Delay-Dependent Dynamics of Switched Cohen-Grossberg Neural Networks with Mixed Delays

Abstract

This paper aims at studying the problem of the dynamics of switched Cohen-Grossberg neural networks with mixed delays by using Lyapunov functional method, average dwell time (ADT) method, and linear matrix inequalities (LMIs) technique. Some conditions on the uniformly ultimate boundedness, the existence of an attractors, the globally exponential stability of the switched Cohen-Grossberg neural networks are developed. Our results extend and complement some earlier publications.

1. Introduction

In recent years, much attention has been devoted to the study of neural networks due to the fact that they have been fruitfully applied in classification of patterns and associative memories, image processing, parallel computation, optimization, and so on [1–3]. These applications rely crucially on the analysis of the dynamical behavior [4–7]. Various neural networks, such as Hopfield neural networks, cellular neural networks, bidirectional associative neural networks, and Cohen-Grossberg neural networks, have been successfully applied. Among them, the Cohen-Grossberg neural network (CGNN) [8] is an important one, which can be described as follows: 𝑥̇_𝑖(𝑡)=−𝛼_𝑖𝑥_𝑖(𝑡)𝛽_𝑖𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+𝐽_𝑖, (1) where 𝑡≥0, 𝑛≥2;  𝑛 corresponds to the number of units in a neural network; 𝑥_𝑖(𝑡) denotes the potential (or voltage) of cell 𝑖 at time 𝑡; 𝑓_𝑗(⋅) denotes a nonlinear output function; 𝛼_𝑖(⋅)>0 represents an amplification function; 𝛽_𝑖(⋅) represents an appropriately behaved function; the 𝑛×𝑛 connection matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 denotes the strengths of connectivity between cells, and if the output from neuron 𝑗 excites (resp., inhibits) neuron 𝑖, 𝑎_𝑖𝑗≥0 (resp., 𝑎_𝑖𝑗≤0); 𝐽_𝑖 denotes an external input source.

Neural network is nonlinearity; in the real world, nonlinear problems are not exceptional, but regular phenomena. Nonlinearity is the nature of matter and its development [9, 10]. In many practical cases, time delays are common phenomenon encountered in the implementation of neural networks, and they may cause the undesirable dynamic behaviors such as oscillation, divergence, or other poor performances. Time delay exists due to the finite speeds of the switching and transmission of signals in a network, which is degenerate to the instability of networks furthermore. For model (1), Ye et al. [11] introduced delays by considering the following differential equation: 𝑥̇_𝑖(𝑡)=−𝛼_𝑖𝑥_𝑖(𝑡)𝛽_𝑖𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑗+𝐽_𝑖,𝑖=1,…,𝑛. (2) Then, the dynamics of delayed neural networks has been widely studied; see [1, 11–18] for some recent results concerning mixed delays. The CGNN models with discrete delays and distributed delays can be characterized as follows: 𝑥̇_𝑖(𝑡)=−𝛼_𝑖𝑥_𝑖(𝑡)×𝛽_𝑖𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑗−^𝑛_𝑗=1𝑐_𝑖𝑗^𝑡_{𝑡−ℎ𝑗}𝑓_𝑗𝑥_𝑗(𝑠)d𝑠−𝐽_𝑖. (3)

System (3) for convenience can be rewritten as the following compact matrix form: 𝑥̇(𝑡)𝛼(𝑥(𝑡×𝛽(𝑥(𝑡(𝑥(𝑡(𝑥(=−))))−𝐴𝐹))−𝐵𝐹𝑡−𝜏))−𝐶^𝑡_𝑡−ℎ𝐹(𝑥(𝑠))d𝑠−𝐽, (4) where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑅^𝑛 is the neural state vector, 𝛼(𝑥(𝑡))=diag(𝛼₁(𝑥₁(𝑡)),…,𝛼_𝑛(𝑥_𝑛(𝑡)))∈𝑅^𝑛×𝑛, 𝛽(𝑥(𝑡)), 𝐹(𝑥(𝑡)) are appropriate dimensions functions, 𝜏=(𝜏₁,…,𝜏_𝑛)^𝑇, ℎ=(ℎ₁,…,ℎ_𝑛)^𝑇, and 𝐽=(𝐽₁,…,𝐽_𝑛)^𝑇 is the constant external input.

With the rapid development of intelligent control, hybrid systems have been investigated extensively for their significance, which is in theory and application. As one of the most important classes of hybrid systems, switched systems have drawn increasing attention in the last decade [19–21]. A typical switched systems are composed of a set of subsystems and a logical switching rule whose subsystem will be activated at each instant of time and orchestrates the switching among the subsystems [22]. In general, the switching rule is a piecewise constant function dependent on the state or time. The logical rule that orchestrates switching between these subsystems generates switching signals [23]. Recently, many results on the stability of switched system with time delay and parametric uncertainties have been reported [24, 25]. Switched system in which all subsystems are stable was studied in [26], and Hu and Michel used dwell time approach to analyse the local asymptotic stability of non-linear switched systems in [27]. Hespanha and Morse [28] extended this concept to develop the average dwell time approach subsequently. Furthermore, in [29], the stability results of switched system extended to the case when subsystems are both stable and unstable have been reported, and therefore we derive less conservative results. So, average dwell time (ADT) approach turns out to be an effective tool to study the stability of switched systems [28] and specially when not all subsystems are stable [29].

Meanwhile, neural networks as a special kind of complex networks, the connection topology of networks may change frequently and often lead to link failure or new link creation during the hardware implemtation. Hence, the abrupt changes in the network structure often occur, and switchings between some different topologies are inevitable [30]. Thus, the switched neural network was proposed and has successful applications in the field of high-speed signal processing and artificial intelligence [31, 32], and switched neural networks are also used to perform the gene selection in a DNA microarray analysis in [33]. Thus, it is of great meaning to discuss the switched neural networks. Recently, the stability of switching neural networks has been intensively investigated [34–36]. Robust exponential stability and 𝐻_∞ control for switched neutral-type neural networks were discussed in [34].

In [35], delay-dependent stability analysis for switched neural networks with time-varying delay was analyzed. In [36], by employing nonlinear measure and LMI techniques, some new sufficient conditions were obtained to ensure global robust asymptotical stability and global robust exponential stability of the unique equilibrium for a class of switched recurrent neural networks with time-varying delay.

By combining the theories of switched systems and Cohen-Grossberg neural networks, the mathematical model of the switched Cohen-Grossberg neural networks is discussed in detail, which can be written as follows: 𝑥̇(𝑡)𝛼(𝑥(𝑡×𝛽(𝑥(𝑡=−))))−𝐴_𝜎(𝑡)𝐹(𝑥(𝑡))−𝐵_𝜎𝐹(𝑡)(𝑥(𝑡−𝜏))−𝐶_𝜎(𝑡)^𝑡_𝑡−ℎ𝐹(𝑥(𝑠))d𝑠−𝐽. (5)

The function 𝜎(𝑡)∶[𝑡₀,+∞)→𝑁={1,2…,𝑁} is a piece-wise constant function of time, called a switching signal, which specifies that subsystem will be activated. 𝑁 denotes the number of subsystems. The switched sequence can be described as {𝜎(𝑡)∶(𝑡₀,𝜎(𝑡₀)),…,(𝑡_𝑘,𝜎(𝑡_𝑘)),…,∣𝜎(𝑡_𝑘)∈𝑁,𝑘=0,1⋯}, where 𝑡₀ denotes the initial time and 𝑡_𝑘 is the 𝑘th switching instant. Moreover, 𝜎(𝑡)=𝑖 means that the 𝑖th subsystem is activated. For any 𝑖∈𝑁, this means that the matrices (𝐴_𝜎,𝐵_𝜎,𝐶_𝜎) can taken values in the finite set {(𝐴₁,𝐵₁,𝐶₁),…,(𝐴_𝑁,𝐵_𝑁,𝐶_𝑁)}. Meanwhile, we assume that the state of the switched CGNN does not jump at the switching instants; that is, the trajectory 𝑥_𝑡 is everywhere continuous.

However, these available literatures mainly consider the stability property of switching neural networks. In fact, except for stability property, boundedness and attractor are also foundational concepts of dynamical neural networks, which play an important role in investigation of the uniqueness of equilibrium point (periodic solutions), stability and synchronization and so on [13, 14]. To the best of the author’s knowledge, few authors have considered the uniformly ultimate boundedness and attractors for switched CGNN with discrete delays and distributed delays.

As is well known, compared with linear matrix inequalities (LMIs) result, algebraic result is more conservative, and criteria in terms of LMI can be easily checked by using the powerful Matlab LMI toolbox. This motivates us to investigate the problems of the uniformly ultimate boundedness and the existence of an attractor for switched CGNN in this paper. The illustrative examples are given to demonstrate the validity of the theoretical results.

The paper is organized as follows. In Section 2, preliminaries and problem formulation are introduced. Section 3 gives the sufficient conditions of uniformly ultimate boundedness (UUB) and the existence of an attractor for switched CGNN. It is the main result of this paper. In Section 4, an example is given to illustrate the effectiveness of the proposed approach. The conclusions are summarized in Section 5.

2. Problem Formulation

Throughout this paper, we use the following notations. The superscript “𝑇” stands for matrix transposition; 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space; the notation 𝑃>0 means that 𝑃 is real symmetric and positive definite; 𝐼 and 𝑂 represent the identity matrix and a zero matrix; diag{⋯} stands for a block-diagonal matrix; 𝜆_min(𝑃) denotes the minimum eigenvalue of symmetric matrix 𝑃; in symmetric block matrices or long matrix expressions, “∗” is used to represent a term that is induced by symmetry. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

Consider the following Cohen-Grossberg neural network model with mixed delays (discrete delays and distributed delays): 𝑥̇(𝑡)𝛼(𝑥(𝑡×𝛽(𝑥(𝑡(𝑥(𝑡(𝑥(=−))))−𝐴𝐹))−𝐵𝐹𝑡−𝜏))−𝐶^𝑡_𝑡−ℎ𝐹(𝑥(𝑠))d𝑠−𝐽≜−𝛼(𝑥(𝑡))𝐻(𝑡), (6) where 𝐻(𝑡)=𝛽(𝑥(𝑡))−𝐴𝐹(𝑥(𝑡))−𝐵𝐹(𝑥(𝑡−𝜏))−𝐶^𝑡_𝑡−ℎ𝐹(𝑥(𝑠))−𝐽. (7)

The discrete delays and distributed delays are bounded as follows: 0≤𝜏_𝑖,𝜏^∗=max_{1≤𝑖≤𝑛}𝜏_𝑖;0≤ℎ_𝑖,ℎ^∗=max_{1≤𝑖≤𝑛}ℎ_𝑖;𝛿=max𝜏^∗,ℎ^∗, (8) where 𝜏^∗, ℎ^∗, 𝛿 are scalars.

As usual, the initial conditions associated with system (6) are given in the form 𝑥(𝑡)=𝜑(𝑡),−𝛿≤𝑡≤0, (9) where 𝜑(𝑡) is a differentiable vector-valued function.

Throughout this paper, we make the following assumptions. (H₁) For any 𝑗∈{1,2,…,𝑛}, there exist constants 𝑙_𝑗 and 𝐿_𝑗, such that 𝑙_𝑗≤𝑓_𝑗(𝑥)−𝑓_𝑗𝑦𝑥−𝑦≤𝐿_𝑗,∀𝑥,𝑦∈𝑅,𝑥≠𝑦. (10)

Remark 1. The constants 𝑙_𝑗 and 𝐿_𝑗 can be positive, negative, or zero. Therefore, the activation functions 𝑓(⋅) are more general than the forms |𝑓_𝑗(𝑢)|≤𝐾_𝑗|𝑢|, 𝐾_𝑗>0, 𝑗=1,2,…,𝑛. (H₂) For continuously bounded function 𝛼_𝑗(⋅), there exist positive constants 𝛼_𝑗, 𝛼_𝑗, such that 𝛼_𝑗≤𝑎_𝑗𝑥_𝑗(𝑡)≤𝛼_𝑗. (11)(H₃) There exist positive constants 𝑏_𝑗, such that 𝑥_𝑗(𝑡)𝛽_𝑗𝑥_𝑗(𝑡)≥𝑏_𝑗𝑥²_𝑗(𝑡). (12)So, to obtain main results of this paper, the following definitions and lemmas are introduced.

Definition 2 (see [15]). System (6) is uniformly ultimately bounded; if there is 𝐵>0, for any constant 𝜚>0, there is 𝑡^′=𝑡^′(𝜚)>0, such that ‖𝑥(𝑡,𝑡₀,𝜑)‖<𝐵 for all 𝑡≥𝑡₀+𝑡^′, 𝑡₀>0, ‖𝜑‖<𝜚, where the supremum norm ‖𝑥(𝑡,𝑡₀,𝜑)‖=max_{1≤𝑖≤𝑛}sup_{−𝛿≤𝑠≤0}|𝑥_𝑖(𝑡+𝑠,𝑡₀,𝜑)|.

Definition 3 (see [37]). The nonempty closed set 𝐴⊂𝑅^𝑛 is called the attractor for the solution 𝑥(𝑡;𝜑) of system (6) if the following formula holds: lim_𝑡→∞𝑑𝑥𝑡;𝜑;𝔸=0a.s. (13) in which 𝜑∈𝐶([−𝛿,0],𝑅^𝑛), 𝑑(𝑥,𝔸)=inf_𝑦∈𝔸‖𝑥−𝑦‖.

Definition 4 (see [28]). For a switching signal 𝜎(𝑡) and each 𝑇>𝑡≥0, let 𝑁_𝜎(𝑡,𝑇) denote the number of discontinuities of 𝜎(𝑡) in the interval (𝑡,𝑇). If there exist 𝑁₀>0 and 𝑇_𝑎>0 such that 𝑁_𝜎(𝑡,𝑇)≤𝑁₀+(𝑇−𝑡)/𝑇_𝑎 holds, then 𝑇_𝑎 is called the average dwell time. 𝑁₀ is the chatter bound.

Remark 5. In Definition 4, it is obvious that there exists a positive number 𝑇_𝑎 such that a switched signal has the ADT property, which means that the average dwell time between any two consecutive switchings is no less than a specified constant 𝑇_𝑎, Hespanha and Morse have proved that if 𝑇_𝑎 is sufficiently large, then the switched system is exponentially stable. In addition, in [18], one can choose 𝑁₀=0, but in our paper, we assume that 𝑁₀>0, this is more preferable.

Lemma 6 (see [16]). For any positive definite constant matrix 𝑊∈𝑅^𝑛×𝑛, scalar 𝑟>0, and vector function 𝑢(𝑡)∶[𝑡−𝑟,𝑡]→𝑅^𝑛, 𝑡≥0, then ^𝑟₀𝑢(𝑠)d𝑠^𝑇𝑊^𝑟₀𝑢(𝑠)d𝑠≤𝑟^𝑟₀𝑢^𝑇(𝑠)𝑊𝑢(𝑠)d𝑠. (14)

Lemma 7 (see [38]). For any given symmetric positive definite matrix 𝑋∈𝑅^𝑛×𝑛 and scalars 𝛼>0, 0≤𝑑₁<𝑑₂, if there exists a vector function 𝑥̇(𝑡)∶[−𝑑₂,0]→𝑅^𝑛 such that the following integration is well defined, then −^−𝑑1_−𝑑2𝑥̇^𝑇(𝑡+𝜃)𝑒^𝛼𝜃𝑋𝑥̇()≤𝛼𝑡+𝜃d𝜃𝑒^𝛼𝑑1−𝑒^𝛼𝑑2𝑥𝑡−𝑑₁𝑥𝑡−𝑑₂^𝑇×𝑥𝑋−𝑋−𝑋𝑋𝑡−𝑑₁𝑥𝑡−𝑑₂. (15)

3. Main Results

Theorem 8. For a given constant 𝑎>0, if there is positive definite matrix 𝑃=diag(𝑝₁,𝑝₂…,𝑝_𝑛), 𝐷_𝑖=diag(𝐷_𝑖1,𝐷_𝑖2…,𝐷_𝑖𝑛), 𝑖=1,2,…,𝑄,𝑆^𝑖, such that the following condition holds: Φ△=¹¹Φ₁₂Φ₁₃𝑃𝐵0𝑃𝐶0∗Φ₂₂0Φ₂₄000∗∗Φ₃₃0000∗∗∗Φ₄₄000∗∗∗∗Φ₅₅Φ₅₆0∗∗∗∗∗Φ₆₆0∗∗∗∗∗∗Φ₇₇<0, (16) where 𝑄𝑄=¹¹𝑄₁₂∗𝑄₂₂≥0,𝐷_𝑖Φ≥0,𝑖=1,2,₁₁=𝑎Ω₁𝑃−2𝑎Ω₂𝑃+𝑃−𝛿𝑒^{−𝑎𝜏∗}𝜏^∗𝑆⁽²⁾−Ω₃𝐷₁+14𝑎²𝐼+𝑆⁽¹⁾−𝑒^{−𝑎𝜏∗}𝑆⁽¹⁾+ℎ^∗𝑄₁₁+𝛿𝑎1−𝑒^𝑎𝜏∗𝑆⁽²⁾,Φ₁₂=−𝛿𝑎1−𝑒^𝑎𝜏∗𝑆⁽²⁾,Φ₁₃=𝑃𝐴+Ω₄𝐷₁+ℎ^∗𝑄₁₂,Φ₂₂=−Ω₃𝐷₂−𝛿𝑒^{−𝑎𝜏∗}𝜏^∗𝑆⁽²⁾−𝑒^{−𝑎𝜏∗}𝑆⁽¹⁾+𝛿𝑎1−𝑒^𝑎𝜏∗𝑆⁽²⁾+14𝑎²Φ𝐼,₂₄=Ω₄𝐷₂,Φ₃₃=−𝐷₁+1𝑎²𝐼+ℎ^∗𝑄₂₂,Φ₄₄=−𝐷₂+1𝑎²𝐼,Φ₅₅𝑒=−^{−𝑎ℎ∗}ℎ^∗𝑄₁₁,Φ₅₆𝑒=−^{−𝑎ℎ∗}ℎ^∗𝑄₁₂,Φ₆₆𝑒=−^{−𝑎ℎ∗}ℎ^∗𝑄₂₂,Φ₇₇=𝛿𝜏^∗𝛼²𝑆⁽²⁾,Ω₁1=diag𝛼₁,1𝛼₂1,…,𝛼_𝑛,Ω₂=diag𝑏₁,𝑏₂,…,𝑏_𝑛,Ω₃=diag𝑙₁𝐿₁,𝑙₂𝐿₂,…,𝑙_𝑛𝐿_𝑛,Ω₄𝑙=diag¹+𝐿₁2,𝑙₂+𝐿₂2𝑙,…,^𝑛+𝐿_𝑛2,𝜌=max_{1≤𝑖≤𝑛}𝑙²_𝑖,𝐿²_𝑖+1; (17) the symbol “∗” within the matrix represents the symmetric term of the matrix, and then system (6) is uniformly ultimately bounded.

Proof. Let us consider the following Lyapunov-Krasovskii functional: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (18) where 𝑉₁(𝑡)=^𝑛_𝑗=1𝑒^𝑎𝑡𝑝_𝑗^𝑥𝑗₀^(𝑡)2𝑠𝛼_𝑗(𝑠)𝑉d𝑠,₂(𝑡)=^𝑡_𝑡−ℎ𝑒^𝑎𝑠(𝑠−(𝑡−ℎ))𝜉^𝑇(𝑠)(𝑠)𝜉(𝑡)=𝑥𝑄𝜉d𝑠,_𝑇(𝑡),𝐹^𝑇(𝑥(𝑡))^𝑇,𝑉₃(𝑡)=^𝑡_𝑡−𝜏𝑒^𝑎𝑠𝑥^𝑇(𝑠)𝑆⁽¹⁾𝑥(𝑠)d𝑠d𝜃+𝛿⁰_−𝜏^𝑡_𝑡+𝜃𝑒^𝑎𝑠𝑥̇^𝑇(𝑠)𝑆⁽²⁾𝑥̇(𝑠)d𝑠d𝜃. (19) We proceed to evaluate the time derivative of 𝑉₁(𝑡) along the trajectory of system (6), and one can get 𝑉̇₁(𝑡)=^𝑛_𝑗=12𝑎𝑝_𝑗𝑒^{𝑎𝑡𝑥𝑗}₀^(𝑡)𝑠𝛼_𝑗(𝑠)d𝑠−𝑝_𝑗𝑒^𝑎𝑡𝑥_𝑗(𝑡)𝛽_𝑗𝑥_𝑗(𝑡+)2𝑥^𝑇(𝑡)𝑃𝐴𝐹(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑃𝐽+2𝑥^𝑇(𝑡)𝑃𝐵𝐹(𝑥(𝑡−𝜏))+2𝑥^𝑇(𝑡)𝑃𝐶^𝑡_𝑡−ℎ𝐹(𝑥(𝑠))d𝑠𝑒^𝑎𝑡. (20) According to assumption (H₂), we obtain the following inequality: 2𝑎𝑝_𝑗^𝑥𝑗₀^(𝑡)𝑠𝛼_𝑗(𝑠)𝑎d𝑠≤𝛼_𝑗𝑝_𝑗𝑥²_𝑗(𝑡). (21) From assumption (H₃) and inequality (21), we obtain 𝑉̇₁(𝑡)≤𝑎𝑒^𝑎𝑡𝑥^𝑇(𝑡)Ω₁𝑃𝑥(𝑡)−2𝑥^𝑇(𝑡)Ω₂(𝑡+𝑃𝑥)2𝑥^𝑇(𝑡)𝑃𝐴𝐹(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑃𝐵𝐹(𝑥(𝑡−𝜏))+2𝑥^𝑇(𝑡)𝑃𝐶^𝑡_𝑡−ℎ𝐹(𝑥(𝑠))d𝑠+𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝐽^𝑇𝑃𝐽𝑒^𝑎𝑡. (22)
Similarly, taking the time derivative of 𝑉₂(𝑡) along the trajectory of system (6), we obtain 𝑉̇₂(𝑡)=𝑡𝑒^𝑎𝑡𝜉^𝑇(𝑡)(𝑡)−()𝑒𝑄𝜉𝑡−ℎ^{𝑎(𝑡−ℎ)}𝜉^𝑇()()−𝑡−ℎ×𝑄𝜉𝑡−ℎ_{𝑡𝑡−ℎ}𝑒^𝑎𝑠𝜉^𝑇(𝑠)𝑄𝜉(𝑠)d𝑠+𝑡𝑒^𝑎𝑡𝜉^𝑇(𝑡)𝑄𝜉(𝑡)−𝑡𝑒^{𝑎(𝑡−ℎ)}𝜉^𝑇(𝑡−ℎ)×𝑄𝜉(𝑡−ℎ)+ℎ𝑒^𝑎𝑡𝜉^𝑇(𝑡)𝑄𝜉(𝑡)−ℎ𝑒^{𝑎(𝑡−ℎ)}𝜉^𝑇(𝑡−ℎ)𝑄𝜉(𝑡−ℎ)≤ℎ^∗𝑒^𝑎𝑡×𝑥^𝑇(𝑡)𝑄₁₁𝑥(𝑡)+𝐹^𝑇(𝑥(𝑡))𝑄^𝑇₁₂𝑥(𝑡)+𝑥^𝑇(𝑡)𝑄₁₂𝐹(𝑥(𝑡))+𝐹^𝑇(𝑥(𝑡))𝑄₂₂𝐹(𝑥(𝑡))−𝑒^{𝑎(𝑡−ℎ∗)𝑡}_𝑡−ℎ𝜉^𝑇(𝑠)𝑄𝜉(𝑠)d𝑠. (23) According to Lemma 6, we can conclude that −𝑒^{𝑎(𝑡−ℎ∗)𝑡}_𝑡−ℎ𝜉^𝑇(𝑠)(𝑠)𝑒𝑄𝜉d𝑠≤−^{𝑎(𝑡−ℎ∗)}ℎ^∗𝑡_𝑡−ℎ𝜉(𝑠)d𝑠^𝑇𝑄^𝑡_𝑡−ℎ𝜉(𝑠)𝑒d𝑠≤−^{−𝑎ℎ∗}ℎ^∗𝑡_𝑡−ℎ𝜉(𝑠)d𝑠^𝑇𝑄^𝑡_𝑡−ℎ𝜉(𝑠)d𝑠. (24)
Computing the derivative of 𝑉₃(𝑡) along the trajectory of system (6) turns out to be 𝑉̇₃(𝑡)=𝑒^𝑎𝑡𝑥^𝑇(𝑡)𝑆⁽¹⁾𝑥(𝑡)−𝑒^{𝑎(𝑡−𝜏)}𝑥^𝑇(𝑡−𝜏)𝑆⁽¹⁾𝑥(𝑡−𝜏)+𝛿⁰_−𝜏𝑒^𝑎𝑡𝑥̇^𝑇(𝑡)𝑆⁽²⁾𝑥̇(𝑡)−𝑒^{𝑎(𝑡+𝜃)}𝑥̇^𝑇(𝑡+𝜃)𝑆⁽²⁾𝑥̇(𝑡+𝜃)d𝜃≤𝑒^𝑎𝑡𝑥^𝑇(𝑡)𝑆⁽¹⁾𝑥(𝑡)−𝑒^{−𝑎𝜏∗}𝑥^𝑇(𝑡−𝜏)𝑆⁽¹⁾𝑥(𝑡−𝜏)+𝛿𝜏^∗𝑒^𝑎𝑡𝑥̇^𝑇(𝑡)𝑆⁽²⁾𝑥̇(𝑡)−𝛿⁰_−𝜏𝑒^{𝑎(𝑡+𝜃)}𝑥̇^𝑇(𝑡+𝜃)𝑆⁽²⁾𝑥̇(𝑡+𝜃)d𝜃, (25) where 𝜏^∗=max_{1≤𝑖≤𝑛}{𝜏_𝑖}.
Denoting that 𝛼=max{𝛼₁,𝛼₂,…,𝛼_𝑛}, we obtain 𝛿𝜏^∗𝑒^𝑎𝑡𝑥̇^𝑇(𝑡)𝑆⁽²⁾𝑥̇(𝑡)=𝛿𝜏^∗𝑒^𝑎𝑡[𝛼(𝑥(𝑡))𝐻(𝑡)]^𝑇𝑆⁽²⁾𝛼(𝑥(𝑡))𝐻(𝑡)≤𝛿𝜏^∗𝛼²𝑒^𝑎𝑡𝐻^𝑇(𝑡)𝑆⁽²⁾𝐻(𝑡). (26)
Using Lemma 7, the following inequality is easily obtained: −𝛿⁰_−𝜏𝑒^𝑎𝜃𝑥̇^𝑇(𝑡+𝜃)𝑆⁽²⁾𝑥̇()≤𝑡+𝜃d𝜃𝛿𝑎1−𝑒^𝑎𝜏𝑥(𝑡)𝑥(𝑡−𝜏)^𝑇𝑆⁽²⁾−𝑆⁽²⁾−𝑆⁽²⁾𝑆⁽²⁾×𝑥(𝑡)𝑥()≤𝑡−𝜏𝛿𝑎1−𝑒^𝑎𝜏∗𝑥(𝑡)𝑥(𝑡−𝜏)^𝑇𝑆⁽²⁾−𝑆⁽²⁾−𝑆⁽²⁾𝑆⁽²⁾×𝑥(𝑡)𝑥().𝑡−𝜏 (27)
From assumption (H₁), it follows that, for 𝑗=1,2,…,𝑛, 𝐹_𝑗𝑥_𝑗(𝑡)−𝐹_𝑗(0)−𝑙_𝑗𝑥_𝑗(𝑡×𝐹_𝑗)𝑥_𝑗(𝑡)−𝐹_𝑗(0)−𝐿_𝑗𝑥_𝑗(𝑡𝐹)≤0,_𝑗𝑥_𝑗(𝑡−𝜏)−𝐹_𝑗(0)−𝑙_𝑗𝑥_𝑗(×𝐹𝑡−𝜏)_𝑗𝑥_𝑗(𝑡−𝜏)−𝐹_𝑗(0)−𝐿_𝑗𝑥_𝑗(𝑡−𝜏)≤0. (28) Then, let Υ₁=−^𝑛_𝑗=1𝐷_1𝑗𝐹_𝑗𝑥_𝑗(𝑡)−𝐹_𝑗(0)−𝑙_𝑗𝑥_𝑗(𝑡×𝐹_𝑗)𝑥_𝑗(𝑡)−𝐹_𝑗(0)−𝐿_𝑗𝑥_𝑗(𝑡,Υ₂)=−^𝑛_𝑗=1𝐷_2𝑗𝐹_𝑗𝑥_𝑗(𝑡−𝜏)−𝐹_𝑗(0)−𝑙_𝑗𝑥_𝑗(×𝐹𝑡−𝜏)_𝑗𝑥_𝑗(𝑡−𝜏)−𝐹_𝑗(0)−𝐿_𝑗𝑥_𝑗(𝑡−𝜏). (29) So, Υ₁+Υ₂=−^𝑛_𝑗=1𝐷_1𝑗𝐹_𝑗𝑥_𝑗(𝑡)−𝐹_𝑗(0)−𝑙_𝑗𝑥_𝑗(𝑡×𝐹_𝑗)𝑥_𝑗(𝑡)−𝐹_𝑗(0)−𝐿_𝑗𝑥_𝑗(𝑡−_𝑛𝑗=1)𝐷_2𝑗𝐹_𝑗𝑥_𝑗(𝑡−𝜏)−𝐹_𝑗(0)−𝑙_𝑗𝑥_𝑗(×𝐹𝑡−𝜏)_𝑗𝑥_𝑗(𝑡−𝜏)−𝐹_𝑗(0)−𝐿_𝑗𝑥_𝑗(𝑡−𝜏)=−^𝑛_𝑗=1𝐷_1𝑗𝐹_𝑗𝑥_𝑗(𝑡)−𝑙_𝑗𝑥_𝑗(𝑡×𝐹_𝑗)𝑥_𝑗(𝑡)−𝐿_𝑗𝑥_𝑗(𝑡−_𝑛𝑗=1)𝐷_2𝑗𝐹_𝑗𝑥_𝑗(𝑡−𝜏)−𝑙_𝑗𝑥_𝑗(×𝐹𝑡−𝜏)_𝑗𝑥_𝑗(𝑡−𝜏)−𝐿_𝑗𝑥_𝑗(−𝑡−𝜏)_𝑛𝑗=1𝐷_1𝑗𝐹²_𝑗(0)+^𝑛_𝑗=1𝐷_1𝑗𝐹_𝑗(0)2𝐹_𝑗𝑥_𝑗(𝑡)−𝐿_𝑗+𝑙_𝑗𝑥_𝑗(𝑡−_𝑛𝑗=1)𝐷_2𝑗𝐹²_𝑗(0)+^𝑛_𝑗=1𝐷_2𝑗𝐹_𝑗(0)2𝐹_𝑗𝑥_𝑗(𝑡−𝜏)−𝐿_𝑗+𝑙_𝑗𝑥_𝑗(𝑡−𝜏)≤−^𝑛_𝑗=1𝐷_1𝑗𝐹_𝑗𝑥_𝑗(𝑡)−𝑙_𝑗𝑥_𝑗(𝑡×𝐹_𝑗)𝑥_𝑗(𝑡)−𝐿_𝑗𝑥_𝑗(𝑡−_𝑛𝑗=1)𝐷_2𝑗𝐹_𝑗𝑥_𝑗(𝑡−𝜏)−𝑙_𝑗𝑥_𝑗(×𝐹𝑡−𝜏)_𝑗𝑥_𝑗(𝑡−𝜏)−𝐿_𝑗𝑥_𝑗(+𝑡−𝜏)_𝑛𝑗=11𝑎²𝐹²_𝑗𝑥_𝑗(𝑡)+𝑎²𝐷²_1𝑗𝐹²_𝑗(0)+14𝑎²𝑥²_𝑗(𝑡)+𝑎²𝐷²_1𝑗𝐹²_𝑗(0)𝐿_𝑗+𝑙_𝑗²+^𝑛_𝑗=11𝑎²𝐹²_𝑗𝑥_𝑗(𝑡−𝜏)+𝑎²𝐷²_2𝑗𝐹²_𝑗(0)+14𝑎²𝑥²_𝑗(𝑡−𝜏)+𝑎²𝐷²_2𝑗𝐹²_𝑗(0)𝐿_𝑗+𝑙_𝑗². (30)
Using (20)–(27) and adding (29), we can derive 𝑉̇(𝑡)≤𝑉̇₁(𝑡)+𝑉̇₂(𝑡)+𝑉̇₃(𝑡)+𝑉̇₄(𝑡)+𝑒^𝑎𝑡Υ₁+Υ₂≤𝑒^𝑎𝑡𝑀^𝑇(𝑡)Δ₁𝑀(𝑡)+𝑒^𝑎𝑡𝐽^𝑇𝑃𝐽+𝑒^𝑛𝑎𝑡_𝑗=1𝑎²𝐷²_1𝑗𝐹²_𝑗(0)+𝑎²𝐷²_1𝑗𝐹²_𝑗(0)𝐿_𝑗+𝑙_𝑗²+𝑎²𝐷²_2𝑗𝐹²_𝑗(0)+𝑎²𝐷²_2𝑗𝐹²_𝑗(0)𝐿_𝑗+𝑙_𝑗². (31) Denote that 𝑀^𝑇(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏),𝐹^𝑇(𝑥(𝑡)),𝐹^𝑇(𝑥(𝑡−𝜏))_{𝑡𝑡−ℎ},𝑥(𝑠)d𝑠^𝑇,^𝑡_𝑡−ℎ𝐹(𝑥(𝑠))d𝑠^𝑇,𝐻^𝑇(𝑥(𝑡))^𝑇. (32)
By integrating both sides of (31) in time interval 𝑡∈[𝑡₀,𝑡], then we can obtain 𝐾𝑒^𝑎𝑡‖𝑥(𝑡)‖²(𝑥(𝑡𝑥𝑡≤𝑉))≤𝑉₀+𝑎⁻¹𝑒^𝑎𝑡𝐽^𝑇𝑃𝐽+𝑒^𝑛𝑎𝑡_𝑗=1𝑎𝐷²_1𝑗𝐹²_𝑗(0)+𝑎𝐷²_1𝑗𝐹²_𝑗(0)𝐿_𝑗+𝑙_𝑗²+𝑎𝐷²_2𝑗𝐹²_𝑗(0)+𝑎𝐷²_2𝑗𝐹²_𝑗(0)𝐿_𝑗+𝑙_𝑗², (33) which implies that ‖𝑥(𝑡)‖²≤𝑒^−𝑎𝑡𝑉𝑥𝑡₀+𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ𝐾, (34) where 𝐾=min_{1≤𝑖≤𝑛}{𝜆_min(𝑃)/𝛼_𝑖}, and Υ=^𝑛_𝑗=1𝑎𝐷²_1𝑗𝐹²_𝑗(0)+𝑎𝐷²_1𝑗𝐹²_𝑗(0)(𝐿_𝑗+𝑙_𝑗)²+𝑎𝐷²_2𝑗𝐹²_𝑗(0)+𝑎𝐷²_2𝑗𝐹²_𝑗(0)𝐿_𝑗+𝑙_𝑗². (35)
If one chooses 𝐵=(1+𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ)/𝐾>0, then for any constant 𝜚>0 and ‖𝜑‖<𝜚, there is 𝑡^′=𝑡^′(𝜚)>0, such that 𝑒^−𝑎𝑡𝑉(𝑥(𝑡₀))²<1 for all 𝑡≥𝑡^′. According to Definition 2, we have ‖𝑥(𝑡,𝑥(𝑡₀),𝜑)‖<𝐵 for all 𝑡≥𝑡^′. That is to say, system (6) is uniformly ultimately bounded. This completes the proof.

From (18), we know that there is a positive constant 𝐿₀, such that 𝑉𝑥𝑡₀≤𝐿₀𝑥𝑡₀²𝑒^−𝑎𝑡0. (36)

Thus, considering (34) and (36), we have the following result: ‖𝑥(𝑡)‖²≤𝑒^−𝑎𝑡𝑉𝑥𝑡₀+𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ𝐾=𝑒^−𝑎𝑡𝑉𝑥𝑡₀𝐾+𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ𝐾≤𝐿₀𝑥𝑡₀²𝑒^{−𝑎(𝑡−𝑡0)}𝐾+𝑁, (37) where 𝑁=(𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ)/𝐾.

Theorem 9. If all of the conditions of Theorem 8 hold, then there exists an attractor 𝔸𝐵 for the solutions of system (6), where 𝔸𝐵={𝑥(𝑡)∶‖𝑥(𝑡)‖≤𝐵,𝑡≥𝑡₀}.

Proof. If one chooses 𝐵=(1+𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ)/𝐾>0, Theorem 8 shows that for any 𝜙, there is 𝑡^′>0, such that ‖𝑥(𝑡,𝑡₀,𝜙)‖<𝐵 for all 𝑡≥𝑡^′. Let 𝔸𝐵={𝑥(𝑡)∶‖𝑥(𝑡)‖≤𝐵,𝑡≥𝑡₀}. Clearly, 𝔸𝐵 is closed, bounded, and invariant. Furthermore, lim_𝑡→∞supinf_𝑦∈𝔸𝐵‖𝑥(𝑡;𝑡₀,𝜙)−𝑦‖=0. Therefore, 𝔸𝐵 is an attractor for the solutions of system (6).

Corollary 10. In addition to the fact that all of the conditions of Theorem 8 hold, if 𝐽=0, and 𝐹_𝑗(0)=0, then system (6) has a trivial solution 𝑥(𝑡)≡0, and the trivial solution of system (6) is globally exponentially stable.

Proof. If 𝐽=0, and 𝐹_𝑗(0)=0, then it is obvious that system (6) has a trivial solution 𝑥(𝑡)≡0. From Theorem 8, one has 𝑥𝑡;0,𝜙²≤𝐾₁𝑒^−𝑎𝑡,∀𝜙, (38) where 𝐾₁=𝑉(𝑥(0))/𝐾.
Therefore, the trivial solution of system (6) is globally exponentially stable. This completes the proof.

In this section, we will present conditions for uniformly ultimate boundedness and the existence of an attractor of the switching CGNN by applying the average dwell time.

Now, we can consider the switched Cohen-Grossberg neural networks with discrete delays and distributed delays as follows: 𝑥̇(𝑡)𝛼(𝑥(𝑡×𝛽(𝑥(𝑡=−))))−𝐴_𝜎(𝑡)𝐹(𝑥(𝑡))−𝐵𝐹_𝜎(𝑡)(𝑥(𝑡−𝜏))−𝐶_𝜎(𝑡)^𝑡_𝑡−ℎ𝐹(𝑥(𝑠,𝑥(𝑡)(𝑡)[].))d𝑠+𝐽=𝜑,when𝑡∈−𝛿,0 (39)(40)

Theorem 11. For a given constant 𝑎>0, if there is positive definite matrix 𝑃=diag(𝑝_𝑖1,𝑝_𝑖2…,𝑝_𝑖𝑛), 𝐷_𝑖=diag(𝐷_𝑖1,𝐷_𝑖2…,𝐷_𝑖𝑛), 𝑖=1,2,𝑄_𝑖,𝑆_𝑖^(𝑖), such that the following condition holds: Φ△=^𝑖11Φ_𝑖12Φ_𝑖13𝑃_𝑖𝐵_𝑖0𝑃_𝑖𝐶_𝑖0∗Φ_𝑖220Φ_𝑖24000∗∗Φ_𝑖330000∗∗∗Φ_𝑖44000∗∗∗∗Φ_𝑖55Φ_𝑖560∗∗∗∗∗Φ_𝑖660∗∗∗∗∗∗Φ_𝑖77<0, (41) where 𝑄𝑄=^𝑖11𝑄_𝑖12∗𝑄_𝑖22≥0,𝐷_𝑖Φ≥0,𝑖=1,2,_𝑖11=𝑎Ω₁𝑃_𝑖−2𝑎Ω₂𝑃_𝑖+𝑃_𝑖−𝛿𝑒^{−𝑎𝜏∗}𝜏^∗𝑆_𝑖⁽²⁾−Ω₃𝐷₁+14𝑎²𝐼+𝑆_𝑖⁽¹⁾−𝑒^{−𝑎𝜏∗}𝑆_𝑖⁽¹⁾+ℎ^∗𝑄_𝑖11+𝛿𝑎1−𝑒^𝑎𝜏∗𝑆_𝑖⁽²⁾,Φ_𝑖12=−𝛿𝑎1−𝑒^𝑎𝜏∗𝑆_𝑖⁽²⁾,Φ_𝑖13=𝑃_𝑖𝐴_𝑖+ℎ^∗2𝑅_𝑖12+Ω₄𝐷₁+ℎ^∗𝑄_𝑖12,Φ_𝑖22=−Ω₃𝐷₂−𝛿𝑒^{−𝑎𝜏∗}𝜏^∗𝑆_𝑖⁽²⁾−𝑒^{−𝑎𝜏∗}𝑆_𝑖⁽¹⁾+𝛿𝑎1−𝑒^𝑎𝑡∗𝑆_𝑖⁽²⁾+14𝑎²Φ𝐼,_𝑖24=Ω₄𝐷₂,Φ_𝑖33=−𝐷₁+1𝑎²𝐼+ℎ^∗𝑄_𝑖22,Φ_𝑖44=−𝐷₂+1𝑎²𝐼,Φ_𝑖55𝑒=−^{−𝑎ℎ∗}ℎ^∗𝑄_𝑖11,Φ_𝑖56𝑒=−^{−𝑎ℎ∗}ℎ^∗𝑄_𝑖12,Φ_𝑖66𝑒=−^{−𝑎ℎ∗}ℎ^∗𝑄_𝑖22,Φ_𝑖77=𝛿𝜏^∗𝛼²𝑆_𝑖⁽²⁾. (42) Then, system (39) is uniformly ultimately bounded for any switching signal with average dwell time satisfying 𝑇_𝑎>𝑇^∗_𝑎=ln𝑀_max𝑎, (43) where 𝑀_max=𝐿_max/𝐾_min, 𝐿_max=max_{𝑘∈𝑁,1≤𝑖≤𝑛}{𝐿_𝑖𝑘}, 𝐾_min=min_𝑖𝑘{𝐾_𝑖𝑘}.

Proof. Define the Lyapunov functional candidate of the form 𝑉_𝜎(𝑡)=^𝑛_𝑗=12𝑝_𝜎(𝑡)𝑒^{𝑎𝑡𝑥𝑗}₀^(𝑡)𝑠𝛼_𝑗(𝑠)+d𝑠_{𝑡𝑡−ℎ}𝑒^𝑎𝑠(𝑠−(𝑡−ℎ))𝜉^𝑇(𝑠)𝑄_𝜎(𝑡)𝜉(𝑠)+d𝑠_{𝑡𝑡−𝜏}𝑒^𝑎𝑠𝑥̇^𝑇(𝑠)𝑆_𝜎(𝑡)⁽¹⁾𝑥̇(𝑠)d𝑠d𝜃+𝛿⁰_−𝜏^𝑡_𝑡+𝜃𝑒^𝑎𝑠𝑥̇^𝑇(𝑠)𝑆_𝜎(𝑡)⁽²⁾𝑥̇(𝑠)d𝑠d𝜃. (44) When 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), the 𝑖_𝑘th subsystem is activated, and from Theorem 8 and (34), we can conclude that there is a positive constant 𝐿_𝑖𝑘, such that ‖𝑥(𝑡)‖²≤𝐿_𝑖𝑘𝑥(𝑡_𝑘)²𝑒^{−𝑎(𝑡−𝑡𝑘)}+𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ𝐾_𝑖𝑘=𝑀_𝑖𝑘𝑥(𝑡_𝑘)²𝑒^{−𝑎(𝑡−𝑡𝑘)}+𝑁_𝑖𝑘, (45) where 𝑀_𝑖𝑘=𝐿_𝑖𝑘𝐾_𝑖𝑘,𝐾_𝑖𝑘=min_{𝑘∈𝑁,1≤𝑖≤𝑛}𝜆_min𝑃_𝑖𝛼_𝑖,𝑁_𝑖𝑘=𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ𝐾_𝑖𝑘. (46)
The system state is continuous. Therefore, it follows that ‖𝑥(𝑡)‖²≤𝐿_𝑖𝑘𝑥𝑡_𝑘²𝑒^{−𝑎(𝑡−𝑡𝑘)}+𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ𝐾_𝑖𝑘=𝑀_𝑖𝑘𝑥(𝑡_𝑘)²𝑒^{−𝑎(𝑡−𝑡𝑘)}+𝑁_𝑖𝑘≤⋯≤𝑒^{∑𝑘𝑣=0ln𝑀𝑖𝑣−𝑎(𝑡−𝑡0)}𝑥(𝑡₀)²+𝑀_𝑖𝑘𝑒^{−𝑎(𝑡−𝑡𝑘)}𝑁_𝑖𝑘+𝑀_𝑖𝑘𝑀_𝑖𝑘−1𝑒^{−𝑎(𝑡−𝑡𝑘−1)}𝑁_𝑖𝑘−1+𝑀_𝑖𝑘𝑀_𝑖𝑘−1𝑀_𝑖𝑘−2𝑒^{−𝑎(𝑡−𝑡𝑘−2)}𝑁_𝑖𝑘−2++⋯𝑀_𝑖𝑘𝑀_𝑖𝑘−1𝑀_𝑖𝑘−2⋯𝑀_𝑖1𝑒^{−𝑎(𝑡−𝑡1)}𝑁_𝑖1+𝑁_𝑖𝑘≤𝑒^{(𝑘+1)ln𝑀max−𝑎(𝑡−𝑡0)}𝑥(𝑡₀)²+𝑀^𝑘_max𝑁_max+𝑀^(𝑘−1)_max𝑁_max+𝑀^(𝑘−2)_max𝑁_max+⋯+𝑀²_max𝑁_max+𝑀_max𝑁_max+𝑁_max≤𝑀_max𝑒^{𝑘ln𝑀max−𝑎(𝑡−𝑡0)}𝑥(𝑡₀)²+𝑁_max1−𝑀^(𝑘+1)_max1−𝑀_max≤𝑀_max𝑒^{ln𝑀max𝑁𝜎(𝑡−𝑡0)−𝑎(𝑡−𝑡0)}𝑥(𝑡₀)²+𝑁_max1−𝑀^(𝑘+1)_max1−𝑀_max≤𝑀_max𝑒^{𝑁0ln𝑀max−(𝑎−ln𝑀max/𝑇𝑎)(𝑡−𝑡0)}𝑥(𝑡₀)²+𝑁_max1−𝑀^(𝑘+1)_max1−𝑀_max≤𝐿_max𝑒^{𝑁0ln𝑀max−(𝑎−ln𝑀max/𝑇𝑎)(𝑡−𝑡0)}𝑥(𝑡₀)²𝑘_min+𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ1−𝐿^(𝑘+1)_max/𝐾^(𝑘+1)_min𝐾_min−𝐿_max. (47) If one chooses 𝐵=1/𝐾_min+(𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ)(1−𝐿^(𝑘+1)_max/𝐾^(𝑘+1)_min)/(𝐾_min−𝐿_max)>0, then for any constant 𝜚>0 and ‖𝜑‖<𝜚, there is 𝑡^′=𝑡^′(𝜚)>0, such that 𝐿_max𝑒^{𝑁0ln𝑀max−(𝑎−ln𝑀max/𝑇𝑎)(𝑡−𝑡0)}‖𝑥(𝑡₀)‖²<1 for all 𝑡≥𝑡^′. According to Definition 2, we have ‖𝑥(𝑡,𝑡₀,𝜑)‖<𝐵 for all 𝑡≥𝑡^′. That is to say, the switched Cohen-Grossberg neural networks system (39) is uniformly ultimately bounded. This completes the proof.

Theorem 12. If all of the conditions of Theorem 11 hold, then there exists an attractor 𝔸𝐵 for the solutions of system (39), where 𝔸𝐵={𝑥(𝑡)∶‖𝑥(𝑡)‖≤𝐵,𝑡≥𝑡₀}.

Proof. If we choose 𝐵=1/𝐾_min+(𝑎⁻¹𝐽^𝑇𝑃𝐽+Υ)(1−𝐿^(𝑘+1)_max/𝐾^(𝑘+1)_min)/(𝐾_min−𝐿_max)>0, Theorem 11 shows that for any 𝜙, there is 𝑡^′>0, such that ‖𝑥(𝑡,𝑡₀,𝜙)‖<𝐵 for all 𝑡≥𝑡^′. Let 𝔸𝐵={𝑥(𝑡)∶‖𝑥(𝑡)‖≤𝐵,𝑡≥𝑡₀}. Clearly, 𝔸𝐵 is closed, bounded, and invariant. Furthermore, lim_𝑡→∞supinf_𝑦∈𝔸𝐵‖𝑥(𝑡;𝑡₀,𝜙)−𝑦‖=0.
Therefore, 𝔸𝐵 is an attractor for the solutions of system (39).

Corollary 13. In addition to the fact that all of the conditions of Theorem 8 hold, if 𝐽=0 and 𝐹_𝑖(0)=0, then system (39) has a trivial solution 𝑥(𝑡)≡0, and the trivial solution of system (39) is globally exponentially stable.

Proof. If 𝐽=0 and 𝐹_𝑖(0)=0, then it is obvious that the switched system (39) has a trivial solution 𝑥(𝑡)≡0. From Theorem 8, one has 𝑥𝑡;𝑡₀,𝜙²≤𝐾₂𝑒^{−𝑎(𝑡−𝑡0)},∀𝜙, (48) where 𝐾₂=𝑀_max𝑒^{𝑁0ln𝑀max−(𝑎−ln𝑀max/𝑇𝛼)(𝑡−𝑡0)}‖𝑥(𝑡₀)‖².
It means that the trivial solution of the switched Cohen-Grossberg neural networks (39) is globally exponentially stable. This completes the proof.

Remark 14. It is noted that common Lyapunov function method requires all the subsystems of the switched system to share a positive definite radially unbounded common Lyapunov function. Generally speaking, this requirement is difficult to achieve. So, in this paper, we select a novel multiple Lyapunov function to study the uniformly ultimate boundedness and the existence of an attractor for switched Cohen-Grossberg neural networks. furthermore, this type of Lyapunov function enables us to establish less conservative results.

Remark 15. When 𝑁=1, we have 𝑃_𝑖=𝑃_𝑗, 𝑄_𝑖=𝑄_𝑗, 𝑆_𝑖⁽¹⁾=𝑆_𝑗⁽¹⁾, 𝑆_𝑖⁽²⁾=𝑆_𝑗⁽²⁾, 𝑖,𝑗∈Σ, then the switched Cohen-Grossberg neural networks (4) degenerates into a general Cohen-Grossberg neural networks with time-delay [15, 17]. Obviously, our result generalizes the previous result.

Remark 16. It is easy to see that 𝜏_𝑎=0 is equivalent to existence of a common function for all subsystems, which implies that switching signals can be arbitrary. Hence, the results reported in this paper are more effective than arbitrary switching signal in the previous literature [16].

Remark 17. The constants 𝑙_𝑖, 𝐿_𝑖 in assumption (H₁) are allowed to be positive, negative, or zero, whereas the constant 𝑙_𝑖 is restricted to be the zero in [1, 15], and the non-linear output function in [5, 18, 34–37] is required to satisfy 𝐹_𝑗(0)=0. However, in our paper, the assumption condition was deleted. Therefore, assumption (H₁) of this paper is weaker than those given in [1, 5, 15, 18, 34–37].

4. Illustrative Examples

In this section, we present an example to show the effectiveness and advantages of the proposed method and consider the switched neural networks with two subsystems.

Example. Consider the following switched Cohen-Grossberg neural network with discrete delays and distributed delays: 𝑥̇(𝑡)𝛼(𝑥(𝑡×𝛽(𝑥(𝑡=−))))−𝐴_𝜎(𝑡)𝐹(𝑥(𝑡))−𝐵_𝜎𝐹(𝑡)(𝑥(𝑡−𝜏))−𝐶_𝜎(𝑡)^𝑡_𝑡−ℎ𝐹(𝑥(𝑠))d𝑠−𝐽, (49) where the behaved function is described by 𝛽_𝑖(𝑥_𝑖(𝑡))=𝑥_𝑖(𝑡), and 𝐹_𝑖(𝑥_𝑖(𝑡))=0.5tanh(𝑥_𝑖(𝑡))(𝑖=1,2); let 𝛼(𝑥(𝑡))=1+sin²𝑥₁(𝑡)001+cos²𝑥₁(𝑡). (50) Take the parameters as follows: 𝐴₁=−0.1−0.40.2−0.5,𝐵₁=,𝐶−0.1−11.4−0.4₁=−0.1−0.20.2−0.1,𝐴₂=,𝐵−0.3−0.50.2−0.1₂=−0.25−0.70.9−0.5,𝐶₂=.−0.15−0.31.60.25 (51)

From assumptions H₁, H₂, we can gain 𝑙_𝑖=0.5, 𝐿_𝑖=1, 𝛼=1, 𝛼=1.5, 𝛼=2, 𝑏_𝑖=1.2, 𝜏^∗=0.15, ℎ^∗=0.3, and 𝛿=0.3 and 𝑖=1,2.

Therefore, for 𝑎=2 and 𝐹_𝑖(0)=0, by solving the inequality (41), we get 𝑃₁=,𝑆7.2667007.2667₁₍₁₎=,𝑆56.59210.20540.205456.1324₁₍₂₎=,𝑃12.2582−0.1936−0.193611.9901₂=,𝑆7.3794007.3794₂₍₁₎=,𝑆55.5355−0.0809−0.080955.8300₂₍₂₎=,𝑄11.4579−0.5681−0.568113.4627₁=,𝑄17.8905−0.1476−3.04620.0962−0.147618.04340.0962−2.5606−3.04620.096215.91170.10660.0962−2.56060.106616.2315₂=.17.80040.0091−2.64440.14270.009117.59860.1427−2.9014−2.64440.142717.0233−0.09530.1427−2.9014−0.095315.8802 (52) Using (41), we can get the average dwell time 𝜏_𝑎>𝜏^∗_𝑎=2.0889.

5. Conclusion

In this paper, the dynamics of switched Cohen-Grossberg neural networks with average dwell time is investigated. A novel multiple Lyapunov-Krasovskii functional is designed to get new sufficient conditions guaranteeing the uniformly ultimate boundedness, the existence of an attractor, and the globally exponential stability. The derived conditions are expressed in terms of LMIs, which are more relaxed than algebraic formulation and can be easily checked by the effective LMI toolbox in Matlab in practice. Based on the method provided in this paper, stochastic disturbance, impulse, and reaction diffusion for switched systems will be considered in the future works.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant no. 11101053 and 11101435, the Key Project of Chinese Ministry of Education under Grant no. 211118, the Excellent Youth Foundation of Educational Committee of Hunan Provincial no. 10B002, the Hunan Provincial NSF no. 11JJ1001, and National Science and Technology Major Projects of China no. 2012ZX10001001-006, the Scientific Research Fund of Hunan Provincial Education Department of China no. 12C0028.