Novel Criteria on Global Robust Exponential Stability to a Class of Reaction-Diffusion Neural Networks with Delays

Abstract

The global exponential robust stability is investigated to a class of reaction-diffusion Cohen-Grossberg neural network (CGNNs) with constant time-delays, this neural network contains time invariant uncertain parameters whose values are unknown but bounded in given compact sets. By employing the Lyapunov-functional method, several new sufficient conditions are obtained to ensure the global exponential robust stability of equilibrium point for the reaction diffusion CGNN with delays. These sufficient conditions depend on the reaction-diffusion terms, which is a preeminent feature that distinguishes the present research from the previous research on delayed neural networks with reaction-diffusion. Two examples are given to show the effectiveness of the obtained results.

1. Introduction

In recent years, considerable attention has been paid to study the dynamics of artificial neural networks with fixed parameters because of their potential applications in the areas such as signal and image processing, pattern recognition, parallel computations, and optimization problems [1–10]. However, during the implementation on very scale integration chips, the stability of a well-designed system may often be destroyed by its unavoidable uncertainty due to the existence of modelling error, external disturbance, and parameter fluctuation. In general, on other hand, a mathematical description is only an approximation of the actual physical system and deals with fixed nominal parameters. Usually, these parameters are not known exactly due to the imperfect identification or measurement, aging of components and/or changes in the environmental condition. Thus, it is almost impossible to get an exact model for the system due to the existence of various parameter uncertainties. So it is essential to introduce the robust technique to design a system with such uncertainty [11, 12]. If the uncertainty of a system is only due to the deviations and perturbations of its parameters, and if those deviations and perturbations are all bounded, then the system is called an interval system [13–23]. Recently, Chen and Rong [23] considered a class of Cohen-Grossberg neural networks (CGNNs) with time-varying delays. Several sufficient conditions were given to ensure global exponential robust stability.

In a real world, strictly speaking, the diffusion phenomena could not be ignored in neural networks and electric circuits once electrons transport in a nonuniform electromagnetic field. Hence, it is essential to consider the state variables varying with the time and space variables. The neural networks with diffusion terms can commonly be expressed by partial differential equations. Recently, some authors have devoted to the study of reaction-diffusion neural networks, for instance see [24–29] and references therein. In particular, more recently, Liu et al. [27] and Wang et al. [28], considered the global exponential robust stability of a class of reaction-diffusion Hopfield neural networks with distributed delays and time-varying delay, respectively. Song and Cao [29] have obtained the criteria to guarantee the global exponential robust stability of a class of reaction-diffusion CGNNs with time-varying delays and Neumann boundary condition. In [27–29], unfortunately, owing to the divergence theorem employed, a negative integral term with gradient is left out in their deduction. As a result, the global exponential robust stability criteria acquired by them do not contain a diffusion term. In other words, the diffusion term does not take effect in their deduction and sufficient conditions. The same case appears also in the other literatures [24–26].

Motivated by the above discussions, in this paper we will consider a class of reaction-diffusion CGNNs with constant time delays and a boundary condition. We will construct a appropriate Lyapunov functional to derive some new criteria ensuring the global exponential robust stability for an equilibrium point of the delayed reaction-diffusion CGGNs with the boundary condition. The present work differs from the paper [27–29] since (i) the diffusion terms play an important role in the global exponential robust stability criteria in the paper, (ii) the boundary condition of CGNNs model considered includes the Neumann type and the Dirichlet type while the boundary condition of model in [27, 28] is the Neumann type. The work will have significance impact on the design and applications of globally exponentially robustly stable reaction-diffusion neural network with delays and is of great interest in many applications.

The rest of this paper is organized as follows. In Section 2, model description and preliminaries are given. In Section 3, several criteria are derived for the global exponential robust stability for an equilibrium point of reaction-diffusion CGNNs with delays and the boundary condition. Then, we give two examples and comparison to illustrate our criteria in Section 4. Finally, in Section 5, some conclusions are made.

2. Model Description and Preliminaries

To begin with, we introduce some notations.

(i)Ω is an open bounded domain in ℝ^𝑚 with smooth boundary 𝜕Ω, and mesΩ>0 as mesΩ denotes the measure of Ω. Ω=Ω∪𝜕Ω. (ii)𝐿²(Ω) is the space of real Lebesgue measurable functions on Ω which is a Banach space. Define the inner product ⟨𝑢,𝑣⟩=∫_Ω𝑢𝑣𝑑𝑥, for any 𝑢,𝑣∈𝐿²(Ω) and the 𝐿²-norm ‖𝑢‖₂∶=⟨𝑢,𝑢⟩^1/2, for 𝑢∈𝐿²(Ω). (iii)𝐻¹(Ω)∶={𝑤∈𝐿²(Ω),𝐷_𝑖𝑤∈𝐿²(Ω)}, where 𝐷_𝑖𝑤=𝜕𝑤/𝜕𝑥_𝑖, 1≤𝑖≤𝑚. 𝐻¹₀(Ω)∶= the closure of 𝐶^∞₀(Ω) in 𝐻¹(Ω). (iv)Let 𝒞∶=𝒞([0,+∞)×Ω,ℝ^𝑛) be the Banach space of continuous functions which map [0,+∞)×Ω into ℝ^𝑛 with the norm ‖𝑢‖₂=(∑^𝑛_𝑖=1‖𝑢_𝑖‖²₂)^1/2 for 𝑢=(𝑢₁,…,𝑢_𝑛)^𝑇∈𝒞, where ‖𝑢_𝑖‖₂=(∫_Ω|𝑢_𝑖|²𝑑𝑥)^1/2, 𝑖=1,…,𝑛. (v)Let 𝒞₁∶=𝒞([−𝜏,0]×Ω,ℝ^𝑛) be the Banach space of bounded continuous functions which map [−𝜏,0]×Ω into ℝ^𝑛 with the following norm: ‖𝜙‖_𝜏∶=sup_{𝑠∈[−𝜏,0]}‖𝜙(𝑠)‖₂, for any 𝜙(𝑠)=(𝜙₁(𝑠),…,𝜙_𝑛(𝑠))^𝑇∈𝒞₁, where ‖𝜙(𝑠)‖_𝒞1=(∑^𝑛_𝑖=1∫_Ω|𝜙_𝑖(𝑠)|²𝑑𝑥)^1/2.

Consider the following reaction-diffusion CGNNs with interval coefficients and delays on Ω:

𝜕𝑢_𝑖(𝑡,𝑥)=𝜕𝑡^𝑚_𝑙=1𝜕𝜕𝑥_𝑙(𝑑_𝑖𝑙𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑥_𝑙)−𝑎_𝑖(𝑢_𝑖(𝑡,𝑥))×[𝑏_𝑖(𝑢_𝑖(𝑡,𝑥))−^𝑛_𝑗=1𝑠_𝑖𝑗𝑓_𝑗(𝑢_𝑗(𝑡,𝑥))−^𝑛_𝑗=1𝑡_𝑖𝑗𝑔_𝑗(𝑔_𝑗(𝑡−𝜏_𝑖𝑗,𝑥))+𝐼_𝑖],(𝑡,𝑥)∈[0,+∞)×Ω, (2.1) for 𝑖=1,…,𝑛, 𝑥=(𝑥₁,…,𝑥_𝑚)^𝑇∈Ω is space variable, 𝑢_𝑖(𝑡,𝑥) corresponds to the state of the 𝑖th unit at time 𝑡 and in space 𝑥; 𝑑_𝑖𝑙>0, for 𝑖=1,…,𝑛, 𝑙=1,…,𝑚, corresponds to the transmission diffusion coefficient along the 𝑖th neuron, 𝑑_𝑖=min_{1≤𝑙≤𝑚}{𝑑_𝑖𝑙} for 𝑖=1,…,𝑛; 𝑎_𝑖(𝑢_𝑖(𝑡,𝑥)) represents an amplification function; 𝑏_𝑖(𝑢_𝑖(𝑡,𝑥)) is an appropriate behavior function; 𝑠_𝑖𝑗, 𝑡_𝑖𝑗 denote the connection strengths of the 𝑗th neuron on the 𝑖th neuron, respectively; 𝑔_𝑗(𝑢_𝑗(𝑡,𝑥)), 𝑓_𝑗(𝑢_𝑗(𝑡,𝑥)) denote the activation functions of 𝑗th neuron at time 𝑡 and in space 𝑥; 𝜏_𝑖𝑗(0<𝜏_𝑖𝑗≤𝜏) corresponds to the transmission delay along the axon of the 𝑗th unit from the 𝑖th unit. 𝐼_𝑖 is the constant input from outside of the network.

Throughout this paper, we assume the following.

(H1)Each function 𝑎_𝑖(𝜉) is positive, continuous, and bounded, that is, there exist constants 𝑎_𝑖, 𝑎_𝑖 such that 0<𝑎_𝑖≤𝑎_𝑖(𝜉)≤𝑎_𝑖<∞, for 𝜉∈ℝ, 𝑖=1,…,𝑛.(H2)Each function 𝑏_𝑖(𝜉)∈𝒞¹(ℝ,ℝ) and 𝑏^′_𝑖(𝜉)≥𝑏_𝑖≥0 is locally Lipschitz continuous.(H3)The activation functions 𝑓_𝑗(𝜉) and 𝑔_𝑗(𝜉) satisfy Lipschitz condition, that is, there exist two positive diagonal matrices 𝐹=diag(𝐹₁,…,𝐹_𝑛) and 𝐺=diag(𝐺₁,…,𝐺_𝑛) such that |𝑓_𝑗(𝜉₁)−𝑓_𝑗(𝜉₂)|≤𝐹_𝑗|𝜉₁−𝜉₂|,|𝑔_𝑗(𝜉₁)−𝑔_𝑗(𝜉₂)|≤𝐺_𝑗|𝜉₁−𝜉₂|, (2.2) for all 𝜉₁,𝜉₂∈𝑅(𝜉₁≠𝜉₂), 𝑗=1,…,𝑛.

Remark 2.1. The activation functions 𝑓_𝑗 and 𝑔_𝑗, 𝑗=1,…,𝑛, are typically assumed to be sigmoidal which implies that they are monotone, bounded, and smooth. However, in this paper, we only need the previous weaker assumptions.

We assume that the nonlinear delayed systems (2.1) are supplemented with the boundary condition:

𝐵[𝑢_𝑖(𝑡,𝑥)]=0,for(𝑡,𝑥)∈[−𝜏,+∞)×𝜕Ω,𝑖=1,…,𝑛, (2.3) where 𝐵[𝑢_𝑖(𝑡,𝑥)]=𝑢_𝑖(𝑡,𝑥) is said the Dirichlet boundary condition, 𝐵[𝑢_𝑖(𝑡,𝑥)]=𝜕𝑢_𝑖(𝑡,𝑥)/𝜕𝑚 is said the Neumann boundary condition, where 𝜕𝑢_𝑖(𝑡,𝑥)/𝜕𝑚 = (𝜕𝑢_𝑖(𝑡,𝑥)/𝜕𝑥₁, …, 𝜕𝑢_𝑖(𝑡,𝑥)/𝜕𝑥_𝑚)^𝑇 denotes the outward normal derivative on 𝜕Ω.

Systems (2.1) are equipped with the initial condition:

𝑢_𝑖(𝑠,𝑥)=𝜙_𝑖(𝑠,𝑥),for(𝑠,𝑥)∈[−𝜏,0]×Ω,𝑖=1,…,𝑛, (2.4) where 𝜙∶=(𝜙₁,…,𝜙_𝑛)^𝑇∈𝒞.

Given boundary condition (2.3) and initial function (2.4), the existence on the solutions of systems (2.1), the reader can refer to [18]. We denote the solution by 𝑢(𝑡,𝜙,𝑥)∶=(𝑢₁(𝑡,𝜙,𝑥), …,𝑢_𝑛(𝑡,𝜙,𝑥))^𝑇 and sometimes it is denoted by 𝑢(𝑡,𝑥), 𝑢(𝑡) or 𝑢 for short when there is no risk of confusion.

Lemma 2.2. Under assumptions (H1)–(H3), system (2.1) has a unique equilibrium point, if (H4)𝑏_𝑖>Σ^𝑛_𝑗=1(𝑠^∗_𝑖𝑗𝐹_𝑗+𝑡^∗_𝑖𝑗𝐺_𝑗), for 𝑖=1,…,𝑛.

As for the proof of Lemma 2.2, the reader can refer to [21, 28]. Here, we omit it.

Definition 2.3. An equilibrium point 𝑢^∗ of system (2.1)–(2.4) is said to be globally exponentially stable on 𝐿²-norm, if there exist constant 𝜂>0 and 𝑀≥1 such that ‖𝑢(𝑡,𝑥)−𝑢^∗‖₂≤𝑀‖𝜙−𝑢^∗‖_𝜏𝑒^−𝜂𝑡∀𝑡≥0, (2.5) where ‖𝜙−𝑢^∗‖_𝜏=sup_{−𝜏≤𝑠≤0}‖𝜙(𝑠,𝑥)−𝑢^∗‖₂.

Definition 2.4. Let 𝑠_𝑖𝑗≤𝑠_𝑖𝑗≤𝑠_𝑖𝑗, 𝑠^∗_𝑖𝑗=max(|𝑠_𝑖𝑗|,|𝑠_𝑖𝑗|), 𝑡_𝑖𝑗≤𝑡_𝑖𝑗≤𝑡_𝑖𝑗, 𝑡^∗_𝑖𝑗=max(|𝑡_𝑖𝑗|,|𝑡_𝑖𝑗|), 𝐼_𝑖≤𝐼_𝑖≤𝐼_𝑖, 𝐼^∗_𝑖=max(|𝐼_𝑖|,|𝐼_𝑖|), 𝜏_𝑖=max(|𝜏_𝑖𝑗|). An equilibrium point 𝑢^∗ of system (2.1)–(2.4) is said to be globally exponentially robustly stable if its equilibrium point 𝑢^∗ is globally exponentially stable for all 𝑠_𝑖𝑗≤𝑠_𝑖𝑗≤𝑠_𝑖𝑗, 𝑡_𝑖𝑗≤𝑡_𝑖𝑗≤𝑡_𝑖𝑗, 𝐼_𝑖≤𝐼_𝑖≤𝐼_𝑖, 𝜏_𝑖𝑗≤𝜏_𝑖𝑗≤𝜏_𝑖𝑗, for 𝑖,𝑗=1,…,𝑛.

Lemma 2.5 (Poincaré inequality [30–32]). Let Ω be a bounded domain of ℝ^𝑚 with a smooth boundary 𝜕Ω of class 𝒞² by Ω. 𝑣(𝑥) is a real-valued function belonging to 𝐻¹₀(Ω) and 𝐵[𝑣(𝑥)]|_𝜕Ω=0. Then 𝜆_1Ω|𝑣(𝑥)|²𝑑𝑥≤_Ω|∇𝑣(𝑥)|²𝑑𝑥, (2.6) which 𝜆₁ is the lowest positive eigenvalue of the Laplacian with boundary condition −Δ𝜓(𝑥)=𝜆𝜓(𝑥),𝑥∈Ω,𝐵[𝜓(𝑥)]=0,𝑥∈𝜕Ω. (2.7)

Regarding the proof of Lemma 2.5, we refer to any textbook on partial differential equations. For example, [30, 31] or [32] are good standard references.

Remark 2.6. (i) When Ω is bounded or at least bounded in one direction, not only limited to a rectangle domain, inequality (2.6) holds. (ii) The lowest positive eigenvalue 𝜆₁ of the Laplacian is sometimes known as the first eigenvalue. Determining the lowest eigenvalue 𝜆₁ is, in general, a very hard task that depends upon the geometry of the domain Ω. Certain special cases are tractable, however. For example, let the Laplacian on Ω={(𝑥₁,𝑥₂)^𝑇∈ℝ²∣0<𝑥₁<𝑎,0<𝑥₂<𝑏}, if 𝐵[𝑣(𝑥)]=𝑣(𝑥) or 𝐵[𝑣(𝑥)]=𝜕𝑣(𝑥)/𝜕𝑚, then 𝜆₁=(𝜋/𝑎)²+(𝜋/𝑏)² or 𝜆₁=min{(𝜋/𝑎)²,(𝜋/𝑏)²}, respectively. (iii) Although the eigenvalue 𝜆₁ of the laplacian with the Dirichlet boundary condition on a generally bounded domain Ω cannot be determined exactly, a lower bound of it may nevertheless be estimated by 𝜆₁≥(𝑚²/(𝑚+2))((2𝜋)²/𝜔_𝑚−1)(1/𝑉)^2/𝑚, where 𝜔_𝑚−1 is a surface area of the unit ball in ℝ^𝑚, 𝑉 is a volume of domain Ω [33].

3. Main Results

Theorem 3.1. Let hypotheses (H1)–(H4) hold. Assume further that (A1)2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)>𝑎_𝑖∑^𝑛_𝑗=1(𝑠^∗_𝑖𝑗𝐹_𝑖+𝑡^∗_𝑖𝑗𝐺_𝑖)+∑^𝑛_𝑗=1𝑎_𝑗(𝑠^∗_𝑗𝑖𝐹_𝑗+𝑡^∗_𝑗𝑖𝐺_𝑗), for 𝑖=1,…,𝑛, then equilibrium point 𝑢^∗ of system (2.1) with (2.3) and (2.4) is globally exponentially robust stable for each constant input 𝐼∈ℝ^𝑛.

Proof. Let 𝑦_𝑖(𝑡)=𝑢_𝑖(𝑡)−𝑢^∗. 𝑦_𝑖(𝑡) is denoted by 𝑦_𝑖 for short. From (2.1), we obtain 𝜕𝑦_𝑖=𝜕𝑡^𝑚_𝑙=1𝜕𝜕𝑥_𝑙(𝑑_𝑖𝑙𝜕𝑦_𝑖𝜕𝑥_𝑙)−𝑎_𝑖(𝑢_𝑖)[𝑏̃_𝑖(𝑦_𝑖)−^𝑛_𝑗=1𝑠_𝑖𝑗𝑓_𝑗(𝑦_𝑗)−^𝑛_𝑗=1𝑡_𝑖𝑗𝑔_𝑗(𝑦_𝑗(𝑡−𝜏_𝑖𝑗))], (3.1) for (𝑡,𝑥)∈[0,+∞)×Ω, 𝑖=1,…,𝑛, where 𝑏̃_𝑖(𝑦_𝑖)=𝑏_𝑖(𝑦_𝑖+𝑢^∗_𝑖)−𝑏_𝑖(𝑢^∗_𝑖),𝑓_𝑗(𝑦_𝑗)=𝑓_𝑗(𝑦_𝑗+𝑢^∗_𝑗)−𝑓_𝑗(𝑢^∗_𝑗𝑔),_𝑗(𝑦_𝑗)=𝑔_𝑗(𝑦_𝑗+𝑢^∗_𝑗)−𝑔_𝑗(𝑢^∗_𝑗), (3.2) for 𝑖,𝑗=1,…,𝑛.
Taking the inner product of both sides of (3.1) with 𝑦_𝑖, we get
12𝑑𝑑𝑡‖𝑦_𝑖‖²₂=_Ω𝑦_𝑖^𝑚_𝑙=1𝜕𝜕𝑥_𝑙(𝑑_𝑖𝑙𝜕𝑦_𝑖𝜕𝑥_𝑙)𝑑𝑥+_Ω𝑦_𝑖𝑎_𝑖(𝑢_𝑖)^𝑛_𝑗=1𝑠_𝑖𝑗𝑓_𝑗(𝑦_𝑗−)𝑑𝑥_Ω𝑦_𝑖𝑎_𝑖(𝑢_𝑖)𝑏̃_𝑖(𝑦_𝑖)𝑑𝑥+_Ω𝑦_𝑖𝑎_𝑖(𝑢_𝑖)^𝑛_𝑗=1𝑡_𝑖𝑗𝑔_𝑗(𝑦_𝑗(𝑡−𝜏_𝑖𝑗))𝑑𝑥, (3.3) for 𝑡∈[0,+∞), 𝑖=1,…,𝑛.
From the boundary condition (2.3), Gauss formula and Lemma 2.2, we have
_Ω𝑦_𝑖^𝑚_𝑙=1𝜕𝜕𝑥_𝑙(𝑑_𝑖𝑙𝜕𝑦_𝑖𝜕𝑥_𝑙)𝑑𝑥=−_Ω^𝑚_𝑙=1𝑑_𝑖𝑙(𝜕𝑦_𝑖𝜕𝑥_𝑙)²𝑑𝑥≤−𝑑_𝑖Ω^𝑚_𝑙=1(𝜕𝑦_𝑖𝜕𝑥_𝑙)²𝑑𝑥=−𝑑_𝑖Ω|∇𝑦_𝑖|²𝑑𝑥≤−𝜆₁𝑑_𝑖Ω𝑦²_𝑖𝑑𝑥=−𝜆₁𝑑_𝑖‖𝑦_𝑖‖²₂. (3.4) From assumption (H2), we get _Ω𝑦_𝑖𝑎_𝑖(𝑢_𝑖)𝑏̃_𝑖(𝑦_𝑖)𝑑𝑥≥_Ω𝑎_𝑖𝑏_𝑖|𝑦_𝑖|²𝑑𝑥≥𝑎_𝑖𝑏_𝑖‖𝑦_𝑖‖²₂. (3.5) From assumptions (H1) and (H3), we obtain _Ω𝑦_𝑖𝑎_𝑖(𝑢_𝑖)^𝑛_𝑗=1𝑠_𝑖𝑗𝑓_𝑗(𝑦_𝑗)𝑑𝑥≤^𝑛_𝑗=1Ω𝑠^∗_𝑖𝑗𝑎_𝑖𝐹_𝑗|𝑦_𝑖||𝑦_𝑗|𝑑𝑥≤^𝑛_𝑗=1𝑠^∗_𝑖𝑗𝑎_𝑖𝐹_𝑗‖𝑦_𝑖‖₂‖𝑦_𝑗‖₂. (3.6) By the same way, we have _Ω𝑦_𝑖𝑎_𝑖(𝑢_𝑖)^𝑛_𝑗=1𝑡_𝑖𝑗𝑔_𝑗(𝑦_𝑗(𝑡−𝜏_𝑖𝑗))𝑑𝑥≤^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝑎_𝑖𝐺_𝑗‖𝑦_𝑖‖₂‖𝑦_𝑗(𝑡−𝜏_𝑖𝑗)‖₂. (3.7) Combining (3.4)–(3.7) into (3.3), we obtain 𝑑𝑑𝑡‖𝑦_𝑖‖²₂≤−2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)‖𝑦_𝑖‖²₂+2^𝑛_𝑗=1𝑠^∗_𝑖𝑗𝑎_𝑖𝐹_𝑗‖𝑦_𝑖‖₂‖𝑦_𝑗‖₂+2^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝑎_𝑖𝐺_𝑗‖𝑦_𝑖‖₂‖𝑦_𝑗(𝑡−𝜏_𝑖𝑗)‖₂, (3.8) for 𝑡∈[0,+∞).
According to (A1), we can choose a sufficiently small 𝜇>0 such that
2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)−𝜇−^𝑛_𝑗=1𝑎_𝑖(𝑠^∗_𝑖𝑗𝐹_𝑖+𝑡^∗_𝑖𝑗𝐺_𝑖)−^𝑛_𝑗=1𝑎_𝑗(𝑠^∗_𝑗𝑖𝐹_𝑗+𝑡^∗_𝑗𝑖𝐺_𝑗𝑒^𝜇𝜏)>0. (3.9) Now consider the Lyapunov functional 𝑉(𝑡) defined by 𝑉(𝑡)=^𝑛_𝑖=1[‖𝑦_𝑖‖²₂𝑒^𝜇𝑡+𝑎_𝑖^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗^𝑡_{𝑡−𝜏𝑖𝑗}𝑒^{𝜇(𝑠+𝜏𝑖𝑗)}‖𝑦_𝑗(𝑠)‖²₂𝑑𝑠]. (3.10) By calculating the upper right Dini derivative 𝐷⁺𝑉(𝑡) of 𝑉(𝑡) along the solutions of (3.1), we get 𝐷⁺𝑉(𝑡)=𝑒^𝑛𝜇𝑡_𝑖=1{𝜇‖𝑦_𝑖‖²₂+𝑑𝑑𝑡‖𝑦_𝑖‖²₂+𝑎_𝑖^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗𝑒^{𝜇𝜏𝑖𝑗}‖𝑦_𝑗(𝑡)‖²₂−𝑎_𝑖^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗‖𝑦_𝑗(𝑡−𝜏_𝑖𝑗)‖²₂}≤𝑒^𝑛𝜇𝑡_𝑖=1{[−2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)+𝜇]‖𝑦_𝑖‖²₂+2^𝑛_𝑗=1𝑠^∗_𝑖𝑗𝑎_𝑖𝐹_𝑗‖𝑦_𝑖‖₂‖𝑦_𝑗‖₂+2^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝑎_𝑖𝐺_𝑗‖𝑦_𝑖‖₂‖𝑦_𝑗(𝑡−𝜏_𝑖𝑗)‖²₂+𝑎_𝑖^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗𝑒^{𝜇𝜏𝑖𝑗}‖𝑦_𝑗‖²₂−𝑎_𝑖^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗‖𝑦_𝑗(𝑡−𝜏_𝑖𝑗)‖²₂}≤𝑒^𝑛𝜇𝑡_𝑖=1{[−2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)+𝜇]‖𝑦_𝑖‖²₂+^𝑛_𝑗=1𝑠^∗_𝑖𝑗𝑎_𝑖𝐹_𝑗‖𝑦_𝑖‖²₂+^𝑛_𝑗=1𝑠^∗_𝑖𝑗𝑎_𝑖𝐹_𝑗‖𝑦_𝑗‖²₂+^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝑎_𝑖𝐺_𝑗‖𝑦_𝑖‖²₂+^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝑎_𝑖𝐺_𝑗‖𝑦_𝑗(𝑡−𝜏_𝑖𝑗)‖₂+𝑎_𝑖^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗𝑒^{𝜇𝜏𝑖𝑗}‖𝑦_𝑗‖²₂−𝑎_𝑖^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗‖𝑦_𝑗(𝑡−𝜏_𝑖𝑗)‖²₂}≤𝑒^𝑛𝜇𝑡_𝑖=1{[−2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)+𝜇+𝑎_𝑖(^𝑛_𝑗=1𝑠^∗_𝑖𝑗𝐹_𝑗+^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗)]‖𝑦_𝑖‖²₂+(^𝑛_𝑗=1𝑠^∗_𝑖𝑗𝑎_𝑖𝐹_𝑗+^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝑎_𝑖𝐺_𝑗𝑒^{𝜇𝜏𝑖𝑗})‖𝑦_𝑗‖²₂}≤𝑒^𝑛𝜇𝑡_𝑖=1{[−2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)+𝜇+𝑎_𝑖(^𝑛_𝑗=1𝑠^∗_𝑖𝑗𝐹_𝑗+^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗)]‖𝑦_𝑖‖²₂+(^𝑛_𝑗=1𝑠^∗_𝑗𝑖𝑎_𝑗𝐹_𝑖+^𝑛_𝑗=1𝑡^∗_𝑗𝑖𝑎_𝑗𝐺_𝑖𝑒^𝜇𝜏)‖𝑦_𝑖‖²₂}=𝑒^𝑛𝜇𝑡_𝑖=1{−2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)+𝜇+𝑎_𝑖^𝑛_𝑗=1(𝑠^∗_𝑖𝑗𝐹_𝑗+^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗+_𝑛𝑗=1)𝑎_𝑗(𝑠^∗_𝑗𝑖𝐹_𝑖+𝑡^∗_𝑗𝑖𝐺_𝑖𝑒^𝜇𝜏)}‖𝑦_𝑖‖²₂, (3.11)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) for 𝑡∈[0,+∞). Hence ‖𝑦(𝑡)‖²₂𝑒^𝜇𝑡≤𝑉(𝑡)≤𝑉(0),for𝑡∈[0,+∞). (3.12) Note that 𝑉(0)=^𝑛_𝑖=1[‖𝑦_𝑖(0)‖²₂+𝑎_𝑖^𝑛_𝑗=1𝑡^∗_𝑖𝑗𝐺_𝑗⁰_{−𝜏𝑖𝑗}𝑒^{𝜇(𝑠+𝜏𝑖𝑗)}‖𝑦_𝑗(𝑠)‖²₂1𝑑𝑠]≤[1+𝜇max_{1≤𝑖≤𝑛}{^𝑛_𝑗=1𝑎_𝑗𝑡^∗_𝑗𝑖𝐺_𝑖𝜏_𝑗𝑖𝑒^{𝜇𝜏𝑗𝑖}}]^𝑛_𝑖=1‖𝑦_𝑖‖²_𝜏. (3.13) Denote 𝑀>0 and 𝑀²1=1+𝜇max_{1≤𝑖≤𝑛}{^𝑛_𝑗=1𝑎_𝑗𝑡^∗_𝑗𝑖𝐺_𝑖𝜏_𝑗𝑖𝑒^{𝜇𝜏𝑗𝑖}}, (3.14) where 𝑀>0, then 𝑀≥1. So ‖𝑦(𝑡)‖²₂≤𝑀²‖𝜙−𝑢^∗‖²_𝜏𝑒^−𝜇𝑡for𝑡∈[0,+∞), (3.15) that is, ‖𝑢(𝑡)−𝑢^∗‖₂≤𝑀‖𝜙−𝑢^∗‖_𝜏𝑒^{−(1/2𝜇𝑡)}for𝑡∈[0,+∞). (3.16) Since all the solutions of system (2.1)–(2.4) tend to 𝑢^∗ exponentially as 𝑡→+∞ for any values of the coefficients in system (2.1) with (2.3)-(2.4), that is, the system described by (2.1) with (2.3)- (2.4) has a unique equilibrium which is globally exponentially robust stable on 𝐿²-norm and the theorem is proved.

Remark 3.2. In the deduction for Theorem 3.1, by Lemma 2.5, we have obtained −𝑑_𝑖∫_Ω|∇𝑦_𝑖|²𝑑𝑥≤−𝜆₁𝑑_𝑖‖𝑦_𝑖(𝑡)‖²₂ (see (3.4)). This is an important step. As a result, the condition of Theorem 3.1 includes the diffusion terms.
Changing a little the Lyapunov functional (3.10) by
𝑉(𝑡)=^𝑛_𝑖=1[‖𝑦_𝑖‖²₂𝑒^𝜇𝑡+𝑎_𝑖^𝑛_𝑗=1𝑡_𝑖𝑗𝐺²_𝑗^𝑡_{𝑡−𝜏𝑖𝑗}𝑒^{𝜇(𝑠+𝜏𝑖𝑗)}‖𝑦_𝑗(𝑠)‖²₂𝑑𝑠], (3.17) and using the similar way of the proof of Theorem 3.1, we derive another new criterion.

Theorem 3.3. Under assumptions (H1)–(H4), if, in addition (A2)2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)>∑^𝑛_𝑗=1𝑎_𝑖(𝑠^∗_𝑖𝑗+𝑡^∗_𝑖𝑗)+∑^𝑛_𝑗=1𝑎_𝑗(𝑠^∗_𝑗𝑖𝐹²_𝑗+𝑡^∗_𝑗𝑖𝐺²_𝑗),   for 𝑖=1,…,𝑛, then equilibrium point 𝑢^∗ of system (2.1) with (2.3)-(2.4) is globally exponentially robust stable for each constant input 𝐼∈ℝ^𝑛.

For system (2.1), when the strength of the neuron interconnections 𝑠_𝑖𝑗 and 𝑡_𝑖𝑗 (𝑖,𝑗=1,…,𝑛) is fixed constant matrices, the following result is obvious from Theorems 3.1 and 3.3.

Corollary 3.4. Under assumptions (H1)–(H4), if any one of the following condition is true: (A3)2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)>𝑎_𝑖∑^𝑛_𝑗=1(𝑠_𝑖𝑗𝐹_𝑖+𝑡_𝑖𝑗𝐺_𝑖)+∑^𝑛_𝑗=1𝑎_𝑗(𝑠_𝑗𝑖𝐹_𝑗+𝑡_𝑗𝑖𝐺_𝑗),(A4)2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)>𝑎_𝑖∑^𝑛_𝑗=1(𝑠_𝑖𝑗+𝑡_𝑖𝑗)+∑^𝑛_𝑗=1𝑎_𝑗(𝑠_𝑗𝑖𝐹²_𝑗+𝑡_𝑗𝑖𝐺²_𝑗),   for 𝑖=1,…,𝑛, then equilibrium point 𝑢^∗ of system (2.1) with (2.3)-(2.4) is globally exponentially stable.

Remark 3.5. When 𝑎_𝑖(𝑢_𝑖(𝑡,𝑥))=1, 𝑏_𝑖(𝑢_𝑖(𝑡,𝑥))=𝑏_𝑖𝑢_𝑖(𝑡,𝑥), 𝑖=1,…,𝑛, then system (2.1) reduces to the following reaction-diffusion cellular neural network: 𝜕𝑢_𝑖(𝑡,𝑥)=𝜕𝑡^𝑚_𝑙=1𝜕𝜕𝑥_𝑙(𝑑_𝑖𝑙𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑥_𝑙)−[𝑏_𝑖𝑢_𝑖(𝑡,𝑥)−^𝑛_𝑗=1𝑠_𝑖𝑗𝑓_𝑗(𝑢_𝑗(𝑡,𝑥))−^𝑛_𝑗=1𝑡_𝑖𝑗𝑔_𝑗(𝑔_𝑗(𝑡−𝜏_𝑖𝑗,𝑥))+𝐼_𝑖],(𝑡,𝑥)∈[0,+∞)×Ω, (3.18) for 𝑖=1,…,𝑛.

From Theorems 3.1 and 3.3, we have the following results.

Corollary 3.6. Under assumptions (H3) and (H4), if, in addition, any one of the following condition is true: (A3)2(𝑑_𝑖𝜆₁+𝑏_𝑖)>∑^𝑛_𝑗=1(𝑠^∗_𝑖𝑗𝐹_𝑖+𝑡^∗_𝑖𝑗𝐺_𝑖)+∑^𝑛_𝑗=1(𝑠^∗_𝑗𝑖𝐹_𝑗+𝑡^∗_𝑗𝑖𝐺_𝑗),(A4)2(𝑑_𝑖𝜆₁+𝑏_𝑖)>∑^𝑛_𝑗=1(𝑠^∗_𝑖𝑗+𝑡^∗_𝑖𝑗)+∑^𝑛_𝑗=1(𝑠^∗_𝑗𝑖𝐹²_𝑗+𝑡^∗_𝑗𝑖𝐺²_𝑗),   for 𝑖=1,…,𝑛, then equilibrium point 𝑢^∗ of system (3.18) with (2.3) and (2.4) is globally exponentially robust stable for each constant input 𝐼∈ℝ^𝑛.

Remark 3.7. When 𝑑_𝑖𝑙=0, then system (2.1) reduces to the following system without diffusive terms: 𝜕𝑢_𝑖(𝑡)𝜕𝑡=−𝑎_𝑖(𝑢_𝑖(𝑡))[𝑏_𝑖(𝑢_𝑖(𝑡))−^𝑛_𝑗=1𝑠_𝑖𝑗𝑓_𝑗(𝑢_𝑗(𝑡))^𝑛_𝑗=1𝑡_𝑖𝑗𝑔_𝑗(𝑔_𝑗(𝑡−𝜏_𝑖𝑗))+𝐼_𝑖], (3.19) for 𝑡≥0, 𝑖=1,…,𝑛.

From Theorems 3.1 and 3.3, we have the following results.

Corollary 3.8. Under assumptions (H1)–(H4), if, in addition, any one of the following condition holds: (A5)2𝑎_𝑖𝑏_𝑖>∑^𝑛_𝑗=1𝑎_𝑖(𝑠_𝑖𝑗𝐹_𝑖+𝑡_𝑖𝑗𝐺_𝑖)+∑^𝑛_𝑗=1𝑎_𝑗(𝑠_𝑗𝑖𝐹_𝑗+𝑡_𝑗𝑖𝐺_𝑗),(A6)2𝑎_𝑖𝑏_𝑖>∑^𝑛_𝑗=1𝑎_𝑖(𝑠_𝑖𝑗+𝑡_𝑖𝑗)+∑^𝑛_𝑗=1𝑎_𝑗(𝑠_𝑗𝑖𝐹²_𝑗+𝑡_𝑗𝑖𝐺²_𝑗),   for 𝑖=1,…,𝑛, then equilibrium point 𝑢^∗ of system (3.19) with (2.4) is globally exponentially robust stable for each constant input 𝐼∈ℝ^𝑛.

Remark 3.9. From Theorems 3.1 and 3.3, Corollary 3.8, we see that condition (A5) or condition (A6) imply (A1) and (A2), respectively, conversely, if conditions (A1) and (A2) hold, (A5) and (A6) do not certainly hold. This show that the reaction-diffusion terms have play an important role in the globally exponentially robust stability to a reaction-diffusion neural network.

4. Examples and Comparison

In order to illustrate the feasibility of the previous established criteria in the preceding sections, we provide concrete two examples. Although the selection of the coefficients and functions in the examples is somewhat artificial, the possible application of our theoretical theory is clearly expressed.

Example 4.1. Consider the following reaction-diffusion CGNNs on Ω={𝑥=(𝑥₁,𝑥₂,𝑥₃)^𝑇∈ℝ³∣𝑥²₁+𝑥²₂+𝑥²₃<1}: 𝜕𝑢𝜕𝑡𝑢¹(𝑡)₂=(𝑡)0.65𝜕𝑢₁(𝑡)𝜕𝑥₁0.72𝜕𝑢₁(𝑡)𝜕𝑥₂0.65𝜕𝑢₁(𝑡)𝜕𝑥₃0.82𝜕𝑢₂(𝑡)𝜕𝑥₁0.65𝜕𝑢₂(𝑡)𝜕𝑥₂0.71𝜕𝑢₂(𝑡)𝜕𝑥₃𝜕𝜕𝑥₁𝜕𝜕𝑥₂𝜕𝜕𝑥₃−1+0.2cos𝑢₁(𝑡,𝑥)001+0.2sin𝑢₂𝐼(𝑡,𝑥)×{¹𝐼₂+1.20𝑢01.2𝑢¹(𝑡)₂−𝑠(𝑡)¹¹𝑠₁₂𝑠₂₁𝑠₂₂sin𝑢₁(𝑡)cos𝑢₂−𝑡(𝑡)₁₁𝑡₁₂𝑡₂₁𝑡₂₂tanh(𝑢₁(𝑡−𝜏_1𝑗))tanh(𝑢₂(𝑡−𝜏_2𝑗𝑢))},(𝑡,𝑥)∈[0,+∞)×Ω,_𝑖𝑢(𝑡)=0,(𝑡,𝑥)∈[0,+∞)×𝜕Ω,𝑖=1,2,_𝑖(𝑠)=𝜙_𝑖(𝑠),(𝑠,𝑥)∈[−1,0]×Ω,𝑖=1,2, (4.1) where tanh(𝑥)=(𝑒^𝑥−𝑒^−𝑥)/(𝑒^𝑥+𝑒^−𝑥), 𝑠₁₁∈[1/5,1/4], 𝑠₁₂∈[1/20,1/16], 𝑠₂₁∈[−1/4,1/17], 𝑠₂₂∈[−1/4,−1/12], 𝑡₁₁∈[1/6,1/4], 𝑡₁₂∈[1/20,1/6], 𝑡₂₁∈[−1/4,−1/12], 𝑡₂₂∈[1/3,1/2], 𝐼₁∈[−1/5,1], 𝐼₁∈[−1/5,3/5], 𝜏₁₁∈[0.3,0.8], 𝜏₁₂∈[0.4,1], 𝜏₂₁∈[0.1,0.6], and 𝜏₂₂∈[0.2,0.9].
This model satisfies assumptions (H1)–(H4) in this paper with 𝜆₁≥0.5387, 𝑑₁=𝑑₂=0.65, 𝑎₁=𝑎₂=1.2, 𝑎₁=𝑎₂=0.8, 𝑏₁=𝑏₂=1.2, 𝐹₁=𝐹₂=𝐺₁=𝐺₂=1, 𝑆^∗=(𝑠^∗_𝑖𝑗)_𝑛×𝑛=[^1/41/16_1/41/4], 𝑇^∗=(𝑡^∗_𝑖𝑗)_𝑛×𝑛=[^1/41/16_1/41/2], 𝜏=1, 𝐼^∗₁=1, 𝐼^∗₂=3/5. It is easily computed that
2.62042.6204=2(𝑑_𝑖𝜆₁+𝑎_𝑖𝑏_𝑖)>𝑎_𝑖^𝑛_𝑗=1(𝑠^∗_𝑖𝑗𝐹_𝑖+𝑡^∗_𝑖𝑗𝐺_𝑖)+^𝑛_𝑗=1𝑎_𝑗(𝑠^∗_𝑗𝑖𝐹_𝑗+𝑡^∗_𝑗𝑖𝐺_𝑗)=1.9500,𝑖=1,2.5500,𝑖=2. (4.2) From Theorem 3.1, we know that model (4.1) has a unique equilibrium point which is globally exponentially robustly stable.

Remark 4.2. It should be noted that 2.4=2𝑎₂𝑏₂0𝑥0008𝑓>𝑎̸₂^𝑛_𝑗=1(𝑠^∗_2𝑗𝐹₂+𝑡^∗_2𝑗𝐺₂)+^𝑛_𝑗=1𝑎_𝑗(𝑠^∗_𝑗2𝐹_𝑗+𝑡^∗_𝑗2𝐺_𝑗)=2.5500. (4.3) From Corollary 3.8, the corresponding delayed differential equation of system (4.1) without reaction-diffusion terms is not certainly robustly stable, as we can see in Example  4.1, reaction-diffusion terms do contribute to the exponentially robust stability of system (4.1).

Example 4.3. For the model in Example 4.1, if the diffusion operator, Ω, and the boundary condition are replaced by, respectively, 2𝜕𝑢₁(𝑡)𝜕𝑥₁1.2𝜕𝑢₁(𝑡)𝜕𝑥₂1.2𝜕𝑢₂(𝑡)𝜕𝑥₁2𝜕𝑢₂(𝑡)𝜕𝑥₂𝜕𝜕𝑥₁𝜕𝜕𝑥₂,Ω={(𝑥₁,𝑥₂)^𝑇∈ℝ²∣0<𝑥_𝑖<𝜋,𝑖=1,2}, (4.4) and the Neumann boundary condition 𝜕𝑢_𝑖(𝑡)𝜕𝑚=0,(𝑡,𝑥)∈[0,+∞)×𝜕Ω,𝑖=1,2, (4.5) the remainder parameters unchanged. According to Remark 2.1, we see that 𝜆₁=2. By Theorem 3.1, using the same way with Example  4.1, we see that model (4.1) has a unique equilibrium point which is globally exponentially robustly stable.

Remark 4.4. Song and Cao have considered reaction-diffusion CGNNs with the Neumann boundary condition and obtained the criteria of the globally exponentially robust stability unique equilibrium point for CGNN, that is, [29, Theorem  1]. We notice that [29, Theorem  1] is irrelevant to the reaction-diffusion terms. In principal, [29, Theorem  1] could be applied to analyze the globally exponentially robust stability for the system in Example  4.2. Unfortunately, [29, Theorem  1] is not applicable to ascertain the globally exponentially robust stability for the system in Example  4.2, since (according to the symbols in this paper) 𝑎₁𝑏₁𝑎₁00𝑎₂𝑏₂𝑎₁−𝑆^∗𝐹−𝑇^∗1𝐺=6−18−12−112 (4.6) is not an 𝑀-matrix, where 𝐹=diag{𝐹₁,𝐹₂}, 𝐺=diag{𝐺₁,𝐺₂}.

5. Conclusion

In this paper, we have proposed several sufficient condition for the globally exponentially robustly stability of equilibrium point for the reaction-diffusion CGNNs with constant time delays. All the criteria are established by constructing suitable Lyapunov functionals, without assuming the monotonicity and differentiability of activation functions and the symmetry of connection matrices. The space domain that CGNNs model is on is relatively general, the boundary condition of CGNNs model includes the Dirichlet and the Neumann. In particular, Poincaré inequality is used and all the criteria obtained depend on reaction-diffusion terms, this is a preeminent feature that distinguishes our research from the previous research on delayed neural network with reaction diffusion. Numerical examples are presented to illustrate the feasibility of this method.

Acknowledgment

The authors would like to thank the editor and the reviewers for their detailed comments and valuable suggestions which have led to a much improved paper.