡)+²_𝑗=1𝑤⁽¹⁾_{𝑖𝑗𝜎(𝑡)}𝑓_𝑗𝑦_𝑗(𝑡)+²_𝑗=1𝑤⁽²⁾_{𝑖𝑗𝜎(𝑡)}𝑓_𝑗𝑦_𝑗(𝑡−𝜏(𝑡))+²_𝑗=1𝑤⁽³⁾_{𝑖𝑗𝜎(𝑡)}^𝑡_{(𝑡)𝑡−𝜏}𝑓_𝑗𝑦_𝑗(𝑠+)𝑑𝑠_2𝑗=1𝑤⁽⁴⁾_{𝑖𝑗𝜎(𝑡)}𝑦_𝑗̇𝑡−𝜇(𝑡),𝑎_{𝑖𝜎(𝑡)}∈𝑎_{𝑖𝜎(𝑡)},𝑎_{𝑖𝜎(𝑡)},𝑤^(𝑘)_{𝑖𝑗𝜎(𝑡)}∈𝑤^(𝑘)_{𝑖𝑗𝜎(𝑡)},𝑤^(𝑘)_{𝑖𝑗𝜎(𝑡)},𝑘=1,2,3,4,𝑦_𝑖(𝑡)=𝜑_𝑖(𝑡),𝑡∈[−ℎ,0],𝑖,𝑗=1,2, (4.1) where the switching signal 𝜎(𝑡)∶[0,+∞)→Γ={1,2}, the activation functions 𝑓_𝑖(𝑥)=(2/3)sin𝑥+(1/3)𝑥,𝑖=1,2, the discrete and distributed delays 𝜏(𝑡)=(1/2)cos𝑡+(1/2), and the neural type delay 𝜇(𝑡)=(1/2)sin𝑡+(1/2). Obviously, the assumptions ℋ₁ and ℋ₂ are satisfied with 𝐾=diag(1,1) and ℎ=1,𝜏=𝜇=1/2. The neural network system parameters are defined as 𝐴₁=,3.99002.99𝐴₁=,𝑊4.01003.01₁₁=,1.1880.090.091.188𝑊₁₁=,𝑊1.2080.110.111.208₂₁=,0.090.140.050.09𝑊₂₁=,𝑊0.110.160.070.11₃₁=,0.44−0.210.290.41𝑊₃₁=,𝑊0.46−0.190.310.43₄₁=,0.09−0.01−0.010.09𝑊₄₁=,𝐴0.110.110.110.11₂=,1.99002.99𝐴₂=,𝑊2.01003.01₁₂=,−0.070.03−0.010.02𝑊₁₂=,𝑊−0.050.05−0.040.04₂₂=,−0.47−0.150.11−0.54𝑊₂₂=,𝑊−0.45−0.130.13−0.54₃₂=,−0.310.09−0.51−0.61𝑊₃₂=,𝑊−0.290.11−0.49−0.59₄₂=,0.03−0.22−0.21−0.44𝑊₄₂=.0.05−0.2−0.19−0.42 (4.3) Solving the LMI in (3.4) by using appropriate LMI solver in the Matlab, the feasible positive definite matrices 𝑃,𝑄_𝑖,𝑖=1,2,3,4 and the matrices 𝑁_𝑖,𝑖=1,2,…,7 could be as 𝑃=6.0192−0.5039−0.50394.8686,𝑄₁=3.2386003.2386,𝑄₂=,𝑄4.70090.33130.33134.9146₃=3.0347−0.0788−0.07882.8395,𝑄₄=1.9859−0.2388−0.23881.4177,𝑁₁=,𝑁−4.11980.49920.7261−2.7597₂=1.9399−0.0997−0.19621.5020,𝑁₃=0000,𝑁₄=,𝑁−0.2957−0.02630.0959−0.0107₅=−0.01470.06060.12520.2426,𝑁₆=0.0734−0.1495−0.0498−0.0592,𝑁₇=,,.𝑀1𝑀2𝑀1=0.0850.085−0.0068−0.00680.0850.085−0.0068−0.00680.0080.0080.08920.08920.0080.0080.08920.0892𝑀2=0.0850.085−0.0068−0.00680.0850.085−0.0068−0.00680.0080.0080.08920.08920.0080.0080.08920.0892 (4.4) By Theorem 3.1, this switched interval neural network with discrete and distributed delays of neural type is globally robustly asymptotically stable under any switching rules.
Let 𝐴₁=𝐴₁,𝑊₁₁=𝑊₁₁,𝑊₂₁=𝑊₂₁,𝑊₃₁=𝑊₃₁,𝑊₄₁=𝑊₄₁ and 𝐴₂=𝐴₂,𝑊₁₂=𝑊₁₂,𝑊₂₂=𝑊₂₂,𝑊₃₂=𝑊₃₂,and𝑊₄₂=𝑊₄₂. For numerical simulation, assume that the two subsystems are switched every four seconds. Figure 1 displays the state trajectories of this network with initial value (𝜑₁(𝑡),𝜑₂(𝑡))^𝑇=((cos2𝑡)²−0.5,(sin3𝑡)²−0.4)^𝑇,𝑡∈[−1,0]. It can be seen that these trajectories asymptotically converge to the unique equilibrium 𝑥^∗=(0,0)^𝑇 of the network. This is in accordance with the conclusion of Theorem 3.1.
Let 𝐴₁=𝐴₁,𝑊₁₁=𝑊₁₁,𝑊₂₁=𝑊₂₁,𝑊₃₁=𝑊₃₁,𝑊₄₁=𝑊₄₁ and 𝐴₂=𝐴₂,𝑊₁₂=𝑊₁₂,𝑊₂₂=𝑊₂₂,𝑊₃₂=𝑊₃₂,and𝑊₄₂=𝑊₄₂. For numerical simulation, assume that the two subsystems are switched every four seconds. Figure 2 displays the state trajectories of this network with initial value (𝜑₁(𝑡),𝜑₂(𝑡))^𝑇=((cos3𝑡)²−0.5,(sin2𝑡)²−0.6)^𝑇,𝑡∈[−1,0]. It can be seen that these trajectories asymptotically converge to the unique equilibrium 𝑥^∗=(0,0)^𝑇 of the network. This is in accordance with the conclusion of Theorem 3.1.

(a)

(b)

Figure 1 

The state trajectories 𝑦1 and 𝑦2 of the network with initial value (𝜑1(𝑡),𝜑2(𝑡))𝑇=((cos2𝑡)2−0.5,(sin3𝑡)2−0.4)𝑇,𝑡∈[−1,0].

(a)

(b)

Figure 2 

The state trajectories 𝑦1 and 𝑦2 of the network with initial value (𝜑1(𝑡),𝜑2(𝑡))𝑇=((cos3𝑡)2−0.5,(sin2𝑡)2−0.6)𝑇,𝑡∈[−1,0].

5. Conclusion

In this paper, a novel class of switched interval neural networks with discrete and distributed delays of neural type has been presented by combing the theories of the switched systems and the interval neural networks with discrete and distributed delays of neural type. Furthermore, a delay-dependent criterion expressed in the form of LMIs has been obtained to guarantee the proposed neural networks to be globally asymptotically robustly stable under interval parameter uncertainties. An illustrative example has been also given to demonstrate the effectiveness of the proposed LMI-based stability criteria.

Acknowledgments

The authors are extremely grateful to the Editor and the Reviewers for their valuable comments and suggestions, which help to enrich the content and improve the presentation of this paper. This work was supported by the Natural Science Foundation of Hebei Province of China (A2011203103) and the Hebei Province Education Foundation of China (2009157).