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Analysis and discontinuous control for finite-time synchronization of delayed complex dynamical networks
Chaos, Solitons and Fractals 114 (2018) 291–305
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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
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Analysis and discontinuous control for finite-time synchronization of delayed complex dynamical networksR Jiarong Li, Haijun Jiang∗, Cheng Hu, Juan Yu College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 6 April 2018 Revised 17 July 2018 Accepted 18 July 2018
Keywords: Complex dynamical networks Discontinuous control Finite-time synchronization Fixed-time synchronization
a b s t r a c t This paper addresses the finite-time and fixed-time synchronization issue of delayed complex dynamical networks (CDNs). Firstly, as an important preliminary, an improved and generalized finite-time stability theory is established for delayed nonlinear systems to prove the finite-time synchronization mainly through the reduction to absurdity. Different from some existing results, a more detailed discussion of the setting time function for finite-time synchronization is given. Besides, a novel feedback controller is firstly proposed to unify finite-time and fixed-time synchronization just by adjusting the key control parameter. Furthermore, several new criteria are derived to ensure the finite-time and fixed-time synchronization based on inequality analysis method and constructing appropriate Lyapunov functional. Finally, some numerical simulations are presented to support the theoretical results. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Complex dynamical networks (CDNs), which present a high degree of complexity, have attracted wide attention because they are widespread in real world, such as the World Wide Web, power grid, patent use network, gene network, transportation network, metabolic pathways and so on. Complex network, composed of a large number of nodes and links, is a nonlinear dynamical system. Where the nodes denote the individuals in the network and the edges represent the connections among them. Over the past decades, complex networks with different structures have been studied and some results also have been yielded [1–3]. Although the results obtained in [1–3] are effective and convenient, none of these articles considered the effect of time delay on the system. Time delay is often the main cause of system instability and poor performance. In realistic CDNs, time delay is ubiquitous, such as the flow of steam and fluid in pipes, the transmission of electrical signals over long lines. Especially in the control system [4]. Nowadays, inspired by the effect of time delays, the CDNs with time delays have attracted increasingly attention [5–8].
R This work was supported in part by the Excellent Doctor Innovation Program of Xinjiang Uyghur Autonomous Region (XJGRI2017001), in part by the Excellent Doctor Innovation Program of Xinjiang University (XJUBSCX-2017003) and the National Natural Science Foundation of People’s Republic of China (Grants No. 61473244, No. 61563048, No. 11402223, No. U1703262). ∗ Corresponding author. E-mail address: [email protected] (H. Jiang).
https://doi.org/10.1016/j.chaos.2018.07.019 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
One of the hot topics in the field of CDNs is synchronization due to its practical application in human heartbeat regulation, secure communication, information science and image processing and so on [9–13]. However, it is worthy of noting that most of existing results including the articles mentioned above are actually asymptotic synchronization. These types of synchronization can be implemented only when the time is near infinity. In practical application, however, the synchronization aims are usually expected to be realized within a finite time. For this purpose, finite-time synchronization, which means that the time it takes for the system to be synchronized can be estimated, has been proposed and extensively studied in recent years [14–23]. In [16,17], the problem of finite-time synchronization of a class of CDNs with time-varying delays and coupling time-varying delays was studied. Although the synchronization results obtained in above articles has the optimal convergence time, the setting time function has to depend on the initial conditions. In practical systems, however, the initial conditions of the system are difficult to obtain in advance, which makes it difficult to estimate the setting time. In order to solve the problem that the finite-time synchronization is heavily dependent on the initial conditions, the concept of fixed-time stability was proposed by Polyakov [18]. Fixed-time stability is proposed based on the finite-time stability by demanding the boundness of the settling time function of the system to achieve synchronization. The fixed-time stability not only has better robustness and disturbance rejection properties, but also ensures that the settling time independent of the initial conditions of the system. These merits can improve the efficiency and quality
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of engineering management greatly. Research on fixed-time synchronization, however, is just beginning to germinate and the related theories are still scarce. There are fewer articles on fixed-time synchronization in neural networks, let alone complex dynamical networks. In [19], the authors provided a general approach to show that the essence of finite-time stability and fixed-time convergence, and some conditions were acquired to ensure that the setting time function was bounded. In recent work [20], for the synchronous convergence time of the system, the author gave a more accurate estimate. Furthermore, investigations of fixed-time synchronization issues of nonlinear dynamical networks have been presented in [24–27]. As we known, many papers have considered the finite-time and fixed-time stability and consensus problems, but there are few works on the fixed-time synchronization. Therefore, it is interesting and important to investigate the fixed-time synchronization of CDNs with coupled delays. Another hot topic in the research of CDNs is control. We all know that most system synchronization can only be achieved with a suitable controller. At present, synchronization control has been investigated widely [1,3,12,21–31]. In [21,28], based on pinning impulsive control and pinning control, the problem of the system’s finite time synchronization was investigated. In [22,23,29], Some sufficient conditions were received to realize the finite-time or fixed-time synchronization of chaotic systems under the sliding mode control protocol. In [30,31], the authors considered the finite-time synchronization of CDNs through designing periodically intermittent control (PIC). Different from the controllers in [28,29], the output of the system is intermittent rather than continuous, based on this principle, the design of the PIC is divided into the control interval and the rest interval, which makes the controller more economical and effective, and can also avoid prolonged work on the controller cause some damages. To the best of our knowledge, however, delayed CDNs are relatively unexplored in finite-time synchronization with periodic intermittent control. From this, an investigation of finite-time synchronization under PIC is important in both theoretical analysis and real applications. From [20,24,32,33] we can find that fixed-time synchronization is a special case of finite-time synchronization. discontinuous feedback controllers designed in [34] were simple and efficient, easy to implement. The results obtained can only be achieved finite-time synchronization. To the best of my knowledge, no author has yet thought of implementing both types of synchronization by designing a uniform form of controller. Based on the above discussions, this paper will discuss the finite-time and fixed-time synchronization of CDNs with timevarying delays. The main contributions of this paper in comparison to the existing ones can be reflected as follows: (1) A novel theory of improvement on accurate analysis of setting time function of the finite-time stable delayed CDNs is developed in this paper compared with [30,31]. (2) A more general intermittent control scheme, which can be employed to achieve synchronization within finite time for time delay systems, is designed compared with the works [15,28,34–36]. (3) An extended unified feedback controller is proposed, which can realize both finite-time and fixed-time synchronization. Until now, few articles consider these two synchronism simultaneously. Therefore, these two synchronism should be investigated, which can help us obtain more general results. (4) By regulating the main control parameters in the controller, not only both the finite-time and fixed-time synchronization can be achieved, but also the convergence rate of synchronization can be adjusted. It is of engineering interest to develop the efficiency of the controller by adjusting the appropriate parameters. The rest of this paper is organized as follows. Some necessary preliminaries and model description are given in Section 2. In Section 3 the finite-time synchronization conditions are presented
for delayed CDNs via periodic intermittent control, and finite-time and fixed-time synchronization are analyzed based on a unified control framework. simulations are given in Section 4 to verify the effectiveness of the obtained results. In Section 5, conclusion is summarized. Notations. Let Rn be the space of n-dimensional real column vectors. Denote x = ( x1 , · · · , xN )T ∈ RN , μ μ μ μ T |ei (t )| = (|ei1 (t )| , |ei2 (t )| , · · · , |ein (t )| ) and sign(ei (t )) = (sign(ei1 (t )), · · · , sign(ein (t )))T ∈ Rn . ||x|| denotes the vector norm 1 defined by ||x|| = ( ni=1 x2i ) 2 . λmax (P)(λmin (P)) is defined as the maximum (minimum) eigenvalue of the positive definite diagonal matrix P. IN is the identity matrix with N-dimensions. 2. Model description and preliminaries Let us consider a general delayed complex network consisting of N dynamical nodes, which is described by
x˙ i (t ) = f (xi (t )) + c
N
ai j x j (t − τ (t )),
i = 1, 2, · · · , N
(1)
j=1
where xi (t ) = (xi1 (t ), · · · , xin (t ))T ∈ Rn is the state variable. f: Rn → Rn is a continuous nonlinear vector function. The nonnegative constant c is the coupling strength. = diag(γ1 , γ2 , · · · , γn ) is a positive definite diagonal matrix, which denotes the inner-coupling matrix between each pair of nodes. A = (ai j )N×N is the coupling configuration matrix. If there is a connection from the node i to the node j(j = i), then aij > 0. Otherwise, ai j = 0( j = i ) and the di agonal elements of matrix A is defined as aii = − Nj=1, j=i ai j . The coupling time-varying delay τ (t) is a bounded and continuously differentiable function, i.e., there exists a positive constant τ satisfying 0 ≤ τ˙ (t ) ≤ τ < 1. Remark 1. In [30,31,37], authors introduced a finite-time intermittent control method to realize the synchronization of CDNs. The influence of time-varying delays on the system is not considered. However, in this paper, we concern the finite-time synchronization problem for CDNs with time-varying delays. The models considered in the paper are more general and more reasonable. Without loss of generality, we refer to system (1) as the drive system, and consider a response system described as follows:
y˙ i (t ) = f (yi (t )) + c
N
ai j y j (t − τ (t )) + Ui (t ), i = 1, 2, · · · , N,
j=1
(2) where yi (t ) = (yi1 (t ), · · · , yin (t ))T ∈ Rn is the state variable. U (t ) = (U1 (t ), · · · , UN (t ))T is the control input to be designed later. The other parameters are the same as in system (1). Assumption 1 (QUAD). Assume that there exists a positive definite diagonal matrix P = diag( p1 , p2 , · · · , pn ) and a diagonal matrix H = diag(h1 , h2 , · · · , hn ), such that f( · ) satisfies the following inequality:
(u − v )T P ( f (u ) − f (v ) − H (u − v )) ≤ −ξ (u − v )T (u − v ), for all u, v ∈ Rn , ξ > 0. Remark 2. The vector field f is required to satisfy the Lipschitz condition in many CDNs [1,2,38–40]. There are also many articles require vector field f which satisfies the QUAD condition [29,30]. Lipschitz condition and QUAD condition, which usually made in the literature to prove network synchronization, are assumptions on the vector field f. Actually, if f is globally Lipschitz with a Lipschitz constant l > 0, then f is QUAD(H, ξ ), with P H − ξ IN ≥ l ||P ||IN
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[41]. Assumption 1, therefore, can ensure the uniqueness and existence of the solutions for CDNs, also helps to analyze the dynamical behaviors of such networks. Most chaotic systems, including Ro¨ssler’s systems, Chua’s circuit and Lorenz system etc., satisfy QUAD condition. To obtain the main results of this paper, we will design the following periodic intermittent controller, and introduce some Definitions and Lemmas. Firstly, let ei (t ) = yi (t ) − xi (t ), i = 1, 2, · · · , N be the synchronization errors, and design the following periodic intermittent control scheme:
Ui (t ) =
⎧
1+2μ ⎪ N t ⎪ ⎪ λ ⎪ −β ei (t ) − α eTi (s )Pei (s )ds ⎪ ⎪ 1−τ ⎪ t −τ (t ) ⎪ i =1 ⎪ ⎨ −1 ei (t ) P
||e(t )||2
1+μ ⎪ (λmax (P )) 2 ⎪ ⎪ −α ⎪ ⎪ λmin (P ) ⎪ ⎪ μ ⎪ ⎪ ⎩sign(ei (t ))|ei (t )| ,
0,
lT ≤ t ≤ lT + θ T , lT + θ T < t < (l + 1 )T ,
1 −ω xω i ≥n
i=1
n
a+b υ a + b υ1 a + b υ2 ) = min{( ) ,( ) }, 2 2 2 max{υ1 , υ2 }, min{υ1 , υ2 },
a+b 2
0≤
a+b 2
a+b 2
≤ 1, > 1.
υ .
Lemma 3. Suppose that function V(t) is non-negative when t ∈ [−τ , +∞ ) and satisfies the following conditions:
D+V (t ) ≤ −αV η (t ), D+V (t ) ≤ β V (t ),
lT ≤ t ≤ lT + θ T , lT + θ T ≤ t < (l + 1 )T ,
( )
0 ≤ t ≤ t ∗.
t ∗ = T1 =
1
β (1 − θ )(1 − η )
,
xqi
ω
≥ n 1 −ω
a+b 2
,
αθ ≥ 1, where T2 is the smallest soluβ (1 − θ )( sup V (s ))1−μ −τ ≤s≤0
−τ ≤s≤0
Proof. Take M0 = ( sup V (s ))1−η and W (t ) = V 1−η (t ) + hα (1 − −τ ≤s≤0
n
ω2 x2i
1 h
η )t, where 0 < h < 1, t ≥ −τ . Let Q (t ) = W (t ) − M0 . It’s clear that Q (t ) < 0,
for all t ∈ [−τ , 0].
(A1)
In the following, we will prove
for all t ∈ [0, θ T ).
(A2)
By using reduction to absurdity, assume that there exists a t0 ∈ [0, θ T) such that
Q (t0 ) = 0, .
i=1
υ
as ln
Q (t ) < 0, ,
xi
t ∗ = T2 ,
( sup V (s ))1−η exp{(1 − η )β (1 − θ )t } − αθ (1 − η )t = 0.
Lemma 2. If a ≥ 0, b ≥ 0, 0 < υ1 < 1, 0 < υ2 < 1, then the following inequality holds.
aυ1 + bυ2 ≥
(
tion of
1q
i=1
Let
or
i=1
≤
The constant t∗ is the setting time given by
Lemma 1. ([42]) Let xi (t) ≥ 0 for i = 1, 2, · · · , n, 0 < p < q, ω > 1, then the following inequalities are true
n
a+b 2
−τ ≤s≤0
is called the setting time. If there exist a fixed-time Tmax ≥ 0 such that T(e(0)) ≤ Tmax for any e(0) ∈ RnN , systems (1) and (2) are said to be fixed-time synchronized.
i=1
b υ1 ≤ a, then ( a+ ≤ aυ1 . When a ≤ b, 2 )
≤ bυ2 . Hence,
−αθ (1 − η )t,
T (e(0 )) = inf{T˜ (e(0 )) ≥ 0 : ||e(t )|| = 0, ∀t ≥ T˜ (e(0 ))}
n
a+b 2
≤ 1, > 1.
V 1−η (t ) ≤ ( sup V (s ))1−η exp{(1 − η )β (1 − θ )t }
−τ ≤s≤0
≥
a+b 2
a+b 2
where α , β > 0, T > 0, 0 < η < 1, 0 < θ < 1, then the following inequality holds:
||ei (t )|| = 0, ||ei (t )|| ≡ 0 for any t ≥ T (e(0 )),
xip
0≤
aυ1 + bυ2 ≥ min{( a+2 b )υ1 , ( a+2 b )υ2 }.
aυ1 + bυ2 ≥
where e(0 ) = sup e(s ) and
n
b, then
( a+2 b )υ2
then,
Definition 1. ([20]) Systems (1) and (2) are said to be finite-time synchronized, if there exists a function T(e(0)) ≥ 0, such that
max{υ1 , υ2 }, min{υ1 , υ2 },
Proof. When a ≥ b,
(3)
for i = 1, 2, · · · , N.
1p
υ=
υ=
(4)
⎧ N ⎪ ⎪ ⎪ f ( y ( t )) − f ( x ( t )) + c ai j e j (t − τ (t )) + Ui (t ), i ⎪ i ⎪ ⎪ j=1 ⎪ ⎨ lT ≤ t ≤ lt + θ T , e˙ i (t ) = N ⎪ ⎪ ⎪ f (yi (t )) − f (xi (t )) + c ai j e j (t − τ (t )), ⎪ ⎪ ⎪ j=1 ⎪ ⎩ lT + θ T < t < (l + 1 )T ,
t→T (e(0 ))
where
obviously,
where ei (t) = 0 as lT ≤ t ≤ lT + θ T , i = 1, 2, · · · , N. α , β , λ are positive constants called control gain. the real number μ satisfies 0 < μ < 1. T > 0 is the control period. 0 < θ < 1 is called the control rate. P is defined in Assumption 1. Then based on the intermittent controller (3), from systems (1) and (2), the error system can be derived as
lim
293
Q (t ) < 0,
D+ Q (t0 ) ≥ 0, 0 ≤ t < t0 .
(A3) (A4)
Using ( ) and (A1), one has
D+ Q (t0 ) ≤ −α (1 − η ) + α h(1 − η ) < 0. This contradicts the second inequality in (A3). So, (A2) holds.
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Now, we prove that for t ∈ [θ T, T)
By ( ) and (A12), one has
H (t ) = W (t ) exp{−(1 − η )β (t − θ T )} − 1h M0 −hα (1 − η )(t − θ T ) exp{−(1 − η )β (t − θ T )} < 0,
(A5)
Otherwise, there exists a t1 ∈ [θ T, T) such that
H (t1 ) = 0, H (t ) < 0,
D+ H (t1 ) ≥ 0,
(A6)
D+ H1 (t3 )≤−β (1−η )hα (1−η )2θ T exp{−(1−η )β (t3 −2θ T )} < 0, which contradicts the second inequality in Eq. (A12). So, (A11) holds. On the one hand, for t ∈ [T , T + θ T )
W (t ) <
θ T ≤ t < t1 .
(A7)
Using Eqs. ( ) and (A6), one can obtain
D+ H (t1 ) ≤ −hα (1 − η )θ T (1 − η )β exp{−(1 − η )β (t1 − θ T )} < 0. This is in contradiction with the second inequality in (A6). Therefore, (A5) holds. On the one hand, for t ∈ [θ T, T),
1 M0 exp{(1 − η )β (t − θ T )} + hα (1 − η )(t − θ T ) h 1 ≤ M0 exp{(1 − η )β (1 − θ )T } + hα (1 − η )(1 − θ )T . h
On the other hand, for t ∈ [T + θ T , 2T ),
W (t ) <
W (t ) <
W (t ) <
1 M0 h 1 < M0 exp{(1 − η )β (1 − θ )T } + hα (1 − η )(1 − θ )T . h
W (t ) <
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )T } h −hα (1 − η )(1 − θ )T < 0.
Q1 (t ) = W (t ) −
(A8) <
Otherwise, there exists a t2 ∈ [T , T + θ T ), such that
D+ Q1 (t2 ) ≥ 0,
(A9) ≤ (A10)
From ( ) and (A9)
W (t ) <
H1 (t ) = W (t ) exp{−(1 − η )β (1 − θ )T 1 −(1 − η )β (t − T − θ T )} − M0 h −h α (1 − η )(1 − θ )T + α (1 − η )(t − θ T − T )
W (t ) <
(A11)
Otherwise, there exists a t3 ∈ [T + θ T , 2T ), such that
T ≤ t < t3 .
1 M0 exp{(1 − η )β (1 − θ )kT } + hα (1 − η )(1 − θ )kT . h
1 M0 exp{(1 − η )β (1 − θ )kT } + hα (1 − η )(1 − θ )kT , h (A16)
1 M0 exp{(1 − η )β (1 − θ )kT } + hα (1 − η )(1 − θ )kT , h
For (k + θ )T ≤ t < (k + 1 )T ,
× exp{−(1 − η )β (1 − θ )T − (1 − η )β (t − T − θ T )}
H1 (t ) < 0,
1 M0 exp{(1 − η )β (l + 1 )(1 − θ )T } h + hα (1 − η )(l + 1 )(1 − θ )T
For kT ≤ t < (k + θ )T ,
this contradicts the second inequality of (A9). So, (A8) holds. Then, we prove that for t ∈ [T + θ T , 2T )
D+ H1 (t3 ) ≥ 0,
1 M0 exp{(1 − η )β (t − (l + 1 )θ T )} h + hα (1 − η )[t − (l + 1 )θ T ]
Then, together with (A1), for any t ∈ [−τ , kT ),
W (t ) <
D+ Q1 (t2 ) = (1 − η )V −η (t2 )D+V (t2 ) + hα (1 − η ) ≤ −α (1 − η ) + hα (1 − η ) < 0,
H1 (t3 ) = 0,
Suppose that (A14) and (A15) are true for m ≤ k − 1, where k is a positive integer. Then, for any integer 0 ≤ l ≤ k − 1, if lT ≤ t < (l + θ )T ,
and if (l + θ )T ≤ t < (l + 1 )T ,
The same we can prove that for t ∈ [T , T + θ T )
< 0.
(A15)
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )T } h +hα (1 − η )(1 − θ )T , for t ∈ [−τ , T ).
T ≤ t < t2 .
1 M0 exp{(1 − η )β (t − (m + 1 )θ T )} h +hα (1 − η )(t − (m + 1 )θ T ).
1 M0 exp{(1 − η )β (1 − θ )lT } + hα (1 − η )(1 − θ )lT h 1 < M0 exp{(1 − η )β (1 − θ )kT } + hα (1 − η )(1 − θ )kT , h
So, we have
Q1 (t ) < 0,
1 M0 exp{(1 − η )β (1 − θ )mT } + hα (1 − η )(1 − θ )mT , h (A14)
For (m + θ )T ≤ t < (m + 1 )T ,
On the other hand, for t ∈ [−τ , θ T )
Q1 (t2 ) = 0,
1 M0 exp{(1 − η )β (t − 2θ T )} + hα (1 − η )(t − 2θ T ). h
In the following, we can get the estimation of W(t) for any integer m. For mT ≤ t < (m + θ )T ,
W (t ) <
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )T } + hα (1 − η )(1 − θ )T , h
(A12) (A13)
1 M0 exp{(1 − η )β [t − (k + 1 )θ T ]} h +hα (1 − η )[t − (k + 1 )θ T ].
Hence, by induction, the inequalities (A14) and (A15) can be obtained for any positive integer n. t For nT ≤ t < (n + θ )T , one has n < , then T
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )t } + hα (1 − η )(1 − θ )t. h
For (n + θ )T ≤ t < (n + 1 )T , one has n + 1 >
t , then T
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )t } + hα (1 − η )(1 − θ )t. h
Let h → 1, one can obtain V 1−η (t ) < M0 exp{(1 − η )β (1 − θ )t } − α (1 − η )θ t = ( sup V (s ))1−η exp{(1 − η )β (1 − θ )t } − α (1 − η )θ t, −τ ≤s≤0
t ≥ 0.
Next, Let
( sup V (s ))1−η exp{(1 − η )β (1 − θ )t } − α (1 − η )θ t = 0. (A17) −τ ≤s≤0
Then discuss the value of t, we can easy to check that l1 = α ( 1 − η )θ t and l2 = exp{(1 − η )β (1 − θ )t } have the same ( sup V (s ))1−η −τ ≤s≤0
Remark 3. The proof method of Lemma 3 is different from the proof technique in the previous works. The Lemma 3, which improves and generalizes the previous conclusions [30,31], plays an important role in the finite-time synchronization analysis of dynamical networks via intermittent control in this paper, because it shows application of finite-time intermittent control. Lemma 4. ([20]). If there exists a regular positive definite and radially unbounded function V: Rn → R such that any solution x(t) of (1) satisfies the inequality
V˙ (x(t )) ≤ −aV δ (x(t )) − bV θ (x(t )),
1
β (1 − θ )(1 − η )
ln
αθ
β (1 − θ )( sup V (s ))
. 1 −η
−τ ≤s≤0
The function values of l1 and l2 at this point are
yl1 =
αθ αθ ln , β (1 − θ )( sup V (s ))1−η β (1 − θ )( sup V (s ))1−η −τ ≤s≤0
−τ ≤s≤0
and
yl2 =
αθ
β (1 − θ )( sup V (s ))1−η
.
−τ ≤s≤0
(1) If yl1 < yl2 , there is no solution of Eq. (A17). For the case of no solution, we don’t discuss it in detail. (2)
If
yl1 ≥ yl2 ,
i.e.,
ln
αθ ≥ 1, β (1 − θ )( sup V (s ))1−η −τ ≤s≤0
Eq. (A17) having solutions
t ∗ = T1 =
1
β (1 − θ )(1 − η )
as ln
αθ = 1, β (1 − θ )( sup V (s ))1−η −τ ≤s≤0
Tmax ≤
αθ
> 1, β (1 − θ )( sup V (s ))1−η −τ ≤s≤0
where T2 is the smallest solution of Eq. (A17). 1 . When t ∗ = T1 , yl1 and yl2 have only one intersection point, one can easy to see that
V (t ) → 0 as t → T1 and V (t ) ≡ 0 as t > T1 .
2 . When t ∗ = T2 , from the first inequality of ( ), V(t) is monotone decreasing for t ∈ [l T , l T + θ T ]. V (t ) → 0 as t ∈ [lT , T2 ] and t → T2 , V (t ) ≡ 0 as t ∈ (T2 , lT + θ T ]. From the second inequality of ( ) and the continuity of V(t), one has
V (t ) → 0 as t ∈ [lT , T2 ] and t → T2 , V (t ) ≡ 0 as t ∈ (T2 , (l + 1 )T ). and for all t ∈ [(l + 1 )T , (l + 1 )T + θ T ], V(t) ≡ 0. To summarize, for all t ∈ [0, +∞ ), when t → t∗ , V(t) → 0 and when t > t∗ , V(t) ≡ 0. The proof is completed.
1
δ−1
).
Lemma 5. ([43]). Assume that V: Rn → R is a regular, positive definite function, x(t ) : [0, +∞ ) → Rn is absolutely continuous . If there exists a continuous function χ : (0, +∞ ) → R, with χ (ι) > 0 for ι ∈ V (0 ) 1 (0, +∞ ), such that V˙ (x(t )) ≤ −χ (V (x(t ))) and ( )d ι = T , 0
χ (ι )
T < +∞, then we have V (x(t )) = 0 for all t ≤ T. In particular, (1) If χ (ι ) = M1 ι + M2 ιμ for all ι ∈ (0, +∞ ), where 0 < μ < 1, M1 , M2 are positive constants. The settling time can be estimated by
T ≤
1 M 1 V 1 −μ ( 0 ) + M 2 ln . M1 ( 1 − μ ) M2
(2) If χ (ι ) = M2 ιμ for all ι ∈ (0, +∞ ), where 0 < μ < 1, M2 > 0, then the settling time can be estimated by
T ≤
V 1 −μ ( 0 ) . M2 ( 1 − μ )
3. Main result 3.1. Finite-time synchronization via periodically intermittent control In this section, we consider the finite-time synchronization of general CDNs via periodic intermittent control. Based on Lemma 3, some novel finite-time synchronization criteria are obtained. The main results are stated as follows. Theorem 1. Suppose that Assumption 1 holds, systems (1) and (2) are synchronized in finite-time under the periodic intermittent controller (3), if there exist constants α , β , λ and 0 < μ < 1 such that the following conditions hold:
λIN − cγ j A ≥ 0,
V (t ) ≡ 0 for all t ∈ (lT + θ T , (l + 1 )T ). (b) Assume that T2 ∈ (lT + θ T , (l + 1 )T ). From ( ) and the continuity of V(t), we can also obtain
1 b 1−θ 1 ( ) δ−θ ( + b a 1−θ
Remark 4. Many fixed-time synchronization articles ([32,33]) are mostly based on the fixed-time stability theory proposed in [18]. The Lemma 4 in this paper is proposed by Hu et al. Compared with the estimation bound of the settling time given in [18], the results obtained in [20] were more effective and more accurate, and the detailed theoretical evidence is given in [20].
or
t ∗ = T2 as ln
x(t ) ∈ Rn {0},
where a, b > 0, δ > 1, 0 ≤ θ < 1, then the origin of system (1) is fixedtime stable and the setting time function
slope at
t=
295
(5)
2(
ξ λ − hj + β − )IN − cγ j A ≥ 0, λmax (P ) 1−τ
(6)
ln
αθ ≥ 1, 1−μ β (1 − θ )( sup V (s )) 2
(7)
−τ ≤s≤0
where j = 1, 2, · · · , N. The setting time can be estimated as
t ∗ = T1 ≤
1
β (1 − μ )(1 − θ )
,
or
t ∗ ≤ T2 ,
296
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
where T2 is the smallest solution of
( sup V (s ))
1−μ 2
exp{β (1 − μ )(1 − θ )t } − α (1 − μ )θ t = 0.
−τ ≤s≤0
N
λ
eTi (t )Pei (t ) +
N
1−τ
i=1
t
t −τ (t )
i=1
= (8)
D V (t ) ≤ 2
N
+c
−2α (
(t )P f (yi (t )) − f (xi (t )) − Hei (t ) + Hei (t )
−β ei (t ) − α ( P
−1
λ
i=1
t −τ (t )
eTi (s )Pei (s )ds )
ei (t ) ||e(t )||2
(λmax (P )) −α λmin (P )
sign(ei (t ))|ei (t )|
μ
+
1+μ 2
1−τ
N
λ
1−τ
eTi (t − τ (t ))Pei (t − τ (t )).
N n
ξ
λmax (P )
Pei (s )ds )
−2α ≤
1−τ
p j e2i j (t ) + 2
i=1 j=1
N n
p j h j e2i j (t ) + 2c
i=1 j=1
t
t −τ (t )
i=1
eTi (s )Pei (s )ds ) 1+μ 2
1+μ 2
(9) sign(ei (t ))|ei (t )|μ .
ei j (t )
≤ =
β p j e2i j (t ) − 2α
i=1 j=1
N
λ
1−τ
N
eTi (t )P
i=1
N n
1−τ
p j e2i j (t ) − λ
i=1 j=1
−2
j=1
i=1
1+μ 2
−2α (λmax (P ))
1+μ 2
N n
−2α (
t
t −τ (t )
eTi (s )
1+μ 2
(λmax (P )) λmin (P )
−2α (
(
N
λ
1−τ N
i=1
(10)
|ei j (t )|2 ) 1+μ 2
1+μ 2
.
+
N
t
t −τ (t )
eTi (s )Pei (s )ds )
λmax (P )eTi (t )ei (t ))
i=1
p j e2i j (t − τ (t ))
λ
N
1−τ
t −τ (t )
i=1
eTi (t )Pei (t ))
t
1+μ 2
1+μ 2
eTi (s )Pei (s )ds
1+μ 2
i=1 1+μ 2
= −2αV
(t ).
Similarly, when lT + θ T < t < (l + 1 )T , one obtains that
(t − τ (t )) p j γ j Ae˜ j (t − τ (t )) − 2β (t ) p j e˜ j (t ) λ T + e˜ (t ) p j e˜ j (t ) 1−τ j λ −λe˜Tj (t − τ (t )) p j e˜ j (t − τ (t )) − 2α ( 1−τ t −τ (t )
i=1 j=1 N n
λmax (P )eTi (t )ei (t ))
≤ −2α (
e˜Tj
eTi (s )Pei (s )ds )
N
D+V (t ) ≤ −2α (
ξ e˜T (t ) p j e˜ j (t ) + 2e˜Tj (t ) p j h j e˜ j (t ) λmax (P ) j
t
|ei j (t )|1+μ
Substituting (10) into (9), it follows that
i=1 j=1
+ce˜Tj
N
N n
sign(ei (t ))|ei (t )|μ
i=1
(t ) p j γ j Ae˜ j (t )
+ce˜Tj
1+μ 2
(λmax (P )) λmin (P )
eTi (t )P
i=1 j=1
1+μ 2
N n
λ
N
−2α (λmax (P ))
sign(ei (t ))|ei (t )|μ
i=1
N
λ
i=1
×p j aik γ j ek j (t − τ (t )) − 2α
≤
sign(ei (t ))|ei (t )|μ .
It follows from Lemma 1 that N
i=1 k=1 j=1
n
1+μ 2
(λmax (P )) −2α eTi (t )P λmin (P ) i=1
eTi (t )Pei (t )
N N n
+
1+μ 2
(λmax (P )) λmin (P )
N
Based on Assumption 1, one can obtain
−2
eTi (s )Pei (s )ds )
where e˜ j (t ) = [e˜ j1 (t ), e˜ j2 (t ), · · · , e˜ jN (t )]T is a column vector of ej (t). According to the conditions (5) and (6) of Theorem 1, one derives that
i=1
D+V (t ) ≤ −2
i=1
eTi (t )P
t
t −τ (t )
D+V (t ) ≤ −2α (
i=1
−λ
N
λ
i=1
N t
1−τ
1+μ 2
N
−2α
aik ek (t − τ (t ))
k=1
λ
i=1 N
ξ I + 2h j IN + cγ j A − 2β IN λmax (P ) N
e˜Tj (t ) p j − 2
j=1
eTi
sign(ei (t ))|ei (t )|μ
IN e˜ j (t ) 1−τ n + e˜Tj (t − τ (t )) p j cγ j A − λIN e˜ j (t − τ (t )) +
Then the right upper Dini derivative of V(t) with respect to time t along the solutions of (4) can be calculated as follows. When lT ≤ t ≤ lT + θ T , +
n j=1
eTi (s )Pei (s )ds.
1+μ 2
(λmax (P )) λmin (P )
eTi (t )P
i=1
Proof. Construct the following Lyapunov functional
V (t ) =
N
−2α
1+μ 2
D+V (t ) ≤
n
−2
j=1
ξ e˜T (t ) p j e˜ j (t ) + 2e˜Tj (t ) p j h j e˜ j (t ) λmax (P ) j
+ce˜Tj (t ) p j γ j Ae˜ j (t ) +ce˜Tj (t − τ (t )) p j γ j Ae˜ j (t − τ (t )) − 2β e˜Tj (t ) p j e˜ j (t ) +2β
n j=1
e˜Tj (t ) p j e˜ j (t )
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
+
n
λ
1−τ
j=1
e˜Tj (t ) p j e˜ j (t ) −
n
λe˜Tj
j=1
(t − τ (t )) p j e˜ j (t − τ (t )) n ξ = e˜Tj (t ) p j − 2 IN + 2h j IN + cγ j A − 2β IN λ max (P ) j=1 +
λ
IN e˜ j (t )
1−τ n + e˜Tj (t − τ (t )) p j cγ j A − λIN e˜ j (t − τ (t ))
where j = 1, 2, · · · , n. The setting time can be estimated as
t2∗ ≤
+2β
Consider the following form of impulsive differential equation as the response CDNs:
y˙ i (t ) = f (yi (t )) + c
e˜Tj (t ) p j e˜ j (t ) = 2β
N
eTi (t )Pei (t ) ≤ 2β V (t ).
i=1
j=1
Namely, one has
1+μ
D+V (t ) ≤ −2αV 2 (t ), lT ≤ t ≤ lT + θ T , D+V (t ) ≤ 2β V (t ), lT + θ T < t < (l + 1 )T .
It can be derived from Lemma 3 that
V
1−μ 2
(t ) ≤ ( sup V (s ))
1−μ 2
−τ ≤s≤0
exp{β (1 − μ )(1 − θ )t } − α (1 − μ )θ t.
This implies that the systems (1) and (2) are finite-time synchronized. The finite-time t∗ can be estimated based on the condition (7) of Theorem 1. The proof of Theorem 1 is completed. If τ (t ) = 0, the system (1) becomes a general complex network, then, we can derive the following Corollary. Corollary 1. Assume that Assumption 1 holds, systems (1) and (2) are synchronization in finite-time under the controller (11), if there exist control parameters α , β and 0 < μ < 1 such that
(
ξ
λmax (P )
ln
i = 1, 2, · · · , N, t = tk . yi = yi (tk+ ) − yi (tk− ) = Bik ei ,
− h j + β )IN − cγ j A ≥ 0,
1
,
or
t1∗ ≤ T˜2 ,
where T˜2 is the smallest solution of
V
1−μ 2
(0 ) exp{β (1 − μ )(1 − θ )t } − α (1 − μ )θ t = 0.
Remark 5. In particular, when θ = 1, the periodic intermittent control problem, essentially translates into continuous control problem, which has made a lot of results [44,45]. In view of the condition (2) of Lemma 5 and Theorem 1, the following Corollary can be obtained via continuous control. Corollary 2. Under Assumption 1, systems (1) and (2) are synchronized in finite-time, if there exist control parameters α , β , λ and 0 < μ < 1 such that
λIN − cγ j A ≥ 0, 2(
= limt →t + yi (t ), k
t = tk , k ∈ . yi (tk− )
= yi (tk ),
(11) =
Then the following Corollary can be obtained.
Corollary 3. Under Assumption 1, systems (1) and (12) are synchronized in finite-time under the controller (13), if there exist control parameters α , β , λ and 0 < μ < 1 such that
λIN − cγ j A ≥ 0, 2(
ξ λ − hj + β − )IN − cγ j A ≥ 0, λmax (P ) 1−τ
βk = λmax [(IN + Bik )T (IN + Bik )] ≤ 1, where j = 1, 2, · · · , n. The setting time can be estimated as 1−μ 2
(0 ) . α (1 − μ ) V
3.2. Finite-time and fixed-time synchronization based on the same control scheme
The setting time can be estimated by
β (1 − μ )(1 − θ )
yi (tk+ )
Remark 6. When θ → 1− , the controllers (4) become a special impulsive control scheme: ⎧ N t 1+μ λ ei (t ) ⎪ ⎪ −β ei (t ) − α ( eTi (s )Pei (s )ds ) 2 P −1 ⎪ ⎪ 1−τ || e(t )||2 ⎨ t −τ (t ) i=1 1+μ Ui (t ) = (12) (λmax (P )) 2 ⎪ μ ⎪ t = tk , ||ei (t )|| = 0 ⎪−α λ (P ) sign(ei (t ))|ei (t )| , ⎪ ⎩ min 0, t = tk , ||ei (t )|| = 0.
t3∗ ≤
αθ ≥ 1. 1−μ β (1 − θ )V 2 (0 )
t1∗ = T˜1 ≤
ai j y j (t − τ (t )) + Ui (t ),
{1, 2, · · · , m1 , m2 , · · · , mk } is a finite natural number set.
It follows from the conditions (5) and (6) of Theorem 1 that n
N j=1
where
e˜Tj (t ) p j e˜ j (t ).
j=1
D+V (t ) ≤ 2β
1−μ 2
(0 ) . α (1 − μ ) V
j=1 n
297
ξ λ − hj + β − )IN − cγ j A ≥ 0, λmax (P ) 1−τ
In this section, we will give some novel finite-time and fixedtime synchronization criteria for CDNs (1) by designing a new unified control scheme. In order to obtain the main results, the controllers are designed as follows:
Ui (t ) = −β ei (t ) − λ
N
ek (t − τ (t )) − ρ1
k=1
(λmax (P )) λmin (P )
1+ε 2
sign(ei (t ))|ei (t )|ε −ρ2
1+ω 2
(λmax (P )) λmin (P )
sign(ei (t ))|ei (t )|ω ,
(13)
where i = 1, 2, · · · , N, 0 < ε < 1, β , λ, ρ 1 , ρ 2 are positive constants. Remark 7. In [32,46], to realize fixed-time synchronization, the controller designed by the authors contains two power exponent items. Based on the proposed method, we also have added two power exponent items into controller. However, the difference between the controllers we design is that we can just by adjusting the power exponent (0 < ω < 1, ω = 1, ω > 1) to achieve finite-time and fixed-time synchronization.
298
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
Based on controller (13), the error system can be rewritten as N
ei (t ) = f (yi (t )) − f (xi (t )) + c
ai j e j (t − τ (t )) + Ui (t ).
sign(ei (t ))|ei (t )|ε
(14) − ρ2
j=1
Theorem 2. Suppose that Assumption 1 holds, under the controller (13), if there exist control parameters β , λ > 0 such that
λIN − cγ j A ≥ 0, (
ξ
λmax (P )
(15)
− h j + β )IN ≥ 0,
(16)
where j = 1, 2, · · · , N. Then, the following conclusions hold. (1) When 0 ≤ ω ≤ 1, systems (1) and (2) are finite-time synchronization. The settling time can be estimated by
⎧ 1−η ⎨ Vρ (1−(η0)) , 0 ≤ ω < 1, 1−ε t1 ≤ 2 ⎩ ρ (12−ε ) ln ρ2V ρ1(0)+ρ1 , ω = 1,
max{ 1+2 ε , 1+2ω }, 0 ≤ V (t ) ≤ 1, min{ 1+2 ε , 1+2ω }, V (t ) > 1.
1 ρ1 1−ε 2 2 ( ) ω−ε ( + ), ρ1 ρˆ2 1−ε ω−1 2
j=1
+
n
V (t ) =
eTi
j=1
− ρ1
N
.
−ρ2
aik ek (t − τ (t )) ek (t − τ (t )) − ρ1
k=1
(λmax (P )) λmin (P )
sign(ei (t ))|ei (t )|ε
≤−
+
1+ω 2
(λmax (P )) λmin (P )
ξ
λmax (P )
N n
sign(ei (t ))|ei (t )|ω
N n
N N n
×p j aik γ j ek j (t − τ (t )) − N N n
λmax (P )eTi (t )ei (t ))
1+ε 2
λmax (P )eTi (t )ei (t ))
1+ω 2
1+ε 2
(t ) − ρ2V
1+ω 2
(t ).
Then, from the condition (2) of Lemma 5, which shows that systems (1) and (2) are synchronized in finite-time t1 . When ω = 1, N
λmax (P )eTi (t )ei (t ))
i=1
−ρ2 (nN )− 2 ( 1
N
1+ε 2
N
λmax (P )eTi (t )ei (t ))
i=1
(t ) − ρ2 V (t ).
λmax (P )eTi (t )ei (t )) ω
−ρ2 (nN )− 2 (
N
λmax (P )eTi (t )ei (t ))
≤ −ρ1 (
N
eTi (t )Pei (t ))
1+ε 2
i=1 k=1 j=1
i=1
(λmax (P )) λmin (P )
1+ω 2
1+ε 2
1+ε 2
ω
− ρ2 (nN )− 2 (
i=1
eTi (t )P
1+ε 2
i=1
β p j e2i j (t )
= −ρ1V N
1+ε 2
i=1
ei j (t )
ei j (t )
×p j ek j (t − τ (t )) − ρ1
sign(ei (t ))|ei (t )|ω .
V˙ (t ) ≤ −ρV η (t ).
V˙ (t ) ≤ −ρ1 (
i=1 j=1
−λ
1+ω 2
sign(ei (t ))|ei (t )|ε
From the condition (1) of Lemma 5, one can easy to see that systems (1) and (2) are synchronized in finite-time t2 . When ω > 1,
i=1 k=1 j=1 N n
(λmax (P )) λmin (P )
i=1
≤ −ρ1V
p j e2i j (t )
i=1 j=1
eTi (t )P
N
1+ε 2
i=1 j=1
p j h j e2i j (t ) + c
sign(ei (t ))|ei (t )|ω ,
Based on Lemma 2, one derives that
V˙ (t ) ≤ −ρ1 (
k=1
−ρ2
(λmax (P )) λmin (P )
N
≤ −ρ1V
N
1+ε 2
eTi (t )P
i=1 N
−ρ2 (
eTi (t )P f (yi (t )) − f (xi (t )) − Hei (t ) + Hei (t )
−β ei (t ) − λ
1+ω 2
i=1
(t )Pei (t ).
N
(λmax (P )) λmin (P )
N
V˙ (t ) ≤ −ρ1 (
i=1
+c
eTi (t )P
sign(ei (t ))|ei (t )|ε
When 0 ≤ ω < 1,
By differentiating V(t) along the trajectories of (14) and based on Assumption 1, one has
V˙ (t ) =
(λmax (P )) λmin (P )
i=1 N
1+ε 2
eTi (t )P
i=1
i=1
N
e˜Tj (t ) p j cγ j A − λIN e˜ j (t − τ (t ))
Proof. Construct the following Lyapunov function N
ξ I + h j IN − β IN e˜ j (t ) λmax (P ) N
e˜Tj (t ) p j −
V˙ (t ) ≤ −ρ1
(2) When ω > 1, systems (1) and (2) are fixed-time synchronization. The settling time can be estimated by
where ρˆ2 = ρ2 (nN )
sign(ei (t ))|ei (t )|ω
where e˜ j (t ) = [e˜ j1 (t ), e˜ j2 (t ), · · · , e˜ jN (t )]T is a column vector of ej (t). By virtue of the second inequality of Lemma 1 and the conditions (15) and (16) of Theorem 2, it follows that
1
−ω
=
i=1
where ρ = min{ρ1 , ρ2 }, ρ2 = ρ2 (nN )− 2 and
Tmax ≤
1+ω 2
(λmax (P )) λmin (P )
eTi (t )P
i=1 n
− ρ2
2
η=
N
N
eTi (t )Pei (t ))
1+ω 2
i=1
(t ) − ρˆ2V
1+ω 2
(t ).
Hence, from Lemma 4, the fixed-time synchronization of systems (1) and (2) are achieved.
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
The proof of Theorem 1 is completed.
If τ (t ) = 0, The controller (13) is simplified into the following form: 1+ε
(λmax (P )) 2 Ui (t ) = −β ei (t ) − ρ1 sign(ei (t ))|ei (t )|ε λ min (P ) 1+ω (P )) 2 ω −ρ2 (λmax λmin (P ) sign(ei (t ))|ei (t )| ,
λmax (P )
− h j + β )IN ≥ 0,
(18)
where j = 1, 2, · · · , N. Then, the following conclusions hold. (1) When 0 ≤ ω ≤ 1, systems (1) and (2) are finite-time synchronization. The settling time can be estimated by
V 1−η (0)
t2 ≤
0 ≤ ω < 1,
ρ ( 1 −η ) ,
2 ρ2 (1−ε )
ln
ρ2 V
1−ε 2
( 0 )+ ρ 1
ρ1
ω = 1,
1 ρ1 1−ε 2 2 ( ) ω−ε ( + ). ρ1 ρˆ2 1−ε ω−1
4. Numerical examples In this section, two examples are presented to demonstrate the validity and effectiveness of our proposed theoretical results for finite-time synchronization via periodically intermittent control and feedback control. Example 1. Consider a network with 10 nodes and each node is a simple three dimensional nonlinear system described as:
x˙ i (t ) = f (xi (t )) + c
10
ai j x j (t − τ (t )),
i = 1, 2, · · · , 10,
(19)
j=1
where xi (t ) = (xi1 (t ), xi2 (t ), xi3 (t ))T , and
f (xi (t )) =
−10 28 0
10 −1 0
0 0 − 83
xi1 (t ) xi2 (t ) xi3 (t )
+
0 −xi1 xi3 . xi1 xi2
The single Lorenz system has a chaotic attractor(see Fig. 1.). Taking P = diag(0.5, 0.4, 0.2 ) and H = diag(50, 50, 50 ) such that the Lorenz system satisfies Assumption 1 [52]. = diag(1, 1, 1 ) , 1 et c = 1, τ (t ) = 1+ and τ = . A = (ai j )10×10 is a symmetrically difet 4 fusive coupling matrix given by
⎛
−2 ⎜1 ⎜1 ⎜ ⎜0 ⎜ ⎜0 A=⎜ ⎜0 ⎜0 ⎜ ⎜0 ⎝0 0
1 −4 0 1 1 0 0 0 1 0
1 0 −3 1 0 1 0 0 0 0
0 1 1 −3 0 0 1 0 0 0
10
ai j y j (t − τ (t )) + Ui (t ),
i = 1, 2, · · · , 10, (20)
For the sake of simplicity of computation, the initial conditions of each nodes are chosen as follows: xi (0 ) = (3 + 0.3i, 0.2 + 0.1i, 0.3 + 0.2i )T , yi (0 ) = (−2 + 0.5i, −3 + 0.3i, −4 + 0.6i )T , i = 1, 2, · · · , 10. The simulation results with control input are calculated by different control scheme as follows: (a) Finite-time synchronization with periodic intermittent control
⎧ 10 ⎪ 8 t ei (t ) 3 ⎪ ⎪ −ei (t ) − 8( eTi (s )Pei (s )) 4 P −1 ⎪ ⎨ 3 ||e(t )||2 t −τ (t ) i=1
Ui (t )=
(λmax (P )) 4 1 ⎪ −8 sign(ei (t ))|ei (t )| 2 , lT ≤ t ≤ lt + θ T , ⎪ ⎪ λmin (P ) ⎪ ⎩ 0, lT + θ T < t < (l + 1 )T , 3
(21) ,
(2) When ω > 1, systems (1) and (2) are fixed-time synchronization. The settling time can be estimated by 1 Tmax ≤
y˙ i (t ) = f (yi (t )) + c
(17)
Corollary 4. Suppose that Assumption 1 holds, under the controller (17), if there exist control parameters β > 0 such that
ξ
Consider Eq. (19) as the drive system, the response system described as
j=1
where i = 1, 2, · · · , N, 0 < ε < 1, β , ρ 1 , ρ 2 are positive constants. Then, the following Corollary holds.
(
299
0 1 0 0 −4 0 1 1 1 0
0 0 1 0 0 −3 1 0 0 1
0 0 0 1 1 1 −5 1 0 1
0 0 0 0 1 0 1 −4 1 1
0 1 0 0 1 0 0 1 −3 0
⎞
0 0⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟. 1⎟ 1⎟ ⎟ 1⎟ ⎠ 0 −3
Then under the above parameters, system (19) have a chaotic attractor (see Fig 2.)
1 . By computing, sup V (s ) = 2 −1≤s≤0 252.28. Further, select T = 2, θ = 0.8, under the above conditions, the assumptions of Theorem 1 are satisfied, then systems (19) and (20) are synchronized in finite-time t ∗ = 1.4381 and the synchronization errors are shown in Fig. 3. (b) Finite-time and fixed-time synchronization with state feedback control where β = 1, α = 8, λ = 2, μ =
Ui (t ) = −ei (t ) − 0.01
10
λmax (P ) 4 λmin (P )
3
ek (t − τ (t )) − 9
k=1
sign(ei (t ))|ei (t )| 2
1
1+ω
−9
λmax (P ) 2 sign(ei (t ))|ei (t )|ω λmin (P )
(22)
1 . It is easy to check 2 that the conditions of Theorem 2 are satisfied. The key parameter ω in the controller (22) is chosen as follows: 1. Choosing ω = 0.15, from the result (1) of Theorem 2, one has the systems (19) and (20) are synchronized in finite-time t1 = 1.9133s (see Fig. 4). 2. Choosing ω = 1, from the result (2) of Theorem 2, systems (19) and (20) are synchronized in finite-time t2 = 1.1269s (see Fig. 5). 3. Choosing ω = 1.5, from the result (3) of Theorem 2, systems (19) and (20) are synchronized in fixed-time Tmax = 3.1825s (see Fig. 6). where β = 1, λ = 0.01, ρ1 = ρ2 = 9, ε =
Example 2. To verify the generality of our results, we consider the following CDNs without time-varying delays:
x˙ i (t ) = f (xi (t )) + c
10
ai j x j (t ),
i = 1, 2, · · · , 10,
(23)
j=1
assume that the parameters in (23) are the same as in Example 1 and the corresponding chaotic behavior is shown in Fig. 7. The corresponding response system is described as:
y˙ i (t ) = f (yi (t )) + c
10
ai j y j (t ) + Ui (t ),
i = 1, 2, · · · , 10.
j=1
(24) The controllers are designed as the following two forms:
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Fig. 1. The chaotic behavior of Lorenz system.
Fig. 2. The chaotic behavior of system (19).
(a ) Periodic intermittent control
⎧ 3 ⎪ ⎨−ei (t ) − 8 (λmax (P )) 4 sign(ei (t ))|ei (t )| 21 , λmin (P ) Ui (t ) = lT ≤ t ≤ lt + θ T , ⎪ ⎩ 0, lT + θ T < t < (l + 1 )T . (25)
(b ) State feedback control
λmax (P ) 4 1 sign(ei (t ))|ei (t )| 2 λmin (P ) 1+ω λmax (P ) 2 −9 sign(ei (t ))|ei (t )|ω . λmin (P ) 3
102.118. Similarly, we choose T = 2, θ = 0.8. Through verification, the assumptions of Corollary 1 are satisfied, the systems (23) and (24) are synchronized in finite-time t1∗ = 1.11s based on controller (25) and the corresponding synchronization errors are shown in Fig. 8. Based on control scheme (26), we can verify the conditions of Corollary 2 hold. By simple computation, the setting time can be estimated as t21 = 1.8674s when ω = 0.15, t22 = 1.1142s when ω = 1 1 and Tmax = 3.1825s when ω = 1.5. From Figs. 9–11, we can see that the systems (23) and (24) are synchronized in finite time or fixed time.
Ui (t ) = −ei (t ) − 9
(26)
The values of the parameters of control schemes (25) and (26) are the same as in (21) and (22). By simple computing, V (0 ) =
Remark 8. By computing, we have t ∗ > t1∗ , which shows that time delays have a negative effect on the estimation of the setting time. Moreover, from Fig. 3 and Fig. 8, it can be seen that time delays have a positive effect on the synchronization of the system. This is in accordance with the controllers (21) and (25).
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
Fig. 3. The Finite-time synchronization of systems (19) and (20) under (21) with (T, θ , μ)=(2,0.8,0.5).
Fig. 4. The Finite-time synchronization of systems (19) and (20) under (22) with (λ, ε , ω)=(10,0.5,0.15).
Fig. 5. The Finite-time synchronization of systems (19) and (20) under (22) with (λ, ε , ω)=(10,0.5,1).
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Fig. 6. The Fixed-time synchronization of systems (19) and (20) under (22) with (λ, ε , ω)=(10,0.5,1.5).
Fig. 7. The chaotic behavior of system (23).
Remark 9. By simple computing and from Figs. 4–6, Figs. 9–11, 1 , which show that the setwe have t11 > t21 , t12 > t22 , Tmax = Tmax ting time function depend on the initial conditions in estimating the finite-time synchronization. However, the setting time in estimating the fixed-time synchronization is not affected by the initial conditions. Additionally, we can conclude that under the controller (26), time delays have a negative effect both on the estimation of the settling time and the convergence rate of systems. Remark 10. From Remarks 8 and 9, we know that by constructing different controllers, time delays can accelerate or restrain the convergence rate of the system. From Figs. 4–6 and Figs. 9–11, we conclude that by regulating the main control parameters, not only both the finite-time and fixed-time synchronization can be achieved, but also the convergence rate of synchronization can be adjusted. It is of engineering interest to develop the efficiency of the controller by adjusting the appropriate parameters. Remark 11. In [47–49], the authors considered the stability and control issues of nonlinear dynamic systems with impulse or
stochastic perturbations. Finite-time and fixed-time synchronization of complex networks with time delays are considered in this paper. In fact, the issues about the finite-time and fixed-time synchronization for complex networks with impulse or stochastic perturbations are of great significance in practical applications, which are our direction for future work. 5. Conclusion In this paper, the finite-time and fixed-time synchronization for a class of CDNs with time-varying coupled delays are investigated by means of periodically intermittent control and feedback control. To derive the main results, a new inequality is proposed and proved. The finite time stability in this paper is not only improved the proof method, but also extended the conclusions in the previous literature. Based on the Lyapunov functionals, inequality techniques, finite-time stability theory, some new easy-verified conditions are established to ensure synchronization for the target CDNs within a settling time. Finally, numerical simulations are given to
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
Fig. 8. The Finite-time synchronization of systems (23) and (24) under (25) with (T, θ , μ)=(2,0.8,0.5).
Fig. 9. The Finite-time synchronization of systems (23) and (24) under (26) with (λ, ε , ω)=(10,0.5,0.15).
Fig. 10. The Finite-time synchronization of systems (23) and (24) under (26) with (λ, ε , ω)=(10,0.5,1).
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Fig. 11. The Fixed-time synchronization of systems (19) and (20) under (22) with (λ, ε , ω)=(10,0.5,1.5).
show the effectiveness and validity of the theoretical results. For the complex networks in this paper, impulse or stochastic perturbations have not been considered. It would be of interest to investigate the finite-time and fixed-time synchronization of delayed complex networks with impulsive effects or stochastic perturbations in the near future.
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos
Review
Analysis and discontinuous control for finite-time synchronization of delayed complex dynamical networksR Jiarong Li, Haijun Jiang∗, Cheng Hu, Juan Yu College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 6 April 2018 Revised 17 July 2018 Accepted 18 July 2018
Keywords: Complex dynamical networks Discontinuous control Finite-time synchronization Fixed-time synchronization
a b s t r a c t This paper addresses the finite-time and fixed-time synchronization issue of delayed complex dynamical networks (CDNs). Firstly, as an important preliminary, an improved and generalized finite-time stability theory is established for delayed nonlinear systems to prove the finite-time synchronization mainly through the reduction to absurdity. Different from some existing results, a more detailed discussion of the setting time function for finite-time synchronization is given. Besides, a novel feedback controller is firstly proposed to unify finite-time and fixed-time synchronization just by adjusting the key control parameter. Furthermore, several new criteria are derived to ensure the finite-time and fixed-time synchronization based on inequality analysis method and constructing appropriate Lyapunov functional. Finally, some numerical simulations are presented to support the theoretical results. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction Complex dynamical networks (CDNs), which present a high degree of complexity, have attracted wide attention because they are widespread in real world, such as the World Wide Web, power grid, patent use network, gene network, transportation network, metabolic pathways and so on. Complex network, composed of a large number of nodes and links, is a nonlinear dynamical system. Where the nodes denote the individuals in the network and the edges represent the connections among them. Over the past decades, complex networks with different structures have been studied and some results also have been yielded [1–3]. Although the results obtained in [1–3] are effective and convenient, none of these articles considered the effect of time delay on the system. Time delay is often the main cause of system instability and poor performance. In realistic CDNs, time delay is ubiquitous, such as the flow of steam and fluid in pipes, the transmission of electrical signals over long lines. Especially in the control system [4]. Nowadays, inspired by the effect of time delays, the CDNs with time delays have attracted increasingly attention [5–8].
R This work was supported in part by the Excellent Doctor Innovation Program of Xinjiang Uyghur Autonomous Region (XJGRI2017001), in part by the Excellent Doctor Innovation Program of Xinjiang University (XJUBSCX-2017003) and the National Natural Science Foundation of People’s Republic of China (Grants No. 61473244, No. 61563048, No. 11402223, No. U1703262). ∗ Corresponding author. E-mail address: [email protected] (H. Jiang).
https://doi.org/10.1016/j.chaos.2018.07.019 0960-0779/© 2018 Elsevier Ltd. All rights reserved.
One of the hot topics in the field of CDNs is synchronization due to its practical application in human heartbeat regulation, secure communication, information science and image processing and so on [9–13]. However, it is worthy of noting that most of existing results including the articles mentioned above are actually asymptotic synchronization. These types of synchronization can be implemented only when the time is near infinity. In practical application, however, the synchronization aims are usually expected to be realized within a finite time. For this purpose, finite-time synchronization, which means that the time it takes for the system to be synchronized can be estimated, has been proposed and extensively studied in recent years [14–23]. In [16,17], the problem of finite-time synchronization of a class of CDNs with time-varying delays and coupling time-varying delays was studied. Although the synchronization results obtained in above articles has the optimal convergence time, the setting time function has to depend on the initial conditions. In practical systems, however, the initial conditions of the system are difficult to obtain in advance, which makes it difficult to estimate the setting time. In order to solve the problem that the finite-time synchronization is heavily dependent on the initial conditions, the concept of fixed-time stability was proposed by Polyakov [18]. Fixed-time stability is proposed based on the finite-time stability by demanding the boundness of the settling time function of the system to achieve synchronization. The fixed-time stability not only has better robustness and disturbance rejection properties, but also ensures that the settling time independent of the initial conditions of the system. These merits can improve the efficiency and quality
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of engineering management greatly. Research on fixed-time synchronization, however, is just beginning to germinate and the related theories are still scarce. There are fewer articles on fixed-time synchronization in neural networks, let alone complex dynamical networks. In [19], the authors provided a general approach to show that the essence of finite-time stability and fixed-time convergence, and some conditions were acquired to ensure that the setting time function was bounded. In recent work [20], for the synchronous convergence time of the system, the author gave a more accurate estimate. Furthermore, investigations of fixed-time synchronization issues of nonlinear dynamical networks have been presented in [24–27]. As we known, many papers have considered the finite-time and fixed-time stability and consensus problems, but there are few works on the fixed-time synchronization. Therefore, it is interesting and important to investigate the fixed-time synchronization of CDNs with coupled delays. Another hot topic in the research of CDNs is control. We all know that most system synchronization can only be achieved with a suitable controller. At present, synchronization control has been investigated widely [1,3,12,21–31]. In [21,28], based on pinning impulsive control and pinning control, the problem of the system’s finite time synchronization was investigated. In [22,23,29], Some sufficient conditions were received to realize the finite-time or fixed-time synchronization of chaotic systems under the sliding mode control protocol. In [30,31], the authors considered the finite-time synchronization of CDNs through designing periodically intermittent control (PIC). Different from the controllers in [28,29], the output of the system is intermittent rather than continuous, based on this principle, the design of the PIC is divided into the control interval and the rest interval, which makes the controller more economical and effective, and can also avoid prolonged work on the controller cause some damages. To the best of our knowledge, however, delayed CDNs are relatively unexplored in finite-time synchronization with periodic intermittent control. From this, an investigation of finite-time synchronization under PIC is important in both theoretical analysis and real applications. From [20,24,32,33] we can find that fixed-time synchronization is a special case of finite-time synchronization. discontinuous feedback controllers designed in [34] were simple and efficient, easy to implement. The results obtained can only be achieved finite-time synchronization. To the best of my knowledge, no author has yet thought of implementing both types of synchronization by designing a uniform form of controller. Based on the above discussions, this paper will discuss the finite-time and fixed-time synchronization of CDNs with timevarying delays. The main contributions of this paper in comparison to the existing ones can be reflected as follows: (1) A novel theory of improvement on accurate analysis of setting time function of the finite-time stable delayed CDNs is developed in this paper compared with [30,31]. (2) A more general intermittent control scheme, which can be employed to achieve synchronization within finite time for time delay systems, is designed compared with the works [15,28,34–36]. (3) An extended unified feedback controller is proposed, which can realize both finite-time and fixed-time synchronization. Until now, few articles consider these two synchronism simultaneously. Therefore, these two synchronism should be investigated, which can help us obtain more general results. (4) By regulating the main control parameters in the controller, not only both the finite-time and fixed-time synchronization can be achieved, but also the convergence rate of synchronization can be adjusted. It is of engineering interest to develop the efficiency of the controller by adjusting the appropriate parameters. The rest of this paper is organized as follows. Some necessary preliminaries and model description are given in Section 2. In Section 3 the finite-time synchronization conditions are presented
for delayed CDNs via periodic intermittent control, and finite-time and fixed-time synchronization are analyzed based on a unified control framework. simulations are given in Section 4 to verify the effectiveness of the obtained results. In Section 5, conclusion is summarized. Notations. Let Rn be the space of n-dimensional real column vectors. Denote x = ( x1 , · · · , xN )T ∈ RN , μ μ μ μ T |ei (t )| = (|ei1 (t )| , |ei2 (t )| , · · · , |ein (t )| ) and sign(ei (t )) = (sign(ei1 (t )), · · · , sign(ein (t )))T ∈ Rn . ||x|| denotes the vector norm 1 defined by ||x|| = ( ni=1 x2i ) 2 . λmax (P)(λmin (P)) is defined as the maximum (minimum) eigenvalue of the positive definite diagonal matrix P. IN is the identity matrix with N-dimensions. 2. Model description and preliminaries Let us consider a general delayed complex network consisting of N dynamical nodes, which is described by
x˙ i (t ) = f (xi (t )) + c
N
ai j x j (t − τ (t )),
i = 1, 2, · · · , N
(1)
j=1
where xi (t ) = (xi1 (t ), · · · , xin (t ))T ∈ Rn is the state variable. f: Rn → Rn is a continuous nonlinear vector function. The nonnegative constant c is the coupling strength. = diag(γ1 , γ2 , · · · , γn ) is a positive definite diagonal matrix, which denotes the inner-coupling matrix between each pair of nodes. A = (ai j )N×N is the coupling configuration matrix. If there is a connection from the node i to the node j(j = i), then aij > 0. Otherwise, ai j = 0( j = i ) and the di agonal elements of matrix A is defined as aii = − Nj=1, j=i ai j . The coupling time-varying delay τ (t) is a bounded and continuously differentiable function, i.e., there exists a positive constant τ satisfying 0 ≤ τ˙ (t ) ≤ τ < 1. Remark 1. In [30,31,37], authors introduced a finite-time intermittent control method to realize the synchronization of CDNs. The influence of time-varying delays on the system is not considered. However, in this paper, we concern the finite-time synchronization problem for CDNs with time-varying delays. The models considered in the paper are more general and more reasonable. Without loss of generality, we refer to system (1) as the drive system, and consider a response system described as follows:
y˙ i (t ) = f (yi (t )) + c
N
ai j y j (t − τ (t )) + Ui (t ), i = 1, 2, · · · , N,
j=1
(2) where yi (t ) = (yi1 (t ), · · · , yin (t ))T ∈ Rn is the state variable. U (t ) = (U1 (t ), · · · , UN (t ))T is the control input to be designed later. The other parameters are the same as in system (1). Assumption 1 (QUAD). Assume that there exists a positive definite diagonal matrix P = diag( p1 , p2 , · · · , pn ) and a diagonal matrix H = diag(h1 , h2 , · · · , hn ), such that f( · ) satisfies the following inequality:
(u − v )T P ( f (u ) − f (v ) − H (u − v )) ≤ −ξ (u − v )T (u − v ), for all u, v ∈ Rn , ξ > 0. Remark 2. The vector field f is required to satisfy the Lipschitz condition in many CDNs [1,2,38–40]. There are also many articles require vector field f which satisfies the QUAD condition [29,30]. Lipschitz condition and QUAD condition, which usually made in the literature to prove network synchronization, are assumptions on the vector field f. Actually, if f is globally Lipschitz with a Lipschitz constant l > 0, then f is QUAD(H, ξ ), with P H − ξ IN ≥ l ||P ||IN
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
[41]. Assumption 1, therefore, can ensure the uniqueness and existence of the solutions for CDNs, also helps to analyze the dynamical behaviors of such networks. Most chaotic systems, including Ro¨ssler’s systems, Chua’s circuit and Lorenz system etc., satisfy QUAD condition. To obtain the main results of this paper, we will design the following periodic intermittent controller, and introduce some Definitions and Lemmas. Firstly, let ei (t ) = yi (t ) − xi (t ), i = 1, 2, · · · , N be the synchronization errors, and design the following periodic intermittent control scheme:
Ui (t ) =
⎧
1+2μ ⎪ N t ⎪ ⎪ λ ⎪ −β ei (t ) − α eTi (s )Pei (s )ds ⎪ ⎪ 1−τ ⎪ t −τ (t ) ⎪ i =1 ⎪ ⎨ −1 ei (t ) P
||e(t )||2
1+μ ⎪ (λmax (P )) 2 ⎪ ⎪ −α ⎪ ⎪ λmin (P ) ⎪ ⎪ μ ⎪ ⎪ ⎩sign(ei (t ))|ei (t )| ,
0,
lT ≤ t ≤ lT + θ T , lT + θ T < t < (l + 1 )T ,
1 −ω xω i ≥n
i=1
n
a+b υ a + b υ1 a + b υ2 ) = min{( ) ,( ) }, 2 2 2 max{υ1 , υ2 }, min{υ1 , υ2 },
a+b 2
0≤
a+b 2
a+b 2
≤ 1, > 1.
υ .
Lemma 3. Suppose that function V(t) is non-negative when t ∈ [−τ , +∞ ) and satisfies the following conditions:
D+V (t ) ≤ −αV η (t ), D+V (t ) ≤ β V (t ),
lT ≤ t ≤ lT + θ T , lT + θ T ≤ t < (l + 1 )T ,
( )
0 ≤ t ≤ t ∗.
t ∗ = T1 =
1
β (1 − θ )(1 − η )
,
xqi
ω
≥ n 1 −ω
a+b 2
,
αθ ≥ 1, where T2 is the smallest soluβ (1 − θ )( sup V (s ))1−μ −τ ≤s≤0
−τ ≤s≤0
Proof. Take M0 = ( sup V (s ))1−η and W (t ) = V 1−η (t ) + hα (1 − −τ ≤s≤0
n
ω2 x2i
1 h
η )t, where 0 < h < 1, t ≥ −τ . Let Q (t ) = W (t ) − M0 . It’s clear that Q (t ) < 0,
for all t ∈ [−τ , 0].
(A1)
In the following, we will prove
for all t ∈ [0, θ T ).
(A2)
By using reduction to absurdity, assume that there exists a t0 ∈ [0, θ T) such that
Q (t0 ) = 0, .
i=1
υ
as ln
Q (t ) < 0, ,
xi
t ∗ = T2 ,
( sup V (s ))1−η exp{(1 − η )β (1 − θ )t } − αθ (1 − η )t = 0.
Lemma 2. If a ≥ 0, b ≥ 0, 0 < υ1 < 1, 0 < υ2 < 1, then the following inequality holds.
aυ1 + bυ2 ≥
(
tion of
1q
i=1
Let
or
i=1
≤
The constant t∗ is the setting time given by
Lemma 1. ([42]) Let xi (t) ≥ 0 for i = 1, 2, · · · , n, 0 < p < q, ω > 1, then the following inequalities are true
n
a+b 2
−τ ≤s≤0
is called the setting time. If there exist a fixed-time Tmax ≥ 0 such that T(e(0)) ≤ Tmax for any e(0) ∈ RnN , systems (1) and (2) are said to be fixed-time synchronized.
i=1
b υ1 ≤ a, then ( a+ ≤ aυ1 . When a ≤ b, 2 )
≤ bυ2 . Hence,
−αθ (1 − η )t,
T (e(0 )) = inf{T˜ (e(0 )) ≥ 0 : ||e(t )|| = 0, ∀t ≥ T˜ (e(0 ))}
n
a+b 2
≤ 1, > 1.
V 1−η (t ) ≤ ( sup V (s ))1−η exp{(1 − η )β (1 − θ )t }
−τ ≤s≤0
≥
a+b 2
a+b 2
where α , β > 0, T > 0, 0 < η < 1, 0 < θ < 1, then the following inequality holds:
||ei (t )|| = 0, ||ei (t )|| ≡ 0 for any t ≥ T (e(0 )),
xip
0≤
aυ1 + bυ2 ≥ min{( a+2 b )υ1 , ( a+2 b )υ2 }.
aυ1 + bυ2 ≥
where e(0 ) = sup e(s ) and
n
b, then
( a+2 b )υ2
then,
Definition 1. ([20]) Systems (1) and (2) are said to be finite-time synchronized, if there exists a function T(e(0)) ≥ 0, such that
max{υ1 , υ2 }, min{υ1 , υ2 },
Proof. When a ≥ b,
(3)
for i = 1, 2, · · · , N.
1p
υ=
υ=
(4)
⎧ N ⎪ ⎪ ⎪ f ( y ( t )) − f ( x ( t )) + c ai j e j (t − τ (t )) + Ui (t ), i ⎪ i ⎪ ⎪ j=1 ⎪ ⎨ lT ≤ t ≤ lt + θ T , e˙ i (t ) = N ⎪ ⎪ ⎪ f (yi (t )) − f (xi (t )) + c ai j e j (t − τ (t )), ⎪ ⎪ ⎪ j=1 ⎪ ⎩ lT + θ T < t < (l + 1 )T ,
t→T (e(0 ))
where
obviously,
where ei (t) = 0 as lT ≤ t ≤ lT + θ T , i = 1, 2, · · · , N. α , β , λ are positive constants called control gain. the real number μ satisfies 0 < μ < 1. T > 0 is the control period. 0 < θ < 1 is called the control rate. P is defined in Assumption 1. Then based on the intermittent controller (3), from systems (1) and (2), the error system can be derived as
lim
293
Q (t ) < 0,
D+ Q (t0 ) ≥ 0, 0 ≤ t < t0 .
(A3) (A4)
Using ( ) and (A1), one has
D+ Q (t0 ) ≤ −α (1 − η ) + α h(1 − η ) < 0. This contradicts the second inequality in (A3). So, (A2) holds.
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Now, we prove that for t ∈ [θ T, T)
By ( ) and (A12), one has
H (t ) = W (t ) exp{−(1 − η )β (t − θ T )} − 1h M0 −hα (1 − η )(t − θ T ) exp{−(1 − η )β (t − θ T )} < 0,
(A5)
Otherwise, there exists a t1 ∈ [θ T, T) such that
H (t1 ) = 0, H (t ) < 0,
D+ H (t1 ) ≥ 0,
(A6)
D+ H1 (t3 )≤−β (1−η )hα (1−η )2θ T exp{−(1−η )β (t3 −2θ T )} < 0, which contradicts the second inequality in Eq. (A12). So, (A11) holds. On the one hand, for t ∈ [T , T + θ T )
W (t ) <
θ T ≤ t < t1 .
(A7)
Using Eqs. ( ) and (A6), one can obtain
D+ H (t1 ) ≤ −hα (1 − η )θ T (1 − η )β exp{−(1 − η )β (t1 − θ T )} < 0. This is in contradiction with the second inequality in (A6). Therefore, (A5) holds. On the one hand, for t ∈ [θ T, T),
1 M0 exp{(1 − η )β (t − θ T )} + hα (1 − η )(t − θ T ) h 1 ≤ M0 exp{(1 − η )β (1 − θ )T } + hα (1 − η )(1 − θ )T . h
On the other hand, for t ∈ [T + θ T , 2T ),
W (t ) <
W (t ) <
W (t ) <
1 M0 h 1 < M0 exp{(1 − η )β (1 − θ )T } + hα (1 − η )(1 − θ )T . h
W (t ) <
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )T } h −hα (1 − η )(1 − θ )T < 0.
Q1 (t ) = W (t ) −
(A8) <
Otherwise, there exists a t2 ∈ [T , T + θ T ), such that
D+ Q1 (t2 ) ≥ 0,
(A9) ≤ (A10)
From ( ) and (A9)
W (t ) <
H1 (t ) = W (t ) exp{−(1 − η )β (1 − θ )T 1 −(1 − η )β (t − T − θ T )} − M0 h −h α (1 − η )(1 − θ )T + α (1 − η )(t − θ T − T )
W (t ) <
(A11)
Otherwise, there exists a t3 ∈ [T + θ T , 2T ), such that
T ≤ t < t3 .
1 M0 exp{(1 − η )β (1 − θ )kT } + hα (1 − η )(1 − θ )kT . h
1 M0 exp{(1 − η )β (1 − θ )kT } + hα (1 − η )(1 − θ )kT , h (A16)
1 M0 exp{(1 − η )β (1 − θ )kT } + hα (1 − η )(1 − θ )kT , h
For (k + θ )T ≤ t < (k + 1 )T ,
× exp{−(1 − η )β (1 − θ )T − (1 − η )β (t − T − θ T )}
H1 (t ) < 0,
1 M0 exp{(1 − η )β (l + 1 )(1 − θ )T } h + hα (1 − η )(l + 1 )(1 − θ )T
For kT ≤ t < (k + θ )T ,
this contradicts the second inequality of (A9). So, (A8) holds. Then, we prove that for t ∈ [T + θ T , 2T )
D+ H1 (t3 ) ≥ 0,
1 M0 exp{(1 − η )β (t − (l + 1 )θ T )} h + hα (1 − η )[t − (l + 1 )θ T ]
Then, together with (A1), for any t ∈ [−τ , kT ),
W (t ) <
D+ Q1 (t2 ) = (1 − η )V −η (t2 )D+V (t2 ) + hα (1 − η ) ≤ −α (1 − η ) + hα (1 − η ) < 0,
H1 (t3 ) = 0,
Suppose that (A14) and (A15) are true for m ≤ k − 1, where k is a positive integer. Then, for any integer 0 ≤ l ≤ k − 1, if lT ≤ t < (l + θ )T ,
and if (l + θ )T ≤ t < (l + 1 )T ,
The same we can prove that for t ∈ [T , T + θ T )
< 0.
(A15)
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )T } h +hα (1 − η )(1 − θ )T , for t ∈ [−τ , T ).
T ≤ t < t2 .
1 M0 exp{(1 − η )β (t − (m + 1 )θ T )} h +hα (1 − η )(t − (m + 1 )θ T ).
1 M0 exp{(1 − η )β (1 − θ )lT } + hα (1 − η )(1 − θ )lT h 1 < M0 exp{(1 − η )β (1 − θ )kT } + hα (1 − η )(1 − θ )kT , h
So, we have
Q1 (t ) < 0,
1 M0 exp{(1 − η )β (1 − θ )mT } + hα (1 − η )(1 − θ )mT , h (A14)
For (m + θ )T ≤ t < (m + 1 )T ,
On the other hand, for t ∈ [−τ , θ T )
Q1 (t2 ) = 0,
1 M0 exp{(1 − η )β (t − 2θ T )} + hα (1 − η )(t − 2θ T ). h
In the following, we can get the estimation of W(t) for any integer m. For mT ≤ t < (m + θ )T ,
W (t ) <
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )T } + hα (1 − η )(1 − θ )T , h
(A12) (A13)
1 M0 exp{(1 − η )β [t − (k + 1 )θ T ]} h +hα (1 − η )[t − (k + 1 )θ T ].
Hence, by induction, the inequalities (A14) and (A15) can be obtained for any positive integer n. t For nT ≤ t < (n + θ )T , one has n < , then T
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )t } + hα (1 − η )(1 − θ )t. h
For (n + θ )T ≤ t < (n + 1 )T , one has n + 1 >
t , then T
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
W (t ) <
1 M0 exp{(1 − η )β (1 − θ )t } + hα (1 − η )(1 − θ )t. h
Let h → 1, one can obtain V 1−η (t ) < M0 exp{(1 − η )β (1 − θ )t } − α (1 − η )θ t = ( sup V (s ))1−η exp{(1 − η )β (1 − θ )t } − α (1 − η )θ t, −τ ≤s≤0
t ≥ 0.
Next, Let
( sup V (s ))1−η exp{(1 − η )β (1 − θ )t } − α (1 − η )θ t = 0. (A17) −τ ≤s≤0
Then discuss the value of t, we can easy to check that l1 = α ( 1 − η )θ t and l2 = exp{(1 − η )β (1 − θ )t } have the same ( sup V (s ))1−η −τ ≤s≤0
Remark 3. The proof method of Lemma 3 is different from the proof technique in the previous works. The Lemma 3, which improves and generalizes the previous conclusions [30,31], plays an important role in the finite-time synchronization analysis of dynamical networks via intermittent control in this paper, because it shows application of finite-time intermittent control. Lemma 4. ([20]). If there exists a regular positive definite and radially unbounded function V: Rn → R such that any solution x(t) of (1) satisfies the inequality
V˙ (x(t )) ≤ −aV δ (x(t )) − bV θ (x(t )),
1
β (1 − θ )(1 − η )
ln
αθ
β (1 − θ )( sup V (s ))
. 1 −η
−τ ≤s≤0
The function values of l1 and l2 at this point are
yl1 =
αθ αθ ln , β (1 − θ )( sup V (s ))1−η β (1 − θ )( sup V (s ))1−η −τ ≤s≤0
−τ ≤s≤0
and
yl2 =
αθ
β (1 − θ )( sup V (s ))1−η
.
−τ ≤s≤0
(1) If yl1 < yl2 , there is no solution of Eq. (A17). For the case of no solution, we don’t discuss it in detail. (2)
If
yl1 ≥ yl2 ,
i.e.,
ln
αθ ≥ 1, β (1 − θ )( sup V (s ))1−η −τ ≤s≤0
Eq. (A17) having solutions
t ∗ = T1 =
1
β (1 − θ )(1 − η )
as ln
αθ = 1, β (1 − θ )( sup V (s ))1−η −τ ≤s≤0
Tmax ≤
αθ
> 1, β (1 − θ )( sup V (s ))1−η −τ ≤s≤0
where T2 is the smallest solution of Eq. (A17). 1 . When t ∗ = T1 , yl1 and yl2 have only one intersection point, one can easy to see that
V (t ) → 0 as t → T1 and V (t ) ≡ 0 as t > T1 .
2 . When t ∗ = T2 , from the first inequality of ( ), V(t) is monotone decreasing for t ∈ [l T , l T + θ T ]. V (t ) → 0 as t ∈ [lT , T2 ] and t → T2 , V (t ) ≡ 0 as t ∈ (T2 , lT + θ T ]. From the second inequality of ( ) and the continuity of V(t), one has
V (t ) → 0 as t ∈ [lT , T2 ] and t → T2 , V (t ) ≡ 0 as t ∈ (T2 , (l + 1 )T ). and for all t ∈ [(l + 1 )T , (l + 1 )T + θ T ], V(t) ≡ 0. To summarize, for all t ∈ [0, +∞ ), when t → t∗ , V(t) → 0 and when t > t∗ , V(t) ≡ 0. The proof is completed.
1
δ−1
).
Lemma 5. ([43]). Assume that V: Rn → R is a regular, positive definite function, x(t ) : [0, +∞ ) → Rn is absolutely continuous . If there exists a continuous function χ : (0, +∞ ) → R, with χ (ι) > 0 for ι ∈ V (0 ) 1 (0, +∞ ), such that V˙ (x(t )) ≤ −χ (V (x(t ))) and ( )d ι = T , 0
χ (ι )
T < +∞, then we have V (x(t )) = 0 for all t ≤ T. In particular, (1) If χ (ι ) = M1 ι + M2 ιμ for all ι ∈ (0, +∞ ), where 0 < μ < 1, M1 , M2 are positive constants. The settling time can be estimated by
T ≤
1 M 1 V 1 −μ ( 0 ) + M 2 ln . M1 ( 1 − μ ) M2
(2) If χ (ι ) = M2 ιμ for all ι ∈ (0, +∞ ), where 0 < μ < 1, M2 > 0, then the settling time can be estimated by
T ≤
V 1 −μ ( 0 ) . M2 ( 1 − μ )
3. Main result 3.1. Finite-time synchronization via periodically intermittent control In this section, we consider the finite-time synchronization of general CDNs via periodic intermittent control. Based on Lemma 3, some novel finite-time synchronization criteria are obtained. The main results are stated as follows. Theorem 1. Suppose that Assumption 1 holds, systems (1) and (2) are synchronized in finite-time under the periodic intermittent controller (3), if there exist constants α , β , λ and 0 < μ < 1 such that the following conditions hold:
λIN − cγ j A ≥ 0,
V (t ) ≡ 0 for all t ∈ (lT + θ T , (l + 1 )T ). (b) Assume that T2 ∈ (lT + θ T , (l + 1 )T ). From ( ) and the continuity of V(t), we can also obtain
1 b 1−θ 1 ( ) δ−θ ( + b a 1−θ
Remark 4. Many fixed-time synchronization articles ([32,33]) are mostly based on the fixed-time stability theory proposed in [18]. The Lemma 4 in this paper is proposed by Hu et al. Compared with the estimation bound of the settling time given in [18], the results obtained in [20] were more effective and more accurate, and the detailed theoretical evidence is given in [20].
or
t ∗ = T2 as ln
x(t ) ∈ Rn {0},
where a, b > 0, δ > 1, 0 ≤ θ < 1, then the origin of system (1) is fixedtime stable and the setting time function
slope at
t=
295
(5)
2(
ξ λ − hj + β − )IN − cγ j A ≥ 0, λmax (P ) 1−τ
(6)
ln
αθ ≥ 1, 1−μ β (1 − θ )( sup V (s )) 2
(7)
−τ ≤s≤0
where j = 1, 2, · · · , N. The setting time can be estimated as
t ∗ = T1 ≤
1
β (1 − μ )(1 − θ )
,
or
t ∗ ≤ T2 ,
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J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
where T2 is the smallest solution of
( sup V (s ))
1−μ 2
exp{β (1 − μ )(1 − θ )t } − α (1 − μ )θ t = 0.
−τ ≤s≤0
N
λ
eTi (t )Pei (t ) +
N
1−τ
i=1
t
t −τ (t )
i=1
= (8)
D V (t ) ≤ 2
N
+c
−2α (
(t )P f (yi (t )) − f (xi (t )) − Hei (t ) + Hei (t )
−β ei (t ) − α ( P
−1
λ
i=1
t −τ (t )
eTi (s )Pei (s )ds )
ei (t ) ||e(t )||2
(λmax (P )) −α λmin (P )
sign(ei (t ))|ei (t )|
μ
+
1+μ 2
1−τ
N
λ
1−τ
eTi (t − τ (t ))Pei (t − τ (t )).
N n
ξ
λmax (P )
Pei (s )ds )
−2α ≤
1−τ
p j e2i j (t ) + 2
i=1 j=1
N n
p j h j e2i j (t ) + 2c
i=1 j=1
t
t −τ (t )
i=1
eTi (s )Pei (s )ds ) 1+μ 2
1+μ 2
(9) sign(ei (t ))|ei (t )|μ .
ei j (t )
≤ =
β p j e2i j (t ) − 2α
i=1 j=1
N
λ
1−τ
N
eTi (t )P
i=1
N n
1−τ
p j e2i j (t ) − λ
i=1 j=1
−2
j=1
i=1
1+μ 2
−2α (λmax (P ))
1+μ 2
N n
−2α (
t
t −τ (t )
eTi (s )
1+μ 2
(λmax (P )) λmin (P )
−2α (
(
N
λ
1−τ N
i=1
(10)
|ei j (t )|2 ) 1+μ 2
1+μ 2
.
+
N
t
t −τ (t )
eTi (s )Pei (s )ds )
λmax (P )eTi (t )ei (t ))
i=1
p j e2i j (t − τ (t ))
λ
N
1−τ
t −τ (t )
i=1
eTi (t )Pei (t ))
t
1+μ 2
1+μ 2
eTi (s )Pei (s )ds
1+μ 2
i=1 1+μ 2
= −2αV
(t ).
Similarly, when lT + θ T < t < (l + 1 )T , one obtains that
(t − τ (t )) p j γ j Ae˜ j (t − τ (t )) − 2β (t ) p j e˜ j (t ) λ T + e˜ (t ) p j e˜ j (t ) 1−τ j λ −λe˜Tj (t − τ (t )) p j e˜ j (t − τ (t )) − 2α ( 1−τ t −τ (t )
i=1 j=1 N n
λmax (P )eTi (t )ei (t ))
≤ −2α (
e˜Tj
eTi (s )Pei (s )ds )
N
D+V (t ) ≤ −2α (
ξ e˜T (t ) p j e˜ j (t ) + 2e˜Tj (t ) p j h j e˜ j (t ) λmax (P ) j
t
|ei j (t )|1+μ
Substituting (10) into (9), it follows that
i=1 j=1
+ce˜Tj
N
N n
sign(ei (t ))|ei (t )|μ
i=1
(t ) p j γ j Ae˜ j (t )
+ce˜Tj
1+μ 2
(λmax (P )) λmin (P )
eTi (t )P
i=1 j=1
1+μ 2
N n
λ
N
−2α (λmax (P ))
sign(ei (t ))|ei (t )|μ
i=1
N
λ
i=1
×p j aik γ j ek j (t − τ (t )) − 2α
≤
sign(ei (t ))|ei (t )|μ .
It follows from Lemma 1 that N
i=1 k=1 j=1
n
1+μ 2
(λmax (P )) −2α eTi (t )P λmin (P ) i=1
eTi (t )Pei (t )
N N n
+
1+μ 2
(λmax (P )) λmin (P )
N
Based on Assumption 1, one can obtain
−2
eTi (s )Pei (s )ds )
where e˜ j (t ) = [e˜ j1 (t ), e˜ j2 (t ), · · · , e˜ jN (t )]T is a column vector of ej (t). According to the conditions (5) and (6) of Theorem 1, one derives that
i=1
D+V (t ) ≤ −2
i=1
eTi (t )P
t
t −τ (t )
D+V (t ) ≤ −2α (
i=1
−λ
N
λ
i=1
N t
1−τ
1+μ 2
N
−2α
aik ek (t − τ (t ))
k=1
λ
i=1 N
ξ I + 2h j IN + cγ j A − 2β IN λmax (P ) N
e˜Tj (t ) p j − 2
j=1
eTi
sign(ei (t ))|ei (t )|μ
IN e˜ j (t ) 1−τ n + e˜Tj (t − τ (t )) p j cγ j A − λIN e˜ j (t − τ (t )) +
Then the right upper Dini derivative of V(t) with respect to time t along the solutions of (4) can be calculated as follows. When lT ≤ t ≤ lT + θ T , +
n j=1
eTi (s )Pei (s )ds.
1+μ 2
(λmax (P )) λmin (P )
eTi (t )P
i=1
Proof. Construct the following Lyapunov functional
V (t ) =
N
−2α
1+μ 2
D+V (t ) ≤
n
−2
j=1
ξ e˜T (t ) p j e˜ j (t ) + 2e˜Tj (t ) p j h j e˜ j (t ) λmax (P ) j
+ce˜Tj (t ) p j γ j Ae˜ j (t ) +ce˜Tj (t − τ (t )) p j γ j Ae˜ j (t − τ (t )) − 2β e˜Tj (t ) p j e˜ j (t ) +2β
n j=1
e˜Tj (t ) p j e˜ j (t )
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
+
n
λ
1−τ
j=1
e˜Tj (t ) p j e˜ j (t ) −
n
λe˜Tj
j=1
(t − τ (t )) p j e˜ j (t − τ (t )) n ξ = e˜Tj (t ) p j − 2 IN + 2h j IN + cγ j A − 2β IN λ max (P ) j=1 +
λ
IN e˜ j (t )
1−τ n + e˜Tj (t − τ (t )) p j cγ j A − λIN e˜ j (t − τ (t ))
where j = 1, 2, · · · , n. The setting time can be estimated as
t2∗ ≤
+2β
Consider the following form of impulsive differential equation as the response CDNs:
y˙ i (t ) = f (yi (t )) + c
e˜Tj (t ) p j e˜ j (t ) = 2β
N
eTi (t )Pei (t ) ≤ 2β V (t ).
i=1
j=1
Namely, one has
1+μ
D+V (t ) ≤ −2αV 2 (t ), lT ≤ t ≤ lT + θ T , D+V (t ) ≤ 2β V (t ), lT + θ T < t < (l + 1 )T .
It can be derived from Lemma 3 that
V
1−μ 2
(t ) ≤ ( sup V (s ))
1−μ 2
−τ ≤s≤0
exp{β (1 − μ )(1 − θ )t } − α (1 − μ )θ t.
This implies that the systems (1) and (2) are finite-time synchronized. The finite-time t∗ can be estimated based on the condition (7) of Theorem 1. The proof of Theorem 1 is completed. If τ (t ) = 0, the system (1) becomes a general complex network, then, we can derive the following Corollary. Corollary 1. Assume that Assumption 1 holds, systems (1) and (2) are synchronization in finite-time under the controller (11), if there exist control parameters α , β and 0 < μ < 1 such that
(
ξ
λmax (P )
ln
i = 1, 2, · · · , N, t = tk . yi = yi (tk+ ) − yi (tk− ) = Bik ei ,
− h j + β )IN − cγ j A ≥ 0,
1
,
or
t1∗ ≤ T˜2 ,
where T˜2 is the smallest solution of
V
1−μ 2
(0 ) exp{β (1 − μ )(1 − θ )t } − α (1 − μ )θ t = 0.
Remark 5. In particular, when θ = 1, the periodic intermittent control problem, essentially translates into continuous control problem, which has made a lot of results [44,45]. In view of the condition (2) of Lemma 5 and Theorem 1, the following Corollary can be obtained via continuous control. Corollary 2. Under Assumption 1, systems (1) and (2) are synchronized in finite-time, if there exist control parameters α , β , λ and 0 < μ < 1 such that
λIN − cγ j A ≥ 0, 2(
= limt →t + yi (t ), k
t = tk , k ∈ . yi (tk− )
= yi (tk ),
(11) =
Then the following Corollary can be obtained.
Corollary 3. Under Assumption 1, systems (1) and (12) are synchronized in finite-time under the controller (13), if there exist control parameters α , β , λ and 0 < μ < 1 such that
λIN − cγ j A ≥ 0, 2(
ξ λ − hj + β − )IN − cγ j A ≥ 0, λmax (P ) 1−τ
βk = λmax [(IN + Bik )T (IN + Bik )] ≤ 1, where j = 1, 2, · · · , n. The setting time can be estimated as 1−μ 2
(0 ) . α (1 − μ ) V
3.2. Finite-time and fixed-time synchronization based on the same control scheme
The setting time can be estimated by
β (1 − μ )(1 − θ )
yi (tk+ )
Remark 6. When θ → 1− , the controllers (4) become a special impulsive control scheme: ⎧ N t 1+μ λ ei (t ) ⎪ ⎪ −β ei (t ) − α ( eTi (s )Pei (s )ds ) 2 P −1 ⎪ ⎪ 1−τ || e(t )||2 ⎨ t −τ (t ) i=1 1+μ Ui (t ) = (12) (λmax (P )) 2 ⎪ μ ⎪ t = tk , ||ei (t )|| = 0 ⎪−α λ (P ) sign(ei (t ))|ei (t )| , ⎪ ⎩ min 0, t = tk , ||ei (t )|| = 0.
t3∗ ≤
αθ ≥ 1. 1−μ β (1 − θ )V 2 (0 )
t1∗ = T˜1 ≤
ai j y j (t − τ (t )) + Ui (t ),
{1, 2, · · · , m1 , m2 , · · · , mk } is a finite natural number set.
It follows from the conditions (5) and (6) of Theorem 1 that n
N j=1
where
e˜Tj (t ) p j e˜ j (t ).
j=1
D+V (t ) ≤ 2β
1−μ 2
(0 ) . α (1 − μ ) V
j=1 n
297
ξ λ − hj + β − )IN − cγ j A ≥ 0, λmax (P ) 1−τ
In this section, we will give some novel finite-time and fixedtime synchronization criteria for CDNs (1) by designing a new unified control scheme. In order to obtain the main results, the controllers are designed as follows:
Ui (t ) = −β ei (t ) − λ
N
ek (t − τ (t )) − ρ1
k=1
(λmax (P )) λmin (P )
1+ε 2
sign(ei (t ))|ei (t )|ε −ρ2
1+ω 2
(λmax (P )) λmin (P )
sign(ei (t ))|ei (t )|ω ,
(13)
where i = 1, 2, · · · , N, 0 < ε < 1, β , λ, ρ 1 , ρ 2 are positive constants. Remark 7. In [32,46], to realize fixed-time synchronization, the controller designed by the authors contains two power exponent items. Based on the proposed method, we also have added two power exponent items into controller. However, the difference between the controllers we design is that we can just by adjusting the power exponent (0 < ω < 1, ω = 1, ω > 1) to achieve finite-time and fixed-time synchronization.
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Based on controller (13), the error system can be rewritten as N
ei (t ) = f (yi (t )) − f (xi (t )) + c
ai j e j (t − τ (t )) + Ui (t ).
sign(ei (t ))|ei (t )|ε
(14) − ρ2
j=1
Theorem 2. Suppose that Assumption 1 holds, under the controller (13), if there exist control parameters β , λ > 0 such that
λIN − cγ j A ≥ 0, (
ξ
λmax (P )
(15)
− h j + β )IN ≥ 0,
(16)
where j = 1, 2, · · · , N. Then, the following conclusions hold. (1) When 0 ≤ ω ≤ 1, systems (1) and (2) are finite-time synchronization. The settling time can be estimated by
⎧ 1−η ⎨ Vρ (1−(η0)) , 0 ≤ ω < 1, 1−ε t1 ≤ 2 ⎩ ρ (12−ε ) ln ρ2V ρ1(0)+ρ1 , ω = 1,
max{ 1+2 ε , 1+2ω }, 0 ≤ V (t ) ≤ 1, min{ 1+2 ε , 1+2ω }, V (t ) > 1.
1 ρ1 1−ε 2 2 ( ) ω−ε ( + ), ρ1 ρˆ2 1−ε ω−1 2
j=1
+
n
V (t ) =
eTi
j=1
− ρ1
N
.
−ρ2
aik ek (t − τ (t )) ek (t − τ (t )) − ρ1
k=1
(λmax (P )) λmin (P )
sign(ei (t ))|ei (t )|ε
≤−
+
1+ω 2
(λmax (P )) λmin (P )
ξ
λmax (P )
N n
sign(ei (t ))|ei (t )|ω
N n
N N n
×p j aik γ j ek j (t − τ (t )) − N N n
λmax (P )eTi (t )ei (t ))
1+ε 2
λmax (P )eTi (t )ei (t ))
1+ω 2
1+ε 2
(t ) − ρ2V
1+ω 2
(t ).
Then, from the condition (2) of Lemma 5, which shows that systems (1) and (2) are synchronized in finite-time t1 . When ω = 1, N
λmax (P )eTi (t )ei (t ))
i=1
−ρ2 (nN )− 2 ( 1
N
1+ε 2
N
λmax (P )eTi (t )ei (t ))
i=1
(t ) − ρ2 V (t ).
λmax (P )eTi (t )ei (t )) ω
−ρ2 (nN )− 2 (
N
λmax (P )eTi (t )ei (t ))
≤ −ρ1 (
N
eTi (t )Pei (t ))
1+ε 2
i=1 k=1 j=1
i=1
(λmax (P )) λmin (P )
1+ω 2
1+ε 2
1+ε 2
ω
− ρ2 (nN )− 2 (
i=1
eTi (t )P
1+ε 2
i=1
β p j e2i j (t )
= −ρ1V N
1+ε 2
i=1
ei j (t )
ei j (t )
×p j ek j (t − τ (t )) − ρ1
sign(ei (t ))|ei (t )|ω .
V˙ (t ) ≤ −ρV η (t ).
V˙ (t ) ≤ −ρ1 (
i=1 j=1
−λ
1+ω 2
sign(ei (t ))|ei (t )|ε
From the condition (1) of Lemma 5, one can easy to see that systems (1) and (2) are synchronized in finite-time t2 . When ω > 1,
i=1 k=1 j=1 N n
(λmax (P )) λmin (P )
i=1
≤ −ρ1V
p j e2i j (t )
i=1 j=1
eTi (t )P
N
1+ε 2
i=1 j=1
p j h j e2i j (t ) + c
sign(ei (t ))|ei (t )|ω ,
Based on Lemma 2, one derives that
V˙ (t ) ≤ −ρ1 (
k=1
−ρ2
(λmax (P )) λmin (P )
N
≤ −ρ1V
N
1+ε 2
eTi (t )P
i=1 N
−ρ2 (
eTi (t )P f (yi (t )) − f (xi (t )) − Hei (t ) + Hei (t )
−β ei (t ) − λ
1+ω 2
i=1
(t )Pei (t ).
N
(λmax (P )) λmin (P )
N
V˙ (t ) ≤ −ρ1 (
i=1
+c
eTi (t )P
sign(ei (t ))|ei (t )|ε
When 0 ≤ ω < 1,
By differentiating V(t) along the trajectories of (14) and based on Assumption 1, one has
V˙ (t ) =
(λmax (P )) λmin (P )
i=1 N
1+ε 2
eTi (t )P
i=1
i=1
N
e˜Tj (t ) p j cγ j A − λIN e˜ j (t − τ (t ))
Proof. Construct the following Lyapunov function N
ξ I + h j IN − β IN e˜ j (t ) λmax (P ) N
e˜Tj (t ) p j −
V˙ (t ) ≤ −ρ1
(2) When ω > 1, systems (1) and (2) are fixed-time synchronization. The settling time can be estimated by
where ρˆ2 = ρ2 (nN )
sign(ei (t ))|ei (t )|ω
where e˜ j (t ) = [e˜ j1 (t ), e˜ j2 (t ), · · · , e˜ jN (t )]T is a column vector of ej (t). By virtue of the second inequality of Lemma 1 and the conditions (15) and (16) of Theorem 2, it follows that
1
−ω
=
i=1
where ρ = min{ρ1 , ρ2 }, ρ2 = ρ2 (nN )− 2 and
Tmax ≤
1+ω 2
(λmax (P )) λmin (P )
eTi (t )P
i=1 n
− ρ2
2
η=
N
N
eTi (t )Pei (t ))
1+ω 2
i=1
(t ) − ρˆ2V
1+ω 2
(t ).
Hence, from Lemma 4, the fixed-time synchronization of systems (1) and (2) are achieved.
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
The proof of Theorem 1 is completed.
If τ (t ) = 0, The controller (13) is simplified into the following form: 1+ε
(λmax (P )) 2 Ui (t ) = −β ei (t ) − ρ1 sign(ei (t ))|ei (t )|ε λ min (P ) 1+ω (P )) 2 ω −ρ2 (λmax λmin (P ) sign(ei (t ))|ei (t )| ,
λmax (P )
− h j + β )IN ≥ 0,
(18)
where j = 1, 2, · · · , N. Then, the following conclusions hold. (1) When 0 ≤ ω ≤ 1, systems (1) and (2) are finite-time synchronization. The settling time can be estimated by
V 1−η (0)
t2 ≤
0 ≤ ω < 1,
ρ ( 1 −η ) ,
2 ρ2 (1−ε )
ln
ρ2 V
1−ε 2
( 0 )+ ρ 1
ρ1
ω = 1,
1 ρ1 1−ε 2 2 ( ) ω−ε ( + ). ρ1 ρˆ2 1−ε ω−1
4. Numerical examples In this section, two examples are presented to demonstrate the validity and effectiveness of our proposed theoretical results for finite-time synchronization via periodically intermittent control and feedback control. Example 1. Consider a network with 10 nodes and each node is a simple three dimensional nonlinear system described as:
x˙ i (t ) = f (xi (t )) + c
10
ai j x j (t − τ (t )),
i = 1, 2, · · · , 10,
(19)
j=1
where xi (t ) = (xi1 (t ), xi2 (t ), xi3 (t ))T , and
f (xi (t )) =
−10 28 0
10 −1 0
0 0 − 83
xi1 (t ) xi2 (t ) xi3 (t )
+
0 −xi1 xi3 . xi1 xi2
The single Lorenz system has a chaotic attractor(see Fig. 1.). Taking P = diag(0.5, 0.4, 0.2 ) and H = diag(50, 50, 50 ) such that the Lorenz system satisfies Assumption 1 [52]. = diag(1, 1, 1 ) , 1 et c = 1, τ (t ) = 1+ and τ = . A = (ai j )10×10 is a symmetrically difet 4 fusive coupling matrix given by
⎛
−2 ⎜1 ⎜1 ⎜ ⎜0 ⎜ ⎜0 A=⎜ ⎜0 ⎜0 ⎜ ⎜0 ⎝0 0
1 −4 0 1 1 0 0 0 1 0
1 0 −3 1 0 1 0 0 0 0
0 1 1 −3 0 0 1 0 0 0
10
ai j y j (t − τ (t )) + Ui (t ),
i = 1, 2, · · · , 10, (20)
For the sake of simplicity of computation, the initial conditions of each nodes are chosen as follows: xi (0 ) = (3 + 0.3i, 0.2 + 0.1i, 0.3 + 0.2i )T , yi (0 ) = (−2 + 0.5i, −3 + 0.3i, −4 + 0.6i )T , i = 1, 2, · · · , 10. The simulation results with control input are calculated by different control scheme as follows: (a) Finite-time synchronization with periodic intermittent control
⎧ 10 ⎪ 8 t ei (t ) 3 ⎪ ⎪ −ei (t ) − 8( eTi (s )Pei (s )) 4 P −1 ⎪ ⎨ 3 ||e(t )||2 t −τ (t ) i=1
Ui (t )=
(λmax (P )) 4 1 ⎪ −8 sign(ei (t ))|ei (t )| 2 , lT ≤ t ≤ lt + θ T , ⎪ ⎪ λmin (P ) ⎪ ⎩ 0, lT + θ T < t < (l + 1 )T , 3
(21) ,
(2) When ω > 1, systems (1) and (2) are fixed-time synchronization. The settling time can be estimated by 1 Tmax ≤
y˙ i (t ) = f (yi (t )) + c
(17)
Corollary 4. Suppose that Assumption 1 holds, under the controller (17), if there exist control parameters β > 0 such that
ξ
Consider Eq. (19) as the drive system, the response system described as
j=1
where i = 1, 2, · · · , N, 0 < ε < 1, β , ρ 1 , ρ 2 are positive constants. Then, the following Corollary holds.
(
299
0 1 0 0 −4 0 1 1 1 0
0 0 1 0 0 −3 1 0 0 1
0 0 0 1 1 1 −5 1 0 1
0 0 0 0 1 0 1 −4 1 1
0 1 0 0 1 0 0 1 −3 0
⎞
0 0⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟. 1⎟ 1⎟ ⎟ 1⎟ ⎠ 0 −3
Then under the above parameters, system (19) have a chaotic attractor (see Fig 2.)
1 . By computing, sup V (s ) = 2 −1≤s≤0 252.28. Further, select T = 2, θ = 0.8, under the above conditions, the assumptions of Theorem 1 are satisfied, then systems (19) and (20) are synchronized in finite-time t ∗ = 1.4381 and the synchronization errors are shown in Fig. 3. (b) Finite-time and fixed-time synchronization with state feedback control where β = 1, α = 8, λ = 2, μ =
Ui (t ) = −ei (t ) − 0.01
10
λmax (P ) 4 λmin (P )
3
ek (t − τ (t )) − 9
k=1
sign(ei (t ))|ei (t )| 2
1
1+ω
−9
λmax (P ) 2 sign(ei (t ))|ei (t )|ω λmin (P )
(22)
1 . It is easy to check 2 that the conditions of Theorem 2 are satisfied. The key parameter ω in the controller (22) is chosen as follows: 1. Choosing ω = 0.15, from the result (1) of Theorem 2, one has the systems (19) and (20) are synchronized in finite-time t1 = 1.9133s (see Fig. 4). 2. Choosing ω = 1, from the result (2) of Theorem 2, systems (19) and (20) are synchronized in finite-time t2 = 1.1269s (see Fig. 5). 3. Choosing ω = 1.5, from the result (3) of Theorem 2, systems (19) and (20) are synchronized in fixed-time Tmax = 3.1825s (see Fig. 6). where β = 1, λ = 0.01, ρ1 = ρ2 = 9, ε =
Example 2. To verify the generality of our results, we consider the following CDNs without time-varying delays:
x˙ i (t ) = f (xi (t )) + c
10
ai j x j (t ),
i = 1, 2, · · · , 10,
(23)
j=1
assume that the parameters in (23) are the same as in Example 1 and the corresponding chaotic behavior is shown in Fig. 7. The corresponding response system is described as:
y˙ i (t ) = f (yi (t )) + c
10
ai j y j (t ) + Ui (t ),
i = 1, 2, · · · , 10.
j=1
(24) The controllers are designed as the following two forms:
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J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
Fig. 1. The chaotic behavior of Lorenz system.
Fig. 2. The chaotic behavior of system (19).
(a ) Periodic intermittent control
⎧ 3 ⎪ ⎨−ei (t ) − 8 (λmax (P )) 4 sign(ei (t ))|ei (t )| 21 , λmin (P ) Ui (t ) = lT ≤ t ≤ lt + θ T , ⎪ ⎩ 0, lT + θ T < t < (l + 1 )T . (25)
(b ) State feedback control
λmax (P ) 4 1 sign(ei (t ))|ei (t )| 2 λmin (P ) 1+ω λmax (P ) 2 −9 sign(ei (t ))|ei (t )|ω . λmin (P ) 3
102.118. Similarly, we choose T = 2, θ = 0.8. Through verification, the assumptions of Corollary 1 are satisfied, the systems (23) and (24) are synchronized in finite-time t1∗ = 1.11s based on controller (25) and the corresponding synchronization errors are shown in Fig. 8. Based on control scheme (26), we can verify the conditions of Corollary 2 hold. By simple computation, the setting time can be estimated as t21 = 1.8674s when ω = 0.15, t22 = 1.1142s when ω = 1 1 and Tmax = 3.1825s when ω = 1.5. From Figs. 9–11, we can see that the systems (23) and (24) are synchronized in finite time or fixed time.
Ui (t ) = −ei (t ) − 9
(26)
The values of the parameters of control schemes (25) and (26) are the same as in (21) and (22). By simple computing, V (0 ) =
Remark 8. By computing, we have t ∗ > t1∗ , which shows that time delays have a negative effect on the estimation of the setting time. Moreover, from Fig. 3 and Fig. 8, it can be seen that time delays have a positive effect on the synchronization of the system. This is in accordance with the controllers (21) and (25).
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
Fig. 3. The Finite-time synchronization of systems (19) and (20) under (21) with (T, θ , μ)=(2,0.8,0.5).
Fig. 4. The Finite-time synchronization of systems (19) and (20) under (22) with (λ, ε , ω)=(10,0.5,0.15).
Fig. 5. The Finite-time synchronization of systems (19) and (20) under (22) with (λ, ε , ω)=(10,0.5,1).
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Fig. 6. The Fixed-time synchronization of systems (19) and (20) under (22) with (λ, ε , ω)=(10,0.5,1.5).
Fig. 7. The chaotic behavior of system (23).
Remark 9. By simple computing and from Figs. 4–6, Figs. 9–11, 1 , which show that the setwe have t11 > t21 , t12 > t22 , Tmax = Tmax ting time function depend on the initial conditions in estimating the finite-time synchronization. However, the setting time in estimating the fixed-time synchronization is not affected by the initial conditions. Additionally, we can conclude that under the controller (26), time delays have a negative effect both on the estimation of the settling time and the convergence rate of systems. Remark 10. From Remarks 8 and 9, we know that by constructing different controllers, time delays can accelerate or restrain the convergence rate of the system. From Figs. 4–6 and Figs. 9–11, we conclude that by regulating the main control parameters, not only both the finite-time and fixed-time synchronization can be achieved, but also the convergence rate of synchronization can be adjusted. It is of engineering interest to develop the efficiency of the controller by adjusting the appropriate parameters. Remark 11. In [47–49], the authors considered the stability and control issues of nonlinear dynamic systems with impulse or
stochastic perturbations. Finite-time and fixed-time synchronization of complex networks with time delays are considered in this paper. In fact, the issues about the finite-time and fixed-time synchronization for complex networks with impulse or stochastic perturbations are of great significance in practical applications, which are our direction for future work. 5. Conclusion In this paper, the finite-time and fixed-time synchronization for a class of CDNs with time-varying coupled delays are investigated by means of periodically intermittent control and feedback control. To derive the main results, a new inequality is proposed and proved. The finite time stability in this paper is not only improved the proof method, but also extended the conclusions in the previous literature. Based on the Lyapunov functionals, inequality techniques, finite-time stability theory, some new easy-verified conditions are established to ensure synchronization for the target CDNs within a settling time. Finally, numerical simulations are given to
J. Li et al. / Chaos, Solitons and Fractals 114 (2018) 291–305
Fig. 8. The Finite-time synchronization of systems (23) and (24) under (25) with (T, θ , μ)=(2,0.8,0.5).
Fig. 9. The Finite-time synchronization of systems (23) and (24) under (26) with (λ, ε , ω)=(10,0.5,0.15).
Fig. 10. The Finite-time synchronization of systems (23) and (24) under (26) with (λ, ε , ω)=(10,0.5,1).
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Fig. 11. The Fixed-time synchronization of systems (19) and (20) under (22) with (λ, ε , ω)=(10,0.5,1.5).
show the effectiveness and validity of the theoretical results. For the complex networks in this paper, impulse or stochastic perturbations have not been considered. It would be of interest to investigate the finite-time and fixed-time synchronization of delayed complex networks with impulsive effects or stochastic perturbations in the near future.
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