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

Authors: Yonggui Kao; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2014)

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## 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.


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## Body

## 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 Section2, 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 Vk, then the global Lyapunov function for the CSRDSNs can be systematically produced by individual Vk. An example is provided in Section 4. Section 5 is conclusion. Notations: for convenience, we sometimes write v, vk, and vj as v(t,x), vk(t,x), and vj(t,x), respectively.


## 2. Preliminaries

A general stochastic reaction-diffusion system reads(1)dv(t,x)=[ρi(t)Δv(t,x)+f(t,x,v(t,x))]dt+g(t,x,v(t,x))dW(t),(t,x)∈ℝ+×G,
with boundary condition
(2)∂v(t,x)∂𝒩=0,(t,x)∈ℝ+×∂G,
where v=col(y,z)∈ℝn, y∈ℝm,  z∈ℝp(m+p=n), Δv=∑i=1n(∂2v/∂xi2); denote |∫G‍v(·,x)dx| by |v(·,x)|G, G={x,|x|<l<+∞}⊂Rr, and both f∈[ℝ+×G×Γ,ℝn] and g∈[ℝ+×G×Γ,ℝn×m] are Borel measurable functions; here, Γ={v,|y(·,x)|G≤H=const, |z(·,x)|G<+∞}⊂ℝn, |·| stands for vector norm, W(t)=(W1,…,Wm)⊺ is an m-dimension Brown motion with natural flow {ℱt}t≥0 defined on complete probability space (Ω,ℱ,(ℱt)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 functiong(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
(3)|g(t,x,v(t,x))|G≤L(1+|v|G),|g(t,x,v1(t,x))-g(t,x,v2(t,x))|G≤L|v1-v2|G,|f(t,x,v1(t,x))-f(t,x,v2(t,x))|G≤L|v1-v2|G,
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∈Sδ={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∈Sδ={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∈Sδ={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∈Sδ={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.
Ifμ(·)∈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 functionV(t,ξ) is said to be positive-definite if V(t,0)=0 and, for some μ∈𝒦ℛ, V(t,ξ)≥μ(|ξ|). Write C1,2(ℝ+×ℝn;ℝ+) for the family of all nonnegative functions V(t,ξ) on ℝ+×ℝn that are continuously twice differentiable in ξ and once in t. If V(t,ξ)∈C1,2(ℝ+×ℝn;ℝ+), then define an operator ℒV(t,ξ) from ℝ+×ℝn to ℝ with respect to (1) by
(4)ℒV(t,ξ)=Vt(t,ξ)+VξT(t,ξ)f(t,x,ξ)+12Trace[gT(t,x,ξ)Vξξ(x,t)g(t,x,ξ)],
where
(5)Vt(t,ξ)=∂V(t,ξ)∂t,VξT(t,ξ)=(∂V(t,ξ)∂ξ1,…,∂V(t,ξ)∂ξn),Vξξ(x,t)=(∂2V(t,ξ)∂ξi∂ξj)n×n.Applying the Itô formula to∫G‍V(t,v(t,x))dx along system (1) gives for ∀t≥t0(6)(d∫G‍V(t,v(t,x))dx)|(1)=∫G‍[(ℒV(t,v(t,x))+VvT(t,v)ρΔv(t,x))dthhh+VvT(t,v)g(t,x,v(t,x))dW(t)]dx.The existence of functionV(t,v)∈C1,2 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 aij. Here aij>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,…,im} such that its set of arcs is {(ik,ik+1):k=1,2,…,m-1}. If im=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=(aij)n×n whose entry aij 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 𝒞𝒬, let 𝒬~ be the unicyclic graph obtained by replacing 𝒞𝒬 with -𝒞𝒬. Suppose that (𝒢,A) is balanced; then W(𝒬)=W(𝒬~). The Laplacian matrix of (𝒢,A) is defined as
(7)𝒥=(∑i≠1‍a1i-a12⋯-a1n-a21∑i≠2‍a2i⋯-a2n⋮⋮⋱⋮-an1-an2⋯∑i≠n‍aik).
Let ci denote the cofactor of the ith diagonal element of 𝒥.Lemma 4 ([34] Kirchhoffs Matrix Tree Theorem).
Assumen≥2. Then
(8)ci=∑𝒯∈𝕋i‍W(𝒯),i=1,2,…,N,
where 𝕋i 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 ci>0 for ci=1,2,…,n.Lemma 5 (see [11]).
Assumen≥2. Let ck be given in (1). Then the following identity holds:
(9)∑i,j=1n‍ciaijFkj(xi,xj)=∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍Fij(xi,xj).
Here  Fij(xi,xj),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 C𝒬 denotes the directed cycle of 𝒬.


## 3. Main Results

To begin with our main results, we will give an SCEDSN represented by digraph𝒢 with N vertices, N≥2. In ith vertex it is assigned a stochastic reaction-diffusion system
(10)dvi(t,x)=[ρi(t)Δvi(t,x)+fi(t,x,vi(t,x))]dt+gi(t,x,vi(t,x))dW(t),(t,x)∈ℝ+×G,
where vi(t,x)=col(yi,zi)∈ℝni,yi∈ℝmi,zi∈ℝpi(mi+pi=ni),fi:ℝ+×G×ℝni→ℝni, and gi:ℝ+×G×ℝni×mi→ℝni×m. If these systems are coupled, let
(11)Hij:ℝni×ℝnj×ℝ⟶ℝni,Nij:ℝni×ℝnj×ℝ⟶ℝni×m,i,j=1,2,…,N
represent the influence of vertex j on vertex i, and Hij=Nij=0 if there exists no arc from j to i in 𝒢. Then, by replacing fi  and  gi with fi+∑j=1N‍Hij and gi+∑j=1N‍Nij, we get the following stochastic coupled system on graph 𝒢:
(12)dvi(t,x)=[ρiΔvi(t,x)+fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]dt+[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dW(t),hhhhhhhhhhhhhhhhhhhhhhhhhhh(t,x)∈ℝ+×G,(13)∂vi(t,x)∂𝒩=0,(t,x)∈ℝ+×∂G.Without loss of generality, we suppose that functionsfi, gi, Hij, and Nij are such that initial-value problems to (10) and (12) have a unique solution and trivial solution v(t,x)=(v1,…,vn)=0. Functions fi,fi+∑j=1N‍Hij,gi and gi+∑j=1N‍Nij meet Lipschitz condition with Lipschitz constant L>0. Functions gi and gi+∑j=1N‍Nij satisfy integral linear growth condition. Consider y=∑i=1N‍yi. For Vi(t,vi)∈C1,2(ℝ+×ℝni;ℝ+), define a differential operator ℒVi(t,vi) associated with the ith equation of (12) by
(14)ℒVi(t,vi)≜∂Vi(t,vi)∂t+(∂Vi(t,vi)∂vi)T×[fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]+12Trace{[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]Thhhhhhhhhhhhh×(Vi(t,vi))vivj′′hhhhhhhhhhhhh×[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)][gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]T}.


### 3.1. Stability in Mean

In this section, we will discuss stability in mean as to partial variablesy of system (12) and draw some relevant conclusions.Theorem 6.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold.(A1)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+),   functions  Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],  (Sh={ζ∣|ζ(·)|<h}) meeting Vi(t,0)=0.(II)
μ1(|y(t,x)|G)≤V(t,v¯(t)), where V(t,v¯)≜∑i=1NciVi(t,v¯i) and ci is defined as (8); μ1∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤∑j=1n‍aijFij(v¯i,v¯j,t),t≥0, i=1,2,…,N.(A2)
Along each directed cycleℂ of the weighted digraph (𝔾,A) in which A=(aij)n×n there is
(15)∑(i,j)∈E(ℂ)‍Fij(v¯i,v¯j,t)≤0,t≥0.Then function V(t,v¯)≜∑i=1N‍ciVi(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(16)d(∫Gvi(t,x)dx)=∫G‍[∑j=1N‍ρi(t)Δvi(t,x)+fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]dxdt+∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t).
Applying Green formula, we deduce ∫G‍Δvi(t,x)dx=∫∂G‍(∂vi/∂𝒩)ds=0. Hence, (16) can be rewritten as
(17)d(∫G‍vi(t,x)dx)=∫G‍[fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]dxdt+∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t).
Since Vi(t0,v¯i) is continuous and Vi(t0,0)=0, we have V(t,v)≜∑i=1N‍ciVi(t,vi) that is continuous and V(t,0)=0. Hence, there exists δ(t0,ε) such that V(t0,v¯0)<μ1(ε) when |v¯0|<δ. Choosing ∀vi0:|v¯i0|<δ/N and applying Ito^ differential formula to Vi(t,v¯i) along the trajectory of system (17) yields
(18)dVi(t,v¯i)|(17)=∂Vi(t,v¯i)∂tdt+∂V(t,v¯i)∂v¯idv¯i+12Trace[∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhh.h+∑j=1N‍Nij(vi,vj,t)]⊺dx∂2V(t,v¯i)∂v¯i∂v¯i×∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhh.h.iihhhhh+∑j=1N‍Nij(vi,vj,t)]dx∫G‍[∑j=1N‍gi(t,x,vi(t,x))]dt=ℒV(t,v¯i)dt+∂V(t,v¯i)∂v¯i×∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t),
where
(19)ℒVi(t,v¯i)≜∂Vi(t,v¯i)∂t+(∂Vi(t,v¯i)∂vi)T×[fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]+12Trace{[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]Thhhhhhhhhhh×(Vi(t,vi))vivj′′hhhhhhhhhhh×[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)][gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]T}
for ∀t≥0. Hence,
(20)Vi(t,v¯i(t,t0,vi0¯))=Vi(t0,vi0¯)+∫t0t‍ℒVi(s,v¯i(s,t0,vi0¯))ds+∫t0t‍∂Vi(s,v¯i)∂v¯i×∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhh.hhhh+∑j=1N‍Nij(vi,vj,t)]dxdw(s).
As  ∂Vi(t,v¯i)/∂v¯i  is continuous on R+×Sh and gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t) satisfies integral linear growth condition, there must exist constant L1>0 such that |∂Vi(t,v¯i)/∂v¯i|≤L1, so
(21)|∂Vi(s,v¯i)∂v¯i∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dx|≤L1K(1+|∫G‍v(t,x,t0,vi0)dx|)≤L1K(1+h).
By Theorem 2.8 of [42, 43], we obtain
(22)ℰ[∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhih+∑j=1N‍Nij(vi,vj,t)]dxdw(s)∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]]=0.
On the other hand, by (A1)(III), it is derived that
(23)ℒV(t,v¯)=∑i=1N‍ciℒVi(t,v¯i)≤∑i,j=1‍ciaijFij(v¯i,v¯j,t).
Making use of Lemma 5 with weighted digraph (𝒢,A), it yields
(24)∑i,j=1‍ciaijFij(v¯i,v¯j,t)=∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍Fij(v¯i,v¯j,t).
In view of condition (A2) and a fact W(𝒬)>0, we get
(25)ℒV(t,v¯)≤∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍Fij(v¯i,v¯j,t)≤0.
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
(26)ℰ{V(t,x,v¯(t,t0,v0¯))}≤V(t0,v0¯),t≥0.
Combine Jensen inequality and condition (A1)(II),
(27)μ1(ℰ(|y(t,x,t0,v0)|G))≤ℰ(μ1(|y(t,x,t0,v0)|G))≤ℰ{V(t,v¯(t,t0,v¯0))}≤V(t0,v¯0)≤μ1(ε).
Therefore ℰ{|y(t,x,t0,v0)|G}<ε. The proof is complete.Note that if(𝒢,A) is balanced, then
(28)∑i,j=1‍ciaijFij(v¯i,v¯j,t)=12∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍[Fij(v¯j,v¯i,t)+Fji(v¯i,v¯j,t)].
In this case, condition (A2) is replaced by the following:
(29)∑(i,j)∈E(C𝒬)‍[Fij(v¯j,v¯i,t)+Fji(v¯i,v¯j,t)]≤0.
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 Vi, then the Lyapunov function for (12) can be systematically constructed by using individual Vi. 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 Theorem6 is substituted by the following.(A3)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, function Vi(t,v¯i(t)) gets rid of the restriction of Vi(t,0)=0).(II)
μ2(|y(t,x)|G)≤V(t,v¯(t))≤μ3(|v(t,x)|G), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), μ2,μ3∈𝒦, and μ2 is a convex function.(III)
ℒVi(t,v¯i)≤∑j=1n‍aijFij(v¯i,v¯j,t),t≥0,i=1,2,…,N.Other conditions remain the same. Then function V(t,v¯)≜∑i=1N‍ciVi(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.
SinceVi(t,v¯i) is continuous, we have V(t,v)≜∑i=1N‍ciVi(t,vi) 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
(30)ℰ{V(t,v¯(t,t0,v¯0))}≤V(t0,v¯0),t≥0.
Let δ(ε)=μ3-1μ2(ε). It follows from Jensen inequality and condition (A3)(II) that, for v0∈Sδ={v∣|v(·,x)|G<δ(ε)}, we have
(31)μ2(ℰ{|y(t,x,t0,v0)|G})hhhh≤ℰ{V(t,v¯(t,t0,v¯0))}≤V(t0,v¯0)hhhh≤μ3(|v0(·,x)|G)<μ2(ε).
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.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold.(B1)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}) meeting Vi(t,0)=0.(II)
μ4(|y(t,x)|G)≤V(t,v¯(t)), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i) and ci is defined as (8);μ4∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤-biVi(t,v¯i)+∑j=1n‍aijFij(v¯i,v¯j,t), constants bi≥0, t≥0,i=1,2,…,N.(B2)
Condition (A2) holds, if(𝒢,A) is balanced and (29) holds.
Then, functionV(t,v¯)≜∑i=1N‍ciVi(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 Theorem6 that
(32)ℒV(t,v¯)=∑i=1N‍ciℒVi(t,v¯i)≤-bV(t,v¯),
where b=min⁡{b1,b2,…,bN}. 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
(33)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}=0
only. Similar to the proof of Theorem 6, it is not difficult to derive
(34)ℰVi(t,v¯i(t,t0,v¯i0))=Vi(t0,v¯i0)+∫t0t‍ℰℒVi(s,v¯i(s,t0,v¯i0))ds.
Then, we obtain that V(t,v¯) is a Lyapunov-A function for (12) and
(35)ℰ{V(t,v¯(t,t0,v¯0))}≤V(t0,v¯0),t≥0.
Here we need to reduce to absurdity. Suppose
(36)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}≠0.
Instead
(37)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}≜λ^∞>0.
Combining condition (B1)(II) of Theorem 11, we obtain
(38)μ4(ℰ{|y(t,x,t0,v0)|G})hhhh≤ℰ{V(t,v¯(t,t0,v¯0))}hhhh≤V(t0,v¯0)-∫t0t‍μ4(ℰ{|y(s,x,t0,v0)|G})ds.
Therefore,
(39)0<lim⁡t→∞α(ℰ{|y(t,x,t0,v0)|G})≤V(t0,v¯0)-∫t0t‍lim⁡t→∞μ4(ℰ{|y(s,x,t0,v0)|G})ds=V(t0,v¯0)-μ4(λ^∞)(t-t0).
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
(40)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}=0,
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.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold. (B3)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}) meeting Vi(t,0)=0.(II)
μ4(|y(t,x)|G)≤V(t,v¯(t)), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i) and ci is defined as (8); μ4∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤-μ5(|y(t,x)|G)+∑j=1n‍aijFij(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¯)≜∑i=1N‍ciVi(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 Theorems6 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 Theorem12, condition(B3)(III) is replaced by
(41)(III)′ℒVi(t,v¯i)≤-μ5(|v(t,x)|G)+∑j=1n‍aijFij(v¯i,v¯j,t),μ5∈𝒦isaconvexfunction,t≥0,i=1,2,…,N.
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 Theorem12 is substituted by the following.(B4)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, function Vi(t,v¯i(t)) gets rid of the restriction of Vi(t,0)=0).(II)
μ6(|y(t,x)|G)≤V(t,v¯(t))≤μ7(|v(t,x)|G), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), μ7∈𝒦, and μ6∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤-μ8(|y(t,x)|G)+∑j=1n‍aijFij(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¯)≜∑i=1N‍ciVi(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 Theorem15 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
(42)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}=0.
Similar to the proof of Theorem 11, here we need to reduce to absurdity. Suppose
(43)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}≠0.
Instead
(44)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}≜λ^∞>0.
Combining condition (B1)(II) of Theorem 15, we obtain
(45)μ6(ℰ{|y(t,x,t0,v0)|G})hhhh≤ℰ{V(t,v¯(t,t0,v¯0))}hhhh≤V(t0,v¯0)-∫t0t‍μ6(ℰ{|y(s,x,t0,v0)|G})ds.
Therefore,
(46)0<lim⁡t→∞α(ℰ{|y(t,x,t0,v0)|G})≤V(t0,v¯0)-∫t0t‍lim⁡t→∞μ6(ℰ{|y(s,x,t0,v0)|G})ds=V(t0,v¯0)-μ6(λ^∞)(t-t0).
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
(47)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}=0,
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 Theorem15, condition (B4)(III) is replaced by
(48)(III)′ℒVi(t,v¯i)≤-μ8(|v(t,x)|G)+∑j=1n‍aijFij(v¯i,v¯j,t),μ8∈𝒦isaconvexfunction,t≥0,i=1,2,…,N.
Other conditions remain the same. Then the conclusion of Theorem 15 holds.Theorem 17.
Assume that condition (B2) of Theorem11 is substituted by the following.(B5)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, function Vi(t,v¯i(t)) gets rid of the restriction of   Vi(t,0)=0).(II)
μ9(|y(t,x)|G)≤V(t,v¯(t))≤μ10(|v(t,x)|G), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), μ10∈𝒦, and μ9∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤-b-iVi(t,v¯i)+∑j=1n‍aijFij(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¯)≜∑i=1N‍ciVi(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.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold.(C1)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, functionVi(t,v¯i(t)) gets rid of the restriction of  Vi(t,0)=0).(II)
c-1|y(t,x)|G)≤V(t,v¯(t))≤c-2|v(t,x)|G, where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), and c-1 and c-2 are positive constants.(III)
ℒVi(t,v¯i)≤-b^iVi(t,v¯i)+∑j=1n‍aijFij(v¯i,v¯j,t), constants b^i>0, t≥0,i=1,2,…,N.(C2)
Condition (A2) holds, if(𝒢,A) is balanced and (29) holds.
Then, functionV(t,v¯)≜∑i=1N‍ciVi(t,v¯i) is a Lyapunov-B function for (12), and
(49)E{|y(t,x)|G}≤c-2c-1|v0|Ge-b^(t-t0),t≥t0,
where  b^=min⁡{b^1,…,b^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 Theorem6 that
(50)ℒV(t,v¯)=∑i=1N‍ciℒVi(t,v¯i)≤-b^V(t,v¯),
where b^=min⁡{b^1,b^2,…,b^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
(51)d(∫G‍vi(t,x)dx)=∫G[∑j=1N‍ρi(t)Δvi(t,x)+fi(t,x,vi(t,x))hhhhhhh+∑j=1N‍Hij(vi,vj,t)]dxdt‍hhhh+∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t).
Combining Green formula and boundary condition we deduce ∫G‍Δvi(t,x)dx=∫∂G(∂vi/∂𝒩)ds=0. Hence, (51) can be rewritten as
(52)d(∫G‍vi(t,x)dx)=∫G‍[fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]dxdthhhh+∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t).
Since Vi(t0,v¯i) is continuous and Vi(t0,0)=0, we have V(t,v¯)≜∑i=1N‍ciVi(t,vi) that is continuous and V(t,0)=0. Hence, there exists δ(t0,ε) such that V(t0,v¯0)<μ1(ε) when |v¯0|<δ. Choosing ∀vi0:|v¯i0|<δ/N and applying Ito^ differential formula to Vi(t,v¯i) along the trajectory of system (17) yields
(53)dVi(t,v¯i)|(17)=∂Vi(t,v¯i)∂tdt+∂V(t,v¯i)∂v¯idv¯i+12Trace[∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhihhhh+∑j=1N‍Nij(vi,vj,t)]⊺dx∂2V(t,v¯i)∂v¯i∂v¯ihhhhhhhhhhhhh×∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhhhhh+∑j=1N‍Nij(vi,vj,t)]dx∫G‍[∑j=1N‍gi(t,x,vi(t,x))]dt=ℒV(t,v¯i)dt+∂V(t,v¯i)∂v¯i×∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t),
where
(54)ℒVi(t,v¯i)≜∂Vi(t,v¯i)∂t+(∂Vi(t,v¯i)∂vi)T×[fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]+12Trace{[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]Thhhhhhhhhhh×(Vi(t,vi))vivj′′hhhhhhhhhhh×[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)][gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]T},
for ∀t≥0. Hence,
(55)Vi(t,v¯i(t,t0,vi0¯))=Vi(t0,vi0¯)+∫t0t‍ℒVi(s,v¯i(s,t0,vi0¯))ds+∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhhihhi+∑j=1N‍Nij(vi,vj,t)]dxdw(s).
Build a Lyapunov function with the form
(56)V-i=eb^(t-t0)Vi(t,v¯i).
Applying Ito^ formula again, we obtain
(57)dV-i=b^eb^(t-t0)Vi(t,v¯i)dt+eb^(t-t0)dVi=eb^(t-t0)[b^Vi(t,v¯i)+ℒVi(t,v¯i)]dt+eb^(t-t0)×∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhhh+∑j=1N‍Nij(vi,vj,t)]dxdw(t).
For any n≥|v0|G, define a stop-time
(58)τn=inf⁡{t≥t0:|v(t,x)|G≥n}.
It follows from integrating (57) as to t from t0 to t∧τn(59)eb^(t∧τn-t0)Vi(t∧τn,v¯i(t∧τn))=Vi(t0,v0¯)+∫t0t∧τn‍eb^(s-t0)[b^Vi(s,v¯i)+ℒVi(s,v¯i)]ds+∫t0t∧τn‍eb^(s-t0)∂Vi(s,v¯i)∂v¯ihhhhhhhhhh×∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhh+∑j=1N‍Nij(vi,vj,t)]dxdw(s).
As  ∂Vi(t,v¯i)/∂v¯i  is continuous on R+×Sh and gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t) satisfies integral linear growth condition, there must exist constant L1>0 such that |∂Vi(t,v¯i)/∂v¯i|≤L1, so
(60)|∂Vi(s,v¯i)∂v¯i∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dx|≤L1K(1+|∫G‍v(t,x,t0,vi0)dx|)≤L1K(1+h).
By Theorem 2.8 of [35], we obtain
(61)ℰ[∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhihhhhhhh+∑j=1N‍Nij(vi,vj,t)]dxdw(s)∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))]=0.
Thus, it can be derived from taking mathematical expectation at both sides of (59) that
(62)ℰ{eb^(t∧τn-t0)Vi(t∧τn,v¯i(t∧τn))}=Vi(t0,v0¯)+ℰ{∫t0t∧τn‍eb^(s-t0)[b^Vi(s,v¯i)+ℒVi(s,v¯i)]ds}.
On the other hand, by (B5)(III), it is derived that
(63)ℒV(t,v¯)=∑i=1N‍ciℒVi(t,v¯i)≤-∑i=1N‍cib^iVi(t,v¯i)+∑i,j=1‍ciaijFij(v¯i,v¯j,t).
Making use of Lemma 5 with weighted digraph (𝒢,A), it yields
(64)∑i,j=1‍ciaijFij(v¯i,v¯j,t)=∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍Fij(v¯i,v¯j,t).
In view of Condition (A2) and the fact that W(𝒬)>0, we get
(65)ℒV(t,v¯)≤-∑i=1N‍cib^iVi(t,v¯i)+∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍Fij(v¯i,v¯j,t)≤-∑i=1N‍cib^iVi(t,v¯i)<-b^V(t,v¯).
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
(66)c-1eb^(t∧τn-t0)ℰ{|y(t∧τn,x)|G}≤ℰ{eb^(t∧τn-t0)V(t∧τn,v¯)}≤V(t0,v0¯).
Obviously τn→∞ when n→∞, we have
(67)c-1eb^(t-t0)ℰ{|y(t,x)|G}≤c-2(|v0|G).
Therefore, we get
(68)ℰ{|y(t,x)|G}≤c-2c-1|v0|Ge-b^(t-t0),t≥0,
as required.
According to Theorem18, it is easy to reach the following theorem.Theorem 19.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold.(C1)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, function Vi(t,v¯i(t)) gets rid of the restriction of  Vi(t,0)=0).(II)
c^1|y(t,x)|G≤V(t,v¯(t)≤c^2|v(t,x)|G, where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), and c^1 and c^2<1 are positive constants.(III)
ℒVi(t,v¯i)≤-b^iVi(t,v¯i)+∑j=1n‍aijFij(v¯i,v¯j,t), constants b^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¯)≜∑i=1N‍ciVi(t,v¯i) is a Lyapunov-A function for (12), and
(69)E{|y(t,x)|G}≤c-2c-1|v0|Ge-b^(t-t0),t≥t0,
where b^=min⁡{b^1,…,b^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.


## 3.1. Stability in Mean

In this section, we will discuss stability in mean as to partial variablesy of system (12) and draw some relevant conclusions.Theorem 6.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold.(A1)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+),   functions  Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],  (Sh={ζ∣|ζ(·)|<h}) meeting Vi(t,0)=0.(II)
μ1(|y(t,x)|G)≤V(t,v¯(t)), where V(t,v¯)≜∑i=1NciVi(t,v¯i) and ci is defined as (8); μ1∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤∑j=1n‍aijFij(v¯i,v¯j,t),t≥0, i=1,2,…,N.(A2)
Along each directed cycleℂ of the weighted digraph (𝔾,A) in which A=(aij)n×n there is
(15)∑(i,j)∈E(ℂ)‍Fij(v¯i,v¯j,t)≤0,t≥0.Then function V(t,v¯)≜∑i=1N‍ciVi(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(16)d(∫Gvi(t,x)dx)=∫G‍[∑j=1N‍ρi(t)Δvi(t,x)+fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]dxdt+∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t).
Applying Green formula, we deduce ∫G‍Δvi(t,x)dx=∫∂G‍(∂vi/∂𝒩)ds=0. Hence, (16) can be rewritten as
(17)d(∫G‍vi(t,x)dx)=∫G‍[fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]dxdt+∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t).
Since Vi(t0,v¯i) is continuous and Vi(t0,0)=0, we have V(t,v)≜∑i=1N‍ciVi(t,vi) that is continuous and V(t,0)=0. Hence, there exists δ(t0,ε) such that V(t0,v¯0)<μ1(ε) when |v¯0|<δ. Choosing ∀vi0:|v¯i0|<δ/N and applying Ito^ differential formula to Vi(t,v¯i) along the trajectory of system (17) yields
(18)dVi(t,v¯i)|(17)=∂Vi(t,v¯i)∂tdt+∂V(t,v¯i)∂v¯idv¯i+12Trace[∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhh.h+∑j=1N‍Nij(vi,vj,t)]⊺dx∂2V(t,v¯i)∂v¯i∂v¯i×∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhh.h.iihhhhh+∑j=1N‍Nij(vi,vj,t)]dx∫G‍[∑j=1N‍gi(t,x,vi(t,x))]dt=ℒV(t,v¯i)dt+∂V(t,v¯i)∂v¯i×∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t),
where
(19)ℒVi(t,v¯i)≜∂Vi(t,v¯i)∂t+(∂Vi(t,v¯i)∂vi)T×[fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]+12Trace{[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]Thhhhhhhhhhh×(Vi(t,vi))vivj′′hhhhhhhhhhh×[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)][gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]T}
for ∀t≥0. Hence,
(20)Vi(t,v¯i(t,t0,vi0¯))=Vi(t0,vi0¯)+∫t0t‍ℒVi(s,v¯i(s,t0,vi0¯))ds+∫t0t‍∂Vi(s,v¯i)∂v¯i×∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhh.hhhh+∑j=1N‍Nij(vi,vj,t)]dxdw(s).
As  ∂Vi(t,v¯i)/∂v¯i  is continuous on R+×Sh and gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t) satisfies integral linear growth condition, there must exist constant L1>0 such that |∂Vi(t,v¯i)/∂v¯i|≤L1, so
(21)|∂Vi(s,v¯i)∂v¯i∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dx|≤L1K(1+|∫G‍v(t,x,t0,vi0)dx|)≤L1K(1+h).
By Theorem 2.8 of [42, 43], we obtain
(22)ℰ[∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhih+∑j=1N‍Nij(vi,vj,t)]dxdw(s)∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]]=0.
On the other hand, by (A1)(III), it is derived that
(23)ℒV(t,v¯)=∑i=1N‍ciℒVi(t,v¯i)≤∑i,j=1‍ciaijFij(v¯i,v¯j,t).
Making use of Lemma 5 with weighted digraph (𝒢,A), it yields
(24)∑i,j=1‍ciaijFij(v¯i,v¯j,t)=∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍Fij(v¯i,v¯j,t).
In view of condition (A2) and a fact W(𝒬)>0, we get
(25)ℒV(t,v¯)≤∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍Fij(v¯i,v¯j,t)≤0.
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
(26)ℰ{V(t,x,v¯(t,t0,v0¯))}≤V(t0,v0¯),t≥0.
Combine Jensen inequality and condition (A1)(II),
(27)μ1(ℰ(|y(t,x,t0,v0)|G))≤ℰ(μ1(|y(t,x,t0,v0)|G))≤ℰ{V(t,v¯(t,t0,v¯0))}≤V(t0,v¯0)≤μ1(ε).
Therefore ℰ{|y(t,x,t0,v0)|G}<ε. The proof is complete.Note that if(𝒢,A) is balanced, then
(28)∑i,j=1‍ciaijFij(v¯i,v¯j,t)=12∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍[Fij(v¯j,v¯i,t)+Fji(v¯i,v¯j,t)].
In this case, condition (A2) is replaced by the following:
(29)∑(i,j)∈E(C𝒬)‍[Fij(v¯j,v¯i,t)+Fji(v¯i,v¯j,t)]≤0.
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 Vi, then the Lyapunov function for (12) can be systematically constructed by using individual Vi. 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 Theorem6 is substituted by the following.(A3)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, function Vi(t,v¯i(t)) gets rid of the restriction of Vi(t,0)=0).(II)
μ2(|y(t,x)|G)≤V(t,v¯(t))≤μ3(|v(t,x)|G), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), μ2,μ3∈𝒦, and μ2 is a convex function.(III)
ℒVi(t,v¯i)≤∑j=1n‍aijFij(v¯i,v¯j,t),t≥0,i=1,2,…,N.Other conditions remain the same. Then function V(t,v¯)≜∑i=1N‍ciVi(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.
SinceVi(t,v¯i) is continuous, we have V(t,v)≜∑i=1N‍ciVi(t,vi) 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
(30)ℰ{V(t,v¯(t,t0,v¯0))}≤V(t0,v¯0),t≥0.
Let δ(ε)=μ3-1μ2(ε). It follows from Jensen inequality and condition (A3)(II) that, for v0∈Sδ={v∣|v(·,x)|G<δ(ε)}, we have
(31)μ2(ℰ{|y(t,x,t0,v0)|G})hhhh≤ℰ{V(t,v¯(t,t0,v¯0))}≤V(t0,v¯0)hhhh≤μ3(|v0(·,x)|G)<μ2(ε).
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.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold.(B1)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}) meeting Vi(t,0)=0.(II)
μ4(|y(t,x)|G)≤V(t,v¯(t)), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i) and ci is defined as (8);μ4∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤-biVi(t,v¯i)+∑j=1n‍aijFij(v¯i,v¯j,t), constants bi≥0, t≥0,i=1,2,…,N.(B2)
Condition (A2) holds, if(𝒢,A) is balanced and (29) holds.
Then, functionV(t,v¯)≜∑i=1N‍ciVi(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 Theorem6 that
(32)ℒV(t,v¯)=∑i=1N‍ciℒVi(t,v¯i)≤-bV(t,v¯),
where b=min⁡{b1,b2,…,bN}. 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
(33)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}=0
only. Similar to the proof of Theorem 6, it is not difficult to derive
(34)ℰVi(t,v¯i(t,t0,v¯i0))=Vi(t0,v¯i0)+∫t0t‍ℰℒVi(s,v¯i(s,t0,v¯i0))ds.
Then, we obtain that V(t,v¯) is a Lyapunov-A function for (12) and
(35)ℰ{V(t,v¯(t,t0,v¯0))}≤V(t0,v¯0),t≥0.
Here we need to reduce to absurdity. Suppose
(36)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}≠0.
Instead
(37)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}≜λ^∞>0.
Combining condition (B1)(II) of Theorem 11, we obtain
(38)μ4(ℰ{|y(t,x,t0,v0)|G})hhhh≤ℰ{V(t,v¯(t,t0,v¯0))}hhhh≤V(t0,v¯0)-∫t0t‍μ4(ℰ{|y(s,x,t0,v0)|G})ds.
Therefore,
(39)0<lim⁡t→∞α(ℰ{|y(t,x,t0,v0)|G})≤V(t0,v¯0)-∫t0t‍lim⁡t→∞μ4(ℰ{|y(s,x,t0,v0)|G})ds=V(t0,v¯0)-μ4(λ^∞)(t-t0).
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
(40)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}=0,
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.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold. (B3)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}) meeting Vi(t,0)=0.(II)
μ4(|y(t,x)|G)≤V(t,v¯(t)), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i) and ci is defined as (8); μ4∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤-μ5(|y(t,x)|G)+∑j=1n‍aijFij(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¯)≜∑i=1N‍ciVi(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 Theorems6 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 Theorem12, condition(B3)(III) is replaced by
(41)(III)′ℒVi(t,v¯i)≤-μ5(|v(t,x)|G)+∑j=1n‍aijFij(v¯i,v¯j,t),μ5∈𝒦isaconvexfunction,t≥0,i=1,2,…,N.
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 Theorem12 is substituted by the following.(B4)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, function Vi(t,v¯i(t)) gets rid of the restriction of Vi(t,0)=0).(II)
μ6(|y(t,x)|G)≤V(t,v¯(t))≤μ7(|v(t,x)|G), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), μ7∈𝒦, and μ6∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤-μ8(|y(t,x)|G)+∑j=1n‍aijFij(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¯)≜∑i=1N‍ciVi(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 Theorem15 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
(42)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}=0.
Similar to the proof of Theorem 11, here we need to reduce to absurdity. Suppose
(43)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}≠0.
Instead
(44)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}≜λ^∞>0.
Combining condition (B1)(II) of Theorem 15, we obtain
(45)μ6(ℰ{|y(t,x,t0,v0)|G})hhhh≤ℰ{V(t,v¯(t,t0,v¯0))}hhhh≤V(t0,v¯0)-∫t0t‍μ6(ℰ{|y(s,x,t0,v0)|G})ds.
Therefore,
(46)0<lim⁡t→∞α(ℰ{|y(t,x,t0,v0)|G})≤V(t0,v¯0)-∫t0t‍lim⁡t→∞μ6(ℰ{|y(s,x,t0,v0)|G})ds=V(t0,v¯0)-μ6(λ^∞)(t-t0).
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
(47)lim⁡t→∞ℰ{|y(t,x,t0,v0)|G}=0,
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 Theorem15, condition (B4)(III) is replaced by
(48)(III)′ℒVi(t,v¯i)≤-μ8(|v(t,x)|G)+∑j=1n‍aijFij(v¯i,v¯j,t),μ8∈𝒦isaconvexfunction,t≥0,i=1,2,…,N.
Other conditions remain the same. Then the conclusion of Theorem 15 holds.Theorem 17.
Assume that condition (B2) of Theorem11 is substituted by the following.(B5)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, function Vi(t,v¯i(t)) gets rid of the restriction of   Vi(t,0)=0).(II)
μ9(|y(t,x)|G)≤V(t,v¯(t))≤μ10(|v(t,x)|G), where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), μ10∈𝒦, and μ9∈𝒦 is a convex function.(III)
ℒVi(t,v¯i)≤-b-iVi(t,v¯i)+∑j=1n‍aijFij(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¯)≜∑i=1N‍ciVi(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.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold.(C1)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, functionVi(t,v¯i(t)) gets rid of the restriction of  Vi(t,0)=0).(II)
c-1|y(t,x)|G)≤V(t,v¯(t))≤c-2|v(t,x)|G, where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), and c-1 and c-2 are positive constants.(III)
ℒVi(t,v¯i)≤-b^iVi(t,v¯i)+∑j=1n‍aijFij(v¯i,v¯j,t), constants b^i>0, t≥0,i=1,2,…,N.(C2)
Condition (A2) holds, if(𝒢,A) is balanced and (29) holds.
Then, functionV(t,v¯)≜∑i=1N‍ciVi(t,v¯i) is a Lyapunov-B function for (12), and
(49)E{|y(t,x)|G}≤c-2c-1|v0|Ge-b^(t-t0),t≥t0,
where  b^=min⁡{b^1,…,b^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 Theorem6 that
(50)ℒV(t,v¯)=∑i=1N‍ciℒVi(t,v¯i)≤-b^V(t,v¯),
where b^=min⁡{b^1,b^2,…,b^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
(51)d(∫G‍vi(t,x)dx)=∫G[∑j=1N‍ρi(t)Δvi(t,x)+fi(t,x,vi(t,x))hhhhhhh+∑j=1N‍Hij(vi,vj,t)]dxdt‍hhhh+∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t).
Combining Green formula and boundary condition we deduce ∫G‍Δvi(t,x)dx=∫∂G(∂vi/∂𝒩)ds=0. Hence, (51) can be rewritten as
(52)d(∫G‍vi(t,x)dx)=∫G‍[fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]dxdthhhh+∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t).
Since Vi(t0,v¯i) is continuous and Vi(t0,0)=0, we have V(t,v¯)≜∑i=1N‍ciVi(t,vi) that is continuous and V(t,0)=0. Hence, there exists δ(t0,ε) such that V(t0,v¯0)<μ1(ε) when |v¯0|<δ. Choosing ∀vi0:|v¯i0|<δ/N and applying Ito^ differential formula to Vi(t,v¯i) along the trajectory of system (17) yields
(53)dVi(t,v¯i)|(17)=∂Vi(t,v¯i)∂tdt+∂V(t,v¯i)∂v¯idv¯i+12Trace[∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhihhhh+∑j=1N‍Nij(vi,vj,t)]⊺dx∂2V(t,v¯i)∂v¯i∂v¯ihhhhhhhhhhhhh×∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhhhhh+∑j=1N‍Nij(vi,vj,t)]dx∫G‍[∑j=1N‍gi(t,x,vi(t,x))]dt=ℒV(t,v¯i)dt+∂V(t,v¯i)∂v¯i×∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dxdW(t),
where
(54)ℒVi(t,v¯i)≜∂Vi(t,v¯i)∂t+(∂Vi(t,v¯i)∂vi)T×[fi(t,x,vi(t,x))+∑j=1N‍Hij(vi,vj,t)]+12Trace{[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]Thhhhhhhhhhh×(Vi(t,vi))vivj′′hhhhhhhhhhh×[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)][gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]T},
for ∀t≥0. Hence,
(55)Vi(t,v¯i(t,t0,vi0¯))=Vi(t0,vi0¯)+∫t0t‍ℒVi(s,v¯i(s,t0,vi0¯))ds+∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhhihhi+∑j=1N‍Nij(vi,vj,t)]dxdw(s).
Build a Lyapunov function with the form
(56)V-i=eb^(t-t0)Vi(t,v¯i).
Applying Ito^ formula again, we obtain
(57)dV-i=b^eb^(t-t0)Vi(t,v¯i)dt+eb^(t-t0)dVi=eb^(t-t0)[b^Vi(t,v¯i)+ℒVi(t,v¯i)]dt+eb^(t-t0)×∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhhh+∑j=1N‍Nij(vi,vj,t)]dxdw(t).
For any n≥|v0|G, define a stop-time
(58)τn=inf⁡{t≥t0:|v(t,x)|G≥n}.
It follows from integrating (57) as to t from t0 to t∧τn(59)eb^(t∧τn-t0)Vi(t∧τn,v¯i(t∧τn))=Vi(t0,v0¯)+∫t0t∧τn‍eb^(s-t0)[b^Vi(s,v¯i)+ℒVi(s,v¯i)]ds+∫t0t∧τn‍eb^(s-t0)∂Vi(s,v¯i)∂v¯ihhhhhhhhhh×∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhhhhhhh+∑j=1N‍Nij(vi,vj,t)]dxdw(s).
As  ∂Vi(t,v¯i)/∂v¯i  is continuous on R+×Sh and gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t) satisfies integral linear growth condition, there must exist constant L1>0 such that |∂Vi(t,v¯i)/∂v¯i|≤L1, so
(60)|∂Vi(s,v¯i)∂v¯i∫G‍[gi(t,x,vi(t,x))+∑j=1N‍Nij(vi,vj,t)]dx|≤L1K(1+|∫G‍v(t,x,t0,vi0)dx|)≤L1K(1+h).
By Theorem 2.8 of [35], we obtain
(61)ℰ[∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))hhhhhhhhhihhhhhhh+∑j=1N‍Nij(vi,vj,t)]dxdw(s)∫t0t‍∂Vi(s,v¯i)∂v¯i∫G‍[∑j=1N‍gi(t,x,vi(t,x))]=0.
Thus, it can be derived from taking mathematical expectation at both sides of (59) that
(62)ℰ{eb^(t∧τn-t0)Vi(t∧τn,v¯i(t∧τn))}=Vi(t0,v0¯)+ℰ{∫t0t∧τn‍eb^(s-t0)[b^Vi(s,v¯i)+ℒVi(s,v¯i)]ds}.
On the other hand, by (B5)(III), it is derived that
(63)ℒV(t,v¯)=∑i=1N‍ciℒVi(t,v¯i)≤-∑i=1N‍cib^iVi(t,v¯i)+∑i,j=1‍ciaijFij(v¯i,v¯j,t).
Making use of Lemma 5 with weighted digraph (𝒢,A), it yields
(64)∑i,j=1‍ciaijFij(v¯i,v¯j,t)=∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍Fij(v¯i,v¯j,t).
In view of Condition (A2) and the fact that W(𝒬)>0, we get
(65)ℒV(t,v¯)≤-∑i=1N‍cib^iVi(t,v¯i)+∑𝒬∈ℚ‍W(𝒬)∑(i,j)∈E(C𝒬)‍Fij(v¯i,v¯j,t)≤-∑i=1N‍cib^iVi(t,v¯i)<-b^V(t,v¯).
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
(66)c-1eb^(t∧τn-t0)ℰ{|y(t∧τn,x)|G}≤ℰ{eb^(t∧τn-t0)V(t∧τn,v¯)}≤V(t0,v0¯).
Obviously τn→∞ when n→∞, we have
(67)c-1eb^(t-t0)ℰ{|y(t,x)|G}≤c-2(|v0|G).
Therefore, we get
(68)ℰ{|y(t,x)|G}≤c-2c-1|v0|Ge-b^(t-t0),t≥0,
as required.
According to Theorem18, it is easy to reach the following theorem.Theorem 19.
Letv¯i(t)=∫G‍vi(t,x)dx. Suppose that the following conditions hold.(C1)
There exist positive-definite functionsVi(t,ξ)∈C1,2(ℝ+×ℝni;ℝ+), functions Fij(v¯i,v¯j,t), and constants aij≥0 satisfying the following.
(I)
Vi(t,v¯i(t))∈C1,2[ℝ+×Sh,ℝ],(Sh={ζ∣|ζ(·)|<h}). (Here, function Vi(t,v¯i(t)) gets rid of the restriction of  Vi(t,0)=0).(II)
c^1|y(t,x)|G≤V(t,v¯(t)≤c^2|v(t,x)|G, where V(t,v¯)≜∑i=1N‍ciVi(t,v¯i), ci is defined as (8), and c^1 and c^2<1 are positive constants.(III)
ℒVi(t,v¯i)≤-b^iVi(t,v¯i)+∑j=1n‍aijFij(v¯i,v¯j,t), constants b^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¯)≜∑i=1N‍ciVi(t,v¯i) is a Lyapunov-A function for (12), and
(69)E{|y(t,x)|G}≤c-2c-1|v0|Ge-b^(t-t0),t≥t0,
where b^=min⁡{b^1,…,b^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
(70)dv1(t,x)=[Δv1(t,x)+v2(t,x)-∑j=12‍a1j(v1(t,x)-vj(t,x))]dt,dv2(t,x)=[∑j=12‍Δv2(t,x)-v1(t,x)-α(t)v2(t,x)h+∑j=12‍a2j(v2(t,x)-vj(t,x))]dt-α(t)v2(t,x)dW(t).
Construct function V=(∫G‍v1(t,x)dx)2+(∫G‍v2(t,x)dx)2,  and we have
(71)ℒV=(2∫G‍v1dx2∫G‍v2dx)(∫G‍v2dx-∫G‍v1dx-α(t)∫G‍v2dx)+α(t)(∫G‍v2(t,x)dx)2+∑j=12‍akjFkj(vk,vj,t)=-α(t)(∫G‍v2(t,x)dx)2<0.According to Theorem6, 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.



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*Source: 597502-2014-05-07.xml*