Stability in Switched Cohen-Grossberg Neural Networks with Mixed Time Delays and Non-Lipschitz Activation Functions

Abstract

The stability for the switched Cohen-Grossberg neural networks with mixed time delays and 𝛼-inverse Hölder activation functions is investigated under the switching rule with the average dwell time property. By applying multiple Lyapunov-Krasovskii functional approach and linear matrix inequality (LMI) technique, a delay-dependent sufficient criterion is achieved to ensure such switched neural networks to be globally exponentially stable in terms of LMIs, and the exponential decay estimation is explicitly developed for the states too. Two illustrative examples are given to demonstrate the validity of the theoretical results.

1. Introduction

In the past few decades, there has been increasing interest in different classes of neural networks such as Hopfield, Cellular, Cohen-Grossberg, and bidirectional associative neural networks due to their potential applications in many areas such as classification, signal and image processing, parallel computing, associate memories, optimization, and cryptography [1–5]. In the design of practical neural networks, the qualitative analysis of neural network dynamics plays an important role. To solve problems of optimization, neural control, and signal processing, neural networks have to be designed in such a way that, for a given external input, they exhibit only one globally asymptotically/exponentially stable equilibrium point. Hence, much effort has been made in the stability of neural networks, and a number of sufficient conditions have been proposed to guarantee the global asymptotic/exponential stability for neural networks with or without delays in recent years, see, for example, [6–26] and the references therein.

Recently, by combing the theories of the switched systems and neural networks, several classes of mathematics models of switched neural networks have been established. As a special class of switched systems, 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 [27–30].

Stability issues of switched neural networks have received great attention of researchers so far [31–39]. In [31], 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 [32], 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 was established, and the robust stability for such switched Cohen-Grossberg neural networks was analyzed. In [33], 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. In [34], 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 employing free weighting matrices and LMIs. In [35], a new class of switched interval neural networks with discrete and distributed time-varying delays of neural type was developed, and a delay-dependent sufficient criterion was also obtained in terms of LMIs which guarantee the global robust exponential stability for the proposed switched interval neural networks. It should be pointed out that results in [31–35] focused on the stability of switched neural networks under arbitrary switching rule by using common Lyapunov function method. However, common Lyapunov function method requires all the subsystems of the switched system to share a positive definite radially unbounded common Lyapunov function. Generally, this requirement is difficult to achieve.

In past few years, much attention has been paid to making use of the dwell time approach to deal with the analysis and synthesis of switched neural networks [36–39]. It should be pointed out that the average dwell time method is regarded as an important and attractive method to find a suitable switching signal to guarantee switched system stability or improve other performance and has been widely applied to investigate the analysis and synthesis for switched system with or without time-delay, see for example, [40–44]. Very recently, in [36], 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. In [37], by using the average dwell time method, the delay-dependent sufficient conditions were derived towards the robust exponential stability for a class of discrete-time switched Hopfield neural networks with time delay. In [38], by applying a new Lyapunov-Krasovskii functional and the average dwell time method, a delay-range-dependent exponential stability criteria and decay estimation are presented in terms of LMIs for switched Hopfield neural networks.

It should be noted that, all the results reported in [31–39] are concerned with switched neural networks with Lipschitz activation functions. To the best of our knowledge, very little attention has been paid to the problem of delay-dependent stability for switched neural networks without Lipschitz activation functions, which motivates the work of this paper.

In this paper, our aim is to study the delay-dependent exponential stability problem for a class of switched Cohen-Grossberg neural networks with mixed time delays and 𝛼-inverse Hölder activation functions. Here, it should be pointed out that 𝛼-inverse Hölder activation functions are a class of non-Lipschitz functions. By applying Brouwer degree properties, LMI technique and constructing a novel Lyapunov-Krasovskii functional, the existence, uniqueness and global exponential stability of equilibrium point are proved for Cohen-Grossberg neural networks with mixed time delays and 𝛼-inverse Hölder activation functions. By means of the multiple Lyapunov-Krasovskii functional and the average dwell time approach, a delay-dependent sufficient condition in terms of LMIs is presented to ensure to the considered switched neural networks to be globally exponentially stable, and a explicit expression for the exponential decay estimation is also obtained for the states. Two illustrative examples are given to demonstrate the validity of the theoretical results.

The rest of this paper is organized as follows. In Section 2, the model formulation and some preliminaries are given. In Section 3, the existence, uniqueness, and global exponential stability of equilibrium point are proved for Cohen-Grossberg neural networks with mixed time delays and 𝛼-inverse Hölder activation functions. In Section 4, the global exponential stability criterion and state decay estimation are presented for the switched Cohen-Grossberg neural networks with mixed time delays and 𝛼-inverse Hölder activation functions. In Section 5, two numerical examples are presented to demonstrate the validity of the proposed results. Some conclusions are made in Section 6.

Notations 1. Throughout this paper, 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⁻¹ denotes the inverse of 𝐴. If 𝐴 is a real symmetric matrix, 𝐴>0(𝐴<0) means that 𝐴 is positive definite (negative definite). Given the column vectors 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇,𝑦=(𝑦₁,…,𝑦_𝑛)^𝑇∈𝑅^𝑛, 𝑥^𝑇𝑦=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖. ‖𝑥‖=(∑^𝑛_𝑖=1𝑥²_𝑖)^1/2. 𝑥̇(𝑡) denotes the derivative of 𝑥(𝑡), ∗ represents the symmetric form of matrix.

2. Neural Network Model and Preliminaries

The Cohen-Grossberg neural networks with mixed time delays can be described by the following differential equation system 𝑥̇(𝑡)=𝛼(𝑥(𝑡))−𝛽(𝑥(𝑡))+𝑊₀𝑔(𝑥(𝑡))+𝑊₁𝑔(𝑥(𝑡−𝜏))+𝑊₂^𝑡_𝑡−𝜏𝑔(𝑥(𝑠))𝑑𝑠+𝐽, (2.1) where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is the vector of neuron states at time 𝑡; 𝛼(𝑥(𝑡))=diag(𝛼₁(𝑥₁(𝑡)),…,𝛼_𝑛(𝑥_𝑛(𝑡))) represents the amplification function; 𝛽(𝑥(𝑡))=diag(𝛽₁(𝑥₁(𝑡)),…,𝛽_𝑛(𝑥_𝑛(𝑡))) is the behaved function; 𝑔(𝑥(𝑡))=(𝑔₁(𝑥₁(𝑡)),…,𝑔_𝑛(𝑥_𝑛(𝑡)))^𝑇 is called the neuron activation function; 𝑊_𝑖,𝑖=0,1,2, are the connection weight matrices; 𝜏>0 denotes the discrete and distributed time delay; 𝐽=(𝐽₁,…,𝐽_𝑛)^𝑇 denotes the external input. The initial value associated with (2.1) is assumed to be 𝑥(𝑠)=𝜑(𝑠), and 𝜑(𝑠) is a continuous function on [−𝜏,0].

In the following, some definitions and lemmas, which play important roles in the proof of our theorems below, are introduced.

Definition 2.1. The equilibrium point 𝑥^∗ of the neural networks (2.1) is said to be globally exponentially stable, if there exist scalars𝜂>0, 𝒯>0, and 𝛿>0,such that ‖𝑥(𝑡)−𝑥^∗‖≤𝜂𝑒^−𝛿𝑡,𝑡≥𝒯, (2.2) where 𝑥(𝑡) is the solution of the system (2.1) with the initial value 𝑥(𝑠)=𝜑(𝑠), and 𝑠∈[−𝜏,0], 𝛿 is called the exponential convergence rate.

Definition 2.2 (see [23, 24]). A continuous function𝒢∶𝑅→𝑅,is said to be an𝛼-inverse Hölder function, if(i)𝒢 is a monotonic nondecreasing function,(ii)for any 𝜌∈𝑅, there exist constants 𝑞_𝜌>0 and 𝑟_𝜌>0 which are correlated with 𝜌, satisfying 𝒢(𝜃)−𝒢𝜌≥𝑞_𝜌𝜃−𝜌^𝛼,∀𝜃−𝜌≤𝑟_𝜌, (2.3) where 𝛼>0 is a constant.

The class of 𝛼-inverse Hölder functions is denoted by ℐℋ(𝛼). There are a great number of functions which belong to ℐℋ(𝛼). For example, 𝒢(𝜃)=arctan𝜃,𝑔(𝜃)=𝜃³+𝜃∈ℐℋ(1), and 𝒢(𝜃)=𝜃³∈ℐℋ(3).

Remark 2.3. If a continuous function 𝒢∶𝑅→𝑅 satisfies that 𝒢(𝜃)−𝒢𝜌≤ℓ𝜃−𝜌,∀𝜃,𝜌∈𝑅, (2.4) where ℓ>0 is a constant, then 𝒢 is said to be a Lipschitz-continuous function. When 𝛼=1 and 𝑞_𝜌 is independent on 𝜌, 1-inverse Hölder functions are called inverse Lipschitz functions. It is easy to see that 𝛼-inverse Hölder functions are a class of non-Lipschitz functions.

Let function ℱ∶𝑅^𝑛→𝑅^𝑛 be locally Lipschitz continuous. According to Rademacher's theorem [45], ℱ is differentiable almost everywhere. Let 𝐷_ℱ denote the set of those points where ℱ is differentiable, then, ℱ̇(𝑥) is the Jacobian of ℱ at 𝑥∈𝐷_ℱ and the set 𝐷_ℱ is dense in 𝑅^𝑛. The generalized Jacobian of a locally Lipschitz function is defined as follows.

Definition 2.4. For any𝑥∈𝑅^𝑛, the generalized Jacobian𝜕ℱ(𝑥)of a locally Lipschitz continuous  functionℱ∶𝑅^𝑛→𝑅^𝑛is a set of matrices defined by 𝜕ℱ(𝑥)=co𝒲∣thereexistsasequence𝑥^𝑘⊂𝐷_ℱwithlim_{𝑥𝑘→𝑥}ℱ̇𝑥^𝑘=𝒲, (2.5) where co(⋅) denotes the convex hull of a set.

The generalized Jacobian is a natural generalization of the Jacobian for continuously differentiable functions, at those points 𝑥, where ℱ is continuously differentiable, 𝜕ℱ(𝑥) reduces to a single matrix which is Jacobian ℱ̇(𝑥) of ℱ.

Definition 2.5. For any switching signal𝜎(𝑡)and any finite constants𝑇₁,𝑇₂satisfying𝑇₂>𝑇₁≥0, let𝑁_𝜎(𝑇₁,𝑇₂)denote the switching times on the time interval(𝑇₁,𝑇₂). If 𝑁_𝜎(𝑇₁,𝑇₂)≤𝑁₀+(𝑇₂−𝑇₁)/𝑇_𝑎holds for𝑇_𝑎>0,𝑁₀≥0, then𝑇_𝑎is said to be the average dwell time.

Lemma 2.6 (see [23, 24]). If 𝒢(𝜃)∈ℐℋ(𝛼), then for any 𝜌₀∈𝑅, one has _𝜌^+∞₀𝒢(𝜃)−𝒢𝜌₀𝑑𝜃=_𝜌^−∞₀𝒢(𝜃)−𝒢𝜌₀𝑑𝜃=+∞. (2.6)

Lemma 2.7 (see [23, 24]). If 𝒢(𝜃)∈ℐℋ(𝛼) and 𝒢(0)=0, then there exist constants 𝑞₀>0 and 𝑟₀>0, such that |𝒢(𝜃)|≥𝑞₀|𝜃|^𝛼,∀|𝜃|≤𝑟₀. (2.7) Moreover, |𝒢(𝜃)|≥𝑞₀𝑟^𝛼₀,∀|𝜃|≥𝑟₀. (2.8)

Lemma 2.8 (see [25]). Let ℱ∶𝑅^𝑛→𝑅^𝑛 be locally Lipschitz continuous. For any given 𝑥,𝑦∈𝑅^𝑛, there exists an element 𝒲 in the union ∪_{𝑧∈[𝑥,𝑦]𝜕ℱ(𝑧)} such that ℱ𝑦−ℱ(𝑥)=𝒲𝑦−𝑥, (2.9) where [𝑥,𝑦] denotes the segment connecting 𝑥 and 𝑦.

Let Ω be a nonempty, bounded, and open subset of 𝑅^𝑛. The closure and boundary of Ω are denoted by Ω and 𝜕Ω, respectively.

Lemma 2.9 (see [46]). (1) Let ℋ∶[0,1]×Ω→𝑅^𝑛 be a continuous mapping. If 𝑝∈ℋ(𝜆,𝜕Ω)forall𝜆∈[0,1], then Brouwer degree deg(ℋ(𝜆,⋅),Ω,𝑝) is constant (∀𝜆∈[0,1]). In this case, one has deg(ℋ(0,⋅),Ω,𝑝)=deg(ℋ(1,⋅),Ω,𝑝).
(2) Let ℌ∶Ω→𝑅^𝑛 be a continuous mapping. If deg(ℌ,Ω,𝑝)≠0, then the equation ℌ(𝑥)=𝑝 has at least a solution in Ω.

Lemma 2.10. Let 𝑥,𝑦∈𝑅^𝑛 and 𝐺>0, then 2𝑥^𝑇𝑦≤𝑥^𝑇𝐺𝑥+𝑦^𝑇𝐺⁻¹𝑦. (2.10)

Lemma 2.11 (Schur complement). Given constant symmetric matrices Σ₁,Σ₂,Σ₃ where Σ₁=Σ^𝑇₁ and 0<Σ₂=Σ^𝑇₂, then Σ₁+Σ^𝑇₃Σ₂⁻¹Σ₃<0 if and only if Σ₁Σ^𝑇₃Σ₃−Σ₂<0,or−Σ₂Σ^𝑇₃Σ₃Σ₁<0. (2.11)

Lemma 2.12 (Jensen's inequality). For any constant matrix Ω∈𝑅^𝑛×𝑛,andΩ=Ω^𝑇>0,scalar𝛾>0, vector function 𝜔∶[0,𝛾]→𝑅^𝑛, such that the integrations concerned are well defined, then 1𝛾^𝛾₀𝜔(𝑠)𝑑𝑠^𝑇Ω^𝛾₀𝜔(𝑠)𝑑𝑠≤^𝛾₀𝜔(𝑠)^𝑇Ω𝜔(𝑠)𝑑𝑠. (2.12)

To give our main results in the next sections, we need to present the following assumptions.(𝐻₁)𝛼_𝑖(𝑠) is continuous, and 0<𝛼_𝑖<𝛼_𝑖(𝑠)<𝛼_𝑖 for all 𝑠∈𝑅, 𝑖=1,2,…,𝑛.(𝐻₂)𝛽_𝑖(𝑠) is locally Lipschitz continuous, and there exists a constant 𝛽_𝑖>0 such that 𝛽̇_𝑖(𝑠)≥𝛽_𝑖 for all 𝑠∈𝑅 at which 𝛽_𝑖(𝑠) is differentiable, 𝑖=1,2,…,𝑛.(𝐻₃)𝑔_𝑖(𝑠)∈ℐℋ(𝛼), 𝑖=1,2,…,𝑛.

3. Exponential Stability of the Cohen-Grossberg Neural Network

In this section, the Cohen-Grossberg neural network (2.1) is considered the main results on the existence and stability of equilibrium point of the neural network (2.1) will be presented in the following theorem.

Theorem 3.1. Under the assumptions (𝐻₁)–(𝐻₃), if there exist two positive definite matrices 𝑆,𝑇, two positive definite diagonal matrices 𝑀, 𝑃 and a scalar 𝛾>0 such that 𝛾𝑃𝐴⁻¹−𝑃𝐵−𝐵𝑃𝑃𝑊₀+𝛾𝑀𝐴⁻¹−𝑀𝐵𝑃𝑊₁𝑃𝑊₂𝑊^𝑇₀𝑃+𝛾𝐴⁻¹𝑀−𝐵𝑀𝑀𝑊₀+𝑊^𝑇₀𝑀+𝑆+𝜏𝑇𝑀𝑊₁𝑀𝑊₂𝑊^𝑇₁𝑃𝑊^𝑇₁𝑀−e^−𝛾𝜏𝑊𝑆0_𝑇2𝑃𝑊^𝑇₂e𝑀0−^−𝛾𝜏𝜏𝑇<0 (3.1) is satisfied, where 𝐴=diag(𝛼₁,𝛼₂,…,𝛼_𝑛), 𝐵=diag(𝛽₁,𝛽₂,…,𝛽_𝑛), 𝑀=diag(𝑚₁,𝑚₂,…,𝑚_𝑛), 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛), then the Cohen-Grossberg neural network (2.1) has one unique equilibrium point which is globally exponentially stable.

Proof. We should prove this theorem in three steps.
Step 1. In this step, we will prove the existence of the equilibrium point.
Let ℌ(𝑥)=𝛽(𝑥)−(𝑊₀+𝑊₁+𝜏𝑊₂)𝑔(𝑥)−𝐽. By the assumption (𝐻₁), 𝑥^∗∈𝑅^𝑛 is an equilibrium point of the system (2.1) if and only if ℌ(𝑥^∗)=0. Rewrite ℌ(𝑥) as ℌ(𝑥)=𝛽(𝑥)−𝑊₀+𝑊₁+𝜏𝑊₂𝑔(𝑥)+ℌ(0), (3.2) where 𝛽(𝑥)=𝛽(𝑥)−𝛽(0),𝑔(𝑥)=𝑔(𝑥)−𝑔(0). Obviously, 𝑔_𝑖(0)=0, 𝑔_𝑖∈ℐℋ(𝛼) and 𝑥_𝑖𝑔_𝑖(𝑥_𝑖)>0(𝑥_𝑖≠0). By Lemma 2.8 and the assumption (𝐻₂), it can follow that 𝛽(𝑥)=𝛽(𝑥)−𝛽(0)=𝐵𝑥,𝐵∈∪_{𝑧∈[0,𝑥]}𝜕𝛽(𝑧), where 𝐵=diag(𝛽₁,𝛽₂,…,𝛽_𝑛) with 𝛽_𝑖≥𝛽_𝑖,𝑖=1,2,…,𝑛.
Let Ω_ℛ={𝑥∈𝑅^𝑛∶‖𝑥‖<ℛ},ℛ>0. Define the mappingℋ∶[0,1]×Ω→𝑅^𝑛as ℋ(𝜆,𝑥)=𝐵𝑥−𝜆𝑊₀+𝑊₁+𝜏𝑊₂𝑔(𝑥)+𝜆ℌ(0), (3.3) where Ω_ℛ={𝑥∈𝑅^𝑛∶‖𝑥‖≤ℛ}.
From (3.1) and the definition of negative definite matrix, we can obtain 𝑀𝑊₀+𝑊^𝑇₀𝑀+𝑆+𝜏𝑇𝑀𝑊₁𝑀𝑊₂𝑊^𝑇₁𝑀−e^−𝛾𝜏𝑊𝑆0_𝑇2e𝑀0−^−𝛾𝜏𝜏𝑇<0. (3.4) By Lemma 2.11, (3.4) is equivalent to 𝑀𝑊₀+𝑊^𝑇₀𝑀+𝑆+e^𝛾𝜏𝑀𝑊₁𝑆⁻¹𝑊^𝑇₁𝑀+𝜏e^𝛾𝜏𝑀𝑊₂𝑇⁻¹𝑊^𝑇₂𝑀+𝜏𝑇<0. (3.5)
By means of Lemma 2.10, we have 𝑔(𝑥)^𝑇𝑀𝑊₁(𝑥)≤1𝑔2𝑔(𝑥)^𝑇𝑀𝑊₁𝑆⁻¹𝑊^𝑇₁𝑀𝑔(𝑥)+𝑔(𝑥)^𝑇(𝑥,(𝑥)𝑆𝑔)𝑔^𝑇𝑀𝑊₂(𝑥)≤1𝑔2𝑔(𝑥)^𝑇𝑀𝑊₂𝑇⁻¹𝑊^𝑇₂𝑀𝑔(𝑥)+𝑔(𝑥)^𝑇𝑇𝑔(𝑥). (3.6)
By using (3.5) and (3.6), we have 𝑔(𝑥)^𝑇()(𝑥)𝑀ℋ𝜆,𝑥=𝑔^𝑇𝑀𝐵𝑥+𝜆𝑔(𝑥)^𝑇𝑀ℌ(0)−𝜆𝑔(𝑥)^𝑇𝑀𝑊₀+𝑊₁+𝜏𝑊₂(𝑥)(𝑥)𝑔=𝑔^𝑇𝑀𝐵𝑥+𝜆ℌ(0)−𝜆𝑔(𝑥)^𝑇𝑀𝑊₀+𝑊₁+𝜏𝑊₂(𝑥)(𝑥)𝑔≥𝑔^𝑇𝑀(0)−𝜆𝐵𝑥+𝜆ℌ2𝑔(𝑥)^𝑇𝑀𝑊₀+𝑊^𝑇₀𝑀+𝑆+𝑀𝑊₁𝑆⁻¹𝑊^𝑇₁𝑀+𝜏𝑀𝑊₂𝑇⁻¹𝑊^𝑇₂(𝑥(𝑥)𝑀+𝜏𝑇𝑔)≥𝑔^𝑇𝑀(0)≥𝐵𝑥+𝜆ℌ_𝑛𝑖=1𝑚_𝑖𝛽_𝑖𝑔_𝑖𝑥_𝑖𝑥_𝑖−𝑔_𝑖𝑥_𝑖(𝑀ℌ(0))_𝑖=^𝑛_𝑖=1𝑚_𝑖𝛽_𝑖𝑔_𝑖𝑥_𝑖𝑥_𝑖−(𝑀ℌ(0))_𝑖𝑚_𝑖𝛽_𝑖, (3.7) where (𝑀ℌ(0))_𝑖(𝑖=1,2,…,𝑛) denotes the 𝑖th element of vector (𝑀ℌ(0)).
By virtue of Lemma 2.7, there exist constants 𝑞_𝑖0>0 and 𝑟_𝑖0>0, 𝑖=1,2,…,𝑛, such that 𝑔_𝑖𝑥_𝑖≥𝑞_𝑖0𝑟^𝛼_𝑖0,∀𝑥_𝑖≥𝑟_𝑖0,𝑖=1,2,…,𝑛. (3.8)
Let 𝑎=max_{1≤𝑖≤𝑛}(|(𝑀ℌ(0))_𝑖|/𝑚_𝑖𝛽_𝑖), 𝒩_𝑘={𝑛₁,…,𝑛_𝑘}⊂{1,2,…,𝑛},forall𝑘<𝑛. Define Ω_𝒩𝑘={𝑥∶|𝑥_𝑖|≤𝑎,𝑖∈𝒩_𝑘,𝑥∈𝑅^𝑘}, 𝑔_𝒩𝑘(𝑥)=∑_{𝑖∈𝒩𝑘}𝑚_𝑖𝛽_𝑖|𝑔_𝑖(𝑥_𝑖)|[|𝑥_𝑖|−𝑎]. Noting that Ω_𝒩𝑘 is a compact subset of 𝑅^𝑘, and 𝑔_𝒩𝑘 is continuous on Ω_𝒩𝑘, it follows that 𝑔_𝒩𝑘 can reach its the minimum min_{𝑥∈Ω𝒩𝑘}𝑔_𝒩𝑘(𝑥) on Ω_𝒩𝑘.
Let 𝑟₀=max_{1≤𝑖≤𝑛}𝑟_𝑖0, 𝑙=min_{1≤𝑖≤𝑛}{𝑚_𝑖𝛽_𝑖𝑞_𝑖0𝑟^𝛼_𝑖0}, ℳ_𝒩𝑘=min_{𝑥∈Ω𝒩𝑘}𝑔_𝒩𝑘(𝑥), and ℳ=min{ℳ_𝒩𝑘∶𝒩_𝑘⊂{1,2,…,𝑛}}. Set ℛ>max{√𝑛(𝑎−ℳ/𝑙),√𝑛𝑟₀} and 𝑥∈𝜕Ω_𝑅, then there exist two index sets 𝒩 and 𝒩, such that 𝑥_𝑖≤𝑎,𝑖∈𝒩,𝑥_𝑖>𝑎,𝑖∈𝒩, (3.9) where 𝒩⋃𝒩={1,2,…,𝑛}. Furthermore, there exists an index 𝑖₀ in 𝒩 such that 𝑥_𝑖0≥ℛ√𝑛≥max𝑎,𝑟₀. (3.10) By (3.8) and (3.10), for any 𝑥∈𝜕Ω_𝑅 and 𝜆∈[0,1],𝑔(𝑥)^𝑇𝑀ℋ(𝜆,𝑥)≥_𝑖∈𝒩𝑚_𝑖𝛽_𝑖𝑔_𝑖𝑥_𝑖𝑥_𝑖−𝑎+_𝑖∈𝒩𝑚_𝑖𝛽_𝑖𝑔_𝑖𝑥_𝑖𝑥_𝑖−𝑎≥ℳ+𝑚_𝑖0𝛽_𝑖0𝑔_𝑖0𝑥_𝑖0𝑥_𝑖0−𝑎≥ℳ+𝑚_𝑖0𝛽_𝑖0𝑞_𝑖0𝑟^𝛼_𝑖0𝑥_𝑖0−𝑎≥𝑚_𝑖0𝛽_𝑖0𝑞_𝑖0𝑟^𝛼_𝑖0𝑥_𝑖0ℳ−𝑎+𝑙≥𝑚_𝑖0𝛽_𝑖0𝑞_𝑖0𝑟^𝛼_𝑖0ℛ√𝑛ℳ−𝑎+𝑙>0. (3.11) Hence, this implies that ℋ(𝜆,𝑥)≠0 for any 𝑥∈𝜕Ω_𝑅 and 𝜆∈[0,1]. By applying Lemma 2.9 (1), it follows degℋ(0,𝑥),Ω_ℛ,0=degℋ(1,𝑥),Ω_ℛ,0, (3.12) that is, deg(ℌ(𝑥),Ω_ℛ,0)=deg(𝐵𝑥,Ω_ℛ,0)=sgn|𝐵|≠0, where |𝐵| is the determinant of 𝐵. By Lemma 2.9 (2), ℌ(𝑥)=0 has at least one solution in Ω_ℛ, that is, the system (2.1) has at least an equilibrium point.
Step 2. In this step, the proof of the uniqueness of the equilibrium point by the method of contradiction will be given.
Assume that 𝑥^∗₁ and 𝑥^∗₂ are two different equilibrium points of the system (2.1), then 𝛽𝑥^∗₁−𝛽𝑥^∗₂=𝑊₀+𝑊₁+𝜏𝑊₂𝑔𝑥^∗₁−𝑔𝑥^∗₂. (3.13) From Lemma 2.8, it can follow that 𝛽𝑥^∗₁−𝛽𝑥^∗₂=𝐵𝑥^∗₁−𝑥^∗₂, (3.14) where 𝐵∈∪_𝑧∈[𝑥^∗_1,𝑥^∗_2]𝜕𝛽(𝑧), 𝐵=diag(𝛽₁,𝛽₂,…,𝛽_𝑛) with 𝛽_𝑖≥𝛽_𝑖,𝑖=1,2,…,𝑛.
By means of (3.5), (3.6), (3.13), and (3.14), it follows that 0<𝑔𝑥^∗₁−𝑔𝑥^∗₂^𝑇𝑀𝐵𝑥^∗₁−𝑥^∗₂=𝑔𝑥^∗₁−𝑔𝑥^∗₂^𝑇𝑀𝑊₀+𝑊₁+𝜏𝑊₂𝑔𝑥^∗₁−𝑔𝑥^∗₂≤12𝑔𝑥^∗₁−𝑔𝑥^∗₂^𝑇×𝑀𝑊₀+𝑊^𝑇₀𝑀+𝑆+𝑀𝑊₁𝑆⁻¹𝑊^𝑇₁𝑀+𝜏𝑀𝑊₂𝑇⁻¹𝑊^𝑇₂𝑀+𝜏𝑇𝑔𝑥^∗₁−𝑔𝑥^∗₂<0. (3.15) This is a contradiction. Hence, 𝑥^∗₁=𝑥^∗₂. This shows that the equilibrium point of the system (2.1) is unique.
Step 3. In this step, we will prove that the system (2.1) is globally exponentially stable.
Let𝐹(𝑥,𝑡)=𝛼(𝑥)[−𝛽(𝑥)+𝑊₀𝑔(𝑥)+𝑊₁𝑔(𝑥_𝜏)+𝑊₂∫^𝑡_𝑡−𝜏𝑔(𝑥(𝑠))𝑑𝑠+𝐽],where 𝑥_𝜏(𝑡)=𝑥(𝑡−𝜏). Since 𝑔_𝑖∈ℐℋ(𝛼), 𝛼(𝑥),𝛽(𝑥) are continuous functions, 𝐹(𝑥,𝑡) are continuous and locally bounded. Hence, we can obtain the existence of the local solution of the system (2.1) with initial value 𝑥(𝑡)=𝜑(𝑡),𝑡∈[−𝜏,0] on [0,𝑡^∗(𝜑)), where 𝑡^∗(𝜑)∈(0,+∞) or 𝑡^∗(𝜑)=+∞, and [0,𝑡^∗(𝜑)) is the maximal right-side existence interval of the local solution.
Let 𝑥^∗ be the unique equilibrium point of the system (2.1). Make a transformation 𝑢(𝑡)=𝑥(𝑡)−𝑥^∗, then system (2.1) is transformed into 𝑢̇(𝑡)=𝛼(𝑢(𝑡))−𝛽(𝑢(𝑡))+𝑊₀𝑔(𝑢(𝑡))+𝑊₁𝑔(𝑢(𝑡−𝜏))+𝑊₂^𝑡_𝑡−𝜏𝑔(𝑢(𝑠))𝑑𝑠, (3.16) where𝛼(𝑢(𝑡))=diag(𝛼₁(𝑢₁(𝑡)),𝛼₂(𝑢₂(𝑡)),…,𝛼_𝑛(𝑢_𝑛(𝑡))), 𝛽(𝑢(𝑡))=(𝛽₁(𝑢₁(𝑡)),𝛽₂(𝑢₂(𝑡)),…,𝛽_𝑛(𝑢_𝑛(𝑡)))^𝑇, 𝑔(𝑢(𝑡))=(𝑔₁(𝑢₁(𝑡)),𝑔₂(𝑢₂(𝑡)),…,𝑔_𝑛(𝑢_𝑛(𝑡)))^𝑇, 𝛼_𝑖(𝑢_𝑖(𝑡))=𝛼_𝑖(𝑢_𝑖(𝑡)+𝑥^∗_𝑖), 𝛽_𝑖(𝑢_𝑖(𝑡))=𝛽_𝑖(𝑢_𝑖(𝑡)+𝑥^∗_𝑖)−𝛽_𝑖(𝑥^∗_𝑖), 𝑔_𝑖(𝑢_𝑖(𝑡))=𝑔_𝑖(𝑢_𝑖(𝑡)+𝑥^∗_𝑖)−𝑔_𝑖(𝑥^∗_𝑖),𝑖=1,2,…,𝑛. Similarly to (3.14), from Lemma 2.8, we have 𝛽(𝑢)=𝛽𝑢+𝑥^∗−𝛽𝑥^∗=𝐵𝑢, (3.17) where 𝐵∈∪_{𝑧∈[𝑥∗,𝑢+𝑥∗]}𝜕𝛽(𝑧), 𝐵=diag(𝛽₁,𝛽₂,…,𝛽_𝑛) with 𝛽_𝑖≥𝛽_𝑖,𝑖=1,2,…,𝑛.
Consider the following Lyapunov-Krasovskii functional candidate 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡), (3.18) where 𝑉₁(𝑡)=2𝑒^𝑛𝛾𝑡_𝑖=1𝑝_𝑖^𝑢𝑖₀^(𝑡)𝑠𝛼_𝑖(𝑠)𝑉𝑑𝑠,₂(𝑡)=2𝑒^𝑛𝛾𝑡_𝑖=1𝑚_𝑖^𝑢𝑖₀^(𝑡)𝑔_𝑖(𝑠)𝛼_𝑖(𝑠)𝑉𝑑𝑠,₃(𝑡)=^𝑡_𝑡−𝜏𝑔(𝑢(𝜃))^𝑇𝑆𝑔(𝑢(𝜃))𝑒^𝛾𝜃𝑉𝑑𝜃,₄(𝑡)=⁰_−𝜏^𝑡_𝑡+𝜃𝑒^𝛾𝑠𝑔(𝑢(𝑠))^𝑇𝑇𝑔(𝑢(𝑠))𝑑𝑠𝑑𝜃. (3.19) Calculating the time derivative of 𝑉(𝑡) along the trajectories of the system (3.16) on [0,𝑡^∗(𝜑)) with (3.17), by the assumption (𝐻₁) and Lemma 2.12, we have 𝑉̇₁(𝑡)≤𝛾𝑒^𝛾𝑡𝑢(𝑡)^𝑇𝑃𝐴⁻¹𝑢(𝑡)+2𝑒^𝛾𝑡𝑢(𝑡)^𝑇𝑃×−𝐵𝑢(𝑡)+𝑊₀𝑔(𝑢(𝑡))+𝑊₁𝑔(𝑢(𝑡−𝜏(𝑡)))+𝑊₂^𝑡_{(𝑡)𝑡−𝜏}𝑔(𝑢(𝑠))𝑑𝑠≤𝑒^𝛾𝑡𝑢(𝑡)^𝑇𝛾𝑃𝐴⁻¹−2𝑃𝐵𝑢(𝑡)+2𝑒^𝛾𝑡𝑢(𝑡)^𝑇𝑃𝑊₀𝑔(𝑢(𝑡))+2𝑒^𝛾𝑡𝑢(𝑡)^𝑇𝑃𝑊₁𝑔(𝑢(𝑡−𝜏))+2𝑒^𝛾𝑡𝑢(𝑡)^𝑇𝑃𝑊₂^𝑡_𝑡−𝜏𝑔(𝑢(𝑠𝑉̇))𝑑𝑠,₂(𝑡)≤2𝛾𝑒^𝛾𝑡𝑢(𝑡)^𝑇𝑀𝐴⁻¹𝑔(𝑢(𝑡))+2𝑒^𝛾𝑡𝑔(𝑢(𝑡))^𝑇𝑀×−𝐵𝑢(𝑡)+𝑊₀𝑔(𝑢(𝑡))+𝑊₁𝑔(𝑢(𝑡−𝜏(𝑡)))+𝑊₂^𝑡_{(𝑡)𝑡−𝜏}𝑔(𝑢(𝑠))𝑑𝑠≤2𝑒^𝛾𝑡𝑢(𝑡)^𝑇𝛾𝑀𝐴⁻¹−𝑀𝐵𝑔(𝑢(𝑡))+2𝑒^𝛾𝑡𝑔(𝑢(𝑡))^𝑇𝑀𝑊₀𝑔(𝑢(𝑡))+2𝑒^𝛾𝑡𝑔(𝑢(𝑡))^𝑇𝑀𝑊₁𝑔(𝑢(𝑡−𝜏))+2𝑒^𝛾𝑡𝑔(𝑢(𝑡))^𝑇𝑀𝑊₂^𝑡_𝑡−𝜏𝑔(𝑢(𝑠𝑉̇))𝑑𝑠,₃(𝑡)=𝑒^𝛾𝑡𝑔(𝑢(𝑡))^𝑇𝑆𝑔(𝑢(𝑡))−𝑒^𝛾𝑡𝑒^−𝛾𝜏𝑔(𝑢(𝑡−𝜏))^𝑇𝑆𝑔(𝑢(,𝑉̇𝑡−𝜏))₄(𝑡)≤𝜏𝑒^𝛾𝑡𝑔(𝑢(𝑡))^𝑇𝑇𝑔(𝑢(𝑡))−^𝑡_𝑡−𝜏𝑒^𝛾𝑠𝑔(𝑢(𝑠))^𝑇𝑇𝑔(𝑢(𝑠))𝑑𝑠≤𝜏𝑒^𝛾𝑡𝑔(𝑢(𝑡))^𝑇𝑇𝑔(𝑢(𝑡))−𝑒^{𝛾(𝑡−𝜏)𝑡}_𝑡−𝜏𝑔(𝑢(𝑠))^𝑇𝑇𝑔(𝑢(𝑠))𝑑𝑠≤𝜏𝑒^𝛾𝑡𝑔(𝑢(𝑡))^𝑇𝑇𝑔(𝑢(𝑡))−𝑒^{𝛾(𝑡−𝜏)}1𝜏^𝑡_𝑡−𝜏𝑔(𝑢(𝑠))^𝑇𝑑𝑠𝑇^𝑡_𝑡−𝜏𝑔(𝑢(𝑠))𝑑𝑠. (3.20) Let 𝜉(𝑡)=[𝑢(𝑡)^𝑇,𝑔(𝑢(𝑡))^𝑇,𝑔(𝑢(𝑡−𝜏))^𝑇,(∫^𝑡_𝑡−𝜏𝑔(𝑢(𝑠))𝑑𝑠)^𝑇]^𝑇. By (3.17) and (3.20), we can obtain 𝑉̇(𝑡)≤𝑒^𝛾𝜏𝜉^𝑇(𝑡)𝛾𝑃𝐴⁻¹−𝑃𝐵−𝐵𝑃𝑃𝑊₀+𝛾𝑀𝐴⁻¹−𝑀𝐵𝑃𝑊₁𝑃𝑊₂𝑊^𝑇₀𝑃+𝛾𝐴⁻¹𝑀−𝐵𝑀𝑀𝑊₀+𝑊^𝑇₀𝑀+𝑆+𝜏𝑇𝑀𝑊₁𝑀𝑊₂𝑊^𝑇₁𝑃𝑊^𝑇₁𝑀−e^−𝛾𝜏𝑊𝑆0_𝑇2𝑃𝑊^𝑇₂e𝑀0−^−𝛾𝜏𝜏𝑇𝜉(𝑡)<0,forany𝜉^𝑇(𝑡)≠0. (3.21) This implies 𝑉(𝑡)≤𝑉(0), 𝑡∈[0,𝑡^∗(𝜑)). Furthermore, from (3.18), it follows that 2^𝑛_𝑖=1𝑚_𝑖𝛼_𝑖^𝑢𝑖₀^(𝑡)𝑔_𝑖(𝑠)𝑑𝑠≤2^𝑛_𝑖=1𝑚_𝑖^𝑢𝑖₀^(𝑡)𝑔_𝑖(𝑠)𝛼_𝑖(𝑠)𝑑𝑠≤𝑉(0)𝑒^−𝛾𝑡≤𝑉(0). (3.22) By (3.22) and Lemma 2.6, it is easy to derive that 𝑢_𝑖(𝑡),𝑖=1,2,…,𝑛, are bounded on [0,𝑡^∗). By virtue of the continuous theorem [47], 𝑡^∗(𝜑)=+∞.
Since 𝑔_𝑖(𝑠)∈ℐℋ(𝛼),𝑔_𝑖(0)=0, by Lemma 2.7, there exist constants 𝑞_𝑖0>0 and 𝑟_𝑖0>0, such that 𝑔_𝑖(𝑠)≥𝑞_𝑖0|𝑠|^𝛼,∀|𝑠|≤𝑟_𝑖0,𝑖=1,2,…,𝑛. (3.23) Moreover, from (3.22), we can get that lim_𝑡→+∞𝑢_𝑖(𝑡)=0,𝑖=1,2,…,𝑛. (3.24) Thus there exists a scalar 𝒯>0, when 𝑡≥𝒯, 𝑢_𝑖(𝑡)∈[−𝑟₀,𝑟₀], where 𝑟₀=min_{1≤𝑖≤𝑛}𝑟_𝑖0. Let 𝑚=min_{1≤𝑖≤𝑛}𝑚_𝑖, 𝛼=max_{1≤𝑖≤𝑛}𝛼_𝑖, and 𝑞₀=min_{1≤𝑖≤𝑛}𝑞_𝑖0. From (3.22) and (3.23), we have 𝑉(0)2𝑒^−𝛾𝑡≥^𝑛_𝑖=1𝑚_𝑖𝛼_𝑖^𝑢𝑖₀^(𝑡)𝑔_𝑖(𝑠)≥𝑑𝑠_𝑛𝑖=1𝑚_𝑖𝛼_𝑖^|𝑢𝑖₀^(𝑡)|𝑞_𝑖0|𝑠|^𝛼≥𝑑𝑠_𝑛𝑖=1𝑚_𝑖𝛼_𝑖^|𝑢𝑖₀^(𝑡)|𝑞₀|𝑠|^𝛼≥𝑚𝑑𝑠𝑞₀𝛼(𝛼+1)max_{1≤𝑖≤𝑛}𝑢_𝑖(𝑡)^𝛼+1,𝑡≥𝒯. (3.25) That is, when 𝑡≥𝒯, max_{1≤𝑖≤𝑛}𝑢_𝑖(𝑡)≤𝛼(𝛼+1)𝑉(0)2𝑚𝑞₀^1/(1+𝛼)𝑒^{−(𝛾/(1+𝛼))𝑡}, (3.26) where 𝑉(0)=2^𝑛_𝑖=1𝑝_𝑖^{𝜑𝑖(0)−𝑥∗𝑖}₀𝑠𝛼_𝑖(𝑠)𝑑𝑠+2^𝑛_𝑖=1𝑚_𝑖^{𝜑𝑖(0)−𝑥∗𝑖}₀𝑔_𝑖(𝑠)𝛼_𝑖(𝑠)+𝑑𝑠_0−𝜏𝑔𝜑(𝜃)−𝑥^∗𝑇𝑆𝑔𝜑(𝜃)−𝑥^∗𝑒^𝛾𝜃+𝑑𝜃_{0−𝜏0𝜃}𝑒^𝛾𝑠𝑔𝜑(𝑠)−𝑥^∗𝑇𝑇𝑔𝜑(𝑠)−𝑥^∗𝑑𝑠𝑑𝜃. (3.27) Let 𝜂=√𝑛[𝛼(𝛼+1)𝑉(0)/2𝑚𝑞₀]^1/(1+𝛼). By (3.26), ‖𝑥(𝑡)−𝑥^∗‖≤𝜂𝑒^{−(𝛾/(1+𝛼))𝑡}, for all 𝑡≥𝒯. This shows that the equilibrium point of the system (2.1) is globally exponentially stable. This completes the proof of Theorem 3.1.

4. Stability of the Switched Cohen-Grossberg Neural Network

The switched Cohen-Grossberg neural networks with mixed time delays consist of a set of Cohen-Grossberg neural networks with mixed time delays and a switching rule. Each of the Cohen-Grossberg neural networks with mixed time delays is regarded as an individual subsystem. The operation mode of the switched neural networks is determined by the switching rule. In the following, we will develop the switched Cohen-Grossberg neural networks model with mixed time delays.

Suppose that 𝑥^∗ is the unique equilibrium point of the system (2.1). Similar to (3.16), make a transformation 𝑢(𝑡)=𝑥(𝑡)−𝑥^∗, then system (2.1) is transformed into 𝑢̇(𝑡)=𝛼(𝑢(𝑡))−𝛽(𝑢(𝑡))+𝑊₀𝑔(𝑢(𝑡))+𝑊₁𝑔(𝑢(𝑡−𝜏))+𝑊₂^𝑡_𝑡−𝜏𝑔(𝑢(𝑠))𝑑𝑠. (4.1)

The switched Cohen-Grossberg neural networks with mixed time delays can be described as 𝑢̇(𝑡)=𝛼(𝑢(𝑡))−𝛽(𝑢(𝑡))+𝑊_0𝜎(𝑡)𝑔(𝑢(𝑡))+𝑊_1𝜎(𝑡)𝑔(𝑢(𝑡−𝜏))+𝑊_2𝜎(𝑡)^𝑡_𝑡−𝜏𝑔(𝑢(𝑠))𝑑𝑠, (4.2) where 𝜎(𝑡)∶[0,+∞)→Γ={1,2,…,𝑁} is the switching signal, which is a piecewise constant function of time. This means that the matrices (𝑊_0𝜎(𝑡),𝑊_1𝜎(𝑡),𝑊_2𝜎(𝑡),𝐽_𝜎(𝑡)) are allowed to take values, at an arbitrary time, in the finite set {(𝑊₀₁,𝑊₁₁,𝑊₂₁,𝐽₁),(𝑊₀₂,𝑊₁₂,𝑊₂₂,𝐽₂),…,(𝑊_0𝑁,𝑊_1𝑁,𝑊_2𝑁,𝐽_𝑁)}. The initial value associated with (4.3) is assumed to be 𝑢(𝑠)=𝜑(𝑠), 𝜑(𝑠) is a continuous function on [−𝜏,0].

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. Corresponding to the switching signal 𝜎(𝑡), we have the switching sequence {𝑥_𝑡0;(𝑖₀,𝑡₀),…,(𝑖_𝑘,𝑡_𝑘),…,∣𝑖_𝑘∈Γ,𝑘=0,1,…}, which means that the 𝑖_𝑘th subsystem is activated when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1).

In the following, we will consider the switched Cohen-Grossberg neural networks with mixed time delays in (4.2). The average dwell time approach will be used to derive the exponential stability of the network.

Theorem 4.1. Under the assumptions (𝐻₁)–(𝐻₃), if there exist positive definite matrices 𝑆_𝑖,𝑇_𝑖, positive definite diagonal matrices 𝑀_𝑖, 𝑃_𝑖 and scalars 𝛾>0,𝜇≥1 such that 𝛾𝑃𝐴⁻¹−𝑃_𝑖𝐵_𝑖−𝐵_𝑖𝑃_𝑖𝑃_𝑖𝑊_0𝑖+𝛾𝑀_𝑖𝐴⁻¹−𝑀_𝑖𝐵_𝑖𝑃_𝑖𝑊_1𝑖𝑃_𝑖𝑊_2𝑖𝑊^𝑇_0𝑖𝑃_𝑖+𝛾𝐴⁻¹𝑀_𝑖−𝐵_𝑖𝑀_𝑖𝑀_𝑖𝑊_0𝑖+𝑊^𝑇_0𝑖𝑀_𝑖+𝑆_𝑖+𝜏𝑇_𝑖𝑀_𝑖𝑊_1𝑖𝑀_𝑖𝑊_2𝑖𝑊^𝑇_1𝑖𝑃_𝑖𝑊^𝑇_1𝑖𝑀_𝑖−e^−𝛾𝜏𝑆_𝑖0𝑊^𝑇_2𝑖𝑃_𝑖𝑊^𝑇_2𝑖𝑀_𝑖e0−^−𝛾𝜏𝜏𝑇_𝑖<0, (4.3)𝑃_𝑖≤𝜇𝑃_𝑗,𝑀_𝑖≤𝜇𝑀_𝑗,𝑆_𝑖≤𝜇𝑆_𝑗,𝑇_𝑖≤𝜇𝑇_𝑗,∀𝑖,𝑗∈Γ, (4.4) where 𝑀_𝑖=diag(𝑚_𝑖1,𝑚_𝑖2,…,𝑚_𝑖𝑛), 𝑃_𝑖=diag(𝑝_𝑖1,𝑝_𝑖2,…,𝑝_𝑖𝑛), then the switched Cohen-Grossberg neural network (4.2) is globally exponentially stable for any switching signal with average dwell time satisfying 𝑇_𝑎>𝑇^∗_𝑎=(1+𝛼)ln𝜇𝛾. (4.5) Moreover, an estimate of the state decay for the system (4.2) is given by ‖𝑢(𝑡)‖≤√𝑛𝑒^−𝛾𝑡0𝛼(𝛼+1)2𝑚𝑞₀^1/(1+𝛼)max_𝑖∈Γ𝑉_𝑖𝑡₀𝑒^{−(𝛾/(1+𝛼)−(ln𝜇)/𝑇𝑎)(𝑡−𝑡0)},𝑡>𝑡₀. (4.6)

Proof. Consider the multiple Lyapunov-Krasovskii functional candidate 𝑉_𝜎(𝑡)(𝑡)=𝑉_1𝜎(𝑡)(𝑡)+𝑉_2𝜎(𝑡)(𝑡)+𝑉_3𝜎(𝑡)(𝑡)+𝑉_4𝜎(𝑡)(𝑡), (4.7) where 𝑉_1𝜎(𝑡)(𝑡)=2𝑒^𝑛𝛾𝑡_𝑗=1𝑝_{𝜎(𝑡)𝑗}^𝑢𝑗₀^(𝑡)𝑠𝛼_𝑗(𝑠)𝑉𝑑𝑠,_2𝜎(𝑡)(𝑡)=2𝑒^𝑛𝛾𝑡_𝑗=1𝑚_{𝜎(𝑡)𝑗}^𝑢𝑗₀^(𝑡)𝑔_𝑗(𝑠)𝛼_𝑗(𝑠)𝑉𝑑𝑠,_3𝜎(𝑡)(𝑡)=^𝑡_𝑡−𝜏𝑔(𝑢(𝜃))^𝑇𝑆_𝜎(𝑡)𝑔(𝑢(𝜃))𝑒^𝛾𝜃𝑉𝑑𝜃,_4𝜎(𝑡)(𝑡)=⁰_−𝜏^𝑡_𝑡+𝜃𝑒^𝛾𝑠𝑔(𝑢(𝑠))^𝑇𝑇_𝜎(𝑡)𝑔(𝑢(𝑠))𝑑𝑠𝑑𝜃. (4.8) When 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), the 𝑖_𝑘th subsystem is activated. Arguing as in the proof Theorem 3.1, we can get 𝑉̇_𝜎(𝑡)(𝑡)≤0. Thus, 𝑉_𝜎(𝑡)(𝑡)≤𝑉_{𝜎(𝑡𝑘)}(𝑡_𝑘). In the light of (4.4) and (4.7), it follows that 𝑉_{𝜎(𝑡𝑘)}(𝑡_𝑘)≤𝜇𝑉_𝜎(𝑡⁻_𝑘)(𝑡⁻_𝑘). Therefore, when 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), we have 𝑉_𝜎(𝑡)(𝑡)≤𝑉_{𝜎(𝑡𝑘)}𝑡_𝑘≤𝜇𝑉_𝜎(𝑡⁻_𝑘)𝑡⁻_𝑘≤𝜇𝑉_{𝜎(𝑡𝑘−1)}𝑡_𝑘−1≤𝜇²𝑉_𝜎(𝑡⁻_𝑘−1)𝑡⁻_𝑘−1≤𝜇²𝑉_{𝜎(𝑡𝑘−2)}𝑡_𝑘−2≤⋯≤𝜇^𝑘𝑉_𝜎(𝑡0)𝑡₀. (4.9) By (4.9); there exist constants 𝒯>𝑡₀, when 𝑡≥𝒯, we have 𝑉_𝜎(𝑡0)𝑡₀2𝜇^𝑘𝑒^−𝛾𝑡≥^𝑛_𝑗=1𝑚_{𝜎(𝑡)𝑗}𝛼_𝑗^𝑢𝑗₀^(𝑡)𝑔_𝑖(𝑠)≥𝑑𝑠_𝑛𝑗=1𝑚_{𝜎(𝑡)𝑗}𝛼_𝑗^|𝑢𝑖₀^(𝑡)|𝑞_𝑖0|𝑠|^𝛼≥𝑑𝑠_𝑛𝑗=1𝑚_{𝜎(𝑡)𝑗}𝛼_𝑗^|𝑢𝑖₀^(𝑡)|𝑞₀|𝑠|^𝛼≥𝑚𝑑𝑠𝑞₀𝛼(𝛼+1)max_{1≤𝑖≤𝑛}|𝑢_𝑖(𝑡)|^𝛼+1, (4.10) where 𝑚=min_{𝑗∈Γ,1≤𝑖≤𝑛}𝑚_𝑗𝑖, 𝛼=max_{1≤𝑖≤𝑛}𝛼_𝑖 and 𝑞₀=min_{1≤𝑖≤𝑛}𝑞_𝑖0. Hence, max_{1≤𝑖≤𝑛}𝑢_𝑖(𝑡)≤𝛼(𝛼+1)𝑉_𝜎(𝑡0)(𝑡₀)2𝑚𝑞₀^1/(1+𝛼)𝜇^𝑘𝑒^{−(𝛾/(1+𝛼))𝑡}. (4.11) This implies that ‖𝑢(𝑡)‖≤√𝑛𝛼(𝛼+1)𝑉_𝜎(𝑡0)(𝑡₀)2𝑚𝑞₀^1/(1+𝛼)𝜇^𝑘𝑒^{−(𝛾/(1+𝛼))𝑡}. (4.12) Due to 𝑘≤(𝑡−𝑡₀)/𝑇_𝑎 and 𝑇_𝑎>𝑇^∗_𝑎=(1+𝛼)((ln𝜇)/𝛾), then we can get ‖𝑢(𝑡)‖≤√𝑛𝑒^−𝛾𝑡0𝛼(𝛼+1)2𝑚𝑞₀^1/(1+𝛼)max_𝑖∈Γ𝑉_𝑖𝑡₀𝜇^𝑘𝑒^{−(𝛾/(1+𝛼))(𝑡−𝑡0)}=√𝑛𝑒^−𝛾𝑡0𝛼(𝛼+1)2𝑚𝑞₀^1/(1+𝛼)max_𝑖∈Γ𝑉_𝑖𝑡₀𝑒^𝑘ln𝜇𝑒^{−(𝛾/(1+𝛼))(𝑡−𝑡0)}≤√𝑛𝑒^−𝛾𝑡0𝛼(𝛼+1)2𝑚𝑞₀^1/(1+𝛼)max_𝑖∈Γ𝑉_𝑖𝑡₀𝑒^{−(𝛾/(1+𝛼)−(ln𝜇)/𝑇𝑎)(𝑡−𝑡0)},𝑡>𝑡₀. (4.13) This implies that the switched Cohen-Grossberg neural network (4.2) is globally exponentially stable. The proof is completed.

Remark 4.2. It is clear that, according to Theorem 4.1, the delay-dependent stability of the considered neural networks is dependent on 𝜇 for given 𝛾. If 𝜇=1, we have from (4.5) that the switching signal can be arbitrary, and (4.4) reduces to 𝑃_𝑖≤𝑃_𝑗,𝑀_𝑖≤𝑀_𝑗,𝑆_𝑖≤𝑆_𝑗,𝑇_𝑖≤𝑇_𝑗,forall𝑖,𝑗∈Γ, which implies 𝑃_𝑖=𝑃_𝑗,𝑀_𝑖=𝑀_𝑗,𝑆_𝑖=𝑆_𝑗,𝑇_𝑖=𝑇_𝑗,forall𝑖,𝑗∈Γ, which means that it requires a common Lyapunov functional for all subsystems. If 𝜇→+∞, we get from (4.5) that there is no switching, that is, switching signal will have a great dwelltime on the average.

Remark 4.3. In the existing literature [2, 3, 17, 22, 27–29, 34–38], the activation functions of switched neural networks are required to be Lipschitz continuous. However, in this paper, the activation function is inverse Hölder function. It is obvious the results of this paper are different from the results in the above literatures. Hence, the work in this paper is an extension of the scope of the current investigation in this field.

5. Illustrative Examples

In this section, two illustrative examples will be given to check the validity of the results obtained in Theorems 3.1 and 4.1.

Example 5.1. Consider a second-order Cohen-Grossberg neural networks with mixed time delays in (2.1) with the following parameters: 𝑊₀=−99990.0090.0001−9999,𝑊₁=0.250.030.010.25,𝑊₂=()0.050.030.020.45,𝐽=0,0^𝑇. (5.1) Set 𝛼_𝑖(𝜃)=2+sin𝜃, 𝛽_𝑖(𝜃)=^{𝜃,𝜃≥0,}_{2𝜃,𝜃≤0,}. The activation functions are taken as 𝑔_𝑖(𝜃)=𝜃³, 𝑖=1,2, and 𝜏=1.
It is easy to check that assumptions (𝐻₁)–(𝐻₃) hold. 𝛽_𝑖(𝜃) is locally Lipschitz; 𝐴 and 𝐵 is the second-order identity matrix.
Take 𝛾=0.3. Solving the LMI in (3.1) by using appropriate LMI solver in the Matlab, the feasible positive definite matrices 𝑃,𝑀,𝑆,𝑇 could be as ,.𝑃=0.0017000.0017,𝑀=23.33160023.3316𝑆=3.4137−0.0019−0.00193.4137,𝑇=3.4137−0.0019−0.00193.4137 (5.2) All assumptions of Theorem 3.1 hold. Hence, this neural network has one unique equilibrium point, which is globally exponentially stable.
Figures 1 and 2 display the state trajectories of this neural network with initial values 𝜑(𝑡)=(cos𝑡,sin𝑡)^𝑇,(0.5cos𝑡,−0.5sin𝑡)^𝑇,(−1.5cos𝑡,1.5sin𝑡)^𝑇,(2cos𝑡,2sin𝑡)^𝑇 and (−2.5cos𝑡,2.5sin𝑡)^𝑇,𝑡∈[−1,0]. It can be seen that these trajectories converge to the unique equilibrium 𝑥^∗=(0,0)^𝑇 of the network. This is in accordance with the conclusion of Theorem 3.1.

Figure 1 

The convergence of the state 𝑥1(𝑡) of the network in Example 5.1.

Figure 2 

The convergence of the state 𝑥2(𝑡) of the network in Example 5.1.

Example 5.2. Consider a third-order switched Cohen-Grossberg neural networks with mixed time delays in (4.3) with the switching signal 𝜎(t)∶[0,+∞)→Γ={1,2} and the following parameters:𝑊₀₁=−832.6513.130.516.01−807.4393.522.033.05−478.954,𝑊₀₂=,𝑊−49594.033.310.01−80792.828.036.05−9278₁₁=0.250.160.030.010.320.250.210.320.06,𝑊₁₂=,𝑊0.250.160.030.010.324.250.210.320.06₂₁=0.951.040.031.021.031.4520.045.0280.37,𝑊₂₂=.0.050.040.030.560.032.450.040.020.37 (5.3) Set 𝛼_𝑖(𝜃)=2−cos𝜃, 𝛽_𝑖(𝜃) are the functions in Example 5.1, and the activation functions are taken as 𝑔_𝑖(𝜃)=𝜃³+𝜃, 𝑖=1,2,3, and 𝜏=1.

It is easy to check that assumptions (𝐻₁)–(𝐻₃) hold; 𝐴 and 𝐵 is the third-order identity matrix.

Take 𝛾=0.1,𝜇=1.2. Solving the LMIs in (4.3) and (4.4) by using appropriate LMI solver in the Matlab, the feasible positive definite matrices 𝑃₁,𝑃₂,𝑀₁,𝑀₂,𝑆₁,𝑆₂,𝑇₁,𝑇₂ could be as𝑃₁=0.63340000.63340000.6334,𝑀₁=,𝑆0.04180000.04180000.0418₁=0.75000.0182−0.06850.01820.8893−0.0165−0.0685−0.01650.2732,𝑇₁=,𝑃0.78110.02660.06790.02660.88950.01830.06790.01830.8362₂=0.66200000.66200000.6620,𝑀₂=,𝑆0.03650000.03650000.0365₂=0.78550.0191−0.07170.01910.9312−0.0173−0.0717−0.01730.2862,𝑇₂=.0.47100.01610.04110.01610.53680.01110.04110.01110.5045 (5.4) By using (4.5), it follows that the average dwell time 𝑇^∗_𝑎=3.6461. All the assumptions of Theorem 4.1 hold. Hence, this switched neural network is globally exponentially stable.

For numerical simulation, assume that the two subsystems are switched every four seconds. Figures 3, 4, and 5 display the state trajectories of this neural network with initial values𝜑(𝑡)=(−0.5cos𝑡,0.5sin𝑡,0.5cos𝑡)^𝑇,(cos𝑡,−sin𝑡,−cos𝑡)^𝑇,(−1.5cos𝑡,1.5sin𝑡,1.5cos𝑡)^𝑇,

Figure 3 

The convergence of the state 𝑥1(𝑡) of the network in Example 5.2.

Figure 4 

The convergence of the state 𝑥2(𝑡) of the network in Example 5.2.

Figure 5 

The convergence of the state 𝑥3(𝑡) of the network in Example 5.2.

(−2cos𝑡,2sin𝑡,2sin𝑡)^𝑇and (2.5cos𝑡,−2.5sin𝑡,−2.5cos𝑡)^𝑇,𝑡∈[−1,0]. It can be seen that these trajectories converge to the unique equilibrium 𝑢^∗=(0,0,0)^𝑇 of the network. This is in accordance with the conclusion of Theorem 4.1.

6. Conclusion

In this paper, the existence, uniqueness, and global stability of the equilibrium point for Cohen-Grossberg neural networks with mixed time delays,𝛼-inverse Hölder neuron activation functions, and nonsmooth behaved functions have been discussed. By applying multiple Lyapunov-Krasovskii functional, a delay-dependent global exponential stability criterion has been obtained in terms of LMIs for the switched Cohen-Grossberg neural networks with mixed time delays and 𝛼-inverse Hölder neuron activation functions under the switching rule with the average dwell time property. The results obtained are easily checked and applied in practice engineering.

When neuron activation functions are non-Lipschitz functions, it is possible that the neural network system has not the global solution and the equilibrium point. This leads to difficulty in solving the stability problem, particularly exponential stability for the switched neural networks with non-Lipschitz activation functions. In the future, the stability problem for the switched neural networks with other non-Lipschitz activation functions will be expected to be solved.

Acknowledgments

This work was supported by the Natural Science Foundation of Hebei Province of China (A2011203103) and the Hebei Province Education Foundation of China (2009157).