Stability Analysis for Impulsive Stochastic Reaction-Diffusion Differential System and Its Application to Neural Networks

Abstract

This paper is concerned with the stability of impulsive stochastic reaction-diffusion differential systems with mixed time delays. First, an equivalent relation between the solution of a stochastic reaction-diffusion differential system with time delays and impulsive effects and that of corresponding system without impulses is established. Then, some stability criteria for the stochastic reaction-diffusion differential system with time delays and impulsive effects are derived. Finally, the stability criteria are applied to impulsive stochastic reaction-diffusion Cohen-Grossberg neural networks with mixed time delays, and sufficient conditions are obtained for the exponential 𝑝-stability of the zero solution to the neural networks. An example is given to illustrate the effectiveness of our theoretical results. The systems we studied in this paper are more general, and some existing results are improved and extended.

1. Introduction

In recent years, impulsive dynamical systems have attracted considerable attention due to its wide applications in the areas of economics, physics, population dynamics, engineering, biology, and so on. These systems arise because they are subject to abrupt state changes at certain moments of time, and these changes may be related to such phenomena as shocks, harvesting, or other faults. Meanwhile, time delays are frequently encountered in real world, which can cause instability and oscillations in a system. A large number of stability criteria of impulsive delay systems have been reported (see [1–6] and references therein).

Stochastic effects are common phenomena due to disturbances or uncertainties in a system. A lot of dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment switching. Hence, considerable attention has been paid to the study of stochastic systems, and various interesting results have been reported in the literatures; for example, see [2–4, 7, 8] and references therein. In particular, Li et al. [8] considered the following impulsive stochastic differential system with time delay: d𝑦_𝑖(𝑡,𝑥)=𝐹_𝑖𝑡,𝑦₁(𝑡),…,𝑦_𝑛(𝑡),𝑦₁(𝑡−𝜏(𝑡)),…,𝑦_𝑛(𝑡−𝜏(𝑡))d𝑡+𝐺_𝑖𝑡,𝑦₁(𝑡),…,𝑦_𝑛(𝑡),𝑦₁(𝑡−𝜏(𝑡)),…,𝑦_𝑛(𝑡−𝜏(𝑡))d𝑤_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑦_𝑖𝑡⁺_𝑘−𝑦_𝑖𝑡_𝑘=𝐼_𝑘𝑖𝑦₁𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑁,𝑖=1,2,…,𝑛. (1) They showed the stability results of system (1) by transforming (1) into an equivalent system without impulses.

Generally speaking, diffusion effects cannot be avoided in systems modeling many real world phenomena. As a representation example in neural networks, when electrons are moving in an asymmetric electromagnetic field, it inevitably leads to diffusion phenomena. In [5, 9–11], the stabilities of the equilibrium points of some types of neural networks with reaction-diffusion terms have been investigated. On the other hand, distributed delay systems can characterize the cumulative effects of the past values of the dynamic and are often used to model the time lag phenomena in thermodynamics, ecology, epidemiology, and neural networks. For example, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. It is desired to model them by introducing continuously distributed delays over a certain duration of time such that the distant past has less influence compared to the recent behavior of the state.

However, in [8], the authors neglected the effects of diffusion and distributed delays. To the best of our knowledge, there are few results about the stability of impulsive stochastic reaction-diffusion deferential systems (ISRDDSs) with mixed time delays. Motivated by [8] and the previous discussions, we are concerned with the stability of the following ISRDDS with time-varying discrete delays and distributed delays: ()(𝐷(((()d𝑢𝑡,𝑥=∇⋅𝑡,𝑥,𝑢𝑡,𝑥))∘∇𝑢𝑡,𝑥))d𝑡+𝐹𝑡,𝑢𝑡,𝑥,𝑢𝑡−𝜏₁(𝑡)𝑢,𝑥,…,𝑡−𝜏_𝑝(𝑡),𝑥,^𝑡_{(𝑡)𝑡−𝑟}𝑢()()𝑠,𝑥d𝑠d𝑡+𝐺𝑡,𝑢𝑡,𝑥,𝑢𝑡−𝜏₁(𝑡)𝑢,𝑥,…,𝑡−𝜏_𝑝(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟},𝑢(𝑠,𝑥)d𝑠d𝑤(𝑡),𝑡≠𝑡_𝑘,𝑢𝑡⁺_𝑘,𝑥−𝑢𝑡_𝑘,𝑥=𝐼_𝑘𝑢𝑡_𝑘,𝑥,𝑡=𝑡_𝑘𝑢()(),()∈,𝑘∈𝑁,𝑡,𝑥=𝜙𝑡,𝑥𝑡,𝑥−𝛾,0×Ω,Ω∈ℝ^𝑚, (2) in which 𝑢(𝑡,𝑥)=𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥)^T,𝑢𝑡−𝜏_𝑗(𝑡),𝑥=𝑢₁𝑡−𝜏_𝑗(𝑡),𝑢,𝑥₂𝑡−𝜏_𝑗(𝑡)𝑢,𝑥,…,_𝑛𝑡−𝜏_𝑗(𝑡),𝑥^T𝑗=1,2,…,𝑝_{𝑡(𝑡)𝑡−𝑟},𝑢(𝑠,𝑥)d𝑠=^𝑡_{𝑡−𝑟1(𝑡)}𝑢₁(𝑠,𝑥)d𝑠,^𝑡_{𝑡−𝑟2(𝑡)}𝑢₂(𝑠,𝑥)d𝑠,…,^𝑡_{𝑡−𝑟𝑛(𝑡)}𝑢_𝑛(𝑠,𝑥)d𝑠^T,𝐹=𝐹₁,…,𝐹_𝑛^T,𝐺=𝐺₁,…,𝐺_𝑛^T,𝐼_𝑘=𝐼_𝑘1,…,𝐼_𝑘𝑛^T(),𝜙𝑘∈𝑁𝜙=₁,…,𝜙_𝑛^T,𝐷(𝑡,𝑥,𝑢(𝑡,𝑥))=𝐷_𝑖𝑙(𝑡,𝑥,𝑢(𝑡,𝑥))_𝑛×𝑚,∇𝑢=∇𝑢₁,…,∇𝑢_𝑛^T,∇𝑢_𝑖=𝜕𝑢_𝑖𝜕𝑥₁,…,𝜕𝑢_𝑖𝜕𝑥_𝑚,𝐷∘∇𝑢=𝐷_𝑖𝑙(𝑡,𝑥,𝑢(𝑡,𝑥))𝜕𝑢_𝑖𝜕𝑥_{𝑙𝑛×𝑚},𝑌=𝑌₁,…,𝑌_𝑛^T,𝑌_𝑖=𝑦_𝑖1,…,𝑦_𝑖𝑚^T,∇⋅𝑌_𝑖=^𝑚_𝑙=1𝜕𝑦_𝑖𝑙𝜕𝑥_𝑙,∇⋅𝑌=∇⋅𝑌₁,…,∇⋅𝑌_𝑛^T. (3)𝑢_𝑖(𝑡,𝑥) is the state variable, 𝑥_𝑖 is the space variable, 𝐷_𝑖𝑙(𝑡,𝑥,𝑢(𝑡,𝑥))≥0 is a diffusion operator, 𝜏_𝑗(𝑡) and 𝑟_𝑖(𝑡) are time-varying functions, 𝑤(𝑡)=(𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑛(𝑡))^T∈ℝ^𝑛 is a Brownian motion defined on a complete probability space (𝑆,ℱ,𝒫), and 𝐼_𝑘𝑖 is the impulsive function; 𝛾=max_{1≤𝑗≤𝑝,1≤𝑖≤𝑛}{𝜏_𝑗,𝑟_𝑖}, 𝜏_𝑗=sup_𝑡≥0𝜏_𝑗(𝑡), and 𝑟_𝑖=sup_𝑡≥0𝑟_𝑖(𝑡).

The organization of this paper is as follows. In Section 2, some preliminaries are given. In Section 3, by transforming the solutions of the stochastic reaction-diffusion differential system with delay and impulsive effects into that of the corresponding system without impulses, some stability criteria for the stochastic reaction-diffusion differential system with delay and impulsive effects are derived. In Section 4, the stability criteria are applied to impulsive stochastic reaction-diffusion Cohen-Grossberg neural networks (ISRDCGNNs) with mixed time delays, and sufficient conditions are obtained for the exponential 𝑝-stability of the zero solution to the neural networks. In Section 5, a numerical example is provided to illustrate the effectiveness of the theoretical results. A concluding remark is given in Section 6 to end this work.

2. Preliminaries

For convenience, we introduce several notations. Let 𝑃𝐶([−𝛾,0] × Ω,ℝ^𝑛)∶={𝑢(𝑡,𝑥)∶[−𝛾,0]×Ω→ℝ^𝑛|𝑢(𝑡,𝑥) is continuous at 𝑡≠𝑡_𝑘,𝑢(𝑡⁻_𝑘,𝑥)=𝑢(𝑡_𝑘,𝑥), and 𝑢(𝑡⁺_𝑘,𝑥) exists for 𝑡_𝑘}. For 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥))^T∈ℝ^𝑛, we define ‖𝑢(𝑡,𝑥)‖=^𝑛_𝑖=1𝑢_𝑖(𝑡,𝑥)=^𝑛_𝑖=1Ω𝑢_𝑖(𝑡,𝑥)^𝑝d𝑥^1/𝑝, (4) and for 𝜙(𝑠,𝑥)=(𝜙₁(𝑠,𝑥),…,𝜙_𝑛(𝑠,𝑥))^T∈𝑃𝐶([−𝛾,0]×Ω,ℝ^𝑛), we define ‖𝜙(𝑠,𝑥)‖ = sup_{−𝛾≤𝑠≤0}∑^𝑛_𝑖=1‖𝜙_𝑖(𝑠,𝑥)‖. Denote 𝑃𝐶^𝑏_𝐹0([−𝛾,0]×Ω,ℝ^𝑛)∶ = {𝑢(𝑡,𝑥)∈𝑃𝐶([−𝛾,0] × Ω,ℝ^𝑛)|𝑢(𝑡,𝑥) is bounded, 𝐹₀-measurable, and 𝐸‖𝑢(𝑡,𝑥)‖<∞}, 𝑃𝐶^𝑏_𝐹0(𝛿)∶={𝑢(𝑡,𝑥)∈𝑃𝐶^𝑏_𝐹0([−𝛾,0]×Ω,ℝ^𝑛)|𝐸‖𝑢(𝑡,𝑥)‖≤𝛿}. Throughout this paper, we always assume that a product equals to unity if the number of factors is zero.

Definition 1. A function 𝑢(𝑡,𝑥)∶[−𝛾,0]×Ω→ℝ^𝑛 is said to be the solution of system (2) if the following conditions are satisfied:(i)𝑢(𝑡,𝑥) is piecewise continuous with the first kind discontinuity at the points 𝑡_𝑘, 𝑘∈𝑁. Moreover, 𝑢(𝑡,𝑥) is left continuous at each point,(ii)𝑢(𝑡,𝑥) satisfies system (2).

Definition 2. The zero solution of system (2) is said to be as follows.(i)𝑝-stable if, for any 𝜀>0, there exists a 𝛿>0 such that the initial function 𝜙∈𝑃𝐶^𝑏_𝐹0(𝛿) implies 𝐸‖𝑢(𝑡,𝑥)‖^𝑝<𝜀 for (𝑡,𝑥)∈(0,+∞)×Ω. Especially, when 𝑝=1, it is said to be stable. (ii)Exponentially 𝑝-stable if there is a pair of positive constants 𝜆 and 𝐾 such that, for any initial condition 𝜙∈𝑃𝐶^𝑏_𝐹0([−𝛾,0]×Ω,ℝ^𝑛), there holds 𝐸‖𝑢(𝑡,𝑥)‖^𝑝≤𝐾‖𝜙‖^𝑝𝑒^−𝜆𝑡, 𝑡≥0. Here, 𝜆 is called the exponential convergence rate. When 𝑝=2 especially, it is said to be exponentially stable in mean square. (iii)Asymptotically stable if it is stable, and there exists a 𝛿>0 such that the initial function 𝜙∈𝑃𝐶^𝑏_𝐹0(𝛿) implies lim_𝑡→∞𝐸‖𝑢(𝑡,𝑥)‖=0.
For system (2), one makes the following assumptions: (H1)0≤𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯ are fixed impulsive moments such that 𝑡_𝑘→∞ as 𝑘→∞; (H2)𝐹,𝐺∶𝑅₊×𝑅^𝑛×⋯×𝑅^𝑛_𝑝+2→𝑅^𝑛 satisfy 𝐹(𝑡,0,…,0)≡0 and 𝐺(𝑡,0,…,0)≡0; (H3)𝐼_𝑘∶𝑅^𝑛→𝑅^𝑛 satisfies 𝐼_𝑘(0)=0,𝑘∈𝑁;(H4) except for the zero solution of (2), for any solution 𝑢(𝑡,𝑥) of (2), 𝑢(𝑡,𝑥)≠0 and 𝐼_𝑘(𝑢(𝑡_𝑘,𝑥))≠−𝑢(𝑡_𝑘,𝑥), 𝑘∈𝑁.
Denoting 𝐽_𝑘𝑖𝑢∶=^𝑖𝑡_𝑘,𝑥𝑢_𝑖𝑡_𝑘,𝑥+𝐼_𝑘𝑖𝑢₁𝑡_𝑘,𝑥,…,𝑢_𝑛𝑡_𝑘,,𝑥𝑖=1,2,…,𝑛,𝑘∈𝑁, (5) we consider the following delayed stochastic reaction-diffusion differential system without impulses: d𝑣(𝑡,𝑥)=∇⋅𝐷(𝑡,𝑥,𝑣(𝑡,𝑥))∘∇𝑣(𝑡,𝑥)d𝑡+_{0≤𝑡𝑘<𝑡}𝐽_𝑘𝐹𝑡,𝑣(𝑡,𝑥),𝑣𝑡−𝜏₁(𝑡)𝑣,𝑥,…,𝑡−𝜏_𝑝(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟},𝑣(𝑠,𝑥)𝑑𝑠d𝑡+_{0≤𝑡𝑘<𝑡}𝐽_𝑘𝐺𝑡,𝑣(𝑡,𝑥),𝑣𝑡−𝜏₁(𝑡)𝑣,𝑥,…,𝑡−𝜏_𝑝(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟},𝑣()(𝑡),𝑣()(),()∈𝑠,𝑥d𝑠d𝑤𝑡,𝑥=𝜙𝑡,𝑥𝑡,𝑥−𝛾,0×Ω, (6) where 𝐽_𝑘=diag𝐽_𝑘1,𝐽_𝑘2,…,𝐽_𝑘𝑛,𝐷(𝑡,𝑥,𝑣(𝑡,𝑥))=𝐷𝑡,𝑥,_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(),𝐹()𝑡,𝑥𝑡,𝑣𝑡,𝑥,𝑣𝑡−𝜏₁(𝑡)𝑣,𝑥,…,𝑡−𝜏_𝑝(𝑡),𝑥,^𝑡_{(𝑡)𝑡−𝑟}𝑣(𝑠,𝑥)d𝑠=𝐹𝑡,_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(𝑡,𝑥_{0≤𝑡𝑘<𝑡−𝜏1(𝑡)}),𝐽_𝑘⁻¹𝑣𝑡−𝜏₁(𝑡),𝑥,…,_{0≤𝑡𝑘<𝑡−𝜏𝑝(𝑡)}𝐽_𝑘⁻¹𝑣𝑡−𝜏_𝑝(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟0≤𝑡𝑘<𝑠},𝐽_𝑘⁻¹𝑣(),𝐺()𝑠,𝑥d𝑠𝑡,𝑣𝑡,𝑥,𝑣𝑡−𝜏₁(𝑡)𝑣,𝑥,…,𝑡−𝜏_𝑝(𝑡),𝑥,^𝑡_{(𝑡)𝑡−𝑟}𝑣(𝑠,𝑥)d𝑠=𝐺𝑡,_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(𝑡,𝑥_{0≤𝑡𝑘<𝑡−𝜏1(𝑡)}),𝐽_𝑘⁻¹𝑣𝑡−𝜏₁(𝑡),𝑥,…,_{0≤𝑡𝑘<𝑡−𝜏𝑝(𝑡)}𝐽_𝑘⁻¹𝑣𝑡−𝜏_𝑝(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟0≤𝑡𝑘<𝑠},𝐽_𝑘⁻¹𝑣(𝑠,𝑥)d𝑠. (7)
A function vector 𝑣(𝑡,𝑥) is a solution of (6) on [−𝛾,+∞)×Ω if it is absolutely continuous on [−𝛾,+∞)×Ω and satisfies (6) almost everywhere for 𝑡≥0 and 𝑥∈Ω.

3. Stability Criteria

In this section, we first establish an equivalent relation between the solution of system (2) and that of system (6).

Lemma 3. Assume that (H1)–(H4) hold. Then 𝑢(𝑡,𝑥) is a solution of system (2) if and only if 𝑣(𝑡,𝑥) is a solution of system (6), where 𝑢(𝑡,𝑥)=∏_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(𝑡,𝑥) or 𝑣(𝑡,𝑥)=∏_{0≤𝑡𝑘<𝑡}𝐽_𝑘𝑢(𝑡,𝑥).

Proof. First, we prove the sufficiency. Letting 𝑣(𝑡,𝑥) be a solution of system (6), we derive that, for any 𝑡≠𝑡_𝑘, d𝑢(𝑡,𝑥)=d_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(𝑡,𝑥)=_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹d(𝑣(𝑡,𝑥))=_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝐷((()+𝐹()∇⋅𝑡,𝑥,𝑣𝑡,𝑥))∘∇𝑣𝑡,𝑥d𝑡𝑡,𝑣𝑡,𝑥,𝑣𝑡−𝜏₁(𝑡)𝑣,𝑥,…,𝑡−𝜏_𝑝(𝑡),𝑥,^𝑡_{(𝑡)𝑡−𝑟}𝑣()+𝐺()𝑠,𝑥d𝑠d𝑡𝑡,𝑣𝑡,𝑥,𝑣𝑡−𝜏₁(𝑡),𝑥,…,𝑣𝑡−𝜏_𝑝(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟},𝑣(𝑠,𝑥)d𝑠d𝑤(𝑡)=_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹∇⋅𝐷𝑡,𝑥,_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(𝑡,𝑥)∘∇_{0≤𝑡𝑘<𝑡}𝐽_𝑘𝑢(𝑡,𝑥)d𝑡+𝐹𝑡,_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(𝑡,𝑥_{0≤𝑡𝑘<𝑡−𝜏1(𝑡)}),𝐽_𝑘⁻¹𝑣𝑡−𝜏₁(𝑡),𝑥,…,_{0≤𝑡𝑘<𝑡−𝜏𝑝(𝑡)}𝐽_𝑘⁻¹𝑣𝑡−𝜏_𝑝(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟0≤𝑡𝑘<𝑠},𝐽_𝑘⁻¹𝑣(𝑠,𝑥)d𝑠d𝑡+𝐺𝑡,_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(𝑡,𝑥_{0≤𝑡𝑘<𝑡−𝜏1(𝑡)}),𝐽_𝑘⁻¹𝑣𝑡−𝜏₁(𝑡),𝑥,…,_{0≤𝑡𝑘<𝑡−𝜏𝑝(𝑡)}𝐽_𝑘⁻¹𝑣𝑡−𝜏_𝑝(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟0≤𝑡𝑘<𝑠},𝐽_𝑘⁻¹𝑣()(𝑡)(𝐷(((()𝑠,𝑥d𝑠d𝑤=∇⋅𝑡,𝑥,𝑢𝑡,𝑥))∘∇𝑢𝑡,𝑥))d𝑡+𝐹𝑡,𝑢𝑡,𝑥,𝑢𝑡−𝜏₁(𝑡)𝑢,𝑥,…,𝑡−𝜏_𝑝(𝑡),𝑥,^𝑡_{(𝑡)𝑡−𝑟}𝑢()()𝑠,𝑥d𝑠d𝑡+𝐺𝑡,𝑢𝑡,𝑥,𝑢𝑡−𝜏₁(𝑡)𝑢,𝑥,…,𝑡−𝜏_𝑝(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟},𝑢(𝑠,𝑥)d𝑠d𝑤(𝑡). (8)
On the other hand, for any 𝑘∈𝑁, we have 𝑢𝑡⁺_𝑘,𝑥=lim_{𝑡→𝑡+𝑘0≤𝑡𝑗<𝑡}𝐽_𝑗⁻¹𝑣(𝑡,𝑥)=_{0≤𝑡𝑗≤𝑡𝑘}𝐽_𝑗⁻¹𝑣𝑡⁺_𝑘,𝑥=𝐽_𝑘⁻¹_{0≤𝑡𝑗<𝑡𝑘}𝐽_𝑗⁻¹𝑣𝑡_𝑘,𝑥=𝐽_𝑘⁻¹𝑢𝑡_𝑘𝑡,𝑥=𝑢_𝑘,𝑥+𝐼_𝑘𝑢𝑡_𝑘,𝑢𝑡,𝑥_−𝑘,𝑥=lim_{𝑡→𝑡−𝑘0≤𝑡𝑗<𝑡}𝐽_𝑗⁻¹𝑣()=𝑡,𝑥_{0≤𝑡𝑗<𝑡𝑘}𝐽_𝑗⁻¹𝑣𝑡⁻_𝑘,𝑥=_{0≤𝑡𝑗<𝑡𝑘}𝐽_𝑗⁻¹𝑣𝑡_𝑘,𝑥=𝑢𝑡_𝑘,𝑥. (9)(10)
Further, if 𝑣(𝑡,𝑥) is a solution of system (6) with initial condition 𝑣(𝑡,𝑥)=𝜙(𝑡,𝑥), (𝑡,𝑥)∈[−𝛾,0]×Ω, then 𝑢(𝑡,𝑥)=∏_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(𝑡,𝑥)=𝑣(𝑡,𝑥)=𝜙(𝑡,𝑥), (𝑡,𝑥)∈[−𝛾,0]×Ω. Therefore, 𝑢(𝑡,𝑥) is a solution of system (2) with initial condition 𝑢(𝑡,𝑥)=𝜙(𝑡,𝑥), (𝑡,𝑥)∈[−𝛾,0]×Ω, and the sufficiency is proved. Similarly, we can prove the necessity.

Remark 4. Lemma 3 gives the equivalent relation between the solution of a stochastic reaction-diffusion differential delay system with impulsive effects and the solution of a corresponding system without impulses. Based on the “equivalent method,” the existence and uniqueness of the solution of a stochastic reaction-diffusion differential delay system with impulsive effects can be derived by a new way; that is, any conditions that ensure the existence and uniqueness of the solution of system (6) without impulses will also ensure the existence and uniqueness of the system (2) with impulses.
In what follows, we will reduce the stabilities of system (2) to those of corresponding system (6).

Theorem 5. Under assumptions (H1)–(H4), if there exists a constant 𝑀>0 such that, for any 𝑡>0, _{0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖≤𝑀,𝑖=1,2,…,𝑛, (11) and the zero solution of (6) is 𝑝-stable (exponentially 𝑝-stable, asymptotically stable), then the zero solution of (2) is also 𝑝-stable (exponentially 𝑝-stable, asymptotically stable).

Proof. Let 𝑢(𝑡,𝑥) and 𝑣(𝑡,𝑥) be the solutions of systems (2) and (6), respectively. Since the zero solution of (6) is 𝑝-stable, we have that, for any 𝜀>0, there exists a scalar 𝛿>0 such that the initial condition 𝜙∈𝑃𝐶^𝑏_𝐹0(𝛿) implies 𝐸‖𝑣(𝑡,𝑥)‖^𝑝<𝜀/𝑀^𝑝 for (𝑡,𝑥)∈(0,+∞)×Ω. By Lemma 3, 𝑢(𝑡,𝑥)=∏_{0≤𝑡𝑘<𝑡}𝐽_𝑘⁻¹𝑣(𝑡,𝑥) is a solution of (2) on (𝑡,𝑥)∈[−𝛾,+∞)×Ω. Furthermore, it is easy to see that 𝐸‖𝑢(𝑡,𝑥)‖^𝑝=𝐸𝑢₁(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥)^T𝑝=𝐸_{0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘1𝑣₁(𝑡,𝑥),…,_{0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑛𝑣_𝑛(𝑡,𝑥)^T𝑝=𝐸^𝑛_{𝑖=1Ω0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖𝑣_𝑖(𝑡,𝑥)^𝑝d𝑥^1/𝑝𝑝≤𝐸𝑀^𝑛_𝑖=1Ω𝑣_𝑖(𝑡,𝑥)^𝑝d𝑥^1/𝑝𝑝=𝑀^𝑝𝐸‖𝑣(𝑡,𝑥)‖^𝑝<𝑀^𝑝⋅𝜀𝑀^𝑝=𝜀. (12) Hence, the zero solution of (2) is 𝑝-stable. Using similar arguments, we can verify that if the zero solution of (6) is exponentially 𝑝-stable (asymptotically stable), then the zero solution of (2) is also exponentially 𝑝-stable (asymptotically stable). This completes the proof.

In a similar way, we can derive the following results.

Theorem 6. Under assumptions (H1)–(H4), if there exists a constant 𝐿>0 such that, for any 𝑡>0, _{0≤𝑡𝑘<𝑡}𝐽_𝑘𝑖≤𝐿,𝑖=1,2,…,𝑛, (13) and the zero solution of (2) is 𝑝-stable (exponentially 𝑝-stable, asymptotically stable), then the zero solution of (6) is also 𝑝-stable (exponentially 𝑝-stable, asymptotically stable).

Combining Theorems 5 and 6, one can easily obtain the following results.

Theorem 7. Assume that (H1)–(H4) hold and inequalities (11) and (13) are satisfied, then the zero solution of (2) is 𝑝-stable (exponentially 𝑝-stable, asymptotically stable) if and only if the zero solution of (6) is also 𝑝-stable (exponentially 𝑝-stable, asymptotically stable).

Remark 8. For impulsive neural networks, many researchers supposed that the impulsive operators are linear (e.g., [7, 12–15]); that is, 𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡_𝑘=−𝛾_𝑘𝑖𝑥_𝑖𝑡_𝑘,0<𝛾_𝑘𝑖≤2. (14) From the definition of 𝐽_𝑘𝑖 in Section 2, we have |𝐽⁻¹_𝑘𝑖|=|1−𝛾_𝑘𝑖|<1; then |∏_{0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖|≤1. Obviously, the condition (11) in Theorem 5 is less conservative than (14), in which (11) ensures that the stability of the delayed stochastic reaction-diffusion differential system without impulses can be used to judge the stability of the corresponding system with impulses.

Remark 9. In [2, 8], the authors dealt with the stochastic differential systems with delay and nonlinear impulsive effects, the stability results of which were showed by transforming the system into a corresponding system without impulses. However, the distributed delays and diffusion effects were not taken into account in the previous systems. In this paper, we incorporated stochastic perturbations, reaction-diffusion effects, and mixed time delays into impulsive differential system and derived the stability criteria of the system. It is readily seen that our results are more general than those reported in [2, 8].

4. Application to Impulsive Stochastic Reaction-Diffusion Neural Networks

In this section, we apply our previous stability results to analyze the stability of the following ISRDCGNNs with time delays: d𝑢_𝑖(𝑡,𝑥)=^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝐷_𝑖𝑙𝑡,𝑥,𝑢_𝑖(𝑡,𝑥)𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑥_𝑙d𝑡−𝛼_𝑖𝑢_𝑖(𝑡,𝑥)𝛽_𝑖𝑢_𝑖()−𝑡,𝑥_𝑛𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑢_𝑗()−𝑡,𝑥_𝑛𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−,𝑥_𝑛𝑗=1𝑐_𝑖𝑗ℎ_𝑗^𝑡_{(𝑡)𝑡−𝑟}𝑢_𝑗()+𝑠,𝑥d𝑠d𝑡_𝑛𝑗=1𝜎_𝑖𝑗𝑡,𝑢_𝑗(𝑡,𝑥),𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟},𝑢_𝑗(𝑠,𝑥)d𝑠d𝑤_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑢_𝑖𝑡⁺_𝑘,𝑥−𝑢_𝑖𝑡_𝑘,𝑥=𝐼_𝑘𝑖𝑢₁𝑡_𝑘,𝑥,…,𝑢_𝑛𝑡_𝑘,𝑥,𝑡=𝑡_𝑘,𝑘∈𝑁,𝑖=1,2,…,𝑛, (15) where 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥))^T∈ℝ^𝑛 denotes the state vector associated with the neurons, 𝑥∈Ω⊂ℝ^𝑚, and Ω={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚)^T||𝑥_𝑘|<𝑙_𝑘,𝑘=1,2,…,𝑚} is a bounded compact set with smooth boundary 𝜕Ω and mesΩ>0, 𝐷_𝑖𝑙(𝑡,𝑥,𝑢_𝑖(𝑡,𝑥))≥0 denotes the diffusion function, and let 𝐷_𝑖𝑙=sup_{𝑡≥0,𝑥∈Ω}𝐷_𝑖𝑙(𝑡,𝑥,𝑢_𝑖(𝑡,𝑥)); (𝑎_𝑖𝑗)_𝑛×𝑛, (𝑏_𝑖𝑗)_𝑛×𝑛 and (𝑐_𝑖𝑗)_𝑛×𝑛 are the interconnection weight matrices, 𝛼_𝑖(𝑢_𝑖(𝑡,𝑥)) represents an amplification function, 𝛽_𝑖(𝑢_𝑖(𝑡,𝑥)) is an appropriately behaved function, 𝑓_𝑗, 𝑔_𝑗, and ℎ_𝑗 denote the activation functions, and (𝜎_𝑖𝑗)_𝑛×𝑛 is the diffusion coefficient matrix; 𝜎_𝑖=(𝜎_𝑖1,𝜎_𝑖2,…,𝜎_𝑖𝑛), 𝛽_𝑖(0)=𝑓_𝑖(0)=𝑔_𝑖(0)=ℎ_𝑖(0)=𝜎_𝑖𝑗(0)=0, and 𝜏_𝑖𝑗(𝑡)≤𝜏, 𝑟(𝑡)≤𝑟.

The boundary condition and the initial value of system (15) are 𝑢_𝑖(𝑡,𝑥)∣_𝜕Ω()∈[)𝑢=0,𝑡,𝑥−𝛿,+∞×𝜕Ω,𝑖=1,2,…,𝑛,_𝑖(𝑠,𝑥)=𝜙_𝑖(𝑠,𝑥),(𝑠,𝑥)∈[−𝛿,0]×Ω,𝑖=1,2,…,𝑛, (16) where 𝛿=max{𝜏,𝑟}.

Equivalently, we consider the following stochastic reaction-diffusion Cohen-Grossberg neura1 networks without impulses: d𝑣_𝑖()=𝑡,𝑥_𝑚𝑙=1𝜕𝜕𝑥_𝑙𝐷_𝑖𝑙𝑡,𝑥,_{0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖𝑣_𝑖(𝑡,𝑥)𝜕𝑣_𝑖(𝑡,𝑥)𝜕𝑥_𝑙d𝑡−_{0≤𝑡𝑘<𝑡}𝐽_𝑘𝑖⋅𝛼_{𝑖0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖𝑣_𝑖()×𝛽𝑡,𝑥_{𝑖0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖𝑣_𝑖()−𝑡,𝑥_𝑛𝑗=1𝑎_𝑖𝑗𝑓_{𝑗0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑗𝑣_𝑗()−𝑡,𝑥_𝑛𝑗=1𝑏_𝑖𝑗𝑔_{𝑗0≤𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}𝐽⁻¹_𝑘𝑗𝑣_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−,𝑥_𝑛𝑗=1𝑐_𝑖𝑗ℎ_𝑗^𝑡_{(𝑡)𝑡−𝑟0≤𝑡𝑘<𝑠}𝐽⁻¹_𝑘𝑠𝑣_𝑗(𝑠,𝑥)d𝑠d𝑡+_{0≤𝑡𝑘<𝑡}𝐽_𝑘𝑖⋅^𝑛_𝑗=1𝜎_𝑖𝑗𝑡,_{0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑗𝑣_𝑗(𝑡,𝑥_{0≤𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}),𝐽⁻¹_𝑘𝑗𝑣_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑥_{𝑡(𝑡)𝑡−𝑟0≤𝑡𝑘<𝑠},𝐽⁻¹_𝑘𝑗𝑣_𝑗(𝑠,𝑥)d𝑠d𝑤_𝑖(𝑡),𝑖=1,2,…,𝑛. (17)

Throughout this section, we make the following assumptions: (H5)𝛼_𝑖(𝑢) is a continuous function, and 0<𝛼_𝑖≤𝛼_𝑖(𝑢)≤𝛼_𝑖 for all 𝑢∈ℝ, 𝑖=1,2,…,𝑛; (H6) there exists a constant 𝛽_𝑖>0 such that 𝛽_𝑖(𝑢)−𝛽_𝑖(𝑣)𝑢−𝑣≥𝛽_𝑖, (18) for all 𝑢,𝑣∈ℝ(𝑢≠𝑣), 𝑖=1,2,…,𝑛; (H7) there exist positive constants 𝐹_𝑖, 𝐺_𝑖, and 𝐻_𝑖 such that 𝐹_𝑖=sup_𝑢≠𝑣𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)𝑢−𝑣,𝐺_𝑖=sup_𝑢≠𝑣𝑔_𝑖(𝑢)−𝑔_𝑖(𝑣),𝐻𝑢−𝑣_𝑖=sup_𝑢≠𝑣ℎ_𝑖(𝑢)−ℎ_𝑖(𝑣),𝑢−𝑣 (19) for all 𝑢,𝑣∈ℝ(𝑢≠𝑣), 𝑖=1,2,…,𝑛; (H8) there exist positive constants 𝑠⁽¹⁾_𝑖𝑗, 𝑠⁽²⁾_𝑖𝑗, and 𝑠⁽³⁾_𝑖𝑗(𝑖,𝑗=1,2,…,𝑛) such that 𝜎_𝑖𝑡,𝑢⁽¹⁾,𝑢⁽²⁾,𝑢⁽³⁾−𝜎_𝑖𝑡,𝑣⁽¹⁾,𝑣⁽²⁾,𝑣⁽³⁾×𝜎_𝑖𝑡,𝑢⁽¹⁾,𝑢⁽²⁾,𝑢⁽³⁾−𝜎_𝑖𝑡,𝑣⁽¹⁾,𝑣⁽²⁾,𝑣^(3)T≤^𝑛_𝑗=1𝑠⁽¹⁾_𝑖𝑗𝑢_𝑗⁽¹⁾−𝑣_𝑗⁽¹⁾²+𝑠⁽²⁾_𝑖𝑗𝑢_𝑗⁽²⁾−𝑣_𝑗⁽²⁾²+𝑠⁽³⁾_𝑖𝑗𝑢_𝑗⁽³⁾−𝑣_𝑗⁽³⁾², (20) for all 𝑢^(𝑘)=(𝑢₁^(𝑘),𝑢₂^(𝑘),…,𝑢_𝑛^(𝑘)), 𝑣^(𝑘)=(𝑣₁^(𝑘),𝑣₂^(𝑘),…,𝑣_𝑛^(𝑘)), 𝑘=1,2,3.

The following lemmas are useful in proving our main results.

Lemma 10 (see [16]). Let 𝑄 be an 𝑛×𝑛 matrix with nonpositive off-diagonal elements; then 𝑄 is an M-matrix if and only if there exists a vector 𝜉>0 such that 𝑄𝜉>0.

Lemma 11 (see [10]). Let 𝑝≥2 be a positive integer, 𝑙_𝑘(𝑘=1,2,…,𝑚) positive constants, and cube Ω={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚)^T∈ℝ^𝑚||𝑥_𝑘|<𝑙_𝑘,𝑘=1,2,…,𝑚}. Let ℎ(𝑥) be a real-valued function belonging to 𝐶¹(Ω) which vanishes on the boundary 𝜕Ω of Ω, that is, ℎ(𝑥)|_𝜕Ω=0. Then _Ω|ℎ(𝑥)|^𝑝𝑝d𝑥≤²𝑙²_𝑘4_Ω|ℎ(𝑥)|^𝑝−2𝜕ℎ(𝑥)𝜕𝑥_𝑘²d𝑥,𝑘=1,2,…,𝑚. (21)

Lemma 12 (see [17]). Let 𝑎,𝑏≥0, 𝑝≥𝑖>0; then 𝑎^𝑝−𝑖𝑏^𝑖≤𝑝−𝑖𝑝𝑎^𝑝+𝑖𝑝𝑏^𝑝. (22)

Theorem 13. Under assumptions (H5)–(H8), if inequalities (11) and (13) are satisfied and 𝑄+𝑇 is an M-matrix, where 𝑄=𝑞_{𝑖𝑗𝑛×𝑛},𝑞_𝑖𝑗=−𝐿𝑀𝛼_𝑖𝑎_𝑖𝑗𝐹_𝑗+𝑏_𝑖𝑗𝐺_𝑗+𝑟𝑐_𝑖𝑗𝐻_𝑗,𝑇=𝑡_{𝑖𝑗𝑛×𝑛},𝑡_𝑖𝑗=−𝑝−1𝐿²𝑀²𝑠⁽¹⁾_𝑖𝑗+𝑠⁽²⁾_𝑖𝑗+𝑟²𝑠⁽³⁾_𝑖𝑗,𝑡𝑖≠𝑗_𝑖𝑖=4𝑝−1𝑝^𝑚_𝑙=1𝐷_𝑖𝑙𝑙²_𝑖+𝑝𝛼_𝑖𝛽_𝑖−𝑝−1𝐿𝑀𝛼_𝑖×^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑏_𝑖𝑗𝐺_𝑗+𝑟𝑐_𝑖𝑗𝐻_𝑗+𝑝−2𝐿𝑀2𝛼_𝑖𝑠⁽¹⁾_𝑖𝑗+𝑠⁽²⁾_𝑖𝑗+𝑟²𝑠⁽³⁾_𝑖𝑗−𝑝−1𝐿²𝑀²𝑠⁽¹⁾_𝑖𝑖+𝑠⁽²⁾_𝑖𝑖+𝑟²𝑠⁽³⁾_𝑖𝑖, (23) then the zero solution of system (15) is exponentially 𝑝-stable.

Proof. Since 𝑄+𝑇 is an M-matrix, by Lemma 10, there exists 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)^T such that (𝑄+𝑇)𝜉>0; that is, 4𝑝−1𝑝^𝑚_𝑙=1𝐷_𝑖𝑙𝑙²_𝑖+𝑝𝛼_𝑖𝛽_𝑖−𝑝−1𝐿𝑀𝛼_𝑖×^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑏_𝑖𝑗𝐺_𝑗+𝑟𝑐_𝑖𝑗𝐻_𝑗+𝑝−2𝐿𝑀2𝛼_𝑖𝑠⁽¹⁾_𝑖𝑗+𝑠⁽²⁾_𝑖𝑗+𝑟²𝑠⁽³⁾_𝑖𝑗𝜉_𝑖−𝐿𝑀𝛼_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑏_𝑖𝑗𝐺_𝑗+𝑟𝑐_𝑖𝑗𝐻_𝑗+𝑝−1𝐿𝑀𝛼_𝑖𝑠⁽¹⁾_𝑖𝑗+𝑠⁽²⁾_𝑖𝑗+𝑟²𝑠⁽³⁾_𝑖𝑗𝜉_𝑗>0,𝑖=1,2,…,𝑛. (24)
We can choose a sufficiently small constant 𝜖>0 such that 4𝑝−1𝑝^𝑚_𝑙=1𝐷_𝑖𝑙𝑙²_𝑖+𝑝𝛼_𝑖𝛽_𝑖−𝜖−𝑝−1𝐿𝑀𝛼_𝑖×^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑏_𝑖𝑗𝐺_𝑗+𝑟𝑐_𝑖𝑗𝐻_𝑗+𝑝−2𝐿𝑀2𝛼_𝑖𝑠⁽¹⁾_𝑖𝑗+𝑠⁽²⁾_𝑖𝑗+𝑟²𝑠⁽³⁾_𝑖𝑗𝜉_𝑖−𝐿𝑀𝛼_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑝−1𝐿𝑀𝛼_𝑖𝑠⁽¹⁾_𝑖𝑗+𝑒^𝜖𝜏𝑏_𝑖𝑗𝐺_𝑗+𝑝−1𝐿𝑀𝛼_𝑖𝑠⁽²⁾_𝑖𝑗+𝑒^𝜖𝑟−1𝜖𝑐_𝑖𝑗𝐻_𝑗+𝑝−1𝐿𝑀𝑟𝛼_𝑖𝑠⁽³⁾_𝑖𝑗𝜉_𝑗>0. (25) Let 𝑤_𝑖(𝑡,𝑥)=𝑒^𝜖𝑡∫_Ω|𝑣_𝑖(𝑡,𝑥)|^𝑝d𝑥, 𝑝≥2, 𝑖=1,2,…,𝑛. By the Ito differential formula, the stochastic derivative of 𝑤_𝑖(𝑡,𝑥) along (17) can be derived as follows: 𝐿𝑤_𝑖(𝑡,𝑥)=𝜖𝑒^𝜖𝑡_Ω𝑣_𝑖(𝑡,𝑥)^𝑝d𝑥+𝑝𝑒^𝜖𝑡sgn𝑣_𝑖(𝑡,𝑥)_Ω𝑣_𝑖(𝑡,𝑥)^𝑝−1×^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝐷_𝑖𝑙𝑡,𝑥,_{0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖𝑣_𝑖(𝑡,𝑥)𝜕𝑣_𝑖(𝑡,𝑥)𝜕𝑥_𝑙−_{0≤𝑡𝑘<𝑡}𝐽_𝑘𝑖⋅𝛼_{𝑖0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖𝑣_𝑖()×𝛽𝑡,𝑥_{𝑖0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖𝑣_𝑖()−𝑡,𝑥_𝑛𝑗=1𝑎_𝑖𝑗𝑓_{𝑗0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑗𝑣_𝑗(−𝑡,𝑥)_𝑛𝑗=1𝑏_𝑖𝑗𝑔_{𝑗0≤𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}𝐽⁻¹_𝑘𝑗𝑣_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−,𝑥_𝑛𝑗=1𝑐_𝑖𝑗ℎ_𝑗^𝑡_{(𝑡)𝑡−𝑟0≤𝑡𝑘<𝑠}𝐽⁻¹_𝑘𝑠𝑣_𝑗()+𝑝𝑠,𝑥d𝑠d𝑥𝑝−12𝑒^𝜖𝑡_{0≤𝑡𝑘<𝑡}𝐽_𝑘𝑖²_Ω𝑣_𝑖(𝑡,𝑥)^𝑝−2𝜎_𝑖𝜎^T_𝑖d𝑥,𝑖=1,2,…,𝑛. (26)
By Lemma 11, we derive from (16) that 𝑝sgn𝑣_𝑖(𝑡,𝑥)_Ω𝑣_𝑖(𝑡,𝑥)^𝑝−1×^𝑚_𝑙=1𝜕𝜕𝑥_𝑙𝐷_𝑖𝑙𝑡,𝑥,_{0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖𝑣_𝑖()×𝑡,𝑥𝜕𝑣_𝑖(𝑡,𝑥)𝜕𝑥_𝑙d𝑥=𝑝_Ω𝑣_𝑖(𝑡,𝑥)^𝑝−2𝑣_𝑖()∇⋅𝐷𝑡,𝑥_𝑖𝑙𝑡,𝑥,_{0≤𝑡𝑘<𝑡}𝐽⁻¹_𝑘𝑖𝑣_𝑖(𝑡,𝑥)𝜕𝑣_𝑖(𝑡,𝑥)𝜕𝑥_𝑙^𝑚_𝑙=1d𝑥≤−𝑝𝑝−1_Ω^𝑚_𝑙=1𝐷_𝑖𝑙𝑣_𝑖(𝑡,𝑥)^𝑝−2𝜕𝑣_𝑖(𝑡,𝑥)𝜕𝑥_𝑙²4d𝑥≤−𝑝−1𝑝^𝑚_𝑙=1𝐷_𝑖𝑙𝑙²_𝑖^Ω𝑣_𝑖(𝑡,𝑥)^𝑝d𝑥. (27) Applying assumptions (H5)–(H8) and Lemma 12, we can deduce from (11), (13), and (27) that 𝐿𝑤_𝑖(𝑡,𝑥)≤𝑒^𝜖𝑡_Ω4𝜖−𝑝−1𝑝^𝑚_𝑙=1𝐷_𝑖𝑙𝑙²_𝑖−𝑝𝛼_𝑖𝛽_𝑖×𝑣_𝑖(𝑡,𝑥)^𝑝+𝑝𝐿𝑀𝛼_𝑖𝑣_𝑖(𝑡,𝑥)^𝑝−1×^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗𝑣_𝑗(+𝑏𝑡,𝑥)_𝑖𝑗𝐺_𝑗𝑣_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑐,𝑥_𝑖𝑗𝐻_𝑗^𝑡_{(𝑡)𝑡−𝑟}𝑣_𝑗(+𝑝𝑠,𝑥)d𝑠𝑝−12𝐿²𝑀²𝑣_𝑖(𝑡,𝑥)^𝑝−2×^𝑛_𝑗=1𝑠⁽¹⁾_𝑖𝑗𝑣²_𝑗(𝑡,𝑥)+𝑠⁽²⁾_𝑖𝑗𝑣²_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑥+𝑟𝑠⁽³⁾_𝑖𝑗^𝑡_{(𝑡)𝑡−𝑟}𝑣²_𝑗(𝑠,𝑥)d𝑠d𝑥≤𝑒^𝜖𝑡_Ω4𝜖−𝑝−1𝑝^𝑚_𝑙=1𝐷_𝑖𝑙𝑙²_𝑖−𝑝𝛼_𝑖𝛽_𝑖×𝑣_𝑖(𝑡,𝑥)^𝑝+𝑝−1𝐿𝑀𝛼_𝑖𝑣_𝑖(𝑡,𝑥)^𝑝×^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑏_𝑖𝑗𝐺_𝑗+𝑟𝑐_𝑖𝑗𝐻_𝑗+𝐿𝑀𝛼_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗𝑣_𝑗(𝑡,𝑥)^𝑝+𝑏_𝑖𝑗×𝐺_𝑗𝑣_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑥^𝑝+𝑐_𝑖𝑗𝐻_𝑗×^𝑡_{(𝑡)𝑡−𝑟}𝑣_𝑗(𝑠,𝑥)^𝑝+d𝑠𝑝−1𝑝−22𝐿²𝑀²𝑣_𝑖(𝑡,𝑥)^𝑝×^𝑛_𝑗=1𝑠⁽¹⁾_𝑖𝑗+𝑠⁽²⁾_𝑖𝑗+𝑟²𝑠⁽³⁾_𝑖𝑗+𝑝−1𝐿²𝑀²×^𝑛_𝑗=1𝑠⁽¹⁾_𝑖𝑗𝑣_𝑗(𝑡,𝑥)^𝑝+𝑠⁽²⁾_𝑖𝑗𝑣_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑥^𝑝+𝑟𝑠⁽³⁾_𝑖𝑗×^𝑡_{(𝑡)𝑡−𝑟}𝑣_𝑗(𝑠,𝑥)^𝑝≤4d𝑠d𝑥𝜖−𝑝−1𝑝^𝑚_𝑙=1𝐷_𝑖𝑙𝑙²_𝑖−𝑝𝛼_𝑖𝛽_𝑖+𝑝−1𝐿𝑀𝛼_𝑖×^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑏_𝑖𝑗𝐺_𝑗+𝑟𝑐_𝑖𝑗𝐻_𝑗+𝑝−2𝐿𝑀2𝛼_𝑖×𝑠⁽¹⁾_𝑖𝑗+𝑠⁽²⁾_𝑖𝑗+𝑟²𝑠⁽³⁾_𝑖𝑗𝑤_𝑖(𝑡,𝑥)+𝐿𝑀𝛼_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑝−1𝐿𝑀𝛼_𝑖𝑠⁽¹⁾_𝑖𝑗×𝑤_𝑗(𝑡,𝑥)+𝑒^𝜖𝜏𝑏_𝑖𝑗𝐺_𝑗+𝑝−1𝐿𝑀𝛼_𝑖𝑠⁽²⁾_𝑖𝑗×𝑤_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑐,𝑥_𝑖𝑗𝐻_𝑗+𝑝−1𝐿𝑀𝑟𝛼_𝑖𝑠⁽³⁾_𝑖𝑗×^𝑡_{(𝑡)𝑡−𝑟}𝑒^{𝜖(𝑡−𝑠)}𝑤_𝑗(𝑠,𝑥)d𝑠. (28) Further, we can get 𝐷⁺𝐸𝑤_𝑖()≤4𝑡,𝑥𝜖−𝑝−1𝑝^𝑚_𝑙=1𝐷_𝑖𝑙𝑙²_𝑖−𝑝𝛼_𝑖𝛽_𝑖+𝑝−1𝐿𝑀𝛼_𝑖×^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑏_𝑖𝑗𝐺_𝑗+𝑟𝑐_𝑖𝑗𝐻_𝑗+𝑝−2𝐿𝑀2𝛼_𝑖𝑠⁽¹⁾_𝑖𝑗+𝑠⁽²⁾_𝑖𝑗+𝑟²𝑠⁽³⁾_𝑖𝑗𝐸𝑤_𝑖(𝑡,𝑥)+𝐿𝑀𝛼_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑗+𝑝−1𝐿𝑀𝛼_𝑖𝑠⁽¹⁾_𝑖𝑗𝐸𝑤_𝑗(𝑡,𝑥)+𝑒^𝜖𝜏𝑏_𝑖𝑗𝐺_𝑗+𝑝−1𝐿𝑀𝛼_𝑖𝑠⁽²⁾_𝑖𝑗×𝐸𝑤_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑐,𝑥_𝑖𝑗𝐻_𝑗+𝑝−1𝐿𝑀𝑟𝛼_𝑖𝑠⁽³⁾_𝑖𝑗×^𝑡_{(𝑡)𝑡−𝑟}𝑒^{𝜖(𝑡−𝑠)}𝐸𝑤_𝑗(𝑠,𝑥)d𝑠. (29)
Denoting 𝑘₀=‖𝜙‖^𝑝/min_{1≤𝑖≤𝑛}{𝜉_𝑖}, we have 𝐸𝑤_𝑖(𝑡,𝑥)=𝑒^𝜖𝑡𝐸𝑣_𝑖(𝑡,𝑥)^𝑝≤𝐸𝑣_𝑖(𝑡,𝑥)^𝑝≤𝑘₀𝜉_𝑖,𝑡∈[−𝛿,0],𝑖=1,2,…,𝑛. (30) In what follows, we prove that 𝐸𝑤_𝑖(𝑡,𝑥)≤𝑘₀𝜉_𝑖,𝑡≥0,𝑖=1,2,…,𝑛. (31) In fact, if (31) is not true, then there exist 𝑖₀∈{1,2,…,𝑛} and 𝑡^∗∈[0,+∞) such that 𝐸𝑤_𝑖0𝑡^∗,𝑥≤𝑘₀𝜉_𝑖0,𝐷⁺𝐸𝑤_𝑖0𝑡^∗,𝑥>0,𝐸𝑤_𝑗(𝑡,𝑥)≤𝑘₀𝜉_𝑗,𝑡∈−𝛿,𝑡^∗,𝑗=1,2,…,𝑛. (32) However, (25), (29), and (32) imply that 𝐷⁺𝐸𝑤_𝑖0𝑡^∗,𝑥≤𝑘₀4𝜖−𝑝−1𝑝^𝑚_𝑙=1𝐷_𝑖0𝑙𝑙²_𝑖0−𝑝𝛼_𝑖0𝛽_𝑖0+𝑝−1𝐿𝑀𝛼_𝑖0×^𝑛_𝑗=1𝑎_𝑖0𝑗𝐹_𝑗+𝑏_𝑖0𝑗𝐺_𝑗+𝑟𝑐_𝑖0𝑗𝐻_𝑗+𝑝−2𝐿𝑀2𝛼_𝑖0𝑠_𝑖⁽¹⁾_0𝑗+𝑠_𝑖⁽²⁾_0𝑗+𝑟²𝑠_𝑖⁽³⁾_0𝑗𝜉_𝑖0+𝐿𝑀𝛼_𝑖0×^𝑛_𝑗=1𝑎_𝑖0𝑗𝐹_𝑗+𝑝−1𝐿𝑀𝛼_𝑖0𝑠_𝑖⁽¹⁾_0𝑗+𝑒^𝜖𝜏𝑏_𝑖0𝑗𝐺_𝑗+𝑝−1𝐿𝑀𝛼_𝑖0𝑠_𝑖⁽²⁾_0𝑗+𝑒^𝜖𝑟−1𝜖𝑐_𝑖0𝑗𝐻_𝑗+𝑝−1𝐿𝑀𝑟𝛼_𝑖0𝑠_𝑖⁽³⁾_0𝑗𝜉_𝑗<0, (33) which is a contradiction. Hence, (31) holds, which leads to 𝐸𝑣_𝑖(𝑡,𝑥)^𝑝≤𝑘₀𝜉_𝑖𝑒^−𝜖𝑡,𝑡≥0,𝑖=1,2,…,𝑛. (34) Therefore, 𝐸‖𝑣(𝑡,𝑥)‖^𝑝≤𝐾𝜙^𝑝𝑒^−𝜖𝑡,𝑡≥0, (35) where 𝐾=(max_{1≤𝑖≤𝑛}{𝜉_𝑖}/min_{1≤𝑖≤𝑛}{𝜉_𝑖})𝑛^𝑝. This means that the zero solution of system (17) is exponentially 𝑝-stable and the exponential convergence rate equals 𝜖. By Theorem 5, we can obtain that the zero solution of system (15) is exponentially 𝑝-stable. The proof is complete.

Remark 14. The stability of impulsive Cohen-Grossberg neural networks without spacial diffusion or distributed delays or stochastic disturbance, which are special cases of system (15), have been studied in [5, 13, 18–20]. It should be noted that the main result in [13] is a special case of Theorem 13. Further, the stability criteria derived in [5, 18–20] are dependent on the intervals of adjoining impulsive moments, while our results are independent of that. Thus, our results are new, and they effectually complement or improve the previously published results.

Remark 15. In [19, 21–26], the reaction-diffusion neural networks have been investigated. Nevertheless, the diffusion terms were eliminated by inequality analysis techniques, and the derived conditions for the stability of neural networks are the same as those obtained in the cases when there are no reaction-diffusion terms in the systems. Thus, our results including reaction-diffusion terms are less conservative than those in [19, 21–26].

Remark 16. As far as we know, almost all the existing results concerning the stability of neural networks are based on 2-norm (e.g., [5, 11, 19, 21, 22, 26–28]). In this paper, we derived the stability criteria of ISRDCGNNs with mixed time delays in terms of 𝑝-norm. Hence, our results generalize and improve the existing results reported in the previous literature.

5. Numerical Example

In this section, we give an example to illustrate the main theoretical results in Sections 3 and 4.

In system (15), let 𝑛=𝑚=2,𝛼₁(𝑥)𝛼=2.5+0.5cos𝑥,₂(𝑥)=3−sin𝑥,𝛽₁(𝑥)𝛽=8𝑥,₂(𝑥)𝑓=10𝑥,_𝑗(𝑥)=𝑔_𝑗(𝑥)=ℎ_𝑗(𝑥)=12(|𝑥+1|+|𝑥−1|),𝑙_𝑗,𝑟(𝑡)𝐷=1𝑗=1,2=1+sin𝑡,_𝑖𝑙(𝑥)=,𝑎6688_𝑖𝑗=,𝑏0.1−0.2−0.3−0.5^𝑖𝑗=,𝑐0.80.80.40.2_𝑖𝑗=,𝜏0.3−0.20.1−0.2_𝑖𝑗(𝑡)=0.02sin²(𝑡)0.01|cos𝑡|0.03cos²(𝑡)||,𝜎0.02sin2𝑡_𝑖𝑗=,𝐼𝑥,𝑦,𝑧0.1𝑥−0.2𝑦−0.1𝑧0.2𝑥+0.3𝑦+0.1𝑧0.5𝑥+0.4𝑦−0.3𝑧0.3𝑥+0.1𝑦+0.2𝑧_𝑘1𝑢₁𝑡_𝑘,𝑥,𝑢₂𝑡_𝑘=1,𝑥2^𝑘+2𝑢₁𝑡_𝑘+1,𝑥(2𝑘)²𝑢₁𝑡_𝑘,𝑥sin²𝑢₂𝑡_𝑘,𝐼,𝑥_𝑘2𝑢₁𝑡_𝑘,𝑥,𝑢₂𝑡_𝑘=1,𝑥2²3^𝑘𝑢₂𝑡_𝑘+1,𝑥(2𝑘)²𝑢₂𝑡_𝑘,𝑥cos²𝑢₁𝑡_𝑘,𝑥. (36)

By direct calculation, we obtain that 𝛼₁=2,𝛼₁=3,𝛼₂=2,𝛼₂=4,𝛽₁=8,𝛽₂𝐹=10,_𝑖=𝐺_𝑖=𝐻_𝑖𝑠=1,𝜏=0.03,𝑟=2,_(1)𝑖𝑗=,𝑠0.040.120.60.18^(2)𝑖𝑗=,𝑠0.080.180.480.06_(3)𝑖𝑗=,𝐽0.040.060.360.12_−1𝑘11=1+2^𝑘+2+1(2𝑘)²sin²𝑢₂𝑡_𝑘,𝐽,𝑥_−1𝑘21=1+2²3^𝑘+1(2𝑘)²cos²𝑢₁𝑡_𝑘,𝑥,𝐿=1,𝑀=2. (37)(38) Hence, assumptions (H5)–(H8) and inequalities (11) and (13) are satisfied. Taking 𝑝=4, it is not difficult to compute that ,,𝑄=−9.0−8.4−7.2−8.8,𝑇=34.60−6.48−30.2432.48𝑄+𝑇=25.60−14.88−37.4423.68 (39) in which 𝑄+𝑇 is an M-matrix, and all the conditions of Theorem 13 are satisfied. From Theorem 13, we know that the zero solution of system (15) with the parameters and functions above is exponentially 4-stable (see Figure 1).

(a)

(b)

Figure 1 

Transient behaviors of the state variables 𝑢1(𝑡,𝑥) and 𝑢2(𝑡,𝑥) in the example.

6. Concluding Remark

In this paper, we incorporated stochastic perturbations, reaction-diffusion effects, and mixed time delays into impulsive differential systems. First, an equivalent relation between the solution of a stochastic reaction-diffusion differential system with time delays and impulsive effects and that of corresponding system without impulses was established. Second, some stability criteria for the stochastic reaction-diffusion differential system with time delays and impulsive effects were derived by transforming the solutions of the system to those of corresponding one without impulses. Third, the stability criteria were applied to ISRDCGNNs with mixed time delays, and sufficient conditions were obtained for the exponential 𝑝-stability of the zero solution to the neural networks. Lastly, a numerical example was provided to illustrate the effectiveness of our theoretical results. Our stability results provide a new, convenient, and efficient approach to study the stability of stochastic reaction-diffusion differential systems with time delays and impulsive effects, and some previously published results are generalized and improved.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11071254, 11371368) and the Natural Science Foundation of Hebei Province (A2013506012).