Fixed/Preassigned-Time Synchronization Control of Complex Networks With Time Varying Delay

The problem of fixed-time (FXT) and preassigned-time (PAT) synchronization for delayed dynamic complex networks via developing control schemes is concerned. First, by comparing with the existing results, some new results for delayed networks model to ensure the synchronization of FXT and PAT are obtained. Second, some looser conditions are obtained for FXT synchronization, and several more accurate settling time (ST) estimates are established through several special functions. Third, by using some non-trivial control strategy, the PAT synchronization is investigated with limited control gains, and the synchronization time is able to specified in advance on the basis of practical need and is not relate to any parameter and any initial value. In particular, as a distinct form, the essential conditions for the FXT and PAT synchronization are obtained for the coupled neural networks. last, the improved the FXT and PAT synchronization results of dynamic delayed networks are demonstrated by one numerical example.


I. INTRODUCTION
Complex networks are complex aggregates composed of interrelated and interactive basic units (nodes) with certain characteristics and functions. They usually contain a huge number of nodes and complex topological connections, showing rich and diverse statistical characteristics and dynamic evolution behavior [1]- [7]. As a matter of fact, complex networks are everywhere in the actual world, and they all affect and dominate the development process of modern human society at any time. In complex networks dynamic behavior and groups, more and more scholars and experts pay attention to synchronous behavior because of its realistic significance and universality. For various dynamic network, some interesting works on synchronization have been received [8]- [13].
In a word, synchronization is one type of overall harmonized dynamic action formed by the mutual coupling between external forces and dynamic systems. In the present research works of control theory, complex networks synchronization commonly takes an limitless time to attain. However, for The associate editor coordinating the review of this manuscript and approving it for publication was Ton Duc Do . real complex networks, we often hope that the complex networks can synchronize in a limited time as soon as possible. In addition, due to the fractional power term of the finite-time controller, the finite-time control has high robustness and anti-disturbance. At the moment, some scholars have started to investigated the finite-time synchronization problem of chaos system and achieved some worthy works [14]- [18].
Generally speaking, under the same condition, solutions starting from different initial values will achieve synchronization in different finite time which results in different synchronization settling time and estimation. However, due to the influence of external interference and other factors on the actual networks, it is difficult to know the initial value of the system, which will lead to the inability to prove the boundedness of the settling time of this kind of network. For the sake of overcoming the limitation and inconvenience caused by the relationship between initial value and settling time estimation, Polyakov brought forward the idea of FXT stability theory of chaos system in 2012 and provided the relevant criterion [19], which supplies a theoretical principle for investigating the FXT synchronization of dynamic systems. VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ FXT synchronization means that the network is first synchronized in finite-time, and the synchronization ST and its estimation are not rely on the initial value of the system and merely depend on system parameters and control parameters. Obviously, compared with finite-time synchronization, FXT synchronization is more convenient and extensive in practical application because the upper bound estimation of ST has not connection with the initial value [20]- [22]. In [20], Feng et al. considered FXT synchronization problem of complex-valued coupled memristive neural networks. Hu and Jiang studied FXT estimation and stabilization problem for chaos networks based on special functions in [21]. Cao and Li researched FXT synchronization problem of delayed memristive recurrent neural networks in [22].
It is worth noting that the ST is relevant to the parameters of controllers and systems in FXT synchronization problem. Nevertheless, in applications, it may be more preferable for the network to achieve synchronization in PAT which is determined by practical need and is unconcerned of any parameters and any initial value. Hence, PAT synchronization has received increasing attention because it can provide a solution that enables the network to reach synchronization within any specified time recently [23]- [26]. In [23], Hu et al. discussed FXT and PAT synchronization problem of coupled system by improving FXT stability theory. Liu et al. considered PAT synchronization problem of coupled networks via continuous activation function by designing delayed and unbounded control strategy in [24]. In [25], Ren et al. discussed PAT lag consensus control problem for leader-following systems. In [26], Liu et al. investigated PAT control problem for uncertain multi-agent systems of prescribed performance.
As we know, time delay is unavoidable because of the limited communication speed and likely causes undesirable dynamic actions, for instance instability behavior and oscillation. In order to overcome the negative effects of time delay, some practical control methods for complex networks are designed to solve the convergence or stability problems of time-delay systems. Thus, the influence of time varying delay should be considered in the synchronization of dynamic system. Recently, a lot of works for delays complex systems have been received. Yet, to the best of my knowledge, the FXT and PAT synchronization for complex delayed systems via feedback strategy obtain few attention. Therefore, we should make great hard to solve the interesting question in the paper.
Based on the above discussion, the problem of FXT and PAT synchronization for delayed dynamic complex networks via developing control schemes is concerned. FXT and PAT synchronization criteria and some corollaries are obtained in our paper which are very verifiable and useful in application. Compare with the previous results, our main results are more general and less conservative. The innovations of the paper are at least the following aspects. First, in consideration of the significance for time delay, we investigated the complex system with varying delay in this paper. Second, some improved requirements for FXT synchronization are obtained and a number of new estimates for ST are achieved in the light of a few special functions. Third, by designing controller without a linear sign function term, FXT synchronization of delayed systems is talked. Forth, several innovative control protocols with limit control strategy are explored to the PAT synchronization, where the synchronized time is not relevant with parameter and initial value of system and can be prespecified on the basis of actual need. Besides, as a distinct form, the essential conditions for the FXT and PAT synchronization are obtained for a type of coupled neural networks. In particular, some corollaries are given and compared with now available works, our works are less conservative.
The structure of the rest of the article is given as follows. Preliminaries and dynamical delayed complex networks are provided in Section 2. In Section 3, FXT synchronization of the investigated delayed model under a simpler controller is investigated. The PAT synchronization is considered by using some control strategies with finite control gains in Section 4. An example is provided to illustrate the feasibility of the given means in Section 5. Last, in Section 6, a simple summary is obtained.
I n represents the n dimensions identity matrix. λ max (A) represents the maximum eigenvalues of matrix A.

II. PRELIMINARIES
A type delayed complex system is considered.
where y i (t) = (y 1 i (t), y 2 i (t), . . . , y n i (t)) T ∈ R n is the state of the ith node, f : R × R n × R n → R n is a continuous nonlinear function, the time delay τ (t) represents the internal delay, c > 0 is the coupling strength, = diag{γ 1 , γ 2 , . . . , γ n } is the inner matric with γ i > 0, B = (b ij ) N ×N is the coupling connecting matric which satisfies conditions as follows.
In view of the condition (2), model (1) can be rewritten in the following.
16820 VOLUME 10, 2022 Definition 1 ( [14]): The hyperplane = (y T 1 , . . . , y T N ) T ∈ R nN , y 1 (t) = · · · = y N (t) = z(t) ∈ R n is said to be the synchronous manifold of (3), and z(t) is called the synchronization state of the (3). Assume that the directed network (1) is strongly connected and balanced Obviously, on the basis of (3), z(t) satisfies the equation as follows.ż where z(t) may be a periodic orbit, an equilibrium point, or even a chaotic attractor. Assumption 1 ( [12]): For the vector-valued function f (t, y i (t), y i (t − τ (t))), there exist constants l 1 > 0, l 2 > 0 such that for any y i (t), z(t) ∈ R n .

Definition 4 ([27]):
Define the incomplete beta function ratio is as follows.
where 0 ≤ x ≤ 1, p > 0, q > 0, and B(p, q) is the beta function which is given by . . , b n are positive numbers and 0 ≤ r 1 < r 2 , v > 1, then
where u i (t) is controller strategy.

III. FIXED-TIME SYNCHRONIZATION
With the assistance of Lemmas 3 and 4, we will design appropriate η 1 , η 2 , η 3 , η 4 and µ, δ, such that the system (8) can realize FXT synchronization. The major results are presented in this section.
For the sake of obtaining the main works, we design the following feedback control.
According to the controller (6), then the error network is given as follows.
n k=1 |e k i (t)| 1+δ and using Lemma 1, it can be obtained that Then, it can be gotten thaṫ Hence, from Lemmas 3 and 4, the system (5) is FXT synchronization. This proves Theorem 1.
Especially, a type of controlled network is considered.
where i ∈ I , b ij is defined as (2), y i (t) = (y 1 i (t), y 2 i (t) . . . , y n i (t)) T ∈ R n is the state vector of the ith neuron, A = diag{a 1 , a 2 , . . . , a n } is the decay matrix with a i > 0, D = (d ij ) n×n and C = (c ij ) n×n are the configuration matrix and delayed configuration matrix, respectively, g(y i (t)) = (g 1 (y 1 i (t)), g 2 (y 2 i (t)), . . . , g n (y n i (t))) T is the activation nonlinear function.
For the sake of ensuring the FXT synchronization for the network. The statement is provided in the following.
Assume matrix C = 0, the network (11) can be provided as follows.
Hence, the synchronization state z(t) is represented bẏ

IV. PREASSIGNED-TIME SYNCHRONIZATION
For the sake of obtaining the main works, the feedback controller is given in the following.
On the basis of the control strategy (15), the error system is given bẏ wheref t, Theorem 2: Under Assumptions 1-2 and the control strategy (15), then the controlled system (5) is the PAT synchronized within the PAT T p if λ ≤ 0, 2l 2 − η 4 ≤ 0.
Proof: Define the following Lyapunov function as 16824 VOLUME 10, 2022 Then, the derivative of V (t) is represented in the following.
Hence, according to Lemma 5, the system (5) is PAT synchronized under the PAT T p . The proof of Theorem 2 is completed.
With regard to λ > 0, the following control scheme is designed.

Theorem 3: Under Assumptions 1-2 and the controller (18), the controlled system (5) is the PAT synchronized within the PAT
Proof: For e(t) ∈ R nN \{0}, it can be obtained thaṫ As a result of 0 < T p ≤Ť 2 , if λ > 0, Hence, according to Lemma 5, the system (5) is PAT synchronized. This proofs Theorem 3.
Particularly, via the following control scheme where η 1 , η 2 , η 3 , η 4 > 0 are the control strengths, T p > 0 is a PAT, and δ satisfies δ > 1,Ť 3 is defined in Theorem 1, and the PAT synchronization of the system (5) is achieved as follows.

Theorem 4: Under Assumptions 1-2 and the control strategy (21), the controlled system (5) is the PAT synchronized within the PAT
In view of Theorems 2-4, the following corollaries 3-5 can be gotten, respectively.

Corollary 3: Under Assumptions 2-3 and the control strategy (15), then the controlled system (5) is the PAT synchronized within the PAT T p if
Corollary 4: Under Assumptions 2-3 and the control strategy (18), then the controlled system (5) is the PAT synchronized within the PAT 0 < T p ≤Ť 2 if λ > 0, s − η 4 ≤ 0.
Remark 1: As far as we know, time varying delay is unavoidable because of the limited communication speed and may lead to undesirable dynamic behaviors, for instance, instability behavior and oscillation. Thus, the influence of time delay should be taken in the synchronization of complex networks. Otherwise, time delay is not considered in other works [14], [15], [20], [21], [23]. By using special Lyapunov function, the FXT and PAT synchronization of systems have been achieved in this paper. When τ (t) = 0, the results of Li et al. [14], Ji et al. [15] and Hu et al. [23] in Theorem 1 is equivalent to Corollary 1 in the paper. In other words, results in Li et al. [14], Ji et al. [15] and Hu et al. [23] are the special form in our results. Hence, the model we considered in the article is more general and more close to the practice.
Remark 3: In controller design, the linear part −k i sign(·) is essential and indispensable in most previous results of FXT synchronization [28]- [32]. As everyone knows, the sign function will cause the oscillation behavior of the network. Distinct in these control laws, a simpler controller without sign function term (6) is designed to realize FXT complex networks synchronization in this article. Besides, based on the Lyapunov functions as follows: FXT synchronization of coupled systems has been investigated in [13], [17], [33]. Different from these works, a distinct Lyapunov function of (8) is provided in the paper in order to investigate FXT synchronization. Remark 4: So far as we know, PAT synchronization are few investigated of dynamic delayed systems. In [24], by proposing an unbounded and time varying control strategy, Liu studied cluster synchronization with a PAT of complex networks. Distinct in the result [24], new control strategy (15), (18) and (21) are provided to analyze the PAT synchronization. Significantly, the control gains of these controllers can be effectively realized and are limited in practice. Especially, PAT synchronization time can not be dependent of any parameter and any initial value. Therefore, the application prospect of PAT synchronization is more broader than FXT synchronization.
Remark 5: Using Lemmas 3-5, the network can realize fixed-time synchronization under linear feedback control in this paper. Feedback control is continuous control approach. For discontinuous control approach, such as adaptive intermittent control, a lot of results have been obtained. However, the fixed time synchronization of networks cannot be received by using Lemmas 3-5 under adaptive intermittent control. Hence, it is meaningful to concern this problem in our recent research topics.
Remark 6: In [23], the model has been considered. In our paper, the model has been considered. The model in [23] is a special form of the model in this paper. In addition, the model we considered in this paper has time delay. In the proof of Theorem 1, the treatment of time delay is very complex. It is embodied in the construction of Lyapunov function and controller. In the constructed Lyapunov function, the integral e T i (s)e i (s)ds can eliminate the influence of delayed error nonlinearity about the states of delayed complex networks. In order to FXT and PAT synchronization,

V. NUMERICAL SIMULATIONS
In the section, a coupled network is considered to show the availability of the outcomes achieved in the article. Example: The dynamic networks model for variable delay is considered in the following.

VI. CONCLUSION
In this essay, the FXT and PAT synchronization are investigated for a type coupled system with delay. Firstly, the coupled dynamical network model we investigated has time varying delay. Secondly, the established conditions are more relaxed in this paper and comparison with the existing results, the values for ST are more precise. Thirdly, the designed controller without −k i sign(·) in FXT synchronization is simpler and more effective, and the given controllers with limit control gains in PAX synchronization are distinct from the controller in [24] with limit gains. Moreover, when τ (t) = 0, a exceptional case which investigated in [11], [15], [16], [25] can be obtained. In this sense, the works achieved in the paper are more common. Finally, the numerical results are obtained to demonstrate the effectiveness of the existing scheme.
Recently, intermittent control has attracted increasing interest naturally in a variety of applications because this control method is more economic and can reduce the amount of the transmitted information. For adaptive control, the control parameters can automatically adjust themselves according to some proper updating laws. As we know, fixed-time synchronization of delayed dynamic networks under adaptive intermittent control are few investigated. Hence, it is meaningful to concern this problem in our recent research topics. ZHANFENG LI was born in Henan, in 1982. He received the B.S. degree in mathematics and applied mathematics from Zhoukou Normal University, Zhoukou, Henan, China, in 2008. He currently works at Zhoukou Normal University. His current research interest includes mathematics education. VOLUME 10, 2022