Event-triggered synchronization in fixed time for complex dynamical networks with discontinuous nodes and disturbances

Abstract

In this paper, the global robust synchronization in fixed time is discussed for discontinuous complex dynamical networks (CDNs) with uncertain disturbances based on event-triggered states-feedback control schemes. Several new hybrid controllers, which include event-triggered controller and discontinuous state-feedback controller, are designed to realize the global robust synchronization goal. By applying Lyapunov functional method, the common theorem with respect to the stability in fixed time for nonlinear systems, and inequality analysis technique, some sufficient synchronization criteria are addressed in terms of linear matrix inequalities (LMIs). In addition, the upper bound of the settling time, which is independent on initial conditions, can be determined to any desired values in advance on the basis of the configuration of parameters in the proposed control law. Finally, three examples are provided to illustrate the validity of the proposed design method and theoretical results.

Keywords

Complex dynamical networks synchronization in fixed time event-triggered control discontinuous nodes linear matrix inequality

1 Introduction

In recently years, as a special sort of significant instrument to describe and comprehend complex systems, the complex dynamic networks, have attain the widespread attention from a great deal of scholars [1 –4]. Particularly, the global synchronization issues with respect to CDNs, have been extensively studied from many researchers because its potential applications in quantum computation, multi-agent cooperative control, cryptography, nuclear magnetic resonance instrument, and other aspects, see [5 –9] and the references therein.

At present, many important and interesting works with respect to the global synchronization behaviors of discontinuous activations CDNs have been developed due to its extensive application in real life. The outer synchronization between two complex networks with discontinuous coupling are considered in [10], and several sufficient conditions for complete outer synchronization and generalized outer synchronization are obtained based on the stability theory of differential equations. In [11], authors investigated finite-Time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays. The global Mittag-Leffler synchronization for a class of fractional-order neural networks with discontinuous activations has been investigated in [12]. In [13], Song et al. discussed the exponential synchronization of time delayed CDNs with periodical on-off coupling.

It is worth nothing that the networks synchronization all mentioned above, the upper bound of synchronization estimated time can only be calculated under synchronization in finite time, see [15 –17] and the references therein. Haimo firstly pointed out that, global synchronization in finite time means the optimality in convergence time in [14]. In [15, 16], authors put forward the techniques of finite-time control, which can demonstrate better disturbance and robustness rejection properties. It is more meaningful and reasonable to investigate global synchronization in finite time for CDNs due to it’s extensively applications in many practical scenarios [17 –20]. However, there exists a problem on the global finite time synchronization in which the upper bound of the settling time greatly depends on the initial values of the CDNs. The convergence times will be changed while initial values being different, there has a limitation in practical applications because it isn’t easy to obtain the initial values. Thus, the efficient strategy that named global synchronization in fixed time be considered in [21] to solve the problem, which the estimated settling time is independent on the initial conditions of systems.

Up to now, fixed-time control as one of the representative dynamical characteristic of CDNs, providing their wide applications in various research field, such as chaos suppression in power system [22] and consensus of multi-agent networks [23]. Due to the fixed-time control can achieve system stabilization within a upper bounded time and not depends on the initial values. Thus, fixed-time control has received tremendous attention in recent years which as a kind of control strategy for investigated the global synchronization in fixed time for CDNs [24 –27]. In [24], the fixed-time synchronization for a class of CDNs in the presence of discontinuous activation functions and parameter uncertainties was discussed. The global fixed-time synchronization of CDNs with discontinuous activations and nonidentical perturbations under the framework of Filippov solution was investigated in [25]. In [26], authors investigated the fixed-time synchronization for CDNs under the framework of Filippov solution. The fixed-time synchronization of CDNs with discontinuous control protocols under undirected or directed topologies are proposed in [27].

With the rapid development of science and technology, modern control systems tend to be controlled by discrete time controllers. Event triggered controllers and time triggered controllers are the two main types of digital controllers. In a time-triggered approach, the information is sampled at pre-specified time instances as signals ignoring the state of the system and transmitted to the communication network. Since such time-triggered strategy is also a waste of computation resources when many regularly sampled signals are unnecessary for the stability of systems. In an aim to reduce data transmission and save the network resources the novel control technique namely event-triggered control has been introduced and some remarkable results are available in the literature [28 –34].

Inspired by the above discussion, in this paper, we summary three typical fixed-time stability lemmas to investigate the robust synchronization in fixed time for discontinuous CDNs with the presence disturbances respectively. Under the designed hybrid controllers with event-triggered terms and discontinuous factors, by applying Lyapunov functional method, the global synchronization conditions in fixed time are addressed in the terms of LMIs. In addition, the upper bound of the settling time is estimated explicitly. The primary contributions of this paper are listed below:

It is first time to investigate the global robust synchronization in fixed time for CDNs with disturbances via event-triggered control.

The novel hybrid controllers, which includes discontinuous factors and event-triggered term, are designed.

The upper bound of the settling time can be determined to any desired values in advance on the basis of the configuration of parameters in the designed control law.

The global robust synchronization conditions in fixed time are achieved in the form of LMIs.

The rest of this paper is organized as follows. In Section 2, some preliminaries and the CDNs model are given. In Section 3, the hybrid controllers with event-triggered term and discontinuous factors are designed, the global synchronization in fixed time is addressed in the form of LMIs. In Section 4, numerical simulations verifying the theoretic findings are presented. Conclusion is received in Section 5.

Notation. R refers to the set of real numbers. Rⁿ represents the n-dimensional Euclidean space, and R^n×n denotes the set of all n × n real matrices. A > 0 (A < 0) represents A is positive (negative) definite matrix. Set A^T where the superscript T is the transpose operator. Define $∥ η ∥_{q} = (\sum_{i = 1}^{n} | η_{i} |^{q})^{\frac{1}{q}}$ , |η|^α = (|η₁|^α, |η₂|^α, ⋯ , |η_n|^α) ^T, where α and q are positive constants, | · | denotes the absolute real value. For x ∈ Rⁿ, [x] ^θ = (sign (x₁) |x₁|^θ, ⋯ , sign (x_n) |x_n|^θ) ^T. Set $\hat{a} = max_{l, r \in [1, n]} | a_{lr} |$ , $\hat{b} = max_{l, r \in [1, n]} | b_{rl} |$ , $\hat{ν} = max_{r \in [1, n]} ν_{r}$ , $\hat{ω} = max_{r \in [1, n]} ω_{r}$ . $u_{i}^{1} (t)$ denotes the first controller for i node and $u_{i}^{2} (t)$ denotes the second controller for i node. Let A, B, C and D be matrices with appropriate dimensions. ⊗ denotes the Kronecker product. It should be pointed out that, ⊗ has the following properties: 1) (A ⊗ B) (C ⊗ D) = AC ⊗ BD; 2)(A ⊗ B) ^T = A^T ⊗ B^T.

2 Preliminaries

2.1 Preliminaries and model description

Consider an array of hybrid CDNs, which can be described by

$\begin{matrix} {\dot{x}}_{i} (t) = - {Qx}_{i} (t) + c_{1} \sum_{j = 1}^{N} d_{ij} (x_{j} (t) - x_{i} (t)) \\ + Af (x_{i} (t)) + J (t) + u_{i} (t) + Δ_{i} (t, x_{i} (t)), \end{matrix}$ (1) where i = 1, 2, ⋯ , N, x_i (t) = (x_i1 (t) , x_i2 (t) , ⋯, x_in (t)) ^T, is the state variable of the neural network i at time t; $x (t) = (x_{1}^{T} (t), x_{2}^{T} (t), \dots, x_{N}^{T} (t))^{T}$ . Q = diag (q₁, ⋯ , q_n), q_i > 0. c₁ is couple strengthes; A = (a_lr) _n×n represents the connection weight matrix. f (x_i (t)) = (f₁ (x_i1 (t)) , ⋯ , f_n (x_in (t))) ^T denotes a activation function. u_i (t) denotes the control input. J (t) = (J₁ (t) , ⋯, J_n (t)) ^T denotes an external input. D = (d_ij) _N×N represents a couple configuration matrix with the underlying topology. If there exists an edge from node i and node j, then d_ij = d_ji > 0 otherwise, d_ij = d_ji = 0 (i ≠ j). And the Laplacian matrix L = (l_ij) _N×N of a graph corresponding to the D is defined by $l_{ii} = \sum_{j = 1, j \neq i}^{N} d_{ij}, l_{ij} = - d_{ij}$ , i ≠ j. Δ_i represses the uncertain disturbance.

In system (1), the nonlinear function f_l (·) is modeled to satisfy the following assumptions:

f_l (·), is continuous expect on a countable set of isolate points $ρ_{k}^{l}$ , and f_l (·) has at most a limited number of discontinuous on any compact interval of R, in addition, at the discontinuous points $ρ_{k}^{i}$ , the finite right limit $f_{l}^{+} (ρ_{k}^{l})$ and left limit $f_{l}^{-} (ρ_{k}^{l})$ exist.

There exist positive constants ν_l and ω_l for each l = 1, 2, ⋯ , n, ∀ t ≥ 0, such that $sup | γ_{il} (t) - {\tilde{γ}}_{l} (t) | \leq ν_{l} | x_{il} (t) - y_{l} (t) | + ω_{l}$ holds for ∀ x_il (t) , y_l (t) ∈ R, where ${\tilde{γ}}_{l} (t) \in \bar{co} [f (y_{l} (t))]$ , $γ_{il} \in \bar{co} [f (x_{il} (t))]$ ,i = 1, 2, ⋯ , N;

Based the assumption (A1), it follows that, $\bar{co} [f_{i} (\cdot)] = (\bar{co} [f_{1} (\cdot)], \bar{co} [f_{2} (\cdot)], \dots,$ $\bar{co} [f_{n} (\cdot)])^{T}$ , where

$\bar{co} [f_{i} (\cdot)] = [min f_{i}^{-} (\cdot), f_{i}^{+} (\cdot), max f_{i}^{-} (\cdot), f_{i}^{+} (\cdot)]$ .

It should be pointed that, under the assumption (A1), system (1) is a functional differential equation with discontinuous right-hand side. The classical definition of solution is not suitable for dynamic system (1). Here, analogous to [35, 36], we take advantage of solution in Filippov sense to deal with system (1).

2.2 Preliminaries

Consider the following differential equation with discontinuous right-hand sides

$\begin{matrix} \dot{x} (t) = g (x (t)), \end{matrix}$ (2) where g (x) is discontinuous function.

Define the set-valued map G : Rⁿ → Rⁿ by

$G (x) = ⋂_{ρ > 0} ⋂_{ι (Ω) = 0} \bar{co} [g (\hat{B} (x, ρ) \ Ω)],$ where ι (Ω) denotes the Lebesgue measure of the set Ω and $\hat{B} (x, ρ) = {δ \in R^{n} : ∥ x - δ ∥ \leq ρ}$ .

Definition 2.1. ([35]) Set T ∈ (0, + ∞), x : [0, T) → Rⁿ, is called as a solution of system (2) on interval [0, T) in the sense of Filippov, if

x (t) is absolutely continuous on [0, T);

x (t) satisfies $\dot{x} (t) \in G (x (t))$ for a . a . t ∈ [0, T).

By the measurable selection theorem [35], there are measurable functions γ_i (t) = (γ_i1 (t) , ⋯ , γ_in (t)) ^T : [0, T) → Rⁿ, $\in \bar{co} [f (x_{i} (t)]$ , and ξ_i (t) = (ξ_i1 (t) , ξ_i2 (t), ⋯, ξ_in (t)) ^T : [0, T) → Rⁿ, $\in \bar{co} [u_{i}$ (t)], for a . e . t ∈ [0, T), such that

$\begin{matrix} {\dot{x}}_{i} (t) = - {Qx}_{i} (t) + c_{1} \sum_{j = 1}^{N} d_{ij} (x_{j} (t) - x_{i} (t)) \\ + A γ_{i} (t) + ξ_{i} (t) + Δ_{i} (t, x_{i} (t)) + J (t) . \end{matrix}$ (3) Consider the following isolated node,

$\begin{matrix} \dot{y} (t) = - Qy (t) + Af (y (t)) + J (t) . \end{matrix}$ (4) By the measurable selection theorem [35], there are measurable functions $\tilde{γ} (t) : [0, T) \to R^{n}$ , $\in \bar{co} [f (y (t)]$ , for a . e . t ∈ [0, T), such that $\begin{matrix} \dot{y} (t) = - Qy (t) + A \tilde{γ} (t) + J (t) . \end{matrix}$ In this paper, our objective is to design fixed-time hybrid controllers to guide CDNs (1) to robustly synchronize with isolated node (4).

Set e_i (t) = x_i (t) - y (t), the error system can be written as

$\begin{matrix} {\dot{e}}_{i} (t) = & - {Qe}_{i} (t) + c_{1} \sum_{j = 1}^{N} d_{ij} (e_{j} (t) - e_{i} (t)) \\ + A \tilde{f} (e_{i} (t)) + u_{i} (t) + Δ_{i} (t, e_{i} (t)), \end{matrix}$ (5) where $\tilde{f_{i}} (t) = f (x_{i} (t)) - f (y (t))$ , i = 1, 2, ⋯ , N.

By the measurable selection theorem, the error system (5) with initial value $(\tilde{e}, \tilde{η})$ can be described as

$\begin{matrix} \dot{e_{i}} (t) = & - {Qe}_{i} (t) + c_{1} \sum_{j = 1}^{N} d_{ij} (e_{j} (t) - e_{i} (t)) \\ + A {\hat{γ}}_{i} (t) + ξ_{i} (t) + Δ_{i} (t, e_{i} (t)) \end{matrix}$ (6) where ${\hat{γ}}_{i} (t) = γ_{i} (t) - \tilde{γ} (t) \in \bar{co} [f (x_{i} (t))] - \bar{co} [f (y$ $(t))] \in \bar{co} [\tilde{f} (e (t))]$ .

In order to derive the robust synchronization results of CDNs (1), for the terms Δ_i (t, x), we also need to make the following assumption:

The disturbances Δ_i (t, x) are bounded by $| Δ_{i} (t, x) | \leq Δ_{max},$ (7) where Δ_max is known nonnegative constant.

Now, we design the following hybrid controllers:

u_{i} (t) = u_{i}^{1} (t) + u_{i}^{2} (t),

(8) where

u_{i}^{1} (t)

is the event-triggered sampling controller for i node, and

u_{i}^{2} (t)

is the state-feedback controller for i node.

In the event-triggered mechanism, in order to solve the problem what based on the time-scheduled control synchronization approaches may be conservative in practice, we focus on the design of event-triggered sampling control scheme. The sampling instants can be characterized as $0 = T_{0_{i}}^{i} < T_{1_{i}}^{i} < \dots < T_{k_{i}}^{i} < \dots < lim_{k_{i} \to + \infty} T_{k_{i}}^{i} = + \infty$ , where $T_{k_{i}}^{i} = k_{i} d$ for each node i, d denotes the maximum sampling interval. It satisfies the following condition, which controls the time instants in a way where to transmit the sampled control data:

$\begin{matrix} (e (T_{k + j}^{i}) - & e (T_{k}^{i}))^{T} \hat{F} (e (T_{k + j}^{i}) - e (T_{k}^{i})) \leq \\ κ e^{T} (T_{k + j}^{i}) \hat{F} e (T_{k + j}^{i}), \end{matrix}$ (9) where $\hat{F} \in R^{N \times N}$ denotes a positive definite matrix, κ is a positive scalar. Then, the considered control inputs can be defined by the sample-data estimates and isolated neural states: $u_{i}^{1} (t) = {Me}_{i} (t_{k_{i}}^{i}),$ (10) where M is the gain matrix, $t_{k_{i}}^{i}$ denotes the even-triggering instants for i node. Denote h (t) = t - t_k and z (t) =0 for $t \in [t_{k_{i}}^{i} + d_{k_{i}}^{i}, t_{(k + 1)_{i}}^{i} + d_{k_{i} + 1}^{i}]$ .

Similar to [28], one can get $u_{i}^{1} (t) = {Mz}_{i} (t) + {Me}_{i} (t - h (t)),$ (11) where 0 ≤ h (t) < h_M, h_M = d + h and z_i (t) satisfies $z_{i}^{T} (t) \hat{F} z_{i} (t) \leq κ e_{i}^{T} (t - h (t)) \hat{F} e_{i} (t - h (t)) .$ (12) Note that $u_{i}^{2} (t)$ will be designed afterwards.

Remark 2.1. In this paper, we adopt the the event-triggered control scheme in order to reduce the number of transmitted sampled data. We design the triggering condition that is dependent on the error between the latest transmitted data and newly sampled one. Obviously, we can adjust the amount of transmission in the communication channel by setting different triggering parameter κ under the event-triggered condition. The bigger value of κ, the more data can be sent to the controller. When κ = 0, the event-triggered scheme reduces to time-triggered scheme.

Definition 2.2. Under the suitable designed controller u_i (t), if there exists a a time function $T (\tilde{e}, \tilde{η}) > 0$ , such that $lim_{t \to T (\tilde{e}, \tilde{η})} ∥ e_{i} (t) ∥ = 0$ , and ∥e_i (t) ∥ ≡0, $t \geq T (\tilde{e}, \tilde{η})$ , then CDNs (1) is said to achieve the global synchronization in finite time. In addition, if there exists a constant T_max > 0, such that $T (\tilde{e}, \tilde{η}) \leq T_{max}$ , then CDNs (1) is said to achieve the global synchronization in fixed time. $T (\tilde{e}, \tilde{η})$ is called as the settling time, and T_max is the upper bound of the settling time.

Definition 2.3. ([38]) For function $V (x)$ satisfies

$V (x)$ is regular in Rⁿ;

for x = 0, $V (0) = 0$ and for x ≠ 0, $V (x) > 0$ ;

function $V (x) \to + \infty$ as ∥x∥ → + ∞, then, $V (x) : R^{n} \to R$ is called as C-regular.

Lemma 2.1. (Chain rule [35]) If function $V (x)$ is C-regular, and x (t) is absolutely continuous on [0, + ∞), then $V (x (t))$ is differentiable for a . a . t ∈ [0, + ∞), $\frac{d}{dt} V (x (t)) = υ^{T} \dot{x} (t), \forall υ \in \partial V (x),$ where $\partial V (x)$ is the Clarke generalized gradient of $V$ at x.

Lemma 2.2. ([21]) Suppose that $V : R^{n} \to R$ is a C-regular function, and e_i (t) : [0, + ∞) → Rⁿ is the solution with initial value $(\tilde{e}, \tilde{η})$ for error system (6). If there exist constants $\tilde{a} > 0, \tilde{b} > 0, \tilde{δ} > 1, 0 < \tilde{θ} < 1, \tilde{k} > 0$ , and $\tilde{δ} \tilde{k} > 1$ , $\tilde{θ} \tilde{k} < 1$ such that $\frac{d}{dt} V (e (t)) \leq - (\tilde{a} V^{\tilde{δ}} (e (t)) + \tilde{b} V^{\tilde{θ}} (e (t)))^{\tilde{k}},$ then the origin is fixed-time stable, and the upper bound settling time $T (\tilde{e}, \tilde{η})$ is estimated by $T_{1} (\tilde{e}, \tilde{η}) \leq T_{1}^{*} ≜ \frac{1}{{\tilde{a}}^{\tilde{k}} (\tilde{δ} \tilde{k} - 1)} + \frac{1}{{\tilde{b}}^{\tilde{k}} (1 - \tilde{θ} \tilde{k})} .$

Lemma 2.3. ([26]) Suppose that $V : R^{n} \to R$ is a C-regular function, and e_i (t) : [0, + ∞) → Rⁿ is the solution with initial value $(\tilde{e}, \tilde{η})$ for error system (6). If, there exist constants $\tilde{a} > 0, \tilde{b} > 0, \tilde{δ} > 0, \tilde{k} > 0$ , and $\tilde{δ} \tilde{k} > 1$ , such that $\frac{d}{dt} V (e (t)) \leq - (\tilde{a} V^{\tilde{δ}} (e (t)) + \tilde{b})^{\tilde{k}}, e (t) \in R^{n} ∖ {0},$ fora . e . t > 0, then the origin is fixed-time stable, and the upper bound settling time $T (\tilde{e}, \tilde{η})$ is estimated by $T_{2} (\tilde{e}, \tilde{η}) \leq T_{2}^{*} ≜ \frac{1}{{\tilde{b}}^{\tilde{k}}} (\frac{\tilde{a}}{\tilde{b}})^{\frac{1}{\tilde{δ}}} (1 + \frac{1}{\tilde{δ} \tilde{k} - 1}) .$

Lemma 2.4. ([37]) Suppose that $V : R^{n} \to R$ is a C-regular function, and e_i (t) : [0, + ∞) → Rⁿ is the solution with initial value $(\tilde{e}, \tilde{η})$ for error system (6). If, there exist constants $\tilde{a} > 0, \tilde{b} > 0, \tilde{c} > 0, 0 < \tilde{θ} < 1$ , and $\tilde{δ} > 1$ , such that $\frac{d}{dt} V (e (t)) \leq - \tilde{a} V^{\tilde{θ}} (e (t)) - \tilde{b} V^{\tilde{δ}} (e (t)) - c,$ for a . e . t > 0, then the origin is fixed-time stable, and the upper bound settling time $T (\tilde{e}, \tilde{η})$ is estimated by $\begin{matrix} \begin{matrix} T_{3} (\tilde{e}, \tilde{η}) \leq T_{3}^{*} ≜ & \frac{1}{{\tilde{a}}^{\frac{1}{\tilde{θ}}} (1 - \tilde{θ})} [({\tilde{a}}^{\frac{1}{\tilde{θ}}} + {\tilde{c}}^{\frac{1}{\tilde{θ}}})^{1 - \tilde{θ}} - {\tilde{c}}^{\frac{1 - \tilde{θ}}{\tilde{θ}}}] \\ + \frac{2^{\tilde{δ} - 1}}{{\tilde{b}}^{\frac{1}{\tilde{δ}}} (\tilde{δ} - 1)} ({\tilde{b}}^{\frac{1}{\tilde{δ}}} + {\tilde{c}}^{\frac{1}{\tilde{δ}}})^{1 - \tilde{δ}} \end{matrix} \end{matrix}$

Remark 2.2. In [21], Polyakov A, firstly put forward the fixed-time stability lemma and provided the estimated of fixed-time synchronization settling time. It was provided global finite-time stability of the closed-loop system and allowed to adjust a guaranteed settling time independently on initial conditions.

Remark 2.3. In [26], authors provide the estimation of synchronization settling time, i.e., $T (\tilde{e}, \tilde{η}) \leq T_{2}^{*} ≜ \frac{1}{a^{k} (δ k - 1)} + \frac{1}{b^{k}}$ . Comparing $T_{1}^{*}$ with $T_{2}^{*}$ in Lemma 2.3, we can see that the estimation expression $T_{2}^{*}$ is more accurate.

Remark 2.4. In [37], Chen et al. proposed a new fixed-time stability theorem based on strict theoretical derivations and provided the estimated of fixed-time synchronization settling time $T_{3}^{*}$ that was much smaller than $T_{1}^{*}$ , $T_{2}^{*}$ in the existing fixed-time stability lemmas.

Remark 2.5. In this paper, we aim at presenting the summary of recent advances in fixed-time synchronization conditions of CDNs systems and preserve some appealing properties about the fixed-time stable systems with a event-triggered control input.

Lemma 2.5. ([10]) Let δ₁, δ₂, ⋯ , δ_n ≥ 0, q > 1 and p ∈ (0, 1]. And then, $\sum_{i = 1}^{n} δ_{i}^{q} \geq n^{1 - q} (\sum_{i = 1}^{n} δ_{i})^{q}, \sum_{i = 1}^{n} δ_{i}^{p} \geq (\sum_{i = 1}^{n} δ_{i})^{p} .$

3 Main results

In this section, the global synchronization in fixed time issue is addressed for a class of CDNs (1) with constant coupling matrix. By designing three different controllers and using different lemmas, different dwelling times are obtained. The conditions of synchronization are given. To do so, the state-feedback controller is designed as follow:

Case 1:

$u_{i}^{2} (t) = - r sign (e_{i} (t)) - ɛ [e_{i} (t)]^{δ} - λ [e_{i} (t)]^{θ},$ (13) where r > 0, ɛ > 0, λ > 0 are scalar, δ > 1, 0 < θ < 1, sign (e_i (t)) = diag {sign (e_i1 (t)) , ⋯ , sign (e_in (t))}.

Note that controller (13) is discontinuous, we have $\bar{c} o [u_{i} (t)] = - r \bar{c} o [s i g n (e_{i} (t))] - ε {[e_{i} (t)]}^{δ} - λ {[e_{i} (t)]}^{θ},$ where $e_{i} (t) > 0, \bar{co} [sign (e_{i} (t))] = 1; e_{i} (t) = 0,$ $\bar{co}$ [sign (e_i (t))] = [- 1, 1]; e_i (t) <0, $\bar{co} [sign (e_{i} (t))] = - 1$ .

Let $ξ_{i} (t) \in \bar{co} [u_{i} (t)]$ , then there exists $Sign (e_{i} (t)) \in \bar{co} [sign (e_{i} (t))]$ , such that $ξ_{i} (t) = - r Sign (e_{i} (t)) - ɛ [e_{i} (t)]^{δ} - λ [e_{i} (t)]^{θ} .$

Case 2: we assume θ = 0, and design the following state-feedback controller $u_{i}^{2} (t) = - r sign (e_{i} (t)) - ɛ [e_{i} (t)]^{δ} - K \frac{e_{i} (t)}{∥ e_{i} (t) ∥^{2}},$ (14) where K > 0, other parameters are given in Case 1.

Case 3: we make the integration of Case 1 and Case 2, and developed the robust global synchronization conditions in fixed time for CDNs (1). We design the following controller: $u_{i}^{2} (t) = - r s i g n (e_{i} (t)) - ε {[e_{i} (t)]}^{δ} - λ {[e_{i} (t)]}^{θ} - K \frac{e_{i}}{(t)} ‖ e_{i} (t) ‖^{2},$ (15) where the parameters are given in Case 1 and Case 2.

Theorem 3.1. Suppose that (A₁) , (A₂) and (A₃) are satisfied. The couple interaction topology is undirected and connected. The CDNs (1) can achieve the global synchronization in fixed time under the event-triggered controller (11) and state-feedback controllers. It satisfies the conditions: $Θ = (\begin{matrix} Φ_{11} & Φ_{12} & Φ_{13} \\ Φ_{21} & Φ_{22} & Φ_{23} \\ Φ_{31} & Φ_{32} & Φ_{33} \end{matrix}) < 0,$ (16) $r - △_{\max} - n \hat{ω} \hat{a} > 0,$ (17) where $\begin{array}{l} Φ_{11} = - c_{1} (L \otimes I_{n}) - I_{N} \otimes Q + n \hat{ν} \hat{a} (I_{N} \otimes I_{n}), \\ Φ_{12} = \frac{M}{\otimes} I_{n} 2, Φ_{13} = \frac{M}{\otimes} I_{n} 2, \\ Φ_{22} = - \hat{F} \otimes I_{n}, Φ_{33} = κ \hat{F} \otimes I_{n}, \end{array}$ and $ρ = ɛ (nN)^{\frac{1 - δ}{2}}$ . There are different settling time with different controllers.

Case 1: The upper bound settling time $T_{1}^{*}$ is estimated by $T_{1}^{*} = \frac{2}{ρ δ - ρ} + \frac{2}{λ - λ θ},$ (18)Proof. Construct the Lyapunov functional

$\begin{matrix} V (t) = \frac{1}{2} \sum_{i = 1}^{N} e_{i}^{T} (t) e_{i} (t), \end{matrix}$ (19) calculating the derivative of $V (t)$ at time t along the trajectories of error dynamic system (6) gives

$\begin{matrix} \dot{V} (t) = - \sum_{i = 1}^{N} e_{i}^{T} (t) {Qe}_{i} (t) + \sum_{i = 1}^{N} e_{i}^{T} (t) A {\hat{γ}}_{i} (t) \\ + \sum_{i = 1}^{N} e_{i}^{T} (t) {Mz}_{i} (t) + \sum_{i = 1}^{N} e_{i}^{T} (t) {Me}_{i} (t - h (t)) \\ - r \sum_{i = 1}^{N} e_{i}^{T} (t) Sign (e_{i} (t)) - ɛ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{δ} \\ - λ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{θ} + \sum_{i = 1}^{N} e_{i}^{T} (t) Δ_{i} (t, e_{i} (t)) \\ + c_{1} \sum_{i = 1}^{N} \sum_{j = 1}^{N} d_{ij} e_{i}^{T} (t) (e_{j} (t) - e_{i} (t)) . \end{matrix}$ (20) Since d_ij = d_ji, we obtain

$\begin{matrix} \sum_{i = 1}^{N} \sum_{j = 1}^{N} d_{ij} e_{i}^{T} (t) (e_{j} (t) - e_{i} (t)) \\ = \frac{1}{2} \sum_{i = 1}^{N} \sum_{j = 1}^{N} d_{ij} e_{i}^{T} (t) (e_{j} (t) - e_{i} (t)) \\ + \frac{1}{2} \sum_{i = 1}^{N} \sum_{j = 1}^{N} d_{ji} e_{j}^{T} (t) (e_{i} (t) - e_{j} (t)) \\ = \frac{1}{2} \sum_{i = 1}^{N} \sum_{j = 1}^{N} d_{ij} e_{i}^{T} (t) (e_{j} (t) - e_{i} (t)) \\ - \frac{1}{2} \sum_{i = 1}^{N} \sum_{j = 1}^{N} d_{ij} e_{j}^{T} (t) (e_{j} (t) - e_{i} (t)) \\ = - \frac{1}{2} \sum_{i = 1}^{N} \sum_{j = 1}^{N} d_{ij} (e_{j} (t) - e_{i} (t))^{T} (e_{j} (t) - e_{i} (t)) \\ = - \frac{1}{2} \sum_{i = 1}^{N} \sum_{j = 1}^{N} \sum_{l = 1}^{n} d_{ij} (e_{jl} (t) - e_{il} (t))^{2} \\ = - e^{T} (t) (L \otimes I_{n}) e (t) . \end{matrix}$ (21) By means of assumption (A₁) and (A₂), we obtain $\begin{matrix} \sum_{i = 1}^{N} e_{i}^{T} (t) A {\hat{γ}}_{i} (t) = \sum_{i = 1}^{N} \sum_{r = 1}^{n} \sum_{l = 1}^{n} e_{il} (t) a_{lr} {\tilde{γ}}_{ir} (t) \\ \leq \sum_{i = 1}^{N} \sum_{r = 1}^{n} \sum_{l = 1}^{n} | e_{il} (t) | | a_{lr} | | {\tilde{γ}}_{ir} (t) | \\ \leq \sum_{i = 1}^{N} \sum_{r = 1}^{n} \sum_{l = 1}^{n} | e_{il} (t) | | a_{lr} | (ν_{r} | e_{ir} (t) | + ω_{r}) \\ \leq n \hat{ν} \hat{a} \sum_{i = 1}^{N} e_{i}^{T} (t) e_{i} (t) \\ + n \hat{ω} \hat{a} \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | . \end{matrix}$ (22) Under assumption (A₃), substituting (22),(23) into (21), it follows that

$\begin{matrix} \dot{V} (t) \leq - e^{T} (t) (I_{N} \otimes Q) e (t) + n \hat{ν} \hat{a} \sum_{i = 1}^{N} e_{i}^{T} (t) e_{i} (t) \\ + n \hat{ω} \hat{a} \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | - c_{1} e^{T} (t) (L \otimes I_{n}) e (t) \\ + Δ_{max} \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | + \sum_{i = 1}^{N} e_{i}^{T} (t) {Mz}_{i} (t) \\ + \sum_{i = 1}^{N} e_{i}^{T} (t) {Me}_{i} (t - h (t)) - r \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | \\ - ɛ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{δ} - λ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{θ} \\ - z_{i}^{T} (t) \hat{F} z_{i} (t) + κ e_{i}^{T} (t) (t - h (t)) \hat{F} e_{i} (t) (t - h (t)) . \end{matrix}$ (24) The inequality above can be rewritten $\begin{matrix} \dot{V} (t) \leq η Θ η^{T} - (r - △_{\max} - n \hat{ω} \hat{a}) \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | \\ - ɛ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{δ} - λ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{θ}, \end{matrix}$ where η = [e_i (t) , z_i (t) , e_i (t - h (t)] ^T.

Together (16) and (17), one has $\dot{V} (t) \leq - ɛ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{δ} - λ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{θ} .$ Using Lemma 2.5 yields $\begin{matrix} \begin{matrix} \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{δ} \geq (nN)^{\frac{1 - δ}{2}} (V (t))^{\frac{δ + 1}{2}}, \\ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{θ} \geq (\sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) |^{2})^{\frac{θ + 1}{2}} = (V (t))^{\frac{θ + 1}{2}} . \end{matrix} \end{matrix}$ Then, we have $\dot{V} (t) \leq - ρ (V (t))^{\frac{δ + 1}{2}} - λ (V (t))^{\frac{θ + 1}{2}} .$ (25) Form Lemma 2.2, this shows that the error system (6) is global fixed-time stable. Therefore, we conclude that the CNNs system (1) can achieve the global robust synchronization in fixed time under the event-triggered controller (11) and state-feedback controller (13).

Case 2: The upper bound settling time $T_{2}^{*}$ is estimated by $T_{2}^{*} = \frac{1}{K} (\frac{K}{ρ})^{\frac{1}{δ}} (1 + \frac{1}{δ - 1}) .$ (26) Analogous as the proof process of Case 1, we can get that $\begin{matrix} \dot{V} (t) & \leq - e^{T} (t) (I_{N} \otimes Q) e (t) + n \hat{ν} \hat{a} \sum_{i = 1}^{N} e_{i}^{T} (t) e_{i} (t) \\ + n \hat{ω} \hat{a} \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | - c_{1} e^{T} (t) (L \otimes I_{n}) e (t) \\ + Δ_{max} \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | + \sum_{i = 1}^{N} e_{i}^{T} (t) {Mz}_{i} (t) \\ + \sum_{i = 1}^{N} e_{i}^{T} (t) {Me}_{i} (t - h (t)) - r \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | \\ - ɛ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{δ} - K - z_{i}^{T} (t) \hat{F} z_{i} (t) \\ + κ e_{i}^{T} (t) (t - h (t)) \hat{F} e_{i} (t) (t - h (t)) \\ \leq η Θ η^{T} - (r - △_{\max} - n \hat{ω} \hat{a}) \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | \\ - ɛ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{δ} - K . \end{matrix}$ (27) By (16), (17) and (28), it follows that $\dot{V} (t) \leq - ρ (V (t))^{\frac{δ + 1}{2}} - K .$ (28) Applying Lemma 2.3, this shows that the error system (6) is global fixed-time stable. Therefore, we conclude that the CDNs system (1) can achieve the global robust synchronization in fixed time under the controller (11) and (14).

Case 3: The upper bound settling time $T_{3}^{*}$ is estimated by $T_{3}^{*} = \frac{1}{λ^{\frac{1}{θ}} (1 - θ)} [{(λ^{\frac{1}{θ}} + K^{\frac{1}{θ}})}^{1 - θ} - K^{\frac{1 - θ}{θ}}] + \frac{2^{δ - 1}}{ρ^{\frac{1}{δ}} (δ - 1)} {(ρ^{\frac{1}{δ}} + K^{\frac{1}{δ}})}^{1 - δ} .$ (29) Analogous as the proof process of Case 1, we can get that $\begin{matrix} \begin{matrix} \dot{V} (t) \leq - e^{T} (t) (I_{N} \otimes Q) e (t) + n \hat{ν} \hat{a} \sum_{i = 1}^{N} e_{i}^{T} (t) e_{i} (t) \\ + n \hat{ω} \hat{a} \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | - c_{1} e^{T} (t) (L \otimes I_{n}) e (t) \\ + Δ_{max} \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | + \sum_{i = 1}^{N} e_{i}^{T} (t) {Mz}_{i} (t) \\ + \sum_{i = 1}^{N} e_{i}^{T} (t) {Me}_{i} (t - h (t)) - r \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | \\ - ɛ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{δ} - λ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{θ} \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} - K - z_{i}^{T} (t) \hat{F} z_{i} (t) + κ e_{i}^{T} (t) (t - h (t)) \hat{F} e_{i} (t) (t - h (t)) . \end{matrix} \end{matrix}$ The inequality above can be rewritten $\begin{matrix} \begin{matrix} \dot{V} (t) & \leq η Θ η^{T} - (r - △_{\max} - n \hat{ω} \hat{a}) \sum_{i = 1}^{N} \sum_{l = 1}^{n} | e_{il} (t) | \\ - ɛ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{δ} - λ \sum_{i = 1}^{N} e_{i}^{T} (t) [e_{i} (t)]^{θ} - K \\ \leq - ρ (V (t))^{\frac{δ + 1}{2}} - λ (V (t))^{\frac{θ + 1}{2}} - K . \end{matrix} \end{matrix}$ Applying Lemma 2.4, this shows that the error system (6) is global fixed-time stable. Therefore, we conclude that the CNNs system (1) can achieve the global robust synchronization in fixed time under the hybrid controllers (11) and (15). The proof is complete.

Remark 3.1. It should be pointed out that, the upper bound of the settling time, $T_{1}^{*}$ , is independent on initial conditions. In addition, it is easy to see that, on the basis of the configuration for parameters λ and δ in the proposed control law, the upper bound $T_{1}^{*}$ can be determined in advance to any desired values. In general, in the practical engineering process, people hope that the synchronization can be obtained in a finite time and the time is the upper bound, which is independent on initial conditions rather than merely finitely. Therefore, the results of this paper extend and improve the previous results [26].

4 Numerical examples

Example 1. Consider an array CDNs (1) with 5 two-dimensional nodes in Case 1. The parameters are described as follow: $A = (\begin{matrix} 2 .4 & 1 .1 \\ 3 .1 & 2 .5 \end{matrix}), Q = (\begin{matrix} 2 .4 & 0 \\ 0 & 5 \end{matrix}) .$ We take $\begin{array}{l} D = (\begin{matrix} 2 & - 1 & - 1 & - 1 & - 1 \\ - 3 & 2 & - 1 & 0 & - 1 \\ - 1 & - 1 & 2 & - 1 & 0 \\ - 1 & 0 & - 1 & 2 & - 1 \\ - 1 & - 1 & 0 & - 1 & 1 \end{matrix}), \\ L = (\begin{matrix} -4 & 1 & 1 & 1 & 1 \\ 1 & - 3 & 1 & 0 & 1 \\ 1 & 1 & - 3 & 1 & 0 \\ 1 & 0 & 1 & - 3 & 1 \\ 1 & 1 & 0 & 1 & - 3 \end{matrix}), \end{array}$ c₁ = 0.4. The initial conditions are x₁ (s) = (5.7, 4) ^T, x₂ (s) = (1, 1.5) ^T, x₃ (s) = (1, 2) ^T, x₄ (s) = (-1.7, 3.9) ^T, x₅ (s) = (-1, 8) ^T, y (s) = (1.5, 1.2) ^T.

We take the activation functions f as f (x_il (t)) =0.7x_il (t) +0.02sign (x_il (t)), i = 1, 2, ⋯ , 5, l = 1, 2. It is easy to see that $\hat{ν} = 0.7, \hat{ω} = 0.02$ .

Set the disturbances Δ (x_i (t)) =0.7 sin(x_i1 (t)) , i = 1, 2, ⋯ , 5, which implies Δ_max = 0.7.

Next, the hybrid controllers (11) and (13) that gain is discussed, and the control gain is supposed to have the following parameters: ɛ = (0.4, 1.3) ^T, λ = (4, 2.4) ^T, and r = (1.3, 1.5) ^T.

Besides, the sampling period is taken as h = 0.15s, and κ = 0.6 with the controller gain and the triggering matrix as $\begin{array}{l} M = (\begin{array}{l} 0 .14 0 .21 0 .36 0 .78 1 .02 \\ 0 .24 0 .67 1 .06 1 .01 0 .39 \\ 0 .17 0 .28 0 .57 0 .65 0 .16 \\ 1 .05 0 .18 0 .17 0 .14 0 .19 \\ 0 .19 0 .26 0 .16 0 .34 0 .04 \end{array}), \\ \tilde{F} = (\begin{array}{l} 2 .01 1 .08 1 .35 1 .07 0 .18 \\ 0 .04 1 .2 0 .47 0 .68 0 .49 \\ 0 .32 0 .16 1 .47 0 .56 0 .31 \\ 0 .25 0 .17 0 .26 1 .16 0 .16 \\ 0 .16 0 .17 0 .47 1 .05 1 .24 \end{array}) . \end{array}$ With the above parameters, by simple computation, $r - △_{\max} - n \hat{ω} \hat{a} = 1.161 > 0 .$

Then, we calculate $T_{1}^{*} = 5.509 s$ . Moreover, it is so easy to verify that the condition (17) in Case 1 is also satisfied. As shown in Figs.1, the simulation results agree well with the theoretical analysis. Moreover, the event-triggered intervals are given in Figs. 2-3, which are shown that the Zone behaviors are excluded.

Fig.1

Time evolutions of e_ij (t) , i, j = 1, ⋯ , 5

Fig.2

Release instants and intervals for node 1

Fig.3

Release instants and intervals for node 2

Example 2. Consider an array CDNs (1) with 3 three-dimensional nodes in Case 2. The parameters are described as follow: $A = (\begin{array}{l} 1 .01 2 .1 2 .05 \\ 1 .15 .0 .85 0 .2 \\ 5 .1 0 .02 2 .21 \end{array}), Q = (\begin{array}{l} 3 .7 0 0 \\ 0 2 .2 0 \\ 0 0 1 .5 \end{array}) .$ We take $D = (\begin{array}{l} 2 1 0 \\ -1 3 -1 \\ -1 -1 3 \end{array}), L = (\begin{array}{l} 1 -1 0 \\ 1 2 1 \\ 1 1 -2 \end{array}) .$ c₁ = 0.04. The initial conditions are x₁ (s) = (-2, -4, 10) ^T, x₂ (s) = (-4, - 6, 3) ^T, x₃ (s) = (2, - 2, 7) ^T, y (s) = (7, 8, - 10) ^T.

We take the activation functions f as f (x_il (t)) =0.8x_il (t) +0.02sign (x_il (t)), i = 1, 2, 3, l = 1, 2, 3. It is easy to see that $\hat{ν} = 0.8, \hat{ω} = 0.02$ .

Set the disturbances Δ (x_i (t)) =0.2 cos(x_i1 (t)) , i = 1, 2, 3, which implies Δ_max = 0.2.

Next, the feedback controller (14) that gain is discussed, and the control gain is supposed to have the following parameters: ɛ = (4, 2.4, 1.5) ^T, and K = (0.2, 0.2, 0.2) ^T, r = (1.5, 1.5, 1.5) ^T.

Besides, the sampling period is taken as h = 0.2s, and κ = 0.6 with the controller gain and the triggering matrix as $M = (\begin{array}{l} 0 .7 0 .143 0 .78 \\ 1 .053 6 .3 0 .2 \\ 0 .01 0 .3 0 .2 \end{array}), \tilde{F} = (\begin{array}{l} 1 .01 3 1 .3 \\ 0 .8 1 .4 3 .8 \\ 1 .7 5 .2 3 .4 \end{array}) .$

With the above parameters, by simple computation, $r - △_{\max} - n \hat{ω} \hat{a} = 1.67 > 0 .$

Then, we calculate $T_{2}^{*} = 2.942 s$ . Fig. 4 shows the evolutions of e_ij, i, j = 1, 2, 3. Moreover, the event-triggered intervals are given in Figs. 5-7, which are shown that the Zone behaviors are excluded.

Fig.4

Time evolutions of e_ij, i, j = 1, 2, 3

Fig.5

Release instants and intervals for node 1

Fig.6

Release instants and intervals for node 2

Fig.7

Release instants and intervals for node 3

Example 3. Consider an array CDNs (1) with 5 three-dimensional nodes in Case 3. The parameters are described as follow: $A = (\begin{array}{l} 1 .4 1 .1 1 .5 \\ 1 .1 1 .2 1 .4 \\ 1 .5 1 .6 1 .51 \end{array}), Q = (\begin{array}{l} 4 .3 0 0 \\ 0 3 .5 0 \\ 0 0 5 .2 \end{array}) .$ We take

$D = (\begin{array}{l} 2 −1 −1 −1 −1 \\ 1 1 −1 0 −1 \\ −1 −1 4 −1 0 \\ −1 0 −1 1 −1 \\ −1 −1 0 −1 2 \end{array}), L = (\begin{array}{l} −4 1 1 1 1 \\ 1 −3 1 0 1 \\ 1 1 −3 1 0 \\ 1 0 1 −3 1 \\ 1 1 0 1 −3 \end{array}) .$

c₁ = 0.05. The initial conditions are x₁ (s) = (5.7, 4, 4) ^T, x₂ (s) = (-10, - 1.5, 0.12) ^T, x₃ (s) = (1, 0.2, 24) ^T, x₄ (s) = (-1.7, 3.9, 0.05) ^T, x₅ (s) = (0.5, 18, 3.4) ^T, y (s) = (1.5, 1.2, 4) ^T.

We take the activation functions f as f (x_il (t)) =0.3x_il (t) +0.5sign (x_il (t)) , i = 1, 2, ⋯ , 5, l = 1, 2, 3. It is easy to see that $\hat{ν} = 0.3, \hat{ω} = 0.5$ .

Set the disturbances Δ (x_i (t)) =0.3 sin(x_i1 (t)) , i = 1, 2, ⋯ , 5, which implies Δ_max = 0.3.

Next, the feedback controller (15) that gain is discussed, and the control gain is supposed to have the following parameters: ɛ = (0.1, 0.3, 1.3) ^T, λ = (1.4, - 0.04, 0.4) ^T, K = (0.4, 0.3, 0.5) ^T, and r = (3, 1.4, 0.5) ^T.

Besides, the sampling period is taken as h = 0.1s, and κ = 0.6 with the controller gain and the triggering matrix as $\begin{array}{l} M = (\begin{array}{l} 1 .001 0 .45 0 .36 0 .14 0 .38 \\ 0 .01 1 .06 0 .45 0 .25 0 .15 \\ 0 .18 0 .17 1 .7 0 .74 0 .26 \\ 0 .19 0 .07 0 .25 1 .13 0 .39 \\ 0 .14 0 .38 0 .27 0 .16 −1 \end{array}), \\ \tilde{F} = (\begin{array}{l} 1 .05 0 .21 0 .37 0 .14 0 .39 \\ 0 .76 1 .14 0 .17 0 .25 0 .47 \\ 0 .13 0 .56 1 .4 0 .34 0 .17 \\ 0 .16 0 .04 0 .12 1 0 .1 \\ 0 .14 0 .02 0 .01 0 .2 0 .3 \end{array}) . \end{array}$ With the above parameters, by simple computation, $r - △_{\max} - n \hat{ω} \hat{a} = 0.648 > 0 .$

Then, we calculate $T_{3}^{*} = 4.971 s$ . Fig. 8 shows the evolutions of e_ij, i, j = 1, 2, ⋯ , 5. Moreover, the event-triggered intervals are given in Figs. 9-11, which are shown that the Zone behaviors are excluded.

Fig.8

Time evolutions of e_ij, i, j = 1, ⋯ , 5

Fig.9

Release instants and intervals for node 1

Fig.10

Release instants and intervals for node 2

Fig.11

Release instants and intervals for node 3

5 Conclusion

In this paper, we considered the global robust synchronization in fixed time for CDNs with discontinuous activation functions. Under the hybrid controllers included the event-triggered term and discontinuity, the global synchronization conditions have been presented, and the settling time, which is independent on initial conditions, has been also evaluated.

Future work will be focused on how to remove the chatter of the designed controller, and to extend the results here obtained for stochastic pining synchronization control for CDNs with delays and discontinuous activations by designing appropriate feedback controllers.

Footnotes

Acknowledgements

This work was supported by the Natural Science Foundation of Hebei Province of China (A2018203288).

References

Abdel-Aty

, Quantum information entropy and multi-qubit entanglement, Progress in Quantum Electronics 31 (2007), 1–49.

Bergman

and Siegal

, Evolutionary capacitance as a general feature of complex gene networks, Nature 424 (2003), 549–552.

Boccaletti

, Latora

, Moreno

, Chavez

and Hwang

D.U.

, Complex networks: structure and dynamics, Physcial, Reports 424 (2006), 175–308.

Jiang

and Yang

, Complex dynamics in a food chain with slow and fast processes, Chaos Soliton Fract 42 (2009), 3160–3618.

and Chua

L.O.

, Synchronization in an array of linearly coupled dynamical systems, IEEE Transactions on Circuits and Systems I: Fundamental Theory and, Applications 42 (1995), 430–447.

Peng

, Wu

and Cao

, Global nonfragile synchronization in finite time for fractional-order discontinuous neural networks with nonlinear growth activations, IEEE Transactions on Neural Networks and Learning Systems 30 (2019), 2123–2137.

Batle

, Ooi

C.R.

, Farouk

, Alkhambashi

M.S.

and Abdalla

, Global versus local quantum correlations in the Grover search algorithm, Quantum Information Processing 15 (2016), 833–849.

Batle

, Ooi

C.R.

, Farouk

, Abutali

and Abdalla

, Do multipartite correlations speed up adiabatic quantum computation or quantum annealing? Quantum Information Processing 15 (2016), 3081–3099.

Liu

and Wu

, Stochastic finite-time synchronization for discontinuous semi-Markovian switching neural networks with time delays and noise disturbance, Neurocomputing 310 (2018), 246–264.

10.

Sun

, Li

and Zhao

, Outer synchronization between two complex dynamical networks with discontinuous coupling, Chaos 22 (2012), 440–450.

11.

Yang

, Finite-Time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays, IEEE Transactions on Fuzzy Systems 23 (2015), 2302–2316.

12.

Ding

, Shen

and Wang

, Global Mittag-Leffer synchronization of fractional-order neural networks with discontinuous activations, Neural Networks 73 (2016), 77–85.

13.

Song

, Li

and Li

, Synchronization transition of time delayed complex dynamical networks with discontinuous coupling, Communications in Theoretical Physics 67 (2017), 54–65.

14.

Haimo

V.T.

, Finite time controllers, Control Optim 24 (1986), 760–770.

15.

Bhat

and Bernstein

, Finite-time stability of homogeneous systems, Proceedings of ACC 4 (1997), 2513–2514.

16.

Bowong

and Kakmeni

, Chaos control and duration time of a class of uncertain chaotic systems, Physical 316 (2003), 206–217.

17.

and Shen

, Reach control problem for linear differential inclusion systems on simplices, IEEE Transactions on Automatic Control 61 (2016), 1403–1408.

18.

, Xia

, Cao

and Sun

, Reach control problem for affine multi-agent systems on simplices, Automatica 107 (2019), 264–271.

19.

Batle

, Ciftja

, Naseri

, Ghoranneviss

, Farou

and Elhoseny

, Equilibrium and uniform charge distribution of a classical two-dimensional system of point charges with hard-wall confinement, Physica Scripta 92 (2017), 055801.

20.

, Zhang

and Xue

, LMI conditions to global Mittag-Leffler stability of fractional-order neural networks with impulses, Neurocomputing 193 (2016), 148–154.

21.

Polyakov

, Nonlinear feedback design for fixed-time stabilization of linear control systems, IEEE Trans Autom Control 57 (2012), 2106–2110.

22.

, Liu

and Liu

, Fast fixed-time nonsingular terminal sliding mode control and its application to chaos suppression in power system, IEEE Transactions on Circuits and Systems II: Express Briefs (2016). 10.1109/TCSII.2016.2551539

23.

Defoort

, Veluvolu

and Djemai

, Leader-follower fixedtime consensus for multi-agent systems with unknown nonlinear inherent dynamics, IET Journals 9 (2015), 2165–2170.

24.

Ding

, Cao

, Alsaedi

, Alsaadi

F.E.

and Hayat

, Robust fixed-time synchronization for uncertain complex-valued neural networks with discontinuous activation functions, Neural Networks 90 (2017), 42–55.

25.

Zhu

, Yang

, Alsaadi

F.-E.

and Hayat

, Fixed-time synchronization of coupled discontinuous neural networks with nonidentical perturbations, Neural Processing Letters 48 (2017), 1161–1174.

26.

, Yu

and Chen

, Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks, Neural Networks 89 (2017), 74–83.

27.

Hui

and Wangli

, Fixed-time synchronization for coupled delayed neural networks with discontinuous or continuous activations, Neurocomputing 314 (2018), 143–153.

28.

Zhao

and Wu

, Fixed-time synchronization of semi-Markovian jumping neural networks with time-varying delays, Advances in Difference Equations (2018). DOI: 10.1186/s13662-018-1666-z

29.

Batle

, Naseri

, Ghoranneviss

, Farouk

, Alkhambashi

and Elhoseny

, Shareability of correlations in multiqubit states: optimization of nonlocal monogamy inequalities, Physical Review A 95 (2017), 032123.

30.

Sivaranjani

, Rakkiyappan

and Cao

, Synchronization of nonlinear singularly perturbed complex networks with uncertain inner coupling via event triggered control, Applied Mathematics and Computation 311 (2017), 283–299.

31.

Wang

and Wu

, Global synchronization in fixed time for semi-Markovian switching complex dynamical networks with hybrid couplings and time-varying delays, Nonlinear Dynamics 95 (2019), 2031–2062.

32.

Batle

, Farouk

, Tarawneh

and Abdalla

, Multipartite quantum correlations among atoms in QED cavities, Frontiers of Physics 13 (2018), 130305.

33.

Nagata

, Nakamura

, Geurdes

, Batle

, Abdalla

and Farouk

, Creating very true quantum algorithms for quantum energy based computing, International Journal of Theoretical Physics 57 (2018), 973–980.

34.

Gao

, Guirao

J.L.G.

, Abdel-Aty

and Xi

, An independent set degree condition for fractional critical deleted graphs, Discrete and Continuous Dynamical Systems-Series 12 (2019), 877–886.

35.

Forti

and Nistri

, Global convergence of neural networks with discontinuous neuron activations, Circuits and Systems I Regular Papers IEEE Transactions on 50 (2003), 1421–1435.

36.

Forti

, Nistri

and Papini

, Global exponential stability and global convergence in finite time of delayed neural networks with infinite gain, IEEE Transactions on Neural Networks 16 (2005), 1449–1463.

37.

Chen

and Li

, A new fixed-time stability theorem and its application to the synchronization control of memristive neural networks, Neurocomputing (2019). DOI: 10.1016/j.neucom.2019.03.040

38.

Peng

and Wu

, Non-fragile robust finite-time stabilization and H1 performance analysis for fractional-order delayed neural networks with discontinuous activations under the asynchronous switching, Neural Computing and Applications (2018). DOI: 10.1007/s00521-018-3682-z