Global Convergence for Cohen-Grossberg Neural Networks with Discontinuous Activation Functions

Abstract

Cohen-Grossberg neural networks with discontinuous activation functions is considered. Using the property of 𝑀-matrix and a generalized Lyapunov-like approach, the uniqueness is proved for state solutions and corresponding output solutions, and equilibrium point and corresponding output equilibrium point of considered neural networks. Meanwhile, global exponential stability of equilibrium point is obtained. Furthermore, by contraction mapping principle, the uniqueness and globally exponential stability of limit cycle are given. Finally, an example is given to illustrate the effectiveness of the obtained results.

1. Introduction

Recently, different types of neural networks with or without time delays have been widely investigated due to their wide applicability [1–32]. Obviously, considerable research interests are focused on the studies of Cohen-Grossberg neural networks (CGNNs) with their various generalizations due to their potential applications in classification, associative memory, and parallel computation and their ability to solve optimization problems. This class of neural networks is proposed by Cohen and Grossberg [1] in 1983, and can be modeled as 𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝑎_𝑖𝑢_𝑖(𝑡)𝑏_𝑖𝑢_𝑖(𝑡)−^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡)−𝐼_𝑖,𝑖=1,2,…,𝑛, (1.1) where 𝑛≥2 is the number of neurons in the network, 𝑢_𝑖 denotes the state variable associated with the 𝑖th neuron, 𝑎_𝑖 represents an amplification function, and 𝑏_𝑖 is an appropriately behaved function. 𝑤_𝑖𝑗 represents the connection strengths between neurons, and if the output from neuron 𝑗 excites (resp., inhibits) neuron 𝑖, then 𝑤_𝑖𝑗≥0 (resp., 𝑤_𝑖𝑗≤0). The activation function 𝑓_𝑗 shows how neurons respond to each other. CGNNs include a lot of famous ecological systems and neural networks as special cases such as the Lotka-Volterra system, the Gilpia-Analg competition system, the Eingen-Schuster system, and the Hopfield neural networks [1–3], where the Hopfield neural networks can be described as follows: 𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝑏_𝑖𝑢_𝑖(𝑡)+^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡)+𝐼_𝑖,𝑖=1,2,…,𝑛. (1.2)

For CGNNs, dynamics behavior have been studied in literature; we refer to [4–10, 27–29] and the references cited therein. In [4], by using the concept of Lyapunov diagonally stable (LDS) and linear matrix inequality approach, some criteria were given to ensure global stability and global exponential stability. Yuan and Cao in [5] considered global asymptotic stability of delayed Cohen-Grossberg neural networks via nonsmooth analysis. Robust exponentially stability of delayed Cohen-Grossberg neural networks is discussed in [10]. In [27], the authors studied the stochastic stability of a class of Cohen-Grossberg neural networks, in which the interconnections and delays are time varying.

In the above papers, a common feature is that the activation functions are assumed to be continuous and even Lipschitz continuous. However, in [11], Forti and Nistri pointed out that neural networks modeled by differential equations with discontinuous right-hand side are important and do frequently arise in practice. In order to model discrete-time cellular neural networks, a conceptually analogous model based on hard comparators was used [12]. The class of neural networks introduced in [13] to deal with linear and nonlinear programming problems can be considered as another important example. Those networks make use of constraint neurons with a diode-like input-output activations. Once again, in order to ensure satisfaction of the constraints, the diodes are required to have a very high slope in the conducting region; that is, they should approximate the discontinuous characteristic of an ideal diode [14]. When treating with dynamical systems with high-slope nonlinear elements, a system of differential equations with discontinuous right-hand side is often used, rather than the model with high but finite slope [15]. The reason of analyzing the ideal discontinuous case is that such analysis is able to reveal crucial features of the dynamics, such as the possibility that trajectories be confined for some time intervals on discontinuity surfaces. Another interesting phenomenon which is peculiar to discontinuous systems is the possibility that trajectories converge toward an equilibrium point in finite time [16, 17], which is of special interest for designing real-time neural optimization solvers.

In [11], Forti and Nistri discussed the global convergence of neural networks with discontinuous neuron activations by means of the concepts and results of differential equations with discontinuous right-hand side introduced by Filippov [21]. In [18], they extended the results in [11] under the assumption that the interconnection matrix is an 𝑀-matrix or 𝐻-matrix. In [19], without assuming the boundedness and the continuity of the neuron activations, the authors presented sufficient conditions for the global stability of neural networks with time delay based on linear matrix inequality. Also, in [20], they present some sufficient conditions for the global stability and exponential stability of a class of the CGNNs by using the LDS, and provided an estimate of the convergence rate. In [24–26], the authors discussed the stability or multistability of the neural networks with discontinuous activation functions. However, [11, 24–26] have shown that convergence of the state does not imply convergence of the outputs. In addition, in the practical applications, the result of the neural computation is usually the steady-state neuron output, rather than the asymptotic value of the state. Hence, in this paper, we will study global convergence of CGNNs with discontinuous activation functions, where the interconnection matrix is assumed to be an 𝑀-matrix or 𝐻-matrix. Firstly, using the property of 𝑀-matrix and a generalized Lyapunov-like approach, we prove the uniqueness of state solutions and corresponding output solutions, and equilibrium point and corresponding output equilibrium point for the considered neural networks. Then, global exponential stability of unique equilibrium point is discussed and exponential convergence rate is estimated. Also, by contraction mapping principle, the globally exponential stability of limit cycle is given. Finally, we use a numerical example to illustrate the effectiveness of the theoretical results. The rest of the paper is organized as follows. In Section 2, model description and preliminaries are presented. The main results are stated in Section 3. In Section 4, an example is given to show the validity of the obtained results. Finally, in Section 5, the conclusions are drawn.

Notations. Throughout the paper, the transpose of and inverse of any square matrix 𝐴 are expressed as 𝐴^𝑇 and 𝐴⁻¹, respectively. For 𝛼=(𝛼₁,…,𝛼_𝑛)^𝑇∈ℝ^𝑛,𝛼>0 denotes 𝛼_𝑖>0 for 𝑖=1,2,…,𝑛. For 𝑥,𝑦∈ℝ^𝑛,   ⟨𝑥,𝑦⟩=∑^𝑛_𝑖=1𝑥_𝑖𝑦_𝑖 denotes the scalar product of 𝑥,𝑦.

2. Model Description and Preliminaries

In this paper, we consider the CGNNs (1.1) with discontinuous right-hand side. The compact form of model (1.1) is expressed as follows: 𝑑𝑢(𝑡)(𝑢(𝑡(𝑡)(𝑢(𝑡,𝑑𝑡=−𝐴))𝐵𝑢−𝑊𝑓))−𝐼 (2.1) where  𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡))^𝑇∈ℝ^𝑛,𝐴(𝑢(𝑡))=diag(𝑎₁(𝑢₁(𝑡)),𝑎₂(𝑢₂(𝑡)),…,𝑎_𝑛(𝑢_𝑛(𝑡))),𝐵=diag(𝑏₁,𝑏₂,…,𝑏_𝑛),𝑊=(𝑤_𝑖𝑗)_𝑛×𝑛,𝐼=(𝐼₁,𝐼₂,…,𝐼_𝑛)^𝑇∈ℝ^𝑛, and 𝑓(𝑢(𝑡))=(𝑓₁(𝑢₁(𝑡)),…,𝑓_𝑛(𝑢_𝑛(𝑡)))^𝑇.

Throughout this paper, we make the following assumptions.(A1) The function 𝑎_𝑖(𝑟) is continuous, 0<𝑎̌_𝑖≤𝑎_𝑖(𝑟)≤𝑎_𝑖 for all 𝑟∈ℝ, where 𝑎̌_𝑖 and 𝑎_𝑖 are positive constants, 𝑖=1,2,…,𝑛.(A2) The matrix 𝑊=(𝑤_𝑖𝑗)_𝑛×𝑛 is nonsingular, that is, det 𝑊≠0.

Moreover, 𝑓=(𝑓₁,…,𝑓_𝑛) is supposed to belong to the following class of discontinuous functions.

Definition 2.1 (see [18] (Function Class ℱ_𝐷)). 𝑓(𝑥)∈ℱ_𝐷 if and only if for 𝑖=1,2,…,𝑛, the following conditions hold:(i)𝑓_𝑖 is bounded on ℝ;(ii)𝑓_𝑖 is piecewise continuous on ℝ; namely, 𝑓_𝑖 is continuous on ℝ except a countable set of points of discontinuity 𝑝_𝑘𝑖, where there exist finite right and left limits 𝑓_𝑖(𝑝⁺_𝑘𝑖) and 𝑓_𝑖(𝑝⁻_𝑘𝑖), respectively; moreover, 𝑓_𝑖 has finite discontinuous points in any compact interval of ℝ;(iii)𝑓_𝑖 is nondecreasing on ℝ.
Denote the set of discontinuous points of 𝑓_𝑖,𝑖=1,2,…,𝑛, by 𝑆_𝑖=𝑝_𝑘𝑖∈ℝ∶𝑓_𝑖𝑝⁺_𝑘𝑖>𝑓_𝑖𝑝⁻_𝑘𝑖. (2.2)
Sometimes, 𝑓=(𝑓₁,…,𝑓_𝑛) is supposed to belong to the next class of discontinuous functions, which is included in ℱ_𝐷.

Definition 2.2 (see [18] (Function Class ℱ_𝐷𝐿)). 𝑓(𝑥)∈ℱ_𝐷𝐿 if and only if 𝑓(𝑥)∈ℱ_𝐷 and for 𝑖=1,2,…,𝑛,𝑓_𝑖 is locally Lipschitz with Lipschitz constant 𝑙_𝑖(𝑥_𝑖)≥0 for all 𝑥_𝑖∈ℝ⧵𝑆_𝑖. Furthermore, we have 𝑙_𝑖(𝑥_𝑖)≥𝐿_𝑖<+∞ for all 𝑥_𝑖∈ℝ⧵𝑆_𝑖.
For model (1.1) or model (2.1) with discontinuous right-hand side, a solution of Cauchy problem need to be explained. In this paper, solutions in the sense of Filippov [21] are considered whose definition will be given next.
Let 𝐾[𝑓(𝑢)]=(𝐾[𝑓₁(𝑢₁),𝐾[𝑓₂(𝑢₂)],…,𝐾[𝑓_𝑛(𝑢_𝑛)])^𝑇, where 𝐾[𝑓_𝑖(𝑢_𝑖)]=[𝑓_𝑖(𝑢⁻_𝑖),𝑓_𝑖(𝑢⁺_𝑖)].

Definition 2.3. A function 𝑢(𝑡),𝑡∈[𝑡₁,𝑡₂], where 𝑡₁<𝑡₂≤+∞ is a solution (in the sense of Filippov) of (2.1) in the interval [𝑡₁,𝑡₂], with initial condition 𝑢(𝑡₁)=𝑢₀∈ℝ^𝑛, if 𝑢(𝑡) is absolutely continuous on [𝑡₁,𝑡₂] and 𝑢(𝑡₁)=𝑢₀, and for almost all (a.a.) 𝑡∈[𝑡₁,𝑡₂] we have 𝑑𝑢(𝑡)(𝑢(𝑡(𝑡)𝑓(𝑢(𝑡.𝑑𝑡∈−𝐴))𝐵𝑢−𝑊𝐾))−𝐼 (2.3) Let 𝑢(𝑡),𝑡∈[𝑡₁,𝑡₂], be a solution of model (2.1). For a.a. 𝑡∈[𝑡₁,𝑡₂], one can obtain 𝑑𝑢(𝑡)(𝑢(𝑡(𝑡)(𝑡),𝑑𝑡=−𝐴))𝐵𝑢−𝑊𝛾−𝐼 (2.4) where 𝛾(𝑡)=𝑊⁻¹𝐴⁻¹(𝑢)𝑢̇(𝑡)+𝐵𝑢(𝑡)−𝐼∈𝐾𝑓(𝑢(𝑡)) (2.5) is the output solution of model (2.1) corresponding to 𝑢(𝑡). And 𝛾(𝑡) is a bounded measurable function [11], which is uniquely defined by the state solution 𝑢(𝑡) for a.a. 𝑡∈[𝑡₁,𝑡₂].

Definition 2.4 (equilibrium point). 𝑢^∗∈ℝ^𝑛 is an equilibrium point of model (2.1) if and only if the following algebraic inclusion is satisfied: 0∈𝐴𝑢^∗−𝐵𝑢^∗+𝑊𝐾𝑓𝑢^∗+𝐼. (2.6)

Definition 2.5 (output equilibrium point). Let 𝑢^∗ be an equilibrium point of model (1.1); 𝛾^∗=𝑊⁻¹𝐵𝑢^∗−𝐼∈𝐾𝑓𝑢^∗ (2.7) is the output equilibrium point of model (2.1) corresponding to 𝑢^∗.

In this paper, we also need the following definitions and lemma.

Definition 2.6 (see [18]). Let 𝑄∈ℝ^𝑛×𝑛 be a square matrix. Matrix 𝑄 is said to be an 𝑀-matrix if and only if 𝑄_𝑖𝑗≤0 for each 𝑖≠𝑗, and all successive principal minors of 𝑄 are positive.

Definition 2.7 (see [18]). Let 𝑄∈ℝ^𝑛×𝑛 be a square matrix. Matrix 𝑄 is said to be an 𝐻-matrix if and only if the comparison matrix of 𝑄, which is defined by [ℳ(𝑄)]_𝑖𝑗=𝑄_𝑖𝑖−𝑄,𝑖=𝑗,_𝑖𝑗,𝑖≠𝑗, (2.8) is an 𝑀-matrix.

Lemma 2.8 (see [18]). Suppose that 𝑄 is an 𝑀-matrix. Then, there exists a vector 𝜉>0 such that 𝑄^𝑇𝜉>0.

All results of this paper are under one of the following assumptions: (a)−𝑊 is an 𝑀-matrix; (b)−𝑊 is an 𝐻-matrix such that 𝑊_𝑖𝑖<0.

(a) and (b) can be applied to cooperative neural networks [22] and cooperative-competitive neural networks, respectively.

From [23], the result that −𝑊 is LDS under (a) or (b) can be obtained; hence, all results in [20] hold. So, for any 𝑢₀∈ℝ^𝑛, model (2.1) has a bounded absolutely continuous solution 𝑢(𝑡) for 𝑡≥0 which satisfies 𝑢(0)=𝑢₀. Meanwhile, there exists an equilibrium point 𝑢^∗∈ℝ^𝑛 of model (2.1).

If −𝑊 is an 𝑀-matrix, then, there exists 𝜉=(𝜉₁,…,𝜉_𝑛)^𝑇>0 such that (−𝑊)^𝑇𝜉=𝛽>0. (2.9) If −𝑊 is an 𝐻-matrix, then, there exists 𝜉=(𝜉₁,…,𝜉_𝑛)^𝑇>0 such that [ℳ(−𝑊)]^𝑇𝜉=𝛽>0. (2.10) Using the positive vector 𝜉, we define the distance in ℝ^𝑛 as follows: for any 𝑥,𝑦∈ℝ^𝑛, define 𝑥−𝑦_𝜉=^𝑛_𝑖=1𝜉_𝑖𝑥_𝑖−𝑦_𝑖. (2.11)

Definition 2.9. The equilibrium point 𝑢^∗ of (2.1) is said to be globally exponentially stable, if there exist constants 𝛼>0 and 𝑀>0 such that for any solution 𝑢(𝑡) of model (2.1), we have 𝑢(𝑡)−𝑢^∗_𝜉≤𝑀𝑢₀−𝑢^∗_𝜉exp{−𝛼𝑡}. (2.12) Also, we can consider the CGNNs with periodic input: 𝑑𝑢(𝑡)(𝑢(𝑡(𝑡)(𝑢(𝑡(𝑡,𝑑𝑡=−𝐴))𝐵𝑢−𝑊𝑓))−𝐼) (2.13) where 𝐼(𝑡)=(𝐼₁(𝑡),𝐼₂(𝑡),…,𝐼_𝑛(𝑡))^𝑇 is periodic input vectors with period 𝜔.

Definition 2.10. A periodic orbit 𝑢^∗(𝑡) of Cohen-Grossberg networks is said to be globally exponentially stable, if there exist constants 𝛼>0 and 𝑀>0 such that such that for any solution 𝑢(𝑡) of model (2.13), we have 𝑢(𝑡)−𝑢^∗(𝑡)≤𝑀𝑢₀−𝑢^∗_0𝜉exp{−𝛼𝑡}. (2.14)

3. Main Results

In this section, we shall establish some sufficient conditions to ensure the uniqueness of solutions, equilibrium point, output equilibrium point, and limit cycle as well as the global exponential stability of the state solutions.

Because Filippov solution includes set-valued function, in the general case, for a given initial condition, a discontinuous differential equation has multiple solutions starting at it [16]. Next, it will be shown that the uniqueness of solutions of model (2.1) can be obtained under the assumptions (A1) and (A2).

Theorem 3.1. Under the assumptions (A1) and (A2), if 𝑓∈ℱ_𝐷 and −𝑊 is an 𝑀-matrix or −𝑊 is an 𝐻-matrix such that 𝑊_𝑖𝑖<0, then, for any 𝑢₀ there is a unique solution 𝑢(𝑡) of model (2.1) with initial condition 𝑢(0)=𝑢₀, which is defined and bounded for all 𝑡≥0. Meanwhile, the corresponding output solution 𝛾(𝑡) of model (2.1) is uniquely defined and bounded for a.a. 𝑡≥0.

Proof. We only need to prove the uniqueness. Let 𝑢(𝑡) and 𝑢(𝑡),𝑡≥0 are two solutions of model (2.1) with the initial condition 𝑢(0)=𝑢(0)=𝑢₀.
Define 𝑉(𝑢−𝑢)=^𝑛_𝑖=1𝜉_𝑖^𝑢𝑖𝑢_𝑖𝑑𝑠𝑎_𝑖(𝑠). (3.1) Computing the time derivative of 𝑉 along the solutions of (2.1) gives 𝑑𝑉(𝑢−𝑢)=𝑑𝑡^𝑛_𝑖=1𝜉_𝑖sgn𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)−𝑏_𝑖𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)+^𝑛_𝑗=1𝑤_𝑖𝑗𝛾_𝑗(𝑡)−𝛾_𝑗(𝑡), (3.2) where sgn(𝑠)=1,𝑠>0,0,𝑠=0,−1,𝑠<0. (3.3) From 𝑓∈ℱ_𝐷 and 𝛾_𝑗(𝑡)∈𝐾[𝑓_𝑗(𝑢_𝑗(𝑡))],𝛾_𝑗(𝑡)∈𝐾[𝑓_𝑗(𝑢_𝑗(𝑡))], one can have sgn𝑢_𝑗(𝑡)−𝑢_𝑗(𝑡)𝛾_𝑗(𝑡)−𝛾_𝑗(𝑡)=𝛾_𝑗(𝑡)−𝛾_𝑗(𝑡). (3.4) Hence 𝑑𝑉(𝑢−𝑢)𝑑𝑡=−^𝑛_𝑖=1𝜉_𝑖𝑏_𝑖𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)+^𝑛_𝑖=1𝜉_𝑖𝑤_𝑖𝑖𝛾_𝑖(𝑡)−𝛾_𝑖(𝑡+_𝑛𝑖=1)𝜉_𝑖sgn𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)^𝑛_{𝑗=1,𝑗≠𝑖}𝑤_𝑖𝑗𝛾_𝑗(𝑡)−𝛾_𝑗(𝑡)≤−^𝑛_𝑖=1𝜉_𝑖𝑏_𝑖𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)−^𝑛_𝑖=1𝜉_𝑖𝑤_𝑖𝑖𝛾_𝑖(𝑡)−𝛾_𝑖(𝑡+_𝑛𝑖=1)𝜉_𝑖^𝑛_{𝑗=1,𝑗≠𝑖}𝑤_𝑖𝑗𝛾_𝑗(𝑡)−𝛾_𝑗(𝑡)=−^𝑛_𝑖=1𝜉_𝑖𝑏_𝑖𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)−^𝑛_𝑖=1𝜉_𝑖𝑤_𝑖𝑖𝛾_𝑖(𝑡)−𝛾_𝑖(𝑡+_{𝑛𝑗=1,𝑗≠𝑖})−𝑤_𝑖𝑗𝛾_𝑗(𝑡)−𝛾_𝑗(𝑡)=−^𝑛_𝑖=1𝜉_𝑖𝑏_𝑖𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)−^𝑛_𝑖=1𝜉_𝑖^𝑛_𝑗=1[ℳ(−𝑊)]_𝑖𝑗𝛾_𝑗(𝑡)−𝛾_𝑗(𝑡)=−^𝑛_𝑖=1𝜉_𝑖𝑏_𝑖𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)−𝜉,ℳ(−𝑊)𝑣(𝑡)=−^𝑛_𝑖=1𝜉_𝑖𝑏_𝑖𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)−[ℳ(−𝑊)]^𝑇𝜉,𝑣(𝑡)=−^𝑛_𝑖=1𝜉_𝑖𝑏_𝑖𝑢_𝑖(𝑡)−𝑢_𝑖(𝑡)−𝛽,𝑣(𝑡)≤0, (3.5) where 𝑣(𝑡)=(|𝛾₁(𝑡)−𝛾₁(𝑡)|,…,|𝛾_𝑛(𝑡)−𝛾_𝑛(𝑡)|)^𝑇. Integrating (3.1) between 0 and 𝑡₀>0, we have 𝑉𝑢𝑡₀−𝑢𝑡₀≤𝑉(𝑢(0)−𝑢(0))=𝑉𝑢₀−𝑢₀=0, (3.6) and hence, 𝑢(𝑡₀)=𝑢(𝑡₀) for any 𝑡₀>0; that is, the solution of model (2.1) with initial condition 𝑢(0)=𝑢₀ is unique.
From (2.5), the output solution 𝛾(𝑡) corresponding to 𝑢(𝑡) is uniquely defined and bounded for a.a. 𝑡≥0. The proof of Theorem 3.1 is completed.

Remark 3.2. Under the assumptions (A1) and (A2), if 𝑓∈ℱ_𝐷 and −𝑊 is an 𝑀-matrix or −𝑊 is an 𝐻-matrix such that 𝑊_𝑖𝑖<0, then, for any 𝐼∈ℝ^𝑛, model (2.1) has a unique equilibrium point and a unique corresponding output equilibrium point. Because from the assumptions, we have −𝑊 is LDS, hence, from Theorem 6 in [20], model (2.1) has a unique equilibrium point. By Definition 2.5, it is easily obtained that corresponding output equilibrium point is unique.
Next, global exponential stability of the equilibrium point of model (2.1) and the uniqueness and global exponential stability of limit cycle of model (2.13) are addressed. The results are given in following theorems.

Theorem 3.3. Under the assumptions (𝐴1) and (𝐴2), if 𝑓∈ℱ_𝐷 and −𝑊 is an 𝑀-matrix or −𝑊 is an 𝐻-matrix such that 𝑊_𝑖𝑖<0, then, for any 𝐼∈ℝ^𝑛, model (2.1) has a unique equilibrium point which is globally exponentially stable.

Proof. Let 𝑢(𝑡),𝑡≥0, be the solution of model (2.1) such that 𝑢(0)=𝑢₀, and for a.a. 𝑡≥0, let 𝛾(𝑡) be the corresponding output solution. For equilibrium point 𝑢^∗, 𝛾^∗ is corresponding output equilibrium point.
Since 𝑏_𝑖>0, we can choose a small 𝜀>0 such that 𝑏_𝑖−𝜀𝑎̌_𝑖>0. (3.7) Define 𝑉𝑢(𝑡)−𝑢^∗=𝑒^𝑛𝜀𝑡_𝑖=1𝜉_𝑖^𝑢𝑖_𝑢^(𝑡)_∗𝑖𝑑𝑠𝑎_𝑖(𝑠). (3.8) Computing the time derivative of 𝑉 along the solutions of (2.1), it follows that 𝑑𝑉𝑢(𝑡)−𝑢^∗𝑑𝑡≤𝑒^𝜀𝑡−^𝑛_𝑖=1𝜉_𝑖𝑏_𝑖−𝜀𝑎̌_𝑖𝑢_𝑖(𝑡)−𝑢^∗_𝑖−𝛽,𝑣(𝑡)≤0, (3.9) where 𝑣(𝑡)=(|𝛾₁(𝑡)−𝛾^∗₁|,…,|𝛾_𝑛(𝑡)−𝛾^∗_𝑛|)^𝑇.
Hence, 𝑉𝑢(𝑡)−𝑢^∗≤𝑉𝑢₀−𝑢^∗≤1𝑎̌𝑢₀−𝑢^∗_𝜉, (3.10) where 𝑎̆=min{𝑎̌₁,𝑎̌₂,…,𝑎̌_𝑛}.
On the other hand, 𝑉𝑢(𝑡)−𝑢^∗≥𝑒^𝜀𝑡1𝑎𝑢(𝑡)−𝑢^∗_𝜉, (3.11) where 𝑎=max{𝑎₁,𝑎₂,…,𝑎_𝑛}.
So, the following inequality holds: 𝑢(𝑡)−𝑢^∗_𝜉≤𝑎𝑎̌𝑢₀−𝑢^∗_𝜉𝑒^−𝜀𝑡, (3.12) that is, 𝑢^∗ is globally exponentially stable.

Remark 3.4. Since 𝑏_𝑖−𝜀/𝑎̌_𝑖>0, the exponential convergence rate 𝜀 can be estimated by means of the maximal allowable value by virtue of inequality 𝜀<𝑎̌_𝑖𝑏_𝑖,𝑖=1,2,…,𝑛. From this, one can see that amplification functions have key effect on the convergence rate of the stability for the considered model.
Next, the uniqueness and the exponentially stability of limit cycle for model (2.13) is given.

Theorem 3.5. Under the assumptions (A1) and (A2), if 𝑓∈ℱ_𝐷 and −𝑊 is an 𝑀-matrix or −𝑊 is an 𝐻-matrix such that 𝑊_𝑖𝑖<0, then model (2.13) has a unique globally exponentially stable limit cycle.

Proof. Let 𝑢(𝑡), 𝑢(𝑡) are two solutions of model (2.13), such that 𝑢(0)=𝑢₀,𝑢(0)=𝑢₀ respectively.
Define 𝑉(𝑢(𝑡)−𝑢(𝑡))=𝑒^𝑛𝜀𝑡_𝑖=1𝜉_𝑖^{𝑢𝑖(𝑡)}𝑢_𝑖(^𝑡₎𝑑𝑠𝑎_𝑖(𝑠). (3.13) Similar to the proof of Theorem 3.3, the following inequality holds: ‖𝑢(𝑡)−𝑢(𝑡)‖_𝜉≤𝑎𝑎̌𝑢₀−𝑢_0𝜉𝑒^−𝜀𝑡, (3.14) Define 𝑢^(𝑡)(𝜃)=𝑢(𝑡+𝜃). Define a mapping 𝐿∶𝑅^𝑛→𝑅^𝑛 by 𝐿(𝑢₀)=𝑢₀^(𝜔), then 𝐿^𝑘(𝑢₀)=𝑢₀^(𝑘𝜔). We can choose a positive integer 𝑘, such that for a positive constant 𝜌<1, 𝑎𝑎̌exp{−𝜀𝑘𝜔}≤𝜌<1. (3.15) And, from (3.14), we have 𝐿^𝑘(𝑢₀)−𝐿^𝑘(𝑢₀)_𝜉≤𝑎𝑎̌𝑢₀−𝑢_0𝜉exp{−𝜀(𝑘𝜔)}≤𝜌𝑢₀−𝑢_0𝜉. (3.16) By contraction mapping principle, there exists a unique fixed point 𝑢^∗₀ such that 𝐿^𝑘(𝑢^∗₀)=𝑢^∗₀. In addition, 𝐿^𝑘(𝐿(𝑢^∗₀))=𝐿(𝐿^𝑘(𝑢^∗₀))=𝐿(𝑢^∗₀); that is, 𝐿(𝑢^∗₀) is also a fixed point of 𝐿^𝑘. By the uniqueness of the fixed point of the mapping 𝐿^𝑘,𝐿(𝑢^∗₀)=𝑢^∗₀; that is, 𝑢₀^∗(𝜔)=𝑢^∗₀. Let 𝑢^∗(𝑡) be a state of model (1.1) with initial condition 𝑢^∗₀; we obtain for all 𝑖∈{1,2,…,𝑛}, 𝑑𝑢^∗_𝑖(𝑡)𝑑𝑡=−𝑎_𝑖𝑢^∗_𝑖(𝑡)𝑢^∗_𝑖(𝑡)−^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝑢^∗_𝑗(𝑡)−𝐼_𝑖(𝑡). (3.17) Then, for all 𝑖∈{1,2,…,𝑛}, 𝑑𝑢^∗_𝑖(𝑡+𝜔)𝑑𝑡=−𝑎_𝑖𝑢^∗_𝑖(𝑡+𝜔)𝑢^∗_𝑖(𝑡+𝜔)−^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝑢^∗_𝑗(𝑡+𝜔)−𝐼_𝑖(𝑡+𝜔)=−𝑎_𝑖𝑢^∗_𝑖(𝑡+𝜔)𝑢^∗_𝑖(𝑡+𝜔)−^𝑛_𝑗=1𝑤_𝑖𝑗𝑓_𝑗𝑢^∗_𝑗(𝑡+𝜔)−𝐼_𝑖(𝑡), (3.18) That is, 𝑢^∗(𝑡+𝜔)^𝑇 is also a state of the model (2.13) with initial condition 𝑢₀^∗(𝜔); here, 𝑢₀^∗(𝜔)=𝑢^∗₀; hence, for all 𝑡≥0, from Theorem 3.1, 𝑢^∗(𝑡+𝜔)=𝑢^∗(𝑡). (3.19) Hence, 𝑢^∗(𝑡) is an isolated periodic orbit of model (2.13) with period 𝜔, that is, a limit cycle of model (2.13). From (3.14), we can obtain that it is globally exponentially stable. The proof of Theorem 3.5 is completed.

Remark 3.6. Similar to those that are given in [18], global convergence of the output solutions in finite time also can be discussed, which can be embodied in the following example, and the detailed results are omitted.

4. Illustrative Example

In this section, we shall give an example to illustrate the effectiveness of our results.

Example 4.1. Consider the following CGNN model: 𝑑𝑢₁(𝑡)=𝑢𝑑𝑡2+0.4cos₁(𝑡)−𝑢₁(𝑡)−4sgn𝑢₁(𝑡)−2sgn𝑢₂(𝑡)+𝐼₁(𝑡),𝑑𝑢₂(𝑡)=𝑢𝑑𝑡2+0.4cos₂(𝑡)−𝑢₂(𝑡)+3sgn𝑢₁(𝑡)−2sgn𝑢₂(𝑡)+𝐼₂(𝑡), (4.1) where sgn(𝑠)=1,𝑠>0,undefined,𝑠=0,−1,𝑠<0. (4.2) Obviously, −𝑊 is an 𝐻-matrix with 𝑤_𝑖𝑖<0 and 1𝜉=541,𝛽=414. (4.3) Also, the subsets Π^𝐶, Π^𝐷, and Π^𝐶𝐷 in this example are the same as those in example 1 in [18] which are depicted in detail in Figure 3 in [18].

Firstly, we choose 𝐼=(0,0)^𝑇∈Π^𝐷,𝑢₀=(6,−6)^𝑇. The equilibrium point 𝑢^∗ of model (4.1) is (0,0)^𝑇, and the corresponding output equilibrium point 𝛾^∗=(0,0)^𝑇. Global convergence of 𝑢(𝑡) and 𝛾(𝑡) in finite time can be obtained. Figure 1 depicts the behavior of state solution 𝑢(𝑡) and output solution 𝛾(𝑡) with 𝐼=(0,0)^𝑇,𝑢₀=(6,−6)^𝑇.

(a)

(b)

(c)

(d)

Figure 1 

Transient behavior of (𝑢1,𝑢2) and (𝛾1,𝛾2) for 𝐼=(0,0)𝑇,𝑢0=(6,−6)𝑇.

Secondly, we choose 𝐼=(4,−6)^𝑇∈Π^𝐶,𝑢₀=(−6,6)^𝑇. Model (4.1) has a unique equilibrium point 𝑢^∗=(2,−1)^𝑇 and a unique output equilibrium point 𝛾^∗=(1,−1)^𝑇. Behavior of state solution and output solution is depicted in Figure 2.

(a)

(b)

(c)

(d)

Figure 2 

Transient behavior of (𝑢1,𝑢2) and (𝛾1,𝛾2) for 𝐼=(0,0)𝑇,𝑢0=(6,−6)𝑇.

(a)

(b)

(c)

(d)

Figure 3 

Transient behavior of (𝑢1,𝑢2) and (𝛾1,𝛾2) for 𝐼=(0,0)𝑇,𝑢0=(6,−6)𝑇.

Then, we choose 𝐼=(0,5)^𝑇∈Π^𝐶𝐷,𝑢₀=(4,−2)^𝑇.𝑢^∗=(0,1.5)^𝑇 and 𝛾^∗=(−0.5,1)^𝑇 are equilibrium point and output equilibrium point of model (4.1), respectively. Simulation results with 𝐼=(0,5)^𝑇,𝑢₀=(4,−2)^𝑇 about global convergence in finite time of the state solution 𝑢(𝑡) and corresponding output solution 𝛾(𝑡) are depicted in Figure 3.

5. Conclusions

In this paper, by using the property of 𝑀-matrix and a generalized Lyapunov-like approach, global convergence of CGNNs possessing discontinuous activation functions is investigated under the condition that neuron interconnection matrix belongs to the class of 𝑀-matrices or 𝐻-matrices. The uniqueness is proved for equilibrium point and corresponding output equilibrium point of considered neural networks. It is also proved that for considered model, the solution starting at a given initial condition is unique. Meanwhile, global exponential stability of equilibrium point is obtained for any input. Furthermore, by contraction mapping principle, the uniqueness and the globally exponential stability of limit cycle are given.

Acknowledgments

The authors appreciate the editor's work and the reviewers' insightful comments and constructive suggestions. This work is supported by the Excellent Youthful Talent Foundation of Colleges and Universities of Anhui Province of China under Grant no. 2011SQRL030ZD.