Synchronization of Switched Interval Networks and Applications to Chaotic Neural Networks

Abstract

This paper investigates synchronization problem of switched delay networks with interval parameters uncertainty, based on the theories of the switched systems and drive-response technique, a mathematical model of the switched interval drive-response error system is established. Without constructing Lyapunov-Krasovskii functions, introducing matrix measure method for the first time to switched time-varying delay networks, combining Halanay inequality technique, synchronization criteria are derived for switched interval networks under the arbitrary switching rule, which are easy to verify in practice. Moreover, as an application, the proposed scheme is then applied to chaotic neural networks. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

1. Introduction

It is well known that switched system, as a special class of hybrid system [1], is a dynamical system that consists of a finite number of subsystems and a logical rule that orchestrates switching between these subsystems [2–4] has attracted significant attention and successfully been applied to many fields such as artificial intelligence, high speed signal processing, and gene selection in a DNA microarray analysis. Recently, the stability and synchronization problem of switched networks have gained much attention [5–7]. In [6], based on switching analysis techniques and the comparison principle, the exponential synchronization criteria were derived for coupled switched neural networks with mode-dependent impulsive effects and delay. Authors considered the problems of passivity and pacification for a class of uncertain switched systems with stochastic disturbance and time-varying delay, based on average dwell time approach, free-weighting matrix method, and Jensen’s integral inequality, delay-dependent sufficient conditions were obtained to guarantee that the proposed switched systems were robustly mean-square exponentially stable and stochastically passive in terms of linear matrix inequalities [7]. Most of these works results on switched system are based on the Lyapunov theory; however, as we all know, it is difficult to construct a proper common or multiple Lyapunov function for a switched system; hence, in this paper, we will adopt matrix measure theory and Halanay inequality technique instead of constructing Lyapunov function to study the global exponential synchronization of switched networks, what is more, it is easy to verify the proposed conditions.

On the other hand, time delay often exists in nature, which may lead to instability, and it should be considered in mathematical model. Due to unavoidable factors, such as bifurcation and chaos, the networks model certainly involved uncertainties such as perturbations and component variations, which can greatly affect the dynamical behaviors of networks. Robust stability analysis of delayed networks with parameter uncertainties have been widely studied [8–10]. In [8], authors investigated switched recurrent neural networks (SRNNs) with time-varying norm bounded uncertainties, global asymptotic stability of periodic solution for all admissible parametric uncertainties are derived by taking the relationship between the terms in the Leibniz-Newton formula into account. It should be emphasized that almost all results treated of the robust stability for switched networks with norm-bounded uncertainty in the existing literature. To the best of our knowledge, few researchers deal with robust synchronization of switched networks with intervals parameters despite its potential and practical importance. Therefore, it is of great importance to study the global exponential synchronization of switched delay networks with interval parameters uncertainty.

Chaos implies extreme sensitivity to initial conditions; it can be observed in many real-world and has been widely application in secure communication, telecommunications, biological networks, artificial neural networks, and so forth [11, 12]. To synchronize two chaotic systems were mistakenly considered to be impossible before the pioneer work of Pecora and Carrol [13] on chaos control, and they proposed the drive-response concept to reach the synchronization of coupled chaotic systems. Recently, chaos control and synchronization attract more and more researchers’ attentions from various fields [14–18]. So far, kinds of effective approaches and techniques have been proposed for synchronization of chaotic systems including adaptive control [14], feedback control [15], switching control [16], impulsive control [17, 18], and others. In [18], global exponential synchronization stability in an array of linearly diffusively coupled reaction-diffusion neural networks with time-varying delays is investigated by adding impulsive controller to a small fraction of nodes, and a new analysis method is developed to overcome the difficulty resulting from the fact that the impulsive controller affects only the dynamical behaviors of the controlled nodes; the proposed results show that designing an appropriate pinning-impulsive controller to realize the synchronization goal as long as a conventional state feedback pinning controller or an adaptive pinning controller can achieve the synchronization goal by controlling the same nodes. In spite of these advances in studying synchronization of chaotic system, synchronization of switched systems with chaotic system as its subsystem under arbitrary switching rule has not been investigated in the literature.

Motivated by the preceding discussion, the aim of this paper is to study the global exponential drive-response synchronization problem for a class of switched interval networks with time-varying delay, instead of Lyapunov-Krasovskii methods, based on matrix measure theory, Halanay inequality technique, designing the coupling control gain matrix, several synchronization criteria are presented for switched interval networks under the arbitrary switching rule, which are easy to verify in practice. Simulations are given to demonstrate the validity of proposed results. The main contributions of this paper can be highlighted as follows. (1) Consider the interval parameters fluctuation; a new mathematical model of the switched coupled networks with parameters in interval is established, which presents more practical significance of our current research. (2) Introducing matrix measure method and Halanay inequality technique to switched system, without constructing Lyapunov function, the proposed results can easily be verified. (3) The proposed results can be applied to chaotic system; an interesting example shows that the switched networks can reach synchronization even if each subsystem of switched systems is a chaotic system; this further means that our results can generalize the previous results. (4) When 𝑁=1, the switched interval networks change as interval networks and the synchronization criteria of interval networks can be seen as a by-product.

The rest of this paper is organized as follows. In Section 2, the model description and preliminaries are given. Section 3 treats global exponential synchronization problems for switched interval networks with discrete time-varying delay. In Section 4, exponential synchronization criteria for interval networks are developed. An example is presented to demonstrate that the proposed results can be applied to chaotic system in Section 5. Some conclusions are drawn in Section 6.

Notations. Throughout this paper, for any matrix 𝐴, 𝐴>0(𝐴<0) means that 𝐴 is positive definite (negative definite), and 𝐴^𝑇 denotes the transpose of 𝐴. 𝜆_max(𝐴) and 𝜆_min(𝐴) denote the maximum and minimum eigenvalue of 𝐴, respectively. 𝐼 is the identity matrix. 𝑃𝐶([𝑡₀−𝜏,𝑡₀];𝑅^𝑛) denote the class of piecewise right continuous function 𝜂∶[𝑡₀−𝜏,𝑡₀]→𝑅^𝑛 with 𝑝-norm ‖𝜂‖_𝑝=sup_{𝑡0−𝜏≤𝑠≤𝑡0}‖𝜂(𝑠)‖_𝑝. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations.

2. Model Description and Preliminaries

Consider a general class of interval networks with time-varying delay described as follows: 𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝐵₁𝑔₁(𝑥(𝑡))+𝐵₂𝑔₂(𝑥(𝑡−𝜏(𝑡)))+𝐽,𝐴∈𝐴_𝑙,𝐵_𝑘∈𝐵_𝑙^(𝑘),𝑘=1,2, (1) where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑅^𝑛 is the vector of neuron states; 𝑔_𝑖(𝑥)=(𝑔_𝑖1(𝑥₁),…,𝑔_𝑖𝑛(𝑥_𝑛))^𝑇∶𝑅^𝑛→𝑅^𝑛, 𝑖=1,2, are the vector-valued neuron activation functions; 𝜏(𝑡) is the transmission time-varying delay; 𝐽=(𝐽₁,…,𝐽_𝑛)^𝑇 is a constant external input vector. 𝐴=diag(𝑎₁,…,𝑎_𝑛) is 𝑛×𝑛 constant diagonal matrices, 𝑎_𝑖>0, 𝑖=1,…,𝑛, are the neural self-inhibitions; 𝐵_𝑘=(𝑏^(𝑘)_𝑖𝑗)∈𝑅^𝑛×𝑛, 𝑘=1,2, are the connection weight matrices, and 𝐴_𝑙=[𝐴,𝐴]={𝐴=diag(𝑎_𝑖)∶0<𝑎_𝑖≤𝑎_𝑖≤𝑎_𝑖,𝑖=1,2,…,𝑛}, 𝐵_𝑙^(𝑘)=[𝐵_𝑘,𝐵_𝑘]={𝐵_𝑘=(𝑏^(𝑘)_𝑖𝑗): 𝑏^(𝑘)_𝑖𝑗≤𝑏^(𝑘)_𝑖𝑗≤𝑏^(𝑘)_𝑖𝑗,𝑖,𝑗=1,2,…,𝑛} with 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛), 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛), 𝐵_𝑘=(𝑏^(𝑘)_𝑖𝑗)_𝑛×𝑛,𝐵_𝑘=(𝑏^(𝑘)_𝑖𝑗)_𝑛×𝑛.

Throughout this paper, the following assumptions are made on 𝑔_𝑖(⋅), 𝑖=1,2, and 𝜏(𝑡).(ℋ₁):For any two different 𝑠, 𝑡∈𝑅, there exist constants 𝑙_𝑖𝑗>0, 𝑖=1,2, 𝑗=1,2,…,𝑛, such that 𝑔_𝑖𝑗(𝑠)−𝑔_𝑖𝑗(𝑡)≤𝑙_𝑖𝑗|𝑠−𝑡|,𝑖=1,2,𝑗=1,…,𝑛. (2)(ℋ₂):For any two different 𝑠, 𝑡∈𝑅, there exist constants 𝑙_𝑖𝑗>0, 𝑖=1,2, 𝑗=1,2,…,𝑛, such that 𝑔0≤^𝑖𝑗(𝑠)−𝑔_𝑖𝑗(𝑡)𝑠−𝑡≤𝑙_𝑖𝑗,𝑖=1,2,𝑗=1,…,𝑛. (3)(ℋ₃):Time-varying delay 𝜏(𝑡) satisfies 0≤𝜏(𝑡)≤𝜏, (4)where 𝜏 is a positive constant.

Using the coupling feedback control to synchronize system (1), the response (slave) system can be designed as 𝑦̇(𝑡)=−𝐴𝑦(𝑡)+𝐵₁𝑔₁𝑦(𝑡)+𝐵₂𝑔₂𝑦(𝑡−𝜏(𝑡))+𝐽+𝑈(𝑡),𝐴∈𝐴_𝑙,𝐵_𝑘∈𝐵_𝑙^(𝑘),𝑘=1,2, (5) where 𝑦(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))^𝑇 is the neuron state of response system; we choose coupling controller as follows: 𝑈(𝑡)=𝐾𝑦(𝑡)−𝑥(𝑡), (6) where the matrix 𝐾∈𝑅^𝑛×𝑛 is the control gain matrix to be designed.

Let error state be 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡); then error dynamical system between the states of drive system (1) and response system (5) can be derived: 𝑒̇(𝑡)=−𝐴𝑒(𝑡)+𝐵₁𝑓₁(𝑒(𝑡))+𝐵₂𝑓₂(𝑒(𝑡−𝜏(𝑡)))+𝐾𝑒(𝑡),𝐴∈𝐴_𝑙,𝐵_𝑘∈𝐵_𝑙^(𝑘),𝑘=1,2, (7) where 𝑒(𝑡)=(𝑒₁(𝑡),…,𝑒_𝑛(𝑡))^𝑇, 𝑓₁(𝑒(𝑡))=𝑔₁(𝑒(𝑡)+𝑥(𝑡))−𝑔₁(𝑥(𝑡)), 𝑓₂(𝑒(𝑡))=𝑔₂(𝑒(𝑡)+𝑥(𝑡))−𝑔₂(𝑥(𝑡)).

Based on some transformations [19], the interval error system (7) can be equivalently written as 𝑒̇(𝑡)=−𝐴₀+𝐸_𝐴Σ_𝐴𝐹_𝐴𝑒(𝑡)+𝐵₁₀+𝐸₁Σ₁𝐹₁𝑓₁(𝑒(𝑡+𝐵))₂₀+𝐸₂Σ₂𝐹₂𝑓₂(𝑥(𝑒(𝑡−𝜏(𝑡))))+𝐾𝑒(𝑡), (8) where Σ_𝐴∈Σ, Σ_𝑘∈Σ, 𝑘=1,2.

Consider the following: Σ=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−𝐴2𝐵_𝑘0=𝐵_𝑘+𝐵_𝑘2,𝐻_𝐵^(𝑘)=𝛽_{𝑖𝑗𝑛×𝑛}=𝐵_𝑘−𝐵_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, (9) where 𝑒_𝑖∈𝑅^𝑛 denotes the column vector with 𝑖th element to be 1 and others to be 0.

System (8) has an equivalent form by the following: 𝑒̇(𝑡)=−𝐴₀𝑒(𝑡)+𝐵₁₀𝑓₁(𝑒(𝑡))+𝐵₂₀𝑓₂(𝑒((𝑡(𝑡)(𝑡),𝑡−𝜏)))+𝐸Δ+𝐾𝑒 (10) where 𝐸=[𝐸_𝐴,𝐸₁,𝐸₂], Δ(𝑡)=−Σ_𝐴𝐹_𝐴𝑒(𝑡)Σ₁𝐹₁𝑓₁(𝑒(𝑡Σ))₂𝐹₂𝑓₂(𝑒((𝑡Σ𝑡−𝜏)))=diag_𝐴,Σ₁,Σ₂−𝐹_𝐴𝑒(𝑡)𝐹₁𝑓₁(𝑒(𝑡𝐹))₂𝑓₂(𝑒((𝑡,𝑡−𝜏))) (11) and Δ(𝑡) satisfies the following matrix quadratic inequality: Δ^𝑇(𝑡)Δ(𝑡)≤𝑓𝑒(𝑡)₁𝑓(𝑒(𝑡))₂(𝑒(𝑡−𝜏(𝑡)))^𝑇𝐹^𝑇_𝐴𝐹^𝑇₁𝐹^𝑇₂×𝐹^𝑇_𝐴𝐹^𝑇₁𝐹^𝑇₂^𝑇𝑒(𝑡)𝑓₁(𝑒(𝑡𝑓))₂(𝑒((𝑡.𝑡−𝜏))) (12)

The switched interval networks with time-varying delay consist of a set of interval networks with discrete time-varying delay and a switching rule [20]. Each of the interval networks is regarded as an individual subsystem. The operation mode of the switched networks is determined by the switching signal. According to (1), the switched interval networks with discrete time-varying delay can be represented as follows: 𝑥̇(𝑡)=−𝐴_𝜎(𝑡)𝑥(𝑡)+𝐵_1𝜎(𝑡)𝑔₁(𝑥(𝑡))+𝐵_2𝜎(𝑡)𝑔₂(𝑥(𝑡−𝜏(𝑡)))+𝐽,𝐴_𝜎(𝑡)∈𝐴_{𝑙𝜎(𝑡)},𝐵_{𝑘𝜎(𝑡)}∈𝐵_𝑙^(𝑘)_𝜎(𝑡),𝑘=1,2, (13) 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}, 𝐵_𝑙^(𝑘)_𝜎(𝑡)=[𝐵_{𝑘𝜎(𝑡)},𝐵_{𝑘𝜎(𝑡)}]={𝐵_{𝑘𝜎(𝑡)}=[𝑏^(𝑘)_{𝑖𝑗𝜎(𝑡)}]∶𝑏^(𝑘)_{𝑖𝑗𝜎(𝑡)}≤𝑏^(𝑘)_{𝑖𝑗𝜎(𝑡)}≤𝑏^(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)}), 𝐵_{𝑘𝜎(𝑡)}=[𝑏^(𝑘)_{𝑖𝑗𝜎(𝑡)}]_𝑛×𝑛,𝐵_{𝑘𝜎(𝑡)}=[𝑏^(𝑘)_{𝑖𝑗𝜎(𝑡)}]_𝑛×𝑛.

Consider the following: 𝐴_0𝜎(𝑡)=𝐴_𝜎(𝑡)+𝐴_𝜎(𝑡)2,𝐻_{𝐴𝜎(𝑡)}=𝛼_{𝑖𝑗𝜎(𝑡)𝑛×𝑛}=𝐴_𝜎(𝑡)−𝐴_𝜎(𝑡)2,𝐵_{𝑘0𝜎(𝑡)}=𝐵_{𝑘𝜎(𝑡)}+𝐵_{𝑘𝜎(𝑡)}2,𝐻_𝐵^(𝑘)_𝜎(𝑡)=𝛽_{𝑖𝑗𝜎(𝑡)𝑛×𝑛}=𝐵_{𝑘𝜎(𝑡)}−𝐵_{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. (14)

𝜎(𝑡)∶[0,+∞)→Γ={1,2,…,𝑁} is the switching signal, which is a piecewise constant function of time. For any 𝑖∈{1,2,…,𝑁}, 𝐴_𝑖=𝐴_0𝑖+𝐸_𝐴𝑖Σ_𝐴𝑖𝐹_𝐴𝑖, 𝐵_𝑘𝑖=𝐵_𝑘0𝑖+𝐸_𝑘𝑖Σ_𝑘𝑖𝐹_𝑘𝑖, and Σ_𝐴𝑖∈Σ, Σ_𝑘𝑖∈Σ, 𝑘=1,2. This means that the matrices (𝐴_𝜎(𝑡),𝐵_1𝜎(𝑡),𝐵_2𝜎(𝑡)) are allowed to take values, at an arbitrary time, in the finite set {(𝐴₁,𝐵₁₁,𝐵₂₁),(𝐴₂,𝐵₁₂,𝐵₂₂),…,(𝐴_𝑁,𝐵_1𝑁,𝐵_2𝑁)}. In 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 value associated with the switched interval networks is assumed to be 𝑥(𝑠)=𝜑(𝑠), 𝜑(𝑠)∈𝐶([𝑡₀−𝜏,𝑡₀];𝑅^𝑛).

Analogously, slave (response) system [21] of switched interval networks should be defined as 𝑦̇(𝑡)=−𝐴_𝜎(𝑡)𝑦(𝑡)+𝐵_1𝜎(𝑡)𝑔₁𝑦(𝑡)+𝐵_2𝜎(𝑡)𝑔₂𝑦((𝑡(𝑡),𝐴𝑡−𝜏))+𝐽+𝑈_𝜎(𝑡)∈𝐴_{𝑙𝜎(𝑡)},𝐵_{𝑘𝜎(𝑡)}∈𝐵_𝑙^(𝑘)_𝜎(𝑡),𝑘=1,2, (15)

The initial value associated with switched response system is assumed to be 𝑦(𝑠)=𝜓(𝑠), 𝜓(𝑠)∈𝐶([𝑡₀−𝜏,𝑡₀];𝑅^𝑛).

From (10), we have the switched interval drive-response error dynamical system as follows: 𝑒̇(𝑡)=−𝐴_0𝜎(𝑡)𝑒(𝑡)+𝐵_10𝜎(𝑡)𝑓₁(𝑒(𝑡))+𝐵_20𝜎(𝑡)𝑓₂(𝑒(𝑡−𝜏(𝑡)))+𝐸_𝜎(𝑡)Δ_𝜎(𝑡)(𝑡)+𝐾_𝜎(𝑡)𝑒(𝑡), (16) where 𝐸_𝜎(𝑡)=[𝐸_{𝐴𝜎(𝑡)},𝐸_1𝜎(𝑡),𝐸_2𝜎(𝑡)] and Δ_𝜎(𝑡) satisfies the following quadratic inequality: Δ^𝑇_𝜎(𝑡)Δ_𝜎(𝑡)≤𝑓𝑒(𝑡)₁𝑓(𝑒(𝑡))₂(𝑒(𝑡−𝜏(𝑡)))^𝑇𝐹^𝑇_{𝐴𝜎(𝑡)}𝐹^𝑇_1𝜎(𝑡)𝐹^𝑇_2𝜎(𝑡)×𝐹^𝑇_{𝐴𝜎(𝑡)}𝐹^𝑇_1𝜎(𝑡)𝐹^𝑇_2𝜎(𝑡)^𝑇𝑒(𝑡)𝑓₁(𝑒(𝑡𝑓))₂(𝑒((𝑡.𝑡−𝜏))) (17)

Define the indicator function 𝜉(𝑡)=[𝜉₁(𝑡),𝜉₂(𝑡),…,𝜉_𝑁(𝑡)]^𝑇, where 𝜉_𝑖(𝑡)=𝐴1,whentheswitchedsystemisdescribedbythe𝑖thmode_0𝑖,𝐵_𝑘0𝑖,𝑘=1,2,𝐸_𝑖,𝐾_𝑖0,otherwise, (18) where 𝑖=1,2,…,𝑁. Therefore, the switched interval error system (16) can also be represented as 𝑒̇(𝑡)=^𝑁_𝑖=1𝜉_𝑖(𝑡)−𝐴_0𝑖𝑒(𝑡)+𝐵_10𝑖𝑓₁(𝑒(𝑡))+𝐵_20𝑖𝑓₂(𝑒(𝑡−𝜏(𝑡)))+𝐸_𝑖Δ_𝑖(𝑡)+𝐾_𝑖𝑒(𝑡), (19) where ∑^𝑁_𝑖=1𝜉_𝑖(𝑡)=1 is satisfied under any switching rules.

To obtain the main results of this paper, the following definitions and lemmas are introduced.

Definition 1. The switched interval system (13) and the corresponding response system (15) are said to be globally exponentially synchronized if there exist scalars 𝑀>0 and 𝛼>0, such that 𝑦(𝑡)−𝑥(𝑡)_𝑝≤𝑀𝜓−𝜑𝑒^{−𝛼(𝑡−𝑡0)},𝑡>𝑡₀. (20)

Definition 2 (see [22]). The matrix measure of a real square matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 is denoted as follows: 𝜇_𝑝(𝐴)=lim_𝜀→0+𝐼_𝑛+𝜀𝐴_𝑝−1𝜀, (21) where ‖⋅‖_𝑝 is an induced matrix norm and 𝑝=1,2,∞.

Remark 3. The matrix norm ‖𝐴‖₁=max_𝑗∑^𝑛_𝑖=1|𝑎_𝑖𝑗|, ‖𝐴‖₂=𝜆_max(𝐴^𝑇𝐴), ‖𝐴‖_∞=max_𝑖∑^𝑛_𝑖=1|𝑎_𝑖𝑗|, the corresponding matrix measure 𝜇₁(𝐴)=max_𝑗{𝑎_𝑗𝑗+∑^𝑛_{𝑖=1,𝑖≠𝑗}|𝑎_𝑖𝑗|}, 𝜇₂(𝐴)=(1/2)𝜆_max(𝐴^𝑇+𝐴), 𝜇_∞(𝐴)=max_𝑖{𝑎_𝑖𝑖+∑^𝑛_{𝑗=1,𝑗≠𝑖}|𝑎_𝑖𝑗|}.

Lemma 4 (see [22]). The matrix measure 𝜇_𝑝(⋅) has the following basic properties:(i)−‖𝐴‖_𝑝≤𝜇_𝑝(𝐴)≤‖𝐴‖_𝑝,∀𝐴∈𝑅^𝑛×𝑛; (ii)𝜇_𝑝(𝛼𝐴)=𝛼𝜇_𝑝(𝐴),∀𝛼>0,𝐴∈𝑅^𝑛×𝑛; (iii)𝜇_𝑝(𝐴+𝐵)≤𝜇_𝑝(𝐴)+𝜇_𝑝(𝐵),∀𝐴,𝐵∈𝑅^𝑛×𝑛.

Lemma 5 (see [23]). Let 𝑠(𝑡)∶[𝑡₀−𝜏,∞)→[0,∞) be a continuous function, and, for all 𝑡≥𝑡₀, we have 𝐷⁺𝑠(𝑡)≤−𝑎𝑠(𝑡)+𝑏sup_{𝑡−𝜏≤𝜃≤𝑡}𝑠(𝜃). (22) If 𝑎>𝑏>0, then 𝑠(𝑡)≤sup_{𝑡0−𝜏≤𝜃≤𝑡0}𝑠(𝜃)𝑒^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀, (23) where 𝜆>0 is the unique positive solution of the equation 𝜆−𝑎+𝑏𝑒^𝜆𝜏=0.

Lemma 6 (see [24]). Under assumption (ℋ₂), for any matrix 𝐴∈𝑅^𝑛×𝑛 we have the following inequality: 𝜇_𝑝(𝐴𝐹(𝑒(𝑡)))≤𝜇_𝑝𝐴^∗𝐿, (24) where 𝐹(𝑒(𝑡))=diag(𝑓₁₁(𝑒₁(𝑡))/𝑒₁(𝑡),…,𝑓_1𝑛(𝑒_𝑛(𝑡))/𝑒_𝑛(𝑡)), 𝐿=diag(𝑙₁₁,𝑙₁₂,…,𝑙_1𝑛), 𝑝=1,∞, 𝐴^∗=𝑎^∗_{𝑖𝑗𝑛×𝑛}max0,𝑎_𝑖𝑖𝑎,𝑖=𝑗_𝑖𝑗,𝑖≠𝑗. (25)

3. Synchronization Criteria for Switched Interval Networks

In this section, we will consider the global exponential synchronization of switched interval networks (13), without constructing Lyapunov-Krasovskii functional, by using matrix measure and Halanay inequality, designing suitable control gain matrix 𝐾_𝑖, global exponential stability criteria are derived for switched interval drive-response error system (16) under any arbitrary switched rule; that is to say, the switched interval networks (13) synchronize with the response system (15).

Theorem 7. Under the assumptions (ℋ₁) and (ℋ₃), the switched interval networks (13) will globally exponentially synchronize with the response system (15) under arbitrary switched rule, if control gain matrix 𝐾_𝑖 satisfies −𝜇_𝑝−𝐴_0𝑖+𝐾_𝑖−𝑙𝐵_10𝑖𝑝−𝑙𝐸_𝑖𝑝𝐹_𝐴𝑖𝑝−𝑙𝐸_𝑖𝑝𝐹_1𝑖𝑝≥𝑙𝐵_20𝑖𝑝+𝑙𝐸_𝑖𝑝𝐹_2𝑖𝑝>0, (26) where 𝑖=1,2,…,𝑁, 𝑝=1,2,∞, 𝑙=max_{1≤𝑗≤𝑛}{𝑙_𝑘𝑗}, 𝑘=1,2.

Proof. Calculating the time derivative of ‖𝑒(𝑡)‖_𝑝 along the solution of the system (19), it can follow that lim_ℎ→0+‖𝑒(𝑡+ℎ)‖_𝑝−‖𝑒(𝑡)‖_𝑝ℎ=lim_ℎ→0+‖𝑒(𝑡)+ℎ𝑒(̇𝑡)+𝑜(ℎ)‖_𝑝−‖𝑒(𝑡)‖_𝑝ℎ=lim_ℎ→0+1ℎ𝑒(𝑡)+ℎ^𝑁_𝑖=1𝜉_𝑖(𝑡)−𝐴_0𝑖𝑒(𝑡)+𝐵_10𝑖𝑓₁(𝑒(𝑡))+𝐵_20𝑖𝑓₂(𝑒(𝑡−𝜏(𝑡)))+𝐸_𝑖Δ_𝑖(𝑡)+𝐾_𝑖𝑒(𝑡(ℎ)+𝑜)_𝑝−‖𝑒(𝑡)‖_𝑝≤lim_ℎ→0+^𝑁_𝑖=1𝜉_𝑖(𝑡)𝐼+ℎ−𝐴_0𝑖+𝐾_𝑖𝑝−1ℎ×‖𝑒(𝑡)‖_𝑝+𝐵_10𝑖𝑝×𝑓₁(𝑒(𝑡))_𝑝+𝐵_20𝑖𝑝×𝑓₂(𝑒(𝑡−𝜏(𝑡)))_𝑝+𝐸_𝑖𝑝Δ_𝑖(𝑡)_𝑝. (27) Using assumption (ℋ₁), we yield 𝑓₁(𝑒(𝑡))_𝑝≤𝑙‖𝑒(𝑡)‖_𝑝,𝑓₂(𝑒(𝑡−𝜏(𝑡)))_𝑝≤𝑙‖𝑒(𝑡−𝜏(𝑡))‖_𝑝Δ_𝑖(𝑡)_𝑝≤𝐹_𝐴𝑖𝑝‖𝑒(𝑡)‖_𝑝+𝐹_1𝑖𝑝𝑓₁(𝑒(𝑡))_𝑝+𝐹_2𝑖𝑝𝑓₂(𝑒(𝑡−𝜏(𝑡)))_𝑝≤𝐹_𝐴𝑖𝑝‖𝑒(𝑡)‖_𝑝+𝑙𝐹_1𝑖𝑝‖𝑒(𝑡)‖_𝑝+𝑙𝐹_2𝑖𝑝‖𝑒(𝑡−𝜏(𝑡))‖_𝑝. (28)
In the light of (27)-(28), for any 𝑖=1,2,…,𝑁, we obtain 𝐥𝐢𝐦_𝐡→𝟎+‖𝐞(𝐭+𝐡)‖_𝐩−‖𝐞(𝐭)‖_𝐩𝐡≤𝐥𝐢𝐦_𝐡→𝟎+^𝐍_𝐢=𝟏𝝃_𝐢(𝐭)𝐈+𝐡(−𝐀_𝟎𝐢+𝐊_𝐢)_𝐩−𝟏𝐡‖𝐞(𝐭)‖_𝐩+𝐁_{𝟏𝟎𝐢𝐩}𝐟_𝟏(𝐞(𝐭))_𝐩+𝐁_{𝟐𝟎𝐢𝐩}𝐟_𝟐(𝐞(𝐭−𝝉(𝐭)))_𝐩+𝐄_𝐢𝐩Δ_𝐢(𝐭)_𝐩≤^𝐍_𝐢=𝟏𝝃_𝐢(𝐭)𝝁_𝐩−𝐀_𝟎𝐢+𝐊_𝐢‖𝐞(𝐭)‖_𝐩+𝐥𝐁_{𝟏𝟎𝐢𝐩}‖𝐞(𝐭)‖_𝐩+𝐥𝐁_{𝟐𝟎𝐢𝐩}‖𝐞(𝐭−𝝉(𝐭))‖_𝐩+𝐄_𝐢𝐩𝐅_𝐀𝐢𝐩‖𝐞(𝐭)‖_𝐩+𝐥𝐄_𝐢𝐩𝐅_𝟏𝐢𝐩‖𝐞(𝐭)‖_𝐩+𝐥𝐄_𝐢𝐩𝐅_𝟐𝐢𝐩‖𝐞(𝐭−𝝉(𝐭))‖_𝐩≤^𝐍_𝐢=𝟏𝝃_𝐢(𝐭)𝝁_𝐩−𝐀_𝟎𝐢+𝐊_𝐢+𝐥𝐁_{𝟏𝟎𝐢𝐩}+𝐄_𝐢𝐩𝐅_𝐀𝐢𝐩+𝐥𝐄_𝐢𝐩𝐅_𝟏𝐢𝐩×‖𝐞(𝐭)‖_𝐩+𝐥𝐁_{𝟐𝟎𝐢𝐩}+𝐥𝐄_𝐢𝐩𝐅_𝟐𝐢𝐩×‖𝐞(𝐭−𝝉(𝐭))‖_𝐩≤𝝁_𝐩−𝐀_𝟎𝐢+𝐊_𝐢+𝐥𝐁_{𝟏𝟎𝐢𝐩}+𝐄_𝐢𝐩𝐅_𝐀𝐢𝐩+𝐥𝐄_𝐢𝐩𝐅_𝟏𝐢𝐩‖𝐞(𝐭)‖_𝐩+𝐥𝐁_{𝟐𝟎𝐢𝐩}+𝐥𝐄_𝐢𝐩𝐅_𝟐𝐢𝐩‖𝐞(𝐭−𝝉(𝐭))‖_𝐩. (29) According to definition of upper-right Dini derivative, we have 𝐷⁺‖𝑒(𝑡)‖_𝑝≤𝜇_𝑝−𝐴_0𝑖+𝐾_𝑖+𝑙𝐵_10𝑖𝑝+𝐸_𝑖𝑝𝐹_𝐴𝑖𝑝+𝑙𝐸_𝑖𝑝𝐹_1𝑖𝑝×‖𝑒(𝑡)‖_𝑝+𝑙𝐵_20𝑖𝑝+𝑙𝐸_𝑖𝑝𝐹_2𝑖𝑝×sup_{𝑡−𝜏≤𝑠≤𝑡}‖𝑒(𝑠)‖_𝑝. (30) Let 𝑎=−𝜇_𝑝(−𝐴_0𝑖+𝐾_𝑖)−𝑙‖𝐵_10𝑖‖_𝑝−‖𝐸_𝑖‖_𝑝‖𝐹_𝐴𝑖‖_𝑝−𝑙‖𝐸_𝑖‖_𝑝‖𝐹_1𝑖‖_𝑝 and 𝑏=𝑙‖𝐵_20𝑖‖_𝑝+𝑙‖𝐸_𝑖‖_𝑝‖𝐹_2𝑖‖_𝑝, from condition (26) and Lemma 5, one can obtain ‖𝑒(𝑡)‖_𝑝≤sup_{𝑡0−𝜏≤𝑠≤𝑡0}‖𝑒(𝑠)‖_𝑝𝑒^{−𝑟(𝑡−𝑡0)},𝑡≥𝑡₀, (31) where 𝑟>0 is the unique positive solution of the equation 𝑟−𝑎+𝑏𝑒^𝑟𝜏=0.
Therefore, 𝑒(𝑡) converges exponentially to zero with a convergence rate of 𝑟, and the formula (31) is equivalent to ‖𝑦(𝑡)−𝑥(𝑡)‖_𝑝≤sup_{𝑡0−𝜏≤𝑠≤𝑡0}‖𝜓−𝜑‖_𝑝𝑒^{−𝑟(𝑡−𝑡0)}, from Definition 1; this completes the proof of the theorem.

Theorem 8. Under assumptions (ℋ₂) and (ℋ₃), global synchronization of the switched interval networks (13) can be achieved under arbitrary switched rule, if designing suitable control gain matrix 𝐾_𝑖 satisfies −𝜇_𝑝−𝐴_0𝑖+𝐾_𝑖−𝜇_𝑝𝐵^∗_10𝑖𝐿−𝑙𝐸_𝑖𝑝𝐹_𝐴𝑖𝑝−𝑙𝐸_𝑖𝑝𝐹_1𝑖𝑝≥𝑙𝐵_20𝑖𝑝+𝑙𝐸_𝑖𝑝𝐹_2𝑖𝑝>0, (32) where 𝑖=1,2,…,𝑁, 𝑝=1,∞, 𝑙=max_{1≤𝑗≤𝑛}{𝑙_𝑘𝑗}(𝑘=1,2), and 𝐿=diag(𝑙₁₁,𝑙₁₂,…,𝑙_1𝑛), 𝐵^∗_10𝑖=𝑏^(10)∗_{𝑚𝑛𝑖𝑛×𝑛}max0,𝑏⁽¹⁰⁾_𝑚𝑛𝑖𝑏,𝑚=𝑛,_{(10)𝑚𝑛𝑖},𝑚≠𝑛. (33)

Proof. Differentiating ‖𝑒(𝑡)‖_𝑝 with respect to time along the solution of (19), one has lim_ℎ→0+‖𝑒(𝑡+ℎ)‖_𝑝−‖𝑒(𝑡)‖_𝑝ℎ=lim_ℎ→0+‖𝑒(𝑡)+ℎ𝑒(̇𝑡)+𝑜(ℎ)‖_𝑝−‖𝑒(𝑡)‖_𝑝ℎ=lim_ℎ→0+1ℎ𝑒(𝑡)+ℎ^𝑁_𝑖=1𝜉_𝑖(𝑡)−𝐴_0𝑖𝑒(𝑡)+𝐵_10𝑖𝑓₁(𝑒(𝑡))+𝐵_20𝑖𝑓₂(𝑒(𝑡−𝜏(𝑡)))+𝐸_𝑖Δ_𝑖(𝑡)+𝐾_𝑖𝑒(𝑡)+𝑜(ℎ)_𝑝−‖𝑒(𝑡)‖_𝑝≤lim_ℎ→0+^𝑁_𝑖=1𝜉_𝑖(𝑡)𝑒(𝑡)+ℎ−𝐴_0𝑖+𝐾_𝑖𝑒(𝑡)+ℎ𝐵_10𝑖𝑓₁(𝑒(𝑡))_𝑝−‖𝑒(𝑡)‖_𝑝×(ℎ)⁻¹+𝐵_20𝑖𝑝𝑓₂(𝑒(𝑡−𝜏(𝑡)))_𝑝+𝐸_𝑖𝑝Δ_𝑖(𝑡)_𝑝≤𝜇_𝑝−𝐴_0𝑖+𝐾_𝑖+𝐵_10𝑖𝐹(𝑒(𝑡))‖𝑒(𝑡)‖_𝑝+𝐵_20𝑖𝑝𝑓₂(𝑒(𝑡−𝜏(𝑡)))_𝑝+𝐸_𝑖𝑝Δ_𝑖(𝑡)_𝑝. (34)
According to (28), Lemmas 4 and 6, we have lim_ℎ→0+‖𝑒(𝑡+ℎ)‖_𝑝−‖𝑒(𝑡)‖_𝑝ℎ≤𝜇_𝑝−𝐴_0𝑖+𝐾_𝑖+𝐵_10𝑖𝐹(𝑒(𝑡))‖𝑒(𝑡)‖_𝑝+𝐵_20𝑖𝑝𝑓₂(𝑒(𝑡−𝜏(𝑡)))_𝑝+𝐸_𝑖𝑝Δ_𝑖(𝑡)_𝑝≤𝜇_𝑝−𝐴_0𝑖+𝐾_𝑖+𝜇_𝑝𝐵^∗_10𝑖𝐿‖𝑒(𝑡)‖_𝑝+𝑙𝐵_20𝑖𝑝‖𝑒(𝑡−𝜏(𝑡))‖_𝑝+𝐸_𝑖𝑝𝐹_𝐴𝑖𝑝‖𝑒(𝑡)‖_𝑝+𝑙𝐸_𝑖𝑝𝐹_1𝑖𝑝‖𝑒(𝑡)‖_𝑝+𝑙𝐸_𝑖𝑝𝐹_2𝑖𝑝‖𝑒(𝑡−𝜏(𝑡))‖_𝑝≤𝜇_𝑝−𝐴_0𝑖+𝐾_𝑖+𝜇_𝑝𝐵^∗_10𝑖𝐿+𝐸_𝑖𝑝𝐹_𝐴𝑖𝑝+𝑙𝐸_𝑖𝑝𝐹_1𝑖𝑝‖𝑒(𝑡)‖_𝑝+𝑙𝐵_20𝑖𝑝+𝑙𝐸_𝑖𝑝𝐹_2𝑖𝑝‖𝑒(𝑡−𝜏(𝑡))‖_𝑝; (35) then we get the upper-right Dini derivative of ‖𝑒(𝑡)‖_𝑝 along the solution of system (19) as follows: 𝐷⁺‖𝑒(𝑡)‖_𝑝≤𝜇_𝑝−𝐴_0𝑖+𝐾_𝑖+𝜇_𝑝𝐵^∗_10𝑖𝐿+𝐸_𝑖𝑝𝐹_𝐴𝑖𝑝+𝑙𝐸_𝑖𝑝𝐹_1𝑖𝑝‖𝑒(𝑡)‖_𝑝+𝑙𝐵_20𝑖𝑝+𝑙𝐸_𝑖𝑝𝐹_2𝑖𝑝sup_{𝑡−𝜏≤𝑠≤𝑡}‖𝑒(𝑠)‖_𝑝. (36)
Let 𝑎=−𝜇_𝑝(−𝐴_0𝑖+𝐾_𝑖)−𝜇_𝑝(𝐵^∗_10𝑖𝐿)−‖𝐸_𝑖‖_𝑝‖𝐹_𝐴𝑖‖_𝑝−𝑙‖𝐸_𝑖‖_𝑝‖𝐹_1𝑖‖_𝑝 and 𝑏=𝑙‖𝐵_20𝑖‖_𝑝+𝑙‖𝐸_𝑖‖_𝑝‖𝐹_2𝑖‖_𝑝, from the condition in Theorem 8 and Lemma 5, we get the following: ‖𝑒(𝑡)‖_𝑝≤sup_{𝑡0−𝜏≤𝑠≤𝑡0}‖𝑒(𝑠)‖_𝑝𝑒^{−𝑟(𝑡−𝑡0)},𝑡≥𝑡₀, (37) where 𝑟>0 is the unique positive solution of the equation 𝑟=𝑎−𝑏𝑒^𝑟𝜏.
Therefore, from Definition 1, switched interval networks (13) globally exponentially synchronize to its respond system (15).

4. Synchronization Criteria of Interval Networks

It can be observed that only one subsystem is activated when 𝑁=1, the switched interval networks (13) and corresponding response system (15) degenerate interval networks (1) and its response system (5), respectively. It should be noted that the global exponential synchronization of interval networks can be a by-product; that is, global exponential synchronization criteria for interval networks (1) can be easily derived from Theorems 7 and 8; then we have the following corollaries.

Corollary 9. Under assumptions (ℋ₁) and (ℋ₂), when control gain matrix 𝐾 satisfies −𝜇_𝑝−𝐴₀+𝐾−𝑙𝐵_10𝑝−𝑙‖𝐸‖_𝑝𝐹_𝐴𝑝−𝑙‖𝐸‖_𝑝𝐹_1𝑝≥𝑙𝐵_20𝑝+𝑙‖𝐸‖_𝑝𝐹_2𝑝, (38) where 𝑝=1,2,∞, 𝑙=max_{1≤𝑗≤𝑛}{𝑙_𝑖𝑗}, 𝑖=1,2.
Then the interval networks (1) will globally exponentially synchronize with the response system (5).

Proof. Similar to Theorem 7, it is not difficult to get the proof of Corollary 9.

Corollary 10. Under assumptions (ℋ₂) and (ℋ₃), global synchronization of the interval networks (1) can be achieved, if designing suitable control gain matrix 𝐾 satisfies −𝜇_𝑝−𝐴₀+𝐾−𝜇_𝑝𝐵^∗₁₀𝐿−𝑙‖𝐸‖_𝑝𝐹_𝐴𝑝−𝑙‖𝐸‖_𝑝𝐹_1𝑝≥𝑙𝐵_20𝑝+𝑙‖𝐸‖_𝑝𝐹_2𝑝>0, (39) where 𝑝=1,∞, 𝑙=max_{1≤𝑗≤𝑛}{𝑙_𝑘𝑗}(𝑘=1,2), and 𝐿=diag(𝑙₁₁,𝑙₁₂,…,𝑙_1𝑛), 𝐵^∗₁₀=𝑏^(10)∗_{𝑚𝑛𝑛×𝑛}max0,𝑏⁽¹⁰⁾_𝑚𝑛𝑏,𝑚=𝑛,_(10)𝑚𝑛,𝑚≠𝑛. (40)

Proof. The proof of Corollary 10 can be easily obtained from Theorem 8, omitted here.

Remark 11. Matrix measure method has been introduced to switched system, and it is a very useful tool to deal with the stability and synchronization problems of networks and avoid constructing a proper Lyapunov function, which sometimes cannot be found. Moreover, most of the previous results on stability or synchronization are in form of algebra or norm, which limit the scope of nonnegative constants, however, from the definition of matrix measure, one easily knows that matrix measure can balance the effect of positive values and negative values of the matrix. Therefore, method based on matrix measure can obtain the more general synchronization criteria.

5. An Illustrative Example

In this section, an example is presented to prove the effectiveness of the theoretical results obtained in Theorem 7. In addition, it shows that the obtained synchronization criteria can be applied to chaotic neural networks, and even when each subsystem is chaotic neural networks, the switched networks can be reached to synchronization.

Example 1. Consider the second-order switched interval networks with discrete delay in (13) described by 𝑥̇_𝑖(𝑡)=−𝑎_{𝑖𝜎(𝑡)}𝑥_𝑖(𝑡)+²_𝑗=1𝑏⁽¹⁾_{𝑖𝑗𝜎(𝑡)}𝑔_1𝑗𝑥_𝑗(𝑡+_2𝑗=1)𝑏⁽²⁾_{𝑖𝑗𝜎(𝑡)}𝑔_2𝑗𝑥_𝑗(𝑡−𝜏(𝑡)𝑎_{𝑖𝜎(𝑡)}∈𝑎_{𝑖𝜎(𝑡)},𝑎_{𝑖𝜎(𝑡)},𝑏^(𝑘)_{𝑖𝑗𝜎(𝑡)}∈𝑏^(𝑘)_{𝑖𝑗𝜎(𝑡)},𝑏^(𝑘)_{𝑖𝑗𝜎(𝑡)}𝑥,𝑘=1,2,_𝑖(𝑡)=𝜑_𝑖(𝑡),𝑡∈[−𝜏,0],𝑖,𝑗=1,2. (41)

Consider 𝜎(𝑡)∶[0,+∞)→Γ={1,2}, 𝑔_𝑖(𝑥)=tanℎ(𝑥), 𝑖=1,2, 𝜏(𝑡)=1, 𝐽=(0,0)^𝑇. Obviously, the assumptions ℋ₁ and ℋ₃ are satisfied with 𝑙=1. The networks system parameters are defined as 𝐴₁=,1.000.000.001.00𝐴₁=,𝐵1.020.000.001.02₁₁=,2.00−0.10−5.004.50𝐵₁₁=,𝐵2.02−0.98−4.984.52₂₁=,−1.50−0.10−0.20−4.00𝐵₂₁=,𝐴−1.48−0.98−0.18−3.98₂=,1.000.000.001.00𝐴₂=,𝐵1.020.000.001.02₁₂=,3.005.000.102.00𝐵₁₂=,𝐵3.025.020.122.02₂₂=,−2.500.200.10−1.50𝐵₂₂=.−2.480.220.12−1.48 (42)

In the following, we will design control gain matrices 𝐾_𝑖(𝑖=1,2) for the switched interval networks in this example, chosen as follows: 𝐾₁=−1244−20,𝐾₂=−1466−26. (43)

By calculating, we have −𝜇_𝑝(−𝐴₀₁+𝐾₁)−𝑙‖𝐵₁₀₁‖_𝑝−𝑙‖𝐸₁‖_𝑝‖𝐹_𝐴1‖_𝑝−𝑙‖𝐸₁‖_𝑝‖𝐹₁₁‖_𝑝=4.3132≥𝑙‖𝐵₂₀₁‖_𝑝+𝑙‖𝐸₁‖_𝑝‖𝐹₂₁‖_𝑝=4.0694, −𝜇_𝑝(−𝐴₀₂+𝐾₂)−𝑙‖𝐵₁₀₂‖_𝑝−𝑙‖𝐸₂‖_𝑝‖𝐹_𝐴2‖_𝑝−𝑙‖𝐸₂‖_𝑝‖𝐹₁₂‖_𝑝=6.2811≥𝑙‖𝐵₂₀₂‖_𝑝+𝑙‖𝐸₂‖_𝑝‖𝐹₂₂‖_𝑝=5.0638, all the assumptions of Theorem 7 hold; therefore, switched drive system (13) can synchronize exponentially toward with response system (15) under any switching rules.

For numerical simulation, let 𝐴₁=𝐴₁, 𝐵₁₁=𝐵₁₁, 𝐵₂₁=𝐵₂₁, 𝐴₂=𝐴₂, 𝐵₁₂=𝐵₁₂, and 𝐵₂₂=𝐵₂₂. In this case, the two subsystems are all chaotic neural networks [25]; Figure 1 displays two chaotic attractors with the initial conditions (𝑥₁₁(𝑡),𝑥₁₂(𝑡))^𝑇=(0.1,0.1)^𝑇, (𝑥₂₁(𝑡),𝑥₂₂(𝑡))^𝑇=(2.3,0.3)^𝑇, respectively; Figure 2 shows the state trajectories of each variable of the first subsystem, and Figure 3 depicts the temporal evolution of each variable of the second subsystem.

Figure 1 

Chaotic attractor of two subsystems of switched networks.

Figure 2 

The state trajectories 𝑥11 and 𝑥12 of the first subsystem.

Figure 3 

The state trajectories 𝑥21 and 𝑥22 of the second subsystem.

For making numerical simulation of the switched drive-response error-state system, Figure 4 depicts the time responses of state variables 𝑒₁(𝑡) and 𝑒₂(𝑡) from the 30 random constant initial states in the set [−2,4]×[−2,4] with step ℎ=0.01; it reveals that the trajectory of the switched interval error-state system globally exponentially converges to a unique equilibrium 𝑒^∗=(0,0)^𝑇. This is in accordance with the conclusion of Theorem 7.

Figure 4 

Time responses of state variables 𝑒1 and 𝑒2 of switched error-state system.

6. Conclusion

In this paper, without constructing complex Lyapunov-Krasovskii functions, we have proposed a new method to study the global exponential synchronization of switched interval delayed networks and designed the coupling control gain matrices 𝐾_𝑖; the derived synchronization criteria for switched interval networks under the arbitrary switching rule are easy to verify in practice. The synchronization criteria for delayed networks with uncertain parameters can be a special case. Additionally, synchronization criteria can be applied to chaotic systems. An interesting example has shown that switched drive systems can synchronize with its response systems even if each subsystem is chaotic neural networks; this further shows that our results improve and extend the existing works. In the near future, we will discuss quasi-synchronization control, dissipativity, and finite-time stochastic stabilization of the switched interval delayed networks [26, 27].

Acknowledgments

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University (KAU), under Grant 3-130/1434/HiCi. The authors, therefore, acknowledge the technical and financial support of KAU.