LMI Approach to Exponential Stability and Almost Sure Exponential Stability for Stochastic Fuzzy Markovian-Jumping Cohen-Grossberg Neural Networks with Nonlinear 𝑝-Laplace Diffusion

Abstract

The robust exponential stability of delayed fuzzy Markovian-jumping Cohen-Grossberg neural networks (CGNNs) with nonlinear 𝑝-Laplace diffusion is studied. Fuzzy mathematical model brings a great difficulty in setting up LMI criteria for the stability, and stochastic functional differential equations model with nonlinear diffusion makes it harder. To study the stability of fuzzy CGNNs with diffusion, we have to construct a Lyapunov-Krasovskii functional in non-matrix form. But stochastic mathematical formulae are always described in matrix forms. By way of some variational methods in 𝑊^1,𝑝(Ω), Itô formula, Dynkin formula, the semi-martingale convergence theorem, Schur Complement Theorem, and LMI technique, the LMI-based criteria on the robust exponential stability and almost sure exponential robust stability are finally obtained, the feasibility of which can efficiently be computed and confirmed by computer MatLab LMI toolbox. It is worth mentioning that even corollaries of the main results of this paper improve some recent related existing results. Moreover, some numerical examples are presented to illustrate the effectiveness and less conservatism of the proposed method due to the significant improvement in the allowable upper bounds of time delays.

1. Introduction

It is well known that in 1983, Cohen-Grossberg [1] proposed originally the Cohen-Grossberg neural networks (CGNNs). Since then the CGNNs have found their extensive applications in pattern recognition, image and signal processing, quadratic optimization, and artificial intelligence [2–6]. However, these successful applications are greatly dependent on the stability of the neural networks, which is also a crucial feature in the design of the neural networks. In practice, time delays always occur unavoidably due to the finite switching speed of neurons and amplifiers [2–8], which may cause undesirable dynamic network behaviors such as oscillation and instability. Besides delay effects, stochastic effects also exist in real systems. In fact, many dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as sudden environment changes, repairs of the components, changes in the interconnections of subsystems, and stochastic failures. (see [9] and references therein). The stability problems for stochastic systems, in particular the Itô-type stochastic systems, become important in both continuous-time case and discrete-time case [10]. In addition, neural networks with Markovian jumping parameters have been extensively studied due to the fact that systems with Markovian jumping parameters are useful in modeling abrupt phenomena, such as random failures, operating in different points of a nonlinear plant, and changing in the interconnections of subsystems [11–15].

Remark 1. Deterministic system is only the simple simulation for the real system. Indeed, to model a system realistically, a degree of randomness should be incorporated into the model due to various inevitable stochastic factors. For example, in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. It is showed that the above-mentioned stochastic factors likewise cause undesirable dynamic network behaviors and possibly lead to instability. So it is of significant importance to consider stochastic effects for neural networks. In recent years, the stability of stochastic neural networks has become a hot study topic [3, 16–21].

On the other hand, diffusion phenomena cannot be unavoidable in real world. Usually diffusion phenomena are simply simulated by linear Laplace diffusion in much of the previous literature [2, 22–24]. However, diffusion behavior is so complicated that the nonlinear reaction-diffusion models were considered in several papers [3, 25–28]. Very recently, the nonlinear 𝑝-Laplace diffusion (𝑝>1) is applied to the simulation of some diffusion behaviors [3]. But almost all of the above mentioned works were focused on the traditional neural networks models without fuzzy logic. In the factual operations, we always encounter some inconveniences such as the complicity, the uncertainty and vagueness. As far as we know, vagueness is always opposite to exactness. To a certain degree, vagueness cannot be avoided in the human way of regarding the world. Actually, vague notations are often applied to explain some extensive detailed descriptions. As a result, fuzzy theory is regarded as the most suitable setting to taking vagueness and uncertainty into consideration. In 1996, Yang and his coauthor [29] originally introduced the fuzzy cellular neural networks integrating fuzzy logic into the structure of traditional neural networks and maintaining local connectedness among cells. Moreover, the fuzzy neural network is viewed as a very useful paradigm for image processing problems since it has fuzzy logic between its template input and/or output besides the sum of product operation. In addition, the fuzzy neural network is a cornerstone in image processing and pattern recognition. And hence, investigations on the stability of fuzzy neural networks have attracted a great deal of attention [30–37]. Note that stochastic stability for the delayed 𝑝-Laplace diffusion stochastic fuzzy CGNNs have never been considered. Besides, the stochastic exponential stability always remains the key factor of concern owing to its importance in designing a neural network, and such a situation motivates our present study. Moreover, the robustness result is also a matter of urgent concern [10, 38–46], for it is difficult to achieve the exact parameters in practical implementations. So in this paper, we will investigate the stochastic global exponential robust stability criteria for the nonlinear reaction-diffusion stochastic fuzzy Markovian-jumping CGNNs by means of linear matrix inequalities (LMIs) approach.

Both the non-linear 𝑝-Laplace diffusion and fuzzy mathematical model bring a great difficulty in setting up LMI criteria for the stability, and stochastic functional differential equations model with nonlinear diffusion makes it harder. To study the stability of fuzzy CGNNs with diffusion, we have to construct a Lyapunov-Krasovskii functional in non-matrix form (see, e.g., [4]). But stochastic mathematical formulae are always described in matrix forms. Note that there is no stability criteria for fuzzy CGNNs with 𝑝-Laplace diffusion, let alone Markovian-jumping stochastic fuzzy CGNNs with 𝑝-Laplace diffusion. Only the exponential stability of Itô-type stochastic CGNNs with 𝑝-Laplace diffusion was studied by one literature [3] in 2012. Recently, Ahn use the passivity approach to derive a learning law to guarantee that Takagi-Sugeno fuzzy delayed neural networks are passive and asymptotically stable (see, e.g., [47, 48] and related literature [49–57]). Especially, LMI optimization approach for switched neural networks (see, e.g., [53]) may bring some new edificatory to our studying the stability criteria of Markovian jumping CGNNs. Muralisankar, Gopalakrishnan, Balasubramaniam, and Vembarasan investigated the LMI-based robust stability for Takagi-Sugeno fuzzy neural networks [36, 38–41]. Mathiyalagan et al. studied robust passivity criteria and exponential stability criteria for stochastic fuzzy systems [10, 37, 42–46]. Motivated by some recent related works ([9, 10, 36–57], and so on), particularly, Zhu and Li [4], Zhang et al. [2], Pan and Zhong [58], we are to investigate the exponential stability and robust stability of Itô-type stochastic Markovian jumping fuzzy CGNNs with 𝑝-Laplace diffusion. By way of some variational methods in 𝑊^1,𝑝(Ω) (Lemma 6), Itô formula, Dynkin formula, the semi-martingale convergence theorem, Schur Complement Theorem, and LMI technique, the LMI-based criteria on the (robust) exponential stability and almost sure exponential (robust) stability are finally obtained, the feasibility of which can efficiently be computed and confirmed by computer matlab LMI toolbox. When 𝑝=2, or ignoring some fuzzy or stochastic effects, the simplified system may be investigated by existing literature (see, e.g., [2–4, 58]). Another purpose of this paper is to verify that some corollaries of our main results improve some existing results in the allowable upper bounds of time delays, which may be illustrated by numerical examples (see, e.g., Examples 30 and 36).

The rest of this paper is organized as follows. In Section 2, the new 𝑝-Laplace diffusion fuzzy CGNNs models are formulated, and some preliminaries are given. In Section 3, new LMIs are established to guarantee the stochastic global exponential stability and almost sure exponential stability of the above-mentioned CGNNs. Particularly in Section 4, the robust exponential stability criteria are given. In Section 5, Examples 28, 30, 32, 35, 36, and 38 are presented to illustrate that the proposed methods improve significantly the allowable upper bounds of delays over some existing results ([4, Theorem 1], [4, Theorem 3], [58, Theorem 3.1], [58, Theorem 3.2]). Finally, some conclusions are presented in Section 6.

2. Model Description and Preliminaries

In 2012, Zhu and Li [4] consider the following stochastic fuzzy Cohen-Grossberg neural networks: 𝑑𝑥_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)𝑏_𝑖𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)−^𝑛_𝑗=1𝑐̆_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗𝑔_𝑗𝑥_𝑗()−𝑡−𝜏_𝑛𝑗=1𝑑̆_𝑖𝑗𝑔_𝑗𝑥_𝑗()+𝑡−𝜏𝑑𝑡_𝑛𝑗=1𝜎_𝑖𝑗𝑥_𝑗(𝑡),𝑥_𝑗(𝑡−𝜏)𝑑𝑤_𝑗(𝑡),𝑥_𝑖(𝑡)=𝜙_𝑖(𝑡),−𝜏⩽𝑡⩽0, (1) where each 𝑤_𝑗(𝑡) is scalar standard Brownian motion defined on a complete probability space (𝛀,ℱ,𝒫) with a natural filtration {ℱ_𝑡}_𝑡⩾0. The noise perturbation 𝜎_𝑖𝑗∶𝑅×𝑅→𝑅 is a Borel measurable function. ⋀ and ⋁ denote the fuzzy AND and OR operation, respectively. Under several inequalities conditions and the following five similar assumptions on System (1), some exponential stability results are obtained in [4]. Of course, in this paper, we may present the following conditions which are more flexible than those of [4].(A1) There exists a positive definite diagonal matrix 𝐴=diag(𝑎₁,𝑎₂,…,𝑎_𝑛) such that 0<𝑎_𝑖(𝑟)⩽𝑎_𝑖, (2) for all 𝑟∈𝑅, 𝑖=1,2,…,𝑛.(A2) There exist positive definite diagonal matrix 𝐵=diag(𝐵₁,𝐵₂,…,𝐵_𝑛) such that 𝑎_𝑗(𝑟)𝑏_𝑗(𝑟)𝑟⩾𝐵_𝑗,∀𝑗=1,2,…,𝑛,0≠𝑟∈𝑅. (3)(A3) For any given 𝑗=1,2,…,𝑛, 𝑓_𝑗 is locally Lipschitz continuous, and there exists a constant 𝐹_𝑗 such that |𝑓^′_𝑗(𝑟)|⩽|𝐹_𝑗| for all 𝑟∈𝑅 at which 𝑓_𝑗 is differentiable; 𝑔_𝑗 is locally Lipschitz continuous, and there exists a constant 𝐺_𝑗 such that |𝑔^′_𝑗(𝑟)|⩽|𝐺_𝑗| at which 𝑔_𝑗 is differentiable.(A4) There exist nonnegative matrices 𝓤=(𝜇_𝑖𝑗)_𝑛×𝑛 and 𝓥=(𝜈_𝑖𝑗)_𝑛×𝑛 such that trace𝜎^𝑇(𝑢,𝑣)𝜎(𝑢,𝑣)⩽𝑢^𝑇𝓤𝑢+𝑣^𝑇𝓥𝑣, (4) where 𝑢,𝑣∈𝑅^𝑛, 𝜎(𝑢,𝑣)=(𝜎_𝑖𝑗(𝑢,𝑣))_𝑛×𝑛.(A5)𝑏_𝑗(0)=𝑓_𝑗(0)=𝑔_𝑗(0)=0, 𝜎_𝑖𝑗(0,0)≡0, 𝑖,𝑗=1,2,…,𝑛.

Remark 2. The condition (A3) is different from that of some existing literature (e.g., [2–4]). In those previous literature, 𝑓_𝑗 and 𝑔_𝑗 are always assumed to be globally Lipschitz continuous. Here, we relax this assumption, for 𝑓_𝑗 and 𝑔_𝑗 are only the local Lipschitz continuous functions. From Rademacher's theorem [59], a locally Lipschitz continuous function 𝑓∶𝑅^𝑛→𝑅^𝑛 is differentiable almost everywhere. Let 𝔇_𝑓 be the set of those points where 𝑓 is differentiable, then 𝑓^′(𝑥) is the Jacobian of 𝑓 at 𝑥∈𝔇_𝑓 and the set 𝔇_𝑓 is dense in 𝑅^𝑛. The generalized Jacobian 𝜕𝑓(𝑥) of a locally Lipschitz continuous function 𝑓∶𝑅^𝑛→𝑅^𝑛 is a set of matrices defined by 𝜕𝑓(𝑥)=co𝔴∣thereexistsasequence𝑥^𝑘⊂𝔇_𝑓withlim_{𝑥𝑘→𝑥}𝑓^′𝑥^𝑘=𝔴, (5)
where co(⋅) denotes the convex hull of a set.

Remark 3. The conditions (A1) and (A2) relax the corresponding ones in some previous literature (e.g., [2–4]).

The condition (A5) guarantees zero-solution is an equilibrium of stochastic fuzzy system (1). Throughout this paper, we always assume that all assumptions (A1)–(A5) hold. In addition, we assume that 𝓤 and 𝓥 are symmetric matrices in consideration of LMI-based criteria presented in this paper.

Besides delays, stochastic effects, the complexity, the vagueness and diffusion behaviors always occur in real nervous systems. So in this paper, we are to consider the following delays stochastic fuzzy Markovian-jumping Cohen-Grossberg neural networks with nonlinear 𝑝-Laplace diffusion (𝑝>1): 𝑑𝑣_𝑖()=𝑡,𝑥_𝑚𝑘=1𝜕𝜕𝑥_𝑘𝒟_𝑖𝑘(𝑡,𝑥,𝑣)∇𝑣_𝑖(𝑡,𝑥)^𝑝−2𝜕𝑣_𝑖𝜕𝑥_𝑘−𝑎_𝑖𝑣_𝑖()×𝑏𝑡,𝑥_𝑖𝑣_𝑖()−𝑡,𝑥_𝑛𝑗=1𝑐_𝑖𝑗(𝑟(𝑡))𝑓_𝑗𝑣_𝑗()−𝑡,𝑥_𝑛𝑗=1𝑐̆_𝑖𝑗(𝑟(𝑡))𝑓_𝑗𝑣_𝑗()−𝑡,𝑥_𝑛𝑗=1𝑑_𝑖𝑗(𝑟(𝑡))𝑔_𝑗𝑣_𝑗()−𝑡−𝜏,𝑥_𝑛𝑗=1𝑑̆_𝑖𝑗(𝑟(𝑡))𝑔_𝑗𝑣_𝑗()+𝑡−𝜏,𝑥𝑑𝑡_𝑛𝑗=1𝜎_𝑖𝑗𝑣_𝑗(𝑡,𝑥),𝑣_𝑗(𝑡−𝜏,𝑥)𝑑𝑤_𝑗(𝑡),∀𝑡⩾𝑡₀𝑣()(),()∈[],𝑥∈Ω,𝜃,𝑥=𝜙𝜃,𝑥𝜃,𝑥−𝜏,0×Ω (6) with boundary condition𝔅𝑣_𝑖(𝑡,𝑥)=0,(𝑡,𝑥)∈[−𝜏,+∞)×𝜕Ω,𝑖=1,2,…,𝑛, (6a)where 𝑝>1 is a given scalar, Ω∈𝑅^𝑚 is a bounded domain with a smooth boundary 𝜕Ω of class 𝒞² by Ω, 𝑣(𝑡,𝑥)=(𝑣₁(𝑡,𝑥), 𝑣₂(𝑡,𝑥),…,𝑣_𝑛(𝑡,𝑥))^𝑇∈𝑅^𝑛, and 𝑣_𝑖(𝑡,𝑥) is the state variable of the 𝑖th neuron and the 𝑗th neuron at time 𝑡 and in space variable 𝑥. The smooth nonnegative functions 𝒟_𝑗𝑘(𝑡,𝑥,𝑣) are diffusion operators. Time delay 𝜏⩾0. 𝑎_𝑗(𝑣_𝑗(𝑡,𝑥)) represents an amplification function, and 𝑏_𝑗(𝑣_𝑗(𝑡,𝑥)) is an appropriately behavior function. 𝑓_𝑗(𝑣_𝑗(𝑡,𝑥)), 𝑔_𝑗(𝑣_𝑗(𝑡,𝑥)) are neuron activation functions of the 𝑗th unit at time 𝑡 and in space variable 𝑥. {𝑟(𝑡),𝑡⩾0} is a right-continuous Markov process on the probability space which takes values in the finite space 𝑆={1,2,…,𝑁} with generator Π={𝜋_𝑖𝑗} given by 𝒫𝑟()(𝑡)=𝜋𝑡+𝛿=𝑗∣𝑟=𝑖^𝑖𝑗𝛿+𝑜(𝛿),𝑗≠𝑖,1+𝜋_𝑖𝑗𝛿+𝑜(𝛿),𝑗=𝑖, (7) where 𝜋_𝑖𝑗⩾0 is transition probability rate from 𝑖 to 𝑗(𝑗≠𝑖) and 𝜋_𝑖𝑖=−∑^𝑠_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗, 𝛿>0 and lim_𝛿→0𝑜(𝛿)/𝛿=0. In mode 𝑟(𝑡)=𝑘, we denote 𝑐_𝑖𝑗(𝑟(𝑡))=𝑐^(𝑘)_𝑖𝑗, 𝑑_𝑖𝑗(𝑟(𝑡))=𝑑^(𝑘)_𝑖𝑗, 𝑐̆_𝑖𝑗(𝑟(𝑡))=𝑐̆^(𝑘)_𝑖𝑗 and 𝑑̆_𝑖𝑗(𝑟(𝑡))=𝑑̆^(𝑘)_𝑖𝑗, which imply the connection strengths of the ith neuron on the 𝑗th neuron, respectively.

The boundary condition (6a) is called Dirichlet boundary condition if 𝔅[𝑣_𝑖(𝑡,𝑥)]=𝑣_𝑖(𝑡,𝑥), and Neumann boundary condition if 𝔅[𝑣_𝑖(𝑡,𝑥)]=𝜕𝑣_𝑖(𝑡,𝑥)/𝜕𝜈, where 𝜕𝑣_𝑖(𝑡,𝑥)/𝜕𝜈=(𝜕𝑣_𝑖(𝑡,𝑥)/𝜕𝑥₁,𝜕𝑣_𝑖(𝑡,𝑥)/𝜕𝑥₂,…,𝜕𝑣_𝑖(𝑡,𝑥)/𝜕𝑥_𝑚)^𝑇 denotes the outward normal derivative on 𝜕Ω. It is well known that the stability of neural networks with Neumann boundary condition has been widely studied. The Dirichlet boundary conditions describe the situation where the space is totally surrounded by a region in which the states of the neuron equal zero on the boundary. And the stability analysis of delayed reaction-diffusion neural networks with the Dirichlet boundary conditions is very important in theories and applications, and also has attracted much attention [2, 3, 29, 58]. So in this paper, we consider the CGNNs under Neumann boundary condition and Dirichlet boundary condition, respectively.

If the complexity and the vagueness of CGNNs are ignored, the stochastic fuzzy system (6) is simplified to the following stochastic system: 𝑑𝑣(𝑡,𝑥)=∇⋅𝒟(𝑡,𝑥,𝑣)∘∇_𝑝𝑣((𝑣(𝐵(𝑣((𝑟(𝑡𝑓(𝑣((𝑟(𝑡𝑔(𝑣((()((𝑡),𝑡,𝑥)−𝐴𝑡,𝑥))[𝑡,𝑥))−𝐶))𝑡,𝑥))−𝐷))𝑡−𝜏,𝑥))𝑑𝑡+𝜎𝑡,𝑣𝑡,𝑥,𝑣𝑡−𝜏,𝑥))𝑑𝑤∀𝑡⩾𝑡₀𝑣()(),()∈[],𝑥∈Ω,𝜃,𝑥=𝜙𝜃,𝑥𝜃,𝑥−𝜏,0×Ω, (8) where matrices 𝐶_𝑟=(𝑐^(𝑟)_𝑖𝑗)_𝑛×𝑛, 𝐷_𝑟=(𝑑^(𝑟)_𝑖𝑗)_𝑛×𝑛. In 2012, Wang et al. [3] studied the stability of System (8) without Markovian-jumping.

Finally, we consider the global robust exponential stability for the following uncertain fuzzy CGNNs with 𝑝-Laplace diffusion: 𝑑𝑣_𝑖()=𝑡,𝑥_𝑚𝑘=1𝜕𝜕𝑥_𝑘𝒟_𝑖𝑘(𝑡,𝑥,𝑣)∇𝑣_𝑖(𝑡,𝑥)^𝑝−2𝜕𝑣_𝑖𝜕𝑥_𝑘−𝑎_𝑖𝑣_𝑖()×𝑏𝑡,𝑥_𝑖𝑣_𝑖()−𝑡,𝑥_𝑛𝑗=1𝑐_𝑖𝑗(𝑟(𝑡),𝑡)𝑓_𝑗𝑣_𝑗()−𝑡,𝑥_𝑛𝑗=1𝑐̆_𝑖𝑗(𝑟(𝑡),𝑡)𝑓_𝑗𝑣_𝑗()−𝑡,𝑥_𝑛𝑗=1𝑑_𝑖𝑗(𝑟(𝑡),𝑡)𝑔_𝑗𝑣_𝑗()−𝑡−𝜏,𝑥_𝑛𝑗=1𝑑̆_𝑖𝑗(𝑟(𝑡),𝑡)𝑔_𝑗𝑣_𝑗()+𝑡−𝜏,𝑥𝑑𝑡_𝑛𝑗=1𝜎_𝑖𝑗𝑣_𝑗(𝑡,𝑥),𝑣_𝑗(𝑡−𝜏,𝑥)𝑑𝑤_𝑗(𝑡),∀𝑡⩾𝑡₀𝑣()(),()∈[],𝑥∈Ω,𝜃,𝑥=𝜙𝜃,𝑥𝜃,𝑥−𝜏,0×Ω. (9)

For any mode 𝑟∈𝑆, we denote 𝑐_𝑖𝑗(𝑟(𝑡),𝑡), 𝑐̆_𝑖𝑗(𝑟(𝑡),𝑡), 𝑑_𝑖𝑗(𝑟(𝑡),𝑡), 𝑑̆_𝑖𝑗(𝑟(𝑡),𝑡) by 𝑐^(𝑟)_𝑖𝑗(𝑡), 𝑐̆^(𝑟)_𝑖𝑗(𝑡), 𝑑^(𝑟)_𝑖𝑗(𝑡), 𝑑̆^(𝑟)_𝑖𝑗(𝑡), and matrices 𝐶_𝑟(𝑡)=(𝑐^(𝑟)_𝑖𝑗(𝑡))_𝑛×𝑛, 𝐶̆_𝑟(𝑡)=(𝑐̆^(𝑟)_𝑖𝑗(𝑡))_𝑛×𝑛, 𝐷_𝑟(𝑡)=(𝑑^(𝑟)_𝑖𝑗(𝑡))_𝑛×𝑛, 𝐷̆_𝑟(𝑡)=(𝑑̆^(𝑟)_𝑖𝑗(𝑡))_𝑛×𝑛. Assume 𝐶_𝑟(𝑡)=𝐶_𝑟+Δ𝐶_𝑟(𝑡);𝐷_𝑟(𝑡)=𝐷_𝑟+Δ𝐷_𝑟(𝑡);𝐶̆_𝑟(𝑡)=𝐶̆_𝑟+Δ𝐶̆_𝑟(𝑡);𝐷̆_𝑟(𝑡)=𝐷̆_𝑟+Δ𝐷̆_𝑟(𝑡). (10) The Δ𝐶_𝑟(𝑡), Δ𝐷_𝑟(𝑡), Δ𝐶̆_𝑟(𝑡), and Δ𝐷̆_𝑟(𝑡) are parametric uncertainties, satisfying Δ𝐶_𝑟(𝑡)Δ𝐶̆_𝑟(𝑡)Δ𝐷_𝑟(𝑡)Δ𝐷̆_𝑟(𝑡)=𝐸_1𝑟𝐸_2𝑟𝓕(𝑡)𝓝_1𝑟𝓝_2𝑟, (11) where 𝓕(𝑡) is an unknown matrix with |𝓕^𝑇(𝑡)||𝓕(𝑡)|⩽𝐼, and 𝐸_1𝑟, 𝐸_2𝑟, 𝓝_1𝑟, 𝓝_2𝑟 are known real constant matrices for all 𝑟∈𝑆.

Throughout this paper, we denote matrices 𝐴(𝑣(𝑡,𝑥))=diag(𝑎₁(𝑣₁(𝑡,𝑥)), 𝑎₂(𝑣₂(𝑡,𝑥)),…,𝑎_𝑛(𝑣_𝑛(𝑡,𝑥))), 𝐵(𝑣(𝑡,𝑥))=(𝑏₁(𝑣₁(𝑡,𝑥)), 𝑏₂(𝑣₂(𝑡,𝑥)),…,𝑏_𝑛(𝑣_𝑛(𝑡,𝑥)))^𝑇, 𝑓(𝑣(𝑡,𝑥))=(𝑓₁(𝑣₁(𝑡,𝑥)), 𝑓₂(𝑣₂(𝑡,𝑥)),…,𝑓_𝑛(𝑣_𝑛(𝑡,𝑥)))^𝑇, 𝑔(𝑣(𝑡,𝑥))=(𝑔₁(𝑣₁(𝑡,𝑥)),…,𝑔_𝑛(𝑣_𝑛(𝑡,𝑥)))^𝑇. For the sake of simplicity, let 𝜎(𝑡)=𝜎(𝑡,𝑣(𝑡,𝑥),𝑣(𝑡−𝜏,𝑥)), and 𝑤(𝑡)=(𝑤₁(𝑡), 𝑤₂(𝑡),…,𝑤_𝑛(𝑡))^𝑇. Matrix 𝒟(𝑡,𝑥,𝑣)=(𝒟_𝑗𝑘(𝑡,𝑥,𝑣))_𝑛×𝑚 satisfies 𝒟_𝑗𝑘(𝑡,𝑥,𝑣)⩾0 for all 𝑗, 𝑘, (𝑡,𝑥,𝑣). Denote ∇_𝑝𝑣=(∇_𝑝𝑣₁,…,∇_𝑝𝑣_𝑛)^𝑇 with ∇_𝑝𝑣_𝑖=(|∇𝑣_𝑖|^𝑝−2(𝜕𝑣_𝑖/𝜕𝑥₁),…,|∇𝑣_𝑖|^𝑝−2(𝜕𝑣_𝑖/𝜕𝑥_𝑚))^𝑇. And 𝒟(𝑡,𝑥,𝑣)∘∇_𝑝𝑣=(𝒟_𝑗𝑘(𝑡,𝑥,𝑣)|∇𝑣_𝑖|^𝑝−2(𝜕𝑣_𝑖/𝜕𝑥_𝑘))_𝑛×𝑚 denotes the Hadamard product of matrix 𝒟(𝑡,𝑥,𝑣) and ∇_𝑝𝑣 (see, [60] or [3]).

For convenience’s sake, we need introduce some standard notations.(i)𝐿²(𝑅×Ω)∶ The space of real Lebesgue measurable functions of 𝑅×Ω, it is a Banach space for the 2-norm ‖𝑣(𝑡)‖₂=(∑^𝑛_𝑖=1‖𝑣_𝑖(𝑡)‖)^1/2 with ‖𝑣_𝑖(𝑡)‖=(∫_Ω|𝑣_𝑖(𝑡,𝑥)|²𝑑𝑥)^1/2, where |𝑣_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 matrix, that is, 𝑦^𝑇𝑄𝑦>0  (<0) for any 0≠𝑦∈𝑅^𝑛.(iv)𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛⩾0  (⩽0): A semi-positive (semi-negative) definite matrix, that is, 𝑦^𝑇𝑄𝑦⩾0  (⩽0) for any 𝑦∈𝑅^𝑛.(v)𝑄₁⩾𝑄₂(𝑄₁⩽𝑄₂): This means 𝑄₁−𝑄₂ is a semi-positive (semi-negative) definite matrix.(vi)𝑄₁>𝑄₂(𝑄₁<𝑄₂): This means 𝑄₁−𝑄₂ is a positive (negative) definite matrix.(vii)𝜆_max(Φ),𝜆_min(Φ) denotes the largest and smallest eigenvalue of matrix Φ, respectively.(viii)Denote |𝐶|=(|𝑐_𝑖𝑗|)_𝑛×𝑛 for any matrix 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛; |𝑢(𝑡,𝑥)|=(|𝑢₁(𝑡,𝑥)|,|𝑢₂(𝑡,𝑥)|,…,|𝑢_𝑛(𝑡,𝑥)|)^𝑇 for any 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥))^𝑇.(ix)𝐼: Identity matrix with compatible dimension.(x)The symmetric terms in a symmetric matrix are denoted by ∗.(xi)The Sobolev space 𝑊_1,𝑝(Ω)={𝑢∈𝐿^𝑝∶𝐷𝑢∈𝐿^𝑝} (see [61] for detail). Particularly in the case of 𝑝=2, then 𝑊^1,𝑝(Ω)=𝐻¹(Ω).(xii)Denote by 𝜆₁ the lowest positive eigenvalue of the boundary value problem ()()𝔅𝜑(−Δ𝜑𝑡,𝑥=𝜆𝜑𝑡,𝑥,𝑥∈Ω,𝑡,𝑥)=0,𝑥∈𝜕Ω. (12)

Let 𝑣(𝑡,𝑥;𝜙,𝑖₀) be the state trajectory from the initial condition 𝑟(0)=𝑖₀, 𝑣(𝜃,𝑥;𝜙)=𝜙(𝜃,𝑥) on −𝜏⩽𝜃⩽0 in 𝐿²_ℱ0([−𝜏,0]×Ω;𝑅^𝑛). Below, we always assume (𝑣(𝑡,𝑥;𝜙,𝑖₀) is a solution of System (6).

Definition 4. For any given scalar 𝑝>1, the null solution of system (6) is said to be stochastically globally exponentially stable in the mean square if for every initial condition 𝜙∈𝐿²_ℱ0([−𝜏,0]×Ω;𝑅^𝑛), 𝑟(0)=𝑖₀, there exist scalars 𝛽>0 and 𝛾>0 such that for any solution 𝑣(𝑡,𝑥;𝜙,𝑖₀), 𝔼𝑣𝑡;𝜙,𝑖₀²₂⩽𝛾𝑒^−𝛽𝑡sup_{−𝜏⩽𝜃⩽0}𝔼𝜙(𝜃)²₂,𝑡⩾𝑡₀. (13)

Definition 5. The null solution of System (6) is said to be almost sure exponentially stable if for every 𝜙∈𝐿²_ℱ0([−𝜏,0]×Ω;𝑅^𝑛), there exists a positive scalar 𝜆>0 such that the following inequality holds: limsup_𝑡→∞log‖𝑣(𝑡)‖²₂⩽−𝜆,𝒫−a.s. (14)

Lemma 6. Let 𝑃=diag(𝑝₁,𝑝₂,…,𝑝_𝑛) be a positive definite matrix, and 𝑣 be a solution of system (6) with the boundary condition (6a). Then one has _Ω𝑣^𝑇𝑃∇⋅𝒟(𝑡,𝑥,𝑣)∘∇_𝑝𝑣𝑑𝑥=−^𝑚𝑛_{𝑘=1𝑗=1Ω}𝑝_𝑗𝒟_𝑗𝑘()×𝑡,𝑥,𝑣∇𝑣_𝑗^𝑝−2𝜕𝑣_𝑗𝜕𝑥_𝑘²𝑑𝑥=_Ω∇⋅𝒟(𝑡,𝑥,𝑣)∘∇_𝑝𝑣^𝑇𝑃𝑣𝑑𝑥. (15)

Proof. Since 𝑣 is a solution of system (6), we can derive it by Guass formula and the boundary condition (6a) that _Ω𝑣^𝑇𝑃∇⋅𝐷(𝑡,𝑥,𝑣)∘∇_𝑝𝑣=𝑑𝑥_Ω𝑣^𝑇𝑃^𝑚_𝑘=1𝜕𝜕𝑥_𝑘𝐷_1𝑘∇𝑣₁^𝑝−2𝜕𝑣₁𝜕𝑥_𝑘,…,_𝑚𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑛𝑘∇𝑣_𝑛^𝑝−2𝜕𝑣_𝑛𝜕𝑥_𝑘^𝑇=𝑑𝑥_Ω𝑛𝑗=1𝑝_𝑗𝑣_𝑗^𝑚_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑗𝑘∇𝑣_𝑗^𝑝−2𝜕𝑣_𝑗𝜕𝑥_𝑘𝑑𝑥=−^𝑚𝑛_{𝑘=1𝑗=1Ω}𝑝_𝑗𝐷_𝑗𝑘∇𝑣_𝑗^𝑝−2𝜕𝑣_𝑗𝜕𝑥_𝑘²𝑑𝑥. (16)
Then the other three equalities can be proved similarly.

Remark 7. Lemma 9 actually generalizes the conclusion of [62, Lemma 3.1] from Hilbert space 𝐻¹(Ω) to Banach space 𝑊^1,𝑝(Ω).

Lemma 8 (nonnegative semi-martingale convergence theorem [63]). Let 𝐴(𝑡) and 𝑈(𝑡) be two continuous adapted increasing processes on 𝑡⩾0 with 𝐴(0)=𝑈(0)=0,𝑎.𝑠. Let 𝑀(𝑡) be a real-valued continuous local martingale with 𝑀(0)=0,𝑎.𝑠. Let 𝜉 be a nonnegative ℱ₀-measurable random variable with 𝐸𝜉<∞. Define 𝑋(𝑡)=𝜉+𝐴(𝑡)−𝑈(𝑡)+𝑀(𝑡) (17)
for 𝑡⩾0. If 𝑋(𝑡) is nonnegative, then lim_𝑡→∞𝐴(𝑡)<∞⊂lim_𝑡→∞𝑋(𝑡)∩<∞lim_𝑡→∞𝑈(𝑡)<∞,a.s., (18)
where 𝐵⊂𝐷𝑎.𝑠. means 𝑃(𝐵∪𝐷^𝑐)=0. In particular, if lim_𝑡→∞𝐴(𝑡)<∞𝑎.𝑠., then for almost all 𝜔∈𝛀, lim_𝑡→∞𝑋(𝑡)<∞ and lim_𝑡→∞𝑈(𝑡)<∞, that is, both 𝑋(𝑡) and 𝑈(𝑡) converge to finite random variables.

Lemma 9 (see [64]). 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, (19)
where [𝑥,𝑦] denotes the segment connecting 𝑥 and 𝑦.

Lemma 10 (see [65]). Let 𝜀>0 be any given scalar, and ℳ,𝔈 and 𝒦 be matrices with appropriate dimensions. If 𝒦^𝑇𝒦⩽𝐼, then one has ℳ𝒦𝔈+𝔈^𝑇𝒦^𝑇ℳ^𝑇⩽𝜀⁻¹ℳℳ^𝑇+𝜀𝔈^𝑇𝔈. (20)

3. Main Results

Theorem 11. Assume that 𝑝>1. In addition, there exist a sequence of positive scalars 𝛼_𝑟(𝑟∈𝑆) and positive definite diagonal matrices 𝑃_𝑟=diag(𝑝_𝑟1,𝑝_𝑟2,…,𝑝_𝑟𝑛)(𝑟∈𝑆) and 𝑄=diag(𝑞₁,𝑞₂,…,𝑞_𝑛) such that the following LMI conditions hold: Θ_𝑟𝒜≜−^𝑟𝑃_𝑟𝐴𝐷_𝑟+𝐷̆_𝑟𝐺∗−𝑒^−𝜆𝜏𝑄+𝛼_𝑟𝓥>0,𝑟∈𝑆,𝑃_𝑟<𝛼_𝑟𝐼,𝑟∈𝑆, (21)(22)
where matrices 𝐶_𝑟=(𝑐^(𝑟)_𝑖𝑗)_𝑛×𝑛,𝐷_𝑟=(𝑑^(𝑟)_𝑖𝑗)_𝑛×𝑛, 𝐶̆_𝑟=(𝑐̆^(𝑟)_𝑖𝑗)_𝑛×𝑛, 𝐷̆_𝑟=(𝑑̆^(𝑟)_𝑖𝑗)_𝑛×𝑛, and 𝒜_𝑟=𝜆𝑃_𝑟−2𝑃_𝑟𝐵+𝑃_𝑟𝐴𝐶_𝑟+𝐶̆_𝑟𝐹+𝐹𝐶^𝑇_𝑟+𝐶̆^𝑇_𝑟𝐴𝑃_𝑟+𝛼_𝑟𝓤+𝑄+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗, (23)
then the null solution of Markovian jumping stochastic fuzzy system (6) is stochastically exponentially stable in the mean square.

Proof. Consider the Lyapunov-Krasovskii functional: 𝑉(𝑡,𝑣(𝑡),𝑟)=𝑒^𝜆𝑡_Ω^𝑛_𝑖=1𝑝_𝑟𝑖𝑣²_𝑖()+𝑡,𝑥𝑑𝑥_{Ω𝑡𝑡−𝜏}𝑒^𝑛𝜆𝑠_𝑖=1𝑞_𝑖𝑣²_𝑖(𝑠,𝑥)𝑑𝑠𝑑𝑥,∀𝑟∈𝑆, (24)
where 𝑣(𝑡,𝑥)=(𝑣₁(𝑡,𝑥),𝑣₂(𝑡,𝑥),…,𝑣_𝑛(𝑡,𝑥))^𝑇 is a solution for stochastic fuzzy system (6). Sometimes we may denote 𝑣(𝑡,𝑥) by 𝑣, 𝑣_𝑖(𝑡,𝑥) by 𝑣_𝑖, and 𝜎(𝑣(𝑡,𝑥), 𝑣(𝑡−𝜏,𝑥)) by 𝜎(𝑡) for simplicity.
Let 𝓛 be the weak infinitesimal operator. Then it follows by Lemma 6 that 𝓛𝑉(𝑡,𝑣(𝑡),𝑟)=𝜆𝑒^𝜆𝑡_Ω𝑣^𝑇𝑃_𝑟𝑣𝑑𝑥−2𝑒^𝜆𝑡×^𝑚𝑛_{𝑘=1𝑖=1Ω}𝑝_𝑟𝑖𝒟_𝑖𝑘()×𝑡,𝑥,𝑣∇𝑣_𝑖^𝑝−2𝜕𝑣_𝑖𝜕𝑥_𝑘²𝑑𝑥−2𝑒^𝑛𝜆𝑡_𝑖=1Ω𝑝_𝑟𝑖𝑣_𝑖×𝑎_𝑖𝑣_𝑖𝑏_𝑖𝑣_𝑖−^𝑛_𝑗=1𝑐^(𝑟)_𝑖𝑗𝑓_𝑗𝑣_𝑗−^𝑛_𝑗=1𝑐̆^(𝑟)_𝑖𝑗𝑓_𝑗𝑣_𝑗−^𝑛_𝑗=1𝑑^(𝑟)_𝑖𝑗𝑔_𝑗𝑣_𝑗()−𝑡−𝜏,𝑥_𝑛𝑗=1𝑑̆^(𝑟)_𝑖𝑗×𝑔_𝑗𝑣_𝑗(𝑡−𝜏,𝑥)𝑑𝑥+𝑒^𝜆𝑡_Ω𝑣^𝑇_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗𝑣𝑑𝑥+𝑒^𝜆𝑡_Ωtrace𝜎^𝑇(𝑡)𝑃_𝑟𝜎(𝑡)𝑑𝑥+_Ω𝑒^𝜆𝑡𝑣^𝑇𝑄𝑣−𝑒^{𝜆(𝑡−𝜏)}×𝑣^𝑇(𝑡−𝜏,𝑥)𝑄𝑣(𝑡−𝜏,𝑥)𝑑𝑥. (25)
Moreover, we get by A4 and A5 𝓛𝑉(𝑡,𝑣(𝑡),𝑟)⩽𝑒^𝜆𝑡_Ω𝑣^𝑇𝜆𝑃_𝑟+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗𝑣𝑑𝑥+0−2^𝑛_𝑖=1Ω𝑝_𝑟𝑖𝑏_𝑖𝑣²_𝑖𝑑𝑥+2^𝑛_𝑖=1Ω𝑝_𝑟𝑖𝑣_𝑖𝑎_𝑖^𝑛_𝑗=1𝑐^(𝑟)_𝑖𝑗×𝑓_𝑗𝑣_𝑗−𝑓_𝑗(0)+𝑝_𝑟𝑖𝑣_𝑖𝑎_𝑖^𝑛_𝑗=1𝑐̆^(𝑟)_𝑖𝑗×𝑓_𝑗𝑣_𝑗−𝑓_𝑗(0)+𝑝_𝑟𝑖𝑣_𝑖𝑎_𝑖^𝑛_𝑗=1𝑑^(𝑟)_𝑖𝑗×𝑔_𝑗𝑣_𝑗(𝑡−𝜏,𝑥)−𝑔_𝑗(0)+𝑝_𝑟𝑖𝑣_𝑖𝑎_𝑖^𝑛_𝑗=1𝑑̆^(𝑟)_𝑖𝑗×𝑔_𝑗𝑣_𝑗(𝑡−𝜏,𝑥)−𝑔_𝑗(0+)𝑑𝑥𝛼_𝑟Ω𝑣^𝑇𝓤𝑣+𝑣^𝑇()()+𝑡−𝜏,𝑥×𝓥𝑣𝑡−𝜏,𝑥𝑑𝑥_Ω𝑒^𝜆𝑡𝑣^𝑇𝑄𝑣−𝑒^{𝜆(𝑡−𝜏)}×𝑣^𝑇(𝑡−𝜏,𝑥)𝑄𝑣(𝑡−𝜏,𝑥)𝑑𝑥. (26)
From A3 and Lemma 9, we know 𝑓(𝑣((0=|𝔉|⋅|𝑣()||𝑣(;𝑔(𝑣((0=|𝔊|⋅|𝑣()||𝑣(,𝑡,𝑥))−𝑓)𝑡,𝑥−0⩽𝐹𝑡,𝑥)|𝑡−𝜏,𝑥))−𝑔)𝑡−𝜏,𝑥−0⩽𝐺𝑡−𝜏,𝑥)| (27)
where 𝔉∈∪_{𝑧∈[0,𝑣(𝑡,𝑥)]}𝜕𝑓(𝑧), and 𝔊∈∪_{𝑧∈[0,𝑣(𝑡−𝜏,𝑥)]}𝜕𝑓(𝑧).
So it follows by A1–A5 that 𝓛𝑉(𝑡,𝑣(𝑡),𝑟)⩽𝑒^𝜆𝑡_Ω𝑣^𝑇𝜆𝑃_𝑟+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗𝑣𝑑𝑥−2^𝑛_𝑖=1Ω𝑝_𝑟𝑖𝑏_𝑖𝑣²_𝑖𝑑𝑥+2^𝑛_𝑖=1Ω𝑝_𝑟𝑖𝑣_𝑖𝑎_𝑖^𝑛_𝑗=1𝑐^(𝑟)_𝑖𝑗𝐹_𝑗𝑣_𝑗+𝑝_𝑟𝑖𝑣_𝑖𝑎_𝑖^𝑛_𝑗=1𝑐̆^(𝑟)_𝑖𝑗𝐹_𝑗𝑣_𝑗+𝑝_𝑟𝑖𝑣_𝑖𝑎_𝑖^𝑛_𝑗=1𝑑^(𝑟)_𝑖𝑗×𝐺_𝑗𝑣_𝑗(𝑡−𝜏,𝑥)+𝑝_𝑟𝑖𝑣_𝑖𝑎_𝑖^𝑛_𝑗=1𝑑̆^(𝑟)_𝑖𝑗×𝐺_𝑗𝑣_𝑗(+𝑡−𝜏,𝑥)𝑑𝑥𝛼_𝑟Ω𝑣^𝑇𝓤𝑣+𝑣^𝑇()()+𝑡−𝜏,𝑥×𝓥𝑣𝑡−𝜏,𝑥𝑑𝑥_Ω𝑒^𝜆𝑡𝑣^𝑇𝑄𝑣−𝑒^{𝜆(𝑡−𝜏)}×𝑣^𝑇(𝑡−𝜏,𝑥)𝑄𝑣(𝑡−𝜏,𝑥)𝑑𝑥, (28)
or 𝓛𝑉(𝑡,𝑣(𝑡),𝑟)⩽𝑒^𝜆𝑡𝜆_Ω𝑣^𝑇𝜆𝑃_𝑟+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗|𝑣|𝑑𝑥−2_Ω𝑣^𝑇𝑃_𝑟𝐵|𝑣|𝑑𝑥+2_Ω𝑣^𝑇𝑃_𝑟𝐴𝐶_𝑟+𝐶̆_𝑟𝐹|𝑣|+𝑣^𝑇𝑃_𝑟𝐴𝐷_𝑟+𝐷̆_𝑟|𝑣(+×𝐺𝑡−𝜏,𝑥)|𝑑𝑥_Ω𝛼_𝑟𝑣^𝑇𝓤|𝑣|+𝑣^𝑇(|𝑣(+𝑡−𝜏,𝑥)×𝓥𝑡−𝜏,𝑥)|𝑑𝑥_Ω𝑣^𝑇𝑄|𝑣|−𝑒^−𝜆𝜏×𝑣^𝑇(𝑡−𝜏,𝑥)𝑄|𝑣(𝑡−𝜏,𝑥)|𝑑𝑥. (29)

Remark 12. In (28), we employ a new method, which is different from that of [4, (3)]. Therefore, our LMI conditions in Theorem 11 may be more feasible and effective than [4, Theorem 1] to some extent, which may be illustrated by a numerical example below (see, Example 30).
Denote 𝜁^𝑇(𝑡,𝑥)=(|𝑣^𝑇(𝑡,𝑥)|,|𝑣^𝑇(𝑡−𝜏,𝑥)|). Then we get by (21) 𝓛𝑉(𝑡,𝑣(𝑡),𝑟)⩽−_Ω𝜁^𝑇(𝑡,𝑥)Θ_𝑟𝜁(𝑡,𝑥)𝑑𝑥⩽0,𝑟∈𝑆. (30)
Then we can obtain by the Dynkin formula 𝔼𝑉(𝑡,𝑣(𝑡),𝑟)−𝔼𝑉(0,𝑣(0),𝑟)=𝔼^𝑡₀𝓛𝑉(𝑠,𝑣(𝑠),𝑟)𝑑𝑠⩽0,𝑟∈𝑆. (31)
Hence, we have 𝔼_Ω𝑒^𝑛𝜆𝑡_𝑖=1𝑝_𝑟𝑖𝑣²_𝑖()((𝑡))((0))𝑡,𝑥𝑑𝑥⩽𝔼𝑉𝑡,𝑣,𝑟⩽𝔼𝑉0,𝑣,𝑟=𝔼_Ω^𝑛_𝑖=1𝑝_𝑟𝑖𝑣²_𝑖(0,𝑥)𝑑𝑥+𝔼_Ω⁰_−𝜏𝑒^𝑛𝜆𝑠_𝑖=1𝑞_𝑖𝑣²_𝑖()⩽𝑠,𝑥𝑑𝑠𝑑𝑥max_𝑟∈𝑆max_{1⩽𝑗⩽𝑛}𝑝_𝑟𝑗+𝜏max_{1⩽𝑗⩽𝑛}𝑞_𝑗sup_{−𝜏⩽𝑠⩽0}𝔼𝜙(𝑠)²₂. (32)
On the other hand, 𝔼_Ω𝑒^𝑛𝜆𝑡_𝑖=1𝑝_𝑟𝑖𝑣²_𝑖(𝑡,𝑥)𝑑𝑥⩾𝑒^𝜆𝑡min_𝑟∈𝑆min_{1⩽𝑗⩽𝑛}𝑝_𝑟𝑗𝔼‖𝑣(𝑡)‖²₂. (33)
Combining the above two inequalities, we obtain 𝔼‖𝑣(𝑡)‖²₂≤max_𝑟∈𝑆max_{1⩽𝑗⩽𝑛}𝑝_𝑟𝑗+𝜏max_{1⩽𝑗⩽𝑛}𝑞_𝑗min_𝑟∈𝑆min_{1⩽𝑗⩽𝑛}𝑝_𝑟𝑗𝑒^−𝜆𝑡×sup_{−𝜏⩽𝑠⩽0}𝔼𝜙(𝑠)²₂. (34)
Therefore, we can see it by Definition 4 that the null solution of stochastic fuzzy system (6) is globally stochastically exponentially stable in the mean square.

Corollary 13. If there exist a positive scalar 𝛼 and positive definite diagonal matrices 𝑃 and 𝑄 such that the following LMI conditions hold: Θ≜−𝒜𝑃𝐴𝐷+𝐷̆𝐺∗−𝑒^−𝜆𝜏𝑄+𝛼𝓥>0,𝑃<𝛼𝐼, (35)
where matrices 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛,𝐷=(𝑑_𝑖𝑗)_𝑛×𝑛,𝐶̆=(𝑐̆_𝑖𝑗)_𝑛×𝑛,𝐷̆=(𝑑̆_𝑖𝑗)_𝑛×𝑛, and 𝒜=𝜆𝑃−2𝑃𝐵+𝑃𝐴𝐶+𝐶̆𝐹+𝐹𝐶^𝑇+𝐶̆^𝑇𝐴𝑃+𝛼𝓤+𝑄, (36)
then the null solution of stochastic fuzzy system (1) is stochastically exponentially stable in the mean square.

Remark 14. It is obvious from Remark 12 that our Corollary 13 is more feasible and effective than [4, Theorem 1]. In addition, the LMI-based criterion of Corollary 13 has its practical value in real work, for it is available to computer matlab calculation.

Corollary 15. Assume that 𝑝>1. In addition, there exist a sequence of positive scalars 𝛼_𝑟(𝑟∈𝑆) and positive definite diagonal matrices 𝑃_𝑟(𝑟∈𝑆) and 𝑄 such that the following LMI conditions hold: Θ_𝑟ℬ≜−^𝑟𝑃_𝑟𝐴𝐷_𝑟𝐺∗−𝑒^−𝜆𝜏𝑄+𝛼_𝑟𝓥𝑃>0,𝑟∈𝑆,_𝑟<𝛼_𝑟𝐼,𝑟∈𝑆, (37)
where ℬ_𝑟=𝜆𝑃_𝑟−2𝑃_𝑟𝐵+𝑃_𝑟𝐴𝐶_𝑟𝐹+𝐹𝐶^𝑇_𝑟𝐴𝑃_𝑟+𝛼_𝑟𝓤+𝑄+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗, (38) then the null solution of Markovian jumping stochastic system (8) is stochastically exponentially stable in the mean square.
Particularly for the case of 𝑝=2, we get from the Poincaré inequality (see, e.g., [58, Lemma 2.4]) that 𝜆_1Ω𝑣_𝑖(𝑡,𝑥)²𝑑𝑥⩽_Ω∇𝑣_𝑖(𝑡,𝑥)²𝑑𝑥. (39)
Denote 𝒟=min_𝑗,𝑘(inf_{𝑡,𝑥,𝑣}𝒟_𝑗𝑘(𝑡,𝑥,𝑣)). Then Lemma 6 derives that _Ω𝑣^𝑇𝑃_𝑟∇⋅𝒟(𝑡,𝑥,𝑣)∘∇_𝑝𝑣𝑑𝑥=−^𝑚𝑛_{𝑘=1𝑗=1Ω}𝑝_𝑟𝑗𝒟_𝑗𝑘(𝑡,𝑥,𝑣)∇𝑣_𝑗²⁻²𝜕𝑣_𝑗𝜕𝑥_𝑘²𝑑𝑥⩽−𝜆₁𝒟𝛼_𝑟‖𝑣‖²₂, (40)
where 𝑃_𝑟=diag(𝑝_𝑟1,𝑝_𝑟2,…,𝑝_𝑟𝑛)>0, and 𝛼_𝑟 is a positive scalar, satisfying 𝛼_𝑟𝐼<𝑃_𝑟,∀𝑟∈𝑆. (41) Moreover, one can conclude the following Corollary from (40) and the proof of Theorem 11.

Corollary 16. Assume that 𝑝=2. In addition, there exist a sequence of positive scalars 𝛼_𝑟,𝛼_𝑟(𝑟∈𝑆) and positive definite diagonal matrices 𝑃_𝑟(𝑟∈𝑆) and 𝑄 such that the following LMI conditions hold: Θ_𝑟ℬ≜−^𝑟−2𝜆₁𝒟𝛼_𝑟𝐼𝑃_𝑟𝐴𝐷_𝑟𝐺∗−𝑒^−𝜆𝜏𝑄+𝛼_𝑟𝓥𝑃>0,𝑟∈𝑆,_𝑟<𝛼_𝑟𝛼𝐼,𝑟∈𝑆,_𝑟𝐼<𝑃_𝑟,∀𝑟∈𝑆, (42)
where ℬ_𝑟 satisfies (38), then the null solution of Markovian jumping stochastic system (8) with 𝑝=2 is stochastically exponentially stable in the mean square.

Remark 17. Corollary 16 not only extends [58, Theorem  3.2] into the case of Markovian jumping, but also improves its complicated conditions by presenting the efficient LMI-based criterion.

Below, we denote 𝜈=max_𝑖,𝑗𝜈_𝑖𝑗 for convenience's sake.

Theorem 18. Assume 𝑝>1. The null solution of Markovian jumping stochastic fuzzy system (6) is almost sure exponentially stable if there exist positive scalars 𝜆, 𝛼_𝑟(𝑟∈𝑆), 𝛽 and positive definite matrices 𝑃_𝑟=diag(𝑝_𝑟1,𝑝_𝑟2,…,𝑝_𝑟𝑛),(𝑟∈𝑆) such that Θ_𝑟𝒜≜−^𝑟𝑃_𝑟𝐴𝐷_𝑟+𝐷̆_𝑟𝐺∗−𝛼_𝑟𝑃𝛽𝓥>0,𝑟∈𝑆,_𝑟<𝛼_𝑟𝐼,𝑟∈𝑆, (43)
where 𝒜_𝑟=𝜆𝑃_𝑟−2𝑃_𝑟𝐵+𝑃_𝑟𝐶_𝑟+𝐶̆_𝑟𝐹+𝐹𝐶^𝑇_𝑟+𝐶̆^𝑇_𝑟𝐴𝑃_𝑟+𝛼_𝑟𝓤+𝑛𝜈𝑒^𝜆𝜏𝛼_𝑟1+𝛽𝐼+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗. (44)

Proof. Consider the Lyapunov-Krasovskii functional: 𝒱(𝑡,𝑣(𝑡),𝑟)=𝑒^𝜆𝑡_Ω^𝑛_𝑖=1𝑝_𝑟𝑖𝑣²_𝑖(𝑡,𝑥)𝑑𝑥,𝑟∈𝑆. (45)
By applying Itô formula (see, e.g., [3, (2.7)]) and Lemma 6, we can get 𝒱((𝑡))((0))=𝑡,𝑣,𝑟−𝒱0,𝑣,𝑟_𝑡0𝜆𝑒^𝜆𝑠_Ω𝑣^𝑇(𝑠,𝑥)𝑃_𝑟𝑣(𝑠,𝑥)𝑑𝑥𝑑𝑠−2^𝑡₀𝑒^{𝑚𝜆𝑠𝑛}_{𝑘=1𝑖=1Ω}𝑝_𝑖𝒟_𝑖𝑘()×𝑠,𝑥,𝑣∇𝑣_𝑖(𝑠,𝑥)^𝑝−2×𝜕𝑣_𝑖(𝑠,𝑥)𝜕𝑥_𝑘²𝑑𝑥𝑑𝑠−2^𝑡₀𝑒^𝑛𝜆𝑠_𝑖=1Ω𝑝_𝑟𝑖𝑣_𝑖()×𝑎𝑠,𝑥_𝑖𝑣_𝑖()×𝑏𝑠,𝑥_𝑖𝑣_𝑖()−𝑠,𝑥_𝑛𝑗=1𝑐_𝑖𝑗𝑓_𝑗𝑣_𝑗()−𝑠,𝑥_𝑛𝑗=1𝑐̆_𝑖𝑗𝑓_𝑗𝑣_𝑗(−𝑠,𝑥)_𝑛𝑗=1𝑑_𝑖𝑗𝑔_𝑗𝑣_𝑗(−𝑠−𝜏,𝑥)_𝑛𝑗=1𝑑̆_𝑖𝑗×𝑔_𝑗𝑣_𝑗()+𝑠−𝜏,𝑥𝑑𝑥𝑑𝑠_𝑡0𝑒^𝜆𝑠_Ω𝑣^𝑇(𝑠,𝑥)_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗𝑣(𝑠,𝑥)𝑑𝑥𝑑𝑠+2^𝑡₀𝑒^𝑛𝜆𝑠_𝑖=1Ω𝑝_𝑟𝑖𝑣_𝑖(𝑠,𝑥)^𝑛_𝑗=1𝜎_𝑖𝑗(𝑠)𝑑𝑤_𝑗(𝑠)+𝑑𝑥_𝑡0𝑒^𝜆𝑠_Ωtrace𝜎^𝑇(𝑠)𝑃_𝑟𝜎(𝑠)𝑑𝑥𝑑𝑠, (46)
where 𝜎_𝑖𝑗(𝑠)=𝜎_𝑖𝑗(𝑣_𝑗(𝑠,𝑥), 𝑣_𝑗(𝑠−𝜏,𝑥)), and 𝜎(𝑠)=(𝜎_𝑖𝑗(𝑠))_𝑛×𝑛.
Similarly as (26)–(29), we can derive by A1–A5 and Lemma 9𝒱((𝑡))⩽𝑡,𝑣,𝑟_Ω𝑛𝑖=1𝑝_𝑟𝑖𝑣²_𝑖(0,𝑥)𝑑𝑥+2^𝑡₀𝑒^𝜆𝑠_Ω𝑣^𝑇(×𝑠,𝑥)𝜆𝑃_𝑟+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗|𝑣(−𝑣𝑠,𝑥)|_𝑇(𝑠,𝑥)𝑃_𝑟𝐵|𝑣(+𝑣𝑠,𝑥)|_𝑇(𝑠,𝑥)𝑃_𝑟𝐴𝐶_𝑟+𝐶̆_𝑟|𝑣(+𝑣×𝐹𝑠,𝑥)|_𝑇(𝑠,𝑥)𝑃_𝑟𝐴𝐷_𝑟+𝐷̆_𝑟|𝑣(+1×𝐺𝑠−𝜏,𝑥)|2𝑣^𝑇(𝑠,𝑥)𝛼_𝑟𝓤|𝑣(+𝑠,𝑥)|𝑑𝑥𝑑𝑠_𝑡0𝑒^𝜆𝑠_Ω𝑣^𝑇(𝑠−𝜏,𝑥)𝛼_𝑟𝓥𝑣(𝑠−𝜏,𝑥)𝑑𝑥𝑑𝑠+2^𝑡₀𝑒^𝑛𝜆𝑠_𝑖=1Ω𝑝_𝑟𝑖𝑣_𝑖(𝑠,𝑥)^𝑛_𝑗=1𝜎_𝑖𝑗(𝑠)𝑑𝑤_𝑗(𝑠)𝑑𝑥. (47)
On the other hand, ^𝑡₀𝑒^𝜆𝑠_Ω𝑣^𝑇(𝑠−𝜏,𝑥)𝛼_𝑟()=𝓥𝑣𝑠−𝜏,𝑥𝑑𝑥𝑑𝑠𝛼_𝑟^𝑡₀𝑒^𝜆𝑠_Ω^𝑛𝑛_𝑖=1𝑗=1𝜈_𝑖𝑗𝑣_𝑖(𝑠−𝜏,𝑥)𝑣_𝑗()⩽1𝑠−𝜏,𝑥𝑑𝑥𝑑𝑠2𝛼_𝑟^𝑡₀𝑒^𝜆𝑠_Ω^𝑛𝑛_𝑖=1𝑗=1𝜈_𝑖𝑗×𝑣²_𝑖(𝑠−𝜏,𝑥)+𝑣²_𝑗(𝑠−𝜏,𝑥)𝑑𝑥𝑑𝑠⩽𝑛𝜈𝛼_𝑟^𝑡₀𝑒^𝜆𝑠_Ω𝑣^𝑇(𝑠−𝜏,𝑥)𝑣(𝑠−𝜏,𝑥)𝑑𝑥𝑑𝑠=𝑛𝜈𝛼_𝑟^𝑡−𝜏_−𝜏𝑒^{𝜆(𝜃+𝜏)}_Ω𝑣^𝑇(𝜃,𝑥)𝑣(𝜃,𝑥)𝑑𝑥𝑑𝜃=𝑛𝜈𝛼_𝑟^𝑡_−𝜏𝑒^{𝜆(𝜃+𝜏)}_Ω𝑣^𝑇()𝑣()−𝜃,𝑥𝜃,𝑥𝑑𝑥𝑑𝜃_{𝑡𝑡−𝜏}𝑒^{𝜆(𝜃+𝜏)}_Ω𝑣^𝑇(𝜃,𝑥)𝑣(𝜃,𝑥)𝑑𝑥𝑑𝜃⩽𝑛𝜈𝛼_𝑟^𝑡_−𝜏𝑒^{𝜆(𝑠+𝜏)}_Ω𝑣^𝑇(𝑠,𝑥)|𝑣(𝑠,𝑥)|𝑑𝑥𝑑𝑠. (48)
Denote 𝑣_𝜏=𝑣(𝑠−𝜏,𝑥) for convenience’s sake. Then we have ^𝑡₀𝑒^𝜆𝑠_Ω𝑣^𝑇(𝑠−𝜏,𝑥)𝛼_𝑟()=𝓥𝑣𝑠−𝜏,𝑥𝑑𝑥𝑑𝑠−𝛽+1+𝛽^𝑡₀𝑒^𝜆𝑠_Ω𝑣^𝑇_𝜏𝛼_𝑟𝓥𝑣_𝜏𝑑𝑥𝑑𝑠⩽−𝛽^𝑡₀𝑒^𝜆𝑠_Ω𝑣^𝑇_𝜏𝛼_𝑟𝓥𝑣_𝜏𝑑𝑥+𝑛𝜈𝛼_𝑟×1+𝛽_𝑡−𝜏𝑒^{𝜆(𝑠+𝜏)}_Ω𝑣^𝑇(𝑠,𝑥)|𝑣(𝑠,𝑥)|𝑑𝑥𝑑𝑠. (49)
Combining (47) and (49) results in 𝒱((𝑡))⩽𝑡,𝑣,𝑟_Ω𝑛𝑖=1𝑝_𝑟𝑖𝑣²_𝑖()+0,𝑥𝑑𝑥_𝑡0𝑒^𝜆𝑠_Ω𝑣^𝑇(×𝑠,𝑥)𝜆𝑃_𝑟+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗−2𝑃_𝑟𝐵+2𝑃_𝑟𝐴𝐶+𝐶̆𝐹+𝛼𝓤+𝑛𝜈𝑒^𝜆𝜏1+𝛽𝛼_𝑟𝐼|𝑣(𝑣𝑠,𝑥)|+2^𝑇(𝑠,𝑥)𝑃_𝑟𝐴𝐷+𝐷̆×𝐺|𝑣(𝑠−𝜏,𝑥)|𝑑𝑥𝑑𝑠−𝛽^𝑡₀𝑒^𝜆𝑠_Ω𝑣^𝑇(𝑠−𝜏,𝑥)𝛼_𝑟𝓥𝑣(𝑠−𝜏,𝑥)𝑑𝑥𝑑𝑠+𝑛𝜈1+𝛽𝛼_𝑟×⁰_−𝜏𝑒^{𝜆(𝑠+𝜏)}_Ω𝑣^𝑇(𝑠,𝑥)|𝑣(𝑠,𝑥)|𝑑𝑥𝑑𝑠+2^𝑡₀𝑒^𝑛𝜆𝑠_𝑖=1Ω𝑝_𝑟𝑖𝑣_𝑖(𝑠,𝑥)^𝑛_𝑗=1𝜎_𝑖𝑗(𝑠)𝑑𝑤_𝑗(𝑠)=𝑑𝑥_Ω𝑛𝑖=1𝑝_𝑟𝑖𝑣²_𝑖()−0,𝑥𝑑𝑥_𝑡0𝑒^𝜆𝑠_Ω𝜁^𝑇(𝑠,𝑥)Θ_𝑟𝜁(𝑠,𝑥)𝑑𝑥𝑑𝑠+𝑛𝜈1+𝛽𝛼_𝑟⁰_−𝜏𝑒^{𝜆(𝑠+𝜏)}_Ω𝑣^𝑇(𝑠,𝑥)|𝑣(𝑠,𝑥)|𝑑𝑥𝑑𝑠+2^𝑡₀𝑒^𝑛𝜆𝑠_𝑖=1Ω𝑝_𝑟𝑖𝑣_𝑖(𝑠,𝑥)^𝑛_𝑗=1𝜎_𝑖𝑗(𝑠)𝑑𝑤_𝑗(𝑠)𝑑𝑥 (50)
which together with (43) implies 𝒱(𝑡,𝑣(𝑡),𝑟)⩽max_𝑟∈𝑆𝛼_𝑟Ω^𝑛_𝑖=1𝑣²_𝑖(0,𝑥)𝑑𝑥+𝑛𝜈𝑒^𝜆𝜏1+𝛽max_𝑟∈𝑆𝛼_𝑟×⁰_−𝜏𝑒^𝜆𝑠_Ω𝑣^𝑇(𝑠,𝑥)𝑣(𝑠,𝑥)𝑑𝑥𝑑𝑠+2max_𝑟∈𝑆𝛼_𝑟^𝑡₀𝑒^𝑛𝜆𝑠_𝑖=1Ω𝑣_𝑖(𝑠,𝑥)^𝑛_𝑗=1𝜎_𝑖𝑗(𝑠)𝑑𝑤_𝑗(𝑠)𝑑𝑥. (51)

Remark 19. The methods employed in (38)–(50) are different from ones in the proof of [4, Theorem 3] so that our efficient LMI criterion can be constructed. In large numerical calculations, LMI-based criterion in Theorem 18 is more effective than the complicated condition (8) in [4, Theorem 3]. To some extent, Theorem 18 is more effective than [58, Theorem 3.1] to some extent if fuzzy system (6) is simplified to system (8) without Markovian jumping (see, e.g., Example 38).

It is obvious that the right-hand side of (51) is a nonnegative semi-martingale. And hence the semi-martingale convergence theorem derives limsup_𝑡→∞𝒱(𝑡,𝑣(𝑡),𝑟)<∞,ℙ−a.s. (52)

Note that 𝑒^𝜆𝑡min_𝑟∈𝑆min_𝑖𝑝_𝑟𝑖Ω𝑣^𝑇𝑣𝑑𝑥⩽𝑒^𝜆𝑡_Ω^𝑛_𝑖=1𝑝_𝑟𝑖𝑣²_𝑖()((𝑡)).𝑡,𝑥𝑑𝑥=𝒱𝑡,𝑣,𝑟 (53)

Then we can conclude from (52) limsup_𝑡→∞1𝑡log‖𝑣‖²₂⩽−𝜆,ℙ−𝑎.𝑠. (54)

So we can see it from Definition 5 that the null solution of stochastic fuzzy system (6) is almost sure exponentially stable.

Corollary 20. The null solution of stochastic fuzzy system (1) is almost sure exponentially stable if there exist positive scalars 𝜆, 𝛼, 𝛽 and positive definite diagonal matrices 𝑃 such that Θ≜−𝒜𝑃𝐴𝐷+𝐷̆𝐺∗−𝛼𝛽𝓥>0,𝑃<𝛼𝐼, (55)
where 𝒜=𝜆𝑃−2𝑃𝐵+𝑃𝐴𝐶+𝐶̆𝐹+𝐹𝐶^𝑇+𝐶̆^𝑇𝐴𝑃+𝛼𝓤+𝑛𝜈𝑒^𝜆𝜏𝛼1+𝛽𝐼. (56)

Remark 21. It seems from Remark 19 that Corollary 20 is obviously more effective than [4, Theorem 3] and [58, Theorem 3.1], which may shown by numerical examples below.

Corollary 22. Assume 𝑝>1. The null solution of Markovian jumping stochastic system (8) is almost sure exponentially stable if there exist positive scalars 𝜆, 𝛼_𝑟(𝑟∈𝑆), 𝛽 and positive definite matrices 𝑃_𝑟=diag(𝑝_𝑟1,𝑝_𝑟2,…,𝑝_𝑟𝑛), (𝑟∈𝑆) such that Θ̆_𝑟𝓑≜−^𝑟𝑃_𝑟𝐴𝐷_𝑟𝐺∗−𝛼_𝑟𝑃𝛽𝓥>0,𝑟∈𝑆,_𝑟<𝛼_𝑟𝐼,𝑟∈𝑆, (57)
where 𝓑_𝑟=𝜆𝑃_𝑟−2𝑃_𝑟𝐵+𝑃_𝑟𝐴𝐶_𝑟𝐹+𝐹𝐶^𝑇_𝑟𝐴𝑃_𝑟+𝛼_𝑟𝓤+𝑛𝜈𝑒^𝜆𝜏𝛼_𝑟1+𝛽𝐼+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗. (58)

Corollary 23. Assumed 𝑝=2. The null solution of Markovian jumping stochastic system (8) is almost sure exponentially stable if there exist positive scalars 𝜆, 𝛼_𝑟, 𝛼_𝑟(𝑟∈𝑆), 𝛽 and positive definite matrices 𝑃_𝑟=diag(𝑝_𝑟1,𝑝_𝑟2,…,𝑝_𝑟𝑛), (𝑟∈𝑆) such that Θ_𝑟𝓑≜−^𝑟−2𝜆₁𝒟𝛼_𝑟𝐼𝑃_𝑟𝐴𝐷_𝑟𝐺∗−𝛼_𝑟𝑃𝛽𝓥>0,𝑟∈𝑆,_𝑟<𝛼_𝑟𝑃𝐼,𝑟∈𝑆,_𝑟>𝛼_𝑟𝐼,𝑟∈𝑆, (59)
where 𝓑_𝑟 satisfies (58).

Remark 24. As pointed out in Remark 19, Corollary 23 is obviously more effective than [58, Theorem 3.1] if Markovian jumping stochastic system (8) is simplified to a stochastic system without Markovian juamping.

4. The Robust Exponential Stability Criteria

Theorem 25. Assume that 𝑝>1. In addition, there exist a sequence of positive scalars 𝛼_𝑟(𝑟∈𝑆) and positive definite diagonal matrices 𝑃_𝑟=diag(𝑝_𝑟1,𝑝_𝑟2,…,𝑝_𝑟𝑛)(𝑟∈𝑆) and 𝑄=diag(𝑞₁,𝑞₂,…,𝑞_𝑛) such that for all 𝑟∈𝑆, 𝔸_𝑟𝑃_𝑟𝐴𝐷_𝑟+𝐷̆_𝑟𝐺𝑃_𝑟𝐴𝐸_1𝑟𝐹𝓝^𝑇_1𝑟+𝓝^𝑇_2𝑟𝑃_𝑟𝐴𝐸_2𝑟0∗−𝑒^−𝜆𝜏𝑄+𝛼_𝑟𝓥000𝐺𝓝^𝑇_1𝑟+𝓝^𝑇_2𝑟∗∗−𝐼000∗∗∗−𝐼00∗∗−𝐼0∗∗∗∗∗−𝐼<0,𝑃_𝑟<𝛼_𝑟𝐼,𝑟∈𝑆, (60)(61)
where matrices 𝐶_𝑟=(𝑐^(𝑟)_𝑖𝑗)_𝑛×𝑛, 𝐷_𝑟=(𝑑^(𝑟)_𝑖𝑗)_𝑛×𝑛, 𝐶̆_𝑟=(𝑐̆^(𝑟)_𝑖𝑗)_𝑛×𝑛, 𝐷̆_𝑟=(𝑑̆^(𝑟)_𝑖𝑗)_𝑛×𝑛, and 𝔸_𝑟=𝜆𝑃_𝑟−2𝑃_𝑟𝐵+𝑃_𝑟𝐴𝐶_𝑟+𝐶̆_𝑟𝐹+𝐹𝐶^𝑇_𝑟+𝐶̆^𝑇_𝑟𝐴𝑃_𝑟+𝛼_𝑟𝓤+𝑄+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗, (62)then the null solution of Markovian jumping stochastic fuzzy system (9) is stochastically exponentially robust stable in the mean square.

Proof. Similarly as the proof of Theorem 11, we consider the same Lyapunov-Krasovskii functional 𝑉(𝑡,𝑣(𝑡),𝑟)=𝑒^𝜆𝑡_Ω^𝑛_𝑖=1𝑝_𝑟𝑖𝑣²_𝑖()+𝑡,𝑥𝑑𝑥_{Ω𝑡𝑡−𝜏}𝑒^𝑛𝜆𝑠_𝑖=1𝑞_𝑖𝑣²_𝑖(𝑠,𝑥)𝑑𝑠𝑑𝑥,∀𝑟∈𝑆, (63)
where 𝑣(𝑡,𝑥)=(𝑣₁(𝑡,𝑥),𝑣₂(𝑡,𝑥),…,𝑣_𝑛(𝑡,𝑥))^𝑇 is a solution for stochastic fuzzy system (9). Then it follows by the proof of Theorem 11 that 𝓛𝑉(𝑡,𝑣(𝑡),𝑟)⩽𝑒^𝜆𝑡𝜆_Ω𝑣^𝑇𝜆𝑃_𝑟+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗|𝑣|𝑑𝑥−2_Ω𝑣^𝑇𝑃_𝑟𝐵|𝑣|𝑑𝑥+2_Ω𝑣^𝑇𝑃_𝑟𝐴𝐶_𝑟(𝑡)+𝐶̆_𝑟(𝑡|𝑣|+𝑣)×𝐹^𝑇𝑃_𝑟𝐴×𝐷_𝑟(𝑡)+𝐷̆_𝑟(𝑡|𝑣(+)×𝐺𝑡−𝜏,𝑥)|𝑑𝑥_Ω𝛼_𝑟𝑣^𝑇𝓤|𝑣|+𝑣^𝑇(|𝑣(+𝑡−𝜏,𝑥)×𝓥𝑡−𝜏,𝑥)|𝑑𝑥_Ω𝑣^𝑇𝑄|𝑣|−𝑒^−𝜆𝜏𝑣^𝑇(|𝑣(.𝑡−𝜏,𝑥)×𝑄𝑡−𝜏,𝑥)|𝑑𝑥 29^∗
Denote 𝜁^𝑇(𝑡,𝑥)=(|𝑣^𝑇(𝑡,𝑥)|,|𝑣^𝑇(𝑡−𝜏,𝑥)|), and 𝔹_𝑟(𝑡)=^{𝔸𝑟(𝑡)𝑃𝑟𝐷𝐴(|𝑟̆𝐷(𝑡)|+|𝑟(𝑡)|)𝐺}_{∗−𝑒−𝜆𝜏𝑄+𝛼𝑟𝓥}, where 𝔸_𝑟(𝑡)=𝜆𝑃_𝑟−2𝑃_𝑟𝐵+𝑃_𝑟𝐴(|𝐶_𝑟(𝑡)|+|𝐶̆_𝑟(𝑡)|)𝐹+𝐹(|𝐶^𝑇_𝑟(𝑡)|+|𝐶̆^𝑇_𝑟(𝑡)|)𝐴𝑃_𝑟+𝛼_𝑟𝓤+𝑄+∑_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗. It is obvious that 𝜁^𝑇(𝑡,𝑥)𝔹_𝑟(𝑡)𝜁(𝑡,𝑥)⩽𝜁^𝑇()𝔸𝑡,𝑥^𝑟𝑃_𝑟𝐴𝐷_𝑟+𝐷̆_𝑟𝐺∗−𝑒^−𝜆𝜏𝑄+𝛼_𝑟𝓥+𝑃_𝑟𝐴Δ𝐶_𝑟(𝑡)+Δ𝐶̆_𝑟(𝑡)𝐹+𝐹Δ𝐶^𝑇_𝑟(𝑡)+Δ𝐶̆^𝑇_𝑟(𝑡)𝐴𝑃_𝑟𝑃_𝑟𝐴Δ𝐷_𝑟(𝑡)+Δ𝐷̆_𝑟(𝑡𝐺𝜁().)∗0𝑡,𝑥 (64)
Then we can conclude it by 29^∗, (11), Lemma 10, and applying Schur Complement ([66]) to (60) that 𝓛𝑉(𝑡,𝑣(𝑡),𝑟)⩽−_Ω𝜁^𝑇(𝑡,𝑥)𝔹_𝑟(𝑡)𝜁(𝑡,𝑥)𝑑𝑥⩽0,𝑟∈𝑆. (65)
Thereby, we can complete the rest of the proof by the same methods employed in (32)–(34).

Similarly, we can derive the following Theorem by (11), Lemma 10, Schur Complement Theorem and the proof of Theorem 18.

Theorem 26. Assume 𝑝>1. The null solution of Markovian jumping stochastic fuzzy system (9) is the almost sure robust exponential stability if there exist positive scalars 𝜆, 𝛼_𝑟(𝑟∈𝑆), 𝛽 and positive definite matrices 𝑃_𝑟=diag(𝑝_𝑟1,𝑝_𝑟2,…,𝑝_𝑟𝑛), (𝑟∈𝑆) such that for all 𝑟∈𝑆,𝔸_𝑟𝑃_𝑟𝐴𝐷_𝑟+𝐷̆_𝑟𝐺𝑃_𝑟𝐴𝐸_1𝑟𝐹𝓝^𝑇_1𝑟+𝓝^𝑇_2𝑟𝑃_𝑟𝐴𝐸_2𝑟0∗𝛼_𝑟𝛽𝓥000𝐺𝓝^𝑇_1𝑟+𝓝^𝑇_2𝑟𝑃∗∗−𝐼000∗∗∗−𝐼00∗∗−𝐼0∗∗∗∗∗−𝐼<0,_𝑟<𝛼_𝑟𝐼,𝑟∈𝑆, (66)
where 𝔸_𝑟=𝜆𝑃_𝑟−2𝑃_𝑟𝐵+𝑃_𝑟𝐶_𝑟+𝐶̆_𝑟𝐹+𝐹𝐶^𝑇_𝑟+𝐶̆^𝑇_𝑟𝐴𝑃_𝑟+𝛼_𝑟𝓤+𝑛𝜈𝑒^𝜆𝜏𝛼_𝑟1+𝛽𝐼+_𝑗∈𝑆𝜋_𝑟𝑗𝑃_𝑗. (67)

Remark 27. Although the stability of Laplace diffusion stochastic neural networks are studied by previous literature. However, it is the first attempt that the robust stability criteria about the nonlinear 𝑝-Laplace diffusion stochastic fuzzy neural networks with Markovian jumping are obtianed, and the first time that the exponential stability criteria of 𝑝-Laplace diffusion stochastic fuzzy neural networks with Markovian jumping are provided. It is also the first attempt to synthesize the variational methods in 𝑊^1,𝑝(Ω) (see, e.g., Lemma 6, Remark 7), Itô formula, Dynkin formula, the semi-martingale convergence theorem, Schur Complement Theorem, and LMI technique to set up the LMI-based (robust) exponential stability or almost sure exponential (robust) stability criteria.

5. Comparisons and Numerical Examples

Example 28. Consider the following stochastic fuzzy CGNNs: 𝑑𝑣₁()=𝑡,𝑥_2𝑘=1𝜕𝜕𝑥_𝑘𝒟_1𝑘(𝑡,𝑥,𝑣)∇𝑣₁(𝑡,𝑥)^𝑝−2𝜕𝑣₁𝜕𝑥_𝑘−𝑎₁𝑣₁(𝑡,𝑥)𝑏₁𝑣₁(−𝑡,𝑥)_2𝑗=1𝑐_1𝑗(𝑟(𝑡))𝑓_𝑗𝑣_𝑗()−𝑡,𝑥_2𝑗=1𝑐̆_1𝑗(𝑟(𝑡))𝑓_𝑗𝑣_𝑗()−𝑡,𝑥_2𝑗=1𝑑_1𝑗(𝑟(𝑡))×𝑔_𝑗𝑣_𝑗()−𝑡−𝜏,𝑥_2𝑗=1𝑑̆_1𝑗(𝑟(𝑡))×𝑔_𝑗𝑣_𝑗()+𝑡−𝜏,𝑥𝑑𝑡_2𝑗=1𝜎_1𝑗𝑣_𝑗(𝑡,𝑥),𝑣_𝑗(𝑡−𝜏,𝑥)𝑑𝑤_𝑗(𝑡),∀𝑡⩾𝑡₀,𝑥∈Ω,𝑑𝑣₂()=𝑡,𝑥_2𝑘=1𝜕𝜕𝑥_𝑘𝒟_2𝑘(𝑡,𝑥,𝑣)∇𝑣₂(𝑡,𝑥)^𝑝−2𝜕𝑣₂𝜕𝑥_𝑘−𝑎₂𝑣₂(𝑡,𝑥)𝑏₂𝑣₂(−𝑡,𝑥)_2𝑗=1𝑐_2𝑗(𝑟(𝑡))𝑓_𝑗𝑣_𝑗()−𝑡,𝑥_2𝑗=1𝑐̆_2𝑗(𝑟(𝑡))𝑓_𝑗𝑣_𝑗()−𝑡,𝑥_2𝑗=1𝑑_2𝑗(𝑟(𝑡))×𝑔_𝑗𝑣_𝑗()−𝑡−𝜏,𝑥_2𝑗=1𝑑̆_2𝑗(𝑟(𝑡))×𝑔_𝑗𝑣_𝑗()+𝑡−𝜏,𝑥𝑑𝑡_2𝑗=1𝜎_2𝑗𝑣_𝑗(𝑡,𝑥),𝑣_𝑗(𝑡−𝜏,𝑥)𝑑𝑤_𝑗(𝑡),∀𝑡⩾𝑡₀𝑣()(),()∈[],𝑥∈Ω,𝜃,𝑥=𝜙𝜃,𝑥𝜃,𝑥−𝜏,0×Ω (68)
under Dirichlet boundary condition, where the initial value function 𝜙()=𝑠,𝑥0.251−cos5𝜋𝑥²cos¹⁸⁹𝑥²−0.25𝑒^−100𝑠0.2sin²4𝜋𝑥²cos²⁰¹𝑥²−0.55𝑒^−100𝑠,−𝜏≤𝑠≤0, (69)
𝑝=2.011,𝑣=(𝑣₁(𝑡,𝑥), 𝑣₂(𝑡,𝑥))^𝑇∈𝑅², 𝑥=(𝑥₁,𝑥₂)^𝑇∈Ω={(𝑥₁,𝑥₂)^𝑇∈𝑅²∶|𝑥_𝑗|<√2,𝑗=1,2},𝑎₁(𝑣₁)=0.11+0.01sin²(𝑣²₁), 𝑎₂(𝑣₂)=0.12+0.01cos²(𝑣²₂), 𝑏₁(𝑣₁)=0.12𝑣₁+2𝑣₁sin²(𝑣²₁), 𝑏₂(𝑣₂)=0.125𝑣₂+𝑣₂cos²(𝑣²₂), 𝑓(𝑣)=𝑔(𝑣)=(0.16𝑣₁,0.166𝑣₂+0.001𝑣₂sin²(𝑣²₂))^𝑇, and ,𝒟()=𝐴=0.12000.13,𝐵=0.0132000.0150𝑡,𝑥,𝑣0.0030.0050.0040.006,𝐹=0.16000.167=𝐺,𝓤=0.0003000.0003=𝓥,𝐶₁==𝐷0.11−0.003−0.0030.12¹,𝐶̆₁==𝐷̆0.16−0.003−0.0030.18¹,𝐶₂==𝐷0.13−0.003−0.0030.15²,𝐶̆₂==𝐷̆0.17−0.003−0.0030.19²,𝐶₃==𝐷0.12−0.003−0.0030.13³,𝐶̆₃==𝐷̆0.175−0.003−0.0030.196³. (70)(71)
The transition rates matrices are considered as 𝜋=¹¹𝜋₁₂𝜋₁₃𝜋₂₁𝜋₂₂𝜋₂₃𝜋₃₁𝜋₃₂𝜋₃₃=.−0.60.40.20.2−0.70.50.50.3−0.8 (72)
Fix 𝜆=0.001. Let 𝜏=50.78, then we use matlab LMI toolbox to solve LMIs (21) and (22), and obtain 𝑡min=−2.5977∗10⁻⁵<0, 𝛼₁=7025.2, 𝛼₂=7051.3, 𝛼₃=7038.5, and 𝑃₁=7023.4007025.1,𝑃₂=,𝑃7041.5007051.1₃=,.7035.1007038.3𝑄=46.47450052.8358 (73)
Then by Theorem 11 we know that the null solution of Markovian jumping stochastic fuzzy system (68) is stochastically exponentially stable in the mean square with the allowable upper bounds of time delays 𝜏=50.78 (see, Figures 1 and 2).

(a)

(b)

Figure 1 

The state variable 𝑣1(𝑡,𝑥).

(a)

(b)

Figure 2 

The state variable 𝑣2(𝑡,𝑥).

Remark 29. (1) Thanks to some novel techniques employed in this paper (Remark 12), LMI-based criterion of Theorem 11 is more effective and feasible in consideration of its significant improvement in the allowable upper bounds of time delays.
(2) Because the stability of the stochastic fuzzy CGNN with 𝑝-Laplace diffusion is never studied by any previous literature, below we have to compare the corollaries of Theorem 11 with some existing results to demonstrate the advantages of the proposed method (see, e.g., Example 30).
(3) Finding a solution 𝑥 to the LMI system 𝐴(𝑥)<𝐵(𝑥) is called the feasibility problem. In matlab LMI toolbox, the feasibility problem is solved by the so-called feasp solver. And the feasp solver always judges the feasibility of the feasibility problem by solving the following convex optimization problem: min𝑡s.t.𝐴(𝑥)−𝐵(𝑥)𝑡⩽𝐼. (74) The global optimum value of the convex optimization problem is always denoted by 𝑡min, which is the first datum of the output data of the feasp solver. Particularly, the system is feasible if 𝑡min <0, and infeasible if 𝑡min>0.

Example 30. Consider the following fuzzy system: 𝑑𝑥₁(𝑡)=−𝑎₁𝑥₁(𝑡)𝑏₁𝑥₁(𝑡)−²_𝑗=1𝑐_1𝑗𝑓_𝑗𝑥_𝑗(𝑡)−²_𝑗=1𝑐̆_1𝑗𝑓_𝑗𝑥_𝑗(𝑡)−²_𝑗=1𝑑_1𝑗𝑔_𝑗𝑥_𝑗()−𝑡−𝜏_2𝑗=1𝑑̆_1𝑗𝑔_𝑗𝑥_𝑗()+𝑡−𝜏𝑑𝑡_2𝑗=1𝜎_1𝑗𝑥_𝑗(𝑡),𝑥_𝑗(𝑡−𝜏)𝑑𝑤_𝑗(𝑡),𝑑𝑥₂(𝑡)=−𝑎₂𝑥₂(𝑡)𝑏₂𝑥₂(𝑡)−²_𝑗=1𝑐_2𝑗𝑓_𝑗𝑥_𝑗(𝑡−_2𝑗=1)𝑐̆_2𝑗𝑓_𝑗𝑥_𝑗(𝑡−_2𝑗=1)𝑑_2𝑗𝑔_𝑗𝑥_𝑗()−𝑡−𝜏_2𝑗=1𝑑̆_2𝑗𝑔_𝑗𝑥_𝑗()+𝑡−𝜏𝑑𝑡_2𝑗=1𝜎_2𝑗𝑥_𝑗(𝑡),𝑥_𝑗(𝑡−𝜏)𝑑𝑤_𝑗(𝑡),𝑥_𝑖(𝑡)=𝜙_𝑖(𝑡),−𝜏⩽𝑡⩽0,𝑖=1,2 (75)
with all the parameters mentioned in Example 28. In addition, denote 1𝐶=3³_𝑗=1𝐶_𝑗==̆10.1200−0.0030−0.00300.1333𝐷,𝐶=3³_𝑗=1𝐶̆_𝑗==̆0.1683−0.0030−0.00300.1887𝐷. (76)
Let 𝜏=45.37, then we solve LMIs (35), and get 𝑡min=−1.6257∗10⁻⁴<0, 𝛼=194020, and ,.𝑃=19397000194020𝑄=1278.1001453.3 (77)
Then by Corollary 13, the null solution of stochastic fuzzy system (75) is stochastically exponentially stable in the mean square with the allowable upper bounds of time delays 𝜏=45.37.

Remark 31. With all the above parameters in Example 30, we solve the inequalities condition (2) in [4, Theorem 1], and obtain 𝑡min=1.8569∗10⁻¹¹>0 which implies infeasible (see, Remark 29 (3)). However, the inequalities condition (2) in [4, Theorem 1] is only sufficient, not necessary for the stability. In Example 30, we can conclude from LMI-base criterion of Corollary 13 that the null solution of stochastic fuzzy system (75) is stochastically exponentially stable in the mean square. Hence, as pointed out in Remarks 12 and 14, Corollary 13 is more feasible and effective than [4, Theorem 1].

Example 32. Consider the Markovian jumping system (8) with all the parameters mentioned in Example 28. In addition, denote 𝐶₁=12𝐶₁+𝐶̆₁=0.1350−0.0030−0.00300.1500=𝐷₁,𝐶₂=12𝐶₂+𝐶̆₂=0.1500−0.0030−0.00300.1700=𝐷₂,𝐶₃=12𝐶₃+𝐶̆₃=0.1475−0.0030−0.00300.1630=𝐷₃. (78)
Let 𝜏=108.9. Then one can solve LMIs (37), and obtain 𝑡min=−0.0168<0, 𝛼₁=51.4646, 𝛼₂=51.0590, 𝛼₃=50.4225, and 𝑃₁=,𝑃31.28800031.8027₂=,𝑃31.37290031.8989₃=,.31.37850031.8989𝑄=0.3659000.4103 (79)
And hence Corollary 15 derives that the null solution of Markovian jumping stochastic system (8) is stochastically exponentially stable in the mean square with the allowable upper bounds of time delays 𝜏=108.9.
If 𝑝=2, we can solve LMIs (42) with 𝜏=109.88, and get 𝑡min=−0.0348, 𝛼₁=44.8100, 𝛼₂=44.4562, 𝛼₃=43.6935, 𝛼₁=77.9603, 𝛼₂=77.5548, 𝛼₃=76.8509, and 𝑃₁=,𝑃60.66290060.7682₂=,𝑃60.89950061.0238₃=,,60.99160061.1055𝑄=2.1773002.2520 (80)
where 𝜆₁=9.8696 for Ω={(𝑥₁,𝑥₂)^𝑇∈𝑅²∶|𝑥_𝑗|<√2,𝑗=1,2} (see, e.g., [58, Remark 2.5]). Then Corollary 16 yields that the null solution of Markovian jumping stochastic system (8) is stochastically exponentially stable in the mean square with the allowable upper bounds of time delays 𝜏=109.88.

Remark 33. Example 32 illustrates that LMI-based criteria of Corollaries 15 and 16 are more effective and feasible than some existing results (see, e.g., [58, Theorem 3.2]) due to the significant improvement in the allowable upper bounds of time delays.

There are some interesting comparisons among Examples 28, 30, and 32 as follows.

From Table 1, we know that the ambiguity of the fuzzy system affect the analysis and judgement on the stability. The maximum allowable upper bounds decrease when the fuzzy factors occur. In addition, both the randomicity of Markovian jumping and nonlinear 𝑝-Laplace diffusion exercised a malign influence on judging the stability.

Table 1 

Allowable upper bound of 𝜏 for various cases.

Remark 34. Table 1 also illustrates that the diffusion item plays an active role in the LMI-based criterion of Corollary 16.

Example 35. Consider (68) with the parameters (71), (72), and𝐴=,0.32000.31,𝒟()=.𝐴=0.38000.36𝐵=6.433006.61,𝐹=0.1000.1=𝐺,𝓤=0.0003000.0003=𝓥,𝑡,𝑥,𝑣0.0030.0050.0040.006 70^∗
Assume, in addition, 𝑛=2. Fix 𝜆=0.0001 and 𝛽=0.0001. Let 𝜏=27.15, then we solve LMIs (43), and obtain 𝑡min=−3.6996∗10⁻⁹<0, 𝛼₁=49955, 𝛼₂=92134, 𝛼₃=68884, and 𝑃₁=,𝑃24.02790023.8713₂=,𝑃37.84430039.0478₃=.30.88660030.2698 (82)
Hence, we can conclude from Theorem 18 that the null solution of Markovian jumping stochastic fuzzy system (6) is almost sure exponentially stable with the allowable upper bounds of time delays 𝜏=27.15.

Example 36. Consider stochastic fuzzy system (75) with all the parameters in Example 35. In addition, denote 1𝐶=3³_𝑗=1𝐶_𝑗==̆10.1200−0.0030−0.00300.1333𝐷,𝐶=3³_𝑗=1𝐶̆_𝑗==̆0.16830.00300.00300.1887𝐷. 76^∗
Let 𝜏=88.15, then one can solve LMIs (55), and obtain 𝑡min=−7.5392∗10⁻⁷<0, 𝛼=22255 and 𝑃=.12.35300010.6073 (84)
Then by Corollary 20, the null solution of stochastic fuzzy system (75) is almost sure exponentially stable with the allowable upper bounds of time delays 𝜏=88.15.

Remark 37. With all the above data in Example 36, we solve the inequalities condition (8) in [4, Theorem 3], and obtain 𝑡min=8.9843∗10⁻¹²>0 which implies infeasible. However, the inequalities condition (8) in [4, Theorem 3] is only sufficient, not necessary for the stability. In Example 36, we can conclude from LMI-base criterion of Corollary 20 that the null solution of stochastic fuzzy system (1) is almost sure exponentially stable. Hence, as pointed out in Remarks 19 and 21, Corollary 20 is more feasible and effective than [4, Theorem 3].

Example 38. Consider Markovian jumpimg stochastic system (8) with all the parameters in Example 35. In addition, denote 𝐶₁=12𝐶₁+𝐶̆₁=0.1350−0.0030−0.00300.1500=𝐷₁,𝐶₂=12𝐶₂+𝐶̆₂=0.1500−0.0030−0.00300.1700=𝐷₂,𝐶₃=12𝐶₃+𝐶̆₃=0.1475−0.0030−0.00300.1630=𝐷₃. 78^∗
Let 𝜏=98.85, then we solve LMIs (43), and obtain 𝑡min=−7.9520∗10⁻⁷<0, 𝛼₁=131.8414, 𝛼₂=130.0644, 𝛼₃=131.3207, and 𝑃₁=,𝑃0.4182000.3723₂=,𝑃0.3328000.2830₃=.0.3525000.3155 (86)
Hence, we can conclude from Corollary 22 that the null solution of Markovian jumping stochastic system (8) is almost sure exponentially stable with the allowable upper bounds of time delays 𝜏=98.85.
If 𝑝=2, then one can solve LMIs (59) with 𝜏=99.89, and get 𝑡min=−3.5692∗10⁻⁷<0, 𝛼₁=0.8469, 𝛼₂=0.6080, 𝛼₃=0.6509, 𝛼₁=452.9764, 𝛼₂=457.1149, 𝛼₃=459.5402, and 𝑃₁=,𝑃1.6566001.4943₂=,𝑃1.3220001.1359₃=,1.3732001.2426 (87)
where 𝜆₁=9.8696 for Ω={(𝑥₁,𝑥₂)^𝑇∈𝑅²∶|𝑥_𝑗|<√2,𝑗=1,2}.
Hence, Corollary 23 yields that the null solution of Markovian jumping stochastic system (8) is almost sure exponentially stable with the allowable upper bounds of time delays 𝜏=99.89.

Remark 39. Example 38 illustrates that LMI-based criteria of Corollaries 22 and 23 are more effective and feasible than some existing results (see, e.g., [58, Theorem 3.1]) due to the significant improvement in the allowable upper bounds of time delays.

There are some interesting comparisons among Examples 35, 36, and 38 as follows.

From Table 2, we know that the ambiguity of the fuzzy system affect the analysis and judgement on the stability. The maximum allowable upper bounds decrease when the fuzzy factors occur. In addition, both the randomicity of Markovian jumping and nonlinear 𝑝-Laplace diffusion exercised a malign influence on judging the stability.

Table 2 

Allowable upper bound of 𝜏 for various cases.

Remark 40. Table 2 also illustrates that the diffusion item plays an active role in the LMI-based criterion of Corollary 23.

Remark 41. From Examples 28, 30, 32, 35, 36, and 38 we learn that owing to some novel techniques employed in this paper (see, Remarks 12 and 19), LMI-based criteria of Theorems 11 and 18, and their corollaries are more effective and feasible than recent related results (Remarks 12, 14, 17, 19, 21, and 24), and improve significantly the allowable upper bounds of time delays (Remarks 29, 31, 33, 34, 37, and 39).

6. Conclusions

In this paper, the stability for delayed nonlinear reaction-diffusion Markovian jumping stochastic fuzzy Cohen-Grossberg neural networks is investigated. The fuzzy factors and the nonlinear 𝑝-Laplace diffusion bring a great difficulty in setting up the LMI-based criteria for the stability. By way of some variational methods in 𝑊^1,𝑝(Ω), Itô formula, Dynkin formula, the semi-martingale convergence theorem, Schur Complement Theorem and LMI technique, the LMI-based criteria on the (robust) exponential stability and almost sure exponential (robust) stability are finally obtained, the feasibility of which can efficiently be computed and confirmed by computer matlab LMI toolbox. As the stability of the nonlinear 𝑝-Laplace diffusion fuzzy CGNNs has never been studied before, we compare some corollaries of our main results with existing results in numerical examples. In Examples 30 and 36, Corollaries 13 and 20 judge what existing criteria cannot do. Numerical examples and simulations illustrate the effectiveness and less conservatism of the proposed method due to the significant improvement in the allowable upper bounds of time delays (see, Remarks 29, 31, 33 and Remarks 34, 37, 39). Tables 1 and 2 show that fuzzy factors and stochastic factors give some difficulties to judge the stability, for the allowable upper bounds of time delays decrease when fuzzy factors and stochastic factors occur. In addition, when 𝑝=2, the diffusion item plays a positive role. So for the future work, the 𝑝-Laplace diffusion item (𝑝>1 and 𝑝≠2) should play its role in stability criteria, which still remains open and challenging.

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 (2010JY0057, 2012JY010), and by Scientific Research Fund of Sichuan Provincial Education Department (12ZB349).