Asymptotic Behavior of Switched Stochastic Delayed Cellular Neural Networks via Average Dwell Time Method

Abstract

The asymptotic behavior of a class of switched stochastic cellular neural networks (CNNs) with mixed delays (discrete time-varying delays and distributed time-varying delays) is investigated in this paper. Employing the average dwell time approach (ADT), stochastic analysis technology, and linear matrix inequalities technique (LMI), some novel sufficient conditions on the issue of asymptotic behavior (the mean-square ultimate boundedness, the existence of an attractor, and the mean-square exponential stability) are established. A numerical example is provided to illustrate the effectiveness of the proposed results.

1. Introduction

Since Chua and Yang’s seminal work on cellular neural networks (CNNs) in 1988 [1, 2], it has witnessed the successful applications of CNN in various areas such as signal processing, pattern recognition, associative memory, and optimization problems (see, e.g., [3–5]). From a practical point of view, both in biological and man-made neural networks, processing of moving images and pattern recognition problems require the introduction of delay in the signals transmitted among the cells [6, 7]. After the widely use of discrete delays, distributed delays arise because that neural networks usually have a spatial extent due to the presences of a multitude of parallel pathway with a variety of axon sizes and lengths. The mathematical model can be described by the following differential equations: 𝑑𝑥_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗^𝑡_{𝑡−ℎ𝑖(𝑡)}𝑓_𝑗𝑥_𝑗(𝑠)d𝑠+𝐽_𝑖,𝑖=1,…,𝑛, (1) where 𝑡≥0, 𝑛≥2 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; 𝑑_𝑖>0 denotes the rate with which the cell 𝑖 resets its potential to the resting state when isolated from other cells and external inputs; 𝑎_𝑖𝑗, 𝑏_𝑖𝑗, 𝑐_𝑖𝑗 denote the strengths of connectivity between cell 𝑖 and 𝑗 at time 𝑡, respectively; 𝜏_𝑖(𝑡) and ℎ_𝑖(𝑡) correspond to the discrete time-varying delays and distributed time-varying delays, respectively.

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 [8, 9]. Although discrete delays combined with distributed delays can usually provide a good approximation for prime model, most real models are often affected by so many external perturbations which are of great uncertainty. For instance, in electronic implementations, it was realized that stochastic disturbances are mostly inevitable owing to thermal noise. Just as Haykin [10] point out that in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. Consequently, noise is unavoidable and should be taken into consideration in modeling. Moreover, it has been well recognized that a CNN could be stabilized or destabilized by certain stochastic inputs. Therefore, it is of significant importance to consider stochastic effects to the delayed neural networks. One approach to the mathematical incorporation of such effects is to use probabilistic threshold models. However, the previous literatures all focus on the stability of stochastic neural networks with delays [11–14]. Actually, studies on dynamical systems involve not only a discussion of the stability property, but also other dynamic behaviors such as the ultimate boundedness and attractor. However, there are very few results on the ultimate boundedness and attractor for stochastic neural networks [15–17]. Hence, discussing the asymptotic behavior of neural networks with mixed delays is valuable and meaningful.

On the other hand, neural networks often exhibit a special characteristic of network mode switching; that is, a neural network sometimes has finite modes that switch from one to another at different times according to a switching law generated from a switching logic. As an important class of hybrid systems, switched systems arise in many practical processes. In current papers, the analysis of switched systems has drawn considerable attention since they have numerous applications in control of mechanical systems, computer communities, automotive industry, electric power systems and many other fields [18–22]. Most recently, the stability analysis of switched neural systems has been further investigated which was mainly based on Lyapunov functions [23, 24]. It is worth noting that the average dwell time (ADT) approach is an effective method for the switched systems, which avoid the common Lyapunov function and can be adopted to obtain less conservative stability conditions. For instance, based on the average dwell time method, the problems of stability have been discussed for uncertain switched Cohen-Grossberg neural networks with interval time-varying delay and distributed time-varying delay in [25]. In [26], the average dwell time method has been utilized to get some sufficient conditions for the exponential stability and the weighted 𝐿₂ gain for a class of switched systems.

However, it is worth emphasizing that when the activation functions are unbounded in some special applications, the existence of equilibrium point cannot be guaranteed [27]. Therefore, in these circumstances, the discussing of stability of equilibrium point for switched neural networks turns to be unreachable, which motivated us to consider the ultimate boundedness and attractor for the switched neural networks. Unfortunately, as far as we know, the issue of asymptotic behavior of switched systems with mixed time delays has not been investigated yet, let alone studying the asymptotic behavior of switched stochastic systems. Therefore, these researches are challenging and interesting since they integrate the switched hybrid system into the stochastic system and are thus theoretically and practically significant. Notice that the asymptotic behavior of switched stochastic neural networks with mixed delays should be studied intensively.

Motivated by the above analysis, the main purpose of this paper is to get sufficient conditions on the asymptotic behavior (the mean-square ultimate boundedness, the existence of an attractor, and mean-square exponential stability) for the switched stochastic system. This paper is organized as follows. In Section 2, the considered model of switched stochastic CNN with mixed delays is presented. Some necessary assumptions, definitions and lemmas are also given in this section. In Section 3, mean-square ultimate boundedness and attractor for the proposed model are studied. A numerical example is arranged to demonstrate the effectiveness of the theoretical results in Section 4, and we conclude this paper in Section 5.

2. Problem Formulation

In general, a stochastic cellular neural network with mixed delays can be described as follows: 𝑑𝑥(𝑡)=−𝐷𝑥(𝑡)+𝐴𝐹(𝑥(𝑡))+𝐵𝐹(𝑥(𝑡−𝜏(𝑡)))+𝐶^𝑡_{(𝑡)𝑡−ℎ}𝐹(𝑥(𝑠(𝑥(𝑡)((𝑡(𝑡),))d𝑠+𝐽𝑑𝑡+𝐺,𝑥𝑡−𝜏)))𝑑𝑤 (2) where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑅^𝑛, 𝐹(𝑥(𝑡))=(𝑓₁(𝑥₁(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡)))^𝑇, 𝐷=diag(𝑑₁,…,𝑑_𝑛), 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛, 𝐽=(𝐽₁,…,𝐽_𝑛)^𝑇, 𝜏(𝑡)=(𝜏₁(𝑡),…,𝜏_𝑛(𝑡))^𝑇, ℎ(𝑡)=(ℎ₁(𝑡),…,ℎ_𝑛(𝑡))^𝑇, 𝐺(⋅,⋅) is a 𝑛×𝑛 matrix valued function, and 𝑤(𝑡)=(𝑤₁(𝑡),…,𝑤_𝑛(𝑡))^𝑇 is an 𝑛-dimensional Brownian motion defined on a complete probability space (𝛀,ℱ,𝐏) with a natural filtration {ℱ_𝑡}_𝑡≥0 (i.e., ℱ_𝑡=𝜎{𝑤(𝑠)∶0≤𝑠≤𝑡}).

By introducing switching signal into the system (2) and taking a set of neural networks as the individual subsystems, the switched system can be obtained, which is described as 𝑑𝑥(𝑡)=−𝐷_𝜎(𝑡)𝑥(𝑡)+𝐴_𝜎(𝑡)𝐹(𝑥(𝑡))+𝐵_𝜎(𝑡)𝐹(𝑥(𝑡−𝜏(𝑡)))+𝐶_𝜎(𝑡)^𝑡_{(𝑡)𝑡−ℎ}𝐹(𝑥(𝑠))d𝑠+𝐽𝑑𝑡+𝐺_𝜎(𝑡)(𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))𝑑𝑤(𝑡), (3) where 𝜎(𝑡)∶[0,+∞)→𝚺={1,2…𝑚} is the switching signal. At each time instant 𝑡, the index 𝜎(𝑡)∈𝚺 (i.e., 𝜎(𝑡)=𝑖) of the active subsystem means that the 𝑖th subsystem is activated.

For the convenience of discussion, it is necessary to introduce some notations. 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space. 𝑋≤𝑌 (𝑋<𝑌) means that each pair of corresponding elements of 𝑋 and 𝑌 satisfies the inequality “≤(<)”. 𝑋 is especially called a positive (negative) matrix if 𝑋>0 (<0). 𝑋^𝑇 denotes the transpose of any square matrix 𝑋, and the symbol “*” within the matrix represents the symmetric term of the matrix. 𝜆_min(𝑋) means the minimum eigenvalue of matrix 𝑋, and 𝜆_max(𝑋) means the maximum eigenvalue of matrix 𝑋. 𝐼 denotes unit matrix.

Let 𝒞([−𝜏^∗,0],𝑅^𝑛) denote the Banach space of continuous functions which mapping from [−𝜏^∗,0] to 𝑅^𝑛 with is the topology of uniform convergence. For any ‖𝜑‖∈𝒞([−𝜏^∗,0],𝑅^𝑛), we define ‖𝜑‖=max_{1≤𝑖≤𝑛}sup_{𝑡−𝜏∗<𝑠≤𝑡}|𝜑_𝑖(𝑠)|.

The initial conditions for system (3) are given in the form: 𝑥(𝑡)=𝜑,𝜑∈𝒞_ℱ0−𝜏^∗,0,𝑅^𝑛, (4) where 𝒞_ℱ0([−𝜏^∗,0],𝑅^𝑛) is the family of all ℱ₀-measurable bounded 𝒞([−𝜏^∗,0],𝑅^𝑛)-valued random variables.

Throughout this paper, we assume the following assumptions are always satisfied. (𝐻₁) The discrete time-varying delay 𝜏(𝑡) and distributed time-varying delay ℎ(𝑡) are satisfying 0≤𝜏(𝑡)≤𝜏,0≤ℎ(𝑡)≤ℎ,𝜏^∗=max_{1≤𝑖≤𝑛}{𝜏,ℎ}, (5) where 𝜏, ℎ, 𝜏^∗ are scalars.(𝐻₂) There exist constants 𝑙_𝑗 and 𝐿_𝑗, 𝑖=1,2,…,𝑛, such that 𝑙_𝑗≤𝑓_𝑗(𝑥)−𝑓_𝑗𝑦𝑥−𝑦≤𝐿_𝑗,∀𝑥,𝑦∈𝑅,𝑥≠𝑦. (6) Moreover, we define Σ₁=diag𝑙₁𝐿₁,𝑙₂𝐿₂,…,𝑙_𝑛𝐿_𝑛,Σ₂=diag𝑙₁+𝐿₁,𝑙₂+𝐿₂,…,𝑙_𝑛+𝐿_𝑛. (7)(𝐻₃) We assume that 𝐺(𝑡,𝑥,𝑦)∶𝑅⁺×𝑅^𝑛×𝑅^𝑛→𝑅^𝑛×𝑚 is locally Lipschitz continuous and satisfies the following condition: trace𝐺^𝑇𝑡,𝑥,𝑦𝐺𝑡,𝑥,𝑦≤𝑥^𝑇𝑈^𝑇₁𝑈₁𝑥+𝑦^𝑇𝑈^𝑇₂𝑈₂𝑦+2𝑥^𝑇𝑈^𝑇₁𝑈₂𝑦, (8) where 𝑈₁>0, 𝑈₂>0 are constant matrices with appropriate dimensions.

Some definitions and lemmas are introduced as follows.

Definition 1 (see [15]). System (2) is called mean-square ultimate boundedness if there exists a constant vector 𝐵^∼>0, such that, for any initial value 𝜑∈𝒞_ℱ0, there is a 𝑡^′=𝑡^′(𝜑)>0, for all 𝑡≥𝑡^′, the solution 𝑥(𝑡,𝜑) of system (2) satisfies 𝐸𝑥𝑡,𝜑²≤𝐵^∼. (9)
In this case, the set 𝔸={𝜑∈𝒞_ℱ0∣𝐸‖𝜑(𝑠)‖²≤𝐵^∼} is said to be an attractor of system (2) in mean square sense.
Clearly, proposition above equals to lim_𝑡→∞sup𝐸‖𝑥(𝑡)‖²≤𝐵^∼.

Definition 2 (see [28]). For any switching signal 𝜎(𝑡), corresponding a switching sequence {(𝜎(𝑡₀),𝑡₀),…(𝜎(𝑡_𝑘),𝑡_𝑘),…,∣𝑘=0,1,…}, where (𝜎(𝑡_𝑘),𝑡_𝑘) means the 𝜎(𝑡_𝑘)th subsystem, is activated during 𝑡∈[𝑡_𝑘,𝑡_𝑘−1), and 𝑘 denotes the switching ordinal number. Given any finite constants 𝑇₁, 𝑇₂ satisfying 𝑇₂>𝑇₁≥0 denotes the number of discontinuity of a switching signal 𝜎(𝑡) over the time interval (𝑇₁,𝑇₂) by 𝑁_𝜎(𝑇₁,𝑇₂). If 𝑁_𝜎(𝑇₁,𝑇₂)≤𝑁₀+(𝑇₂−𝑇₁)/𝑇_𝛼 holds for 𝑇_𝛼>0, 𝑁₀>0, then 𝑇_𝛼>0 is called the average dwell time. 𝑁₀ is the chatter bound.

Lemma 3. Let 𝑋 and 𝑌 be any 𝑛-dimensional real vectors, 𝑃 be a positive semidefinite matrix and a scalar 𝜀>0. Then the following inequality holds: 2𝑋^𝑇𝑃𝑌≤𝜀𝑋^𝑇𝑃𝑋+𝜀⁻¹𝑌^𝑇𝑃𝑌. (10)

Lemma 4 (see [29]). For any positive definite constant matrix 𝑀∈ℛ^𝑛×𝑛, and a scalar 𝑟, if there exists a vector function 𝜂∶[0,𝑟]→ℛ^𝑛 such that the integrals ∫^𝑟₀𝜂^𝑇(𝑠)𝑀𝜂(𝑠)d𝑠 and ∫^𝑟₀𝜂(𝑠)d𝑠 are well defined, then ^𝑟₀𝜂^𝑇(𝑠)(𝑠)1𝑀𝜂d𝑠≥𝑟^𝑟₀𝜂^𝑇(𝑠)d𝑠𝑀^𝑟₀𝜂(𝑠)d𝑠. (11)

3. Main Results

Let 𝒞^2,1∶(𝑅^𝑛×𝑅⁺;𝑅+) denote the family of all nonnegative functions 𝑉(𝑡,𝑥) on 𝑅^𝑛×𝑅⁺ which are continuously twice differentiable in 𝑥 and once differentiable in 𝑡. If 𝑉∈𝒞^2,1∶(𝑅^𝑛×𝑅⁺;𝑅+), define an operator ℒ𝑉 associated with general stochastic system 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑡)𝑑𝑡+𝐺(𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))𝑑𝑤(𝑡) as ℒ𝑉(𝑡,𝑥)=𝑉_𝑡(𝑡,𝑥)+𝑉_𝑥()𝑓(𝑥(𝑡))+1𝑡,𝑥,𝑡2trace𝐺^𝑇(𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))𝑉_𝑥𝑥()(𝑥(𝑡)((𝑡,𝑡,𝑥×𝐺,𝑥𝑡−𝜏))) (12) where 𝑉_𝑡()=()𝑡,𝑥𝜕𝑉𝑡,𝑥𝜕𝑡,𝑉_𝑥()=()𝑡,𝑥𝜕𝑉𝑡,𝑥𝜕𝑥₁(),…,𝜕𝑉𝑡,𝑥𝜕𝑥_𝑛^𝑇,𝑉_𝑥𝑥()=()𝑡,𝑥𝜕𝑉𝑡,𝑥𝜕𝑥_𝑖𝜕𝑥_{𝑗𝑛×𝑛}. (13)

Theorem 5. If there are constants 𝜇, 𝜈 such that 𝜏̇(𝑡)≤𝜇, ℎ̇(𝑡)≤𝜈, we denote 𝑔(𝜇), 𝑘(𝜈) as: 𝑔𝜇=1−𝜇𝑒^−𝛼𝜏𝑘(𝜈)=()𝑒,𝜇≤1;1−𝜇,𝜇≥1,1−𝜈^−𝛼ℎ,𝜈≤1;1−𝜈,𝜈≥1. (14) For a given constant 𝛼>0, if there exist positive definite matrixes 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛), 𝑄, 𝑅, 𝑆, 𝑍, 𝑈₁, 𝑈₂, 𝑌_𝑖=diag(𝑌_𝑖1,𝑌_𝑖2,…,𝑌_𝑖𝑛),𝑖=1,2, such that the following condition holds: Δ₁=Φ₁₁Φ₁₂Φ₁₃Φ₁₄0Φ₁₆∗Φ₂₂0Φ₂₄00∗∗Φ₃₃000∗∗∗Φ₄₄Φ₅₅0∗∗∗∗0Φ₆₆Φ<0,₁₁=2𝛼𝑃−2𝐷𝑃+𝑄+𝜏²𝑆−2Σ₁𝑌₁+𝛼𝐼+𝑈^𝑇₁𝑃𝑈₁,Φ₁₂=𝑈^𝑇₁𝑃𝑈₂,Φ₁₃=𝑃𝐴+Σ₂𝑌₁,Φ₁₄=𝑃𝐵,Φ₁₆Φ=𝑃𝐶,₂₂=−𝑔𝜇𝑄−2Σ₁𝑌₂+𝛼𝐼+𝑈^𝑇₂𝑃𝑈₂,Φ₂₄=Σ₂𝑌₂,Φ₃₃=𝑅+ℎ²𝑍−2𝑌₁Φ+𝛼𝐼,₄₄=−𝑘(𝜈)𝑅−2𝑌₂Φ+𝛼𝐼,₅₅=−𝑔𝜇𝑆,Φ₆₆=−𝑘(𝜈)𝑍, (15) then system (2) is mean-square ultimate boundedness.

Proof. Consider the positive definite Lyapunov functional as follows: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡)+𝑉₅(𝑡), (16) where 𝑉₁(𝑡)=𝑒^𝛼𝑡𝑥^𝑇(𝑡)(𝑡),𝑉𝑃𝑥₂(𝑡)=^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)𝑄𝑒^𝛼𝑠𝑥(𝑠)𝑉d𝑠,₃(𝑡)=^𝑡_{(𝑡)𝑡−ℎ}𝐹^𝑇(𝑥(𝑠))𝑅𝑒^𝛼𝑠𝐹(𝑥(𝑠𝑉))d𝑠,₄(𝑡)=𝜏⁰_{(𝑡)−𝜏}^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑆𝑒^𝛼𝑠𝑥(𝑠)𝑉d𝑠d𝜃,₅(𝑡)=ℎ⁰_(𝑡)−ℎ^𝑡_𝑡+𝜃𝐹^𝑇(𝑥(𝑠))𝑍𝑒^𝛼𝑠𝐹(𝑥(𝑠))d𝑠d𝜃. (17) Then, by Ito’s formula, the stochastic derivative of 𝑉(𝑥,𝑡) is 𝑑𝑉(𝑥,𝑡)=ℒ𝑉(𝑥,𝑡)𝑑𝑡+𝑉_𝑥(𝑥,𝑡)𝐺(𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))𝑑𝑤(𝑡), (18) the operator ℒ𝑉 along the trajectory of system (2) can be obtained ℒ𝑉₁(𝑡)=𝜕𝑉₁(𝑥(𝑡),𝑡)+𝜕𝑡𝜕𝑉₁(𝑥(𝑡),𝑡)×(𝑡)(𝑥(𝑡(𝑥((𝑡𝜕𝑥−𝐷𝑥+𝐴𝐹))+𝐵𝐹𝑡−𝜏)))+𝐶^𝑡_{(𝑡)𝑡−ℎ}𝐹(𝑥(𝑠+1))d𝑠+𝐽2trace𝐺^𝑇(𝑥(𝑡)((𝑡𝜕,𝑥𝑡−𝜏)))²𝑉₁(𝑥(𝑡),𝑡)𝜕𝑥²×𝐺(𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))=𝛼𝑒^𝛼𝑡𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+2𝑒^𝛼𝑡𝑥^𝑇(𝑡)𝑃×−𝐷𝑥(𝑡)+𝐴𝐹(𝑥(𝑡))+𝐵𝐹(𝑥(𝑡−𝜏(𝑡)))+𝐶^𝑡_{(𝑡)𝑡−ℎ}𝐹(𝑥(𝑠))d𝑠+𝐽+𝑒^𝛼𝑡trace𝐺^𝑇(𝑥(𝑡)((𝑡𝑃(𝑥(𝑡)((𝑡.,𝑥𝑡−𝜏)))×𝐺,𝑥𝑡−𝜏))) (19) From Assumption (𝐻₃), Lemma 3, and (19), we can get ℒ𝑉₁(𝑡)≤2𝛼𝑒^𝛼𝑡𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+2𝑒^𝛼𝑡𝑥^𝑇(𝑡)𝑃×−𝐷𝑥(𝑡)+𝐴𝐹(𝑥(𝑡))+𝐵𝐹(𝑥(𝑡−𝜏(𝑡)))+𝐶^𝑡_{(𝑡)𝑡−ℎ}𝐹(𝑥(𝑠))d𝑠+𝑒^𝛼𝑡𝛼⁻¹𝐽^𝑇𝑃𝐽+𝑒^𝛼𝑡𝑥^𝑇(𝑡)𝑈^𝑇₁𝑃𝑈₁𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏(𝑡))𝑈^𝑇₂𝑃𝑈₂𝑥(𝑡−𝜏(𝑡))+2𝑥^𝑇(𝑡)𝑈^𝑇₁𝑃𝑈₂𝑥(𝑡−𝜏(𝑡)). (20)
Similarly, calculating the operator ℒ𝑉_𝑖 (𝑖=2,3,4,5), along the trajectory of system (2), one can get ℒ𝑉₂=𝑒^𝛼𝑡𝑥^𝑇(𝑡)(𝑡)−(𝜏̇(𝑡𝑒𝑄𝑥1−))^{𝛼(𝑡−𝜏(𝑡))}𝑥^𝑇(𝑡−𝜏(𝑡))𝑄𝑥(𝑡−𝜏(𝑡))≤𝑒^𝛼𝑡𝑥^𝑇(𝑡)(𝑡)−𝑒𝑄𝑥1−𝜇^{𝛼(𝑡−𝜏)}𝑥^𝑇(𝑡−𝜏(𝑡))𝑄𝑥(𝑡−𝜏(𝑡))≤𝑒^𝛼𝑡𝑥^𝑇(𝑡)(𝑡)𝜇𝑒𝑄𝑥−𝑔^𝛼𝑡𝑥^𝑇(𝑡−𝜏(𝑡))𝑄𝑥(𝑡−𝜏(𝑡)),ℒ𝑉₃≤𝑒^𝛼𝑡𝐹^𝑇(𝑥(𝑡(𝑥(𝑡(𝜈)𝑒))𝑅𝐹))−𝑘^𝛼𝑡𝐹^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑅𝐹(𝑥(𝑡−𝜏(𝑡))),ℒ𝑉₄=𝜏𝜏(𝑡)𝑒^𝛼𝑡𝑥^𝑇(𝑡)(𝑡)−(𝜏̇(𝑡𝑒𝑆𝑥1−))^{𝛼(𝑡−𝜏(𝑡))𝑡}_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)𝑆𝑥(𝑠)d𝑠≤𝜏²𝑒^𝛼𝑡𝑥^𝑇(𝑡)(𝑡)𝜇𝑒𝑆𝑥−𝜏𝑔^𝛼𝑡𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)𝑆𝑥(𝑠)d𝑠,ℒ𝑉₅≤ℎ²𝑒^𝛼𝑡𝐹^𝑇(𝑥(𝑡(𝑥(𝑡(𝜈)𝑒))𝑍𝐹))−ℎ𝑘^𝛼𝑡𝑡_{(𝑡)𝑡−ℎ}𝐹^𝑇(𝑥(𝑠))𝑍𝐹(𝑥(𝑠))d𝑠. (21)
According to Lemma 4, the following inequalities can be obtained: ^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)(𝑠)≥1𝑆𝑥d𝑠𝜏^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)d𝑠𝑆^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)d𝑠,_{𝑡(𝑡)𝑡−ℎ}𝐹^𝑇(𝑥(𝑠(𝑥(𝑠≥1))𝑍𝐹))d𝑠ℎ^𝑡_{(𝑡)𝑡−ℎ}𝐹^𝑇(𝑥(𝑠))d𝑠𝑍^𝑡_{(𝑡)𝑡−ℎ}𝐹(𝑥(𝑠))d𝑠. (22) Then, we can get ℒ𝑉₄≤𝜏²𝑒^𝛼𝑡𝑥^𝑇(𝑡)(𝑡)𝜇𝑒𝑆𝑥−𝑔^𝛼𝑡𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)d𝑠𝑆^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)d𝑠,ℒ𝑉₅≤ℎ²𝑒^𝛼𝑡𝐹^𝑇(𝑥(𝑡(𝑥(𝑡(𝜈)𝑒))𝑍𝐹))−𝑘^𝛼𝑡𝑡_{(𝑡)𝑡−ℎ}𝐹^𝑇(𝑥(𝑠))d𝑠𝑍^𝑡_{(𝑡)𝑡−ℎ}𝐹(𝑥(𝑠))d𝑠. (23)
On the other hand, it follows from Assumption (𝐻₂) that we can easily obtain 𝑓_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖(0)−𝐿_𝑖𝑥_𝑖(𝑡×𝑓_𝑖)𝑥_𝑖(𝑡)−𝑓_𝑖(0)−𝑙_𝑖𝑥_𝑖(𝑡𝑓)≤0,_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝑖(0)−𝐿_𝑖𝑥_𝑖((𝑡×𝑓𝑡−𝜏))_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝑖(0)−𝑙_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))≤0,𝑖=1,2,…,𝑛. (24)
Then we obtain 0≤𝛿₁=−2^𝑛_𝑖=1𝑦_1𝑖𝑓_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖(0)−𝐿_𝑖𝑥_𝑖(𝑡×𝑓_𝑖)𝑥_𝑖(𝑡)−𝑓_𝑖(0)−𝑙_𝑖𝑥_𝑖(𝑡),0≤𝛿₂=−2^𝑛_𝑖=1𝑦_2𝑖𝑓_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝑖(0)−𝐿_𝑖𝑥_𝑖((𝑡×𝑓𝑡−𝜏))_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝑖(0)−𝑙_𝑖𝑥_𝑖((𝑡,𝛿𝑡−𝜏))₁=−2^𝑛_𝑖=1𝑦_1𝑖𝑓_𝑖𝑥_𝑖(𝑡)−𝐿_𝑖𝑥_𝑖(𝑡)𝑓_𝑖𝑥_𝑖(𝑡)−𝑙_𝑖𝑥_𝑖(𝑡)−2^𝑛_𝑖=1𝑦_1𝑖𝑓²_𝑖(0)+2^𝑛_𝑖=1𝑦_1𝑖𝑓_𝑖(0)2𝑓_𝑖𝑥_𝑖(𝑡)−𝐿_𝑖+𝑙_𝑖𝑥_𝑖(𝑡)≤−2^𝑛_𝑖=1𝑦_1𝑖𝑓_𝑖𝑥_𝑖(𝑡)−𝐿_𝑖𝑥_𝑖(𝑡)𝑓_𝑖𝑥_𝑖(𝑡)−𝑙_𝑖𝑥_𝑖(𝑡+_𝑛𝑖=1)𝛼𝑓²_𝑖𝑥_𝑖(𝑡)+4𝛼⁻¹𝑓²_𝑖(0)𝑦²_1𝑖+𝛼𝑥²_𝑖(𝑡)+𝛼⁻¹𝑓²_𝑖(0)𝑦²_1𝑖𝐿_𝑖+𝑙_𝑖². (25) Similarly, one can get 𝛿₂≤−2^𝑛_𝑖=1𝑦_2𝑖𝑓_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝐿_𝑖𝑥_𝑖((𝑡×𝑓𝑡−𝜏))_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑙_𝑖𝑥_𝑖((𝑡+𝑡−𝜏))𝛼𝑓²_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))+4𝛼⁻¹𝑓²_𝑖(0)𝑦²_2𝑖+𝛼𝑥²_𝑖(𝑡−𝜏(𝑡))+𝛼⁻¹𝑓²_𝑖(0)𝑦²_2𝑖𝐿_𝑖+𝑙_𝑖². (26)
Denote 𝜁(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏(𝑡)),𝐹^𝑇(𝑥(𝑡,𝐹))_𝑇(𝑥(𝑡−𝜏(𝑡))),^𝑡_{(𝑡)𝑡−𝜏}𝑥(𝑠)d𝑠^𝑇,_{𝑡(𝑡)𝑡−ℎ}𝐹(𝑥(𝑠))d𝑠^𝑇𝑇, (27) and combing with (16)–(26), we can get 𝑑𝑉=ℒ𝑉₁𝑑𝑡+ℒ𝑉₂𝑑𝑡+ℒ𝑉₃𝑑𝑡+ℒ𝑉₄𝑑𝑡+ℒ𝑉₅𝑑𝑡+2𝑃𝑒^𝛼𝑡𝑥^𝑇(𝑡)𝐺(𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))𝑑𝑤(𝑡)≤𝑒^𝛼𝑡𝜁^𝑇(𝑡)Δ₁𝜁(𝑡)𝑑𝑡+𝑒^𝛼𝑡𝒩₁𝑑𝑡+2𝑃𝑒^𝛼𝑡𝑥(𝑡)𝐺(𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))𝑑𝑤(𝑡), (28) where 𝒩₁=𝛼⁻¹𝐽^𝑇+𝑃𝐽_𝑛𝑖=14𝛼⁻¹𝑓²_𝑖(0)𝑦²_2𝑖+𝛼⁻¹𝑓²_𝑖(0)𝑦²_1𝑖𝐿_𝑖+𝑙_𝑖²+4𝛼⁻¹𝑓²_𝑖(0)𝑦²_2𝑖+𝛼⁻¹𝑓²_𝑖(0)𝑦²_2𝑖𝐿_𝑖+𝑙_𝑖². (29)
By integrating both sides of (28) in time interval 𝑡∈[𝑡₀,𝑡] and then taking expectation results in 𝐾𝑒^𝛼𝑡‖𝑥(𝑡)‖²≤𝑉(𝑥(𝑡))≤𝑉𝑥𝑡₀+𝛼⁻¹𝑒^𝛼𝑡𝒩₁+^𝑡_𝑡02𝑃𝑒^𝛼𝑡𝑥(𝑠)𝐺(𝑥(𝑠),𝑥(𝑠−𝜏(𝑠)))d𝑤(𝑠), (30) where 𝐾=𝜆_min(𝑃).
Therefore, one obtains 𝐸{𝑉(𝑥(𝑡))}≤𝐸𝑉𝑥𝑡₀+𝐸𝛼⁻¹𝑒^𝛼𝑡𝒩₁, (31) which implies 𝐸‖𝑥(𝑡)‖²≤𝑒^−𝛼𝑡𝐸𝑉𝑥𝑡₀+𝛼⁻¹𝒩₁𝐾. (32) If one chooses 𝐵^∼=(1+𝛼⁻¹𝒩₁)/𝐾>0, then, for initial value 𝜑∈𝒞_ℱ0, there is 𝑡^′=𝑡^′(𝜑)>0, such that 𝑒^−𝛼𝑡𝐸{𝑉(𝑥(𝑡₀))}≤1 for all 𝑡≥𝑡^′. According to Definition 1, we have 𝐸‖𝑥(𝑡,𝜑)‖²≤𝐵^∼ for all 𝑡≥𝑡^′. That is to say, system (2) is mean-square ultimate boundedness. This completes the proof.

Theorem 6. If all of the conditions of Theorem 5 hold, then there exists an attractor 𝔸^∼_𝐵={𝜑∈𝒞_ℱ0∣𝐸‖𝜑(𝑠)‖²≤𝐵^∼} for the solutions of system (2).

Proof. If one chooses 𝐵^∼=(1+𝛼⁻¹𝒩₁)/𝐾>0, Theorem 5 shows that, for any 𝜑, there is 𝑡^′>0, such that 𝐸‖𝑥(𝑡,𝜑)‖²≤𝐵^∼ for all 𝑡≥𝑡^′. Let 𝔸^∼_𝐵 denote by 𝔸^∼_𝐵={𝜑∈𝒞_ℱ0∣𝐸‖𝜑(𝑠)‖²≤𝐵^∼}. Clearly, 𝔸^∼_𝐵 is closed, bounded, and invariant. Furthermore, lim_𝑡→∞supinf_{𝑦∈𝔸∼𝐵}‖𝑥(𝑡,𝜑)−𝑦‖=0. Therefore, 𝔸^∼_𝐵 is an attractor for the solutions of system (2). This completes the proof.

Corollary 7. In addition to that all of the conditions of Theorem 5 hold, if 𝐽=0, 𝐺(𝑡,0,0)=0, and 𝑓_𝑖(0)=0 for all 𝑖=1,2,…,𝑛, then system (2) has a trivial solution 𝑥(𝑡)≡0, and the trivial solution of system (2) is mean-square exponentially stable.

Proof. If 𝐽=0 and 𝑓_𝑖(0)=0(𝑖=1,2,…,𝑛), then 𝒩₁=0, and it is obvious that system (2) has a trivial solution 𝑥(𝑡)≡0. From Theorem 5, one has 𝐸𝑥𝑡,𝜑²≤𝐾^∗𝑒^−𝛼𝑡,∀𝜑, (33) where 𝐾^∗=𝐸{𝑉(𝑥(𝑡₀))}/𝐾. Therefore, the trivial solution of system (2) is mean-square exponentially stable. This completes the proof.

According to Theorem 5–Corollary 7, we will present conditions of mean-square ultimate boundedness for the switched systems (3) by applying the average dwell time method in the follow-up studies.

Theorem 8. If there are constants 𝜇, 𝜈 such that 𝜏̇(𝑡)≤𝜇, ℎ̇(𝑡)≤𝜈, we denote 𝑔(𝜇), 𝑘(𝜈) as 𝑔𝜇=1−𝜇𝑒^−𝛼𝜏𝑘(𝜈)=()𝑒,𝜇≤1;1−𝜇,𝜇≥1,1−𝜈^−𝛼ℎ,𝜈≤1;1−𝜈,𝜈≥1. (34) For a given constant 𝛼>0, if there exist positive definite matrixs 𝑄_𝑖,𝑅_𝑖, 𝑆_𝑖, 𝑍_𝑖, 𝑈_1𝑖, 𝑈_2𝑖, 𝑃_𝑖=diag(𝑝_𝑖1,𝑝_𝑖2,…,𝑝_𝑖𝑛), 𝑌_𝑖=diag(𝑌_𝑖1,𝑌_𝑖2,…,𝑌_𝑖𝑛),𝑖=1,2, such that the following condition holds Δ_𝑖1=Φ_𝑖11Φ_𝑖12Φ_𝑖13Φ_𝑖140Φ_𝑖16∗Φ_𝑖220Φ_𝑖2400∗∗Φ_𝑖33000∗∗∗Φ_𝑖44Φ_𝑖550∗∗∗∗0Φ_𝑖66<0, (35) where Φ_𝑖11=2𝛼𝑃_𝑖−2𝐷𝑃_𝑖+𝑄_𝑖+𝜏²𝑆_𝑖−2Σ₁𝑌₁+𝛼𝐼+𝑈^𝑇_1𝑖𝑃𝑈_1𝑖,Φ_𝑖12=𝑈^𝑇_1𝑖𝑃𝑈_2𝑖,Φ_𝑖13=𝑃_𝑖𝐴_𝑖+Σ₂𝑌₁,Φ_𝑖14=𝑃_𝑖𝐵_𝑖,Φ₁₆=𝑃_𝑖𝐶_𝑖,Φ_𝑖22=−𝑔𝜇𝑄_𝑖−2Σ₁𝑌₂+𝛼𝐼+𝑈^𝑇_2𝑖𝑃𝑈_2𝑖,Φ_𝑖24=Σ₂𝑌₂,Φ_𝑖33=𝑅_𝑖+ℎ²𝑍_𝑖−2𝑌₁Φ+𝛼𝐼,_𝑖44=−𝑘(𝜈)𝑅_𝑖−2𝑌₂+𝛼𝐼,Φ_𝑖55=−𝑔𝜇𝑆_𝑖,Φ_𝑖66=−𝑘(𝜈)𝑍_𝑖. (36) Then system (3) is mean-square ultimate boundedness for any switching signal with average dwell time satisfying 𝑇_𝛼>𝑇^∗_𝛼=lnℛ_max𝛼, (37) where ℛ_max=max_{𝑘∈𝚺,1≤𝑖≤𝑛}{ℛ_𝑖𝑘}.

Proof. Define the Lyapunov functional candidate 𝑉_𝜎(𝑡)=𝑒^𝛼𝑡𝑥^𝑇(𝑡)𝑃_𝜎(𝑡)𝑥(𝑡)+^𝑡_{(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)𝑄_𝜎(𝑡)𝑒^𝛼𝑠𝑥(𝑠)+d𝑠_{𝑡(𝑡)𝑡−ℎ}𝐹^𝑇(𝑥(𝑠))𝑅_𝜎(𝑡)𝑒^𝛼𝑠𝐹(𝑥(𝑠))d𝑠+𝜏⁰_{(𝑡)−𝜏}^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑆_𝜎(𝑡)𝑒^𝛼𝑠𝑥(𝑠)d𝑠d𝜃+ℎ⁰_(𝑡)−ℎ^𝑡_𝑡+𝜃𝐹^𝑇(𝑥(𝑠))𝑍_𝜎(𝑡)𝑒^𝛼𝑠𝐹(𝑥(𝑠))d𝑠d𝜃. (38)
From (16) and (32), we have the following result: 𝐸‖𝑥(𝑡)‖²≤ℛ₀𝐸𝑥𝑡₀²𝑒^{−𝛼(𝑡−𝑡0)}𝐾+Λ𝐾, (39) where Λ=𝛼⁻¹𝒩₁, ℛ₀ is a positive constant.
When 𝑡∈[𝑡_𝑘,𝑡_𝑘+1], the 𝑖_𝑘th subsystem is activated; from (39) and Theorem 5,we can get 𝐸‖𝑥(𝑡)‖²≤ℛ_𝑖𝑘𝐸𝑥𝑡_𝑘²𝑒^{−𝛼(𝑡−𝑡𝑘)}𝐾_𝑖𝑘+Λ𝐾_𝑖𝑘=𝐻_𝑖𝑘𝐸𝑥𝑡_𝑘²𝑒^{−𝛼(𝑡−𝑡𝑘)}+𝐽_𝑖𝑘, (40) where ℛ_𝑖𝑘 is a positive constant, 𝐾_𝑖𝑘=𝜆_min(𝑃_𝑖), 𝐻_𝑖𝑘=ℛ_𝑖𝑘/𝐾_𝑖𝑘, 𝐽_𝑖𝑘=Λ/𝐾_𝑖𝑘.
Since the system state is continuous, it follows from (40) that 𝐸‖𝑥(𝑡)‖²≤ℛ_𝑖𝑘𝑥𝑡_𝑘²𝑒^{−𝛼(𝑡−𝑡𝑘)}𝐾_𝑖𝑘+Λ𝐾_𝑖𝑘=𝐻_𝑖𝑘𝐸𝑥𝑡_𝑘²𝑒^{−𝛼(𝑡−𝑡𝑘)}+𝐽_𝑖𝑘≤⋯≤𝑒^{∑𝑘𝑣=0ln𝐻𝑖𝑣−𝛼(𝑡−𝑡0)}𝐸𝑥𝑡₀²+𝐻_𝑖𝑘𝑒^{−𝛼(𝑡−𝑡𝑘)}𝐽_𝑖𝑘+𝐻_𝑖𝑘𝐻_𝑖𝑘−1𝑒^{−𝛼(𝑡−𝑡𝑘−1)}𝐽_𝑖𝑘−1+𝐻_𝑖𝑘𝐻_𝑖𝑘−1𝐻_𝑖𝑘−2𝑒^{−𝛼(𝑡−𝑡𝑘−2)}𝐽_𝑖𝑘−2++⋯𝐻_𝑖𝑘𝐻_𝑖𝑘−1𝐻_𝑖𝑘−2⋯𝐻_𝑖1𝑒^{−𝛼(𝑡−𝑡1)}𝐽_𝑖1+𝐽_𝑖𝑘≤𝑒^{(𝑘+1)ln𝐻max−𝛼(𝑡−𝑡0)}𝐸𝑥𝑡₀²+𝐻^𝑘_max𝐽_max+𝐻^𝑘−1_max𝐽_max+𝐻^𝑘−2_max𝐽_max+⋯+𝐻²_max𝐽_max+𝐻_max𝐽_max+𝐽_max≤𝐻_max𝑒^{𝑘ln𝐻max−𝛼(𝑡−𝑡0)}𝐸𝑥𝑡₀²+𝐽_max𝐻_max−1𝐻^𝑘+1_max≤−1𝐻_max𝑒^{ln𝐻max𝑁𝜎(𝑡0,𝑡)−𝛼(𝑡−𝑡0)}𝐸𝑥𝑡₀²+𝐽_max𝐻_max−1𝐻^𝑘+1_max≤ℛ−1_max𝑒^{𝑁0lnℛmax−(𝛼−(lnℛmax/𝑇𝛼))(𝑡−𝑡0)}𝐾^𝑘+1_min𝐸𝑥𝑡₀²+Λℛ^𝑛+1_max/𝐾^𝑛+1_min−1ℛ_max−𝐾_min, (41) where 𝐾_min=min_𝑖𝑘{𝐾_𝑖𝑘}, 𝐻_max=max_𝑖𝑘{𝐻_𝑖𝑘}.
If one chooses 𝐵^∼=(1/𝐾_min)+Λ[(ℛ^𝑛+1_max/𝐾^𝑛+1_min)−1]/(ℛ_max−𝐾_min)>0, then, for initial value 𝜑∈𝒞_ℱ0, there is 𝑡^′=𝑡^′(𝜑)>0, such that ℛ_max𝑒^{𝑁0lnℛmax−(𝛼−(lnℛmax/𝑇𝛼))(𝑡−𝑡0)}𝐸‖𝑥(𝑡₀)‖²≤1 for all 𝑡≥𝑡^′. According to Definition 1, we have 𝐸‖𝑥(𝑡,𝜑)‖²≤𝐵^∼ for all 𝑡≥𝑡^′. That is to say, system (3) is mean-square ultimate boundedness, and the proof is completed.

Remark 9. In this paper, we construct two piecewise functions 𝑔(𝜇), 𝑘(𝜈) to remove the restrictive condition 𝜇<1 and 𝜈<1 in the results, which have reduced the conservatism of the obtained results and also avoid the computational complexity.

Remark 10. The condition (35) is given as in the form of linear matrix inequalities, which are more relaxing than the algebraic formulation. Furthermore, by using the MATLAB LMI toolbox, we can check the feasibility of (35) straightforward without tuning any parameters.

Theorem 11. If all of the conditions of Theorem 8 hold, then there exists an attractor 𝔸^′∼_𝐵 for the solutions of system (3), where 𝔸^′∼_𝐵={𝜑∈𝒞_ℱ0∣𝐸‖𝜑(𝑠)‖²≤𝐵^∼}.

Proof. If one chooses 𝐵^∼=(1/𝐾_min)+Λ[(ℛ^𝑛+1_max/𝐾^𝑛+1_min)−1]/(ℛ_max−𝐾_min)>0, Theorem 8 shows that, for any 𝜑, there is 𝑡^′>0, such that 𝐸‖𝑥(𝑡,𝜑)‖²≤𝐵^∼ for all 𝑡≥𝑡^′. Let 𝔸^′∼_𝐵 denote by 𝔸^′∼_𝐵={𝜑∈𝒞_ℱ0∣𝐸‖𝜑(𝑠)‖²≤𝐵^∼}. Clearly, 𝔸^′∼_𝐵 is closed, bounded, and invariant. Furthermore, lim_𝑡→∞supinf_{𝑦∈𝔸′∼𝐵}‖𝑥(𝑡,𝜑)−𝑦‖=0. Therefore, 𝔸^′∼_𝐵 is an attractor for the solutions of system (3). This completes the proof.

Corollary 12. In addition to all that of the conditions of Theorem 8 hold, if 𝐽=0, 𝐺(𝑡,0,0)=0 and 𝑓_𝑖(0)=0 for all 𝑖=1,2,…,𝑛, then system (3) has a trivial solution 𝑥(𝑡)≡0, and the trivial solution of system (3) is mean-square exponentially stable.

Proof. If 𝐽=0 and 𝑓_𝑖(0)=0 for all 𝑖=1,2,…,𝑛, then it is obvious that system (3) has a trivial solution 𝑥(𝑡)≡0. From Theorem 8, one has 𝐸𝑥𝑡,𝜑²≤𝐾^∼∗𝑒^−𝛼𝑡,∀𝜑, (42) where 𝐾^∼∗=(ℛ_max𝑒^{𝑁0lnℛmax−(𝛼−(lnℛmax/𝑇𝛼))(𝑡−𝑡0)}𝐸‖𝑥(𝑡₀)‖²/𝐾^𝑘+1_min. Therefore, the trivial solution of system (3) is mean-square exponentially stable. This completes the proof.

Remark 13. Assumption (𝐻₃) is less conservative than that in [17] since the constants 𝑙_𝑗 and 𝐿_𝑗 are allowed to be positive, negative, or zero. Hence, the resulting activation functions 𝑓(⋅) could be nonmonotonic and are more general than the usual forms |𝑓_𝑗(𝑢)|≤𝐾_𝑗|𝑢|, 𝐾_𝑗>0, 𝑗=1,2,…,𝑛. Moreover, unlike the bounded case, there will be no equilibrium point for the switched system (3) under the assumption (𝐻₃). For this reason, to investigate the asymptotic behavior (the ultimate boundedness and the existence of attractor) of switched system that contains mixed delays is more complex and challenge.

Remark 14. In this paper, the chatter bound 𝑁₀ is a positive integer, which is more practical in significance and can include the model 𝑁₀=0 in [16, 25, 26] as a special case.

Remark 15. If 𝚺=0, which implies that the switched delay system (3) reduces to the usual stochastic CNN with delays. In this case, attractor and ultimate boundedness are discussed in [17]. And when 𝑈₁=𝑈₂=0, the model in our paper turns out to be a switched CNN with mixed delays; to the best of our knowledge, there are no published results in this aspect yet. Thus, the main results of this paper are novel. Moreover, when uncertainties appear in the switched stochastic CNN system (3), we can obtain the corresponding results, by applying the similar method as in [25].

4. Illustrative Examples

In this section, we shall give a numerical example to demonstrate the validity and effectiveness of our results. Consider the switched cellular neural networks with two subsystems.

Consider the switched stochastic cellular neural network system (3) with 𝑓_𝑖(𝑥_𝑖(𝑡))=0.5tanh(𝑥_𝑖(𝑡)),𝑓_𝑖(0)=0 (𝑖=1,2), 𝜏(𝑡)=0.25sin²(𝑡), ℎ(𝑡)=0.3sin²(𝑡), and the connection weight matrices as follows: 𝐴₁=0.30.10.20.2,𝐵₁=,𝐶0.200.30.5₁=0.2−0.10.30.1,𝑈₁₁=,𝑈0.10−0.10.2₂₁=0.20.100.1,𝐴₂=,𝐵0.20.40.10.3₂=0.10−0.10.2,𝐶₂=,𝑈0.30.20.10.2₁₂=0.20.100.3,𝑈₂₂=.0.100.20.1 (43)

From assumptions (𝐻₁)--(𝐻₃), we can gain 𝑑_𝑖=1,𝑙_𝑖=0,𝐿_𝑖=0.5,(𝑖=1,2), 𝜏=0.25,ℎ=0.3, and 𝜇=0.5,𝜈=0.6.

Therefore, for 𝛼=0.5, by solving LMIs (35), we get 𝑃₁=1.4968001.4851,𝑄₁=,𝑅1.6073−0.0528−0.05281.4567₁=1.86420.46980.46981.5241,𝑆₁=,𝑍2.74670.02250.02251.9941₁=5.43730.06440.06444.5969,𝑃₂=,𝑄1.4316001.4528₂=1.65410.02290.02291.8391,𝑅₂=,𝑆1.08370.45400.45401.2710₂=1.68880.43560.43561.6165,𝑍₂=.4.57360.56980.56984.4524 (44)

Using (37), we can get the average dwell time 𝑇^∗_𝑎=1.3445.

5. Conclusions

In this paper, we studied the switched stochastic cellular neural networks with discrete time-varying delays and distributed time-varying delays. With the help of the average dwell time approach, the novel multiple Lyapunov-Krasovkii functionals methods, and some inequality techniques, we obtain the new sufficient conditions guaranteeing the mean-square ultimate boundedness, the existence of an attractor, and the mean-square exponential stability. A numerical example is also given to demonstrate our results. Furthermore, our derived conditions are presented in the forms of LMIs, which are more relaxing than the algebraic formulation and can be easily checked in practice by the effective LMI toolbox in MATLAB.

Acknowledgments

The authors are extremely grateful to Prof. Zhichun Yang and the anonymous reviewers for their constructive and valuable comments, which have contributed much to the improvement of this paper. This work was jointly supported by the National Natural Science Foundation of China under Grant no. 11101053, 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, National Science and technology Major Projects of China no. 2012ZX10001001-006, the Scientific Research Funds of Hunan Provincial Science and Technology Department of China no. 2012SK3096.