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Finite-time synchronization of delayed dynamical networks via aperiodically intermittent control
Accepted Manuscript
Finite-time synchronization of delayed dynamical networks via aperiodically intermittent control Mei Liu, Haijun Jiang, Cheng Hu PII: DOI: Reference:
S0016-0032(17)30265-X 10.1016/j.jfranklin.2017.05.030 FI 3004
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
16 November 2016 25 March 2017 20 May 2017
Please cite this article as: Mei Liu, Haijun Jiang, Cheng Hu, Finite-time synchronization of delayed dynamical networks via aperiodically intermittent control, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.05.030
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Finite-time synchronization of delayed dynamical networks via
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aperiodically intermittent control Mei Liu, Haijun Jiang∗ and Cheng Hu
College of Mathematics and System Sciences, Xinjiang University,
Abstract.
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Urumqi, 830046, Xinjiang, P.R. China
In this paper, we concern the finite-time synchronization problem for
delayed dynamical networks via aperiodically intermittent control. Compared with some correspondingly previous results, the intermittent control can be aperiodic
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which is more general. Moreover, by establishing a new differential inequality and constructing Lyapunov function, several useful criteria are derived analytically to
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realise finite-time synchronization for delay complex networks. Additionally, as a special case, some sufficient conditions ensuring the finite-time synchronization for a class of coupled neural network are obtained. It is worth noting that the convergence
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time is carefully discussed and does not depend on control widths or rest widths for the proposed aperiodically intermittent control. Finally, a numerical example is
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given to demonstrate the validness of the proposed scheme. Key words: Complex networks; Aperiodically intermittent control; Finite-time
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synchronization; Delay
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Introduction
Complex dynamical networks, which are everywhere in the real world, have drawn
more and more attention from various areas of science and engineering [1–8] because complex networks can describe a variety of real systems, such as ecosystems, the World Wide Web, genetic networks, biological neural networks, social networks and biomolecular ∗
E-mail: [email protected]; [email protected].
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networks [9–11]. As the major collective behavior of dynamical networks, synchronization has been extensively investigated recently attribute to the fact that it not only can explain many natural phenomena [12, 13] but also has a large quantity of promising potential applications in secure communication [14], image processing [15], and so on. Up to now, many different synchronization patterns including finite-time synchronization [3],
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complete synchronization [4], projective synchronization [7], cluster synchronization [8] have been introduced and widely studied. Finite-time synchronization of memristor-based Cohen-Grossberg neural networks with time-varying delays was investigated in [3]. Synchronization of hybrid-coupled delayed dynamical networks via aperiodically intermittent pinning control was considered in [4]. In [7], the authors considered function projective synchronization of impulsive neural networks with mixed time-varying delays. Cluster
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synchronization of non-linearly coupled Lur’e networks with identical and non-identical nodes and an asymmetrical coupling matrix was studied in [8]. As far as we all know, the dynamical network cannot realize synchronization by itself. Furthermore, for the purpose of driving the network to achieve synchronization, control inputs should be imposed on the nodes of dynamical network. Until now, many useful control methods have been focused on this topic, such as adaptive control [16, 17], feedback
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control [18, 19], intermittent control [20, 21] and impulsive control [22, 23]. Recently, discontinuous control approaches including intermittent control and impulsive control have
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attracted more and more interest naturally in a wide variety of applications because these control methods are more economic and can reduce the amount of the transmitted information. Intermittent control used for manufacturing, transportation and communication,
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was first introduced to control linear econometric models. In practice, intermittent con-
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trol is activated during certain nonzero time intervals and off during other time intervals. When the control interval time is reduced to a point, intermittent control become impulsive control. In other words, the impulsive control is activated only at some instants. Hence, when the states are observable, impulsive control is an effective approach. How-
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ever, when the states of controlled systems are unobservable, impulsive control seems to be invalid. Moreover, it may be more reasonable to achieve control process in some time intervals other than some time instants in real control application. Therefore, the intermittent controller is designed instead of the impulsive one in the paper. Nevertheless, the requirement of periodical intermittent control is quite restricted and may be unreasonable and unnecessary in practice. For example, the generation of wind power is obviously aperiodically intermittent. Therefore, for the real applications and the theoretical analysis, it is necessary to consider the synchronization problem under
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aperiodically intermittent control strategy. Based on the above discussion, we will investigate the following aperiodically intermittent control strategy (see Fig. 1), for any time span [tm , tm+1 ), t0 = 0, m = 0, 1, 2, ..., [tm , sm ] is the work time and (sm , tm+1 ) is the rest time. Moreover, sm − tm is called the mth control width, while tm+1 − sm is called the mth rest width, where tm and sm denote the start time and the end time of mth
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control, tm+1 denotes the end time of mth rest. Obviously, this control strategy is more general. Especially, when tm+1 − tm ≡ T, sm − tm ≡ δ, where T, δ are positive constants,
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t0 = 0, m = 0, 1, 2, ..., the intermittent control type becomes the periodic one, which has been investigated in [16, 20, 21].
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Fig. 1. Aperiodically intermittent control strategy.
Since a finite-time stability result is proposed for a continuous-time system, many extended results have been presented in continuous-time networks systems. One of the
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hot research topics is the finite-time synchronization of dynamical networks [3, 24–26].
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Finite-time synchronization of memristor-based Cohen-Grossberg neural networks with time-varying delays was developed in [3] with discontinuous systems. Periodically intermittent controlling for finite-time synchronization of complex dynamical networks was shown in [24]. The paper [25] studied finite-time synchronization of complex delayed
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networks via intermittent control with multiple switched periods. The recently paper [26] investigated finite-time synchronization of drive-response systems via periodically intermittent adaptive control. To the best of our knowledge, there are few results of the finite-time synchronization for dynamical networks via aperiodically intermittent control. Motivated by the above discussion, we will investigate finite-time synchronization of coupled dynamical networks with delay via aperiodically intermittent control. First, the FTS problem is presented for the first time via the aperiodic intermittent control approach. Second, in this paper, a new differential inequality is established to deal with the finitetime synchronization problem and our results are only dependent on the parameter Ψ 3
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in Definition 1, not the control width or the rest width. Third, by using the properties of fundamental functions, we give the value of finite time T . Forth, a kind of novel controller, state-feedback controller which contain integral term and delayed term, is proposed. Finally, aperiodically intermittent control strategy, which is a more general controller, is considered in the paper to realise synchronization. Based on those, finite-time
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synchronization criteria obtained in our paper are very useful and verifiable in application. The rest of this paper is organized as follows. In Section 2, model of dynamical networks with time-varying delays and preliminaries are given. Synchronization of the considered model under the aperiodically intermittent control is investigated based timevarying delay in Section 3. In Section 4, the effectiveness and feasibility of the developed methods are presented by a numerical example. Finally, some conclusions are obtained
k φ k= sup−τ ≤t≤0
N P
i=1
| φi (t) |2
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in Section 5. Notations: Let RN be the space of N -dimensional real column vectors. For x = N 1 P (x1 , · · · , xN )T ∈ RN , kxk denotes a vector norm defined by kxk = x2i 2 . We define i=1
, for all φ = (φ1 (t), φ2 (t), ..., φN (t))T ∈ C([−τ, 0], RN ),
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which denotes the Banach space of all continuous functions mapping [−τ, 0] → RN . The initial conditions of systems (1) and (4) are assumed to be xi (s) = ϕi (s), Π(s) = ψ(s), i ∈
I , respectively, s ∈ [−τ, 0], where ϕi (s), ψ(s) ∈ C([−τ, 0], RN ). IN is the identity matrix
Preliminaries
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2
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with N dimensions. λmax (A) is the maximum eigenvalues of matrix A.
In this section, we consider a general complex dynamical network consisting of N
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linearly identical nodes and each node is an n-dimensional dynamic system. The network
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is described by
x˙ i (t) = f (t, xi (t), xi (t − τ (t))) + c i ∈ I = {1, 2, ..., N },
N X
j=1,j6=i
bij Γ(xj (t) − xi (t)) + ui (t),
(1)
where xi (t) = (x1i (t), x2i (t), ..., xni (t))T ∈ Rn denotes the state variable of the ith node, f : R × Rn × Rn → Rn is a continuously vector-valued function governing the dynamics
of isolated nodes, the time-varying delays τ (t) may be unknown but are bounded by
known constants, i.e., 0 ≤ τ (t) ≤ τ , in which τ (t) represents the internal delay occurring
inside the dynamical node, ui (t) ∈ Rn is intermittent controller shown in Fig. 1, the 4
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positive constant c is the coupling strength, Γ = diag{γ1 , γ2 , ..., γn } is the inner connecting
matric with γi > 0, B = (bij )N ×N is the coupling matric defined to satisfy the following conditions:
bij ≥ 0, i 6= j, bii = −
N P
bij .
(2)
j=1,j6=i
x˙ i (t) = f (t, xi (t), xi (t − τ (t))) + c
N X
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Based on the condition (2), we can rewrite network model (1) as follows: bij Γxj (t) + ui (t),
j=1
i ∈ I.
(3)
In the case that network (3) reaches complete synchronization, that is, lim kxi (t) −
Π(t)k = 0, i ∈ I , we have the following synchronized state equation:
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˙ Π(t) = f (t, Π(t), Π(t − τ (t))),
i ∈ I.
t→∞
(4)
In order to obtain our main results, the following assumptions, lemmas and definitions are necessary: Assumption 1 [27]: For the vector-valued function f (t, x(t), x(t−τ (t))), suppose the uniform semi-Lipschitz condition with respect to the time t holds, i.e., for any x(t), y(t) ∈
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Rn , there exist positive constants l1 > 0 and l2 > 0 such that
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T x(t) − y(t) f (t, x(t), x(t − τ (t))) − f (t, y(t), y(t − τ (t))) T T ≤ l1 x(t) − y(t) x(t) − y(t) + l2 x(t − τ (t)) − y(t − τ (t)) x(t − τ (t)) − y(t − τ (t)) .
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Assumption 2 [27]: For the aperiodically intermittent control strategy, there exist
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two positive scalars 0 < θ < ω < +∞, such that, for m = 0, 1, 2, ..., inf (sm − tm ) = θ, m
sup(tm+1 − tm ) = ω.
(5)
m
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Assumption 3: Time-delay function τ (t) : [0, +∞) → [0, +∞) is real-valued continuous function and satisfies τ˙ (t) ≤ σ < 1.
Definition 1 [27]: For the aperiodically intermittent control, define Ψ = lim sup m→∞
tm+1 − sm . tm+1 − tm
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Obviously, 0 ≤ Ψ < 1, when Ψ = 0, the aperiodically intermittent control becomes
the continuous control.
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θ . ω
Lemma 1 [27]: If Assumption 2 holds, then Ψ ≤ 1 − Lemma 2 [27]: For any m = 0, 1, 2, ..., we denote Ψ(t) =
t − sm , t − tm
t ∈ [sm , tm+1 ].
(7)
tm+1 − sm . tm+1 − tm Definition 2: [25] Drive-response systems (3) and (4) are said to be finite-time syn-
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Then, Ψ(t) is an strictly increasing function and Ψ(t) ≤
chronized if, for a suitable designed feedback controller ui (t), there exists a constant T > 0 such that lim (Π(t) − xi (t)) = 0
t→T
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and Π(t) − xi (t) = 0 for t > T, i ∈ I , where xi (t) and Π(t) are the solutions of drive-response systems (3) and (4) with initial conditions ϕ and ψ, respectively. Lemma 3: [3] Let a1 , a2 , ..., an are positive numbers and 0 ≤ r1 < r2 , then n X
ari 2
i=1
r1
2
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(a1 + a2 + · · · + an )
1+µ 2
i=1
1+µ (0 2
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Especially, if we select r2 = 1 and r1 = following inequality holds:
≤
n X
1+µ
ari 1
r1
1
.
≤ µ < 1), then 0 < r1 < r2 and the 1+µ
1+µ
≤ a1 2 + a2 2 + · · · + an 2 .
Lemma 4. [4] If Y and Z are real matrices with appropriate dimensions, then there exists a positive constant ς > 0 such that
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1 YT Z + ZT Y ≤ ςYT Y + ZT Z. ς
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In the following, we will give a very useful and more general lemma dealing with finite-time synchronization problem for ordinary differential inequalities with aperiodically intermittent control and delay.
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Lemma 5: Suppose that function V (t) is continuous and non-negative when t ∈
[−τ, +∞) and satisfies the following conditions: V˙ (t) ≤ −αV η (t), tm ≤ t ≤ sm , V˙ (t) ≤ βV (t), sm < t < tm+1 ,
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where m = 0, 1, 2, ..., α, β > 0, 0 < η < 1. If there exists a constant Ψ ∈ (0, 1), where Ψ is defined in Definition 1, then the following inequality holds: 1−η V 1−η (t) ≤ sup V (s) exp βΨ(1 − η)t − α(1 − Ψ)(1 − η)t, 0 ≤ t ≤ T, −τ ≤s≤0
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where the constant T is the setting time given by the following: α(1−Ψ) 1 1−η = eβΨ, then T = βΨ(1−η) . If sup
V (s)
−τ ≤s≤0
If
sup
1−η > eβΨ, then T = T1 , where T1 is the smaller solution of
V (s)
1−η sup V (s)
−τ ≤s≤0
exp βΨ(1 − η)t − α(1 − Ψ)(1 − η)t = 0. 1−η Proof : Take M0 = sup V (s) and W (t) = V 1−η (t)+α(1−η)t, where t ≥ −τ . −τ ≤s≤0
−τ ≤s≤0
Let Q(t) = W (t) − M0 . It is easy to see that
Q(t) ≤ 0, for all t ∈ [−τ, 0].
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Next, we will prove that
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α(1−Ψ)
Q(t) ≤ 0, for all t ∈ [0, s0 ]. For ∀ t ∈ [0, s0 ], we can get
˙ Q(t) ≤ (1 − η)V −η (t)(−αV η (t)) + α(1 − η) = 0.
(H1 )
(H2 )
(H3 )
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This is to say that Q(t) is decreasing in [0, s0 ]. Hence, Q(t) ≤ Q(0) ≤ 0. So, (H2 ) holds. Let W1 (t) = W (t) exp − β(1 − η)(t − s0 ) , H(t) = W1 (t) − M0 − α(1 − η)(t − s0 ) exp − β(1 − η)(t − s0 ) . Next, we prove that for t ∈ (s0 , t1 ) H(t) ≤ 0.
(H4 )
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For ∀ t ∈ (s0 , t1 ), we can get
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˙ H(t) = (1 − η)V −η (t)V˙ (t) exp − (1 − η)β(t − s0 ) − (1 − η)β × exp − (1 − η)β(t − s0 ) V 1−η (t) + (1 − η)α exp − (1 − η) ×β(t − s0 ) − (1 − η)2 β exp − (1 − η)β(t − s0 ) αt −(1 − η)α exp − (1 − η)β(t − s0 ) +(1 − η)2 β exp − (1 − η)β(t − s0 ) α(t − s0 ) ≤ −(1 − η)2 αβs0 exp − (1 − η)β(t − s0 ) < 0.
This is to say that H(t) is decreasing in (s0 , t1 ). Hence, H(t) ≤ H(s0 ) = W1 (s0 ) − M0 = W (s0 ) − M0 ≤ 0. So, (H4 ) holds. 7
(H5 )
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On the other hand, together with (H1 ) and (H2 ), we can obtain W (t) ≤ M0 exp (1 − η)β(t1 − s0 ) + α(1 − η)(t1 − s0 ), t ∈ [−τ, t1 ).
(H6 )
Similar to the same approach of Eq. (H2 ), one can prove that for t ∈ [t1 , s1 ],
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W (t) ≤ M0 exp (1 − η)β(t1 − s0 ) + α(1 − η)(t1 − s0 ).
Suppose Q1 (t) = W (t) − hM0 exp (1 − η)β(t1 − s0 ) − α(1 − η)(t1 − s0 ), it is easy to see that Q˙ 1 (t) < 0, for t ∈ [t1 , s1 ]. Similar to the proof of Eq. (H4 ), we can verify W (t) ≤ M0 exp (1 − η)β(t1 − s0 ) + (1 − η)β(t − s1 ) +α(1 − η)(t1 − s0 ) + α(1 − η)(t − s1 ),
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for t ∈ (s1 , t2 ). Take W2 (t) = W (t) exp − β(1 − η)(t1 − s0 ) − β(1 − η)(t − s1 ) and H1 (t) = W2 (t) − M0 − α(1 − η)(t1 − s0 ) + α(1 − η)(t − s1 ) × exp {−β(1 − η)(t1 − s0 ) − β(1 − η)(t − s1 )} .
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Then, according to Eq. (H5 ), we can easy obtain that H˙ 1 (t) < 0, for t ∈ (s1 , t2 ). By induction, we can derive the following estimation of W (t) for any integer m. For tm ≤ t ≤ sm ,
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m o n X (tk − sk−1 ) W (t) ≤ M0 exp β(1 − η)
and for sm < t < tm+1 ,
k=1
m X (tk − sk−1 ), +α(1 − η) k=1
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m hX i W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + (t − sm )
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(H7 )
k=1
m hX i +α(1 − η) (tk − sk−1 ) + (t − sm ) .
(H8 )
k=1
Assume that inequalities (H7 ) and (H8 ) are true for m ≤ p − 1, where p is a positive
integer. Then, for any integer q satisfying 0 ≤ q ≤ p − 1, for tq ≤ t ≤ sq , q n o X W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) k=1
q X +α(1 − η) (tk − sk−1 ), k=1
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and for sq < t < tq+1 ,
q hX i W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + (t − sq ) k=1
q hX i +α(1 − η) (tk − sk−1 ) + (t − sq ) k=1
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k=1 h X q i < M0 exp β(1 − η) (tk − sk−1 ) + (tq+1 − sq ) q hX i +α(1 − η) (tk − sk−1 ) + (tq+1 − sq ) k=1
q+1 o n X (tk − sk−1 ) = M0 exp β(1 − η)
k=1
k=1
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+α(1 − η)
q+1 X
(tk − sk−1 )
p n o X < M0 exp β(1 − η) (tk − sk−1 ) k=1
k=1
(tk − sk−1 ).
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+α(1 − η)
p X
This together with (H1 ), for any t ∈ [−τ, tp ], we have
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p p n o X X W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + α(1 − η) (tk − sk−1 ). k=1
(H9 )
k=1
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Similar to the proof of (H2 ), we can prove that (H9 ) holds for tp ≤ t ≤ sp . And similar to (H4 ), we can verify the fact that
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n p P o W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + (t − sp ) k=1
p hX i +α(1 − η) (tk − sk−1 ) + (t − sp ) ,
(H10 )
k=1
for sp < t < tp+1 . Hence, the process of these proofs are omitted for simplicity. Therefore, from mathematical induction, we know that (H7 ) and (H8 ) hold for any natural number m. Since for any t ≥ 0, there exists a nonnegative integers m, such that tm < t < tm+1 , we can deduce the following estimation of W (t) for any t.
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For tm ≤ t ≤ sm , according to Definition 1 m m n o X X W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + α(1 − η) (tk − sk−1 ) k=1
+α(1 − η)Ψ
m X k=1
k=1
(tk − tk−1 )
(H11 )
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= M0 exp{β(1 − η)Ψtm } + α(1 − η)Ψtm
k=1
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m o n X tk − sk−1 · (tk − tk−1 ) = M0 exp β(1 − η) tk − tk−1 k=1 m X tk − sk−1 +α(1 − η) · (tk − tk−1 ) t k − tk−1 k=1 m n o X ≤ M0 exp β(1 − η)Ψ (tk − tk−1 )
≤ M0 exp{β(1 − η)Ψt} + α(1 − η)Ψt.
In addition, for sm < t < tm+1 , according to Lemma 2 m P (tk − sk−1 ) + (t − sm ) W (t) ≤ M0 exp β(1 − η) k=1
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m hX i +α(1 − η) (tk − sk−1 ) + (t − sm ) k=1
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m hX tk − sk−1 · (tk − tk−1 ) = M0 exp β(1 − η) tk − tk−1 k=1 m i hX t − sm tk − sk−1 + · (t − tm ) + α(1 − η) t − tm tk − tk−1 k=1 i t − sm ·(tk − tk−1 ) + · (t − tm ) t − tm m hX t − s k k−1 ≤ M0 exp β(1 − η) · (tk − tk−1 ) t k − tk−1 k=1 m i hX tm+1 − sm tk − sk−1 + · (t − tm ) + α(1 − η) tm+1 − tm tk − tk−1 i k=1 tm+1 − sm ·(tk − tk−1 ) + · (t − tm ) tm+1 − tm m i hX (tk − tk−1 ) + (t − tm ) ≤ M0 exp β(1 − η)Ψ k=1
m hX i +α(1 − η)Ψ (tk − tk−1 ) + (t − tm ) k=1
= M0 exp{β(1 − η)Ψt} + α(1 − η)Ψt. 10
(H12 )
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From the definition of W (t), we obtain V 1−η (t) ≤ M0 exp{β(1 − η)Ψt} − α(1 − η)t + α(1 − η)Ψt 1−η = sup V (s) exp{βΨ(1 − η)t} − α(1 − Ψ)(1 − η)t, t ≥ 0. −τ ≤s≤0
α(1 − Ψ)(1 − η)T =
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By Lemma 3, the settling time T can be given by
n o 1−η sup V (s) exp βΨ(1 − η)T .
−τ ≤s≤0
In the following, we discuss the value of T . Obviously, T is satisfied to the following equation
(H13 )
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n o α(1 − Ψ)(1 − η) 1−η t = exp βΨ(1 − η)t . sup V (s) −τ ≤s≤0
Based on the properties of linear function and exponential function, when n o α(1 − Ψ)(1 − η) 1−η = βΨ(1 − η) exp βΨ(1 − η)t , sup V (s) −τ ≤s≤0
i.e.
1 ln βΨ(1 − η)
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t=
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the images of two functions are tangent. At this point, we can work out the value of t, α(1 − Ψ) . 1−η sup V (s) βΨ
−τ ≤s≤0
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Take the value of the t into the equation of (H13 ), we can get α(1 − Ψ) 1−η = eβΨ. sup V (s)
−τ ≤s≤0
Hence, we sum up the solution of the (H13 ) as follows: α(1−Ψ) 1−η < eβΨ, there is no solution of the (H13 ). (i): If sup
V (s)
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−τ ≤s≤0
α(1−Ψ)
(ii): If
sup
1−η = eβΨ, there is only one solution of the (H13 ). And the solution
V (s)
−τ ≤s≤0
of the (H13 ) is
T = α(1−Ψ)
(iii): If
sup −τ ≤s≤0
T2 and T1 < T2 .
1 . βΨ(1 − η)
1−η > eβΨ, there are two solutions of the (H13 ). Denoted as T1 ,
V (s)
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For case (i), the state of the master-slave system is uncertain. They may be asymptotically synchronous, may be finite-time synchronization, or even asynchronous. Therefore, we cannot give a clear judgment at this time. 1 For case (ii), we know the value of T = βΨ(1−η) . When t → T , we have V (t) → 0. When t ≥ T , T ∈ [tm , sm ] or (sm , tm+1 ). When T ∈ [tm , sm ], t ∈ [T, sm ] or (sm , tm+1 ).
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When t ∈ [T, sm ], by the first inequality of (6), we can obtain V (t) ≤ V (T ) exp{−α(t −
T )} ≡ 0. When t ∈ (sm , tm+1 ), by the second inequality of (6), we can obtain V (t) ≤ V (sm ) exp{β(t − sm )} ≡ 0. When T ∈ (sm , tm+1 ), for all t ∈ [T, tm+1 ], by the second inequality of (6), we can
obtain V (t) ≤ V (T ) exp{β(t − T )} ≡ 0. For all t ∈ [tm+1 , sm+1 ], by the first inequality of
(6), we can obtain V (t) ≤ V (tm+1 ) exp{−α(t − tm+1 )} ≡ 0.
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For case (iii), take T = T1 , similar to the discussion of the case (ii), we can derived When t → T , we have V (t) → 0. When t ≥ T, V (t) ≡ 0. By analyzing, only the case of (ii) and (iii) can the master-slave system be finite-time
synchronization. Therefore, the settling time T can be given by T =
for the case 2
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(ii) and T = T1 for the case (iii). The proof is completed.
1 βΨ(1−η)
Remark 1: In [27], lemma 5 is given to deal with stability analysis for ordinary differential inequalities with aperiodically intermittent control and delay. However, in
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practical engineering fields, the networks might always be expected to synchronize as quickly as possible under a finite time, that is, the finite-time synchronization. Finitetime synchronization means the optimality in convergence time. Hence, we will investigate
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finite-time synchronization of coupled dynamical networks with delay via aperiodically intermittent control in this paper.
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Let ei (t) = (e1i (t), e2i (t), ..., eni (t))T = xi (t) − Π(t) (i ∈ I ) be synchronization errors. Since our main objective here is to apply aperiodically intermittent control scheme to
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make the states of network (3) finite-time synchronize with Π(t), i.e., lim kxi (t) − Π(t)k = 0, i ∈ I ,
t→∞
then we have the following controlled delayed dynamical network: N X x˙ i (t) = f t, xi (t), xi (t − τ (t)) + c bij Γxj (t) + ui (t), j=1
12
(9)
ACCEPTED MANUSCRIPT
CR IP T
where ui (t) is the aperiodically intermittent feedback control gain defined as follows: η Z t 1+µ ei (t) 2 3 T −η1 ei (t) − η2 e (s)e(s)ds 1 − σ t−τ (t) ||e(t)||2 ui (t) = (10) −η2 sign(ei (t))|ei (t)|µ , k e(t) k6= 0 and tm ≤ t ≤ sm , 0, k e(t) k= 0 or s < t < t m m+1 , T where m = 0, 1, 2, ..., |ei (t)|µ = |e1i (t)|µ , |e2i (t)|µ , ..., |eni (t)|µ , sign(ei (t)) = diag sign(e1i (t)), T sign(e2i (t)), ..., sign(eni (t)) , η1 , η3 are the positive constants control strengths, η2 is a tunable positive constant, and the real number µ satisfies 0 ≤ µ < 1. According to the control law (10), the error dynamical system is then governed as
j=1
(11) = f t, xi (t), xi (t − τ (t)) − f t, Π(t), Π(t − τ (t)) . It is easy to
M
where ˜ f
t, xi , Π, xτi , Πτ
AN US
follows: N X τ τ ˜ e˙ i (t) = f (t, xi , Π, xi , Π ) + c bij Γej (t) − η1 ei (t) − η2 sign(ei (t))|ei (t)|µ j=1 Z t 1+µ ei (t) 2 −η2 η3 eT (s)e(s)ds , k e(t) k6= 0 and tm ≤ t ≤ sm , ||e(t)||2 t−τ (t) N X τ τ ˜ e ˙ (t) = f (t, x , Π, x , Π ) + c bij Γej (t), k e(t) k= 0 or sm < t < tm+1 , i i i
ED
see that the finite-time synchronization of the controlled delayed dynamical network (9) is achieved if the zero solution of the error dynamical system (11) is finite-time stable.
Main results
PT
3
CE
In this section, we will investigate the coupling networks model with aperiodically intermittent control as well as time-varying delay. In the following, with the help of
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Lemma 5, we will show how to design suitable η1 , η2 , η3 and Ψ, such that the controlled delayed dynamical network (9) can realize finite-time synchronization. The main results are stated in the following. Theorem 1. Suppose that Assumptions 1-3 hold, there exist positive constants Ψ ∈
(0, 1) defined in (6) in Definition 1, and η1 , η2 , η3 , l1 , l2 such that T η3 (i) : 2l1 − 2η1 + 1−σ IN + cγk B 2+B < 0, (ii) : 2l2 − η3 ≤ 0, (iii) :
η2 (1−Ψ)
sup −τ ≤s≤0
≥ eη1 Ψ. 1−µ 2
V (s)
Then the controlled delayed dynamical network (9) is the finite-time synchronized. 13
ACCEPTED MANUSCRIPT
T T T Proof: Let e(t) = (eT 1 (t), e2 (t), ..., eN (t)) and define the Lyapunov function as
V (t) =
N X
eT i (t)ei (t)
i=1
η3 + 1−σ
Z
t
eT (s)e(s)ds.
(12)
t−τ (t)
CR IP T
Then, its derivative with respect to time t along with solutions of (12) can be calculated as follows. when tm ≤ t ≤ sm , m = 0, 1, 2, ... V˙ (t) = 2
N X
N h X τ τ ˜ f (t, xi , Π, xi , Π ) + c bij Γej (t) − η1 ei (t)
eT i (t)
i=1
×bij Γej (t) − 2η1
N X i=1
eT i (t)ei (t) − 2η2
M
i=1
AN US
j=1 1+µ η Z t ei (t) i 2 3 T µ e (s)e(s)ds −η2 sign(ei (t))|ei (t)| − η2 1 − σ t−τ (t) ||e(t)||2 N N X η3 X T + ei (t)ei (t) − η3 eT i (t − τ (t))ei (t − τ (t)) 1 − σ i=1 i=1 N N N X N X X X T ≤ 2l1 eT (t)e (t) + 2l e (t − τ (t))e (t − τ (t)) + 2c eT i 2 i i i i (t) i=1
N X
i=1 j=1
µ eT i sign(ei (t))|ei (t)|
i=1
i=1
eki (t) 2l1 − 2η1 +
PT
=
N X n X
ED
N 1+µ η Z t η3 X T 2 3 eT (s)e(s)ds −2η2 + e (t)ei (t) 1 − σ t−τ (t) 1 − σ i=1 i N X −η3 eT i (t − τ (t))ei (t − τ (t))
i=1 k=1
N
N
(13)
n
XXX η3 k ei (t) + 2c γk eki (t) 1−σ i=1 j=1 k=1
N X bji + bij k ej (t) + (2l2 − η3 ) eT i (t − τ (t))ei (t − τ (t)) 2 i=1 N 1+µ η Z t X 2 3 T µ −2η2 ei (t)sign(ei (t))|ei (t)| − 2η2 eT (s)e(s)ds 1 − σ t−τ (t) i=1 n h T X η3 B + Bi k = (˜ ek (t))T 2l1 − 2η1 + IN + cγk ˜ e (t) + (2l2 − η3 ) 1−σ 2 k=1 N N X X T µ × ei (t − τ (t))ei (t − τ (t)) − 2η2 eT i (t)sign(ei (t))|ei (t)| i=1 i=1 η Z t 1+µ 2 3 −2η2 eT (s)e(s)ds , 1 − σ t−τ (t)
AC
CE
×
where ˜ ek = (ek1 , ek2 , ..., ekN )T for k = 1, 2, ..., n. Based on the conditions of (i) and (ii), we 14
ACCEPTED MANUSCRIPT
can get
PN
i=1
µ eT i (t)sign(ei (t))|ei (t)|
i=1
µ eT i (t)sign(ei (t))|ei (t)| =
ing Lemma 3, it can be implied that N X n X i=1 k=1
i=1 k=1
|eki (t)|1+µ
i=1
|eki (t)|1+µ
Hence, N X n X
PN
≥
1+µ η Z t 2 3 T e (s)e(s)ds . − 2η2 1 − σ t−τ (t)
µ |eT i (t)||ei (t)| =
1 1+µ
X N X n i=1 k=1
≥
N X n X i=1 k=1
|eki (t)|2
1+µ 2
=
N hX
eT i (t)ei (t)
i=1
(t).
k=1
|eki (t)|1+µ and us-
|eki (t)|2 N X
21
.
1+µ 2 . eT (t)e (t) i i
η3 + 1−σ
Z
t
t−τ (t)
i 1+µ 2 eT (s)e(s)ds
(15)
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= −2η2 V
1+µ 2
i=1
i=1
Then, combining with Lemma 3, we can obtain V˙ (t) ≤ −2η2
PN Pn
(14)
CR IP T
Since
N X
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V˙ (t) ≤ −2η2
Similarly, for sm < t < tm+1 , we have
BT + B i k 2l1 − 2η1 + η3 IN + cγk ˜ e (t) + (2l2 − η3 ) 2 k=1 N n X X (˜ ek (t))T 2η1˜ ek (t) × eT (t − τ (t))e (t − τ (t)) + i i (˜ ek (t))T
h
ED
n X
PT
V˙ (t) ≤
i=1
(16)
k=1
CE
≤ 2η1 V (t),
AC
Thus, we get
V˙ (t) ≤ −2η V 1+µ 2 (t), 2 V˙ (t) ≤ 2η1 V (t),
tm ≤ t ≤ sm ,
sm < t < tm+1 ,
(17)
By Lemma 5, we obtain
V
1−µ 2
(t) ≤
sup V (s) −τ ≤s≤0
1−µ 2
exp{η1 Ψ(1 − µ)t} − η2 (1 − Ψ)(1 − µ)t, 0 ≤ t ≤ T1 . (18)
and the settling time T can be given by T =
1 or T = T1 , η1 Ψ(1 − µ) 15
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1−µ sup V (s) 2 exp{η1 Ψ(1−µ)t}−η2 (1−Ψ)(1−µ)t =
where T1 is the smaller solution of
−τ ≤s≤0
0.
Hence, the zero solution of the error dynamical system (11) is finite-time stable, that is, the system (9) is finite-time synchronized. This completes the proof of Theorem 1. 2
x˙ i (t) = −Axi (t) + Dg(xi (t)) + Cg(xi (t − τ (t))) + c
CR IP T
As a special case, we consider a class of controlled coupled neural network N X
bij Γxj (t) + ui (t),
(19)
j=1
where i ∈ I , bij is defined in (2), xi (t) = (x1i (t), x2i (t)..., xni (t))T ∈ Rn denotes the state variable associated with the ith neuron, A = diag{a1 , a2 , ..., an } is the decay constant ma-
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trix with ai > 0 for i = 1, 2, ..., n, D = (dij )n×n and C = (cij )n×n are the connection matrix
and delayed connection matrix, respectively, g(xi (t)) = (g1 (x1i (t)), g2 (x2i (t)), ..., gn (xni (t)))T is the activation function of the neurons and satisfies the following condition. 0 Assumption 1 . There exists a positive constant s > 0 such that
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for any x, y ∈ Rn .
T g(x) − g(y) g(x) − g(y) ≤ s(x − y)T (x − y),
Correspondingly, the synchronization state Π(t) associated with (19) is represented by
ED
˙ Π(t) = −AΠ(t) + Dg(Π(t)) + Cg(Π(t)).
(20)
The following statement is provided to ensure the finite-time synchronization of the
PT
coupled network.
0
CE
Theorem 2. Suppose that Assumptions 1 and 2-3 hold, there exist positive constants Ψ ∈ (0, 1) defined in (6) in Definition 1, and η1 , η2 , η3 , λ, s such that T η3 (i) : λ − 2η1 + 1−σ IN + cγk B 2+B < 0, (ii) : s − η3 ≤ 0,
η2 (1−Ψ)
AC
(iii) :
sup
−τ ≤s≤0
≥ eη1 Ψ, 1−µ 2
V (s)
where λ = λmax (−A − AT + DDT + CCT + sIN ). Then the controlled delayed dynamical network (19) is finite-time synchronized in a finite-time T =
where T1 is the smaller solution of 0.
1 or T = T1 , η1 Ψ(1 − µ)
1−µ sup V (s) 2 exp{η1 Ψ(1−µ)t}−η2 (1−Ψ)(1−µ)t =
−τ ≤s≤0
16
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. Proof: From Theorem 1, the network is finite-time synchronized, if assumption 1 0
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CR IP T
holds. Using Lemma 4 and assumption 1 , we can get T (xi (t) − yi (t)) f (t, xi (t), xi (t − τ (t))) − f (t, yi (t), yi (t − τ (t))) h = (xi (t) − yi (t))T − A(xi (t) − yi (t)) + D(g(xi (t)) − g(yi (t))) i + C(g(xi (t − τ (t))) − g(yi (t − τ (t)))) 1 ≤ (xi (t) − yi (t))T − A − AT + DDT + CCT + sIN (xi (t) − yi (t)) 2 T s + xi (t − τ (t)) − yi (t − τ (t))) xi (t − τ (t)) − yi (t − τ (t))) 2 T s λ xi (t − τ (t)) − yi (t − τ (t))) ≤ (xi (t) − yi (t))T (xi (t) − yi (t)) + 2 2 × xi (t − τ (t)) − yi (t − τ (t))) ,
s λ , l2 = . Hence, it follows from Theorem 1 2 2 that the coupled network (19) is finite-time synchronized to (20). The proof of Theorem
which shows assumption 1 holds and l1 = 2 is completed.
2
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Suppose Ψ = 0, the the aperiodically intermittent control becomes a general continuous control. Then based on Theorem 1 and Theorem 2, the following Corollary 1 and
ED
Corollary 2 is immediate, respectively.
Corollary 1. Suppose that Assumptions 1 and 3 hold, there exist positive constants
PT
η1 , η2 , η3 , l1 , l2 such that T η3 (i) : 2l1 − 2η1 + 1−σ IN + cγk B 2+B < 0, (ii) : 2l2 − η3 ≤ 0.
time:
CE
Then the controlled delayed dynamical network (10) is finite-time synchronized in a finite 1−µ
V 2 (0) T ≤ . η2 (1 − µ) 0
AC
Corollary 2. Suppose that Assumptions 1 and 3 hold, there exist positive constants
η1 , η2 , η3 , λ, s such that T η3 (i) : λ − 2η1 + 1−σ IN + cγk B 2+B < 0, (ii) : s − η3 ≤ 0,
where λ = λmax (−A − AT + DDT + CCT + sIN ). Then the controlled delayed dynamical network (19) is finite-time synchronized in a finite time: 1−µ
V 2 (0) T ≤ . η2 (1 − µ) 17
ACCEPTED MANUSCRIPT
Suppose τ (t) = 0, we can rewrite network model (9) as follows: x˙ i (t) = f (t, xi (t)) + c
N X
bij Γxj (t) + ui (t),
(21)
j=1
CR IP T
where ui (t) is the aperiodically intermittent feedback control gain defined as follows: −η e (t) − η sign(e (t))|e (t)|µ , tm ≤ t ≤ sm , 1 i 2 i i ui (t) = (22) 0, sm < t < tm+1 ,
T where m = 0, 1, 2, ..., |ei (t)|µ = |e1i (t)|µ , |e2i (t)|µ , ..., |eni (t)|µ , sign(ei (t)) = diag sign(e1i (t)), T sign(e2i (t)), ..., sign(eni (t)) , η1 , η2 are the positive constants control strengths, and the
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real number µ satisfies 0 ≤ µ < 1.
Correspondingly, the synchronization state Π(t) associated with (21) is represented by ˙ Π(t) = f (t, Π(t)),
i ∈ I.
(23)
Based on Theorem 1 and Theorem 2, we can get the following Corollary 3 and Corollary
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4, respectively. Corollary 3. Suppose that Assumptions 1 and 2 hold, there exist positive constants Ψ ∈ (0, 1) defined in (6) in Definition 1, and η1 , η2 , l1 , l2 such that T +B
η2 (1−Ψ)
(ii) :
sup
< 0,
= eη1 Ψ. 1−µ 2
V (s)
−τ ≤s≤0
2
ED
(i) : (2l1 − 2η1 )IN + cγk B
finite time
PT
Then the controlled delayed dynamical network (21) is the finite-time synchronized in a 1 or T = T1 , η1 Ψ(1 − µ) 1−µ where T1 is the smaller solution of sup V (s) 2 exp{η1 Ψ(1−µ)t}−η2 (1−Ψ)(1−µ)t =
AC
0. .
CE
T =
−τ ≤s≤0
T T T Proof: Let e(t) = (eT 1 (t), e2 (t), ..., eN (t)) and define the Lyapunov function as
V (t) =
N X
eT i (t)ei (t).
(24)
i=1
The other proof is the similar as Theorem 1, here it is omitted. Hence, the proof of Corollary 1 is completed. 0
Corollary 4. Suppose that Assumptions 1 and 2 hold, there exist positive constants Ψ ∈ (0, 1) defined in (6) in Definition 1, and η1 , η2 , λ, s such that 18
ACCEPTED MANUSCRIPT
(i) : (ii) :
η3 1−σ η2 (1−Ψ)
λ − 2η1 + sup
T +B
IN + cγk B
2
< 0,
≥ eη1 Ψ, 1−µ 2
V (s)
−τ ≤s≤0 where λ = λmax − A − AT + DDT + CCT + sIN . Then the controlled delayed dynamical
CR IP T
network (21) is finite-time synchronized in a finite time 1 T = or T = T1 , η1 Ψ(1 − µ) 1−µ where T1 is the smaller solution of sup V (s) 2 exp{η1 Ψ(1−µ)t}−η2 (1−Ψ)(1−µ)t = −τ ≤s≤0
0.
Remark 2: In [24], finite-time synchronization problem for ordinary differential inequalities with periodically intermittent control is condered. But, Lemma 3 in [24] is only
AN US
used to cope with complex networks without delay. However, it is well known that time delay is a very common phenomenon in some fields and may result in undesirable dynamic behaviors, such as oscillation behavior and network instability. Therefore, it is very essential to investigate the influence of time delay on the synchronization. In this paper, by constructing appropriate controller and Lyapunov function, we deal with finite-time
M
synchronization analysis for delayed dynamical networks.
Remark 3: In [24–26], the authors coped with stability analysis for general differential inequalities with periodically intermittent approach. However, we give Lemma 5
ED
to cope with the finite-time synchronization of complex dynamical networks via aperiodically intermittent control. If tm+1 − tm ≡ T, sm − tm ≡ δ, where T, δ are positive
PT
constants, t0 = 0, m = 0, 1, 2, ..., the aperiodically intermittent control strategy becomes the periodic type, i.e. Lemma 5 in this paper is equivalent to Lemma 3 in [24]. This is
to say, results in [24] are the special case of our results. Besides, in [24–26], they did not
CE
discuss the value of the settling time T . Because the value of the settling time T may be
AC
one, may not exist, or may be two. Hence, we give a discussion about the value of the settling time T . By analyzing, the settling time T can be given by: (i): If
α(1−Ψ)
sup
V (s)
−τ ≤s≤0
1−η < eβΨ, there is no solution of the (H13 ). Hence, the value of
the settling time T does not exist. α(1−Ψ) 1−η = eβΨ, there is only one solution of the (H13 ). Hence, the value (ii): If sup
V (s)
−τ ≤s≤0
of the settling time T is
T = (iii): If
α(1−Ψ) sup −τ ≤s≤0
1 . βΨ(1 − η)
1−η > eβΨ, there are two solutions of the (H13 ). Denoted as T1 ,
V (s)
19
ACCEPTED MANUSCRIPT
T2 and T1 < T2 . Here, the value of the settling time T is T = T1 , where T1 is the smaller 1−η solution of sup V (s) exp βΨ(1 − η)t − α(1 − Ψ)(1 − η)t = 0. −τ ≤s≤0
4
Numerical simulations
achieved in this paper.
CR IP T
In this section, a chaotic network is given to present the effectiveness of our results Example. Consider the following 2-D coupled networks model with variable delay as follows:
i = 1, 2, ..., 8,
bij Γxj (t) + ui (t),
j=1
AN US
x˙ i (t) = Axi (t) + Dg1 (xi (t)) + Cg2 (xi (t − τ (t))) + c
2 X
(25)
where xi (t) = (x1i (t), x2i (t)) ∈ R2 , i = 1, 2, ..., 8, g1 (u) = g2 (u) = (tanh(u1 ), tanh(u2 )),
M
τ (t) = et /(1 + et ), c = 1, and −1.5 −0.1 1 0 −1 0 , C = , Γ = . A= −0.1 −1.5 0 1 0 −1
ED
In the case that network (25) reaches complete synchronization, that is, lim kxi (t) −
Π(t)k = 0, i = 1, 2, ..., 8, we have the following synchronized state equation:
PT
˙ Π(t) = AΠ(t) + Dg1 (Π(t)) + Cg2 (Π(t − τ (t))).
t→∞
(26)
AC
CE
We choose different connection matrices D presented by 2 −0.1 2 −0.1 , D2 = . D1 = −4 2 −4 3
The different dynamic properties of (26) with the initial values (Π1 (s), Π2 (s))T = (0.8, 0.6)T with s ∈ [−1, 0] can be emerged, which are revealed in Fig. 2 and the synchronized states are chaotic attractor in those case. In the following, we only consider the case D = D1 . Moreover, it is easy to verify that s = 1, λ = λmax (−A − AT + DDT + CCT + sIN ) =
24.5047. Hereinafter, we will choose suitable control parameters such that (25) achieves the finite-time synchronization.
20
ACCEPTED MANUSCRIPT
Here, we choose B=
−2
0.72
0.2
0.4
0.2
0.3
0.3
0.32 0.28
−4
0.48 0.48 0.72 0.56 0.64 0.16 0.13 0.17 0.15
0.6
−1
0.32
0.48 0.72 0.2
0.1 0.8
0.1
0.64
0.6
0.6
−4
0.68
0.68 0.44
0.3
0.32
−4 0.2
0.64
−2
0.32
0.52 0.72 0.56 0.64 0.64 0.52
0.4
0.3
0.68 0.56 0.64 0.52 0.48 0.36
0.4 0.19 0.4 . 0.4 0.36 0.76 −4
CR IP T
−4
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By computation, let η1 = 14, η3 = 1, σ = 1/4, µ = 7/8, τ = 1, then we can obtain the conditions (i), (ii) of Theorem 2 are satisfied. Moreover, the different dynamic properties of (25) with the initial values x11 = 0.1, x12 = −0.45, x13 = 0.45, x14 = 0.2, x15 = −0.2, x16 =
1, x17 = 2, x18 = −1, x21 = −2, x22 = 2, x23 = −0.5, x24 = 1, x25 = 0.8, x26 = −0.4, x27 =
0.6, x28 = 0.4 with s ∈ [−1, 0]. By computing, we can get sup V (s) = 45.3250. The −1≤s≤0
M
intermittent control exists on time span
[0, 3] ∪ [3.2, 6.4] ∪ [6.5, 9.6] ∪ [9.8, 12.8] ∪ [13, 16] ∪ [16.2, 19.2] ∪ [19.5, 22.6] ∪ [22.8, 25.8] ∪ · · · η2 (1−Ψ)
ED
So, θ = 3, ω = 3.3. By computing, we get Ψ = 1/11. When
sup
= eη1 Ψ, 1−µ 2
V (s)
−τ ≤s≤0
we obtain that η2 = 1.7769e. From Theorem 2, the networks can be synchronized with
PT
finite-time T = 6.286. Fig. 3 and Fig. 4 show the synchronization of dynamics, and Fig. 5 and Fig. 6 show the errors of dynamics.
3
3 2
AC
2
4
CE
4
1 Π2
Π2
1 0
0
−1
−1
−2
−2
−3 −0.8
−0.6
−0.4
−0.2
0 Π1
0.2
0.4
0.6
−3 −1
0.8
(a)
−0.8
−0.6
−0.4
Π1
−0.2
0
0.2
(b)
Fig. 2. The chaotic attractor of neural networks (26) with different D and the same initial conditions Π1 (s) = 0.8, Π2 (s) = 0.6. (a) D = D1 . (b) D = D2 .
21
ACCEPTED MANUSCRIPT
4
4
3
2
2
0 −2
Π2, x2 (i=1,2,...,8) i
0 −1 −2
−4 −6 −8
−3
−10
−4
−12
−5 −6 0
5
t
10
−14 0
15
||e1(t)||
20
5
t
10
15
Fig. 4 The synchronization of state x2i (t) and Π2 (t).
AN US
Fig. 3 The synchronization of state x1i (t) and Π1 (t).
||e1(t)||
20
15
||e1(t)||
15
||e1(t)||
CR IP T
Π1, x1 (i=1,2,...,8) i
1
10
M
5
0
5
10 t
ED
0 15
PT
Fig. 5 The synchronization error ||e1 || =
8 P
i=1
20
10
5
0 0
|x1i − Π1 |.
5
10 t
15
Fig. 6 The synchronization error ||e2 || =
20 8 P
i=1
|x2i − Π2 |.
In the following, we consider the following 2-D coupled networks model without delays
CE
as follows:
AC
x˙ i (t) = Axi (t) + Dg1 (xi (t)) + Cg2 (xi (t)) + c
2 X
bij Γxj (t) + ui (t),
j=1
i = 1, 2, ..., 8,
(27)
where all parameters are the same as the system (25). Corresponding synchronization state equation is given as following: ˙ Π(t) = AΠ(t) + Dg1 (Π(t)) + Cg2 (Π(t)).
(28)
Next, we give the state of system (28) and the synchronization figures of system (27) and system (28).
22
ACCEPTED MANUSCRIPT
1 0
Π2
−1
−3 −4
0.4
0.5
0.6
Π1
0.7
CR IP T
−2
0.8
AN US
Fig. 7. The state of neural networks (28) with the same initial conditions Π1 (s) = 0.8, Π2 (s) = 0.6. 2
2 1
1
0
−1
i
1
0.5
Π ,x2(i=1,2,...,8)
Π1,xi (i=1,2,...,8)
1.5
M
2
0
−1 0
5
t
ED
−0.5
10
−3 −4 0
15
PT
Fig. 8 The synchronization of state x1i (t) and Π1 (t). 8
||e2(t)||
5
3
4 3 2
2
1
1
0 0
15
2
4
10
6
AC
||e1(t)||
5
t
7
||e1(t)||
||e (t)||
6
5
Fig. 9 The synchronization of state x2i (t) and Π2 (t).
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7
−2
5
10 t
15
Fig. 10 The synchronization error ||e1 || =
0 0
20 8 P
i=1
|x1i − Π1 |.
5
10 t
15
Fig. 11 The synchronization error ||e2 || =
20 8 P
i=1
|x2i − Π2 |.
Remark 4: In numerical simulations, it can be seen from the Fig. 2 that D plays a crucial role for the chaotic dynamics of neural networks (26). Eight nodes are chosen 23
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in the slave system and some corresponding parameters, which satisfy conditions (i) and (ii), are given. Under the condition (iii), a feasible aperiodically intermittent control time interval is presented. Hence, all conditions of Theorem 2 are satisfied in the numerical simulation. The states of master and slave systems are given in Fig. 3 and Fig. 4, and the errors trajectories of the systems (25) and (26) are shown in Fig. 5 and Fig. 6. It
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can be seen that the synchronization of systems (25) and (26) is indeed solved and the simulation results demonstrate the theoretical analysis very well.
Remark 5: As can be seen in Figure 7, the system (27) without delay is convergent to the equilibrium point. However, the system (25) with time delay is a chaotic system as figure 2. For the two cases, by applying the controller to the corresponding slave system, we can get that the master system and the corresponding slave system can be
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synchronized. Based on the simulation, it is also shown that the delay not only affects the dynamic behavior of the system, but also affects the rate of synchronization. Remark 6: Over the past decades, much attention has been attracted to the field of Markov jump systems. Much effort has been devoted to such systems and some important results have been obtained [31–33]. However, little attention has been paid to the stability and synchronization problem for nonlinear Markov jump systems. Hence, this is an open
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Conclusion
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problem and will be considered in our future works.
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In this paper, the finite-time synchronization for a general coupled complex networks model with delay is investigated via aperiodic intermittent control. As we know, this is the first time to cope with finite-time synchronization problem with aperiodically inter-
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mittent strategy. By establishing a new differential inequality and constructing Lyapunov function, we present several simple yet useful sufficient criteria of coupled networks under aperiodically intermittent control. Furthermore, when Ψ = 0 or τ (t) = 0, we get two spe-
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cial cases, which considered in [3] or [26], respectively. In this sense, the results derived here are more general. What is more, by using the properties of fundamental functions, we give the value of finite time T . Finally, the numerical example is given to demonstrate the validness of the proposed scheme. Acknowledgments This work was supported by National Natural Science Foundation of Peoples Republic of China (Grants Nos. 61473244, 61563048, 11402223), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20136501120001), Natural Science Foundation of Xinjiang (Grant No. 2014211B002), Natural Science Foun24
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dation of Xinjiang University (Grant No. BS130108), Project funded by China Postdoctoral Science Foundation (Grant No. 2013M540782 and No. 2014T70953).
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[33] Shen, M., Park, Ju H., & Ye, D. (2016). A separated approach to control of Markov jump nonlinear systems with general transition probabilities. IEEE Transactions on
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Cybernetics, 46(9), 2010-2018.
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Finite-time synchronization of delayed dynamical networks via aperiodically intermittent control Mei Liu, Haijun Jiang, Cheng Hu PII: DOI: Reference:
S0016-0032(17)30265-X 10.1016/j.jfranklin.2017.05.030 FI 3004
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
16 November 2016 25 March 2017 20 May 2017
Please cite this article as: Mei Liu, Haijun Jiang, Cheng Hu, Finite-time synchronization of delayed dynamical networks via aperiodically intermittent control, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.05.030
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Finite-time synchronization of delayed dynamical networks via
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aperiodically intermittent control Mei Liu, Haijun Jiang∗ and Cheng Hu
College of Mathematics and System Sciences, Xinjiang University,
Abstract.
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Urumqi, 830046, Xinjiang, P.R. China
In this paper, we concern the finite-time synchronization problem for
delayed dynamical networks via aperiodically intermittent control. Compared with some correspondingly previous results, the intermittent control can be aperiodic
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which is more general. Moreover, by establishing a new differential inequality and constructing Lyapunov function, several useful criteria are derived analytically to
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realise finite-time synchronization for delay complex networks. Additionally, as a special case, some sufficient conditions ensuring the finite-time synchronization for a class of coupled neural network are obtained. It is worth noting that the convergence
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time is carefully discussed and does not depend on control widths or rest widths for the proposed aperiodically intermittent control. Finally, a numerical example is
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given to demonstrate the validness of the proposed scheme. Key words: Complex networks; Aperiodically intermittent control; Finite-time
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synchronization; Delay
1
Introduction
Complex dynamical networks, which are everywhere in the real world, have drawn
more and more attention from various areas of science and engineering [1–8] because complex networks can describe a variety of real systems, such as ecosystems, the World Wide Web, genetic networks, biological neural networks, social networks and biomolecular ∗
E-mail: [email protected]; [email protected].
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networks [9–11]. As the major collective behavior of dynamical networks, synchronization has been extensively investigated recently attribute to the fact that it not only can explain many natural phenomena [12, 13] but also has a large quantity of promising potential applications in secure communication [14], image processing [15], and so on. Up to now, many different synchronization patterns including finite-time synchronization [3],
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complete synchronization [4], projective synchronization [7], cluster synchronization [8] have been introduced and widely studied. Finite-time synchronization of memristor-based Cohen-Grossberg neural networks with time-varying delays was investigated in [3]. Synchronization of hybrid-coupled delayed dynamical networks via aperiodically intermittent pinning control was considered in [4]. In [7], the authors considered function projective synchronization of impulsive neural networks with mixed time-varying delays. Cluster
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synchronization of non-linearly coupled Lur’e networks with identical and non-identical nodes and an asymmetrical coupling matrix was studied in [8]. As far as we all know, the dynamical network cannot realize synchronization by itself. Furthermore, for the purpose of driving the network to achieve synchronization, control inputs should be imposed on the nodes of dynamical network. Until now, many useful control methods have been focused on this topic, such as adaptive control [16, 17], feedback
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control [18, 19], intermittent control [20, 21] and impulsive control [22, 23]. Recently, discontinuous control approaches including intermittent control and impulsive control have
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attracted more and more interest naturally in a wide variety of applications because these control methods are more economic and can reduce the amount of the transmitted information. Intermittent control used for manufacturing, transportation and communication,
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was first introduced to control linear econometric models. In practice, intermittent con-
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trol is activated during certain nonzero time intervals and off during other time intervals. When the control interval time is reduced to a point, intermittent control become impulsive control. In other words, the impulsive control is activated only at some instants. Hence, when the states are observable, impulsive control is an effective approach. How-
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ever, when the states of controlled systems are unobservable, impulsive control seems to be invalid. Moreover, it may be more reasonable to achieve control process in some time intervals other than some time instants in real control application. Therefore, the intermittent controller is designed instead of the impulsive one in the paper. Nevertheless, the requirement of periodical intermittent control is quite restricted and may be unreasonable and unnecessary in practice. For example, the generation of wind power is obviously aperiodically intermittent. Therefore, for the real applications and the theoretical analysis, it is necessary to consider the synchronization problem under
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aperiodically intermittent control strategy. Based on the above discussion, we will investigate the following aperiodically intermittent control strategy (see Fig. 1), for any time span [tm , tm+1 ), t0 = 0, m = 0, 1, 2, ..., [tm , sm ] is the work time and (sm , tm+1 ) is the rest time. Moreover, sm − tm is called the mth control width, while tm+1 − sm is called the mth rest width, where tm and sm denote the start time and the end time of mth
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control, tm+1 denotes the end time of mth rest. Obviously, this control strategy is more general. Especially, when tm+1 − tm ≡ T, sm − tm ≡ δ, where T, δ are positive constants,
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t0 = 0, m = 0, 1, 2, ..., the intermittent control type becomes the periodic one, which has been investigated in [16, 20, 21].
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Fig. 1. Aperiodically intermittent control strategy.
Since a finite-time stability result is proposed for a continuous-time system, many extended results have been presented in continuous-time networks systems. One of the
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hot research topics is the finite-time synchronization of dynamical networks [3, 24–26].
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Finite-time synchronization of memristor-based Cohen-Grossberg neural networks with time-varying delays was developed in [3] with discontinuous systems. Periodically intermittent controlling for finite-time synchronization of complex dynamical networks was shown in [24]. The paper [25] studied finite-time synchronization of complex delayed
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networks via intermittent control with multiple switched periods. The recently paper [26] investigated finite-time synchronization of drive-response systems via periodically intermittent adaptive control. To the best of our knowledge, there are few results of the finite-time synchronization for dynamical networks via aperiodically intermittent control. Motivated by the above discussion, we will investigate finite-time synchronization of coupled dynamical networks with delay via aperiodically intermittent control. First, the FTS problem is presented for the first time via the aperiodic intermittent control approach. Second, in this paper, a new differential inequality is established to deal with the finitetime synchronization problem and our results are only dependent on the parameter Ψ 3
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in Definition 1, not the control width or the rest width. Third, by using the properties of fundamental functions, we give the value of finite time T . Forth, a kind of novel controller, state-feedback controller which contain integral term and delayed term, is proposed. Finally, aperiodically intermittent control strategy, which is a more general controller, is considered in the paper to realise synchronization. Based on those, finite-time
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synchronization criteria obtained in our paper are very useful and verifiable in application. The rest of this paper is organized as follows. In Section 2, model of dynamical networks with time-varying delays and preliminaries are given. Synchronization of the considered model under the aperiodically intermittent control is investigated based timevarying delay in Section 3. In Section 4, the effectiveness and feasibility of the developed methods are presented by a numerical example. Finally, some conclusions are obtained
k φ k= sup−τ ≤t≤0
N P
i=1
| φi (t) |2
12
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in Section 5. Notations: Let RN be the space of N -dimensional real column vectors. For x = N 1 P (x1 , · · · , xN )T ∈ RN , kxk denotes a vector norm defined by kxk = x2i 2 . We define i=1
, for all φ = (φ1 (t), φ2 (t), ..., φN (t))T ∈ C([−τ, 0], RN ),
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which denotes the Banach space of all continuous functions mapping [−τ, 0] → RN . The initial conditions of systems (1) and (4) are assumed to be xi (s) = ϕi (s), Π(s) = ψ(s), i ∈
I , respectively, s ∈ [−τ, 0], where ϕi (s), ψ(s) ∈ C([−τ, 0], RN ). IN is the identity matrix
Preliminaries
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2
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with N dimensions. λmax (A) is the maximum eigenvalues of matrix A.
In this section, we consider a general complex dynamical network consisting of N
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linearly identical nodes and each node is an n-dimensional dynamic system. The network
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is described by
x˙ i (t) = f (t, xi (t), xi (t − τ (t))) + c i ∈ I = {1, 2, ..., N },
N X
j=1,j6=i
bij Γ(xj (t) − xi (t)) + ui (t),
(1)
where xi (t) = (x1i (t), x2i (t), ..., xni (t))T ∈ Rn denotes the state variable of the ith node, f : R × Rn × Rn → Rn is a continuously vector-valued function governing the dynamics
of isolated nodes, the time-varying delays τ (t) may be unknown but are bounded by
known constants, i.e., 0 ≤ τ (t) ≤ τ , in which τ (t) represents the internal delay occurring
inside the dynamical node, ui (t) ∈ Rn is intermittent controller shown in Fig. 1, the 4
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positive constant c is the coupling strength, Γ = diag{γ1 , γ2 , ..., γn } is the inner connecting
matric with γi > 0, B = (bij )N ×N is the coupling matric defined to satisfy the following conditions:
bij ≥ 0, i 6= j, bii = −
N P
bij .
(2)
j=1,j6=i
x˙ i (t) = f (t, xi (t), xi (t − τ (t))) + c
N X
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Based on the condition (2), we can rewrite network model (1) as follows: bij Γxj (t) + ui (t),
j=1
i ∈ I.
(3)
In the case that network (3) reaches complete synchronization, that is, lim kxi (t) −
Π(t)k = 0, i ∈ I , we have the following synchronized state equation:
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˙ Π(t) = f (t, Π(t), Π(t − τ (t))),
i ∈ I.
t→∞
(4)
In order to obtain our main results, the following assumptions, lemmas and definitions are necessary: Assumption 1 [27]: For the vector-valued function f (t, x(t), x(t−τ (t))), suppose the uniform semi-Lipschitz condition with respect to the time t holds, i.e., for any x(t), y(t) ∈
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Rn , there exist positive constants l1 > 0 and l2 > 0 such that
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T x(t) − y(t) f (t, x(t), x(t − τ (t))) − f (t, y(t), y(t − τ (t))) T T ≤ l1 x(t) − y(t) x(t) − y(t) + l2 x(t − τ (t)) − y(t − τ (t)) x(t − τ (t)) − y(t − τ (t)) .
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Assumption 2 [27]: For the aperiodically intermittent control strategy, there exist
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two positive scalars 0 < θ < ω < +∞, such that, for m = 0, 1, 2, ..., inf (sm − tm ) = θ, m
sup(tm+1 − tm ) = ω.
(5)
m
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Assumption 3: Time-delay function τ (t) : [0, +∞) → [0, +∞) is real-valued continuous function and satisfies τ˙ (t) ≤ σ < 1.
Definition 1 [27]: For the aperiodically intermittent control, define Ψ = lim sup m→∞
tm+1 − sm . tm+1 − tm
(6)
Obviously, 0 ≤ Ψ < 1, when Ψ = 0, the aperiodically intermittent control becomes
the continuous control.
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θ . ω
Lemma 1 [27]: If Assumption 2 holds, then Ψ ≤ 1 − Lemma 2 [27]: For any m = 0, 1, 2, ..., we denote Ψ(t) =
t − sm , t − tm
t ∈ [sm , tm+1 ].
(7)
tm+1 − sm . tm+1 − tm Definition 2: [25] Drive-response systems (3) and (4) are said to be finite-time syn-
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Then, Ψ(t) is an strictly increasing function and Ψ(t) ≤
chronized if, for a suitable designed feedback controller ui (t), there exists a constant T > 0 such that lim (Π(t) − xi (t)) = 0
t→T
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and Π(t) − xi (t) = 0 for t > T, i ∈ I , where xi (t) and Π(t) are the solutions of drive-response systems (3) and (4) with initial conditions ϕ and ψ, respectively. Lemma 3: [3] Let a1 , a2 , ..., an are positive numbers and 0 ≤ r1 < r2 , then n X
ari 2
i=1
r1
2
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(a1 + a2 + · · · + an )
1+µ 2
i=1
1+µ (0 2
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Especially, if we select r2 = 1 and r1 = following inequality holds:
≤
n X
1+µ
ari 1
r1
1
.
≤ µ < 1), then 0 < r1 < r2 and the 1+µ
1+µ
≤ a1 2 + a2 2 + · · · + an 2 .
Lemma 4. [4] If Y and Z are real matrices with appropriate dimensions, then there exists a positive constant ς > 0 such that
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1 YT Z + ZT Y ≤ ςYT Y + ZT Z. ς
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In the following, we will give a very useful and more general lemma dealing with finite-time synchronization problem for ordinary differential inequalities with aperiodically intermittent control and delay.
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Lemma 5: Suppose that function V (t) is continuous and non-negative when t ∈
[−τ, +∞) and satisfies the following conditions: V˙ (t) ≤ −αV η (t), tm ≤ t ≤ sm , V˙ (t) ≤ βV (t), sm < t < tm+1 ,
(8)
where m = 0, 1, 2, ..., α, β > 0, 0 < η < 1. If there exists a constant Ψ ∈ (0, 1), where Ψ is defined in Definition 1, then the following inequality holds: 1−η V 1−η (t) ≤ sup V (s) exp βΨ(1 − η)t − α(1 − Ψ)(1 − η)t, 0 ≤ t ≤ T, −τ ≤s≤0
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where the constant T is the setting time given by the following: α(1−Ψ) 1 1−η = eβΨ, then T = βΨ(1−η) . If sup
V (s)
−τ ≤s≤0
If
sup
1−η > eβΨ, then T = T1 , where T1 is the smaller solution of
V (s)
1−η sup V (s)
−τ ≤s≤0
exp βΨ(1 − η)t − α(1 − Ψ)(1 − η)t = 0. 1−η Proof : Take M0 = sup V (s) and W (t) = V 1−η (t)+α(1−η)t, where t ≥ −τ . −τ ≤s≤0
−τ ≤s≤0
Let Q(t) = W (t) − M0 . It is easy to see that
Q(t) ≤ 0, for all t ∈ [−τ, 0].
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Next, we will prove that
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α(1−Ψ)
Q(t) ≤ 0, for all t ∈ [0, s0 ]. For ∀ t ∈ [0, s0 ], we can get
˙ Q(t) ≤ (1 − η)V −η (t)(−αV η (t)) + α(1 − η) = 0.
(H1 )
(H2 )
(H3 )
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This is to say that Q(t) is decreasing in [0, s0 ]. Hence, Q(t) ≤ Q(0) ≤ 0. So, (H2 ) holds. Let W1 (t) = W (t) exp − β(1 − η)(t − s0 ) , H(t) = W1 (t) − M0 − α(1 − η)(t − s0 ) exp − β(1 − η)(t − s0 ) . Next, we prove that for t ∈ (s0 , t1 ) H(t) ≤ 0.
(H4 )
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For ∀ t ∈ (s0 , t1 ), we can get
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˙ H(t) = (1 − η)V −η (t)V˙ (t) exp − (1 − η)β(t − s0 ) − (1 − η)β × exp − (1 − η)β(t − s0 ) V 1−η (t) + (1 − η)α exp − (1 − η) ×β(t − s0 ) − (1 − η)2 β exp − (1 − η)β(t − s0 ) αt −(1 − η)α exp − (1 − η)β(t − s0 ) +(1 − η)2 β exp − (1 − η)β(t − s0 ) α(t − s0 ) ≤ −(1 − η)2 αβs0 exp − (1 − η)β(t − s0 ) < 0.
This is to say that H(t) is decreasing in (s0 , t1 ). Hence, H(t) ≤ H(s0 ) = W1 (s0 ) − M0 = W (s0 ) − M0 ≤ 0. So, (H4 ) holds. 7
(H5 )
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On the other hand, together with (H1 ) and (H2 ), we can obtain W (t) ≤ M0 exp (1 − η)β(t1 − s0 ) + α(1 − η)(t1 − s0 ), t ∈ [−τ, t1 ).
(H6 )
Similar to the same approach of Eq. (H2 ), one can prove that for t ∈ [t1 , s1 ],
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W (t) ≤ M0 exp (1 − η)β(t1 − s0 ) + α(1 − η)(t1 − s0 ).
Suppose Q1 (t) = W (t) − hM0 exp (1 − η)β(t1 − s0 ) − α(1 − η)(t1 − s0 ), it is easy to see that Q˙ 1 (t) < 0, for t ∈ [t1 , s1 ]. Similar to the proof of Eq. (H4 ), we can verify W (t) ≤ M0 exp (1 − η)β(t1 − s0 ) + (1 − η)β(t − s1 ) +α(1 − η)(t1 − s0 ) + α(1 − η)(t − s1 ),
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for t ∈ (s1 , t2 ). Take W2 (t) = W (t) exp − β(1 − η)(t1 − s0 ) − β(1 − η)(t − s1 ) and H1 (t) = W2 (t) − M0 − α(1 − η)(t1 − s0 ) + α(1 − η)(t − s1 ) × exp {−β(1 − η)(t1 − s0 ) − β(1 − η)(t − s1 )} .
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Then, according to Eq. (H5 ), we can easy obtain that H˙ 1 (t) < 0, for t ∈ (s1 , t2 ). By induction, we can derive the following estimation of W (t) for any integer m. For tm ≤ t ≤ sm ,
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m o n X (tk − sk−1 ) W (t) ≤ M0 exp β(1 − η)
and for sm < t < tm+1 ,
k=1
m X (tk − sk−1 ), +α(1 − η) k=1
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m hX i W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + (t − sm )
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(H7 )
k=1
m hX i +α(1 − η) (tk − sk−1 ) + (t − sm ) .
(H8 )
k=1
Assume that inequalities (H7 ) and (H8 ) are true for m ≤ p − 1, where p is a positive
integer. Then, for any integer q satisfying 0 ≤ q ≤ p − 1, for tq ≤ t ≤ sq , q n o X W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) k=1
q X +α(1 − η) (tk − sk−1 ), k=1
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and for sq < t < tq+1 ,
q hX i W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + (t − sq ) k=1
q hX i +α(1 − η) (tk − sk−1 ) + (t − sq ) k=1
CR IP T
k=1 h X q i < M0 exp β(1 − η) (tk − sk−1 ) + (tq+1 − sq ) q hX i +α(1 − η) (tk − sk−1 ) + (tq+1 − sq ) k=1
q+1 o n X (tk − sk−1 ) = M0 exp β(1 − η)
k=1
k=1
AN US
+α(1 − η)
q+1 X
(tk − sk−1 )
p n o X < M0 exp β(1 − η) (tk − sk−1 ) k=1
k=1
(tk − sk−1 ).
M
+α(1 − η)
p X
This together with (H1 ), for any t ∈ [−τ, tp ], we have
ED
p p n o X X W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + α(1 − η) (tk − sk−1 ). k=1
(H9 )
k=1
PT
Similar to the proof of (H2 ), we can prove that (H9 ) holds for tp ≤ t ≤ sp . And similar to (H4 ), we can verify the fact that
AC
CE
n p P o W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + (t − sp ) k=1
p hX i +α(1 − η) (tk − sk−1 ) + (t − sp ) ,
(H10 )
k=1
for sp < t < tp+1 . Hence, the process of these proofs are omitted for simplicity. Therefore, from mathematical induction, we know that (H7 ) and (H8 ) hold for any natural number m. Since for any t ≥ 0, there exists a nonnegative integers m, such that tm < t < tm+1 , we can deduce the following estimation of W (t) for any t.
9
ACCEPTED MANUSCRIPT
For tm ≤ t ≤ sm , according to Definition 1 m m n o X X W (t) ≤ M0 exp β(1 − η) (tk − sk−1 ) + α(1 − η) (tk − sk−1 ) k=1
+α(1 − η)Ψ
m X k=1
k=1
(tk − tk−1 )
(H11 )
AN US
= M0 exp{β(1 − η)Ψtm } + α(1 − η)Ψtm
k=1
CR IP T
m o n X tk − sk−1 · (tk − tk−1 ) = M0 exp β(1 − η) tk − tk−1 k=1 m X tk − sk−1 +α(1 − η) · (tk − tk−1 ) t k − tk−1 k=1 m n o X ≤ M0 exp β(1 − η)Ψ (tk − tk−1 )
≤ M0 exp{β(1 − η)Ψt} + α(1 − η)Ψt.
In addition, for sm < t < tm+1 , according to Lemma 2 m P (tk − sk−1 ) + (t − sm ) W (t) ≤ M0 exp β(1 − η) k=1
M
m hX i +α(1 − η) (tk − sk−1 ) + (t − sm ) k=1
AC
CE
PT
ED
m hX tk − sk−1 · (tk − tk−1 ) = M0 exp β(1 − η) tk − tk−1 k=1 m i hX t − sm tk − sk−1 + · (t − tm ) + α(1 − η) t − tm tk − tk−1 k=1 i t − sm ·(tk − tk−1 ) + · (t − tm ) t − tm m hX t − s k k−1 ≤ M0 exp β(1 − η) · (tk − tk−1 ) t k − tk−1 k=1 m i hX tm+1 − sm tk − sk−1 + · (t − tm ) + α(1 − η) tm+1 − tm tk − tk−1 i k=1 tm+1 − sm ·(tk − tk−1 ) + · (t − tm ) tm+1 − tm m i hX (tk − tk−1 ) + (t − tm ) ≤ M0 exp β(1 − η)Ψ k=1
m hX i +α(1 − η)Ψ (tk − tk−1 ) + (t − tm ) k=1
= M0 exp{β(1 − η)Ψt} + α(1 − η)Ψt. 10
(H12 )
ACCEPTED MANUSCRIPT
From the definition of W (t), we obtain V 1−η (t) ≤ M0 exp{β(1 − η)Ψt} − α(1 − η)t + α(1 − η)Ψt 1−η = sup V (s) exp{βΨ(1 − η)t} − α(1 − Ψ)(1 − η)t, t ≥ 0. −τ ≤s≤0
α(1 − Ψ)(1 − η)T =
CR IP T
By Lemma 3, the settling time T can be given by
n o 1−η sup V (s) exp βΨ(1 − η)T .
−τ ≤s≤0
In the following, we discuss the value of T . Obviously, T is satisfied to the following equation
(H13 )
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n o α(1 − Ψ)(1 − η) 1−η t = exp βΨ(1 − η)t . sup V (s) −τ ≤s≤0
Based on the properties of linear function and exponential function, when n o α(1 − Ψ)(1 − η) 1−η = βΨ(1 − η) exp βΨ(1 − η)t , sup V (s) −τ ≤s≤0
i.e.
1 ln βΨ(1 − η)
ED
t=
M
the images of two functions are tangent. At this point, we can work out the value of t, α(1 − Ψ) . 1−η sup V (s) βΨ
−τ ≤s≤0
CE
PT
Take the value of the t into the equation of (H13 ), we can get α(1 − Ψ) 1−η = eβΨ. sup V (s)
−τ ≤s≤0
Hence, we sum up the solution of the (H13 ) as follows: α(1−Ψ) 1−η < eβΨ, there is no solution of the (H13 ). (i): If sup
V (s)
AC
−τ ≤s≤0
α(1−Ψ)
(ii): If
sup
1−η = eβΨ, there is only one solution of the (H13 ). And the solution
V (s)
−τ ≤s≤0
of the (H13 ) is
T = α(1−Ψ)
(iii): If
sup −τ ≤s≤0
T2 and T1 < T2 .
1 . βΨ(1 − η)
1−η > eβΨ, there are two solutions of the (H13 ). Denoted as T1 ,
V (s)
11
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For case (i), the state of the master-slave system is uncertain. They may be asymptotically synchronous, may be finite-time synchronization, or even asynchronous. Therefore, we cannot give a clear judgment at this time. 1 For case (ii), we know the value of T = βΨ(1−η) . When t → T , we have V (t) → 0. When t ≥ T , T ∈ [tm , sm ] or (sm , tm+1 ). When T ∈ [tm , sm ], t ∈ [T, sm ] or (sm , tm+1 ).
CR IP T
When t ∈ [T, sm ], by the first inequality of (6), we can obtain V (t) ≤ V (T ) exp{−α(t −
T )} ≡ 0. When t ∈ (sm , tm+1 ), by the second inequality of (6), we can obtain V (t) ≤ V (sm ) exp{β(t − sm )} ≡ 0. When T ∈ (sm , tm+1 ), for all t ∈ [T, tm+1 ], by the second inequality of (6), we can
obtain V (t) ≤ V (T ) exp{β(t − T )} ≡ 0. For all t ∈ [tm+1 , sm+1 ], by the first inequality of
(6), we can obtain V (t) ≤ V (tm+1 ) exp{−α(t − tm+1 )} ≡ 0.
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For case (iii), take T = T1 , similar to the discussion of the case (ii), we can derived When t → T , we have V (t) → 0. When t ≥ T, V (t) ≡ 0. By analyzing, only the case of (ii) and (iii) can the master-slave system be finite-time
synchronization. Therefore, the settling time T can be given by T =
for the case 2
M
(ii) and T = T1 for the case (iii). The proof is completed.
1 βΨ(1−η)
Remark 1: In [27], lemma 5 is given to deal with stability analysis for ordinary differential inequalities with aperiodically intermittent control and delay. However, in
ED
practical engineering fields, the networks might always be expected to synchronize as quickly as possible under a finite time, that is, the finite-time synchronization. Finitetime synchronization means the optimality in convergence time. Hence, we will investigate
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finite-time synchronization of coupled dynamical networks with delay via aperiodically intermittent control in this paper.
CE
Let ei (t) = (e1i (t), e2i (t), ..., eni (t))T = xi (t) − Π(t) (i ∈ I ) be synchronization errors. Since our main objective here is to apply aperiodically intermittent control scheme to
AC
make the states of network (3) finite-time synchronize with Π(t), i.e., lim kxi (t) − Π(t)k = 0, i ∈ I ,
t→∞
then we have the following controlled delayed dynamical network: N X x˙ i (t) = f t, xi (t), xi (t − τ (t)) + c bij Γxj (t) + ui (t), j=1
12
(9)
ACCEPTED MANUSCRIPT
CR IP T
where ui (t) is the aperiodically intermittent feedback control gain defined as follows: η Z t 1+µ ei (t) 2 3 T −η1 ei (t) − η2 e (s)e(s)ds 1 − σ t−τ (t) ||e(t)||2 ui (t) = (10) −η2 sign(ei (t))|ei (t)|µ , k e(t) k6= 0 and tm ≤ t ≤ sm , 0, k e(t) k= 0 or s < t < t m m+1 , T where m = 0, 1, 2, ..., |ei (t)|µ = |e1i (t)|µ , |e2i (t)|µ , ..., |eni (t)|µ , sign(ei (t)) = diag sign(e1i (t)), T sign(e2i (t)), ..., sign(eni (t)) , η1 , η3 are the positive constants control strengths, η2 is a tunable positive constant, and the real number µ satisfies 0 ≤ µ < 1. According to the control law (10), the error dynamical system is then governed as
j=1
(11) = f t, xi (t), xi (t − τ (t)) − f t, Π(t), Π(t − τ (t)) . It is easy to
M
where ˜ f
t, xi , Π, xτi , Πτ
AN US
follows: N X τ τ ˜ e˙ i (t) = f (t, xi , Π, xi , Π ) + c bij Γej (t) − η1 ei (t) − η2 sign(ei (t))|ei (t)|µ j=1 Z t 1+µ ei (t) 2 −η2 η3 eT (s)e(s)ds , k e(t) k6= 0 and tm ≤ t ≤ sm , ||e(t)||2 t−τ (t) N X τ τ ˜ e ˙ (t) = f (t, x , Π, x , Π ) + c bij Γej (t), k e(t) k= 0 or sm < t < tm+1 , i i i
ED
see that the finite-time synchronization of the controlled delayed dynamical network (9) is achieved if the zero solution of the error dynamical system (11) is finite-time stable.
Main results
PT
3
CE
In this section, we will investigate the coupling networks model with aperiodically intermittent control as well as time-varying delay. In the following, with the help of
AC
Lemma 5, we will show how to design suitable η1 , η2 , η3 and Ψ, such that the controlled delayed dynamical network (9) can realize finite-time synchronization. The main results are stated in the following. Theorem 1. Suppose that Assumptions 1-3 hold, there exist positive constants Ψ ∈
(0, 1) defined in (6) in Definition 1, and η1 , η2 , η3 , l1 , l2 such that T η3 (i) : 2l1 − 2η1 + 1−σ IN + cγk B 2+B < 0, (ii) : 2l2 − η3 ≤ 0, (iii) :
η2 (1−Ψ)
sup −τ ≤s≤0
≥ eη1 Ψ. 1−µ 2
V (s)
Then the controlled delayed dynamical network (9) is the finite-time synchronized. 13
ACCEPTED MANUSCRIPT
T T T Proof: Let e(t) = (eT 1 (t), e2 (t), ..., eN (t)) and define the Lyapunov function as
V (t) =
N X
eT i (t)ei (t)
i=1
η3 + 1−σ
Z
t
eT (s)e(s)ds.
(12)
t−τ (t)
CR IP T
Then, its derivative with respect to time t along with solutions of (12) can be calculated as follows. when tm ≤ t ≤ sm , m = 0, 1, 2, ... V˙ (t) = 2
N X
N h X τ τ ˜ f (t, xi , Π, xi , Π ) + c bij Γej (t) − η1 ei (t)
eT i (t)
i=1
×bij Γej (t) − 2η1
N X i=1
eT i (t)ei (t) − 2η2
M
i=1
AN US
j=1 1+µ η Z t ei (t) i 2 3 T µ e (s)e(s)ds −η2 sign(ei (t))|ei (t)| − η2 1 − σ t−τ (t) ||e(t)||2 N N X η3 X T + ei (t)ei (t) − η3 eT i (t − τ (t))ei (t − τ (t)) 1 − σ i=1 i=1 N N N X N X X X T ≤ 2l1 eT (t)e (t) + 2l e (t − τ (t))e (t − τ (t)) + 2c eT i 2 i i i i (t) i=1
N X
i=1 j=1
µ eT i sign(ei (t))|ei (t)|
i=1
i=1
eki (t) 2l1 − 2η1 +
PT
=
N X n X
ED
N 1+µ η Z t η3 X T 2 3 eT (s)e(s)ds −2η2 + e (t)ei (t) 1 − σ t−τ (t) 1 − σ i=1 i N X −η3 eT i (t − τ (t))ei (t − τ (t))
i=1 k=1
N
N
(13)
n
XXX η3 k ei (t) + 2c γk eki (t) 1−σ i=1 j=1 k=1
N X bji + bij k ej (t) + (2l2 − η3 ) eT i (t − τ (t))ei (t − τ (t)) 2 i=1 N 1+µ η Z t X 2 3 T µ −2η2 ei (t)sign(ei (t))|ei (t)| − 2η2 eT (s)e(s)ds 1 − σ t−τ (t) i=1 n h T X η3 B + Bi k = (˜ ek (t))T 2l1 − 2η1 + IN + cγk ˜ e (t) + (2l2 − η3 ) 1−σ 2 k=1 N N X X T µ × ei (t − τ (t))ei (t − τ (t)) − 2η2 eT i (t)sign(ei (t))|ei (t)| i=1 i=1 η Z t 1+µ 2 3 −2η2 eT (s)e(s)ds , 1 − σ t−τ (t)
AC
CE
×
where ˜ ek = (ek1 , ek2 , ..., ekN )T for k = 1, 2, ..., n. Based on the conditions of (i) and (ii), we 14
ACCEPTED MANUSCRIPT
can get
PN
i=1
µ eT i (t)sign(ei (t))|ei (t)|
i=1
µ eT i (t)sign(ei (t))|ei (t)| =
ing Lemma 3, it can be implied that N X n X i=1 k=1
i=1 k=1
|eki (t)|1+µ
i=1
|eki (t)|1+µ
Hence, N X n X
PN
≥
1+µ η Z t 2 3 T e (s)e(s)ds . − 2η2 1 − σ t−τ (t)
µ |eT i (t)||ei (t)| =
1 1+µ
X N X n i=1 k=1
≥
N X n X i=1 k=1
|eki (t)|2
1+µ 2
=
N hX
eT i (t)ei (t)
i=1
(t).
k=1
|eki (t)|1+µ and us-
|eki (t)|2 N X
21
.
1+µ 2 . eT (t)e (t) i i
η3 + 1−σ
Z
t
t−τ (t)
i 1+µ 2 eT (s)e(s)ds
(15)
M
= −2η2 V
1+µ 2
i=1
i=1
Then, combining with Lemma 3, we can obtain V˙ (t) ≤ −2η2
PN Pn
(14)
CR IP T
Since
N X
AN US
V˙ (t) ≤ −2η2
Similarly, for sm < t < tm+1 , we have
BT + B i k 2l1 − 2η1 + η3 IN + cγk ˜ e (t) + (2l2 − η3 ) 2 k=1 N n X X (˜ ek (t))T 2η1˜ ek (t) × eT (t − τ (t))e (t − τ (t)) + i i (˜ ek (t))T
h
ED
n X
PT
V˙ (t) ≤
i=1
(16)
k=1
CE
≤ 2η1 V (t),
AC
Thus, we get
V˙ (t) ≤ −2η V 1+µ 2 (t), 2 V˙ (t) ≤ 2η1 V (t),
tm ≤ t ≤ sm ,
sm < t < tm+1 ,
(17)
By Lemma 5, we obtain
V
1−µ 2
(t) ≤
sup V (s) −τ ≤s≤0
1−µ 2
exp{η1 Ψ(1 − µ)t} − η2 (1 − Ψ)(1 − µ)t, 0 ≤ t ≤ T1 . (18)
and the settling time T can be given by T =
1 or T = T1 , η1 Ψ(1 − µ) 15
ACCEPTED MANUSCRIPT
1−µ sup V (s) 2 exp{η1 Ψ(1−µ)t}−η2 (1−Ψ)(1−µ)t =
where T1 is the smaller solution of
−τ ≤s≤0
0.
Hence, the zero solution of the error dynamical system (11) is finite-time stable, that is, the system (9) is finite-time synchronized. This completes the proof of Theorem 1. 2
x˙ i (t) = −Axi (t) + Dg(xi (t)) + Cg(xi (t − τ (t))) + c
CR IP T
As a special case, we consider a class of controlled coupled neural network N X
bij Γxj (t) + ui (t),
(19)
j=1
where i ∈ I , bij is defined in (2), xi (t) = (x1i (t), x2i (t)..., xni (t))T ∈ Rn denotes the state variable associated with the ith neuron, A = diag{a1 , a2 , ..., an } is the decay constant ma-
AN US
trix with ai > 0 for i = 1, 2, ..., n, D = (dij )n×n and C = (cij )n×n are the connection matrix
and delayed connection matrix, respectively, g(xi (t)) = (g1 (x1i (t)), g2 (x2i (t)), ..., gn (xni (t)))T is the activation function of the neurons and satisfies the following condition. 0 Assumption 1 . There exists a positive constant s > 0 such that
M
for any x, y ∈ Rn .
T g(x) − g(y) g(x) − g(y) ≤ s(x − y)T (x − y),
Correspondingly, the synchronization state Π(t) associated with (19) is represented by
ED
˙ Π(t) = −AΠ(t) + Dg(Π(t)) + Cg(Π(t)).
(20)
The following statement is provided to ensure the finite-time synchronization of the
PT
coupled network.
0
CE
Theorem 2. Suppose that Assumptions 1 and 2-3 hold, there exist positive constants Ψ ∈ (0, 1) defined in (6) in Definition 1, and η1 , η2 , η3 , λ, s such that T η3 (i) : λ − 2η1 + 1−σ IN + cγk B 2+B < 0, (ii) : s − η3 ≤ 0,
η2 (1−Ψ)
AC
(iii) :
sup
−τ ≤s≤0
≥ eη1 Ψ, 1−µ 2
V (s)
where λ = λmax (−A − AT + DDT + CCT + sIN ). Then the controlled delayed dynamical network (19) is finite-time synchronized in a finite-time T =
where T1 is the smaller solution of 0.
1 or T = T1 , η1 Ψ(1 − µ)
1−µ sup V (s) 2 exp{η1 Ψ(1−µ)t}−η2 (1−Ψ)(1−µ)t =
−τ ≤s≤0
16
ACCEPTED MANUSCRIPT
. Proof: From Theorem 1, the network is finite-time synchronized, if assumption 1 0
AN US
CR IP T
holds. Using Lemma 4 and assumption 1 , we can get T (xi (t) − yi (t)) f (t, xi (t), xi (t − τ (t))) − f (t, yi (t), yi (t − τ (t))) h = (xi (t) − yi (t))T − A(xi (t) − yi (t)) + D(g(xi (t)) − g(yi (t))) i + C(g(xi (t − τ (t))) − g(yi (t − τ (t)))) 1 ≤ (xi (t) − yi (t))T − A − AT + DDT + CCT + sIN (xi (t) − yi (t)) 2 T s + xi (t − τ (t)) − yi (t − τ (t))) xi (t − τ (t)) − yi (t − τ (t))) 2 T s λ xi (t − τ (t)) − yi (t − τ (t))) ≤ (xi (t) − yi (t))T (xi (t) − yi (t)) + 2 2 × xi (t − τ (t)) − yi (t − τ (t))) ,
s λ , l2 = . Hence, it follows from Theorem 1 2 2 that the coupled network (19) is finite-time synchronized to (20). The proof of Theorem
which shows assumption 1 holds and l1 = 2 is completed.
2
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Suppose Ψ = 0, the the aperiodically intermittent control becomes a general continuous control. Then based on Theorem 1 and Theorem 2, the following Corollary 1 and
ED
Corollary 2 is immediate, respectively.
Corollary 1. Suppose that Assumptions 1 and 3 hold, there exist positive constants
PT
η1 , η2 , η3 , l1 , l2 such that T η3 (i) : 2l1 − 2η1 + 1−σ IN + cγk B 2+B < 0, (ii) : 2l2 − η3 ≤ 0.
time:
CE
Then the controlled delayed dynamical network (10) is finite-time synchronized in a finite 1−µ
V 2 (0) T ≤ . η2 (1 − µ) 0
AC
Corollary 2. Suppose that Assumptions 1 and 3 hold, there exist positive constants
η1 , η2 , η3 , λ, s such that T η3 (i) : λ − 2η1 + 1−σ IN + cγk B 2+B < 0, (ii) : s − η3 ≤ 0,
where λ = λmax (−A − AT + DDT + CCT + sIN ). Then the controlled delayed dynamical network (19) is finite-time synchronized in a finite time: 1−µ
V 2 (0) T ≤ . η2 (1 − µ) 17
ACCEPTED MANUSCRIPT
Suppose τ (t) = 0, we can rewrite network model (9) as follows: x˙ i (t) = f (t, xi (t)) + c
N X
bij Γxj (t) + ui (t),
(21)
j=1
CR IP T
where ui (t) is the aperiodically intermittent feedback control gain defined as follows: −η e (t) − η sign(e (t))|e (t)|µ , tm ≤ t ≤ sm , 1 i 2 i i ui (t) = (22) 0, sm < t < tm+1 ,
T where m = 0, 1, 2, ..., |ei (t)|µ = |e1i (t)|µ , |e2i (t)|µ , ..., |eni (t)|µ , sign(ei (t)) = diag sign(e1i (t)), T sign(e2i (t)), ..., sign(eni (t)) , η1 , η2 are the positive constants control strengths, and the
AN US
real number µ satisfies 0 ≤ µ < 1.
Correspondingly, the synchronization state Π(t) associated with (21) is represented by ˙ Π(t) = f (t, Π(t)),
i ∈ I.
(23)
Based on Theorem 1 and Theorem 2, we can get the following Corollary 3 and Corollary
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4, respectively. Corollary 3. Suppose that Assumptions 1 and 2 hold, there exist positive constants Ψ ∈ (0, 1) defined in (6) in Definition 1, and η1 , η2 , l1 , l2 such that T +B
η2 (1−Ψ)
(ii) :
sup
< 0,
= eη1 Ψ. 1−µ 2
V (s)
−τ ≤s≤0
2
ED
(i) : (2l1 − 2η1 )IN + cγk B
finite time
PT
Then the controlled delayed dynamical network (21) is the finite-time synchronized in a 1 or T = T1 , η1 Ψ(1 − µ) 1−µ where T1 is the smaller solution of sup V (s) 2 exp{η1 Ψ(1−µ)t}−η2 (1−Ψ)(1−µ)t =
AC
0. .
CE
T =
−τ ≤s≤0
T T T Proof: Let e(t) = (eT 1 (t), e2 (t), ..., eN (t)) and define the Lyapunov function as
V (t) =
N X
eT i (t)ei (t).
(24)
i=1
The other proof is the similar as Theorem 1, here it is omitted. Hence, the proof of Corollary 1 is completed. 0
Corollary 4. Suppose that Assumptions 1 and 2 hold, there exist positive constants Ψ ∈ (0, 1) defined in (6) in Definition 1, and η1 , η2 , λ, s such that 18
ACCEPTED MANUSCRIPT
(i) : (ii) :
η3 1−σ η2 (1−Ψ)
λ − 2η1 + sup
T +B
IN + cγk B
2
< 0,
≥ eη1 Ψ, 1−µ 2
V (s)
−τ ≤s≤0 where λ = λmax − A − AT + DDT + CCT + sIN . Then the controlled delayed dynamical
CR IP T
network (21) is finite-time synchronized in a finite time 1 T = or T = T1 , η1 Ψ(1 − µ) 1−µ where T1 is the smaller solution of sup V (s) 2 exp{η1 Ψ(1−µ)t}−η2 (1−Ψ)(1−µ)t = −τ ≤s≤0
0.
Remark 2: In [24], finite-time synchronization problem for ordinary differential inequalities with periodically intermittent control is condered. But, Lemma 3 in [24] is only
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used to cope with complex networks without delay. However, it is well known that time delay is a very common phenomenon in some fields and may result in undesirable dynamic behaviors, such as oscillation behavior and network instability. Therefore, it is very essential to investigate the influence of time delay on the synchronization. In this paper, by constructing appropriate controller and Lyapunov function, we deal with finite-time
M
synchronization analysis for delayed dynamical networks.
Remark 3: In [24–26], the authors coped with stability analysis for general differential inequalities with periodically intermittent approach. However, we give Lemma 5
ED
to cope with the finite-time synchronization of complex dynamical networks via aperiodically intermittent control. If tm+1 − tm ≡ T, sm − tm ≡ δ, where T, δ are positive
PT
constants, t0 = 0, m = 0, 1, 2, ..., the aperiodically intermittent control strategy becomes the periodic type, i.e. Lemma 5 in this paper is equivalent to Lemma 3 in [24]. This is
to say, results in [24] are the special case of our results. Besides, in [24–26], they did not
CE
discuss the value of the settling time T . Because the value of the settling time T may be
AC
one, may not exist, or may be two. Hence, we give a discussion about the value of the settling time T . By analyzing, the settling time T can be given by: (i): If
α(1−Ψ)
sup
V (s)
−τ ≤s≤0
1−η < eβΨ, there is no solution of the (H13 ). Hence, the value of
the settling time T does not exist. α(1−Ψ) 1−η = eβΨ, there is only one solution of the (H13 ). Hence, the value (ii): If sup
V (s)
−τ ≤s≤0
of the settling time T is
T = (iii): If
α(1−Ψ) sup −τ ≤s≤0
1 . βΨ(1 − η)
1−η > eβΨ, there are two solutions of the (H13 ). Denoted as T1 ,
V (s)
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T2 and T1 < T2 . Here, the value of the settling time T is T = T1 , where T1 is the smaller 1−η solution of sup V (s) exp βΨ(1 − η)t − α(1 − Ψ)(1 − η)t = 0. −τ ≤s≤0
4
Numerical simulations
achieved in this paper.
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In this section, a chaotic network is given to present the effectiveness of our results Example. Consider the following 2-D coupled networks model with variable delay as follows:
i = 1, 2, ..., 8,
bij Γxj (t) + ui (t),
j=1
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x˙ i (t) = Axi (t) + Dg1 (xi (t)) + Cg2 (xi (t − τ (t))) + c
2 X
(25)
where xi (t) = (x1i (t), x2i (t)) ∈ R2 , i = 1, 2, ..., 8, g1 (u) = g2 (u) = (tanh(u1 ), tanh(u2 )),
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τ (t) = et /(1 + et ), c = 1, and −1.5 −0.1 1 0 −1 0 , C = , Γ = . A= −0.1 −1.5 0 1 0 −1
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In the case that network (25) reaches complete synchronization, that is, lim kxi (t) −
Π(t)k = 0, i = 1, 2, ..., 8, we have the following synchronized state equation:
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˙ Π(t) = AΠ(t) + Dg1 (Π(t)) + Cg2 (Π(t − τ (t))).
t→∞
(26)
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We choose different connection matrices D presented by 2 −0.1 2 −0.1 , D2 = . D1 = −4 2 −4 3
The different dynamic properties of (26) with the initial values (Π1 (s), Π2 (s))T = (0.8, 0.6)T with s ∈ [−1, 0] can be emerged, which are revealed in Fig. 2 and the synchronized states are chaotic attractor in those case. In the following, we only consider the case D = D1 . Moreover, it is easy to verify that s = 1, λ = λmax (−A − AT + DDT + CCT + sIN ) =
24.5047. Hereinafter, we will choose suitable control parameters such that (25) achieves the finite-time synchronization.
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Here, we choose B=
−2
0.72
0.2
0.4
0.2
0.3
0.3
0.32 0.28
−4
0.48 0.48 0.72 0.56 0.64 0.16 0.13 0.17 0.15
0.6
−1
0.32
0.48 0.72 0.2
0.1 0.8
0.1
0.64
0.6
0.6
−4
0.68
0.68 0.44
0.3
0.32
−4 0.2
0.64
−2
0.32
0.52 0.72 0.56 0.64 0.64 0.52
0.4
0.3
0.68 0.56 0.64 0.52 0.48 0.36
0.4 0.19 0.4 . 0.4 0.36 0.76 −4
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−4
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By computation, let η1 = 14, η3 = 1, σ = 1/4, µ = 7/8, τ = 1, then we can obtain the conditions (i), (ii) of Theorem 2 are satisfied. Moreover, the different dynamic properties of (25) with the initial values x11 = 0.1, x12 = −0.45, x13 = 0.45, x14 = 0.2, x15 = −0.2, x16 =
1, x17 = 2, x18 = −1, x21 = −2, x22 = 2, x23 = −0.5, x24 = 1, x25 = 0.8, x26 = −0.4, x27 =
0.6, x28 = 0.4 with s ∈ [−1, 0]. By computing, we can get sup V (s) = 45.3250. The −1≤s≤0
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intermittent control exists on time span
[0, 3] ∪ [3.2, 6.4] ∪ [6.5, 9.6] ∪ [9.8, 12.8] ∪ [13, 16] ∪ [16.2, 19.2] ∪ [19.5, 22.6] ∪ [22.8, 25.8] ∪ · · · η2 (1−Ψ)
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So, θ = 3, ω = 3.3. By computing, we get Ψ = 1/11. When
sup
= eη1 Ψ, 1−µ 2
V (s)
−τ ≤s≤0
we obtain that η2 = 1.7769e. From Theorem 2, the networks can be synchronized with
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finite-time T = 6.286. Fig. 3 and Fig. 4 show the synchronization of dynamics, and Fig. 5 and Fig. 6 show the errors of dynamics.
3
3 2
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2
4
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4
1 Π2
Π2
1 0
0
−1
−1
−2
−2
−3 −0.8
−0.6
−0.4
−0.2
0 Π1
0.2
0.4
0.6
−3 −1
0.8
(a)
−0.8
−0.6
−0.4
Π1
−0.2
0
0.2
(b)
Fig. 2. The chaotic attractor of neural networks (26) with different D and the same initial conditions Π1 (s) = 0.8, Π2 (s) = 0.6. (a) D = D1 . (b) D = D2 .
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4
4
3
2
2
0 −2
Π2, x2 (i=1,2,...,8) i
0 −1 −2
−4 −6 −8
−3
−10
−4
−12
−5 −6 0
5
t
10
−14 0
15
||e1(t)||
20
5
t
10
15
Fig. 4 The synchronization of state x2i (t) and Π2 (t).
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Fig. 3 The synchronization of state x1i (t) and Π1 (t).
||e1(t)||
20
15
||e1(t)||
15
||e1(t)||
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Π1, x1 (i=1,2,...,8) i
1
10
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5
0
5
10 t
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0 15
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Fig. 5 The synchronization error ||e1 || =
8 P
i=1
20
10
5
0 0
|x1i − Π1 |.
5
10 t
15
Fig. 6 The synchronization error ||e2 || =
20 8 P
i=1
|x2i − Π2 |.
In the following, we consider the following 2-D coupled networks model without delays
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as follows:
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x˙ i (t) = Axi (t) + Dg1 (xi (t)) + Cg2 (xi (t)) + c
2 X
bij Γxj (t) + ui (t),
j=1
i = 1, 2, ..., 8,
(27)
where all parameters are the same as the system (25). Corresponding synchronization state equation is given as following: ˙ Π(t) = AΠ(t) + Dg1 (Π(t)) + Cg2 (Π(t)).
(28)
Next, we give the state of system (28) and the synchronization figures of system (27) and system (28).
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1 0
Π2
−1
−3 −4
0.4
0.5
0.6
Π1
0.7
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−2
0.8
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Fig. 7. The state of neural networks (28) with the same initial conditions Π1 (s) = 0.8, Π2 (s) = 0.6. 2
2 1
1
0
−1
i
1
0.5
Π ,x2(i=1,2,...,8)
Π1,xi (i=1,2,...,8)
1.5
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2
0
−1 0
5
t
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−0.5
10
−3 −4 0
15
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Fig. 8 The synchronization of state x1i (t) and Π1 (t). 8
||e2(t)||
5
3
4 3 2
2
1
1
0 0
15
2
4
10
6
AC
||e1(t)||
5
t
7
||e1(t)||
||e (t)||
6
5
Fig. 9 The synchronization of state x2i (t) and Π2 (t).
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7
−2
5
10 t
15
Fig. 10 The synchronization error ||e1 || =
0 0
20 8 P
i=1
|x1i − Π1 |.
5
10 t
15
Fig. 11 The synchronization error ||e2 || =
20 8 P
i=1
|x2i − Π2 |.
Remark 4: In numerical simulations, it can be seen from the Fig. 2 that D plays a crucial role for the chaotic dynamics of neural networks (26). Eight nodes are chosen 23
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in the slave system and some corresponding parameters, which satisfy conditions (i) and (ii), are given. Under the condition (iii), a feasible aperiodically intermittent control time interval is presented. Hence, all conditions of Theorem 2 are satisfied in the numerical simulation. The states of master and slave systems are given in Fig. 3 and Fig. 4, and the errors trajectories of the systems (25) and (26) are shown in Fig. 5 and Fig. 6. It
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can be seen that the synchronization of systems (25) and (26) is indeed solved and the simulation results demonstrate the theoretical analysis very well.
Remark 5: As can be seen in Figure 7, the system (27) without delay is convergent to the equilibrium point. However, the system (25) with time delay is a chaotic system as figure 2. For the two cases, by applying the controller to the corresponding slave system, we can get that the master system and the corresponding slave system can be
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synchronized. Based on the simulation, it is also shown that the delay not only affects the dynamic behavior of the system, but also affects the rate of synchronization. Remark 6: Over the past decades, much attention has been attracted to the field of Markov jump systems. Much effort has been devoted to such systems and some important results have been obtained [31–33]. However, little attention has been paid to the stability and synchronization problem for nonlinear Markov jump systems. Hence, this is an open
5
Conclusion
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problem and will be considered in our future works.
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In this paper, the finite-time synchronization for a general coupled complex networks model with delay is investigated via aperiodic intermittent control. As we know, this is the first time to cope with finite-time synchronization problem with aperiodically inter-
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mittent strategy. By establishing a new differential inequality and constructing Lyapunov function, we present several simple yet useful sufficient criteria of coupled networks under aperiodically intermittent control. Furthermore, when Ψ = 0 or τ (t) = 0, we get two spe-
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cial cases, which considered in [3] or [26], respectively. In this sense, the results derived here are more general. What is more, by using the properties of fundamental functions, we give the value of finite time T . Finally, the numerical example is given to demonstrate the validness of the proposed scheme. Acknowledgments This work was supported by National Natural Science Foundation of Peoples Republic of China (Grants Nos. 61473244, 61563048, 11402223), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20136501120001), Natural Science Foundation of Xinjiang (Grant No. 2014211B002), Natural Science Foun24
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dation of Xinjiang University (Grant No. BS130108), Project funded by China Postdoctoral Science Foundation (Grant No. 2013M540782 and No. 2014T70953).
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