Pinning Stabilization of Complex Networks Coupled with Time Delay and Disturbed with Stochastic Noise

A pinning stabilization problem of complex networks with time-delay coupling is studied under stochastic noisy circumstances in this paper. Only one controller is used to stabilize the network to the equilibrium point when the network is connected and the minimal number of controllers is used when the network is unconnected, where the structure of complex network is fully used. Some criteria are achieved to control the complex network under stochastic noise in the form of linear matrix inequalities. Several examples are given to show the validity of the proposed control criteria.


Introduction
Complex networks have been a major research topic and attracted increasing attention from various fields including physics, biology, sociology, and engineering.Many real phenomena can be described as complex networks, such as the World Web, telephone call graphs, and social organization.Recently, the stabilization and synchronization problem and stabilization problem of complex network have become more and more important.
In fact, the synchronization problem is a special stabilization one since it can be converted into the stabilization problem of the error system between the complex network and the synchronization manifold [1][2][3].Specially, the synchronization problem is a stabilization one when the synchronization manifold is an equilibrium orbit.So in this paper, we study the stabilization problem of complex networks which can be extended to the synchronization problem.Many contributions on complex network synchronization or stabilization are derived on the basis of the inner coupling strength adjustable [4][5][6]; that is, the whole network can synchronize or stabilize by itself.
However, it is true that the inner coupling strength sometimes cannot be adjustable for a complex network.Consequently, the whole network cannot be synchronized or stabilized by itself [7].Therefore, some additional controllers have to be applied to force the network to be synchronized or stabilized.How many controllers are added to stabilize the complex network?Adding the controllers to all the nodes is the most simple but costly and impossible due to the complexity of network.To reduce the number of controlled nodes, some local feedback injections are applied to a fraction of networks nodes, which is called pinning control [8][9][10][11][12].Wang and Chen found that specific pinning of the nodes with larger degree required a smaller number of controlled nodes than the random pinning for a scale-free network [8].Li et al. proposed the virtual control method for microscopic dynamics throughout the process pinning control to stabilize a complex network to its equilibriums [13].Topology is important for the network synchronization.Chen et al. used a single controller to synchronize the complex network with irreducible topology matrix and a minimal number of controllers for network with reducible topology matrix [9].Lu et al. extended the results obtained in [9] to linearly coupled neural network perturbed by stochastic noise and pinning stabilized the neural networks to homogenous solutions [14].In [9,14], the inner coupled matrix of network must be diagonal and positive.How to deal with the more general coupled way?Zhou et al. proposed a scheme of determining the number of pinning controlled nodes for general complex networks with positive definite inner coupling [15].

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Journal of Control Science and Engineering All of the above-mentioned contributions focus on the complex networks coupled with no time delaying.However, there often exists time delaying when the signals are transmitted over networks [10,11,[16][17][18][19][20].Therefore, we have to consider the complex network coupled with time delaying.For the coupling matrix being irreducible, the pinning stabilization criteria were proposed in the form of liner matrix inequality for a complex network coupling time delaying [10,11].However, it is regretted that the method of selecting the controlled node number is not given.All of the nodes needed to be controlled in a complex network coupled with time delaying [16,17,20].How to select the controlled node for a complex network coupled with time delaying?Motivated by the above discussion, we study the pinning stabilization problem on complex networks coupled with time delaying and disturbed by the stochastic noise from three kinds of topology matrices: symmetrical and irreducible, asymmetrical and irreducible, and reducible.Only one controller is used to stabilize a complex network coupled with irreducible matrix and only  controllers are used to a complex network coupled with -irreducible matrix.Some stabilization criteria are derived by using Llyapunov stability theorem and stochastic analysis.
The rest of this paper will be organized as follows.In Section 2, the model formulation of linearly delaycoupled complex networks with a noise perturbation and some preliminaries will be presented.In Section 3, the pinning stabilization problem of complex networks with, respectively, symmetric irreducible coupling matrix, asymmetric irreducible coupling matrix, and -reducible coupling matrix will be studied.Some criteria will be derived in terms of linear matrix inequalities (LMIs) to guarantee the success of designed controllers in Section 3. In Section 4, several computer numerical simulations will be given to show the effectiveness of the proposed stabilization approach, and Section 5 concludes the investigation.
Notations.The standard notations will be used in this paper.Throughout this paper, for real symmetric matrices  and , the notation  ≤  ( < ) means that the matrix  −  is negative semidefinite (negative definite).  is the identity matrix of order .We use  min (⋅) and  max (⋅) to respectively denote the minimum and maximum Eigen value of a real symmetric matrix.The notation ‖‖ denotes Euclidean norm of vector , and max  {  } means the maximum element of vector .R denotes the set of real numbers.R × denotes the  ×  real matrices.diag{⋅ ⋅ ⋅ } stands for a blockdiagonal matrix.The superscript "" represents the transpose of the matrix.{⋅} denotes the mathematical expectation operator.=     = 1, 2, . . ., .  denotes the time delay of the networks coupling, () = ( 1 (),  2 (), . . .,   ())  ∈ R  is an -dimensional Brownian motion defined on a complete probability space (Ω, , ) with filtration {  } ≥0 satisfying the usual conditions (i.e., the filtration contains all -null sets and is right continuous).Here, the white noise   () is independent of   () for  ̸ = , and : R + × R  × R  → R × is named the noise intensity function matrix.The stabilization controllers   () ( = 1, 2, . . ., ,  ≤ ) will be determined and the number of the controlled nodes will be given in Section 3, which can stabilize the network (1) to equilibrium in mean square.

Model Formulation and Some Preliminaries
For the function (⋅), one has the following assumption.
Remark 1. Network (1) is different from the one considered in [9][10][11].Here, network (1) is disturbed by a stochastic noise and coupled with time delaying.However, the transmitting time delaying is not considered in [9] and the stochastic perturbation is not considered in [10,11].
Assumption 2. The function (⋅) of the complex networks satisfies the following Lipschitz condition: where   is a positive constant for  = 1, 2, . . ., .For convenience, let  = diag{ 1 , where  1 and  2 are known constant matrices with compatible dimensions.
Remark 4. Condition (3) on the noise density function matrix guarantees that the elements of (,   (),   ( − )) are bounded and differential [21].This assumption has been widely used for the stability analysis of stochastic differential equations [22,23].
Remark 6. Assumption 5 has been used in neural networks and complex networks [9,14,20].This diagonal condition is easy and convenient to achieve the controller.And this diagonal condition is extended to the general coupling since a lot of matrix can be diagonalized.
The following definitions and lemmas are required for the derivation of our main results in this paper.
Definition 7 (see [14]).Matrix  is said to be reducible if it can be transformed to a matrix of the form by the same permutation of the rows and columns, where  1 and  2 are square matrices and  is a null matrix.Definition 8 (see [14]).The pinning controlled network ( 1) is said to be globally exponentially stabilized at the original point in mean square, if for any given initial condition, there exists positive constant  0 and  such that where {⋅} is the mathematical expectation.
Definition 10 (see [24]).Consider a reducible matrix  of order .The matrix is -reducible if it is diagonal.For 1 ≤  < , the matrix  is -reducible if it is not ( + 1)reducible and it can be rewritten in the following Frobenius normal form after certain permutations: where   ( = 1, 2, . . .,  + ) are square irreducible matrices, and for each  ≤ , there exists  >  such that   ̸ = 0.

Main Results
In this section, the pinning controllers are designed for complex network according to the structure of the network topologies (resp., symmetric irreducible, asymmetric irreducible, and -reducible).
Case 1 ( is a symmetric irreducible coupling matrix).If the coupling matrix  is a symmetric irreducible matrix, it means that the corresponding coupling network is undirected and strongly connected.The pinning controllers are designed as follows: where  is a positive constant; that is,  > 0.
From (5), we can know that only the first node of the complex network (1) is selected to be controlled to stabilize the complex network (1) with symmetric irreducible coupling matrix in mean square.By using controller (5), the complex network (1) is controlled through pinning.Then the pinned network (1) can be rewritten as follows: where the elements c of matrix C are defined as follows: Then the following theorem gives the stabilization criteria network (1).
From Assumption 3 and LMI (9), the following inequality can be obtained: Equality ( 12) can be simplified to the following due to ( 13)- (15):  = 0. Then we can prove that  max ( C ) < 0 by using Lemmas 11 and 13.From Lemma 15, it is known that all Eigen values of the matrix C are negative.Then  max ( C ) < 0 can be obtained.Since  max ( C ) < 0, LMI (8) is possible.
Since (0) = ∑  =1    (0)  (0), the following holds: Further, the following is obtained: where  0 =  max ()/ min ().By Definition 7, it can be concluded that the pinning controlled complex network (3) with a symmetric irreducible coupling matrix C is globally exponentially stable in mean square, which implies that complex network (1) with symmetric irreducible coupling matrix  has been globally exponentially stabilized to the origin point in mean square by injecting the single controller (7) to the first node of complex network (1).The proof of this Theorem 16 is completed.
Case 2 ( is an asymmetric irreducible coupling matrix).If the coupling matrix  is an asymmetric irreducible matrix, it means that the corresponding coupling network is directed and connected.
Taking the mathematical expectation of both sides of (30), one can obtain the following: By following the proof of Theorem 16 and referring to Definition 7, it can be concluded that the controlled complex network (1) with an asymmetric irreducible coupling matrix C and with controller ( 5) is globally exponentially stable on the origin point in mean square, which implies that complex network (1) with asymmetric irreducible coupling matrix  has been globally exponentially stabilized by using a single controller.The proof of Theorem 17 is completed.
Case 3 ( is an -reducible coupling matrix).If the coupling matrix  is an -reducible matrix, it means that the corresponding coupling network is weakly connected.Therefore, it is not able to control complex network (1) to the origin point in mean square by using only one controller.It is important for us to design the minimum number of controllers to pinning control network (1).
From Definition 10, a complex network with -reducible coupling matrix can be disconnected.For example, let  = 2,  ,+2 = 0 for  = 1, 2, . . ., , and let  ,+1 ̸ = 0 for  = 1, 2, . . ., .Then we can make the corresponding network with such coupling matrix disconnect with two components.Therefore, any network can be represented by an -reducible matrix, which implies that the result in this section is quite general.
Therefore not less than  controllers is needed for the pinning synchronization of the  self-governed error subnetworks   for  + 1 ≤  ≤  + , which implies that at least  controllers are required to stabilize complex network (1) with -reducible coupling matrix.
controllers are designed to exponentially stabilize complex network (1) with -reducible coupling matrix.The first node of each error subnetwork   for  + 1 ≤  ≤  +  is chosen to be controlled, and the  controllers are designed as follows: Then, we can obtain the error networks by using the controllers (33) to control complex network (1), which can be described by (31) and the following: where ,  ∈   .Then, the error subnetworks (36) can be rewritten as where , and ( C  ) is given in Definition 9, then the controlled network (1) with an -reducible coupling matrix  is globally exponentially at the drive network (1) in mean square.
Proof.For  + 1 ≤  ≤  + , since LMIs (38) hold, from Theorem 17, it can be concluded that every controlled subnetwork with symmetric irreducible coupling matrix   or asymmetric irreducible coupling matrix   is globally exponentially at the drive subnetwork in mean square; that is, the error subnetworks   ( + 1 ≤  ≤  + ) can be globally exponentially synchronized.It means that the states of network (1) can be constringed to 0; that is, lim  → ∞ () → 0.
Therefore complex network (1) with -reducible coupling matrix can be stabilized to the origin point by designing at least  controllers.The proof of Theorem 18 is completed.

Numerical Examples
In this section, two examples are given to show the effectiveness of the proposed synchronization scheme.The networks are composed of ten coupled nodes, and each node satisfies the chaotic Lorenz system as its dynamics.A chaotic Lorenz system can be described as follows: where , , and  are parameters.Let  = 10,  = 28, and  = 8/3; the system can have a chaotic attractor.The attractor of the first node is shown in Figure 1. Figure 1(a) shows the attractor of the three states of the first node without the noise perturbation, and Figure 1(b) shows the attractor with the same initial condition and with the noise perturbation.And referring to [13], we can obtain that Assumption 2 is satisfied by using this chaotic Lorenz system.
Example 1.We consider a complex network consisting of ten nodes which are described by Lorenz system.Assume that  complex network, which further means that the nodes 3, 6, and 8 (first node of subnetworks 2, 3, and 4) are to be controlled.And the controlling gains of the three controllers are all taken as  = 0.026, and the time delays of the controllers are also taken as  = 50.Figure 4 shows ten trajectories of the stabilized states   () ( = 1, 2, 3;  = 1, 2, . . ., 10) of the complex network.It can be concluded that all of the states   () ( = 1, 2, 3;  = 1, 2, . . ., 10) tend to zero, which implies that the complex networks coupled with reducible matrix can be stabilized by utilizing the minimize number of controllers.
Remark 19.It can be inferred that the stabilization criteria obtained in Theorems 16-18 are effective for complex networks coupled with time delays and disturbed by the stochastic noise.Only one controller is used for a connected network to stabilize it.For an unconnected network, we can firstly portion it as several connected subnetworks; then only one controller is used for the connected subnetwork and therefore the unconnected network is stabilized by the minimal number of controllers.

Conclusion
The stabilization problem of time-delayed coupling complex networks with a stochastic perturbation by utilizing pinning controllers is studied in this paper.A minimal number of controllers are designed to force the states of complex networks to the origin point in mean square by fully using the structure of the network topology matrix.Some stabilization criteria are achieved to pinning control the complex network, which are described in the form of LMIs.Some examples are given to show the effectiveness of the proposed pinning controller in this paper.

Figure 4 :
Figure 4: The states of complex networks with an -reducible coupling matrix .
or asymmetric irreducible coupling matrix   ; therefore, we can use the same way of Case 1 or Case 2. Similarly to Theorem 16 and Theorem 17, we can obtain the following.  a diagonal positive-definite matrix   = diag{  1 ,   2 , . . .,    } > 0 such that the following LMIs hold