Finite/fixed-time synchronization for complex networks via quantized adaptive control

1.   Introduction

In practice, many real systems can be described by complex networks, which are composed of a large number of interconnected nodes, such as social networks, food Webs, electric power grids, biological networks, and so on[1,9]. Synchronization, which means that all dynamic nodes tend common dynamic behavior, is a typical dynamic phenomenon in complex networks. In recent years, various synchronization control strategies have been provided, which mainly include impulsive control [7], adaptive control [5,12], sliding mode control [10], periodic intermittent control [18], and pinning control [20] etc.

The stability of synchronization error dynamic systems is one of the main research issues of synchronization control for complex networks. Recently, finite-time synchronization has received more and more attention due to its faster convergence rate and better disturbance rejection ability [32,25,24,13]. However, one critical problem is that the settling time of finite-time control is heavily dependent on the initial conditions. Especially for some large complex networks in practical applications, it is often difficult even impossible to get information on the initial conditions, which could directly result in the inaccessibility of the control time. In this case, the finite-time synchronization control technology cannot be appropriately applied. In order to overcome the above-mentioned drawbacks of finite-time control, the concept of fixed-time convergence is proposed in [23]. Recently, according to this idea, the corresponding research results of fixed-time synchronization control have been presented in [30,31,8]. In [30], the continuous pinning controller has been designed for the complex networks, and the sufficient condition has been obtained to ensure the fixed time synchronization of complex networks. In some practical applications, the stability of many systems is always affected by various random perturbations. Therefore, more researches have been focused on the fixed-time synchronization control for a class of complex networks with random noise disturbance, and the It$\hat{o}$ integral principle has been proposed to solve the problem such as [31]. For a type of cluster networks with linear coupling and discontinuous nodes, the differential inclusion principle has been used to obtain the fixed-time cluster synchronization criterion in [8].

On the other hand, signal transmission is usually affected by bandwidth and communication channels. Due to the limited transmission capacity, signal distortion is caused, which affects the control performance of the system. Therefore, in order to solve this problem, signal quantization is an effective method to improve communication efficiency. For example, by designing aperiodically intermittent pinning controllers with logarithmic quantization, the sufficient condition for finite-time synchronization of the nonlinear systems is obtained in [26]. Based on the convex combination technique, the finite-time synchronization criteria for dynamic switching systems via quantized control is proposed in [27]. To the best of our knowledge, a quantized adaptive controller, which can make complex networks achieve synchronization in fixed-time, is not proposed. Furthermore, it is assumed that the internal coupling is known in most of the existing literature for finite-time synchronous control of complex networks. However, complex networks with a large number of nodes always encounter uncertain or unknown internal coupling [16].

Motivated by the discussions mentioned above, the finite/fixed-time synchronization control problem is investigated for complex networks with uncertain inner coupling via quantized adaptive control technology. The main contributions of this paper can be summarized as follows: (1) This paper studies a more extensive complex networks whose inner coupling is uncertain, and the interval matrix method is used to deal with the problem of the uncertain internal coupling; (2) By designing a quantized adaptive controller and adaptive law of parameters, it can effectively solve the problem of bandwidth limitations of networks; (3) A unified method is given for quantized adaptive finite/fixed time synchronization control. By adjusting the controller parameters, two types of setting time, which are dependent and independent on the initial values, can be realized, respectively; (4) Two finite/fixed-time synchronization criteria are established by the linear matrix inequalities technique.

Notations. $\mathbb{R}$ is the space of real number, $\mathbb{R}^{n}$ denotes the $n$ dimensional Euclidean space and $\|\cdot\|$ is the Euclidean norm. $I$ denotes the identity matrix of compatible dimensions. ${\rm min}\{\cdots\}$ and ${\rm max}\{\cdots\}$ are the minimum and maximum values of this set. For real symmetric matrices, $X$ and $Y$ the notation $X>Y$ means that the matrix $X-Y$ is positive define. $B^{T}$ represents the transpose of the matrix $B$. $\otimes$ is the Kronecker product. ${\rm diag}\{\cdots\}$ stands for a block-diagonal matrix. The asterisk $*$ in a matrix is used to denote the term that is induced by symmetry.

2.   Problem formulation and preliminaries

In this paper, consider the following complex networks model:

where $x_{i}(t)\in \mathbb{R}^{n}$ and $u_{i}(t)\in \mathbb{R}^{n}$represents the states vector and control input of the $i$th node, respectively, $f: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is a continuous nonlinear vector-valued function, $A, B\in \mathbb{R}^{n\times n}$ are constant matrices. $L = (l_{ij})_{N\times N}$ denotes outer-coupling matrix, if there exists a link from node $j$ to node $i$, then $l_{ij}>0$, otherwise $l_{ij} = 0$, for $i\neq j$, $l_{ii} = -\sum\limits_{j = 1, j\neq i}^{N}l_{ij}$. $\Gamma = {\rm diag}\{r_{1}, r_{2}, \ldots, r_{n}\}$ is an inner-coupling matrix. Here, the coupling strength $r_{i}(i = 1, 2, \ldots, n)$ is unknown, but it belongs to a certain interval $[\underline{r}_{i}, \overline{r}_{i}]$, where $\underline{r}_{i}$ and $\overline{r}_{i}$ are known constants satisfying $\underline{r}_{i}\leq\overline{r}_{i}$. By denoting

the inner-coupling matrix $\Gamma$ can be written as $\Gamma = \Gamma_{0}+\tilde{\Gamma}$, where $\tilde{\Gamma}\in[-\Gamma_{1}, \Gamma_{1}]$. Defining $F = \tilde{\Gamma}\Gamma_{1}^{-1}$, the matrix $\Gamma$ can be further expressed as follows:

and $F$ satisfies $FF^{T} = F^{T}F\leq I$.

Furthermore, let $s(t)\in \mathbb{R}^{n}$ be a solution of the isolate node, which is regarded as the target state and is assumed to be unique.

where $s(t)$ can be an equilibrium point, a nontrivial periodic orbit, or even a chaotic orbit. Let the synchronization error be $e_{i}(t) = x_{i}(t)-s(t)$ and $f(e_{i}(t)) = f(x_{i}(t))-f(s(t))$. Then, subtracting (3) from (1) gives the dynamic of synchronization error in the following as:

Definition 2.1. [17] The complex networks (1) is said to be finite-timely (or fixed-timely) synchronized onto (3), if there exists a settling time $T$ depending (or independent) on the initial values, such that

Assumption 1. There exists a constant $d>0$ such that

Lemma 2.2. [26] Let $\eta_{1}, \eta_{2}, \ldots, \eta_{n}\geq0$, $0<\zeta\leq1, \;\omega>1$, then one can derive that

Lemma 2.3. [16] For any dimension-compatible matrices $M, H$ and $E$ with $HH^{T}\leq I$ and a scalar $\varepsilon>0$, then the following inequality holds:

Lemma 2.4. [19] Suppose that $V(x(t))$ : $\mathbb{R}^{n}\rightarrow \mathbb{R}$ is C-regular and that $x(t)$ is absolutely continuous on any compact interval of $[0, +\infty)$. If there exists a continuous function $K(V(t)):[0, +\infty)\rightarrow (0, +\infty)$, with $K(\sigma)>0$ for $\sigma\in(0, +\infty)$, such that the derivative $\dot{V}(t)\leq-K(V(t)) = -k_{1}V^{\alpha}(t)-k_{2}V^{\beta}(t)$ and $T = \int_{0}^{V(0)}\frac{1}{K(\sigma)}d\sigma<+\infty$. Then, we have $V(t) = 0$ for $t\geq T$.

1) If $0<\alpha<1, 0<\beta<1$, $V(t)$ will reach zero in a finite time, and the settling time is estimated as

2) If $0<\alpha<1$, $\beta>1$, $V(t)$ will reach zero in a fixed time, and the settling time is estimated as

Remark 1. There are two cases in Lemma 2.4, for a given parameter $\alpha\in (0, 1)$, the value of $\beta$ will determine the settling time is finite or fixed, that is to say, the finite-time convergence can be achieved if $0<\beta<1$ as claimed in case 1 and the fixed-time convergence can be realized if $\beta>1$ as described in case 2.

3.   Quantized synchronization controller design

$\varphi(\cdot):\mathbb{R}\rightarrow\Omega$ is a logarithmic quantizer, where $\Omega = \{\pm \omega_{i}:\omega_{i} = \rho^{i}\omega_{0}, \;i = 0, \;\pm1, \;\pm2, \;\ldots \}\bigcup \{0\}$, with $w_{0}>0$. According to the analysis in [26,6], for $\tau\in \mathbb{R}$, the quantizer $\varphi(\tau)$ is constructed as follows:

where $\delta = \dfrac{1-\rho}{1+\rho}$ and $0<\rho<1$ is the quantized density. From equation (5) there exists $\Delta\in[-\delta, \delta]$ such that

The design of the quantized synchronization controller is as follows:

where $\xi_{i}>0$ to be determined, $\lambda_{i}>0, \chi_{i}>0$ are tunable parameters and $r, \;\theta, \;s$ and $v$ are positive odd integers, $e_{i}(t) = [e_{i1}(t), e_{i2}(t), \ldots, e_{in}(t)]^{T}$, $\varphi(e_{i}(t)) = [\varphi(e_{i1}(t)), \varphi(e_{i2}(t)), \ldots, \varphi(e_{in}(t))]^{T}$ and $[\varphi(e_{i}(t))]^{\sigma} = [ (\varphi(e_{i1}(t)))^{\sigma}, (\varphi(e_{i2}(t)))^{\sigma},$ $\ldots, (\varphi(e_{in}(t)))^{\sigma}]^{T}$, here $\sigma = \frac{r}{\theta}$ or $\sigma = \frac{s}{v}$.

Let the matrix $\Lambda(t) = {\rm diag}\left\{\Lambda_{1}(t), \Lambda_{2}(t), \ldots, \Lambda_{N}(t)\right\}$, $\Lambda_{i}(t) = {\rm diag}\{\Lambda_{i1}(t), \Lambda_{i2}(t),$ $\ldots, \Lambda_{in}(t)\}$, and satisfy $\Lambda_{ij}(t)\in[-\delta, \delta]$, $i = 1, 2, \ldots, N$, $j = 1, 2, \ldots, n$. Therefore, according to (6) we have

Theorem 3.1. Let Assumption 1 hold, the complex networks (1) can be finite/fixed-time synchronization with (3) via the quantized controller (7), if there exist positive definite diagonal matrices $P, \;M$ and scalar $\varepsilon_{1}>0$, such that the following LMI holds:

where $\bar{\Omega}_{1} = \mu d^{2}I+[P\otimes A+(PL)\otimes\Gamma_{0}-(1-\delta)M\otimes I_{n}]+[P\otimes A+(PL)\otimes\Gamma_{0}-(1-\delta)M\otimes I_{n}]^{T}+\varepsilon_{1}\bar{\Gamma}_{1}^{T}\bar{\Gamma}_{1}$, $\bar{L} = (PL)\otimes I_{n}$, $\bar{\Gamma}_{1} = {\rm diag}_{N}\{\Gamma_{1}\}$, $M = P\xi$, $P = {\rm diag}\{p_{1}, p_{2}, \ldots, p_{N}\}$, $\xi = {\rm diag} \{\xi_{1}, \xi_{2}, \ldots, \xi_{N}\}$. Then,

1) when $r<\theta$, $s<v$, finite-time synchronization can be realized and the settling time is estimated as

where $\hat{\lambda} = 2\lambda\eta_{1}(1-\delta)^{\frac{r}{\theta}}$, $\hat{\chi}_{1} = 2\chi\eta_{2}(1-\delta)^{\frac{s}{v}}$, $\lambda = \min\limits_{i}\lambda_{i}$, $\chi = \min\limits_{i}\chi_{i}$, $\eta_{1} = \min\limits_{i} \{p_{i}^{\frac{\theta-r}{2\theta}}\}$, $\eta_{2} = \min\limits_{i}\{p_{i}^{\frac{v-s}{2v}}\}$, $i = 1, \ldots, N$;

2) when $r<\theta$, $s>v$, fixed-time synchronization can be realized and the settling time is estimated as

where $\hat{\chi}_{2} = 2\chi\eta_{2}(1-\delta)^{\frac{s}{v}}(Nn)^{\frac{v-s}{2v}}$.

Proof. Choose Lyapunov function as follows

where $p_{i}>0$ is the scalar. According to the dynamic equation (4) of synchronization error and the controller (7), we can obtain

According to (8) we have

From (13) and (14), we get

where $e(t) = [e_{1}^{T}(t), e_{2}^{T}(t), \ldots, e_{N}^{T}(t)]^{T}$, $f(e(t)) = [f(e_{1}(t)), f(e_{2}(t)), \ldots, f(e_{N}(t))]^{T}$

By Assumption 1, one can derive that

such that

According to (15) and (16), the following inequality can be obtained for $\mu>0$:

where $z(t) = [e^{T}(t), f^{T}(e(t))]^{T}$, $\Pi = \begin{bmatrix} \Omega_{1} & P\otimes B \\ (P\otimes B)^{T} & -\mu I \\ \end{bmatrix}, \;$ $\Omega_{1} = \mu d^{2}I+[P\otimes A+(PL)\otimes \Gamma_{0}+(PL)\otimes (F\Gamma_{1})-(1-\delta)(P\xi)$$\otimes I_{n}]+[P\otimes A+(PL)\otimes \Gamma_{0}+(PL)\otimes (F\Gamma_{1})-(1-\delta)(P\xi)\otimes I_{n}]^{T}$.

Noting that the matrix $(PL)\otimes (F\Gamma_{1})$ contains an unknown parameter $F$, we rewrite the matrix $(PL)\otimes (F\Gamma_{1})$ as

where $\bar{L} = (PL)\otimes I_{n}$, $\bar{F} = {\rm diag}_{N}\{F\}$, $\bar{\Gamma}_{1} = {\rm diag}_{N}\{\Gamma_{1}\}$. Furthermore, by Lemma 2.3, we can get

where $\Pi_{0} = \begin{bmatrix} \Omega_{2} & P\otimes B \\ (P\otimes B)^{T} & -\mu I \\ \end{bmatrix}$, $\Omega_{2} = \mu d^{2}I+[P\otimes A+(PL)\otimes\Gamma_{0}-(1-\delta)(P\xi)\otimes I_{n}]+[P\otimes A+(PL)\otimes\Gamma_{0}-(1-\delta)(P\xi)\otimes I_{n}]^{T}$, $T_1 = [\bar{\Gamma}_{1}\;\;\;\;0]^{T}$, $N_{1} = [\bar{L}^{T}\;\;\;\;0]$. From (18) and by using the Schur complement, it is easily known that $\Pi<0$ is implied by the matrix inequality (9) in Theorem 3.1. Therefore, we can obtain

If $r<\theta$, $s<v$, by Lemma 2.2 and (8), it can be derive that

Similarly, the following inequality can be obtained

It follows from (19)-(21) that

If $r<\theta$, $s>v$, by Lemma 2.2 and (8), it can be derive that

From (19), (20) and (23), it can be obtained

By Lemma 2.4, we can get $V(t)\equiv0$ for $t\geq T$, the synchronization error system (4) can be finite/fixed-time stabilized under quantized controller (7), where the settling time $T$ satisfies (10) and (11), respectively.

4.   Adaptive quantized synchronization controller design

This section discusses the quantized adaptive finite/fixed-time synchronization of complex networks (1). We design a quantized adaptive controller and adaptive parameter updating law as follows

where$\lambda_{i}>0$, $\chi_{i}>0$ are tunable parameters and $r, \;\theta, \;s$ and $v$ are positive odd integers.

where $\xi^{*}>0$ is a constant to be determined, $k_{1}>0$, $k_{2}>0$, $c>0$, $q_{i}>0$ are the constant.

Theorem 4.1. Let Assumption 1 hold, the complex networks (1) can be finite/fixed-time synchronization with (3) via the quantized adaptive controller (25) and adaptive parameter updating law (26), if there exist positive definite diagonal matrices $P, \;M^{*}$ and scalars $\xi^{*}>0$, $\varepsilon_{1}>0$, such that the following LMI holds:

where $\bar{\Omega}_{2} = \mu d^{2}I+[P\otimes A+(PL)\otimes\Gamma_{0}-c(1-\delta)(M^{*}\otimes I_{n})]+$$[P\otimes A+(PL)\otimes\Gamma_{0}-c(1-\delta)(M^{*}\otimes$$I_{n})]^{T}+\varepsilon_{1}\bar{\Gamma}^{T}\bar{\Gamma}$, $M^{*} = \xi^{*}P$.

1) when $r<\theta$, $s<v$, finite-time synchronization can be realized and the settling time is estimated as

where $\hat{k}_1 = \min\{\hat{\lambda}, 2k_1\}$, $\hat{k}_2 = \min\{\hat{\chi}_1, 2k_2\}$;

2) when $r<\theta$, $s>v$, fixed-time synchronization can be realized and the settling time is estimated as

where $\hat{k}_3 = \min\{\hat{\chi}_2, 2k_2\}$.

Proof. Consider Lyapunov function as follows

It follows from (4) and (30) that

According to (16) and (31), for $\mu>0$ it is derived that

where $\tilde{\Pi} = \begin{bmatrix} \Omega_{3} & P\otimes B \\ (P\otimes B)^{T} & -\mu I \\ \end{bmatrix}$, $\Omega_{3} = \mu d^{2}I+[P\otimes A+(PL)\otimes \Gamma_{0}+(PL)$$\otimes (F\Gamma_{1})-c(1-\delta)\xi^{*}(P\otimes I_{n})]$$+[P\otimes A+(PL)\otimes \Gamma_{0}+(PL)\otimes (F\Gamma_{1})-c(1-\delta)\xi^{*}(P\otimes I_{n})]^{T}$.

Using the similar proof method of Theorem 3.1, the following inequality can be obtained

where $\tilde{\Pi}_{0} = \begin{bmatrix} \Omega_{4} & P\otimes B \\ (P\otimes B)^{T} & -\mu I \\ \end{bmatrix}$, $\Omega_{4} = \mu d^{2}I+[P\otimes A+(PL)\otimes\Gamma_{0}-c(1-\delta)(M^{*}\otimes I_{n})]$$+[P\otimes A+(PL)\otimes\Gamma_{0}-c(1-\delta)(M^{*}\otimes I_{n})]^{T}$.

From (33) and by using the Schur complement, it is easily known that $\tilde{\Pi}<0$ is implied by the matrix inequality (27) in Theorem 4.1. Therefore, it can be obtained that

If $r<\theta$, $s<v$, from (20), (21), (34) and Lemma 2.2, we have

If $r<\theta$, $s>v$, From (20), (23), (34) and Lemma 2.2, we can get

By Lemma 2.4, we can get $V(t)\equiv0$ for $t\geq T$, the synchronization error system (4) can be finite/fixed-time stabilized under adaptive quantized controller (25) and adaptive parameter updating law (26). The settling time $T$ satisfies (28) and (29), respectively.

Remark 2. In Theorem 3.1 and Theorem 4.1, by adjusting different parameters of the unified quantized controller, both finite time and fixed time synchronization goals are achieved. By choosing of parameters $r, \theta$, such that $\dfrac{r}{\theta}\in (0, 1)$ and $s$ and $v$ are in different parameter ranges, which results in different settling time. When $s<v$, it can be seen from (10) and (28) that the settling time is dependent of the initial value. While $s>v$, it can be seen from (11) and (29) that the settling time is independent of the initial value, which is only dependent of the parameters of the controller.

Remark 3. From Theorem 4.1, we can see that the parameters $k_1, k_2, \lambda_i$ and $\chi_i (i = 1, \cdots, N)$ can affect the settling time for synchronization. From (28) and (29), we can see that $k_1, k_2, \lambda_i, \chi_i (i = 1, \cdots, N)$ are larger, the settling time $T$ will decrease. Furthermore, from (25) and (26) it is known that the increase of $k_1, k_2, \lambda_i, \chi_i (i = 1, \cdots, N)$ will lead to the controller gain and adaptive updating law gain raising. Therefore, the choice of $k_1, k_2, \lambda_i, \chi_i (i = 1, \cdots, N)$ need a compromise with control performance and settling time.

5.   Simulation examples

Consider the complex networks composed of five Chua's circuits based on [2]. The signal circuit model is depicted in Figure 1. According to [11], the feedback controller and the inductor are connected in series to form the voltage $u_{i}(t)$. So the complex networks can be expressed as follows:

where $x_{i1} = v_{i1}$, $x_{i2} = v_{i2}$, $x_{i3} = i_{i3}$, and $f(v_{1}) = G_{b1}v_{1}+0.5(G_{a1}-G_{b1})(|v_{1}+1|-|v_{1}-1|)$. $v_{1}$ and $v_{2}$ represent the voltage at both sides of capacitor $C_{1}$ and $C_{2}$ in order. $i_{3}$ represents the current through the inductor $L$. $R_{0}$ and $R$ are linear resistance.

The coupling configuration matrix is described by:

From [30], select $p = -\dfrac{19}{7}$, $s = 9$, $v = 14.28$, $G_{a1} = -0.8$, $G_{b1} = -0.5$, and $w = -1$. Let the quantization density as $\rho = 0.7$ and $\Gamma = {\rm diag}\{1+0.7{\rm sin}(t), 1+0.8{\rm cos}(t), 1+0.9{\rm sin}(t){\rm cos}(t)\}$ is an inner-coupling matrix. We can get $\underline{r}_{1} = 0.3$, $\underline{r}_{2} = 0.2$, $\underline{r}_{3} = 0.1$, $\bar{r}_{1} = 1.7$, $\bar{r}_{2} = 1.8$, $\bar{r}_{3} = 1.9$ and calculate $\Gamma_{0} = {\rm diag}\{1, 1, 1\}$, $\Gamma_{1} = {\rm diag}\{0.7, 0.8, 0.9\}$.

The initial states of each node in the networks are $x_{1}(0) = [-2, 2.6, 0]^{T}$, $x_{2}(0) = [-1.5, 1.3, -6]^{T}$, $x_{3}(0) = [-2.3, 6, -3.4]^{T}$, $x_{4}(0) = [3, -4.1, 5.3]^{T}$, $x_{5}(0) = [3, -4.4, -3]^{T}$, and the initial value of the isolated node is $s(0) = [0, 65, 0.2, 0.8]^T$.

First of all, considering the complex network (1) and the target system (3) under uncontrolled, the open-loop responses is shown in Figure 2. It can be seen from Figure 2 that the synchronization error trajectories diverge in the case of open-loop. And then, the quantized adaptive controller (25) is applied to the Chua's circuit network. The parameters of the controller are selected as: $\lambda_{i} = 6$, $\chi_{i} = 6$, $i = 1, 2, \ldots, 5$, $r = 3$, $\theta = 5$, $s = 5$, $v = 7$, $c = 1.6$, $q_{i} = 2$, $k_{1} = 4$, and $k_{2} = 6$. It is obvious that the Assumption 1 is satisfied with $d = 2$. For given $\xi_{0} = \{30.8, 29, 18, 15, 24\}$ and $\mu = 0.075$, the following parameters can be obtained by using MATLAB to solve the LMIs (27):

Figure 3 shows the synchronization error trajectories of the complex network (1) and the target system (3) under the action of the quantized adaptive controller (25). It can be seen from the Figure 3 that the synchronization error system is stable in finite time and the settling time is estimated as follows

Adjust the parameters of the controller such that $r = 3, \theta = 5, s = 5$ and $v = 3$. Figure 4 shows the trajectories of fixed-time synchronization error between the complex network (1) and the target system (3) under the action of a quantized adaptive controller (25). The corresponding settling time is estimated as follows

It can be seen from Figure 4 that fixed-time stability of the synchronization error dynamic system can be achieved. Figure 5 and Figure 6 show the variation curves of adaptive parameter $\xi_{i}(t), i = 1, 2, \ldots, 5$ for the finite/fixed-time stabilization, respectively. It can be seen from the simulation results that the time-varying controller gain parameters $\xi_{i}(t)$ also converges to some constants. It is obvious that the effectiveness of presented method is proved by the simulation.

6.   Conclusions

In this paper, a unified parameterized quantized controller and quantized adaptive controller have been designed to simultaneously realize the finite/fixed-time synchronization of complex networks with uncertain internal coupling. It can be decided just by regulating the power parameters in one common controller that the setting time is either dependent or independent of the initial condition. The interval matrix method is used to describe the uncertainty of coupling in networks. Based on Lyapunov stability theory and linear matrix inequalities technology, the criterion of finite/fixed-time stability for synchronization error systems of complex networks is obtained. Finally, the effectiveness of the proposed control scheme is verified by simulation. Moreover, many factors can influence the dynamic behavior of complex networks, such as random disturbances and actuator faults. Therefore, the corresponding results with the aforementioned factors will realize in the near future. It is brought to our attention that the synchronization control problem in our study is closely related to inverse problems for differential equations, see e.g. [3,4,21,22,28,29] in the deterministic setting and [14,15] in the random setting. It would be interesting for us to consider inverse problem techniques to the synchronization control problem in our future study.