Exponential synchronization of chaotic neural networks with time-varying delay via intermittent output feedback approach

Applied Mathematics and Computation 314 (2017) 121–132

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Exponential synchronization of chaotic neural networks with time-varying delay via intermittent output feedback approach Zhi-Ming Zhang a, Yong He b,c,∗, Min Wu b,c, Qing-Guo Wang d a

School of Information Science and Engineering, Central South University, Changsha, Hunan 410083, China School of Automation, China University of Geosciences, Wuhan, Hubei 430074, China c Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, Wuhan, Hubei 430074, China d Institute for Intelligent Systems, University of Johannesburg, Johannesburg, South Africa b

a r t i c l e

i n f o

Keywords: Exponential synchronization Neural networks Time-varying delay Lyapunov–Krasovskii functional Intermittent output feedback control

a b s t r a c t This paper is dealt with the problem of exponential synchronization for chaotic neural networks with time-varying delay by using intermittent output feedback control. Based on the Lyapunov–Krasovskii functional method and the lower bound lemma for reciprocally convex technique, a novel criterion for existence of the controller is first established to ensure synchronization between the master and slave systems. Moreover, from the delay point of view, the derived criterion is extended to the relaxed case because of introducing an adjustable parameter in the Lyapunov–Krasovskii functional. Finally, a numerical simulation is carried out to demonstrate the effectiveness of the proposed synchronization law. © 2017 Elsevier Inc. All rights reserved.

1. Introduction Since Pecora and Carroll [1] first proposed the master–slave concept to achieve the synchronization for two identical chaotic systems with different initial conditions, chaos synchronization has been extensively investigated due to its potential applications in secure communication and cryptography, harmonic oscillation generation, and some other nonlinear fields [2–4]. In fact, as special complex networks, neural networks such as Hopfield neural networks, recurrent neural networks and BP neural networks have also been found to exhibit complicated dynamic characteristics and even chaotic behavior. Therefore, the study of stability analysis and control for neural networks has become a hot topic in the past decades [5–14]. Up to now, numerous papers on the subject have been published. And then, a number of synchronization and control schemes have been put forward, for example, sampled-data control [15–17], impulsive control [18,19], output feedback control [20,21], and intermittent control [22–26]. In recent years, intermittent control has become a key control strategy due to its broad potential applications in engineering fields and the special role in explaining the mechanism of human physiological and imitating human behavior [27–29]. In comparison with impulsive control, intermittent control is more easily implemented in practical application since it is not activated instantaneously. On the other hand, compared with the continuous control, intermittent control is more economical because the control signal is reset intermittently. Therefore, intermittent control can be regarded as a link between impulsive and continuous feedback control and integrates those merits. Until now, many papers on intermittently controlled systems have been carried out, for example, see [22–25,30–33] and references therein. ∗

Corresponding author at: School of Automation, China University of Geosciences, Wuhan, Hubei 430074, China. E-mail address: [email protected] (Y. He).

http://dx.doi.org/10.1016/j.amc.2017.07.019 0 096-30 03/© 2017 Elsevier Inc. All rights reserved.

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In addition, time delay, especially time-varying delay is inevitable in modeling realistic neural networks due to various reasons, such as the signal transmission speed among the neurons, finite speed of information processing and the finite switching speed of amplifier circuits. It is well known that the existence of time delay also causes poor performance or even chaotic behavior. Therefore, it is very important not only in theory but also in practical applications to study intermittent control systems with time-varying delay. In the available intermittent control literatures, the Lyapunov function method and the Lyapunov–Krasovskii functional method are often introduced to analyze the stabilization and synchronization of the considered systems. Especially, the latter can provide more useful state information and delay information, which has attracted a lot of interests. Meanwhile, some approaches are developed to reduce the possible conservativeness of the Lyapunov– Krasovskii functional method, such as free-weighting matrices method [35], delay-partitioning scheme [36,37] and some integral inequalities technique [39–44]. It should be pointed out that for the derivation upper bound of time-varying delay, many scholars often assume that it is less than 1, that is, the results reported in the existing literature are only applicable to the case of slow time-varying delays, see [15,30,32,33,45,46]. In the framework of intermittent control, some types of the intermittent controllers have been designed to achieve the synchronization and stabilization problems for various dynamic systems, for example, the adaptive intermittent controller [47], the intermittent predictive controller [48], the intermittent pinning controller [49,50], the periodically intermittent feedback controller [33,34,51,52], and the delayed intermittent feedback controller [30,31,53]. It is easy to see that the controllers designed in the above references are based on intermittent state-feedback, in other words, all state information must be known. However, in most real control situations, the system state cannot be fully captured, or the cost of obtaining some state information is huge. Therefore, there is a strong need to design an intermittent output feedback controller instead of an intermittent state-feedback controller for the sake of obtaining a better performance and dynamical behavior of the state response. To the best of our knowledge, no results on synchronization for chaotic neural networks with time-varying delay via intermittent output feedback control have been reported in the literature. Motivated by the preceding discussions, in this paper, we will investigate the problem of exponential synchronization for chaotic neural networks with time-varying delay. First, the intermittent output feedback controller is presented. Second, by constructing a novel Lyapunov–Krasovskii functional and employing the lower bound lemma for reciprocally convex approach, linear transformation technique and linear matrix equality formulation, a novel criterion for existence of the controller is first derived in terms of linear matrix equalities and linear matrix equality, which can guarantee the master system to synchronize with the slave system. Additionally, the traditional assumption that the delay-derivation upper bound of time-varying delay is restricted to be smaller than 1 is removed in this paper. Finally, a simulation example is given to demonstrate the effectiveness and the benefits of the proposed methods. This paper is organized as follows. In Section 2, model description and preliminaries are given. Some new criteria are obtained in Section 3 to ensure the exponential synchronization for chaotic neural networks with time-varying delay. In Section 4, the effectiveness of the theoretical results is shown by a numerical example. Throughout this paper, the superscripts ‘−1’ and ‘T’ stand for the inverse and transpose of a matrix, respectively; n denotes the n-dimensional Euclidean space; p × q is the set of all p × q real matrices; P > 0( < 0, ≤ 0, ≥ 0) means that the matrix is symmetric positive(negative, semi-negative, semi-positive) definite matrix; P symmetric terms in a symmetric matrix are denoted by ‘∗ ’ ; ‘I’ is an appropriately dimensioned identity matrix; λmin (P) strands for the minimum eigenvalue of the matrix P; Sym{X } = X + X T . 2. Problem formulation and preliminaries Consider the following chaotic neural networks with time-varying delay



x˙ m (t ) = −Axm (t ) + D f (xm (t )) + Eg(xm (t − τ (t ))) + ν (t ), t > 0 ym (t ) = Cxm (t ) xm (t ) = ψ (t ), ∀t ∈ [−d, 0]

(1)

where xm (t) ∈ n is the state vector of the master system associated with n neutrons; ym (t) ∈ p is the output of the master system; ν (t) is an external input vector; f( · ), g( · ): n → n , represent the neutron activation functions with respect to the current state xm (t) and delayed state xm (t − τ (t )), respectively; A ∈ n × n is the self-feedback term; C ∈ p × n , D ∈ n × n and E ∈ n × n are known connection weight matrices; with loss of generality, one can assume that rank{C } = p.The initial condition of the master system, ψ (t), denotes a continuous vector-valued function on the interval [−d, 0]. The interval timevarying delay τ (t) satisfies

0 ≤ τ (t ) ≤ d,

τ˙ (t ) ≤ μ < ∞

(2)

where d and μ are constants. Remark 1. It is worth noting that the derivative upper bound of the time-varying delay, μ, plays a key role in analyzing the exponential stability and synchronization of time-varying-delayed systems. Generally, μ is restricted to be less than 1 (see [15,30,33,34], etc.), and such constraint in some papers is not given directly in their conditions, for example, [32,45,46], however, the obtained results can only be used to deal with the case. In this paper, the derivative upper bound is more general since μ may be less or more than 1, namely, the system considered may be slow or fast time-varying delay system. Thus, the criterion to be developed in this paper has the wider range in comparison with the previous ones.

Z.-M. Zhang et al. / Applied Mathematics and Computation 314 (2017) 121–132

123

For the neutron activation functions f( · ) and g( · ) with f (0 ) = g(0 ) = 0, suppose that there exist two positive diagonal matrices Lf and Lg such that for any x, y, xτ , yτ ∈ n , there hold

f ( x ) − f ( y ) 2 ≤ ( x − y )T L f ( x − y )

g( x τ ) − g( y τ ) 2 ≤ ( x τ − y τ ) T L g ( x τ − y τ )

(3)

where xτ = x(t − τ (t )), yτ = y(t − τ (t )). In this paper, we regard system (1) as the master system, and the slave system for (1) can be described by the following equation:



x˙ s (t ) = −Axs (t ) + D f (xs (t )) + Eg(xs (t − τ (t ))) + ν (t ) + Bu(t ), t > 0 ys (t ) = Cxs (t ) xs (t ) = φ (t ), ∀ t ∈ [−d, 0]

(4)

where xs (t) ∈ n is the state vector of the slave system associated with n neutrons; ys (t) ∈ p is the output of the slave system; B ∈ n × q is constant matrix; φ (t) denotes a continuous vector-valued initial function of the slave system on the interval [−d, 0]; u(t) ∈ q is the control input that will be designed in order to obtain a certain control objective. Let the error signal be r (t ) = xs (t ) − xm (t ). The error system can be represented as follows



r˙ (t ) = −Ar (t ) + D fˆ(r (t )) + E gˆ(r (t − τ (t ))) + Bu(t ), t > 0 yr (t ) = Cr (t ) r (t ) = ϕ (t ), ∀ t ∈ [−d, 0]

(5)

where yr (t ) = ys (t ) − ym (t ), fˆ(r (t )) = f (xs (t )) − f (xm (t )), gˆ(r (t − τ (t ))) = g(xs (t − τ (t ))) − g(xm (t − τ (t ))), and ϕ (t ) = φ (t ) − ψ (t ) is the initial condition of error system. The intermittent output feedback controller is adopted in this paper to make the slave system (4) synchronize with the master system (1), and it is given by



u(t ) =

F yr (t ), t ∈ [l T , l T + δ ) 0, t ∈ [lT + δ, (l + 1 )T )

(6)

where l ∈ Z0+ , F ∈ q × p is a control gain matrix, T is the control period, 0 < δ ≤ T, and δ > 0 is the so-called control width. Substituting controller (6) into system (5) gives the following closed-loop system:



r˙ (t ) =

−(A − BF C )r (t ) + D fˆ(r (t )) + E gˆ(r (t − τ (t ))), t ∈ [lT , lT + δ ) −Ar (t ) + D fˆ(r (t )) + E gˆ(r (t − τ (t ))), t ∈ [lT + δ, (l + 1 )T ).

(7)

In development of our main results, we need the following definition and lammas. Definition 1. (Liu et al. [38]). The master system (1) and the slave system (4) are said to be exponential synchronized if there exist k > 0 and N > 0 such that every solution r(t) of the system (7) satisfies

r (t ) ≤ N ϕ e−kt

(8)

where the constant k is defined as the exponential synchronization rate, and ϕ = sup−d2 ≤θ ≤0 r (θ ) , ∀t ≥ 0. Lemma 1. (Gu et al. [39]). For any positive definite matrix M ∈ n × n , a scalar d > 0, and vector function z: [0, d] → n such that the integrations concerned are well defined, the following inequality holds:

 −

t

1 z (s )Mz(s )ds ≤ − d



T

t−d

t

t−d

z(s )ds

T 

t

M t−d



z(s )ds .

(9)

Lemma 2. (Park et al. [54]). Define, for all vector ξ ∈ p , the function (h, R) by:

1 h

(h, R ) = ξ T W1T RW1 ξ +

1 ξ T W2T RW2 ξ 1−h

for positive integers q, p, and a scalar h ∈ (0, 1), a positive definite matrix R ∈ q × p , two matrices W1 and W2 ∈ q × p . If there exists a matrix X ∈ q × q such that



R ∗



X >0 R

then the following inequality holds



min (h, R ) ≥

h∈ ( 0, 1 )

W1 ξ W2 ξ

T 

R ∗

X R



W1 ξ W2 ξ

 .

The objective of this paper is to design a controller with the form (6) to achieve the exponential synchronization of the master system (1) and the slave system (4). In other words, we are interested in finding feedback gain matrix F such that the controlled error system (7) is exponentially stable.

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3. Main results In this section, we use the Lyapunov–Krasovskii functional method and employ lower bound lemma for reciprocally convex approach to derive some sufficient conditions. These conditions ensure that the master system (1) and the slave system (4) are exponential synchronized. 3.1. Stability analysis of the controlled error system Theorem 1. Suppose condition (2) and ρμ < 1(0 < ρ ≤ 1). The controlled error system (7) is exponentially stable, if, for given scalars α > 0 and β > 0, there exist n × n-symmetry matrices P > 0, Q > 0, S > 0, R > 0, n × n-matrix H and q × p-matrix F such that the following conditions hold

⎡ 11 ⎢ ∗ ⎢ ∗ ⎢ =⎢ ⎢ ∗ ⎢ ∗ ⎣ ∗

12 22

13 23 33

∗ ∗ ∗ ∗ ∗

∗

⎡ 11 ⎢ ∗ ⎢ ∗ ⎢ =⎢ ⎢ ∗ ⎢ ∗ ⎣ ∗

12 22 ∗ ∗ ∗ ∗ ∗

∗



0 0 0

15 0 0 0

∗ ∗ ∗ ∗

44 ∗ ∗ ∗

55

13 23 33

0 0 0

15

∗ ∗ ∗

55

44

∗ ∗ ∗ ∗

∗ ∗

0 0 0 ∗ ∗

⎤

PB 0 0 0 PB −I ∗

PD 0⎥ 0⎥ ⎥ 0⎥ ⎥<0 P D⎥ ⎦ 0 −I

PD 0 0 0 PD −I ∗

PE 0⎥ 0⎥ ⎥ 0⎥ ⎥<0 PE ⎥ ⎦ 0 −I

(10)

⎤

(11)



R Rˆ = ∗

H >0 R

(12)

αδ − β (T − δ ) > 0

(13)

where

11 = −PA − AT P + PBF C + C T F T BT P + 2α P + Q + S − R + L f 12 = R − H, 13 = H, 15 = −AT P + C T F T BT P 22 = H + H T − 2R + Lg , 23 = R − H, 33 = −R − e2αd Q 44 = −e−2αρ d (1 − ρμ )S, 55 = d2 e2αd R − 2P 11 = −PA − AT P − 2β P + Q + S − R + L f , 15 = −AT P. Proof. Construct the following candidate Lyapunov–Krasovskii functional for the error system (7),

V (t, rt ) = r T (t )P r (t ) +  +

t

t −ρτ (t )



t

t−d

e2α (s−t ) r T (s )Qr (s )ds

e2α (s−t ) r T (s )Sr (s )ds + d



0

−d



t

t+θ

e2α (s−t+d ) r˙ T (s )Rr˙ (s )dsdθ .

(14)

Calculating the derivative of V(t, rt ) with respect to t along the solutions of (7) yields

V˙ (t, rt ) = 2r T (t )P r˙ (t ) + r T (t )(Q + S )r (t ) − e−2α d r T (t − d )Qr (t − d ) −e−2αρτ (t ) (1 − ρ τ˙ (t ))r T (t − ρτ (t ))Sr (t − ρτ (t ))  t  +d2 e2α d r˙ T (t )Rr˙ (t ) − d e2α (s−t+d ) r˙ T (s )Rr˙ (s )ds − 2α −2α



t−d

t

t −ρτ (t )

e2α (s−t ) r T (s )Sr (s )ds − 2α d



0 −d



t

t−d

t

t+θ

e2α (s−t ) r T (s )Qr (s )ds

e2α (s−t+d ) r˙ T (s )Rr˙ (s )dsdθ .

By Lemmas 1 and 2, there exists a n × n matrix X such that Rˆ > 0, and the following inequality is established,

(15)

Z.-M. Zhang et al. / Applied Mathematics and Computation 314 (2017) 121–132

 −d

t

t−d

e2α (s−t+d ) r˙ T (s )Rr˙ (s )ds ≤ −d



t−d

 = −d ≤− −

t

125

r˙ T (s )Rr˙ (s )ds

t

t −τ (t )

r˙ T (s )Rr˙ (s )ds − d

d r (t ) − r (t − τ (t )) τ (t ) d d − τ (t )



t −τ (t )

t−d

r˙ T (s )Rr˙ (s )ds

T

R r (t ) − r (t − τ (t ))

r (t − τ (t )) − r (t − d )



T

R r (t − τ (t )) − r (t − d )



≤ −ζ T (t )N T RˆNζ (t ) where ζ (t ) = [ξ1T (t ) ξ2T (t ) ξ3T (t )]T , ξ1 (t ) = [r T (t ) [ fˆT (r (t )) gˆT (r (t − τ (t )))]T , and



N=

I 0

−I I

0 −I

0 0

0 0

0 0

r T (t − τ (t ))

(16) r T (t − d )]T , ξ2 (t ) = [r T (t − ρτ (t ))

r˙ T (t )]T , ξ3 (t ) =



0 . 0

It follows from (3) that

0 ≤ r T (t )L f r (t ) − fˆT (r (t )) fˆ(r (t ))

(17)

0 ≤ r T (t − τ (t ))Lg r (t − τ (t )) − gˆT (r (t − τ (t )))gˆ(r (t − τ (t ))).

(18)

According to the operation mechanism of the controller (6), there exist running time and off time during every control period: [l T , l T + δ ) and [l T + δ, (l + 1 )T ). Thus, for t > 0, the upper bound of V(t, rt ) will be estimated into two parts. First, for t ∈ [l T , l T + δ ), by the first subsystem of system (7), the following zero equation is derived

0 = 2r˙ T (t )P [−r˙ (t ) − (A − BF C )r (t ) + D fˆ(r (t )) + E gˆ(r (t − τ (t )))].

(19)

From (15) to (19), we obtain

V˙ (t, rt ) + 2αV (t, rt ) ≤ r T (t )(−PA − AT P + P BF C + C T F T BT P + 2α P +Q + S + L f )r (t ) + 2r T (t )P B fˆ(r (t )) +2r T (t )P Dgˆ(r (t − τ (t ))) − e−2α d r T (t − d )Qr (t − d ) −e−2αρ d (1 − ρμ )r T (t − ρτ (t ))Sr (t − ρτ (t )) +r˙ T (t )(d2 e2α d R − 2P )r˙ (t ) − ζ T (t )N T RˆNζ (t )

+r T (t − τ (t ))Lg r (t − τ (t )) − fˆT (r (t )) fˆ(r (t )) −gˆT (r (t − τ (t )))gˆ(r (t − τ (t ))) +2r˙ T (t )[−(PA − P BF C )r (t ) + P D fˆ(r (t )) + P E gˆ(r (t − τ (t )))] =

ζ T (t )ζ (t ).

(20)

If  < 0, then we get

V˙ (t, rt ) < −2αV (t, rt ).

(21)

Integrating the preceding inequality for t ∈ [lT, t], we have

V (t, rt ) ≤ V (r (lT ))e−2α (t−lT )

(22)

which implies

V (r (lT + δ )) ≤ V (r (lT ))e−2αδ . Second, for t ∈ [lT + δ, (l + 1 )T ), similarly, we can get

V˙ (t, rt ) + 2αV (t, rt ) ≤ r T (t )(−PA − AT P − 2β P + Q + S + L f )r (t ) + 2r T (t )P B fˆ(r (t )) +2r T (t )P Dgˆ(r (t − τ (t ))) − e−2α d r T (t − d )Qr (t − d ) −e−2αρ d (1 − ρμ )r T (t − ρτ (t ))Sr (t − ρτ (t )) +r˙ T (t )(d2 e2α d R − 2P )r˙ (t ) − ζ T (t )N T RˆNζ (t )

+r T (t − τ (t ))Lg r (t − τ (t )) − fˆT (r (t )) fˆ(r (t )) −gˆT (r (t − τ (t )))gˆ(r (t − τ (t ))) +2r˙ T (t )[−PAr (t ) + P D fˆ(r (t )) + P E gˆ(r (t − τ (t )))]

(23)

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Z.-M. Zhang et al. / Applied Mathematics and Computation 314 (2017) 121–132

ζ T (t )ζ (t ) + (2α + 2β )rT (t )Y r (t ) ≤ ζ T (t )ζ (t ) + (2α + 2β )V (t, rt ). ≤

(24)

If  < 0, we have

V˙ (t, rt ) ≤ 2β V (t, rt ).

(25)

Integrating it for t ∈ [lT + δ, t] ⊂ [lT + δ, (l + 1 )T ), we get

V (t, rt ) ≤ V (r (lT + δ ))e2β (t−lT −δ )

(26)

which implies

V (r ((l + 1 )T )) ≤ V (r (lT + δ ))e2β (T −δ ) .

(27)

From (23) and (27), we have

V (r ((l + 1 )T )) ≤ V (r (lT + δ ))e2β (T −δ ) ≤ V (r (lT ))e−2αδ e2β (T −δ ) = V (r (lT ))e−[2αδ −2β (T −δ )]

≤ V (r ((l − 1 )T + δ ))e2β (T −δ ) e−[2αδ −2β (T −δ )]

≤ V (r ((l − 1 )T ))e−2αδ e2β (T −δ ) e−[2αδ −2β (T −δ )] .. . ≤ V (r (0 ))e−[2αδ −2β (T −δ )](l+1) .

(28)

Thus, for t ∈ [l T , l T + δ ), by (22), (28) and (10), we have

V (t, rt ) < V (r (0 ))e−[2αδ −2β (T −δ )]l e−2α (t−lT ) ≤ V (r (0 ))e−[2αδ −2β (T −δ )]l

= V (r (0 ))e2αδ −2β (T −δ ) e ≤e

2αδ −2β (T −δ ) T

−[2αδ −2β (T −δ )][(lT +δ )+(T −δ )] T

δ V (x (0 ))e −[2αδ−2Tβ (T −δ )]t ,

(29)

for t ∈ [lT + δ, (l + 1 )T ), by (10), (23), (26) and (28), we have

V (t, rt ) < V (r (lT + δ ))e2β (t−lT −δ )

≤ V (r (lT ))e−2αδ e2β (t−lT −δ )

≤ V (r (0 ))e−2αδ (l+1)+2β (T −δ ))l e2β (t−lT −δ ) ≤ V (r (0 ))e−[2αδ −2β (T −δ )](l+1) = V (r (0 ))e

−[2αδ −2β (T −δ )](l+1 )T T

≤ V (r (0 ))e

−[2αδ −2β (T −δ )]t T

.

(30)

From (29) and (30), we get

V (t, rt ) ≤ γ V (r (0 ))e

−[2αδ −2β (T −δ )]t T

(31)

2αδ −2β (T −δ ) δ T

where γ = e . Furthermore, it follows from (14) that

V (r (0 )) ≤ κ ϕ 2 , for some

κ >0

(32)

and

V (t, rt ) ≥ r T (t )P r (t ) ≥ λmin (P ) r (t ) 2 .

(33)

So, we get from (31) to (33) that



r (t ) ≤

αδ −β (T −δ ) κγ

ϕ e− T t . λmin (P )

(34)

By Definition 1, the master system (1) is globally exponentially synchronized with the slave system (4) under the controller (6). The proof is completed.  Remark 2. In [33], there is technical disadvantage that the authors considered the relationship between the interval timevarying delay and its interval, because such τ (t) does not exist. In this paper, the lower bound lemma is employed to deal with the relationship, which can reduce the possible conservatism. Remark 3. It is clear that the intermittent output feedback controller (6) will reduce to a continuous output feedback when δ = T . In this case, condition (13) in the Theorem 1 is automatically satisfied.

Z.-M. Zhang et al. / Applied Mathematics and Computation 314 (2017) 121–132

127

Fig. 1. Chaotic attractor of the master system (1).

Remark 4. In many practical processes, time-varying delay may be unknown or the delay is non-differentiable. Setting S = 0 in (14), we can easily obtain a criterion to deal with the case by using a procedure similar to Theorem 1. Remark 5. When ρ = 1, the condition (2) can be reduced to the case in [15,30,33,34], that is

0 ≤ τ (t ) ≤ d,

τ˙ (t ) ≤ μ < 1.

(35)

In order to deal with this case, we define

T ζˆ (t ) = ξ1T (t ) r˙ T (t ) ξ3T (t ) . Then, Corollary 1 can be obtained by applying an approach similar to Theorem 1. Corollary 1. Suppose that (35) is satisfied. For given scalars α > 0 and β > 0, the controlled error system is exponentially stable, if there exist n × n-symmetry matrices P > 0, Q > 0, S > 0, R > 0, n × n-matrix H and q × p-matrix F such that conditions (12) and (13), and the following conditions hold

˜ < 0,  ˜ <0 

(36)

˜ and  ˜ denote the two new matrices whose the fourth rows and columns of  and  are deleted, respectively, and 22 where  ˜ 22 = −2R + X + X − e−2α d (1 − μ )S + Lg in Theorem 1. is replaced by  3.2. The controller design One sufficient criterion for existence of the designed controller is derived in Theorem 1, however, the controller gain matrix F cannot be computed directly due to the fact that it is coupled with the positive definite matrix P in (10). In order to deal with the problem, inspired by [55], a method of changing matrix variables will introduced such that the following sufficient conditions can be solved by the YALMIP toolbox. Theorem 2. Suppose condition (H1 ) and ρμ < 1(0 < ρ ≤ 1). The intermittent output feedback control problem is solvable, if, for given scalars α > 0 and β > 0, there exist n × n-symmetry matrices W > 0, Qˆ > 0, Sˆ > 0, Rˆ > 0, n × n-matrix Hˆ , q × p-matrix X and an inverse p × p-matrix Y such that (13) and the following conditions hold

⎡ ˆ 11  ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ˆ =⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗

ˆ 12  ˆ 22 ∗ ∗ ∗ ∗ ∗ ∗ ∗

ˆ 13  ˆ 23 ˆ 33  ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 ˆ 44  ∗ ∗ ∗ ∗ ∗

ˆ 15  0 0 0 ˆ 55  ∗ ∗ ∗ ∗

D 0 0 0 D −I ∗ ∗ ∗

E 0 0 0 E 0 −I ∗ 0

W 0 0 0 0 0 0 ˆ 88  0

⎤

0 W ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥<0 ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎦ ˆ 99 

(37)

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Fig. 2. Chaotic attractor of the slave system (4) with u(t ) = 0.

⎡ ˆ 11  ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ˆ =⎢  ⎢ ∗ ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗



Rˆ Rˆ = ∗

ˆ 12  ˆ 22  ∗ ∗ ∗ ∗ ∗ ∗ ∗

ˆ 13  ˆ 23  ˆ 33  ∗ ∗ ∗ ∗ ∗ ∗

0 0 0 ˆ 44  ∗ ∗ ∗ ∗ ∗

ˆ 15  0 0 0 ˆ 55  ∗ ∗ ∗ ∗

D 0 0 0 D −I ∗ ∗ ∗

E 0 0 0 E 0 −I ∗ 0

W 0 0 0 0 0 0 ˆ 88  0

⎤

0 W ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥<0 ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎦ ˆ 99 

(38)



Hˆ >0 Rˆ

(39)

CW = Y C

(40)

where

ˆ 11 = Sym{−AW + BXC } + 2αW + Qˆ + Sˆ − Rˆ  ˆ 12 = Rˆ − Hˆ ,  ˆ 13 = Hˆ ,  ˆ 15 = −WAT + C T X BT  T ˆ 22 = −2Rˆ + Hˆ + Hˆ ,  ˆ 23 = −Hˆ + Rˆ  −2α d ˆ ˆ ˆ ˆ 33 = −e Q − R, 44 = −e−2αρ d (1 − ρμ )Sˆ 2 2 α d ˆ 55 = d e Rˆ − 2W,  ˆ 88 = −L−1 ,  ˆ 99 = −L−1  g f ˆ 11 = −AW − WAT − 2β W + Qˆ + Sˆ − Rˆ,  ˆ 15 = −WAT .  Moreover, the desired gain matrix in the controller (6) is given by F = XY −1 . Proof. Denote W = P −1 , T = diag{W , W , W , W , W , I, I}. Based on the concept of congruence transformation, the inequalities (10) and (11) by premultiplying and postmultiplying the matrices T T and T , respectively, after some manipulations, we obtain that

⎡

⎢ ⎢ ⎢ ⎢ ˆ 0 = ⎢ ⎢ ⎢ ⎣

ˇ 11  ∗ ∗ ∗ ∗ ∗ ∗

ˆ 12  ˇ 22  ∗ ∗ ∗ ∗ ∗

ˆ 13  ˆ 23  ˆ 33  ∗ ∗ ∗ ∗

0 0 0 ˆ 44  ∗ ∗ ∗

ˇ 15  0 0 0 ˆ 55  ∗ ∗

D 0 0 0 D −I ∗

⎤

E 0⎥ ⎥ 0⎥ ⎥ 0⎥<0 ⎥ E⎥ ⎦ 0 −I

(41)

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129

Fig. 3. Error signals without the controller u(t).

⎡ ˇ 11  ⎢ ∗ ⎢ ⎢ ∗ ˆ0=⎢  ⎢ ∗ ⎢ ⎢ ∗ ⎣ ∗

ˆ 12  ˇ 22

∗

ˆ 13  ˆ 23 ˆ 33 

∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗

0 0 0 ˆ 44  ∗ ∗ ∗

ˆ 15  0 0 0 ˆ 55  ∗ ∗

D 0 0 0 D −I ∗

⎤

E 0⎥ ⎥ 0⎥ ⎥ 0⎥<0 ⎥ E⎥ ⎦ 0 −I

(42)

ˇ 11 = Sym{−AW + BF CW } + 2αW + Qˆ + Sˆ − Rˆ + W L f W,  ˇ 15 = −WAT + W C T F T BT ,  ˇ 22 = −2Rˆ + Hˆ + Hˆ T + W LgW, where  ˇ 11 = −AW − WAT − 2β W + Qˆ + Sˆ − Rˆ + W L f W, Qˆ = W QW, Sˆ = W SW, Rˆ = W RW, Hˆ = W HW and the others are defined in  Theorem 2. Performing the similar idea in [55], there exist a real matrix Y such that the matrix equality (40) is satisfied. Let us define a new matrix variable

X = FY

(43)

ˆ 0 < 0 and  ˆ 0 < 0 are equivalent to LMIand apply the Schur complementary Lemma, the non-linear matrix inequalities  based conditions in (37) and (38), respectively. The proof is completed.  Remark 6. In this paper, the common equality constraint for intermittent output feedback control is adopted without using the iterative linear matrix inequality algorithm that it usually leads to high computational complexity. Recently, slack solutions in similar works [56–59] have been tried, and a novel method [60] is proposed to construct a lower conservative Lyapunov functional. In the future research, we will consider how to apply these latest results to investigate intermittent control systems with time-varying delay. 4. Numerical example In this section, an example is given to demonstrate the effectiveness of the proposed controller. Example 1. Consider a 3-dimensional chaotic neural networks with time-varying delay as follows



A=

 B=

1 −1 0 1 0 1

−8 1 11





0 −1 , D = 0

0  1 , C= 1 0

1



0.2 −0.5 0.3

0

−0.1 0.2 0.3



−0.2 0 , E= −0.7



2 0.2 0

0 −0.1 0



0 0 , 0.1



the external input vector ν (t ) = [0; 0; 0], the nonlinear function and the time-varying delay are described by f (s ) = [tanh(s1 ) tanh(s2 ) tanh(s3 )]T (s1 , s2 , s3 ∈ ) and τ (t ) = 0.38 + 0.38 sin(4t ), respectively.

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Fig. 4. The response curves of the controlled error system.

Fig. 5. Control signals when T = 2 and δ = 1.9.

For given α = 0.3, β = 3.2, ρ = 0.5, by taking T = 2, δ = 1.9, L f = Lg = diag{1, 1, 1}. Applying Theorem 2, it is found that the intermittent output feedback control problem can be solved by using the YALMIP Toolbox of MATLAB, and a desired gain in the controller (6) is derived



F = −2.7767

T

−4.6921 .

Set the initial conditions of the master–slave system be ψ (t ) = [0.2; 0.1; −0.5] and φ (t ) = [0.3; 0.5; −0.2], respectively. The chaotic behaviors of the master system (1) and the slave system (4) without the controller (6) are shown in Figs. 1 and 2, respectively. In Fig. 3, we plot the error signals of the master system and the slave system without the controller, and one sees easily that the master system (1) and the slave system (4) are not synchronization. The trajectories of r1 (t), r2 (t) and r3 (t) of the error system with the controller are given in Fig. 4. It shows that the slave system (4) synchronize with the master system (1) via the controller (6). And control signals of the designed controller are shown in Fig. 5. Remark 7. In most practical systems, only a part of the system states can be directly measured or used as the output of the system, which limits the application of the existing results on intermittent state feedback control [31–33] and references therein. Thus, in this paper, we design the intermittent output feedback controller to synchronize the master–slave systems. On the other hand, the coefficient matrix of the control input, B, is often assumed that B = I in [4,19,31–33], which leads to the case that the obtained criterion is not applied when the number of inputs and outputs is less than to the number

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of state variables. For this example, our results can be used to ensure synchronization between the master system and the slave system. Therefore, these comparisons show the superiority of the given results in this paper. Remark 8. As pointed out that the conditions obtained in [15,30,32–35,46] are only applied to the case of sup{τ˙ (t )} = μ < 1. However, it is easy to see that sup{τ˙ (t )} = 1.52 > 1 in this example. Therefore, the simulation result shows that the derived results in the delay point of view extend ones in existing literature. 5. Conclusions In this paper, the intermittent output feedback controller was designed to achieve the synchronization for two ideal chaotic neural networks with time-varying delay. By using Lyapunov–Krasovskii functional method and the lower bound lemma for reciprocally convex approach, a novel criterion for existence of the controller to ensure synchronization between the master and slave systems has been derived. 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