Abstract and Applied Analysis

Volume 2014 (2014), Article ID 597502, 13 pages

http://dx.doi.org/10.1155/2014/597502

Research Article

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

Yonggui Kao¹ and Hamid Reza Karimi²

¹Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China

²Department of Engineering, Faculty of Technology and Science, University of Agder, 4898 Grimstad, Norway

Received 24 February 2014; Accepted 11 April 2014; Published 7 May 2014

Academic Editor: Jun Hu

Copyright © 2014 Yonggui Kao and Hamid Reza Karimi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , then the global Lyapunov function for the CSRDSNs can be systematically produced by individual . An example is provided in Section 4. Section 5 is conclusion. Notations: for convenience, we sometimes write , , and as , , and , respectively.

2. Preliminaries

A general stochastic reaction-diffusion system reads

with boundary condition

where , ,  , ; denote by , , and both and are Borel measurable functions; here, , , stands for vector norm, is an -dimension Brown motion with natural flow defined on complete probability space , is the normal vector to , , and .

Throughout this paper, we suppose function satisfies integral linear growth condition and , meet Lipschitz condition; that is, there exists constant such that

where . 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 if, for , , there is such that holds for .

The trivial solution of system (1) is said to be uniformly stable in mean as to partial variables if, for , , there is such that holds for .

The trivial solution of system (1) is said to be asymptotically stable in mean as to partial variables if, for , , there is such that for and .

The trivial solution of system (1) is said to be uniformly asymptotically stable in mean as to partial variables if, for , , there is such that for and .

Definition 2. If is a strictly increasing function and , function is said to be class function. Denote concisely. If and , then .

A continuous function is said to be positive-definite if and, for some , . Write for the family of all nonnegative functions on that are continuously twice differentiable in and once in . If , then define an operator from to with respect to (1) by

where

Applying the Itô formula to along system (1) gives for

The existence of function 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. is called a Lyapunov-A function for (1), if , and is called a Lyapunov-B function for (1), if , in which .

The following basic concepts and theorems on graph theory can be found in [11, 41]. A directed graph contains a set of vertices and a set of arcs leading from initial vertex to terminal vertex . A subgraph of is said to be spanning if and have the same vertex set. A digraph is weighted if each arc is assigned to a positive weight . Here if and only if there exists an arc from vertex to vertex in . The weight of is the product of the weights on all its arcs. A directed path in is a subgraph with distinct vertices such that its set of arcs is . If , we call a directed cycle. A connected subgraph is a tree if it contains no cycles. A tree 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 is strongly connected if, for any pair of distinct vertices, there exists a directed path from one to the other. Given a weighted digraph with vertices, define the weight matrix whose entry equals the weight of arc if it exists and otherwise. Denote the directed graph with weight matrix by . A weighted digraph is said to be balanced if for all directed cycles . Here, denotes the reverse of and is constructed by reversing the direction of all arcs in . For a unicyclic graph with cycle , let be the unicyclic graph obtained by replacing with . Suppose that is balanced; then . The Laplacian matrix of is defined as

Let denote the cofactor of the th diagonal element of .

Lemma 4 ([34] Kirchhoffs Matrix Tree Theorem). Assume . Then

where is the set of all spanning trees of that are rooted at vertex . In particular, if is strongly connected, then for .

Lemma 5 (see [11]). Assume . Let be given in (1). Then the following identity holds:

Here  , are arbitrary functions, is the set of all spanning unicyclic graphs of , is the weight of ,  and denotes the directed cycle of .

3. Main Results

To begin with our main results, we will give an SCEDSN represented by digraph with vertices, . In th vertex it is assigned a stochastic reaction-diffusion system

where ,, and . If these systems are coupled, let

represent the influence of vertex on vertex , and if there exists no arc from to in . Then, by replacing  and  with and , we get the following stochastic coupled system on graph :

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 . Functions and meet Lipschitz condition with Lipschitz constant . Functions and satisfy integral linear growth condition. Consider . For , define a differential operator associated with the th equation of (12) by

3.1. Stability in Mean

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

Theorem 6. Let . Suppose that the following conditions hold.

(A1)There exist positive-definite functions ,   functions  , and constants satisfying the following.

,   meeting ., where and is defined as (8); is a convex function., .

(A2)Along each directed cycle of the weighted digraph in which there is

Then function is a Lyapunov-A function for (12). Furthermore, the trivial solution of (12) is stable in mean as to partial variables .

Proof. It is not difficult to find that

Applying Green formula, we deduce . Hence, (16) can be rewritten as

Since is continuous and , we have that is continuous and . Hence, there exists such that when . Choosing and applying It differential formula to along the trajectory of system (17) yields

where

for . Hence,

As  /  is continuous on and satisfies integral linear growth condition, there must exist constant such that , so

By Theorem 2.8 of [42, 43], we obtain

On the other hand, by (A1)(III), it is derived that

Making use of Lemma 5 with weighted digraph , it yields

In view of condition (A2) and a fact , we get

Thus 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

Combine Jensen inequality and condition (A1)(II),

Therefore . The proof is complete.

Note that if is balanced, then

In this case, condition (A2) is replaced by the following:

Consequently, we get the following corollary.

Corollary 7. Suppose that 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 , 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 , functions , and constants satisfying the following.

. (Here, function gets rid of the restriction of ., where , is defined as (8), , and is a convex function..

Other conditions remain the same. Then function is a Lyapunov-A function for (12). Furthermore, the trivial solution of (12) is uniformly stable in mean as to partial variables .

Proof. Since is continuous, we have that is continuous. Similar to the proof of Theorem 6, we can obtain that is a Lyapunov-A function for (12) and

Let . It follows from Jensen inequality and condition (A3)(II) that, for , we have

Therefore, we derive as required.

Corollary 10. Suppose that 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 . Suppose that the following conditions hold.

(B1)There exist positive-definite functions , functions , and constants satisfying the following.

meeting ., where and is defined as (8) is a convex function., constants , .

(B2)Condition (A2) holds, if is balanced and (29) holds.

Then, function is a Lyapunov-B function for (12). Consequently, the trivial solution of (12) is asymptotically stable in mean as to partial variables .

Proof. We can show in the same way as in the proof of Theorem 6 that

where . Hence, we conclude that function 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 . 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 is a Lyapunov-A function for (12) and

Here we need to reduce to absurdity. Suppose

Instead

Combining condition (B1)(II) of Theorem 11, we obtain

Therefore,

However, it is obvious that (39) can not be satisfied as . Thus, hypothesis does not come into existence. It should be

as required; that is, the trivial solution of system (12) is asymptotically stable in mean as to partial variables . This completes the proof.

Theorem 12. Let . Suppose that the following conditions hold.

(B3)There exist positive-definite functions , functions , and constants satisfying the following.

meeting ., where and is defined as (8); is a convex function., is a convex function, , .

(B2)Condition (A2) holds, if is balanced and (29) holds.

Then, function is a Lyapunov-A function for (12). Consequently, the trivial solution of (12) is asymptotically stable in mean as to partial variables .

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 . 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.

Theorem 15. Assume that condition (B3) of Theorem 12 is substituted by the following.

(B4)There exist positive-definite functions , functions , and constants satisfying the following.

. , function gets rid of the restriction of ., where , is defined as (8), , and is a convex function., is a convex function, .

Other conditions remain the same. Then function is a Lyapunov-A function for (12). Furthermore, the trivial solution of (12) is uniformly asymptotically stable in mean as to partial variables .

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 . Now, we only need to prove

Similar to the proof of Theorem 11, here we need to reduce to absurdity. Suppose

Instead

Combining condition (B1)(II) of Theorem 15, we obtain

Therefore,

However, it is obvious that (46) cannot be satisfied as . Thus, hypothesis does not come into existence. It should be

as required; that is, the trivial solution of system (12) is uniformly asymptotically stable in mean as to partial variables . This completes the proof.

Note the fact that . We can still derive another conclusion as follows.

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.

Theorem 17. Assume that condition (B2) of Theorem 11 is substituted by the following.

(B5)There exist positive-definite functions , functions , and constants satisfying the following.

. , function gets rid of the restriction of   ., where , is defined as (8), , and is a convex function., constants , .

Other conditions remain the same. Then function is a Lyapunov-B function for (12). Furthermore, the trivial solution of (12) is uniformly asymptotically stable in mean as to partial variables .

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 . Suppose that the following conditions hold.

(C1)There exist positive-definite functions , functions , and constants satisfying the following.

. , gets rid of the restriction of  ., where , is defined as (8), and and are positive constants., constants , .

(C2)Condition (A2) holds, if is balanced and (29) holds.

Then, function is a Lyapunov-B function for (12), and

where  ; that is, the trivial solution of system (12) is exponentially stable in mean as to partial variables .

Proof. We can show in the same way as in the proof of Theorem 6 that

where . Hence, we conclude that function is a Lyapunov-B function for (12). Integrating system (12) as to spatial variables gives

Combining Green formula and boundary condition we deduce . Hence, (51) can be rewritten as

Since is continuous and , we have that is continuous and . Hence, there exists such that when . Choosing and applying It differential formula to along the trajectory of system (17) yields

where

for . Hence,

Build a Lyapunov function with the form

Applying It formula again, we obtain

For any , define a stop-time

It follows from integrating (57) as to from to

As  /  is continuous on and satisfies integral linear growth condition, there must exist constant such that , so

By Theorem 2.8 of [35], we obtain

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 , it yields

In view of Condition (A2) and the fact that , we get

Thus 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.

Theorem 19. Let . Suppose that the following conditions hold.

(C1)There exist positive-definite functions , functions , and constants satisfying the following.

. , function gets rid of the restriction of  ., where , is defined as (8), and and are positive constants., constants , , .

(C2)Condition (A2) holds, if is balanced and (29) holds.

Then, function is a Lyapunov-A function for (12), and

where ; that is, the trivial solution of system (12) is exponentially stable in mean as to partial variables .

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 is strongly connected and balanced. Consider

Construct function ,  and we have

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

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.

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