Finite Time Hybrid Synchronization Control for Duplex Complex Networks via Intermittent Control

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Finite Time Hybrid Synchronization Control for Duplex Complex Networks via Intermittent Control

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Cheng-jun Xie,Ying Zhang,

Qiang Wei^*

Cheng-jun Xie,Ying Zhang,

Qiang Wei^*

This version is not peer-reviewed

Submitted:

01 October 2024

Posted:

02 October 2024

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Abstract

This paper is devoted to the finite time internal and the external synchronization (hybrid synchronization) for complex networks by means of intermittent control. Different from most previous research achievements, the finite time synchronization include internal and the external synchronization, this article employs the intermittent control to drive the complex networks to achieve synchronization in finite time. To be specific, the internal and the external synchronization all achieve consistent synchronization in finite time by devise Lyapunov function. Finally, numerical examples are presented to illustrate the validness of theoretical results.

Keywords:

Subject: Computer Science and Mathematics - Other

1. Introduction

Complex networks exists in nature and society widely, in the past decades, complex networks have attracted extensive attention in the system science, the complex networks are divided into various types, such as social networks [1], communication networks [2], neural networks [3], and epidemic spreading networks [4] and so on. The synchronization is one of the most important collective behaviors of complex networks, which has always been a hot issue in the research of complex networks [5,6,7]. To sum up, the synchronization of complex networks is divided in two types: the internal synchronization (synchronization between network and node) and the external synchronization (synchronization between two complex networks). Therefore, in order to achieve synchronization, the controllers have to add to networks. Many effective control methods have been developed for synchronization of complex networks [8,9,10]. These methods can be divided into continuous controls and discontinuous controls based on the controllers work continuously or not. A typical way of discontinuous control is the intermittent control, which the working hours are discontinuous, the control is only active in the work time. The intermittent controls is better than continuous controls, the synchronization have been addressed via intermittent control. For example, in [11], the coupled network is synchronized by aperiodically intermittent controllers with adaptive strategy. In [12,13], outer synchronization of complex networks is investigated via periodically and aperiodically intermittent control, respectively.

The synchronization time is divided into infinite time and finite time. However, in reality, the networks should achieve synchronization as quickly as possible, finite time has better application prospects than infinite time, particularly in many engineering fields. To achieve faster synchronization of networks, finite time control technique is applied in [14]. Combining with the intermittent control scheme, finite time synchronization, whose convergence time can be predicted, has been studied in [15,16,17]. It is well known that many real complex networks, including transportation networks, social network and ecological network, where networks interact with other networks. Therefore, the different part of networks have different property, which lead to the different part of networks with different synchronization types [18,19,20].

However, to the best of our knowledge, there are no results on the finite time internal and the external synchronization for duplex complex networks by intermittent control. More studies are expected to study the variety of finite time synchronization of duplex complex networks by intermittent control. Motivated by the above discussions, this paper investigates the finite time internal and the external synchronization problem for duplex complex networks via intermittent control. The internal and the external synchronization can be synchronized simultaneously in finite time.

2. Network Model and Preliminaries

Assumption 1 [3]. The time-varying delay

τ (t)

is a function which satisfies

0 \leq τ (t) \leq μ,

maximum

μ = 1

in this paper. Derivative of the delay

τ (t)

exists, and

τ (t)

less than 1, which can also be written as

0 \leq \dot{τ} (t) \leq ξ < 1

Due to diversity of network, the different parts of the duplex network have different types of synchronization, the synchronization is mainly divided into two kinds: the internal and the external synchronization. Internal synchronization means that all nodes in the network are consistent with a certain node, external synchronization means that the node states of two or more complex networks are progressively consistent.

Consider a duplex complex networks (1)-(2) and a nonlinear dynamics mathematical model (3) can be expressed as:

\begin{array}{l} {\dot{x}}_{i} (t) = \tilde{A} x_{i} (t) + \tilde{B} f (x_{i} (t - τ_{1} (t)) + \sum_{j = 1}^{N} c_{i j}^{1} g (x_{j} (t)) + \sum_{j = 1}^{N} d_{i j}^{1} h (x_{j} (t - τ_{2} (t)) + u_{i n} (t) + u_{o u t} (t), (1) \\ {\dot{y}}_{i} (t) = \tilde{A} y_{i} (t) + \tilde{B} f (y_{i} (t - τ_{1} (t)) + \sum_{j = 1}^{N} c_{i j}^{2} g (y_{j} (t)) + \sum_{j = 1}^{N} d_{i j}^{2} h (x_{j} (t - τ_{2} (t)), (2) \\ \dot{s} (t) = \tilde{z} (s (t)), (3) \end{array}

where

x_{i} (t) = {(x_{i 1} (t), x_{i 2} (t), \dots, x_{i n} (t))}^{T} \in R^{n}

is the state variables of the ith node in complex network (1),

f : R^{n} \to R^{n}

is the continuously vector function which explain the dynamical equation of the ith node, the

τ_{1} (t) > 0

and

τ_{2} (t) > 0

are coupling time-varying delay. The

g : R^{n} \to R^{n}

and

h : R^{n} \to R^{n}

are couple functons.

G = (g_{i j}) \in R^{N \times N}

is the coupling matrix,

g_{i j} > 0

if there is connection between the ith node and the jth node, otherwise

g_{i j} = g_{j i} = 0

y_{i} (t) = {(y_{i 1} (t), y_{i 2} (t), \dots, y_{i n} (t))}^{T} \in R^{n}

is the state variables of the ith node in complex network (3).

g (\cdot) \in R^{n}

and

h (\cdot) \in R^{n}

are the nonlinear coupling functions.

\tilde{A} = A + Δ A (t) \in R^{n \times n}

\tilde{B} = B + Δ B (t) \in R^{n \times n}

and

\tilde{Z} = Z + Δ Z (t) \in R^{n \times n}

stand for the parametric uncertainties. For

k = 1, 2, C^{k} \in R^{N \times N}

and

D^{k} \in R^{N \times N}

are the coupling configuration matrices which reflect the fundamental topology structure of network. If there is a connection between node

i

and node

j (i \neq j),

then

c_{i j}^{k} > 0 and d_{i j}^{k} > 0;

otherwise,

c_{i j}^{k} = 0

and

d_{i j}^{k} = 0

. The diagonal elements of matrix

C^{k}

and

D^{k}

can be represented as

c_{i i}^{k} = - \sum_{j = 1, j \neq i}^{N} c_{i j}^{k}

and

d_{i i}^{k} = - \sum_{j = 1, j \neq i}^{N} d_{i j}^{k}

, respectively.

s (t) \in R^{n}

is equinoctial point, or chaotic attractors, or a periodic orbit. The

u_{i n} (t) \in R^{n} and u_{o u t} (t) \in R^{n}

are intermittent controllers to be designed. The

u_{i n} (t)

is the internal synchronization controller which control node 1 to d, the

u_{o u t} (t)

is the external synchronization controller which control node

d + 1

to N. Furthermore,

C^{k}

and

D^{k}

are not necessarily assumed to be symmetric.

Definition 1. The hybrid synchronization error is defined as follows,

e (t) = e_{i n} (t) + e_{o u t} (t)

where

e_{i n} (t)

is the internal synchronization error, one has,

e_{i n} (t) = x_{i} (t) - s (t - τ (t)), i = 1, 2, \dots, d,

where

τ (t) > 0

is time-varying delay,

x_{i} (t) \in R^{n}

is a state variable of complex networks (1),

s (t)

is a state variable of system (2). The

e_{o u t} (t)

is the external synchronization error. Defined,

e_{o u t} (t) = x_{j} (t) - y_{j} (t - τ (t)), j = d + 1, d + 2, \dots, N,

where nodes 1 to d implement internal synchronization, the nodes

d + 1

to N implement external synchronization.

Definition 1: Hybrid synchronization is achieved if there exists a setting time

T

such that,

\begin{array}{l} \lim_{t \to T} ‖e (t)‖ \underset{t \to T}{= \lim} ‖e_{i n} (t)‖ + \lim_{t \to T} ‖e_{o u t} (t)‖ \underset{t \to T}{= \lim} ‖e_{i n} (t) + e_{o u t} (t)‖ \\ = \lim_{t \to T} ‖x_{i} (t) - s (t - τ (t)) + x_{j} (t) - y_{j} (t - τ (t))‖ = 0 . \end{array}

Lemma1 [5]: Assuming that a continuous positive definite function

V (t)

satisfying,

\dot{V} (t) \leq - ω V^{r} (t), \forall t \geq t_{0}, V (t_{0}) \geq 0,

where

ω > 0

0 < r < 1

V (t) \equiv 0

for all

t \geq T (x_{0})

, the settling time

T (x_{0}) \leq \frac{V^{1 - r} (x_{0})}{ω (1 - r)}

Definition 2: Given

θ \in R^{n}, r > 0

. Then, one has

s i g n (θ) = d i a g (s i g n (θ_{1}), s i g n (θ_{2}), \dots, s i g n (θ_{n})),

| θ |^{r} = {(| θ_{1} |^{r}, | θ_{2} |^{r}, \dots, | θ_{n} |^{r})}^{T} .

Assumption 1: The

τ_{1} (t)

and

τ_{2} (t)

represent two time-varying delay, which are bounded differential functions, one has

0 \leq τ_{1} (t) \leq τ_{1}^{*}, 0 \leq τ_{2} (t) \leq τ_{2}^{*}

{\dot{τ}}_{1} (t) \leq γ < 1, {\dot{τ}}_{2} (t) \leq γ < 1

for all

t

, where

τ_{1}^{*}

τ_{2}^{*}

and

γ

are positive constants.

Assumption 2: The

j (\cdot)

r (\cdot)

and

h (\cdot)

represent three Lipschitz functions, i.e., there exists positive constants

L_{j}

L_{r}

L_{h}

such that for all

μ, ν \in R^{n}

, one has,

\begin{array}{l} ‖ j (μ) - j (ν) ‖ \leq L_{j} ‖ μ - ν ‖, \\ ‖ r (μ) - r (ν) ‖ \leq L_{r} ‖ μ - ν ‖, \\ ‖ h (μ) - h (ν) ‖ \leq L_{h} ‖ μ - ν ‖, \end{array}

Assumption 3: Given uncertain parameter matrices

Δ A (t)

and

Δ B (t)

satisfying,

[Δ A (t), Δ B (t)] = M F (t) [N_{1}, N_{2}],

where

M, N_{1} N_{2}

are known real matrices,

F (t)

is a unknown time-varying real matrix, which satisfying

F^{T} (t) F (t) \leq I .

Lemma 1 [5]: Given X and Y are two real matrices, then

X^{T} Y + Y^{T} X \leq ζ X^{T} X + \frac{1}{ζ} Y^{T} Y

holds for any

ζ > 0

3. Main Results

In this section, we establish the finite time intermittent control scheme for hybrid synchronization. According to system (1)-(3), the error state equations can be expressed in the compact form,

\begin{array}{l} \dot{e} (t) = (I_{d} \otimes \tilde{A}) e_{i n} (t) + (I_{d} \otimes \tilde{B}) F (e_{i n} (τ_{1})) + (C \otimes I_{d}) G (e_{i n} (t)) + (D \otimes I_{d}) H (e_{i n} (τ_{2})) + u_{i n} (t) \\ + \tilde{A} e_{o u t} (t) + (I_{N - d} \otimes \tilde{B}) F (e_{o u t} (τ_{1})) + I_{N - d} G (e_{o u t} (t)) + (D \otimes I_{N - d}) H (e_{o u t} (τ_{2})) + u_{o u t} (t), \end{array}

(4)

where

The intermittent controllers is designed as

where

ξ > 0

η_{1} > 0

η_{2} > 0

η_{3} > 0

and

η_{4} > 0

are the control strengths.

α > 0

λ > 0

0 \leq σ < 1

and

0 \leq μ < 1

are tunable constants.

V_{i n} (t)

and

V_{o u t} (t)

are two given Lyapunov functions,

V_{i n i} (t)

and

V_{o u t j} (t)

are two nonnegative continuous functions to be designed.

t^{-}

indicates the previous instant of time

t

. From (4), it is easy to see that the work time of controller

u_{1} (t)

and

u_{2} (t)

are decided by the time-varying relation among

V (t)

V_{i n i} (t)

and

V_{o u t j} (t)

. For brevity, take

β = \frac{1 + σ}{2}, χ = \frac{μ}{3}

, one has,

\begin{array}{l} V_{i n i} (t) = {({(k V (t_{0}))}^{1 - β} - α_{i} (1 - β) (t - t_{0}))}^{\frac{1}{1 - β}}, i = 1, 2, \\ V_{o u t j} (t) = {({(ϑ V (t_{0}))}^{1 - χ} - λ_{j} χ (t - t_{0}))}^{\frac{1}{χ}}, j = 1, 2, \end{array}

where

α_{i}

λ_{i}

k

and

ϑ

are positive scalars,

α > α_{1} > α_{2}

λ > λ_{1} > λ_{2}

and

0 < k < 1

0 < ϑ < 1

, respectively.

Based on the relation among

V_{i n} (t)

V_{o u t} (t)

V_{i n i} (t)

and

V_{o u t j} (t)

, the intermittent control process are as follows:

1. If

V_{i n} (t) \geq V_{i n 1} (t) and V_{o u t} (t) \geq V_{o u t 1} (t)

, then the controllers

u_{i n} (t) = u_{1} (t)

and

u_{o u t} (t) = u_{2} (t)

are actived at instant

t

and continuously working.

2. If

V_{i n 2} (t) \leq V_{i n} (t) \leq V_{i n 1} (t) and V_{o u t 2} (t) \leq V_{o u t} (t) \leq V_{o u t 1} (t)

and the controllers

u_{i n} (t) = u_{1} (t)

and

u_{o u t} (t) = u_{2} (t)

are working at the instant

t^{-}

, then the controllers

u_{i n} (t) = u_{1} (t)

and

u_{o u t} (t) = u_{2} (t)

are continuously working after the time

t

3. If

V_{i n 2} (t) \leq V_{i n} (t) \leq V_{i n 1} (t) and V_{o u t 2} (t) \leq V_{o u t} (t) \leq V_{o u t 1} (t)

and the controller

u_{i n} (t) = 0

and

u_{o u t} (t) = 0

are working at the instant

t^{-}

, then the controller

u_{i n} (t) = 0

and

u_{o u t} (t) = 0

are continuously working after the time

t

4. If

V_{i n} (t) \leq V_{i n 2} (t) and V_{o u t} (t) \leq V_{o u t 2} (t)

, then the controller

u_{i n} (t) = 0

and

u_{o u t} (t) = 0

are actived at instant

t

and continuously working.

Remark 1: The two controllers

u_{i n} (t)

and

u_{o u t} (t)

can achieve finite time hybrid synchronization in the whole network, or the controller

u_{i n} (t)

achieve internal synchronization in finite time and controller

u_{o u t} (t)

achieve external synchronization in finite time. This paper focuses on finite time hybrid synchronization in the whole network. The finite time synchronization criterion with the Lyapunov function

V_{i n} (t)

and

V_{o u t} (t)

are designed as follows:

\begin{array}{l} V_{i n} (t) = e_{i n}^{T} (t) e (t) + \frac{η_{1}}{1 - γ} \int_{t - τ_{1} (t)}^{t} e_{i n}^{T} (s) e_{i n} (s) d s + \frac{η_{1}}{1 - γ} \int_{t - τ_{2} (t)}^{t} e_{i n}^{T} (s) e (s) d s, \\ V_{o u t} (t) = \frac{η_{3}}{1 - φ} \int_{t - τ_{1} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s + \frac{η_{4}}{1 - φ} \int_{t - τ_{2} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s . \end{array}

Therefore, the finite time synchronization criterion is given.

Theorem 1: Supposing that Assumptions 1-3 hold, there exist tunable positive constants

ξ

η_{1}

η_{2}

and any positive constants

s_{1}, \dots, s_{5}

such that,

\begin{array}{l} (a) . Δ \leq 0, \\ (b) . s_{2}^{- 1} L_{j}^{2} + s_{3}^{- 1} L_{j}^{2} λ_{m a x} (N_{2}^{T} N_{2}) - s_{1}^{- 1} L_{j}^{2} + η_{1} + η_{3} \leq 0, \\ (c) . s_{5}^{- 1} L_{h}^{2} - η_{2} - η_{2} + η_{4} \leq 0, \end{array}

where

\begin{array}{l} Δ = λ_{m a x} (A^{s}) + s_{1} λ_{m a x} (M M^{T}) + s_{4}^{- 1} L_{r}^{2} + s_{3} L_{r}^{2} + s_{1} L_{r}^{2} - s_{1}^{- 1} λ_{m a x} (N_{1}^{T} N_{1}) + s_{2} λ_{m a x} (B B^{T}) \\ + s_{4} λ_{m a x} (C C^{T}) + s_{5} λ_{m a x} (D D^{T}) + \frac{η_{1} + η_{2} + η_{3}}{1 - γ} - ξ . \end{array}

Then the networks (1)-(3) with the intermittent control scheme (4) and (5) will be synchronized in a finite time

T

, and

T < T_{1} = \frac{{(ϑ V_{o u t} (t_{0}))}^{χ}}{λ_{2} χ} + t_{0} .

Proof: Differentiating

V_{i n} (t)

and

V_{o u t} (t)

along the solution of error system (3), one has

\begin{array}{l} {\dot{V}}_{i n} (t) \leq 2 e_{i n}^{T} (t) {\dot{e}}_{i n} (t) + \frac{η_{1}}{1 - γ} e_{i n}^{T} (t) e_{i n}^{T} (t) - η_{1} e_{i n}^{T} (τ_{1}) e_{i n} (τ_{1}) ​ + \frac{η_{2}}{1 - γ} e_{i n}^{T} (t) e_{i n} (t) - η_{2} e_{i n}^{T} (τ_{2}) e_{i n} (τ_{2}) . \\ {\dot{V}}_{o u t} (t) \leq \frac{η_{3}}{1 - φ} e_{o u t}^{T} (t) e_{o u t}^{T} (t) - η_{3} e_{o u t}^{T} (τ_{1}) e_{o u t} (τ_{1}) ​ + \frac{η_{4}}{1 - φ} e_{o u t}^{T} (t) e_{o u t} (t) - η_{4} e_{o u t}^{T} (τ_{2}) e_{o u t} (τ_{2}) . \end{array}

Based on the error equation (3) and the control scheme (4) and (5), one has

\begin{array}{l} 2_{e}^{T} (t)_{\dot{e}} (t) = 2 e_{i n}^{T} (t) (I_{N} \otimes \tilde{A}) e_{i n} (t) + 2 e_{i n}^{T} (t) (I_{N} \otimes \tilde{B}) F (e_{i n} (τ_{1})) + 2 e_{i n}^{T} (t) (C \otimes I_{n}) G (e_{i n} (t)) \\ + 2 e_{i n}^{T} (t) (D \otimes I_{n}) H (e_{i n} (τ_{2})) - 2 ξ e_{i n}^{T} (t) e_{i n} (t) - α e_{i n}^{T} (t) s i g n (e_{i n} (t)) | e_{i n} (t) |^{σ} \\ - α {(\frac{η_{1}}{1 - γ} \int_{t - τ_{1} (t)}^{t} e_{i n}^{T} (s) e_{i n} (s) d s)}^{\frac{1 + σ}{2}} - α (\frac{η_{2}}{1 - γ} \int_{t - τ_{2} (t)}^{t} e_{i n}^{T} (s) e_{i n} (s) d s)^{\frac{1 + σ}{2}} \\ - λ {(\frac{η_{3}}{1 - γ} \int_{t - τ_{1} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s)}^{\frac{μ}{2}} - λ (\frac{η_{4}}{1 - γ} \int_{t - τ_{2} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s)^{\frac{μ}{2}} . \end{array}

Furthermore, by Lemma1, one has

\begin{array}{l} 2 e_{i n}^{T} (t) (I_{N} \otimes \tilde{A}) e_{i n} (t) + 2 e_{i n}^{T} (t) (I_{N} \otimes \tilde{B}) F (e_{i n} (τ_{1})) \\ \leq 2 e_{i n}^{T} (t) (I_{N} \otimes A) e_{i n} (t) + s_{1} e_{i n}^{T} (t) (I_{N} \otimes M M^{T}) e_{i n}^{T} (t) \\ + s_{1}^{- 1} e_{i n}^{T} (t) (I_{N} \otimes N_{1}^{T} N_{1}) e_{i n}^{T} (t) + s_{2} e_{i n}^{T} (t) (I_{N} \otimes B B^{T}) e_{i n}^{T} (t) + \\ s_{2}^{- 1} F^{T} (e_{o u t} (τ_{1})) F (e_{o u t} (τ_{1})) + s_{3} e_{i n}^{T} (t) (I_{N} \otimes M M^{T}) e_{i n}^{T} (t) \\ + s_{3}^{- 1} F^{T} (e_{o u t} (τ_{1})) (I_{N} \otimes N_{2}^{T} N_{2}) F (e_{o u t} (τ_{1})) \\ \leq (2 λ_{m a x} (A^{s}) + s_{1} λ_{m a x} (M M^{T}) + s_{1}^{- 1} λ_{m a x} (N_{1}^{T} N_{1}) + s_{2} λ_{m a x} (B B^{T}) \\ + s_{3} λ_{m a x} (M M^{T})) e_{i n}^{T} (t) e_{i n} (t) + (s_{2}^{- 1} L_{j}^{2} + s_{3}^{- 1} L_{j}^{2} λ_{m a x} (N_{2}^{T} N_{2})) e_{o u t}^{T} (τ_{1}) e_{o u t} (τ_{1}) . \end{array}

Similarly, we have the estimation for another term,

\begin{array}{l} 2 e_{i n}^{T} (t) (C \otimes I_{n}) G (e (t)) + 2 e_{i n}^{T} (t) (D \otimes I_{n}) H (e_{i n}^{T} (τ_{2})) \\ \leq s_{4} e_{i n}^{T} (t) (C C^{T} \otimes I_{n}) e_{i n}^{T} (t) + s_{4}^{- 1} G^{T} (_{e}^{i n} (t)) G (e_{i n} (t)) s_{5} e_{i n}^{T} (t) (D D^{T} \otimes I_{n}) e_{i n} (t) \\ + s_{5}^{- 1} H^{T} (e_{o u t} (τ_{2})) H (e_{o u t} (τ_{2})) \\ \leq (s_{4} λ_{m a x} (C C^{T}) + s_{4}^{- 1} L_{r}^{2} + s_{5} λ_{m a x} (D D^{T})) e_{i n}^{T} (t) e_{i n} (t) + s_{5}^{- 1} L_{h}^{2} e_{o u t}^{T} (τ_{2}) e_{o u t} (τ_{2}) + s_{3}^{- 1} L_{r}^{2} + s_{5}^{- 1} L_{r}^{2} . \end{array}

Base on Definition 2 and Lemma 2 that, one can get

- α^{e^{T}} (t) s i g n (e (t)) | e (t) |^{σ} = - α | e^{T} (t) | | e (t) |^{σ} \leq - α {(e^{T} (t) e (t))}^{\frac{1 + σ}{2}} .

Together with (6) and (7), one has

\begin{array}{l} {\dot{V}}_{i n} (t) \leq Ψ e_{i n}^{T} (t) e_{i n} (t) + (s_{5}^{- 1} L_{h}^{2} - η_{2}) e_{i n}^{T} (τ_{2}) e_{i n} (τ_{2}) + (s_{2}^{- 1} L_{j}^{2} + s_{3}^{- 1} L_{j}^{2} λ_{m a x} (N_{2}^{T} N_{2}) - η_{1}) e_{i n}^{T} (τ_{1}) e_{i n} (τ_{1}) \\ - α {(e_{i n}^{T} (t) e_{i n} (t))}^{\frac{1 + σ}{2}} - α {(\frac{η_{1}}{1 - γ} \int_{t - τ_{1} (t)}^{t} e_{i n}^{T} (s) e_{i n} (s) d s)}^{\frac{1 + σ}{2}} - α {(\frac{η_{2}}{1 - γ} \int_{t - τ_{2} (t)}^{t} e_{i n}^{T} (s) e_{i n} (s) d s)}^{\frac{1 + σ}{2}}, \\ {\dot{V}}_{o u t} (t) \leq (s_{5}^{- 1} - η_{3}) e_{o u t}^{T} (τ_{2}) e_{o u t} (τ_{2}) + (s_{3}^{- 1} L_{j}^{2} λ_{m a x} (N_{2}^{T} N_{2}) - η_{3}) e_{o u t}^{T} (τ_{1}) e_{o u t} (τ_{1}) \\ - λ {(e_{o u t}^{T} (t) e_{o u t} (t))}^{\frac{μ}{2}} - λ {(\frac{η_{3}}{1 - φ} \int_{t - τ_{1} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s)}^{\frac{μ}{2}} - λ {(\frac{η_{4}}{1 - φ} \int_{t - τ_{2} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s)}^{\frac{μ}{2}} . \end{array}

From Lemma 2, one has

\begin{array}{l} {\dot{V}}_{i n} (t) \leq - α {(e_{i n}^{T} (t) e_{i n} (t))}^{\frac{1 + σ}{2}} - α {(\frac{η_{1}}{1 - γ} \int_{t - τ_{1} (t)}^{t} e_{i n}^{T} (s) e_{i n} (s) d s)}^{\frac{1 + σ}{2}} {(\frac{η_{1}}{1 - γ} \int_{t - τ_{1} (t)}^{t} e_{i n}^{T} (s) e_{i n} (s) d s)}^{\frac{1 + σ}{2}} \\ - α {(\frac{η_{2}}{1 - γ} \int_{t - τ_{2} (t)}^{t} e_{i n}^{T} (s) e_{i n} (s) d s)}^{\frac{1 + σ}{2}} \leq - α V_{i n}^{β} (t) . \\ {\dot{V}}_{o u t} (t) \leq (s_{5}^{- 1} - η_{3}) e_{o u t}^{T} (τ_{2}) e_{o u t} (τ_{2}) + (s_{3}^{- 1} L_{j}^{2} λ_{m a x} (N_{2}^{T} N_{2}) - η_{3}) e_{o u t}^{T} (τ_{1}) e_{o u t} (τ_{1}) \\ - λ (e_{o u t}^{T} (t) e_{o u t} (t))^{\frac{μ}{2}} - λ {(\frac{η_{3}}{1 - φ} \int_{t - τ_{1} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s)}^{\frac{μ}{2}} - λ {(\frac{η_{4}}{1 - φ} \int_{t - τ_{2} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s)}^{\frac{μ}{2}} \\ \leq - λ {(e_{o u t}^{T} (t) e_{o u t} (t))}^{\frac{μ}{2}} - λ {(\frac{η_{3}}{1 - φ} \int_{t - τ_{1} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s)}^{\frac{μ}{2}} - λ {(\frac{η_{4}}{1 - φ} \int_{t - τ_{2} (t)}^{t} e_{o u t}^{T} (s) e_{o u t} (s) d s)}^{\frac{μ}{2}} \\ \leq - λ V_{o u t}^{μ} (t) . \end{array}

Integrating (8) from

t_{0}

t

, one has

V (t) \leq {({(k V (t_{0}))}^{1 - β} - α (1 - β) (t - t_{0}))}^{\frac{1}{1 - β}},

where

0 < k < 1, V (t_{0}) > V_{i n 1} (t_{0}), V (t_{0}) > V_{o u t 1} (t_{0}) .

The controller

u_{i n} (t) = u_{1} (t) and u_{o u t} (t) = u_{2} (t)

are activated at

t_{0}

4. Numerical Example

In this section, numerical examples are provided to demonstrate the effectiveness of the theoretical results. Consider the duplex network is composed of five nodes. The function

f (\cdot)

in networks (1)-(2) is chosen as

f (z (t)) = {(|z_{1} (t) + 1| - |z_{1} (t) - 2|, 1, 0, 0, 0)}^{T}

, the nonlinear inner connecting functions and the time-varying delays are taken as

g (\cdot) = h (\cdot) = \tanh (\cdot)

and

τ_{1} (t) = τ_{2} (t) = \sin t

, respectively.

L_{j} = 1.5, L_{h} = L_{r} = 2, γ = 0.3, φ = 0.1

. Some parameters are selected as follows,

\begin{array}{l} \tilde{A} = [\begin{matrix} - 0.1 & - 0.3 & 0 & 0 \\ 0.2 & - 0.2 & 0.1 & 0 \\ 0 & 0.1 & 0 & 0 \\ 0.1 & 0.1 & 0 & 0 \end{matrix}], \tilde{B} = [\begin{matrix} - 0.1 & - 0.3 & 0 & 0 \\ 0.2 & - 0.2 & 0.1 & 0 \\ 0 & 0.1 & 0 & 0 \\ 0.1 & 0.1 & 0 & 0 \end{matrix}], \\ C^{1} = [\begin{matrix} - 0.1 & 0 & 0.1 & 0 \\ 0 & - 0.1 & 0.2 & 0 \\ 0 & 0 & - 0.1 & 0.1 \\ 0 & 0.1 & 0 & - 0.1 \end{matrix}], C^{2} = [\begin{matrix} - 0.3 & 0.1 & 0 & 0.1 \\ 0.1 & 0 & 0.2 & 0.1 \\ 0.2 & 0 & - 0.2 & 0 \\ 0 & 0.3 & 0 & - 0.3 \end{matrix}], \end{array}

D^{1} = [\begin{matrix} - 0.2 & 0.1 & 0 & 0.1 \\ 0.1 & - 0.2 & 0 & 0 \\ 0 & 0 & - 0.2 & 0.1 \\ 0 & 0.1 & 0 & - 0.1 \end{matrix}], D^{2} = [\begin{matrix} - 0.5 & 0 & 0.3 & 0 \\ 0.4 & - 0.2 & 0 & 0 \\ 0.1 & 0 & - 0.1 & 0.2 \\ 0.2 & 0.1 & 0 & - 0.5 \end{matrix}] .

M = N_{1} = N_{2} = I, F (t) = d i a g (0.2, \sin t, \cos t, 0.1), s_{i} = 2 (i = 1, \dots, 1), ξ = 10, η_{1} = 2, η_{2} = 1, σ = 0.4,

α = 3, α_{1} = 2, α_{2} = 1, k = 0.4, β = 0.3, η_{3} = 0 . 2, η_{4} = 0 . 3, λ = 0 . 1, χ = 0 . 6 .

The initial conditions

x (t_{0}) = {(0.04 i, 0.03 i, 0.02 i, 0.01 i)}^{T}

y (t_{0}) = {(- 0.4 - 0.1 i, - 0.3 - 0.2 i, - 0.2 i, - 0.1 i)}^{T}

(i = 1, \dots, 4) .

It is shown in Figure 1 display the synchronization error

e (t)

diverges when the network is in the absence of control. Figure 2 display the synchronization error

e (t)

tend to zero in finite time.

5. Conclusions

This paper investigated finite time hybrid synchronization of duplex complex networks via intermittent control. Compared with similar work, the internal synchronization and the external all be considered to achieve synchronization within a finite time, which determined by the designed Lyapunov function. A criterion has been proposed to guarantee the synchronization within a finite time T. Numerical simulations show that the control scheme in this brief is of good performance.

Acknowledgments

This research is supported by the 2020 Jilin Province Science and Technology Development Plan project “20200404223YY”, Jilin Province science and technology development plan project “YDZJ202201ZYTS605”, Jilin Province Science and Technology Development Plan Project (No.:20220101137JC).

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Figure 1. synchronization without control.

Figure 2. synchronization with control.

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Finite Time Hybrid Synchronization Control for Duplex Complex Networks via Intermittent Control

Abstract

1. Introduction

2. Network Model and Preliminaries

3. Main Results

4. Numerical Example

5. Conclusions

Acknowledgments

References

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