Stability in Mean of Partial Variables for Coupled Stochastic Reaction-Diffusion Systems on Networks: A Graph Approach

Abstract

This paper is devoted to investigating stability in mean of partial variables for coupled stochastic reaction-diffusion systems on networks (CSRDSNs). By transforming the integral of the trajectory with respect to spatial variables as the solution of the stochastic ordinary differential equations (SODE) and using Itô formula, we establish some novel stability principles for uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, and exponential stability in mean of partial variables for CSRDSNs. These stability principles have a close relation with the topology property of the network. We also provide a systematic method for constructing global Lyapunov function for these CSRDSNs by using graph theory. The new method can help to analyze the dynamics of complex networks. An example is presented to illustrate the effectiveness and efficiency of the obtained results.

1. Introduction

Coupled systems on networks (CSNs), composed of a large number of highly interconnected dynamical nodes [1], have received more and more attention due to its popularity in modelling many large-scale dynamical systems from science and engineering, such as communication networks, social networks, power grids, cellular networks, World Wide Web, metabolic systems, food webs, and disease transmission networks; see for instance [2–6] and the references therein. Stability is one important constituent part of performance investigation for dynamical systems, and it is very necessary to construct a relation between the stability criteria of a CSN and some topology properties of the network [7–11]. Li and Shuai [11] have considered global stability for the general CSNs based on graph theory, without discussing the stochastic effects. Due to the fact that most motions are actually the results of deterministic processes mingling with random processes [12, 13], Kao et al. [14] have investigated stability of coupled stochastic systems with time-delay on networks without reaction diffusion effects. In fact, for many realistic CSNs, the node state is seriously dependent on the time and space [15–20]. Hence, in order to describe more accurately the dynamics changes of CSNs, Kao and Wang put up with stochastic coupled reaction-diffusion systems on networks (SCEDSNs) based on graph theory and probed global stability analysis for SCEDSNs [21].

On the other hand, in real world, it is difficult or even impossible to measure or estimate all the states of the systems due to the factors of expensive cost or technique [22–26]. Partial stability technique (stability of part of the variables) is most useful when a fully stabilized system losses some control engines or some phase variables are not actively controlled. Such situations are most applicable for automatic systems which need to work remotely without a proper access to maintenance, such as satellite or robots. Therefore, stability and stabilization of motion with respect to part of the variables is of great significance [27–38]. Kao et al. [27] have studied stability in mean of partial variables for stochastic reaction-diffusion systems with Markovian switching. Xi et al. [31] have investigated output consensus analysis and design for high-order linear swarm systems by partial stability method. Partial stabilization technology has been applied into the guidance problem by Shafiei and Binazadeh [32]. Ignatyev [34] has probed partial asymptotic stability in probability of stochastic differential equations. Chen et al. [36] have discussed impulsive synchronization of chaotic Lur’e systems via partial states. Stability in mean of partial variables for stochastic reaction-diffusion systems has been considered in [38]. To the best of the authors' knowledge, stability analysis for stability in mean of partial variables for coupled stochastic reaction-diffusion systems on networks (CSRDSNs) has not been properly addressed, which still remains important and challenging.

Motivated by the above discussions, in this paper, we propose the CSRDSNs model. In Section 2, some preliminaries are presented. In Section 3, some new stability principles for uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, and exponential stability in mean of partial variables for CSRDSNs are established. These stability principles have a close relation to the topology property of the network. A systematic method is provided to construct the global Lyapunov function of CSRDSNs by combining graph theory and the Lyapunov second method. The findings show that, if each vertex system has a globally stable equilibrium and possesses a global Lyapunov function 𝑉_𝑘, then the global Lyapunov function for the CSRDSNs can be systematically produced by individual 𝑉_𝑘. An example is provided in Section 4. Section 5 is conclusion. Notations: for convenience, we sometimes write 𝐯, 𝐯_𝐤, and 𝐯_𝐣 as 𝐯(𝑡,𝐱), 𝐯_𝐤(𝑡,𝐱), and 𝐯_𝐣(𝑡,𝐱), respectively.

2. Preliminaries

A general stochastic reaction-diffusion system reads d𝐯(𝑡,𝐱)=𝜌_𝑖(𝑡)()(((((𝑡),()Δ𝐯𝑡,𝐱+𝑓𝑡,𝐱,𝐯𝑡,𝐱))d𝑡+𝑔𝑡,𝐱,𝐯𝑡,𝐱))d𝑊𝑡,𝐱∈ℝ₊×𝐺, (1) with boundary condition 𝜕𝐯(𝑡,𝐱)()𝜕𝒩=0,𝑡,𝐱∈ℝ₊×𝜕𝐺, (2) where 𝐯=col(𝐲,𝐳)∈ℝ^𝑛, 𝐲∈ℝ^𝑚,  𝐳∈ℝ^𝑝(𝑚+𝑝=𝑛), Δ𝐯=∑^𝑛_𝑖=1(𝜕²𝐯/𝜕𝑥²_𝑖); denote |∫_𝐺𝐯(⋅,𝐱)d𝐱| by |𝐯(⋅,𝐱)|_𝐺, 𝐺={𝐱,|𝐱|<𝑙<+∞}⊂𝑅^𝑟, and both 𝑓∈[ℝ₊×𝐺×Γ,ℝ^𝑛] and 𝑔∈[ℝ₊×𝐺×Γ,ℝ^𝑛×𝑚] are Borel measurable functions; here, Γ={𝐯,|𝐲(⋅,𝑥)|_𝐺≤𝐻=const, |𝐳(⋅,𝐱)|_𝐺<+∞}⊂ℝ^𝑛, |⋅| stands for vector norm, 𝑊(𝑡)=(𝑊₁,…,𝑊_𝑚)^⊺ is an 𝑚-dimension Brown motion with natural flow {ℱ_𝑡}_𝑡≥0 defined on complete probability space (Ω,ℱ,(ℱ_𝑡)_𝑡∈𝐼,ℙ), 𝒩 is the normal vector to 𝜕𝐺, 𝑓(𝑡,𝐱,0)≡0, and 𝑔(𝑡,𝐱,0)≡0.

Throughout this paper, we suppose function 𝑔(𝑡,𝐱,𝐯(𝑡,𝐱)) satisfies integral linear growth condition and 𝑓, 𝑔 meet Lipschitz condition; that is, there exists constant 𝐿>0 such that 𝑔(𝑡,𝐱,𝐯(𝑡,𝐱))_𝐺≤𝐿1+|𝐯|_𝐺,𝑔𝑡,𝐱,𝐯_𝟏(𝑡,𝐱)−𝑔𝑡,𝐱,𝐯_𝟐(𝑡,𝐱)_𝐺≤𝐿𝐯_𝟏−𝐯_𝟐𝐺,𝑓𝑡,𝐱,𝐯_𝟏(𝑡,𝐱)−𝑓𝑡,𝐱,𝐯_𝟐(𝑡,𝐱)_𝐺≤𝐿𝐯_𝟏−𝐯_𝟐𝐺, (3) where |𝐯(⋅,𝐱))|_𝐺≜|∫_𝐺𝐯(⋅,𝐱)d𝐱|. The existence of the solution for system (1) can be proved by the common stepwise interactive method and the relevant conclusion can also refer to [39, 40].

Before the start of our discussion, we will first introduce some definitions as to stability in mean of partial variables for stochastic reaction-diffusion systems.

Definition 1. The trivial solution of system (1) is said to be stable in mean as to partial variables 𝐲 if, for ∀𝜀>0, ∀𝑡₀>0, there is 𝛿(𝑡₀,𝜀) such that ℰ{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}<𝜀 holds for ∀𝐯₀∈𝑆_𝛿={𝐯∣|𝐯(⋅,𝐱)|_𝐺<𝛿}.
The trivial solution of system (1) is said to be uniformly stable in mean as to partial variables 𝐲 if, for ∀𝜀>0, ∀𝑡₀>0, there is 𝛿(𝜀) such that ℰ{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}<𝜀 holds for ∀𝐯₀∈𝑆_𝛿={𝐯∣|𝐯(⋅,𝐱)|_𝐺<𝛿}.
The trivial solution of system (1) is said to be asymptotically stable in mean as to partial variables 𝐲 if, for ∀𝜀>0, ∀𝑡₀>0, there is 𝛿(𝑡₀,𝜀) such that for ∀𝐯₀∈𝑆_𝛿={𝐯∣|𝐯(⋅,𝐱)|_𝐺<𝛿}ℰ{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}<𝜀 and lim_𝑡→∞ℰ{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}=0.
The trivial solution of system (1) is said to be uniformly asymptotically stable in mean as to partial variables 𝐲 if, for ∀𝜀>0, ∀𝑡₀>0, there is 𝛿(𝜀) such that for ∀𝐯₀∈𝑆_𝛿={𝐯∣|𝐯(⋅,𝐱)|_𝐺<𝛿}ℰ{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}<𝜀 and lim_𝑡→∞ℰ{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}=0.

Definition 2. If 𝜇(⋅)∈𝐶[[0,𝑟],ℝ] is a strictly increasing function and 𝜇(0)=0, function 𝜇 is said to be class 𝒦 function. Denote 𝜇∈𝒦 concisely. If 𝜇(⋅)∈𝐶[ℝ⁺,ℝ⁺] and 𝜇∈𝒦,lim_𝑟→+∞𝜇(𝑟)=+∞, then 𝜇∈𝒦𝑅.

A continuous function 𝑉(𝑡,𝜉) is said to be positive-definite if 𝑉(𝑡,0)=0 and, for some 𝜇∈𝒦ℛ, 𝑉(𝑡,𝜉)≥𝜇(|𝜉|). Write 𝐶^1,2(ℝ₊×ℝ^𝑛;ℝ₊) for the family of all nonnegative functions 𝑉(𝑡,𝜉) on ℝ₊×ℝ^𝑛 that are continuously twice differentiable in 𝜉 and once in 𝑡. If 𝑉(𝑡,𝜉)∈𝐶^1,2(ℝ₊×ℝ^𝑛;ℝ₊), then define an operator ℒ𝑉(𝑡,𝜉) from ℝ₊×ℝ^𝑛 to ℝ with respect to (1) by ℒ𝑉(𝑡,𝜉)=𝑉_𝑡(𝑡,𝜉)+𝑉^𝑇_𝜉()𝑓()+1𝑡,𝜉𝑡,𝐱,𝜉2Trace𝑔^𝑇(𝑡,𝐱,𝜉)𝑉_𝜉𝜉(𝑥,𝑡)𝑔(𝑡,𝐱,𝜉), (4) where 𝑉_𝑡()=()𝑡,𝜉𝜕𝑉𝑡,𝜉,𝑉𝜕𝑡_𝑇𝜉()=()𝑡,𝜉𝜕𝑉𝑡,𝜉𝜕𝜉₁(),…,𝜕𝑉𝑡,𝜉𝜕𝜉_𝑛,𝑉_𝜉𝜉()=𝜕𝑥,𝑡²𝑉(𝑡,𝜉)𝜕𝜉_𝑖𝜕𝜉_{𝑗𝑛×𝑛}. (5)

Applying the Itô formula to ∫_𝐺𝐕(𝑡,𝐯(𝑡,𝐱))d𝐱 along system (1) gives for ∀𝑡≥𝑡₀d_𝐺𝑉(𝑡,𝐯(𝑡,𝐱))d𝐱₍₁₎=_𝐺ℒ𝑉(𝑡,𝐯(𝑡,𝐱))+𝑉^𝑇_𝐯(𝑡,𝐯)𝜌Δ𝐯(𝑡,𝐱)d𝑡+𝑉^𝑇_𝐯(𝑡,𝐯)𝑔(𝑡,𝐱,𝐯(𝑡,𝐱))d𝑊(𝑡)d𝐱. (6)

The existence of function 𝑉(𝑡,𝐯)∈𝐶^1,2 and another condition in the classical Lyapunov theorem on the stability of (1) are needed [30]. For convenience, similarly, we give the following definitions.

Definition 3. 𝑉∈𝐶^1,2(ℝ₊×ℝ^𝑛;ℝ₊) is called a Lyapunov-A function for (1), if ℒ∫_𝐺𝑉(𝑡,𝐯)d𝐱≤0, and is called a Lyapunov-B function for (1), if ℒ∫_𝐺𝑉(𝑡,𝐯)d𝐱≤−𝑏∫_𝐺𝑉(𝑡,𝐯)d𝐱, in which 𝑏>0.

The following basic concepts and theorems on graph theory can be found in [11, 41]. A directed graph 𝒢=(𝒱,𝐸) contains a set 𝒱={1,2,…,𝑛} of vertices and a set 𝐸 of arcs (𝑖,𝑗) leading from initial vertex 𝑖 to terminal vertex 𝑗. A subgraph ℋ of 𝒢 is said to be spanning if ℋ and 𝒢 have the same vertex set. A digraph 𝒢 is weighted if each arc (𝑗,𝑖) is assigned to a positive weight 𝑎_𝑖𝑗. Here 𝑎_𝑖𝑗>0 if and only if there exists an arc from vertex 𝑗 to vertex 𝑖 in 𝒢. The weight 𝑊(𝒢) of 𝒢 is the product of the weights on all its arcs. A directed path 𝒫 in 𝒢 is a subgraph with distinct vertices {𝑖₁,𝑖₂,…,𝑖_𝑚} such that its set of arcs is {(𝑖_𝑘,𝑖_𝑘+1)∶𝑘=1,2,…,𝑚−1}. If 𝑖_𝑚=𝑖₁, we call 𝒫 a directed cycle. A connected subgraph 𝒯 is a tree if it contains no cycles. A tree 𝒯 is rooted at vertex 𝑖, called the root, if 𝑖 is not a terminal vertex of any arcs, and each of the remaining vertices is a terminal vertex of exactly one arc. A digraph 𝒢 is strongly connected if, for any pair of distinct vertices, there exists a directed path from one to the other. Given a weighted digraph 𝒢 with 𝑛 vertices, define the weight matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 whose entry 𝑎_𝑖𝑗 equals the weight of arc (𝑗,𝑖) if it exists and 0 otherwise. Denote the directed graph with weight matrix 𝐴 by (𝒢,𝐴). A weighted digraph (𝒢,𝐴) is said to be balanced if 𝑊(𝒞)=𝑊(−𝒞) for all directed cycles 𝒞. Here, −𝒞 denotes the reverse of 𝒞 and is constructed by reversing the direction of all arcs in 𝒞. For a unicyclic graph 𝒬 with cycle 𝒞_𝒬, let 𝒬 be the unicyclic graph obtained by replacing 𝒞_𝒬 with −𝒞_𝒬. Suppose that (𝒢,𝐴) is balanced; then 𝑊(𝒬)=𝑊(𝒬). The Laplacian matrix of (𝒢,𝐴) is defined as 𝒥=^𝑖≠1𝑎_1𝑖−𝑎₁₂⋯−𝑎_1𝑛−𝑎_21𝑖≠2𝑎_2𝑖⋯−𝑎_2𝑛⋮⋮⋱⋮−𝑎_𝑛1−𝑎_𝑛2⋯_𝑖≠𝑛𝑎_𝑖𝑘. (7) Let 𝑐_𝑖 denote the cofactor of the 𝑖th diagonal element of 𝒥.

Lemma 4 ([34] Kirchhoffs Matrix Tree Theorem). Assume 𝑛≥2. Then 𝑐_𝑖=_{𝒯∈𝕋𝑖}𝑊(𝒯),𝑖=1,2,…,𝑁, (8) where 𝕋_𝑖 is the set of all spanning trees 𝒯 of (𝒢,𝐴) that are rooted at vertex 𝑖. In particular, if (𝒢,𝐴) is strongly connected, then 𝑐_𝑖>0 for 𝑐_𝑖=1,2,…,𝑛.

Lemma 5 (see [11]). Assume 𝑛≥2. Let 𝑐_𝑘 be given in (1). Then the following identity holds: ^𝑛_𝑖,𝑗=1𝑐_𝑖𝑎_𝑖𝑗𝐹_𝑘𝑗𝑥_𝑖,𝑥_𝑗=_𝒬∈ℚ𝑊(𝒬)_{𝐶𝑖,𝑗∈𝐸𝒬}𝐹_𝑖𝑗𝑥_𝑖,𝑥_𝑗. (9) Here  𝐹_𝑖𝑗(𝑥_𝑖,𝑥_𝑗),1≤𝑖,𝑗≤𝑛, are arbitrary functions, ℚ is the set of all spanning unicyclic graphs of (𝒢,𝐴), 𝑊(𝒬) is the weight of 𝒬,  and 𝐶_𝒬 denotes the directed cycle of 𝒬.

3. Main Results

To begin with our main results, we will give an SCEDSN represented by digraph 𝒢 with 𝑁 vertices, 𝑁≥2. In 𝑖th vertex it is assigned a stochastic reaction-diffusion system d𝐯_𝑖(𝑡,𝐱)=𝜌_𝑖(𝑡)Δ𝐯_𝑖(𝑡,𝐱)+𝑓_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)d𝑡+𝑔_𝑖𝑡,𝐱,𝐯_𝑖()(𝑡),()𝑡,𝐱d𝑊𝑡,𝐱∈ℝ₊×𝐺, (10) where 𝐯_𝑖(𝑡,𝐱)=col(𝐲_𝐢,𝐳_𝐢)∈ℝ^𝑛𝑖,𝐲_𝐢∈ℝ^𝑚𝑖,𝐳_𝐢∈ℝ^𝑝𝑖(𝑚_𝑖+𝑝_𝑖=𝑛_𝑖),𝑓_𝑖∶ℝ₊×𝐺×ℝ^𝑛𝑖→ℝ^𝑛𝑖, and 𝑔_𝑖∶ℝ₊×𝐺×ℝ^{𝑛𝑖×𝑚𝑖}→ℝ^{𝑛𝑖×𝑚}. If these systems are coupled, let 𝐻_𝑖𝑗∶ℝ^𝑛𝑖×ℝ^𝑛𝑗×ℝ⟶ℝ^𝑛𝑖,𝑁_𝑖𝑗∶ℝ^𝑛𝑖×ℝ^𝑛𝑗×ℝ⟶ℝ^{𝑛𝑖×𝑚},𝑖,𝑗=1,2,…,𝑁 (11) represent the influence of vertex 𝑗 on vertex 𝑖, and 𝐻_𝑖𝑗=𝑁_𝑖𝑗=0 if there exists no arc from 𝑗 to 𝑖 in 𝒢. Then, by replacing 𝑓_𝑖  and  𝑔_𝑖 with 𝑓_𝑖+∑^𝑁_𝑗=1𝐻_𝑖𝑗 and 𝑔_𝑖+∑^𝑁_𝑗=1𝑁_𝑖𝑗, we get the following stochastic coupled system on graph 𝒢: d𝐯_𝑖()=𝜌𝑡,𝐱_𝑖Δ𝐯_𝑖(𝑡,𝐱)+𝑓_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝐻_𝑖𝑗𝐯_𝑖,𝐯_𝑗+𝑔,𝑡d𝑡_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗(𝑡),(),𝑡d𝑊𝑡,𝐱∈ℝ₊×𝐺,𝜕𝐯_𝑖(𝑡,𝐱)()𝜕𝒩=0,𝑡,𝐱∈ℝ₊×𝜕𝐺. (12)(13)

Without loss of generality, we suppose that functions 𝑓_𝑖, 𝑔_𝑖, 𝐻_𝑖𝑗, and 𝑁_𝑖𝑗 are such that initial-value problems to (10) and (12) have a unique solution and trivial solution 𝐯(𝑡,𝐱)=(𝐯₁,…,𝐯_𝑛)=0. Functions 𝑓_𝑖,𝑓_𝑖+∑^𝑁_𝑗=1𝐻_𝑖𝑗,𝑔_𝑖 and 𝑔_𝑖+∑^𝑁_𝑗=1𝑁_𝑖𝑗 meet Lipschitz condition with Lipschitz constant 𝐿>0. Functions 𝑔_𝑖 and 𝑔_𝑖+∑^𝑁_𝑗=1𝑁_𝑖𝑗 satisfy integral linear growth condition. Consider 𝐲=∑^𝑁_𝑖=1𝐲_𝐢. For 𝑉_𝑖(𝑡,𝐯_𝑖)∈𝐶^1,2(ℝ₊×ℝ^𝑛𝑖;ℝ₊), define a differential operator ℒ𝑉_𝑖(𝑡,𝐯_𝑖) associated with the 𝑖th equation of (12) by ℒ𝑉_𝑖𝑡,𝐯_𝑖≜𝜕𝑉_𝑖𝑡,𝐯_𝑖+𝜕𝑡𝜕𝑉_𝑖𝑡,𝐯_𝑖𝜕𝐯_𝑖^𝑇×𝑓_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝐻_𝑖𝑗𝐯_𝑖,𝐯_𝑗+1,𝑡2Trace𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡^𝑇×𝑉_𝑖𝑡,𝐯_𝑖𝐯^′′_𝑖𝐯𝑗×𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡. (14)

3.1. Stability in Mean

In this section, we will discuss stability in mean as to partial variables 𝐲 of system (12) and draw some relevant conclusions.

Theorem 6. Let 𝐯_𝑖(𝑡)=∫_𝐺𝐯_𝐢(𝑡,𝐱)d𝐱. Suppose that the following conditions hold.(A1)There exist positive-definite functions 𝑉_𝑖(𝑡,𝜉)∈𝐶^1,2(ℝ₊×ℝ^𝑛𝑖;ℝ₊),   functions  𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), and constants 𝑎_𝑖𝑗≥0 satisfying the following.(I)𝑉_𝑖(𝑡,𝐯_𝑖(𝑡))∈𝐶^1,2[ℝ₊×𝑆_ℎ,ℝ],  (𝑆_ℎ={𝜁∣|𝜁(⋅)|<ℎ}) meeting 𝑉_𝑖(𝑡,0)=0.(II)𝜇₁(|𝐲(𝑡,𝐱)|_𝐺)≤𝑉(𝑡,𝐯(𝑡)), where 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) and 𝑐_𝑖 is defined as (8); 𝜇₁∈𝒦 is a convex function.(III)ℒ𝑉_𝑖(𝑡,𝐯_𝑖)≤∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡),𝑡≥0, 𝑖=1,2,…,𝑁.(A2)Along each directed cycle ℂ of the weighted digraph (𝔾,𝐴) in which 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 there is _{(ℂ)𝑖,𝑗∈𝐸}𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡≤0,𝑡≥0. (15)Then function 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) is a Lyapunov-A function for (12). Furthermore, the trivial solution of (12) is stable in mean as to partial variables 𝐲.

Proof. It is not difficult to find that d_𝐺𝐯_𝐢()=𝑡,𝐱d𝐱_𝐺𝜌_𝑖(𝑡)Δ𝐯_𝑖(𝑡,𝐱)+𝑓_𝑖𝑡,𝐱,𝐯_𝑖(+𝑡,𝐱)_𝑁𝑗=1𝐻_𝑖𝑗𝐯_𝑖,𝐯_𝑗+,𝑡d𝐱d𝑡_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑊(𝑡). (16) Applying Green formula, we deduce ∫_𝐺Δ𝐯_𝐢(𝑡,𝐱)d𝐱=∫_𝜕𝐺(𝜕𝐯_𝐢/𝜕𝒩)d𝑠=0. Hence, (16) can be rewritten as d_𝐺𝐯_𝐢()=𝑡,𝐱d𝐱_𝐺𝑓_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝐻_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑡+_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑊(𝑡). (17) Since 𝑉_𝑖(𝑡₀,𝐯_𝑖) is continuous and 𝑉_𝑖(𝑡₀,0)=0, we have 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) that is continuous and 𝑉(𝑡,0)=0. Hence, there exists 𝛿(𝑡₀,𝜀) such that 𝑉(𝑡₀,𝐯₀)<𝜇₁(𝜀) when |𝐯₀|<𝛿. Choosing ∀𝐯_𝑖0∶|𝐯_𝑖0|<𝛿/𝑁 and applying Ito differential formula to 𝑉_𝑖(𝑡,𝐯_𝑖) along the trajectory of system (17) yields d𝑉_𝑖𝑡,𝐯_𝑖(17)=𝜕𝑉_𝑖𝑡,𝐯_𝑖𝜕𝑡d𝑡+𝜕𝑉𝑡,𝐯_𝑖𝜕𝐯_𝑖𝑑𝐯_𝑖+12Trace_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡^⊺𝜕d𝐱²𝑉𝑡,𝐯_𝑖𝜕𝐯_𝑖𝜕𝐯_𝑖×_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑡=ℒ𝑉𝑡,𝐯_𝑖d𝑡+𝜕𝑉𝑡,𝐯_𝑖𝜕𝐯_𝑖×_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑊(𝑡), (18) where ℒ𝑉_𝑖𝑡,𝐯_𝑖≜𝜕𝑉_𝑖𝑡,𝐯_𝑖+𝜕𝑡𝜕𝑉_𝑖𝑡,𝐯_𝑖𝜕𝐯_𝑖^𝑇×𝑓_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝐻_𝑖𝑗𝐯_𝑖,𝐯_𝑗+1,𝑡2Trace𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡^𝑇×𝑉_𝑖𝑡,𝐯_𝑖𝐯^′′_𝑖𝐯𝑗×𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡 (19) for ∀𝑡≥0. Hence, 𝑉_𝑖𝑡,𝐯_𝑖𝑡,𝑡₀,𝐯_𝐢𝟎=𝑉_𝑖𝑡₀,𝐯_𝐢𝟎+^𝑡_𝑡0ℒ𝑉_𝑖𝑠,𝐯_𝑖𝑠,𝑡₀,𝐯_𝐢𝟎+d𝑠_𝑡𝑡0𝜕𝑉_𝑖𝑠,𝐯_𝑖𝜕𝐯_𝑖×_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑤(𝑠). (20) As  𝜕𝑉_𝑖(𝑡,𝐯_𝑖)/𝜕𝐯_𝑖  is continuous on 𝑅₊×𝑆_ℎ and 𝑔_𝑖(𝑡,𝐱,𝐯_𝑖(𝑡,𝐱))+∑^𝑁_𝑗=1𝑁_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡) satisfies integral linear growth condition, there must exist constant 𝐿₁>0 such that |𝜕𝑉_𝑖(𝑡,𝐯_𝑖)/𝜕𝐯_𝑖|≤𝐿₁, so 𝜕𝑉_𝑖𝑠,𝐯_𝑖𝜕𝐯_𝑖𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱≤𝐿₁𝐾1+_𝐺𝐯𝑡,𝐱,𝑡₀,𝐯_𝑖0d𝐱≤𝐿₁𝐾(1+ℎ). (21) By Theorem 2.8 of [42, 43], we obtain ℰ^𝑡_𝑡0𝜕𝑉_𝑖𝑠,𝐯_𝑖𝜕𝐯_𝑖𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑤(𝑠)=0. (22) On the other hand, by (A1)(III), it is derived that ℒ𝑉(𝑡,𝐯)=^𝑁_𝑖=1𝑐_𝑖ℒ𝑉_𝑖𝑡,𝐯_𝑖≤_𝑖,𝑗=1𝑐_𝑖𝑎_𝑖𝑗𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡. (23) Making use of Lemma 5 with weighted digraph (𝒢,𝐴), it yields _𝑖,𝑗=1𝑐_𝑖𝑎_𝑖𝑗𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡=_𝒬∈ℚ𝑊(𝒬)_{𝐶𝑖,𝑗∈𝐸𝒬}𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡. (24) In view of condition (A2) and a fact 𝑊(𝒬)>0, we get ℒ𝑉(𝑡,𝐯)≤_𝒬∈ℚ𝑊(𝒬)_{𝐶𝑖,𝑗∈𝐸𝒬}𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡≤0. (25) Thus 𝑉(𝑡,𝐯) is a Lyapunov-A function for (12). Taking the mathematical expectation at the two sides of (20) and using (22), (23), and (25) we have ℰ𝑉𝑡,𝐱,𝐯𝑡,𝑡₀,𝐯₀≤𝑉𝑡₀,𝐯₀,𝑡≥0. (26) Combine Jensen inequality and condition (A1)(II), 𝜇₁ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≤ℰ𝜇₁𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≤ℰ𝑉𝑡,𝐯𝑡,𝑡₀,𝐯₀≤𝑉𝑡₀,𝐯₀≤𝜇₁(𝜀). (27) Therefore ℰ{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}<𝜀. The proof is complete.

Note that if (𝒢,𝐴) is balanced, then _𝑖,𝑗=1𝑐_𝑖𝑎_𝑖𝑗𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗=1,𝑡2_𝒬∈ℚ𝑊(𝒬)_{𝐶𝑖,𝑗∈𝐸𝒬}𝐹_𝑖𝑗𝐯_𝑗,𝐯_𝑖,𝑡+𝐹_𝑗𝑖𝐯_𝑖,𝐯_𝑗,𝑡. (28) In this case, condition (A2) is replaced by the following: _{(𝑖,𝑗)∈𝐸(𝐶𝒬)}𝐹_𝑖𝑗𝐯_𝑗,𝐯_𝑖,𝑡+𝐹_𝑗𝑖𝐯_𝑖,𝐯_𝑗,𝑡≤0. (29) Consequently, we get the following corollary.

Corollary 7. Suppose that (𝒢,𝐴) is balanced. Then the conclusion of Theorem 6 holds if  (15) is replaced by (29).

Remark 8. Partial stability technique (stability of part of the variables) is most useful when a fully stabilized system losses some control engines or some phase variables are not actively controlled. However, the CSRDSNs are too complicated to derive the analytical solution. Therefore, it is of importance to work on the qualitative analysis of the system and how to construct an appropriate Lyapunov function is a key step. The proof shows that, if each vertex system of (12) has a globally stable trivial solution and possesses a Lyapunov function 𝑉_𝑖, then the Lyapunov function for (12) can be systematically constructed by using individual 𝑉_𝑖. Our results are new and extend some findings in [38], because our stability principle has a close relation to the topology property of the network.

Theorem 9. Assume that condition (A1) of Theorem 6 is substituted by the following.(A3)There exist positive-definite functions 𝑉_𝑖(𝑡,𝜉)∈𝐶^1,2(ℝ₊×ℝ^𝑛𝑖;ℝ₊), functions 𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), and constants 𝑎_𝑖𝑗≥0 satisfying the following.(I)𝑉_𝑖(𝑡,𝐯_𝑖(𝑡))∈𝐶^1,2[ℝ₊×𝑆_ℎ,ℝ],(𝑆_ℎ={𝜁∣|𝜁(⋅)|<ℎ}). (Here, function 𝑉_𝑖(𝑡,𝐯_𝑖(𝑡)) gets rid of the restriction of 𝑉_𝑖(𝑡,0)=0).(II)𝜇₂(|𝐲(𝑡,𝐱)|_𝐺)≤𝑉(𝑡,𝐯(𝑡))≤𝜇₃(|𝐯(𝑡,𝐱)|_𝐺), where 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖), 𝑐_𝑖 is defined as (8), 𝜇₂,𝜇₃∈𝒦, and 𝜇₂ is a convex function.(III)ℒ𝑉_𝑖(𝑡,𝐯_𝑖)≤∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡),𝑡≥0,𝑖=1,2,…,𝑁. Other conditions remain the same. Then function 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) is a Lyapunov-A function for (12). Furthermore, the trivial solution of (12) is uniformly stable in mean as to partial variables 𝐲.

Proof. Since 𝑉_𝑖(𝑡,𝐯_𝑖) is continuous, we have 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) that is continuous. Similar to the proof of Theorem 6, we can obtain that 𝑉(𝑡,𝐯) is a Lyapunov-A function for (12) and ℰ𝑉𝑡,𝐯𝑡,𝑡₀,𝐯₀≤𝑉𝑡₀,𝐯₀,𝑡≥0. (30) Let 𝛿(𝜀)=𝜇₃⁻¹𝜇₂(𝜀). It follows from Jensen inequality and condition (A3)(II) that, for 𝐯₀∈𝑆_𝛿={𝐯∣|𝐯(⋅,𝐱)|_𝐺<𝛿(𝜀)}, we have 𝜇₂ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≤ℰ𝑉𝑡,𝐯𝑡,𝑡₀,𝐯₀≤𝑉𝑡₀,𝐯₀≤𝜇₃𝐯₀(⋅,𝐱)_𝐺<𝜇₂(𝜀). (31) Therefore, we derive 𝐸{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}<𝜀 as required.

Corollary 10. Suppose that (𝒢,𝐴) is balanced. Then the conclusion of Theorem 9 holds if (15) is replaced by (29).

3.2. Asymptotical Stability in Mean

In this section, some sufficient principles are established for asymptotic stability in mean and uniformly asymptotic stability in mean as to partial variables.

Theorem 11. Let 𝐯_𝑖(𝑡)=∫_𝐺𝐯_𝐢(𝑡,𝐱)d𝐱. Suppose that the following conditions hold.(B1)There exist positive-definite functions 𝑉_𝑖(𝑡,𝜉)∈𝐶^1,2(ℝ₊×ℝ^𝑛𝑖;ℝ₊), functions 𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), and constants 𝑎_𝑖𝑗≥0 satisfying the following.(I)𝑉_𝑖(𝑡,𝐯_𝑖(𝑡))∈𝐶^1,2[ℝ₊×𝑆_ℎ,ℝ],(𝑆_ℎ={𝜁∣|𝜁(⋅)|<ℎ}) meeting 𝑉_𝑖(𝑡,0)=0.(II)𝜇₄(|𝐲(𝑡,𝐱)|_𝐺)≤𝑉(𝑡,𝐯(𝑡)), where 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) and 𝑐_𝑖 is defined as (8);𝜇₄∈𝒦 is a convex function.(III)ℒ𝑉_𝑖(𝑡,𝐯_𝑖)≤−𝑏_𝑖𝑉_𝑖(𝑡,𝐯_𝑖)+∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), constants 𝑏_𝑖≥0, 𝑡≥0,𝑖=1,2,…,𝑁.(B2)Condition (A2) holds, if (𝒢,𝐴) is balanced and (29) holds.
Then, function 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) is a Lyapunov-B function for (12). Consequently, the trivial solution of (12) is asymptotically stable in mean as to partial variables 𝐲.

Proof. We can show in the same way as in the proof of Theorem 6 that ℒ𝑉(𝑡,𝐯)=^𝑁_𝑖=1𝑐_𝑖ℒ𝑉_𝑖𝑡,𝐯_𝑖≤−𝑏𝑉(𝑡,𝐯), (32) where 𝑏=min{𝑏₁,𝑏₂,…,𝑏_𝑁}. Hence, we conclude that function 𝑉(𝑡,𝐯) is a Lyapunov-B function for (12). From Theorem 6, it is easy to derive that the trivial solution of system (12) is stable in mean as to partial variables 𝐲. So the following task is to prove lim_𝑡→∞ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺=0 (33) only. Similar to the proof of Theorem 6, it is not difficult to derive ℰ𝑉_𝑖𝑡,𝐯_𝑖𝑡,𝑡₀,𝐯_𝑖0=𝑉_𝑖𝑡₀,𝐯_𝑖0+^𝑡_𝑡0ℰℒ𝑉_𝑖𝑠,𝐯_𝑖𝑠,𝑡₀,𝐯_𝑖0d𝑠. (34) Then, we obtain that 𝑉(𝑡,𝐯) is a Lyapunov-A function for (12) and ℰ𝑉𝑡,𝐯𝑡,𝑡₀,𝐯₀≤𝑉𝑡₀,𝐯₀,𝑡≥0. (35) Here we need to reduce to absurdity. Suppose lim_𝑡→∞ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≠0. (36) Instead lim_𝑡→∞ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≜𝜆_∞>0. (37) Combining condition (B1)(II) of Theorem 11, we obtain 𝜇₄ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≤ℰ𝑉𝑡,𝐯𝑡,𝑡₀,𝐯₀≤𝑉𝑡₀,𝐯₀−^𝑡_𝑡0𝜇₄ℰ𝐲𝑠,𝐱,𝑡₀,𝐯_0𝐺d𝑠. (38) Therefore, 0<lim_𝑡→∞𝛼ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≤𝑉𝑡₀,𝐯₀−^𝑡_𝑡0lim_𝑡→∞𝜇₄ℰ𝐲𝑠,𝐱,𝑡₀,𝐯_0𝐺𝑡d𝑠=𝑉₀,𝐯₀−𝜇₄𝜆_∞𝑡−𝑡₀. (39) However, it is obvious that (39) can not be satisfied as 𝑡≫𝑡₀. Thus, hypothesis lim_𝑡→∞ℰ{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}≠0 does not come into existence. It should be lim_𝑡→∞ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺=0, (40) as required; that is, the trivial solution of system (12) is asymptotically stable in mean as to partial variables 𝐲. This completes the proof.

Theorem 12. Let 𝐯_𝑖(𝑡)=∫_𝐺𝐯_𝐢(𝑡,𝐱)d𝐱. Suppose that the following conditions hold. (B3)There exist positive-definite functions 𝑉_𝑖(𝑡,𝜉)∈𝐶^1,2(ℝ₊×ℝ^𝑛𝑖;ℝ₊), functions 𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), and constants 𝑎_𝑖𝑗≥0 satisfying the following.(I)𝑉_𝑖(𝑡,𝐯_𝑖(𝑡))∈𝐶^1,2[ℝ₊×𝑆_ℎ,ℝ],(𝑆_ℎ={𝜁∣|𝜁(⋅)|<ℎ}) meeting 𝑉_𝑖(𝑡,0)=0.(II)𝜇₄(|𝐲(𝑡,𝐱)|_𝐺)≤𝑉(𝑡,𝐯(𝑡)), where 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) and 𝑐_𝑖 is defined as (8); 𝜇₄∈𝒦 is a convex function.(III)ℒ𝑉_𝑖(𝑡,𝐯_𝑖)≤−𝜇₅(|𝐲(𝑡,𝐱)|_𝐺)+∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), 𝜇₅∈𝒦 is a convex function, 𝑡≥0, 𝑖=1,2,…,𝑁.(B2)Condition (A2) holds, if (𝒢,𝐴) is balanced and (29) holds.Then, function 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) is a Lyapunov-A function for (12). Consequently, the trivial solution of (12) is asymptotically stable in mean as to partial variables 𝐲.

Remark 13. Similar to the proof of Theorems 6 and 11, we can easily proof Theorem 12. Please note that, in Theorem 11, we can construct a Lyapunov-B function for (12), but in Theorem 12 only a Lyapunov-A function for (12). Further, note the fact that −𝜇₅(|𝐯(𝑡,𝐱)|_𝐺)≤−𝜇₅(|𝐲(𝑡,𝐱)|_𝐺). We can draw the following theorem immediately.

Corollary 14. Suppose that in Theorem 12, condition (B3)(III) is replaced by (III)^′ℒ𝑉_𝑖𝑡,𝐯_𝑖≤−𝜇₅|𝐯(𝑡,𝐱)|_𝐺+^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝜇,𝑡₅∈𝒦isaconvexfunction,𝑡≥0,𝑖=1,2,…,𝑁. (41) Other conditions remain the same. Then the conclusion of Theorem 12 holds.

The foregoing are all concerned with asymptotic stability as to partial variables. The following is focused on uniformly asymptotic stability as to partial variables.

Theorem 15. Assume that condition (B3) of Theorem 12 is substituted by the following.(B4)There exist positive-definite functions 𝑉_𝑖(𝑡,𝜉)∈𝐶^1,2(ℝ₊×ℝ^𝑛𝑖;ℝ₊), functions 𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), and constants 𝑎_𝑖𝑗≥0 satisfying the following.(I)𝑉_𝑖(𝑡,𝐯_𝑖(𝑡))∈𝐶^1,2[ℝ₊×𝑆_ℎ,ℝ],(𝑆_ℎ={𝜁∣|𝜁(⋅)|<ℎ}). (Here, function 𝑉_𝑖(𝑡,𝐯_𝑖(𝑡)) gets rid of the restriction of 𝑉_𝑖(𝑡,0)=0).(II)𝜇₆(|𝐲(𝑡,𝐱)|_𝐺)≤𝑉(𝑡,𝐯(𝑡))≤𝜇₇(|𝐯(𝑡,𝐱)|_𝐺), where 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖), 𝑐_𝑖 is defined as (8), 𝜇₇∈𝒦, and 𝜇₆∈𝒦 is a convex function.(III)ℒ𝑉_𝑖(𝑡,𝐯_𝑖)≤−𝜇₈(|𝐲(𝑡,𝐱)|_𝐺)+∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), 𝜇₈∈𝒦 is a convex function, 𝑡≥0,𝑖=1,2,…,𝑁.Other conditions remain the same. Then function 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) is a Lyapunov-A function for (12). Furthermore, the trivial solution of (12) is uniformly asymptotically stable in mean as to partial variables 𝐲.

Proof. Because the conditions of Theorem 15 cover those of Theorem 9, it is obvious that the trivial solution of system (1) is uniformly stable as to partial variables 𝐲. Now, we only need to prove lim_𝑡→∞ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺=0. (42) Similar to the proof of Theorem 11, here we need to reduce to absurdity. Suppose lim_𝑡→∞ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≠0. (43) Instead lim_𝑡→∞ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≜𝜆_∞>0. (44) Combining condition (B1)(II) of Theorem 15, we obtain 𝜇₆ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≤ℰ𝑉𝑡,𝐯𝑡,𝑡₀,𝐯₀≤𝑉𝑡₀,𝐯₀−^𝑡_𝑡0𝜇₆ℰ𝐲𝑠,𝐱,𝑡₀,𝐯_0𝐺d𝑠. (45) Therefore, 0<lim_𝑡→∞𝛼ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺≤𝑉𝑡₀,𝐯₀−^𝑡_𝑡0lim_𝑡→∞𝜇₆ℰ𝐲𝑠,𝐱,𝑡₀,𝐯_0𝐺𝑡d𝑠=𝑉₀,𝐯₀−𝜇₆𝜆_∞𝑡−𝑡₀. (46) However, it is obvious that (46) cannot be satisfied as 𝑡≫𝑡₀. Thus, hypothesis lim_𝑡→∞ℰ{|𝐲(𝑡,𝐱,𝑡₀,𝐯₀)|_𝐺}≠0 does not come into existence. It should be lim_𝑡→∞ℰ𝐲𝑡,𝐱,𝑡₀,𝐯_0𝐺=0, (47) as required; that is, the trivial solution of system (12) is uniformly asymptotically stable in mean as to partial variables 𝐲. This completes the proof.

Note the fact that −𝜇₈(|𝐯(𝑡,𝐱)|_𝐺)≤−𝜇₈(|𝐲(𝑡,𝐱)|_𝐺). We can still derive another conclusion as follows.

Corollary 16. Suppose that, in Theorem 15, condition (B4)(III) is replaced by (III)^′ℒ𝑉_𝑖𝑡,𝐯_𝑖≤−𝜇₈|𝐯(𝑡,𝐱)|_𝐺+^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝜇,𝑡₈∈𝒦𝑖𝑠𝑎𝑐𝑜𝑛𝑣𝑒𝑥𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛,𝑡≥0,𝑖=1,2,…,𝑁. (48) Other conditions remain the same. Then the conclusion of Theorem 15 holds.

Theorem 17. Assume that condition (B2) of Theorem 11 is substituted by the following.(B5)There exist positive-definite functions 𝑉_𝑖(𝑡,𝜉)∈𝐶^1,2(ℝ₊×ℝ^𝑛𝑖;ℝ₊), functions 𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), and constants 𝑎_𝑖𝑗≥0 satisfying the following.(I)𝑉_𝑖(𝑡,𝐯_𝑖(𝑡))∈𝐶^1,2[ℝ₊×𝑆_ℎ,ℝ],(𝑆_ℎ={𝜁∣|𝜁(⋅)|<ℎ}). (Here, function 𝑉_𝑖(𝑡,𝐯_𝑖(𝑡)) gets rid of the restriction of   𝑉_𝑖(𝑡,0)=0).(II)𝜇₉(|𝐲(𝑡,𝐱)|_𝐺)≤𝑉(𝑡,𝐯(𝑡))≤𝜇₁₀(|𝐯(𝑡,𝐱)|_𝐺), where 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖), 𝑐_𝑖 is defined as (8), 𝜇₁₀∈𝒦, and 𝜇₉∈𝒦 is a convex function.(III)ℒ𝑉_𝑖(𝑡,𝐯_𝑖)≤−𝑏_𝑖𝑉_𝑖(𝑡,𝐯_𝑖)+∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), constants 𝑏_𝑖≥0, 𝑡≥0,𝑖=1,2,…,𝑁.Other conditions remain the same. Then function 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) is a Lyapunov-B function for (12). Furthermore, the trivial solution of (12) is uniformly asymptotically stable in mean as to partial variables 𝐲.

3.3. Exponential Stability in Mean

In this section, we will discuss exponential stability in mean of the trivial solution of system (12) as to partial variables.

Theorem 18. Let 𝐯_𝑖(𝑡)=∫_𝐺𝐯_𝑖(𝑡,𝐱)d𝐱. Suppose that the following conditions hold.(C1)There exist positive-definite functions 𝑉_𝑖(𝑡,𝜉)∈𝐶^1,2(ℝ₊×ℝ^𝑛𝑖;ℝ₊), functions 𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), and constants 𝑎_𝑖𝑗≥0 satisfying the following.(I)𝑉_𝑖(𝑡,𝐯_𝑖(𝑡))∈𝐶^1,2[ℝ₊×𝑆_ℎ,ℝ],(𝑆_ℎ={𝜁∣|𝜁(⋅)|<ℎ}). (Here, function𝑉_𝑖(𝑡,𝐯_𝑖(𝑡)) gets rid of the restriction of  𝑉_𝑖(𝑡,0)=0).(II)𝑐₁|𝐲(𝑡,𝐱)|_𝐺)≤𝑉(𝑡,𝐯(𝑡))≤𝑐₂|𝐯(𝑡,𝐱)|_𝐺, where 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖), 𝑐_𝑖 is defined as (8), and 𝑐₁ and 𝑐₂ are positive constants.(III)ℒ𝑉_𝑖(𝑡,𝐯_𝑖)≤−𝑏̂_𝑖𝑉_𝑖(𝑡,𝐯_𝑖)+∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), constants 𝑏̂_𝑖>0, 𝑡≥0,𝑖=1,2,…,𝑁.(C2)Condition (A2) holds, if (𝒢,𝐴) is balanced and (29) holds.
Then, function 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) is a Lyapunov-B function for (12), and 𝐸|𝐲(𝑡,𝐱)|_𝐺≤𝑐₂𝑐₁𝑣_0𝐺𝑒^{−̂𝑏(𝑡−𝑡0)},𝑡≥𝑡₀, (49) where  𝑏̂=min{𝑏̂₁,…,𝑏̂_𝑁}; that is, the trivial solution of system (12) is exponentially stable in mean as to partial variables 𝐲.

Proof. We can show in the same way as in the proof of Theorem 6 that ℒ𝑉(𝑡,𝐯)=^𝑁_𝑖=1𝑐_𝑖ℒ𝑉_𝑖𝑡,𝐯_𝑖≤−𝑏̂𝑉(𝑡,𝐯), (50) where 𝑏̂=min{𝑏̂₁,𝑏̂₂,…,𝑏̂_𝑁}. Hence, we conclude that function 𝑉(𝑡,𝐯) is a Lyapunov-B function for (12). Integrating system (12) as to spatial variables 𝐱 gives d_𝐺𝐯_𝐢()=𝑡,𝐱d𝐱_𝐺𝜌_𝑖(𝑡)Δ𝐯_𝑖(𝑡,𝐱)+𝑓_𝑖𝑡,𝐱,𝐯_𝑖(+𝑡,𝐱)_𝑁𝑗=1𝐻_𝑖𝑗𝐯_𝑖,𝐯_𝑗+,𝑡d𝐱d𝑡_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑊(𝑡). (51) Combining Green formula and boundary condition we deduce ∫_𝐺Δ𝐯_𝐢(𝑡,𝐱)d𝐱=∫_𝜕𝐺(𝜕𝐯_𝐢/𝜕𝒩)d𝑠=0. Hence, (51) can be rewritten as d_𝐺𝐯_𝐢()=𝑡,𝐱d𝐱_𝐺𝑓_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝐻_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑡+_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑊(𝑡). (52) Since 𝑉_𝑖(𝑡₀,𝐯_𝑖) is continuous and 𝑉_𝑖(𝑡₀,0)=0, we have 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) that is continuous and 𝑉(𝑡,0)=0. Hence, there exists 𝛿(𝑡₀,𝜀) such that 𝑉(𝑡₀,𝐯₀)<𝜇₁(𝜀) when |𝐯₀|<𝛿. Choosing ∀𝐯_𝑖0∶|𝐯_𝑖0|<𝛿/𝑁 and applying It𝑜̂ differential formula to 𝑉_𝑖(𝑡,𝐯_𝑖) along the trajectory of system (17) yields d𝑉_𝑖𝑡,𝐯_𝑖(17)=𝜕𝑉_𝑖𝑡,𝐯_𝑖𝜕𝑡d𝑡+𝜕𝑉𝑡,𝐯_𝑖𝜕𝑣_𝑖d𝐯_𝑖+12Trace_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡^⊺𝜕d𝐱²𝑉𝑡,𝐯_𝑖𝜕𝐯_𝑖𝜕𝐯_𝑖×_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑡=ℒ𝑉𝑡,𝐯_𝑖d𝑡+𝜕𝑉𝑡,𝐯_𝑖𝜕𝐯_𝑖×_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑊(𝑡), (53) where ℒ𝑉_𝑖𝑡,𝐯_𝑖≜𝜕𝑉_𝑖𝑡,𝐯_𝑖+𝜕𝑡𝜕𝑉_𝑖𝑡,𝐯_𝑖𝜕𝐯_𝑖^𝑇×𝑓_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝐻_𝑖𝑗𝐯_𝑖,𝐯_𝑗+1,𝑡2Trace𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡^𝑇×𝑉_𝑖𝑡,𝐯_𝑖𝐯^′′_𝑖𝐯𝑗×𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡, (54) for ∀𝑡≥0. Hence, 𝑉_𝑖𝑡,𝐯_𝑖𝑡,𝑡₀,𝐯_𝐢𝟎=𝑉_𝑖𝑡₀,𝐯_𝐢𝟎+^𝑡_𝑡0ℒ𝑉_𝑖𝑠,𝐯_𝑖𝑠,𝑡₀,𝐯_𝐢𝟎+d𝑠_𝑡𝑡0𝜕𝑉_𝑖𝑠,𝐯_𝑖𝜕𝐯_𝑖𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑤(𝑠). (55) Build a Lyapunov function with the form 𝑉_𝑖=𝑒^{̂𝑏(𝑡−𝑡0)}𝑉_𝑖𝑡,𝐯_𝑖. (56) Applying Ito formula again, we obtain 𝑑𝑉_𝑖=𝑏̂𝑒^{̂𝑏(𝑡−𝑡0)}𝑉_𝑖𝑡,𝐯_𝑖d𝑡+𝑒^{̂𝑏(𝑡−𝑡0)}𝑑𝑉_𝑖=𝑒^{̂𝑏(𝑡−𝑡0)}𝑏̂𝑉_𝑖𝑡,𝐯_𝑖+ℒ𝑉_𝑖𝑡,𝐯_𝑖d𝑡+𝑒^{̂𝑏(𝑡−𝑡0)}×^𝑡_𝑡0𝜕𝑉_𝑖𝑠,𝐯_𝑖𝜕𝐯_𝑖𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑤(𝑡). (57) For any 𝑛≥|𝐯₀|_𝐺, define a stop-time 𝜏_𝑛=inf𝑡≥𝑡₀∶|𝐯(𝑡,𝐱)|_𝐺≥𝑛. (58) It follows from integrating (57) as to 𝑡 from 𝑡₀ to 𝑡∧𝜏_𝑛𝑒^{̂𝑏(𝑡∧𝜏𝑛−𝑡0)}𝑉_𝑖𝑡∧𝜏_𝑛,𝐯_𝑖𝑡∧𝜏_𝑛=𝑉_𝑖𝑡₀,𝑣₀+^{𝑡∧𝜏𝑛}_𝑡0𝑒^{̂𝑏(𝑠−𝑡0)}𝑏̂𝑉_𝑖𝑠,𝐯_𝑖+ℒ𝑉_𝑖𝑠,𝐯_𝑖+d𝑠_{𝑡∧𝜏𝑛𝑡0}𝑒^{̂𝑏(𝑠−𝑡0)}𝜕𝑉_𝑖𝑠,𝐯_𝑖𝜕𝐯_𝑖×_𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑤(𝑠). (59) As  𝜕𝑉_𝑖(𝑡,𝐯_𝑖)/𝜕𝐯_𝑖  is continuous on 𝑅₊×𝑆_ℎ and 𝑔_𝑖(𝑡,𝐱,𝐯_𝑖(𝑡,𝐱))+∑^𝑁_𝑗=1𝑁_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡) satisfies integral linear growth condition, there must exist constant 𝐿₁>0 such that |𝜕𝑉_𝑖(𝑡,𝐯_𝑖)/𝜕𝐯_𝑖|≤𝐿₁, so 𝜕𝑉_𝑖𝑠,𝐯_𝑖𝜕𝐯_𝑖𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖(𝑡,𝐱)+^𝑁_𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱≤𝐿₁𝐾1+_𝐺𝐯𝑡,𝐱,𝑡₀,𝐯_𝑖0d𝐱≤𝐿₁𝐾(1+ℎ). (60) By Theorem 2.8 of [35], we obtain ℰ^𝑡_𝑡0𝜕𝑉_𝑖𝑠,𝐯_𝑖𝜕𝐯_𝑖𝐺𝑔_𝑖𝑡,𝐱,𝐯_𝑖()+𝑡,𝐱_𝑁𝑗=1𝑁_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡d𝐱d𝑤(𝑠)=0. (61) Thus, it can be derived from taking mathematical expectation at both sides of (59) that ℰ𝑒^{̂𝑏(𝑡∧𝜏𝑛−𝑡0)}𝑉_𝑖𝑡∧𝜏_𝑛,𝐯_𝑖𝑡∧𝜏_𝑛=𝑉_𝑖𝑡₀,𝑣₀+ℰ^{𝑡∧𝜏𝑛}_𝑡0𝑒^{̂𝑏(𝑠−𝑡0)}𝑏̂𝑉_𝑖𝑠,𝐯_𝑖+ℒ𝑉_𝑖𝑠,𝐯_𝑖d𝑠. (62) On the other hand, by (B5)(III), it is derived that ℒ𝑉(𝑡,𝐯)=^𝑁_𝑖=1𝑐_𝑖ℒ𝑉_𝑖𝑡,𝐯_𝑖≤−^𝑁_𝑖=1𝑐_𝑖𝑏̂_𝑖𝑉_𝑖𝑡,𝐯_𝑖+_𝑖,𝑗=1𝑐_𝑖𝑎_𝑖𝑗𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡. (63) Making use of Lemma 5 with weighted digraph (𝒢,𝐴), it yields _𝑖,𝑗=1𝑐_𝑖𝑎_𝑖𝑗𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡=_𝒬∈ℚ𝑊(𝒬)_{𝐶𝑖,𝑗∈𝐸𝒬}𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡. (64) In view of Condition (A2) and the fact that 𝑊(𝒬)>0, we get ℒ𝑉(𝑡,𝐯)≤−^𝑁_𝑖=1𝑐_𝑖𝑏̂_𝑖𝑉_𝑖𝑡,𝐯_𝑖+_𝒬∈ℚ𝑊(𝒬)_{𝐶𝑖,𝑗∈𝐸𝒬}𝐹_𝑖𝑗𝐯_𝑖,𝐯_𝑗,𝑡≤−^𝑁_𝑖=1𝑐_𝑖𝑏̂_𝑖𝑉_𝑖𝑡,𝐯_𝑖<−𝑏̂𝑉(𝑡,𝐯). (65) Thus 𝑉(𝑡,𝐯) is a Lyapunov-B function for (12). Taking the mathematical expectation at the two sides of (59) and using (61), (63), and (65) and combining conditions (C1), (I), and (II), we have 𝑐₁𝑒^{̂𝑏(𝑡∧𝜏𝑛−𝑡0)}ℰ𝐲𝑡∧𝜏_𝑛,𝐱_𝐺≤ℰ𝑒^{̂𝑏(𝑡∧𝜏𝑛−𝑡0)}𝑉𝑡∧𝜏_𝑛,𝐯≤𝑉𝑡₀,𝐯₀. (66) Obviously 𝜏_𝑛→∞ when 𝑛→∞, we have 𝑐₁𝑒^{̂𝑏(𝑡−𝑡0)}ℰ|𝐲(𝑡,𝐱)|_𝐺≤𝑐₂𝐯_0𝐺. (67) Therefore, we get ℰ|𝐲(𝑡,𝐱)|_𝐺≤𝑐₂𝑐₁𝑣_0𝐺𝑒^{−̂𝑏(𝑡−𝑡0)},𝑡≥0, (68) as required.
According to Theorem 18, it is easy to reach the following theorem.

Theorem 19. Let 𝐯_𝑖(𝑡)=∫_𝐺𝐯_𝑖(𝑡,𝐱)d𝐱. Suppose that the following conditions hold.(C1)There exist positive-definite functions 𝑉_𝑖(𝑡,𝜉)∈𝐶^1,2(ℝ₊×ℝ^𝑛𝑖;ℝ₊), functions 𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), and constants 𝑎_𝑖𝑗≥0 satisfying the following.(I)𝑉_𝑖(𝑡,𝐯_𝑖(𝑡))∈𝐶^1,2[ℝ₊×𝑆_ℎ,ℝ],(𝑆_ℎ={𝜁∣|𝜁(⋅)|<ℎ}). (Here, function 𝑉_𝑖(𝑡,𝐯_𝑖(𝑡)) gets rid of the restriction of  𝑉_𝑖(𝑡,0)=0).(II)𝑐₁̂|𝐲(𝑡,𝐱)|_𝐺≤𝑉(𝑡,𝐯(𝑡)≤𝑐₂̂|𝐯(𝑡,𝐱)|_𝐺, where 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖), 𝑐_𝑖 is defined as (8), and 𝑐₁̂ and 𝑐₂̂<1 are positive constants.(III)ℒ𝑉_𝑖(𝑡,𝐯_𝑖)≤−𝑏̂_𝑖𝑉_𝑖(𝑡,𝐯_𝑖)+∑^𝑛_𝑗=1𝑎_𝑖𝑗𝐹_𝑖𝑗(𝐯_𝑖,𝐯_𝑗,𝑡), constants 𝑏̂_𝑖>0, 𝑡≥0, 𝑖=1,2,…,𝑁.(C2)Condition (A2) holds, if (𝒢,𝐴) is balanced and (29) holds.Then, function 𝑉(𝑡,𝐯)≜∑^𝑁_𝑖=1𝑐_𝑖𝑉_𝑖(𝑡,𝐯_𝑖) is a Lyapunov-A function for (12), and 𝐸|𝐲(𝑡,𝐱)|_𝐺≤𝑐₂𝑐₁𝑣_0𝐺𝑒^{−̂𝑏(𝑡−𝑡0)},𝑡≥𝑡₀, (69) where 𝑏̂=min{𝑏̂₁,…,𝑏̂_𝑁}; that is, the trivial solution of system (12) is exponentially stable in mean as to partial variables 𝐲.

Remark 20. In this section, we derive some novel findings on stability principles for uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, and exponential stability in mean of partial variables for CSRDSNs. The results in previous literature [38] are special cases of our findings, because our results have a close relation to the topology property of the network. When we employ Lyapunov function method to tackle the stability problems for coupled stochastic large-scale systems, the most difficult thing is to construct a Lyapunov function. We also provide a systematic method for constructing a global Lyapunov function for these CSRDSNs by using graph theory. The new method is helpful to analyze the dynamics of complex networks.

4. Example

Consider the 2-dimensional Itô SRDSMS (1) satisfying the bounded condition (2), and we assume (𝒢,𝐴) is strongly connected and balanced. Consider d𝐯₁()=𝑡,𝐱Δ𝐯₁(𝑡,𝐱)+𝐯₂(𝑡,𝐱)−²_𝑗=1𝑎_1𝑗𝐯₁(𝑡,𝐱)−𝐯_𝑗(𝑡,𝐱)d𝑡,d𝐯₂()=𝑡,𝐱Δ𝐯₂(𝑡,𝐱)−𝐯₁(𝑡,𝐱)−𝛼(𝑡)𝐯₂()+𝑡,𝐱_2𝑗=1𝑎_2𝑗𝐯₂(𝑡,𝐱)−𝐯_𝑗()−𝑡,𝐱d𝑡𝛼(𝑡)𝐯₂(𝑡,𝐱)d𝑊(𝑡). (70) Construct function 𝑉=(∫_𝐺𝐯₁(𝑡,𝐱)d𝐱)²+(∫_𝐺𝐯₂(𝑡,𝐱)d𝐱)²,  and we have ℒ𝑉=2_𝐺𝐯₁d𝐱2_𝐺𝐯₂d𝐱^𝐺𝐯₂−d𝐱_𝐺𝐯₁d𝐱−𝛼(𝑡)_𝐺𝐯₂(𝑡)d𝐱+𝛼_𝐺𝐯₂(𝑡,𝐱)d𝐱²+²_𝑗=1𝑎_𝑘𝑗𝐹_𝑘𝑗𝐯_𝑘,𝐯_𝑗(𝑡),𝑡=−𝛼_𝐺𝐯₂(𝑡,𝐱)d𝐱²<0. (71)

According to Theorem 6, we know the trial solution of system (70) is stable in mean as to partial variable 𝐯₂.

5. Conclusions

In this paper, stability in mean of partial variables for coupled stochastic reaction-diffusion systems on networks (CSRDSNs) is considered. By transforming the integral of the trajectory with respect to spatial variables as the solution of the stochastic ordinary differential equations (SODE) and using Itô formula, some novel stability principles are established for uniform stability in mean, asymptotic stability in mean, uniformly asymptotic stability in mean, and exponential stability in mean of partial variables for CSRDSNs. These stability principles have a close relation with the topology property of the network. A systematic method for constructing global Lyapunov function for these CSRDSNs is also provided by using graph theory. Our methods can be extended to deal with coupled stochastic neutral differential equations on networks. Future work is to give a systematic approach to build a Lyapunov functional for coupled Markovian switching reaction-diffusion systems on networks.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the Natural Science Foundation of Guangxi Autonomous Region (nos. 2012GXNSFBA053003, 2013YB141), the National Natural Science Foundations of China (60973048, 61272077, 60974025, 60673101, and 60939003), National 863 Plan Project (2008 AA04Z401, 2009AA043404), the Natural Science Foundation of Shandong Province (no. ZR2012FM006), the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF. 2001120), and by the Polish-Norwegian Research Programme operated by the National Centre for Research and 24 Development under the Norwegian Financial Mechanism 2009-2014 in the frame of Project Contract no. Pol-Nor/200957/47/2013.