LMI Approach to Stability Analysis of Cohen-Grossberg Neural Networks with 𝑝-Laplace Diffusion

Abstract

The nonlinear 𝑝-Laplace diffusion (𝑝>1) was considered in the Cohen-Grossberg neural network (CGNN), and a new linear matrix inequalities (LMI) criterion is obtained, which ensures the equilibrium of CGNN is stochastically exponentially stable. Note that, if 𝑝=2, 𝑝-Laplace diffusion is just the conventional Laplace diffusion in many previous literatures. And it is worth mentioning that even if 𝑝=2, the new criterion improves some recent ones due to computational efficiency. In addition, the resulting criterion has advantages over some previous ones in that both the impulsive assumption and diffusion simulation are more natural than those of some recent literatures.

1. Introduction and Preparation

It is well known that Cohen-Grossbeg neural network (CGNN) was proposed by Cohen and Grossberg [1] in 1983. Since then there have been a lot of interested results obtained in many literatures (see [2–9]) due to its general applications, such as pattern recognition, image and signal processing, optimization automatic control, and artificial intelligence. Usually, there exist the impulsive effect and time-varying delays phenomenon in various neural networks [3, 5–7, 10–14]. Besides, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields [15–18]. However, diffusion disturbance was always simulated simply by linear Laplace diffusion [15–18]. Few papers involved the nonlinear reaction-diffusion [19]. So in this paper, we investigate the stability of the following stochastic CGNN with nonlinear 𝑝-Laplace diffusion (𝑝>1): 𝑑𝑢(𝑡,𝑥)=∇⋅𝐷(𝑡,𝑥,𝑢)∘∇_𝑝𝑢(𝑢(×𝐵(𝑢((𝑢((𝑢((𝑡)(𝑢((𝑡)𝑡−𝐴𝑡,𝑥))𝑡,𝑥))−𝐶𝑓𝑡,𝑥))+𝐷𝑔𝑡−𝜏,𝑥))𝑑𝑡+𝜎𝑡,𝑥))𝑑𝑤,𝑡∈_𝑘,𝑡_𝑘+1𝑢𝑡⁺_𝑘,𝑥=𝑀𝑢(𝑡⁻,𝑥),𝑡=𝑡_𝑘𝑢𝑡₀+𝜃,𝑥=𝜑(𝜃,𝑥),(𝜃,𝑥)∈[−𝜏,0]×𝜕Ω𝜕𝑢_𝑖(𝑡,𝑥)()∈[)𝜕𝜈=0,𝑡,𝑥−𝜏,+∞×𝜕Ω,𝑖=1,2,…,𝑛, (1.1) where Ω is a bounded subset in 𝑅^𝑚 with smooth boundary 𝜕Ω, and 𝜕𝑢_𝑖(𝑡,𝑥)/𝜕𝜈=((𝜕𝑢_𝑖(𝑡,𝑥)/𝜕𝑥₁),(𝜕𝑢_𝑖(𝑡,𝑥)/𝜕𝑥₂),…,(𝜕𝑢_𝑖(𝑡,𝑥)/𝜕𝑥_𝑚))^𝑇 denotes the outward normal derivative on 𝜕Ω.

Remark 1.1. If 𝑝=2, system (1.1) was studied by [5] though there is a little difference between Dirichlet boundary condition and Neumann boundary condition. However, our impulsive assumption 𝑢(𝑡⁺_𝑘,𝑥)=𝑀𝑢(𝑡⁻_𝑘,𝑥) is more natural than that of [5], which will result in some difference in methods.

Here, 𝑀 is a diagonal matrix, Ω∈𝑅^𝑚 is a bounded compact set with smooth boundary, 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈𝑅^𝑛, and 𝑤(𝑡)=(𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑛(𝑡))^𝑇 is a 𝑛-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,𝑃) with the natural filtration {ℱ_𝑡}_𝑡≥0 generated by the process {𝑤(𝑠)∶0≤𝑠≤𝑡}. We associate Ω with the canonical space generated by all {𝑤_𝑖(𝑡)} and denoted by ℱ the associated 𝜎-algebra generated by 𝑤(𝑡) with the probability measure 𝑝. 𝐴(𝑢(𝑡,𝑥)) presents an amplification function, 𝐵(𝑢(𝑡,𝑥)) is an appropriately behavior function, and 𝑓 and 𝑔 denote the activation function. 𝜏(𝑡)(0≤𝜏(𝑡)≤𝜏) corresponds to the transmission delays at time, and 𝑡_𝑘 is called the impulsive moment with 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯ and lim_𝑘→∞𝑡_𝑘=+∞. We always assume 𝑢(𝑡⁺_𝑘,𝑥)=𝑢(𝑡_𝑘,𝑥). ∇_𝑝𝑢=(∇_𝑝𝑢₁,…,∇_𝑝𝑢_𝑛)^𝑇, ∇_𝑝𝑢_𝑖=(|∇𝑢_𝑖|^𝑝−2(𝜕𝑢_𝑖/𝜕𝑥₁),…,|∇𝑢_𝑖|^𝑝−2(𝜕𝑢_𝑖/𝜕𝑥_𝑚))^𝑇, 𝐷∘∇_𝑝𝑢=(𝐷_𝑖𝑘|∇𝑢_𝑖|^𝑝−2(𝜕𝑢_𝑖/𝜕𝑥_𝑘))_𝑛×𝑚 is Hadamard product of matrix 𝐷 and ∇_𝑝𝑢 [20]. Here, the diffusion parameters matrix 𝐷(𝑡,𝑥,𝑢) is denoted simply as 𝐷=(𝐷_𝑖𝑘)_𝑛×𝑚. Let 𝑌_𝑖=(𝑦_𝑖1,…,𝑦_𝑖𝑚)^𝑇,𝑖=1,2,…,𝑛, and matrix 𝕐=(𝑌₁,…,𝑌_𝑛)^𝑇, and we denote ∇⋅𝑌_𝑖=∑^𝑚_𝑘=1𝜕𝑦_𝑖𝑘/𝜕𝑥_𝑘,∇⋅𝕐=(∇⋅𝑌₁,∇⋅𝑌₂,…,∇⋅𝑌_𝑛)^𝑇. Particularly, ∇_𝑝𝑢=∇𝑢 for the case of 𝑝=2.

Remark 1.2. Diffusion effects always occur in the neural networks when electrons are moving in asymmetric electromagnetic fields [15–18], and diffusion behavior is so complicated that it cannot always be simulated by linear Laplace diffusion. So in this paper, the nonlinear 𝑝-Laplace diffusion is considered in System (1.1).

Assume, in addition, the following.(H1)𝐴(𝑢(𝑡,𝑥)) is a bounded, positive, and continuous diagonal matrix, that is, there exist two positive diagonal matrices 𝐴 and 𝐴 such that 0<𝐴≤𝐴(𝑢(𝑡,𝑥))≤𝐴.(H2)𝐵(𝑢(𝑡,𝑥))=(𝑏₁(𝑢₁(𝑡,𝑥)),𝑏₂(𝑢₂(𝑡,𝑥)),…,𝑏_𝑛(𝑢_𝑛(𝑡,𝑥)))^𝑇 such that there exists a positive diagonal matrix 𝐵=diag(𝐵₁,𝐵₂,…,𝐵_𝑛) satisfying (𝑏_𝑖(𝑢_𝑖(𝑡,𝑥))/𝑢_𝑖(𝑡,𝑥))≥𝐵_𝑖 for all 𝑖.(H3) There exist two positive diagonal matrices 𝐹=diag(𝐹₁,𝐹₂,…,𝐹_𝑛) and 𝐺=diag(𝐺₁,𝐺₂,…,𝐺_𝑛) such that 𝑓0≤^𝑖(𝑟)𝑟≤𝐹_𝑖𝑔,0≤^𝑖(𝑟)𝑟≤𝐺_𝑖,∀𝑖. (1.2)(H4) The null solution is the equilibrium point of system (1.1), that is, the following conditions hold: 𝐵(0)−𝐶𝑓(0)−𝐷𝑔(0)=0,trace𝜎^𝑇(𝑢(𝑡,𝑥))𝜎(𝑢(𝑡,𝑥))≤𝑢^𝑇(𝑡,𝑥)𝑄𝑢(𝑡,𝑥), (1.3)where the symmetrical matrix 𝑄>0.

For convenience's sake, we introduce some standard notations(i)𝐿²(𝑅)(𝑅×Ω): the space of real Lebesgue measurable functions of 𝑅×Ω, it is a Banach space for the 2-norm ‖𝑢(𝑡)‖₂=(∑^𝑛_𝑖=1‖𝑢_𝑖(𝑡)‖)^1/2 with ‖𝑢_𝑖(𝑡)‖=(∫_Ω|𝑢_𝑖(𝑡,𝑥)|²𝑑𝑥)^1/2, where |𝑢_𝑖(𝑡,𝑥)| is Euclid norm.(ii)𝐿²_𝐹0([−𝜏,0]×Ω;𝑅^𝑛): the family of all 𝐹₀-measurable 𝐶([−𝜏,0]×Ω;𝑅^𝑛-) value random variable 𝜉={𝜉(𝜃,𝑥)∶−𝜏≤𝜃≤0,𝑥∈Ω} such that sup_{−𝜏≤𝜃≤0}𝐸‖𝜉(𝜃)‖²₂<∞, where 𝐸{⋅} stands for the mathematical expectation operator with respect to the given probability measure 𝑝.(iii)𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛>0 (<0): a positive (negative) definite symmetrical matrix, that is, 𝑦^𝑇𝑄𝑦>0 (<0) for any 0≠𝑦∈𝑅^𝑛.(iv)𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛≥0 (≤0): a semipositive (semi-negative) definite symmetrical matrix, that is, 𝑦^𝑇𝑄𝑦≥0 (≤0) for any 𝑦∈𝑅^𝑛.(v)𝑄≥𝑄(𝑄≤𝑄): this means 𝑄−𝑄 is a semi-positive (semi-negative) symmetrical definite matrix.(vi)𝑄>𝑄(𝑄<𝑄): this means 𝑄−𝑄 is a positive (negative) symmetrical definite matrix.(vii)𝜆_max(Φ),𝜆_min(Φ) denotes the largest and smallest eigenvalue of symmetrical matrix Φ, respectively.(viii)𝐼: identity matrix with compatible dimension.(ix)Denote |𝐶|=(|𝑐_𝑖𝑗|)_𝑛×𝑛 for any matrix 𝐶_𝑛×𝑛; |𝑢(𝑡,𝑥)|=(|𝑢₁|,|𝑢₂|,…,|𝑢_𝑛|) for any 𝑢∈𝑅^𝑛.

Let 𝑢(𝑡,𝑥;𝜑) denote the state trajectory from the initial data 𝑢(𝑡₀+𝜃,𝑥;𝜑)=𝜑(𝜃,𝑥) on −𝜏≤𝜃≤0 in 𝐿²_𝐹0([−𝜏,0]×Ω;𝑅^𝑛).

Definition 1.3. The null solution of impulsive system (2.2) is globally stochastically exponentially stable in the mean square if for every 𝜑∈𝐿²_𝐹0([−𝜏,0]×Ω;𝑅^𝑛), there exists scalars 𝛽>0 and 𝛾>0 such that 𝐸𝑢𝑡,𝜑²₂≤𝛾𝑒^−𝛽𝑡sup_{−𝜏≤𝜃≤0}𝐸𝜑(𝜃)²₂. (1.4)

Lemma 1.4 (see [11]). Let 𝑈,𝑃 be any matrices, 𝜀>0 is a positive number and matrix 𝐻=𝐻^𝑇>0, then 𝑃^𝑇𝑈+𝑈^𝑇𝑃≤𝜀𝑃^𝑇𝐻𝑃+𝜀⁻¹𝑈^𝑇𝐻⁻¹𝑈. (1.5)

Lemma 1.5 (Schur complement [3]). The LMI 𝑄(𝑡)𝑆(𝑡)𝑆^𝑇(𝑡)𝑅(𝑡)>0, (1.6) where matrix 𝑆(𝑡) and symmetrical matrices 𝑄(𝑡) and  𝑅(𝑡) depend on 𝑡, is equivalent to any one of the following conditions: (L₁)𝑅(𝑡)>0,𝑄(𝑡)−𝑆(𝑡)𝑅⁻¹(𝑡)𝑆^𝑇(𝑡)>0; (L₂)𝑄(𝑡)>0,𝑅(𝑡)−𝑆^𝑇(𝑡)𝑄⁻¹(𝑡)𝑆^𝑇(𝑡)>0.

Lemma 1.6 (see [21]). Consider the following differential inequality: 𝐷⁺𝑣(𝑡)≤−𝑎𝑣(𝑡)+𝑏[𝑣(𝑡)]_𝜏,𝑡≠𝑡_𝑘𝑣𝑡_𝑘≤𝑎_𝑘𝑣𝑡⁻_𝑘+𝑏_𝑘𝑣𝑡⁻_𝑘𝜏, (1.7) where 𝑣(𝑡)≥0,[𝑣(𝑡_𝑘)]_𝜏=sup_{𝑡−𝜏≤𝑠≤𝑡}𝑣(𝑠),[𝑣(𝑡⁻_𝑘)]_𝜏=sup_{𝑡−𝜏≤𝑠≤𝑡}𝑣(𝑠) and 𝑣(𝑡) is continuous except 𝑡_𝑘,𝑘=1,2,…, where it has jump discontinuities. The sequence 𝑡_𝑘 satisfies 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<𝑡_𝑘+1<⋯, and lim_𝑘→∞𝑡_𝑘=∞. Suppose that(1)𝑎>𝑏≥0; (2)𝑡_𝑘−𝑡_𝑘−1>𝛿𝜏, where 𝛿>1, and there exist constants 𝛾>0,𝑀>0 such that 𝜌₁𝜌₂⋯𝜌_𝑘+1≤𝑀𝑒^𝛾𝑡𝑘, (1.8)where 𝜌_𝑖=max{1,𝑎_𝑖+𝑏_𝑖𝑒^𝜆𝜏}, 𝜆>0 is the unique solution of equation 𝜆=𝑎−𝑏𝑒^𝜆𝜏 then 𝑣(𝑡)≤𝑀[𝑣(0)]_𝜏𝑒^{−(𝜆−𝛾)𝑡}. (1.9) In addition, if 𝜃=sup_𝑘∈𝑍{1,𝑎_𝑘+𝑏_𝑘𝑒^𝜆𝜏}, then 𝑣(𝑡)=𝜃[𝑣(0)]_𝜏𝑒^{−(𝜆−(ln(𝜃𝑒𝜆𝜏)/𝛿𝜏))𝑡},𝑡≥0. (1.10)

Computer simulation is shown in Figures 1, 2, 3, and 4.

Figure 1 

Computer simulation of the state 𝑢1(𝑡,𝑥).

Figure 2 

Sectional curve of the state variable 𝑢1(𝑡,𝑥).

Figure 3 

Computer simulation of the state 𝑢2(𝑡,𝑥).

Figure 4 

Sectional curve of the state variable 𝑢2(𝑡,𝑥).

2. Main Results

Theorem 2.1. If assumptions (H1)–(H4) hold, in addition, the following conditions are satisfied:(C1) there exists diagonal matrices 𝑃₁=diag(𝑝₁₁,𝑝₁₂,…,𝑝_1𝑛)>0 and 𝑃₂>0 such that −2𝑃₁𝐴𝐵+𝐹²+𝑃₁𝑄+𝑃₂𝑃₁𝐴|𝐶|𝑃₁𝐴|𝐷|𝐶^𝑇𝐴𝑃₁𝐷−𝐼0_𝑇𝐴𝑃₁0−𝐼<0; (2.1)(C2)min{(𝜆_minΘ/𝜆_max𝑃₁),(1−𝜇)}>(𝜆_max𝐺²/𝜆_min𝑃₂)≥0, where Θ=2𝑃₁𝐴𝐵−𝑃₁𝐴|𝐶||𝐶|^𝑇𝐴𝑃₁−𝑃₁𝐴|𝐷||𝐷|^𝑇𝐴𝑃₁−𝐹²−𝑃₁𝑄−𝑃₂, where 𝜏^′(𝑡)≤𝜇<1 for all 𝑡;(C3) there exists a constant 𝛿>1 such that inf_𝑘∈𝑍(𝑡_𝑘−𝑡_𝑘−1)>𝛿𝜏, 𝛿²𝜏>ln(𝜌𝑒^𝜆𝜏) and 𝜆−(ln(𝜌𝑒^𝜆𝜏)/𝛿𝜏)>0, where 𝜆>0 is the unique solution of the equation 𝜆=𝑎−𝑏𝑒^𝜆𝜏, and 𝜌=max{1,(𝜆_max(𝑀𝑃₁𝑀)/𝜆_min𝑃₁)+𝑒^𝜆𝜏}, 𝑎=min{(𝜆_minΘ/𝜆_max𝑃₁),(1−𝜇)},𝑏=𝜆_max𝐺²/𝜆_min𝑃₂, then the null solution of system (1.1) is stochastically exponentially stable with convergence rate (1/2)(𝜆−(ln(𝜌𝑒^𝜆𝜏)/𝛿𝜏)).

Proof. First, we can get by Guass formula (see [20, Lemma 2.3]) _Ω𝑢^𝑇𝑃₁∇⋅𝐷(𝑡,𝑥,𝑢)∘∇_𝑝𝑢=𝑑𝑥_Ω𝑢^𝑇𝑃₁^𝑚_𝑘=1𝜕𝜕𝑥_𝑘𝐷_1𝑘∇𝑢₁^𝑝−2𝜕𝑢₁𝜕𝑥_𝑘,…,^𝑚_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑛𝑘∇𝑢_𝑛^𝑝−2𝜕𝑢_𝑛𝜕𝑥_𝑘^𝑇=𝑑𝑥_Ω𝑛𝑗=1𝑝_1𝑗𝑢_𝑗^𝑚_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑗𝑘∇𝑢_𝑗^𝑝−2𝜕𝑢_𝑗𝜕𝑥_𝑘𝑑𝑥=−^𝑚𝑛_{𝑘=1𝑗=1Ω}𝑝_1𝑗𝐷_𝑗𝑘∇𝑢_𝑗^𝑝−2𝜕𝑢_𝑗𝜕𝑥_𝑘²𝑑𝑥. (2.2)
Construct the Lyapunov functional as follows: 𝑉=𝑉₁+𝑉₂, (2.3) where 𝑉₁=_Ω𝑢^𝑇(𝑡,𝑥)𝑃₁𝑢(𝑡,𝑥)𝑑𝑥=_Ω𝑢^𝑇(𝑡,𝑥)𝑃₁|𝑢(𝑉𝑡,𝑥)|𝑑𝑥₂=_Ω^𝑡_{(𝑡)𝑡−𝜏}𝑢^𝑇(𝑠,𝑥)𝑃₂𝑢(𝑠,𝑥)𝑑𝑠𝑑𝑥. (2.4)
Then 𝐿𝑉₁=_Ω−^𝑚𝑛_𝑘=1𝑗=12𝑝_1𝑗𝐷_𝑗𝑘∇𝑢_𝑗^𝑝−2𝜕𝑢_𝑗𝜕𝑥_𝑘²𝑑𝑥−2_Ω𝑢^𝑇(𝑡,𝑥)𝑃₁𝐴(𝑢(𝑡,𝑥))𝐵(𝑢(𝑡,𝑥))𝑑𝑥+2_Ω𝑢^𝑇(𝑡,𝑥)𝑃₁𝐴(𝑢(𝑡,𝑥))𝐶𝑓(𝑢(𝑡,𝑥))𝑑𝑥+2_Ω𝑢^𝑇(𝑡,𝑥)𝑃₁𝐴(𝑢((𝑢((𝑡)+𝑡,𝑥))𝐷𝑔𝑡−𝜏,𝑥))𝑑𝑥_Ωtrace𝜎^𝑇(𝑢(𝑡,𝑥))𝑃₁𝜎(𝑢(𝑡,𝑥))𝑑𝑥≤−2_Ω𝑢^𝑇(𝑡,𝑥)𝑃₁𝐴𝐵|𝑢(+𝑡,𝑥)|𝑑𝑥_Ω𝑢^𝑇(𝑡,𝑥)𝑃₁𝐴|𝐶||𝐶|^𝑇𝐴𝑃₁|𝑢(𝑡,𝑥)|+𝑓^𝑇(𝑢(𝑓(𝑢(+𝑡,𝑥))𝑡,𝑥))𝑑𝑥_Ω𝑢^𝑇(𝑡,𝑥)𝑃₁𝐴|𝐷||𝐷|^𝑇𝐴𝑃|𝑢(𝑡,𝑥)|+𝑔^𝑇(𝑢((𝑡)𝑔(𝑢((𝑡)+𝑡−𝜏,𝑥))𝑡−𝜏,𝑥))𝑑𝑥_Ω𝑢^𝑇(𝑡,𝑥)𝑃₁𝑄𝑢(𝑡,𝑥)𝑑𝑥≤−_Ω𝑢^𝑇(𝑡,𝑥)2𝑃₁𝐴𝐵−𝑃₁𝐴|𝐶||𝐶|^𝑇𝐴𝑃₁−𝑃₁𝐴|𝐷||𝐷|^𝑇𝐴𝑃₁−𝐹²−𝑃₁𝑄𝑢(+|𝑡,𝑥)|𝑑𝑥_Ω𝑢^𝑇(𝑡−𝜏(𝑡),𝑥)𝐺²𝑢(𝑡−𝜏(𝑡),𝑥)𝑑𝑥𝐿𝑉₂=_Ω𝑢^𝑇(𝑡,𝑥)𝑃₂𝑢(𝑡,𝑥)−1−𝜏^′(𝑡)𝑢^𝑇(𝑡−𝜏(𝑡))𝑃₂𝑢((𝑡≤𝑡−𝜏))𝑑𝑥_Ω𝑢^𝑇(𝑡,𝑥)𝑃₂|𝑢(𝑡,𝑥)|𝑑𝑥+𝜇−1_Ω𝑢^𝑇(𝑡−𝜏(𝑡))𝑃₂𝑢(𝑡−𝜏(𝑡))𝑑𝑥. (2.5) And then we have 𝐿𝑉≤−_Ω𝑢^𝑇(𝑡,𝑥)Θ|𝑢(𝑡,𝑥)|𝑑𝑥+_Ω𝑢^𝑇(𝑡−𝜏(𝑡),𝑥)𝐺²+𝜇−1𝑃₂𝑢(𝑡−𝜏(𝑡),𝑥)𝑑𝑥. (2.6)
Next, we use the method similar as that of [22]. Since 𝑢(𝑡,𝑥) is the solution of system, and 𝑉(𝑢(𝑡,𝑥))∈𝐶²[𝑅^𝑚,𝑅⁺] for all 𝑡, we can get by 𝐼𝑡𝑜̂ formula 𝑉(𝑡)=𝑉𝑡_𝑘+^𝑡_𝑡𝑘𝐿𝑉(𝑢(𝑠,𝑥))𝑑𝑠+^𝑡_𝑡𝑘𝜕𝑉𝜎(𝑢((𝑠).𝜕𝑢𝑠,𝑥))𝑑𝑤 (2.7)
Then we have 𝐸𝑉(𝑢(𝑡,𝑥))=𝐸𝑉𝑢𝑡_𝑘,𝑥+^𝑡_𝑡𝑘𝐸𝐿𝑉(𝑠,𝑥)𝑑𝑠,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (2.8)
Thus, for small enough Δ𝑡>0, we have 𝐸𝑉(𝑢(𝑡+Δ𝑡,𝑥))=𝐸𝑉𝑢𝑡_𝑘,𝑥+_𝑡^𝑡+Δ𝑡_𝑘𝐸𝐿𝑉(𝑠,𝑥)𝑑𝑠,𝑡∈𝑡_𝑘,𝑡_𝑘+1, (2.9) and then (𝑢((𝑢(=𝐸𝑉𝑡+Δ𝑡,𝑥))−𝐸𝑉𝑡,𝑥))_{𝑡𝑡+Δ𝑡}𝐸𝐿𝑉(𝑠,𝑥)𝑑𝑠≤𝐸_𝑡^𝑡+Δ𝑡−_Ω𝑢^𝑇(Θ|𝑢(+𝑠,𝑥)𝑠,𝑥)|𝑑𝑥_Ω𝑢^𝑇(𝑠−𝜏(𝑠),𝑥)𝐺²+𝜇−1𝑃₂𝑢(𝑠−𝜏(𝑠),𝑥)𝑑𝑥𝑑𝑠,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (2.10) Since _Ω𝑢^𝑇(𝑡,𝑥)Θ|𝑢(𝑡,𝑥)|𝑑𝑥+_Ω𝑢^𝑇(𝑡−𝜏(𝑡),𝑥)1−𝜇𝑃₂𝑢((𝑡))≥𝜆𝑡−𝜏,𝑥𝑑𝑥_minΘ𝜆_max𝑃_1Ω𝑢^𝑇(𝑡,𝑥)𝑃₁|𝑢(𝑡,𝑥)|𝑑𝑥+1−𝜇_Ω𝑢^𝑇(𝑡−𝜏(𝑡),𝑥)𝑃₂𝑢((𝑡))𝜆𝑡−𝜏,𝑥𝑑𝑥≥min^minΘ𝜆_max𝑃₁,1−𝜇_Ω𝑉𝑢^𝑇(𝑡−𝜏(𝑡),𝑥)𝐺²𝑢((𝑡))𝜆𝑡−𝜏,𝑥𝑑𝑥≤^max𝐺²𝜆_min𝑃_2Ω𝑢^𝑇(𝑡−𝜏(𝑡),𝑥)𝑃₂𝑢(𝑡−𝜏(𝑡),𝑥)𝑑𝑥, (2.11)
Then, 𝐷⁺(𝑢(𝜆𝐸𝑉𝑡,𝑥))≤−min^minΘ𝜆_max𝑃₁,(𝑡)+𝜆1−𝜇𝐸𝑉^max𝐺²𝜆_min𝑃₂[𝐸𝑉(𝑡)]_𝜏,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (2.12) Next, we have 𝑉𝑡_𝑘=_Ω𝑢^𝑇𝑡_𝑘,𝑥𝑃₁𝑢𝑡_𝑘,𝑥𝑑𝑥+_Ω𝑢^𝑇𝑡_𝑘−𝜏𝑡_𝑘,𝑥𝑃₂𝑢𝑡_𝑘−𝜏𝑡_𝑘=,𝑥𝑑𝑥_Ω𝑢^𝑇𝑡⁻_𝑘,𝑥𝑀𝑃₁𝑀𝑢𝑡⁻_𝑘,𝑥𝑑𝑥+_Ω𝑢^𝑇𝑡_𝑘−𝜏𝑡_𝑘,𝑥𝑃₂𝑢𝑡_𝑘−𝜏𝑡_𝑘≤𝜆,𝑥𝑑𝑥_max𝑀𝑃₁𝑀𝜆_min𝑃₁𝑉𝑡⁻_𝑘+𝑉𝑡⁻_𝑘𝜏. (2.13)
Now the conditions (C1)–(C3) and Lemma 1.6 deduce 𝐸𝑉(𝑡)≤𝜌[𝑉(0)]_𝜏𝑒^{−(𝜆−(ln(𝜌𝑒𝜆𝜏)/𝛿𝜏))𝑡} (2.14) or 𝐸‖𝑢(𝑡,𝑥)‖²≤𝜌𝜆_max𝑃𝜆_min𝑃sup_{−𝜏≤𝑠≤0}𝐸𝜙(𝑠)²₂𝑒^{−(𝜆−(ln(𝜌𝑒𝜆𝜏)/𝛿𝜏))𝑡} (2.15) which together with Definition 1.3 implies the accomplishment of the proof.

Remark 2.2. The nonlinear 𝑝-Laplace diffusion (𝑝>1) brings a great difficulties in judging the stability. However, even if 𝑝=2, Theorem 2.1 has more computational efficiency than [15, Theorem 3.1] due to LMI criterion.

3. Examples

Consider the following impulsive CGNN: 𝑑𝑢(𝑡,𝑥)=∇⋅0.0005000.0005∘∇_𝑝𝑢−1.65+0.5sin𝑢₁001.58+0.5sin𝑢₂×5.19𝑢₁−0.02𝑢₁cos𝑢₁5.18𝑢₂−0.01𝑢₂cos𝑢₂−𝑓(𝑢(−𝑔(𝑢((𝑡)(𝑢((𝑡)𝑡0.1−0.03−0.030.1𝑡,𝑥))0.1−0.03−0.030.1𝑡−𝜏,𝑥))𝑑𝑡+𝜎𝑡,𝑥))𝑑𝑤,𝑡∈_𝑘,𝑡_𝑘+1𝑢𝑡⁺_𝑘=𝑢(𝑡1.350001.350⁻,𝑥),𝑡=𝑡_𝑘𝑢𝑡₀+𝜃,𝑥=𝜙(𝜃,𝑥),(𝜃,𝑥)∈[−0.65,0]×𝜕Ω𝜕𝑢_𝑖(𝑡,𝑥)()∈[)𝜕𝜈=0,𝑡,𝑥−𝜏,+∞×𝜕Ω,𝑖=1,2, (3.1) where Ω={(𝑥₁,𝑥₂)^𝑇∈𝑅²|−√2≤𝑥₁,𝑥₂≤√2}, and the corresponding matrices 𝐹=.0.100000.100=𝐺,𝑄=0.0001000.0001 (3.2) We might as well assume that 𝑡₀=0,𝑡_𝑘−𝑡_𝑘−1=0.525,𝜏(𝑡)=0.65,𝜏^′(𝑡)≤𝜇=0.99 for all 𝑡≥𝑡₀, and 𝑓(𝑢)(𝑢)=𝑢=𝑔¹+1−𝑢₁−1𝑢20₂+1−𝑢₂−1,𝜙()=𝑥((20𝑠,𝑥1−cos5𝜋𝑥))cos¹⁸⁹(𝑥−0.25)𝑒^−100𝑠(1−𝑥)sin²(4𝜋𝑥)cos²⁰¹(𝑥−0.55)𝑒^−100𝑠𝐴,−0.65≤𝑠≤0,=,1001𝐴=.2002,𝐵=5.160005.160 (3.3) By way of MATLAB LMI Control Toolbox, we can solve the LMI condition in (C1) and get 𝑃₁=0.2392000.2392,𝑃₂=.1.2291001.2291 (3.4) Next, we will prove that such 𝑃₁ and 𝑃₂ make (C2) and (C3) hold. Indeed, by computing directly, we can obtain 𝑎=0.01,𝑏=0.0081, and then (C2) holds. Moreover, we might as well assume 𝛿=82, and then we have 𝜆=0.0019,𝜌=2.8235, thus (C3) is satisfied. Now from Theorem 2.1 we can compute the convergence 9.3809∗10^−0.004.

4. Conclusions

In this paper, we investigate the influence of impulse, time-delays and diffusion behaviors on the stability of stochastic Cohen-Grossberg neural network (CGNN). The LMI conditions of stochastic exponential stability of impulsive CGNN with 𝑝-Laplace reaction-diffusion terms was given, and an illustrate example was also given to show the effectiveness of the obtained result. Besides, the result obtained in this paper is also valid to the Laplace reaction-diffusion (in the case of 𝑝=2) and has more computational efficiency due to the LMI approach even if 𝑝=2 (Remark 2.2).

Acknowledgments

This work was supported by the National Basic Research Program of China (2010CB732501), by Scientific Research Fund of Science Technology Department of Sichuan Province 2011JYZ010, and by Scientific Research Fund of Sichuan Provincial Education Department (11ZA172, 12ZB349).