Finite-Time Synchronization of Markovian Jumping Complex Networks with Non-Identical Nodes and Impulsive Effects

In this paper, we investigate the finite-time synchronization problem for a class of Markovian jumping complex networks (MJCNs) with non-identical nodes and impulsive effects. Sufficient conditions for the MJCNs are presented based on an M-matrix technique, Lyapunov function method, stochastic analysis technique, and suitable comparison systems to guarantee finite-time synchronization. At last, numerical examples are exploited to illustrate our theoretical results, and they testify the effectiveness of our results for complex dynamic systems.


Introduction
Over the past decades, owing to the wide applications in engineering and science, for instance, the Internet, power grid networks and communication networks, the search efforts of complex networks have improved considerably [1][2][3][4][5]. Complex networks consist of a great deal of interconnected nodes. As a basic unit, every node has special dynamic behavior different from other nodes. Different types of complex networks have different types of connections. Besides, whether or not the nodes have weight also determines the types of complex networks. The complexity of complex networks is not only manifested in the topological, but also reflected in dynamic movement of the network nodes. Because of the complexity of the network, it has given rise to wide attention from scholars in different fields. In particular, as a hot topic, the synchronization of network nodes has been studied by researchers in kinds of fields [6][7][8][9].
Many publications have presented results concerning asymptotic synchronization of complex networks [9][10][11][12][13][14][15]. Only when the time reaches infinity can the state of the coupling system reach synchronization. That is called asymptotic synchronization. However, asymptotic synchronization is not keeping with practical logic because apparatus and human beings have a limited life span. Consequently, convergence rate is a greatly important index when we study synchronization of complex networks. Therefore, the faster a complex network achieves synchronization, the less resources it consumes.
A notably effective method for achieving faster synchronization is to use a finite-time control technique. The finite-time control technique has several advantages [16,17]. In a settling time, this method can synchronize all nodes in the network. Finite-time control has not only improved robustness of systems but also ameliorated disturbance rejection properties [18][19][20][21][22]. As a result, the finite-time control technique is a more desirable method for achieving synchronization, and it has been widely studied.
In real applications, it is noted that communication signals may be suddenly significantly changed at one time when signals are transmitted. This phenomenon is called impulsive effects [12,20,23,24].
The effects of impulse may promote synchronization or damage synchronization, or yield no effect on synchronization. Recently, based on the concept of average impulsive interval, in a unified framework, the authors in [12,23] studied the synchronization of complex networks with synchronous and desynchronization impulses, respectively. Because the synchronizing impulses are instrumental in synchronization of dynamic systems, many scholars have designed all kinds of impulsive controllers to achieve synchronization of complex networks [12]. In [25], authors proposed an impulsive control method to realize exponential synchronization of coupled complex networks. However, according to [26,27], the desynchronizing impulses which are called impulsive disturbance should not happen too frequently. Because of the instantaneous perturbations, the influence of impulsive disturbance is not negligible. Actually, based on analysis about the stability of impulsive system, a great of results only relate to the asymptotic synchronization, and the finite-time synchronization is less covered. Therefore, studying the finite-time synchronization of dynamic systems with impulsive effects is of vital importance to understand the behaviour of many dynamic networks in real life.
In addition, as a suitable mathematical model for describing a kind of complex network in which the topology is stochastic variation all of a sudden, Markovian jumping complex networks (MJCNs) have given rise to extensive attention among scholars [27,28]. Furthermore, considering that there is a kind of complex dynamic network with finite modes, these modes can switch from one to another at a certain moment. As a result, we consider that it can be looked upon as the frame of theory of the Markovian switching system. In this framework, the Markovian chain controls the switching between different nodes. Many interesting results about synchronization with Markovian chain networks have been reported [29][30][31][32]. In [30,31], the authors indicate that the complex dynamic networks with the Markov jumping topology structure can realize asymptotic synchronization. In [32], authors investigated a stability problem of a class of discrete-time Markovian jumping system via state and mode feedback control. To our best knowledge, by introducing uncertain parameters, the influence of uncertainties in Markovian chains is overcome by a typical method that results in dependence on uncertain-parameter norm of asymptotic synchronization [29][30][31]. However, we need to synchronize the network in finite time in real world. Hence, it is of great importance that we study the finite-time synchronization of MJCNs to meet the needs of the real world.
Note that amounts of published works involved the synchronization of complex dynamic networks usually assume that a series of nodes in the complex dynamic network are identical. From the view of real world, this assumption is impractical for complex networks including all kinds of nodes that usually have different physical parameters [33]. As far as we know, in the previous literature, authors have extensively studied synchronization problems of complex networks with non-identical nodes [34][35][36]. In [34], impulsive complex networks with delayed nonidentical nodes were investigated by Razumikhin technique, a convex combination technique and time varying Lyapunov function. In [36], the authors designed a set of new controllers to achieve finite-time synchronization of dynamic systems with non-identical discontinuous nodes. However, it is remarkable that the problems of finite-time synchronization of non-identity-nodes systems with impulsive effects and markovian jumping topology has not been appropriately investigated.
Motivated by these ideas, it is our task to discuss the finite-time synchronization of Markovian jumping complex dynamic networks with non-identical nodes and impulsive effects. What's more, a summary of the main contributions in this paper can be listed as follows: • The article considers a class of Markovian jumping complex dynamic networks with non-identical nodes and impulsive effects. The system model is more comprehensive and closer to engineering practice. The finite-time method has such outstanding disturbance rejection capability that the results of this subject are of great significance.

•
We propose a new one-norm-based Lyapunov function to solve the difficult points induced by non-identical nodes and impulsive effects. Also, we use the monotonicity to analyze the finite-time synchronization instead of the traditional theorem, and settling time can be theoretically estimated for a given network.
• Without drawing into any uncertain parameters, the finite time synchronization of dynamic systems is guaranteed by using the stochastic analysis technique, the M-matrix technique and some effective conditions.
The rest of this article is organized as follows. In Section 2, model description and preliminaries are presented. Section 3 introduces the problem of finite-time synchronization of MJCNs with non-identical nodes and impulsive effects, and the convergence analysis is fully derived. In Section 4, we provide a detailed numerical simulations to demonstrate the effectiveness of results. The conclusion is drawn in the last.
Notations: There are some standard notations throughout this paper. 1 n is a column vector with all the elements being invariant constant 1. R n×m denotes the set of all n × m matrices and R n is the n-dimensional Euclidean space. · 1 represents the one-norm of a matrix or a vector, i.e., . , x n ) T ∈ R n and, A 1 = max j ∑ n i=1 a ij for A = a ij n×n ∈ R n×n . sgn (·) denotes the sign function. N + denotes the set of positive integers and real numbers.
Let Ω, F , {F t } t 0 , P be a complete probability space with filtration {F t } t 0 satisfying the usual conditions (i.e., the filtration contains all P-null sets and is right continuous), and E {·} represents the mathematical expectation operator with respect to a given probability measure P.

Model Description and Preliminaries
The following preliminaries and necessary assumptions about the system are used throughout this article in this section. Next, we briefly outline the problem formulation. Let {σ(t), t ≥ 0} be a right continuous Markovian chain on the probability space Ω, F , {F t } t 0 , P which takes values in the finite state space S = 1, 2, . . . , s with generator Θ = θ ij s×s (i, j ∈ S) given by where ∆t > 0 and lim ∆t→0 o(∆t) The matrix Θ is assumed to be irreducible as a typical hypothesis. Another way to think about this amounts to the situation that, for any i, j ∈ S, we could get i 1 , i 2 , ..., i k ∈ S and j 1 , j 2 , ..., j k ∈ S such that θ i 1 j 1 , θ i 2 j 1 , ..., θ i k j 1 , θ i k j 2 , ..., θ i k j k are all positive.
Consider the following nonlinear coupled dynamic systems consisting of N non-identical nodes with Markovian jumping. The ith node is an n-dimensional coupled dynamic system, i.e., where ) T is a continuously nonlinear dynamic function, i = 1, 2, ..., N; and C(σ(t)) = (c ij (σ(t))) N×N ∈ R N×N is the outer coupling matrix of complex dynamic network, which is defined as follows: if there is a connection between node j and node i(i = j) at time t, then c ij (σ(t)) > 0, otherwise, c ij (σ(t)) = 0. We give a definition about the diagonal elements as is the Markovian process in continuous time, which is represented by the evolution of the node at time t. Φ = diag(φ 1 , φ 2 , ..., φ n ) ∈ R n×n is an inner coupling matrix with φ l ≥ 0, l = 1, 2, ..., n.
The main purpose of this article is to synchronize all of the non-identical nodes of system (1) with the following isolate node: with the initial value z(0), where z(t) = (z 1 (t), z 2 (t), . . . , z n (t)) T ∈ R n denotes state of the given isolated node, and g(z(t)) = (g 1 (z(t)), g 2 (z(t)), ..., g n (z(t))) T is a differentiable nonlinear dynamic function with continuity.
Furthermore, in the signal transmission process, signals may be suddenly changed at some discrete time [12,23]. The state x i (t), i = 1, 2, ..., N with step change can be represented as a class of differential equation with impulsive effects. Note that the impulsive effects widely found in many processes may potentially have a significant impact on synchronization. Hence, we consider the controlled complex network with impulsive effects as follows: is an impulsive gain. Furthermore, in this paper, it is assumed that the pulse phenomena does not rely on the Markovian chain.

Remark 1.
Signals may be suddenly changed in the form of impulses during signal transmission at some discrete time t k , and the impulsive gains can be different from node to node. Additionally, at different impulsive instants, they may be diverse. Thus, . Impulsive effects are considered as synchronous impulses and desynchronous impulses in this paper, respectively. For analytical simplicity, we should note the impulsive gains as |α k Before presenting the main results, some basic assumptions and definitions are presented as follows: Assumption 2. There exist positive constants M ij and L j such that, respectively, for i = 1, 2, ..., Definition 1 ([37]). The MJCNs (3) is said to be synchronized onto (2) in finite time by putting in a proper designed controller u i (σ(t), t), if there exists a constant T 1 > 0, where T 1 relies on the initial state x(0) = (x T 1 (0), ..., x T N (0)) T , z(0) and the initial value of the Markovian chain σ(0), such that lim t→T 1 E( e i (t) 1 ) = 0 and E( e i (t) 1 ) ≡ 0 for t ≥ T 1 , i = 1, 2, ..., N. T 1 is called as the settling time in this paper.
there exists a positive integer N 0 and a positive constant T a such that for anyt > t ≥ 0, N ς (t,t) represents the number of impulsive times of the impulsive sequence ς on the interval (t,t).

Definition 3 ([38]).
A nonpositive real matrix A = (a ij ) n×n ∈ R n×n with a ij ≤ 0 for i = j is known as Minkowski matrix (M-matrix for short) if all the eigenvalues of A have positive real parts.
Some significant lemmas are listed in the following part and they are required to obtain the results in this paper.

Lemma 1 ([38]
). The following descriptions are equivalent if A = (a ij ) n×n ∈ R n×n with a ij ≤ 0(i = j): (1) All the eigenvalues of A have positive real parts. (2) A is a nonsingular Minkowski matrix (M-matrix).
A −1 exists and all the elments of A −1 are nonnegative.

Finite-Time Synchronization of the Complex Networks
In this section, a theorem is presented to synchronize the MJCNs (3) with non-identical nodes and impulsive disturbances onto the system (2). Moreover, the setting time is theoretically estimated.
The value α k i will lead to α > 1, α = 1, 0 < α < 1. Next, we give the provement about estimating the settling time T 1 , and there are three cases to be considered in the following part.

Remark 2. As a classic example, the upper bound and lower bound of impulsive intervals are demonstrated by the Example 3 in the reference
where ε and T a denotes positive numbers satisfying ε ≤ T a ; mod(k, N 0 ) represents the remainder of k dividing by N 0 and N 0 ∈ N + . According to (32), we can obtain that inf k∈N + {t k − t k−1 } ∈ ε and sup k∈N + {t k − t k−1 } = N 0 (T a − ε) + ε. The quantity inf k∈N + {t k − t k−1 } will be quite small, and the quantity sup k∈N + {t k − t k−1 } will be very big if ε is small enough and N 0 is sufficiently large. In such a situation, two types of impulsive effects, namely, synchronizing impulses and desynchronizing impulses are considered in this article. In order to ensure the finite-time synchronization desynchronizing impulses shouldn't occur frequently. Given the analysis provided above, our criterion in Theorem 1 is proper for impulsive control signals which have a wider range. Compared with the results deriving by sup k∈N + {t k − t k−1 }, our results are less conservative. To guarantee that the impulses do not occur too frequently, the positive integer N 0 will not be too large. Accordingly, the impulsive interval T a will not be too small.
It is worth noting that our work is affected by the maximum average impulsive intervals and the minimum average impulsive intervals. Are the networks (3) synchronized to the isolate system (2) in finite-time with the maximum average impulsive intervals and the minimum average impulsive intervals? To answer these questions it is worthwhile to investigate the influence of the maximum average and the minimum average impulsive intervals.
Respectively, we replace T a with T andT, and the inequality (6) will be adapted to the following form for any t > s ≥ 0, where respectively. There exists a positive number α such that 1 < |1 + α k i | ≤ α, i ∈ 1, 2, ..., N, and satisfied conditions are listed as follows: for σ ∈ S, i = 1, 2, ..., N, then complex dynamical network (3) with controller (9) is synchronized onto system (2) in a finite time is the initial value of the Markovian chain.
When v(t) > 0, according to (26), we can deduce that In the case of α > 1, it can be deduced on the basis of (33) and (37) that Compute the derivative of f (t) Then, letḟ (t) = 0, we can find thatt =T T ln Then we can obtain that f (t) < 0.
Furthermore,since v(0) > 0, we can get According to the existence theorem of zero point, for t ∈ [0,t], we can get that f (t) has one zero point T 1 .
The proof is similar to that of Corollary 1, so we omit it.
Remark 3. Corollary 2 and Theorem 1 indicate that average impulsive interval does not matter to the finite-time synchronization when 0 < α < 1.

Numerical Examples
In this section, numerical simulations are provided to demonstrate that the applications of the theorem described above are valid. Using the similar example as that given in [33], we also consider two different chaotic node systems as follows: where Figures 1 and 2   In the following, consider a coupled complex network being composed of two types of non-identical chaotic nodes (43) and (44), which are described such that: whereẋ (1, 1, 0) , and the outer coupling matrix is supposed to be Moreover, the transition probability matrix is given as Θ = According to Assumption 1, we can verify the matrix which is similar to [33] satisfying with We synchronize (45) onto the following isolated node system: where z (t) = (z 1 (t) , z 2 (t) , z 3 (t)) T . The initial condition is taken as z (0) = (−0.3, 0.1, 0.12) T . Figure 4 shows the chaotic trajectory of (46).   In the following section, we verify Theorem 1. The parameters in the simulations are taken as step-length at 0.0001. We choose α k 1 = α k 2 = 0.1, α k 3 = 0.2 for any k ∈ N + because of the impulsive disturbance. Accordingly, we use α = 1.2. Additionally, we choose η = 43.8981 and σ (0) = 1. By simple computation, we obtain η * = 10. The condition (12) in Theorem 1 implies that T a > 0.004898 should be satisfied. Consider T a = 0.0049, N 0 = 5 According to the Equation (13), the complex network is synchronized with the isolates node within T 1 = 0.5245. The synchronization trajectories e i (t) 1 , i = 1, 2, 3 are shown in Figures 5-7. The figures depict that the complex network (3) is synchronized with (2) before t = 0.09.

Remark 5.
It is well known that time is not negative, therefore, T a in (12) of Theorem 1 should not be too small. Otherwise, this kind of statements will reflect a real world situation and will not be in synchronization.

Remark 6.
In Corollary 1, we choose the same data for the simulation. We obtain the result T > 0.00114, and T a = 0.0049. From the simulation data, we can get inf k∈N + {t k − t k−1 } ≤ 0.0009. Obviously, our criterion is suitable for a wider range of impulsive signals. Because the impulsive effect discussed in this paper is considered to be disturbance, the impulses should not occur frequently. Comparing the range of T a and T, we see that an average impulsive interval of T a is better than the average impulsive interval of T.

Conclusions
In this paper, we have studied the finite-time synchronization of the MJCNs with non-identical nodes and impulsive effects. Sufficient conditions for the MJCNs are presented based on the M-matrix technique, the Lyapunov function method, the stochastic analysis technique and suitable comparison systems to ensure the finite-time synchronization of the communication systems. We show that the MJCNs with non-identical nodes and impulsive effects can reach synchronization within a finite time through rigorous mathematical derivation and simulation examples.
In the real world, we realize that a more general class of complex networks does not have complete information. Therefore, we would further investigate how to synchronize the complex networks with incomplete information in finite time.