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

and Applied Analysis 3 Applying the Itô formula to ∫ G V(t, v(t, x))dx along system (1) gives for ∀t ≥ t0 (d∫ G V (t, v (t, x)) dx) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨(1) = ∫ G [(LV (t, v (t, x)) + V 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 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 ∈ C1,2(R+ × Rn;R+) is called a LyapunovA function for (1), if L∫ G V(t, v)dx ≤ 0, and is called a Lyapunov-B function for (1), if L∫ G V(t, v)dx ≤ −b ∫ G V(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 subgraphT 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 fromone to the other. Given aweighted 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 matrixA 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 ofC and is constructed by reversing the direction of all arcs in C. For a unicyclic graph Q with cycle CQ, let Q̃ be the unicyclic graph obtained by replacing CQ with −CQ. Suppose that (G, A) is balanced; then W(Q) = W(Q̃). The Laplacian matrix of (G, A) is defined as


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][3][4][5][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][8][9][10][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][16][17][18][19][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][23][24][25][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][28][29][30][31][32][33][34][35][36][37][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
Throughout this paper, we suppose function (, x, v(, x)) satisfies integral linear growth condition and ,  meet Lipschitz condition; that is, there exists constant  > 0 such that      (, x, v (, x)) where |v(⋅, x))|  ≜ | ∫  v(⋅, 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 ∀ > 0, ∀ 0 > 0, there is ( 0 , ) such that E{|y(, x,  0 , v 0 )|  } <  holds for The trivial solution of system ( 1) is said to be uniformly stable in mean as to partial variables y if, for ∀ > 0, ∀ 0 > 0, there is () such that E{|y(, The trivial solution of system ( 1) is said to be asymptotically stable in mean as to partial variables y if, for ∀ > 0, ∀ 0 > 0, there is ( 0 , ) such that for The trivial solution of system ( 1) is said to be uniformly asymptotically stable in mean as to partial variables y if, for ∀ > 0, ∀ 0 > 0, there is () such that for A continuous function (, ) is said to be positivedefinite if (, 0) = 0 and, for some  ∈ KR, (, ) ≥ (||).Write  1,2 (R + × R  ; R + ) for the family of all nonnegative functions (, ) on R + × R  that are continuously twice differentiable in  and once in .If (, ) ∈  1,2 (R + ×R  ; R + ), then define an operator L(, ) from R + × R  to R with respect to (1) by where Abstract and Applied Analysis 3 Applying the Itô formula to ∫  V(, v(, x))dx along system (1) gives for ∀ ≥  0 The existence of function (, v) ∈  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.
The following basic concepts and theorems on graph theory can be found in [11,41].A directed graph G = (V, ) contains a set V = {1, 2, . . ., } of vertices and a set  of arcs (, ) leading from initial vertex  to terminal vertex .
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 (, ) is assigned to a positive weight   .Here   > 0 if and only if there exists an arc from vertex  to vertex  in G.The weight (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 { 1 ,  2 , . . .,   } such that its set of arcs is {(  ,  +1 ) :  = 1, 2, . . .,  − 1}.If   =  1 , 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 , called the root, if  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  vertices, define the weight matrix  = (  )  ×  whose entry   equals the weight of arc (, ) if it exists and 0 otherwise.Denote the directed graph with weight matrix  by (G, ).A weighted digraph (G, ) is said to be balanced if (C) = (−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, ) is balanced; then (Q) = ( Q).The Laplacian matrix of (G, ) is defined as ) .
Let   denote the cofactor of the th diagonal element of J.
where T  is the set of all spanning trees T of (G, ) that are rooted at vertex .In particular, if (G, ) is strongly connected, then   > 0 for   = 1, 2, . . ., .
Lemma 5 (see [11]).Assume  ≥ 2. Let   be given in (1).Then the following identity holds: Here   (  ,   ), 1 ≤ ,  ≤ , are arbitrary functions, Q is the set of all spanning unicyclic graphs of (G, ), (Q) is the weight of Q, and  Q denotes the directed cycle of Q.

Main Results
To begin with our main results, we will give an SCEDSN represented by digraph G with  vertices,  ≥ 2. In th vertex it is assigned a stochastic reaction-diffusion system where v  (, x) = col(y i , z i ) ∈ R   , y i ∈ R   , z i ∈ R   (  +   =   ),  : R + ×  × R   → R   , and   : R + ×  × R   ×  → R   × .If these systems are coupled, let represent the influence of vertex  on vertex , and   =   = 0 if there exists no arc from  to  in G.Then, by replacing   and   with   + ∑  =1   and   + ∑  =1   , we get the following stochastic coupled system on graph G: Without loss of generality, we suppose that functions   ,   ,   , and   are such that initial-value problems to (10) and ( 12) have a unique solution and trivial solution v(, x) = (v 1 , . . ., v  ) = 0. Functions   ,   +∑  =1   ,   and   +∑  =1   meet Lipschitz condition with Lipschitz constant  > 0. Functions   and   + ∑  =1   satisfy integral linear growth condition.Consider y = ∑  =1 y i .For   (, v  ) ∈  1,2 (R + × R   ; R + ), define a differential operator L  (, v  ) associated with the th equation of ( 12) by

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  () = ∫  v i (, x) dx.Suppose that the following conditions hold.
(A1) There exist positive-definite functions ), and constants   ≥ 0 satisfying the following.(12).Furthermore, the trivial solution of (12) is stable in mean as to partial variables y.
Note that if (G, ) is balanced, then Abstract and Applied Analysis In this case, condition (A2) is replaced by the following: Consequently, we get the following corollary.
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   , then the Lyapunov function for ( 12) can be systematically constructed by using individual   .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 Other conditions remain the same.Then function (, v) ≜ ∑  =1     (, v  ) is a Lyapunov-A function for (12).Furthermore, the trivial solution of (12) is uniformly stable in mean as to partial variables y.

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  () = ∫  v i (, x)dx.Suppose that the following conditions hold.
Then, function (, v) ≜ ∑  =1     (, v  ) 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 where  = min{ 1 ,  2 , . . .,   }.Hence, we conclude that function (, 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 only.Similar to the proof of Theorem 6, it is not difficult to derive Then, we obtain that (, v) is a Lyapunov-A function for (12) and Here we need to reduce to absurdity.Suppose lim Combining condition (B1)(II) of Theorem 11, we obtain Therefore, However, it is obvious that (39) can not be satisfied as  ≫  0 .Thus, hypothesis lim  → ∞ E{|y(, x,  0 , v 0 )|  } ̸ = 0 does not come into existence.It should be lim 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  () = ∫  v i (, x)dx.Suppose that the following conditions hold.
Then, function (, v) ≜ ∑  =1     (, v  ) 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 − 5 (|v(, x)|  ) ≤ − 5 (|y(, x)|  ).We can draw the following theorem immediately.

Corollary 14. Suppose that in Theorem 12, condition (B3)(III) is replaced by
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.
Other conditions remain the same.Then function (, v) ≜ ∑  =1     (, v  ) 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 Similar to the proof of Theorem 11, here we need to reduce to absurdity.Suppose lim Combining condition (B1)(II) of Theorem 15, we obtain Therefore, However, it is obvious that (46) cannot be satisfied as  ≫  0 .Thus, hypothesis lim  → ∞ E{|y(, x,  0 , v 0 )|  } ̸ = 0 does not come into existence.It should be lim 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.

Corollary 16. Suppose that, in Theorem 15, condition (B4)(III) is replaced by
Other conditions remain the same.Then the conclusion of Theorem 15 holds.
Other conditions remain the same.Then function (, v) ≜ ∑  =1     (, v  ) is a Lyapunov-B function for (12).Furthermore, the trivial solution of (12) is uniformly asymptotically stable in mean as to partial variables y.

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  () = ∫  v  (, x)dx.Suppose that the following conditions hold.
,   is defined as (8), and  1 and  2 are positive constants.(12), and where b = min{ b1 , . . ., b }; 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 where b = min{ b1 , b2 , . . ., b }.Hence, we conclude that function (, v) is a Lyapunov-B function for (12).Integrating system (12) Since   ( 0 , v  ) is continuous and   ( 0 , 0) = 0, we have (, v) ≜ ∑  =1     (, v  ) that is continuous and (, 0) = 0. Hence, there exists ( 0 , ) such that ( 0 , v 0 ) <  1 () when |v 0 | < .Choosing ∀v 0 : |v 0 | < / and applying Itô differential formula to   (, v  ) along the trajectory of system (17) where for ∀ ≥ 0. Hence, Build a Lyapunov function with the form Applying Itô formula again, we obtain For any  ≥ |v 0 |  , define a stop-time It follows from integrating (57) as to  from Thus, it can be derived from taking mathematical expectation at both sides of (59) that On the other hand, by (B5)(III), it is derived that Making use of Lemma 5 with weighted digraph (G, ), it yields In view of Condition (A2) and the fact that (Q) > 0, we get Thus (, 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 Obviously   → ∞ when  → ∞, we have Therefore, we get as required.
According to Theorem 18, it is easy to reach the following theorem.where b = min{ b1 , . . ., b }; 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.

Example
Consider the 2-dimensional Itô SRDSMS (1) satisfying the bounded condition (2), and we assume (G, ) is strongly connected and balanced.Consider (71) According to Theorem 6, we know the trial solution of system (70) is stable in mean as to partial variable v 2 .

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.