Abstract
This paper is concerned with the global finite-time and fixed-time synchronization for a class of discontinuous complex dynamical networks with semi-Markovian switching and mixed time-varying delays. The novel state-feedback controllers, which include integral terms and discontinuous facts, are designed to realize the global synchronization between the drive system and response system. By applying the Lyapunov functional method and matrix inequality analysis technique, the global finite-time and fixed-time synchronization conditions are addressed in terms of linear matrix inequalities (LMIs). Finally, two numerical examples are provided to illustrate the feasibility of the proposed control scheme and the validity of theoretical results.
1. Introduction
In the past ten years, complex dynamical networks (CDNs) have aroused a great attention owing to their extensive applications in many areas such as the Internet, biological system, the World Wide Web, power grid networks, and so on [1–3]. CDNs consist of many subnetworks, which are called nodes, and each of the nodes has the similar functions. Different connection modes between nodes form different networks. Thus, such a complexity makes that the study of CDNs become a hot topic. Up to now, many significative works with respect to CDNs are reported in the literature, see [4–10] and references therein. It should be mentioned that the analysis of dynamical behaviors of CDNs has become a research hotspot in recent years.
The global synchronization, as a dynamical behavior, is the most important. It means the dynamical behavior of the network systems achieve the identical state in time-spatial. Nowadays, the considerable attention is being devoted to the analysis of the global synchronization of CDNs, and some effective synchronization criteria with respect to CDNs have been established in the existing papers [11–15]. In [11], the synchronization problem of complex dynamical networks was addressed under the way of impulsive control. Reference[12] introduced the synchronization problem of complex dynamical networks with switching topology via adaptive control. Also, global synchronization of complex dynamical networks with nonidentical nodes was considered in [13]. Moreover, synchronization of complex dynamical networks with nonidentical nodes was illustrated in Ref. [14], and the obtained result global synchronization for discrete-time stochastic complex networks in Ref. [15] was complex, which has randomly shown nonlinearities and mixed time delays. To the best of our knowledge, most of the existing works with regard to the synchronization of CDNs is taking the continuous networks system into account. But, in practice, it is not applicable to all the synchronization conditions. Thus, the study of discontinuous CDNs has attracted the interests of many scholars. Reference [16] investigated the finite-time synchronization of complex networks with nonidentical discontinuous nodes. Also, exponential synchronization of a discontinuous complex-valued complex dynamical network is realized in [17]. Reference [18] illustrated the synchronization of time-delayed complex dynamical networks with discontinuous coupling.
As we all know that, among the different types of synchronization, global finite-time synchronization is one of the most optimal synchronization types, only the maximum synchronization time for global finite-time synchronization can be computed. Moreover, some efforts have been spent on solving the global finite-time synchronization of discontinuous CDNs. The finite-time synchronization of chaotic complex networks with stochastic disturbance was obtained in [19]. In Ref. [20], the finite-time hybrid projective synchronization of the drive-response complex networks with distributed delay was considered by the researchers, based on the adaptive intermittent control. Reference [21] introduces finite-time cluster synchronization of Markovian switching complex networks with stochastic perturbations. In Ref. [22], the finite-time synchronization of Markovian jump complex networks with partially unknown transition rates was introduced by the author. It is also necessary to point out that the settling time of finite-time synchronization heavily depends on the initial conditions which would result in different convergence rates under different initial conditions. But, the initial conditions may be ineffective or invalid in practice application. For overcoming these difficulties, a new concept named fixed-time synchronization is firstly taken into consideration in [23]. Then, the future study on fixed-time synchronization can be found in [24,25]. By means of the sliding mode control technique, the fixed-time synchronization of complex dynamical networks is realized in [24]. In [25], the fixed-time cluster synchronization for complex networks is derived under the pinning control scheme. Furthermore, in [25], accompanying with nonidentical nodes and stochastic noise perturbations, the fixed-time synchronization of complex networks was investigated. Also, the fixed-time synchronization of hybrid coupled networks was addressed in Ref. [26].
By adding the Markovian process into the network systems of CDNs, a new network model is developed. Up to now, the study concerning synchronization of CDNs with Markovian switching, especially the global finite-time synchronization, has received wide attention from the scholars, and many efforts have been made for analyzing the synchronization of CDNs with Markovian switching, see [21, 27–30]. Since the sojourn time of the Markovian process obeys exponential distribution, it results in the transition rate to be a constant. However, the sojourn-time of the semi-Markovian process can obey to some other probability distributions, such as Weibull distribution, Gaussian distribution, and so on. Therefore, the investigation for the global finite-time and fixed-time synchronization of CDNs with semi-Markovian switching is of great theoretical value and application value. In [31], the finite-time synchronization for complex networks with semi-Markov jump topology was addressed. The finite-time control for linear systems with semi-Markovian switching was discussed in [32–34]. The finite-time nonfragile synchronization of stochastic complex dynamical networks with semi-Markov switching outer coupling was investigated in Ref. [35]. The global fixed-time synchronization of CDNs with semi-Markovian switching is relatively complex. In addition, it is also difficult to apply the semi-Markovian switching to complex dynamic network (CDNs) and use appropriate control methods to derive the corresponding global synchronization conditions. As a result, the global fixed-time synchronization of CDNs with semi-Markovian switching is seldom studied before.
Motivated by the abovementioned discussions, we intend to consider the global finite-time and fixed-time synchronization for discontinuous CDNs with semi-Markovian switching and mixed time-varying delays. The main contributions of this paper, different from the existing works, can be summarized as follows:(1)The novel discontinuous state-feedback controllers, which include integral terms, are designed(2)The mixed delay term with integral and discontinuous function is handled by the approach of the contraction of inequalities(3)The mixed delay and semi-Markovian switching are introduced in the construction of the discontinuous CDNs model(4)The global finite-time and fixed-time synchronization conditions are addressed in terms of LMIs
The rest of this article is arranged as follows. Some preliminaries and model description are described in Section 2. In Section 3, we introduce the main results: finite time and fixed-time synchronization with different nonlinear controllers. In Section 4, two examples are presented which are in order to show the correctness of our main results. In Section 5, also the last part, the conclusion of this paper is given.
Notation: denotes the identity matrix with proper dimension. represents the sets of real numbers. denotes the set of all matrices, and denotes the n-dimensional Euclidean space. stands for the Euclidean norm of the vector , and . denotes an -dimension matrix , and the superscript means transposition. For all the matrices, stands for the minimum and the maximum eigenvalue of the matrices, respectively., where and are both symmetric matrices, means that is negative (positive) definite. and stand for mathematical expectation. . denotes the infinitesimal generator of . Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operation.
2. Preliminaries and Model Description
2.1. Preliminaries
Let be the complete probability space with filtration satisfying the usual conditions (i.e., it is increasing and right continuous while contain all -null sets), where is the sample space, is the algebra of events, and is the probability measure defined on . Let be a continuous-time semi-Markovian process taking values in a finite state space . The evolution of the semi-Markovian process is governed by the following probability transitions:where is the transition rate from mode to and for any state or mode, and it satisfies
Remark 1. In practice, the transition rate is general bounded, i.e., , where and are real constant scalars. Then, , where and with .
Definition 1. The complex dynamical networks model (17) is said to be synchronized with the networks model (18), if for any initial condition, we havefor .
Definition 2. (See [36]). The Filippov set-valued map of at is defined as follows:where is the closure of the convex hull of the set , and is the Lebesgue measure of set .
Definition 3. (See [37]). The set-valued map of is said to satisfy the basic conditions in the domain , if for any , is upper semicontinuous in .
Definition 4. (See [38]). A function is C-regular if it is
Lemma 1 (See [39]). Given any scalar and matrix , there exists symmetric positive definite matrices , such that
Lemma 2 (See [40]). Let . Then, the following two inequalities hold:
Lemma 3. Let be a symmetric matrix, and let ; then, we have the following inequality:
Lemma 4 (See [41]). Given constant matrices , , and , where and , we haveif and only if
Lemma 5 (See [42]). Let be any vectors and be a real number which satisfies the following condition:
2.2. Model Description
The model we consider in the present paper is the complex dynamical network model which consists of identical nodes with semi-Markovian switching in the probability space . The dynamics of nodes of drive systems is described by the following differential equations:with the initial condition .
Also, the corresponding response system iswith the initial condition .
Here, is the continuous-time semi-Markovian process, and describes the evolution of the mode at time . , denotes the state vector of network at time ; is a positive definite diagonal matrix; and are matrices with real values in mode ; is the number of coupled nodes; is the nonlinear vector-valued function; and is the control input of the node to be determined later. represents the distributively delayed connection weight; the diagonal matrix represents the inner-coupling matrix of the complex networks; and stands for the coupling configuration matrix which represents the topological structures of the considered networks. For matrix , if there is a connection from node to , then ; otherwise, .
The function and denote the discrete delay and distributed delay and satisfywhere , and are some known constants.
Throughout this paper, we list the following assumptions: satisfy the general conditions defined in Definition 3, where : there exists two positive constants and , and then, the following inequalities hold:where and
For notation simplicity, we replace , , and with , for . Then, the complex dynamical systems can be rewritten as follows:where is a measurable function and .where is a measurable function and .
The network model (17) is a differential equation with a discontinuous right-hand side, and the traditional definition of the solution is not applicable here. Hence, we introduce the Filippov solution for system (17).
Definition 5 (See [43]). A function is a solution (in the sense of Filippov) of the discontinuous system (17) on if
In this paper, the error dynamics system between drive-response system (17) and (18) attracts us. Hence, let be the synchronization error system, and it can be expressed aswhere , , and .
In the following, some Lemmas about global finite-time and fixed-time synchronization will be introduced.
Lemma 6 (See [44]). Suppose that R is C-regular and that is absolutely continuous on any compact interval of . Let , if there exists a continuous function with for , such thatfor all such that and is differentiable at t and satisfies the condition
Then, we have for . In particular, if for all , where and , then the setting time is estimated as
Lemma 7 (See [23]). If a continuous radially unbounded function satisfieswhere , then , with the settling time bounded by
Lemma 8 (See [45]). If there exists a continuous radially unbounded function such that(1)(2)For some any solution of (20) admitsthen the origin is global fixed-time stable for (20), and the estimated value of the settling time function satisfies
3. Main Results
3.1. Global Finite-Time Synchronization
Based on the linear matrix inequalities technique [45] and the Lyapunov functional method [38, 46], this section aims to develop some new conditions which can ensure the synchronization between complex dynamical network models (17) and (18) over a finite-time interval.
For the purpose of global finite-time synchronization, a state-feedback controller is designed as follows:in which , , and is a tunable parameter, and is a parameter to be determined.
Theorem 1. Let the assumptions and hold. Then, the drive system (17) is synchronized with the response system (18) in finite time, if there exist symmetric positive definite matrices , and , such thatwhere . , , , , and is the eigenvalue of . , . .
Proof. To develop the finite-time synchronization criterion, we consider a stochastic Lyapunov function as follows:Calculating along the trajectory of the error system (20) based on the condition formula, we havewhere is a small positive number. Hence, for every , it can be deduced thatSubstituting (20) into (33), we haveUnder assumption , we obtain thatand similarly, we haveBased on the condition , we havewhere .
Combining (33), (35), and (36), we obtainThen, based on (37), we have the following inequality:Substituting controller (28) into (39),Noting the condition (29) and Lemma 3, we haveIn view of and , according to Lemma 1, we can obtain thatThen, combining (41) and (42), we getBy virtue of the condition of the theorem, we get from inequality (43) thatAccording to Lemma 3, (44) can be rewritten aswhere .
Exploiting condition (29) and (30), we have the following inequality:By means of Lemma 2, we obtain thatThus,Combining (46), (48), and Lemma 3, we getBy Lemma 5, (49) is equal tothat is,whereAs we all know, , for any . Thus, we have the following inequality:Then, based on Lemma 6, the finite-time synchronization of the complex network is finally achieved. At the same time, the settling time is estimated asThus, we can conclude that the drive system (17) synchronizes with the response system (18) in finite time by adopting the controller (28). Also, the settling time is given in (54). The proof of the global finite-time synchronization is finished.
Remark 2. The global finite-time synchronization of discontinuous CDNs with semi-Markovian switching and mixed time-varying delays is considered for the first time in Theorem 1. The approach we get the global finite-time synchronization condition is the matrix inequalities but not the linear matrix inequalities because has a nonlinear term .
3.2. Global Fixed-Time Synchronization
In this part, the global fixed-time synchronization criteria are developed under the following state-feedback controller:where and are the parameters to be determined. and are tunable parameters. .
Theorem 2. Consider network models (17) and (18) with the hypotheses and . For any given constants scalars , the drive system (17) is said to be synchronized with the response system (18) in fixed-time, if there exist symmetric positive definite matrix and , and matrices such thatwhere . , , , and is the eigenvalue of . .where
Proof. Taking the same stochastic Lyapunov functional with Theorem 1,Then, based on (20), (33), and (42), we can obtain the derivative of as follows:Substituting (55) into (60) and adopting Lemma 3, (60) changes intoFrom (35)–(37), (42), and (61), we haveBased on the (56), it yields thatSimilar to (44), we have the following inequality:Then, (64) can be rewritten as follows:where .
Based on the condition (56), we obtain thatwhereBecause of the nonlinear terms, in is difficult to be handled, and we need to turn it into a linear matrix inequality.
Hence, we defineThen, according to the (56), (30) can be rewritten as follows:We note that Lemma 4 implies thatHence, from (70), (66) turns intoBy using Lemmas 2 and 3, we can obtain thatThen, by exploiting Lemmas 2 and 5, we obtainAlso, (73) is equal towhereTaking the expectation on both sides of (74), we can get the following inequality:Similar with Theorem 1, the differential coefficients of V(t) are given as follows:Because , (77) satisfies Lemma 7, and we are in a position to complete the proof of Theorem 2. In other words, the drive system (17) synchronizes with the response system (18) in fixed-time. Also, the settling time is estimated asThus, the global fixed-time synchronization problem is finally addressed under the controller (55). The proof of this theorem is completed.
Corollary 1. Suppose that the hypotheses hold. For given scalars , the drive system (17) synchronizes with the response system (18) in fixed time based on the controller (55), if there exist symmetric positive definite matrices , and positive definite matrix Q such that
where . . is the eigenvalue of . , .where
Meanwhile, on the condition of Lemma 8, the settling time of global fixed-time synchronization is estimated aswhere and are the same with the parameters in Theorem 2. Also, .
Remark 3. There exist many kinds of chaotic system with discontinuous functions, such as the discontinuous Chua circuit, discontinuous Chen system, and neural network with discontinuous activation functions. In this paper, we choose the discontinuous condition that it is applicable to most of the chaotic systems.
Remark 4. Global fixed-time synchronization is more complex than global finite-time synchronization in the synchronization conditions. For global finite-time synchronization, a term such as is only needed, whereas for fixed-time synchronization, we need , to realize it. Comparing with the settling time of the global finite-time synchronization in Theorem 1, the settling time of the global fixed-time synchronization in Theorem 2 is independent of the initial condition . When the is so large, the is not reasonable in practice application.
4. Numerical Examples
In this section, we perform two examples to demonstrate the effectiveness of the results obtained in this paper.
Example 1. consider the complex dynamical networks models in the form of (17) and (18) with two modes and two nodes, and each node is a 2-dimensional dynamical system. The system parameters are described as follows: .The coupling matrix T is given as follows:The inner-coupling matrix isThe scalars we select in this paper are as follows: . and . The mixed time-varying delay is assumed to be , and the initial value is selected as and . We can see that the upper bound of the delay is . Also, the activation function is taken asThe transition rates of the semi-Markovian jumping system for each mode are given as follows:
For mode 1,For mode 2,Then, according to the transition rates, we can get the parameters , where s, k .Through simple computations, we haveFor mode 1,For mode 2,and the parameter V(0) = 8.6528 and the settling time is .
Then, Figure 1 displays the first state trajectories of the drive-response system with controller (28).The second state trajectories of the drive-response system with controller (28) presented in Figures 2 and 3 depicts the state trajectories of the synchronization error system (20) with the controller (28).
To this, the effectiveness of Theorem 1 is proved. The finite-time synchronization problem of discontinuous semi-Markovian switching complex dynamical networks with mixed time-varying delays is realized.
Example 2. For Theorem 2, we take network models in the form of (17) and (18) with two nodes, and each node is a 2-dimensional dynamical system. The system parameters are described as follows: .The coupling matrix L is given as follows:The inner-coupling matrix we choose isThe transition rates of the semi-Markovian switching system for each mode are given as follows.
For mode 1,For mode 2,Then, according to the transition rates, we can get the parameters , where s, k .For Theorem 2, the corresponding scalars we choose are , , , and . The mixed time-varying delays are assumed to be and , and the initial value is selected as and . We can see that the upper bound of the delay is . Also, the activation function we select isThrough simple computations, we haveMeanwhile, the parameters of the controller we choose are as follows.
For mode 1,For mode 2,and the settling time is .
Then, Figure 4 introduces the first state trajectories of the drive-response system with controller (55). The second state trajectories of drive-response system with controller (55) shown in Figures 5 and 6 display the state trajectories of the synchronization error system (20) with the controller (55).
Hence, the effectiveness of Theorem 2 is verified. In other words, the global fixed-time synchronization of discontinuous complex dynamical networks with semi-Markovian switching and mixed delays can be addressed.
5. Conclusions
In this paper, the global finite-time and fixed-time synchronization of discontinuous complex dynamical networks with semi-Markovian switching and mixed time-varying delays are investigated. Two novel state-feedback controllers are designed which include the integral term and mixed delay-term. Based on the linear matrix inequality technique (LMIs), Lyapunov functional method, and the proposed control schemes, some sufficient conditions are established to guarantee the global finite-time and fixed-time synchronization of complex dynamical networks. Then, two numerical examples are provided to confirm the effectiveness of the main results.
More complex conditions, such as stochastic disturbance, mixed coupling, and the approach of impulsive control, will be taken into consideration in the future study.
Data Availability
The underlying data supporting the results of this study can be found in the manuscript.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation (61827811), National Defense Basic Research Program (JCKY2019407C002), Hebei Provincial Education Departments Support Plan (SLRC2019042), and Hebei Province Funding Project for the Introduction of Overseas Students (C20200364).