Pinning Synchronization of Complex Dynamical Networks with Variable-Delayed Coupling by Periodically Intermittent Control

+is paper studied the adaptive pinning synchronization in complex networks with variable-delay coupling via periodically intermittent control. +eoretical analysis is included by means of Lyapunov functions and linear matrix inequalities (LMI) to make all nodes reach complete synchronization. Moreover, the synchronization criteria do not impose any restriction on the size of time delay. Numerical examples including the regular, Watts–Strogatz and scale-free BA random topological architecture are provided to illustrate the importance of our theoretical analysis.


Introduction
Complex dynamical networks have been intensively studied [1,2] over the last few years due to their potential applications in various fields of the real world, such as in the biological systems, scientific citation web, social networks, and electrical power grids and thus became an important part of our daily life.
Since Pecora and Carroll [3] found chaos synchronization in 1990, it has become a focal research topic recently. Due to the synchronization not only can well explain many natural phenomena observed, such as synchronized firefly flashing and swarming of fishes, but also has many practical potential applications such as image processing, the operation of unmanned aerial vehicle, and secure communication. However, the complex network cannot synchronize by itself, so many control methods have been developed, such as linear and nonlinear feedback control, time-delay feedback control, sliding mode control [4], adaptive control [5], pinning control [6], impulsive control [7,8], and intermittent control [9][10][11][12]. Especially, discontinuous feedback controls, including impulsive control [13] and intermittent control, have been attracting much attention since they are practically and easily implemented in some engineering domains. However, the intermittent control is different from the impulsive control because impulsive control is activated only at some isolated moments, while the intermittent control has a nonzero control width.
Intermittent control was first introduced to control chaos systems by Zochowski [9] in 2000. It is more efficient when the system output is measured intermittently rather than continuously. ere are some novel synchronization criteria in complex networks with delay or nondelay coupling by intermittent control. Li et al. [14] studied the inner synchronization of delayed nonlinear chaotic systems by intermittent control. Further, Mei et al. [10] used diver and response systems to produce the finite-time synchronization for the complex systems without delay. In [12], the authors investigated the synchronization problem of stochastic perturbed complex systems with time-varying delays. On the contrary, complex networks have a large number of nodes in the real world, and it is usually impractical to control a complex network by adding the controllers to all nodes. To reduce the number of controlled nodes, pinning control, in which controllers are only applied to partial nodes, is introduced [15][16][17][18][19][20]. In [21], the authors discussed the synchronization for coupled dynamical networks with mixed delays and uncertain parameters using pinning control and intermittent control. By means of intermittent control, the authors [22] studied finite-time synchronization for a class of reaction-diffusion neural networks by small domains on their spatial boundaries, associated with an interaction graph. Li et al. [23] and Xu et al. [24]discussed the synchronization problem of general complex networks with fractional-order dynamical networks by periodically intermittent pinning control.
Moreover, in [11,15], the authors investigated the synchronization of complex networks with delays by pinning periodically intermittent control; especially, they assumed that the control width needs to be larger than the time delay or the time-varying delays should be differentiable, and their derivatives are less than 1. Motivated by the above discussions, we removed these constraints in our results. In this paper, we give the complex systems with both timevarying delays and nondelay couplings, and using pinning control and periodically intermittent control methods, some novel criteria for pinning synchronization are derived. Numerical examples including the regular, Watts-Strogatz and scale-free BA random topological architecture are provided to illustrate the importance of our theoretical analysis. e rest of the paper is organized as follows: in Section 2, we propose a general complex dynamical network model; some preliminaries and lemmas are given. In Section 3, some pinning adaptive synchronization criteria for the general complex dynamical networks with delay coupling are given. Numerical examples are given in Section 4. Finally, we draw our conclusion in Section5.

Preliminaries and Mathematical Models
Consider a generally controlled complex delayed dynamical system consisting of N nodes, with each node being of n dimensions, which is described by . , x in (t)) T ∈ R n is the state variable of node i, A ∈ R n×n is a given constant matrix, and f: R n ⟶ R n is a continuously differentiable function describing the nonlinear dynamics of the single node. Here, c 0 and c 1 are two parameters of the nondelay and delay coupling strength, respectively, Γ � (c ij ) ∈ R n×n is an inner-coupling matrix, and τ(t) is the coupled delay and bounded by a known constant, i.e., 0 < τ(t) < τ. B � (b ij ) N×N and B � (b ij ) N×N represent the adjacency configuration of the network with the nondelay and delay couplings, respectively; if there is a link from node i to node j, As we know, the complex network cannot synchronize by itself; then, we add the adaptive controller v i (t) as follows: (2) is a synchronous solution of controlled complex delayed dynamical system (2). Note that s(t) may be an equilibrium point, a periodic orbit, or a chaotic attractor.
Define error vectors as According to system (2), the error system is described by Lemma 1 (Schur complement, see [25]). e linear matrix inequality (LMI) is as follows: where is equivalent to one of the following conditions: Lemma 2 (see [26]). Let X and Y be arbitrary n-dimensional real vectors, K be a positive definite matrix, and P ∈ R m×m . en, the following matrix inequality holds:

Main Results
In the following, assume that Γ ≠ 0 and ‖Γ‖ 2 � c > 0. Denote ρ ii as the minimum eigenvalue of the matrix (Γ + Γ T )/2. Let B s � (B + B T )/2, and its eigenvalues are expressed as Note that, generally, B does not possess the property of zero row sums.
To realize the network synchronization, the controllers v i should guide the error vectors e i (t) to approach zero as t goes to infinity as Choose the adaptive controllers as follows: , q i , and a 1 are positive constants. T > 0 denotes the control period, 0 < h < 1 is the rate of control duration called control rate, and m � 0, 1, 2, . . ..
If there exists a positive constant a 1 , a 2 , and k such that where (2) is globally asymptotically stable under periodical intermittent controllers (8).
Proof. Construct the candidate Lyapunov function as follows: and when mT ≤ t ≤ mT + h, calculating the time derivative of V(t) along the trajectories of (4), one has

for all t ∈ [− τ, T). (28)
Similarly, we can prove that By induction, we can derive the following estimation of W(t) for any integer m: (30) Since for any t ≥ 0, there exists a nonnegative integer r such that rT ≤ t ≤ (r + 1)T; then, we have t ∈ [(r + h)T, (r + 1)T).

(31)
Let ε ⟶ 1, and from the definition of W(t), one has (32) In view of ξ − a 2 (1 − h) > 0, we can draw the conclusion. e proof is thus completed. Let where D � diag(p, . .  there exists a positive constant a 1 , a 2 , and k such that where  (2) is globally asymptotically stable under periodical intermittent controllers (8).
e number of control nodes l can be chosen properly to adjust the synchronization efficiency of networks. However, it is noted that l by the theoretical prediction is only a sufficient condition but not a necessary one. In simulations, we will show that a small value of l can also lead to synchronization.

Numerical Simulation
In this section, a numerical example is used to verify the effectiveness of the proposed network synchronization criteria.
Consider the Lorenz oscillator model described by the following equation:

and
A � − 10 10 0 It has been known that the Lorenz oscillators exhibit chaotic behavior, and Figure 1 shows it clearly. en, we consider controlled complex delayed dynamical system (2) consisting of 200 identical Lorenz systems, which are described by Here, the coupling coefficient c 0 � 30.
Please note that the average number of neighbours is the same between the small-world network with the connection probability p � 0.1 and the regular network (p � 0), the number of neighbours becomes rand in the scale-free/smallworld topological structure, and max b ii can be estimated by the statistic method.
As a 1 − 8 < ξ < a 1 − 9, then if we choose a 1 � 45 and a 2 � 325, it is easy to verify that the criteria in eorem 1 are satisfied.
For the regular network (p � 0), one has η � 131.580; from corollary, one can find λ l * < − 3.6531. As λ 25 � − 3.5984 and λ 26 � − 4.0041, we can choose l * � 26. By numerical simulation, the synchronization error quickly tends to zero for l � 15 < l * . is further indicates that l > l * is only a sufficient condition, and a small value of l can also lead to synchronization. e initial conditions of the numerical simulations are as follows: x i (0) � (4 + 0.5i, 5 + 0.5i, 6 + 0.5i) T and s(0) � (4, 5, 6) T . Synchronous errors are shown in Figure 2. And we choose different control periods, and the synchronous errors are shown in Figure 3. It is seen from the figures that smallworld or scale-free networks can reach complete synchronization by pin-controlling fewer nodes than regular systems.

Conclusions
In this paper, we investigate adaptive pinning synchronization in complex delayed dynamical networks with timevarying delays by intermittent control. Based on the Lyapunov stability theory and chaos control method, several adaptive synchronization criteria are obtained. Our results show that the control width does not need to be larger than the time delays, and there is no restriction on the size of time delays. Moreover, we also find that small-world or scale-free networks can reach complete synchronization by pin-controlling fewer nodes than regular systems.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.