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Synchronization of stochastic impulsive discrete-time delayed networks via pinning control
Communicated by Prof. Y. Liu
Accepted Manuscript
Synchronization of stochastic impulsive discrete-time delayed networks via pinning control Yu Lin, Yu Zhang PII: DOI: Reference:
S0925-2312(18)30077-8 10.1016/j.neucom.2018.01.052 NEUCOM 19254
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
16 June 2017 6 November 2017 19 January 2018
Please cite this article as: Yu Lin, Yu Zhang, Synchronization of stochastic impulsive discrete-time delayed networks via pinning control, Neurocomputing (2018), doi: 10.1016/j.neucom.2018.01.052
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ACCEPTED MANUSCRIPT
Highlights • Pinning synchronization of stochastic impulsive discrete delay networks is studied; • Using the Razumikhin technique, synchronization criteria of networks are achieved; • Impulses do contribute to the synchronization of the networks;
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• Synchronization of the networks can be achieved even with enlarging impulses.
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Synchronization of stochastic impulsive discrete-time delayed networks via pinning control✩ Yu Lina , Yu Zhanga,∗ of Mathematical Sciences, Tongji University, 200092, China
Abstract
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a School
This paper is concerned with the synchronization of stochastic impulsive discrete-time delayed networks via pinning control. By using Lyapunov functions together with Razumikhin technique, some sufficient conditions about exponential synchronization of stochastic impulsive discrete-time delayed networks in mean square are developed. Two numerical examples are given to illustrate the effectiveness and superiority of the obtained results.
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Keywords: Delay, Pinning control, Impulse, Discrete-time. 1. Introduction
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causing instability of networks (see, e.g., [3–5, 8–13, 15, 23– 26, 29–37] and the references therein), therefore, it is necIn our realistic world, many real systems can be considered essary to consider dynamical networks with time delay. At as complex systems, such as the Internet, epidemic spread- the same time, the states of real networks are often subject ing networks, the World Wide Web, neural networks and so- to instantaneous perturbations or experience abrupt change at cial networks. A typical complex network is consisted of a certain moments, which can be called impulsive effects, imlarge number of interconnected dynamical nodes, which has pulsive effects can be caused by switching phenomena, frethe property of “scale-free” or “small-world” . Complex net- quency changes or other sudden noises, i.e., impulsive effects works have become a focus of research and have received exist in networks (see, e.g., [2, 4, 7, 8, 15–19, 22–34] and the more and more attention from various fields of science and references therein). Thus, it is necessary to study the discreteengineering. Complex networks tend to exhibit complex and time delayed networks with impulsive effects. interesting dynamical behaviors including synchronization, In the past few years, there have been some literature on the consensus, flocking etc. synchronization of impulsive discrete-time complex networks Synchronization is one of the most interesting and impor- (see, e.g., [16–18, 22–24] and the references therein). For tant collective behaviors in complex dynamical networks, it example, in [16], Zhang etc. studied the impulsive synchrohas attracted special attention of researchers in different fields nization problem of a continuous network composed with the chaotic Chen oscillators and a discrete-time network consist(see, e.g., [1–9, 12–27] and the references therein). There are various types of traditional and new control ing of H´enon map. [17] obtained some sufficient conditions methods have been successfully used to achieve network syn- for asymptotic H-synchronization between the drive system chronization, including impulsive control, adaptive control, and response system by the asymptotic stability criteria of and pinning control etc. Moreover, it is too difficult to con- systems with impulsive effects. In [22], Wu etc. studied the trol all nodes in a large-scale complex network, pinning con- synchronization problem of discrete-time network by impultrol can achieve synchronization of complex dynamical net- sive and pinning control. In [23], Li etc. investigated the exworks by pinning a small fraction of the network nodes, thus ponential synchronization of discrete-time complex networks it is economic and effective to achieve the synchronization with time-varying delay and obtained two synchronization of networks via pinning control. Lately the pinning control criteria by using the average impulsive interval. In [24], Li has stimulated many interesting for the synchronization of etc. obtained several synchronization criteria by using the dynamical networks (see, e.g., [18–20, 22, 25–27] and the Razumikhin technique and the discrete Gronwall inequality. Unfortunately, so far, the synchronization problem for references therein). It is well known that time delay shows as a vital character- stochastic impulsive discrete-time delayed complex networks istic of networks, and becomes one of the main sources for via pinning control has not been investigated yet. Motivated by the above discussions, the purpose of this paper is to investigate the synchronization of stochastic impul∗ Corresponding author at: School of Mathematical Sciences, Tongji Unisive discrete-time delayed networks via pinning control. The versity, 200092, China main contributions of this paper are highlighted as follows: Email address: [email protected], [email protected], Fax:+86-21-65981985. ( Yu Zhang) (1) it is the first time that synchronization of stochastic impulPreprint submitted to Neurocomputing
January 30, 2018
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hii = −
M X
j=1, j,i
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dk ))]T ∈ Rn is the activation function of the networks; c ∈ R (c , 0) is a constant; dk ∈ Z + corresponds to the discrete¯ where d, d¯ are known time delay which satisfies d ≤ dk ≤ d, positive integers and represent the lower and upper delays, respectively; N0 = 0, Nl ∈ Z + for l ∈ Z + , N1 < N2 < · · · < Nl < · · · , Nl → ∞ as l → ∞; κ(k, xi (k)− s(k)) ∈ Rn is the stochastic intensity; Γ ∈ Rn×n represents the inner coupling matrix between the subsystems; matrix H = (hi j ) M×M is the coupling configuration matrix representing the coupling strength and the topological structure of the networks which satisfies
sive discrete-time delayed complex network via pinning control is investigated; (2) based on Lyapunov stability theory, by designing pinning control and using Lyapunov functions together with Razumikhin technique, the synchronization of stochastic impulsive discrete-time delayed complex networks is achieved; (3) the obtained results show that impulses do contribute to the synchronization of stochastic discrete-time delayed networks with the objective state. The obtained results also show that even impulses have enlarging effects on the states of the networks, synchronization of the stochastic impulsive discrete-time delayed networks with the objective state can still be achieved under certain conditions. The rest of this paper is organized as follows. In Section 2, some basic definitions and notations are introduced. In Section 3, some criteria for the stochastic impulsive discretetime delay networks synchronize with the objective state are presented. In Section 4, two numerical examples are given to demonstrate the effectiveness and superiority of the proposed results. Finally, conclusions are drawn in Section 5. Notations: In this paper, Rn and Rn×m denote, respectively, the n-dimensional Euclidean space and the set of all n × m real matrices; In is the identity matrix of order n; the notation X ≥ Y (respectively, X > Y ), where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite); AT represents the transpose of A; the symmetric terms below the main diagonal of a symmetric matrix are denoted by ∗; λmax (A) and λmin (A) denote the maximum and minimum eigenvalue of A, respectively; for matrices A ∈ Rm×n and B ∈ R p×q , their Kronecker product is a matrix in Rmp×nq denoted by A ⊗ B; E{x(k)} stands for the expectation of the stochastic variable x(k); kx(k)k describes the Euclidean norm of a vector x(k); diag{· · · } stands for a block-diagonal matrix; R denotes the set of real numbers; Z + denotes the set of positive integer numbers; N denotes the set of natural numbers, i.e., N = {0, 1, 2, . . .}.
hi j , hi j = h ji , hi j ≥ 0(i , j), i, j = 1, 2, ..., M.
ω(k) is a one-dimensional Gaussian white noise sequence satisfying
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E{ω(k)} = 0, E{ω2 (k)} = 1, E{ω(i)ω( j)} = 0(i , j).
s(k) is the state vector of the isolate node which satisfies s(k + 1) = As(k) + f˜(s(k − dk )),
(3)
s(k) = [s1 (k), s2 (k), ..., sn (k)]T ∈ Rn . We hope that the complex network (1) can be synchronized to (3).
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In this paper, we have the following assumptions: Assumption 1[10, 13]. Each activation function f˜j (t), t ∈ R( j = 1, 2, ..., n) is continuous and bounded, and there exist constants b j1 , b j2 such that b j1 ≤
f˜j (a) − f˜j (b) ≤ b j2 , j = 1, 2, ..., n, a−b
(4)
where a, b ∈ R, and a , b. Assumption 2[9]. The stochastic intensity κ(k, y) satisfies the following inequality
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2. Preliminaries
(2)
κT (k, y)κ(k, y) ≤ yT W T Wy, i = 1, 2, ..., M, k ∈ N,
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Consider the following stochastic impulsive discrete-time delay networks consisting of M coupled identical neural networks in a complete probability space (Ω, F, P) , where each node of the networks is an n-dimensional system. M X ˜ x (k + 1) = Ax (k) + f (x (k − d )) + c hi j Γx j (k) i i i k j=1 + κ(k, xi (k) − s(k))ω(k) + ui (k), k , Nl − 1; (1) x (k + 1) = xi (k) + Bik (xi (k) − s(k)) i + (A − In )s(k) + f˜(s(k − dk )), k = Nl − 1; x (τ) = ϕ (τ) + s(τ), τ ∈ J , {−d, ¯ −d¯ + 1, ..., 0}, i i
(5)
where y ∈ Rn , W is a matrix with appropriate dimension.
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The pinning controllers in the complex network (1) are designed as follows: ( −pi Γ(xi (k) − s(k)), i = 1, 2, ..., m, ui (k) = (6) 0, i = m + 1, ..., M, where pi > 0 is the feedback gain and m is the number of pinning controllers. Let ei (k) = xi (k) − s(k), noting that
where k ∈ N, xi (k) = [xi1 (k), xi2 (k), . . . , xin (k)]T ∈ Rn is the state vector of the i th node, i = 1, 2, .., M; A ∈ Rn×n is a constant matrix; Bi(Nl −1) ∈ Rn×n is a constant matrix, l ∈ Z + ; f˜(xi (k − dk )) = [ f˜1 (xi1 (k − dk )), f˜2 (xi2 (k − dk )), ..., f˜n (xin (k −
c
M X j=1
3
M X
hi j Γs(k) = 0, thus,
j=1
hi j Γx j (k) − pi Γ(xi (k) − s(k))
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=c
M X j=1
where Q = QT , R = RT , is equivalent to either of the following conditions: 1) Q < 0, R − S T Q−1 S < 0, 2) R < 0, Q − S R−1 S T < 0. Lemma 3 [38]. Let Φ ∈ Rn×n be a positive definite matrix and M ∈ Rn×n be a symmetric matrix, then for any x ∈ Rn , the following inequality holds :
hi j Γ(x j (k) − s(k)) − pi Γ(xi (k) − s(k))
= c[hi1 Γe1 (k) + hi2 Γe2 (k) + · · · + hii Γei (k) + · · · + hiM Γe M (k)] − pi Γei (k)
= c[hi1 Γe1 (k) + hi2 Γe2 (k) + · · · + (hii −
where G = (gi j ) M×M , ( h − gi j = ii hi j ,
M X
λmin (Φ−1 M)xT Φx ≤ xT Mx ≤ λmax (Φ−1 M)xT Φx.
gi j Γe j (k),
j=1
3. Main Results
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+ · · · + hiM Γe M (k)] = c
pi )Γei (k) c
In this section, we will investigate the synchronization of stochastic impulsive discrete-time delayed networks (1) and (3) in mean square. Theorem 1. Under Assumptions 1, 2, if for given constants 0 < ηl < 1, l ∈ Z + , α > 0, β > 0, there exist constants q > 0, ξ > 0, ρ > 0, symmetric positive matrix P ∈ Rn×n and positive diagonal matrix U =diag{u1 , u2 , ..., un } such that
pi c ,i
= 1, 2, ..., m, otherwise.
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Therefore, from inequalities (1), (3) and (6), we get the error system as follows M X e (k + 1) = Ae (k) + f (e (k − d )) + c gi j Γe j (k) i i i k j=1 (7) + κ(k, ei (k))ω(k), k , Nl − 1; ei (k + 1) − ei (k) = Bik ei (k), k = Nl − 1; e (τ) = ϕ (τ), τ ∈ J, i
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(8)
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In this paper, the following definition and Lemmas are useful for the derivation of the main results. Definition 1. The complex network (1) is said to be exponentially synchronized to (3) in mean square, if there exist constants θ > 0, M0 > 0 such that (X M
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i=1
) kei (k)k2 ≤ M0 |ϕ|e−θ(k−k0 )
T
α − 1 > 0), (1 − βq)
(12)
Ω1 (i) ∗ ∗
(13)
AT P U Bˇ < 0, Ω3 (i) # (In + Bi(Nl −1) )T P ≤ 0, l ∈ Z + , −P
−ηl P ∗
0 Ω2 ∗
(14)
(15)
M X ρW T W − αP, Ω2 = −U Bˆ − βP, Ω3 (i) = P + c g2i j P − U, j=1
(9)
then the complex network (1) is exponentially synchronized to (3) in mean square.
holds for all k ≥ k0 , where |ϕ| = maxτ∈J {kϕi (τ)k, i = 1, 2, ..., M}. Lemma 1 [12]. Let X and Y be any n-dimensional real vectors, and let P be an n × n positive semidefinite matrix. Then, the following matrix inequality holds: T
(11)
12 , where Bˆ =diag(b11 b12 , b21 b22 , ..., bn1 bn2 ), Bˇ =diag( b11 +b 2 bn1 +bn2 b21 +b22 T , ..., ), i = 1, 2, ..., M, Ω (i) = A PA+ 1 2 2 M M 2 X X c M 2 c α2i j ΓPΓ + ΓPΓ + c g2i j AT PA + 2cMΓPΓ+ 2 2 j=1 j=1
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where a ∈ R and a , 0.
d¯ ≤ Nl+1 − Nl ≤ ξ, l ∈ N,
ln ηl < −µξ − µ(Nl+1 − Nl ), l ∈ Z + ,
It can be verified from Assumption 1 that functions f j (t), t ∈ R( j = 1, 2, ..., n) satisfy the following condition: f j (a) ≤ b j2 , j = 1, 2, ..., n, a
(10)
q > e2µξ , (µ =
where ei (k) = [ei1 (k), ei2 (k), ..., ein (k)]T ∈ Rn , f (ei (k − dk )) = f˜(xi (k − dk )) − f˜(s(k − dk )) = [ f1T (ei1 (k − dk )), f2T (ei2 (k − dk )), ..., fnT (ein (k − dk ))]T ∈ Rn , i = 1, 2, .., M.
b j1 ≤
P ≤ ρI,
Proof. Choose the following Lyapunov function: V(k) = eT (k)(I M ⊗ P)e(k) =
M X
eTi (k)Pei (k),
i=1
where e(k) = [eT1 (k), eT2 (k), ..., eTM (k)]T ∈ R Mn , k ∈ N.
T
2X PY ≤ X PX + Y PY.
From inequalities (2) and (7), for any k , Nl − 1, l ∈ Z + ,
Lemma 2 (Schur Complement). The linear matrix inequality " # Q S < 0, ST R
E{V(k + 1)} = E
(X M h i=1
4
Aei (k) + f (ei (k − dk )) + c
M X j=1
gi j Γe j (k)
iT
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= 2F T (e(k − dk ))(I M ⊗ P)(c(G ⊗ Γ)e(k))
= 2cF T (e(k − dk ))(G ⊗ PΓ)e(k)) M M X X gi j PΓe j (k) = 2c f T (ei (k − dk ))
M X i h ·P Aei (k) + f (ei (k − dk )) + c gi j Γe j (k)
+[κ(k, ei (k))]T P[κ(k, ei (k))]
o
j=1
(X M h =E eTi (k)AT PAei (k) + f T (ei (k − dk ))P f (ei (k − dk )) i=1
+cM
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j=1
[κ(k, ei (k))]T P[κ(k, ei (k))] ≤ ρeTi (k)W T Wei (k).
M i X +2 f T (ei (k − dk ))P c gi j Γe j (k)
(16)
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c
j=1
i=1
T gi j Γe j (k) P c T
gi j Γe j (k)
j=1
= [c(G ⊗ Γ)e(k)] (I M ⊗ P)[c(G ⊗ Γ)e(k)]
· [b j2 ei j (k − dk ) − f j (ei j (k − dk ))] ≥ 0,
M X
c 2
eTi (k)
j=1
i=1
Therefore
0≤
α2i j ΓPΓ + MΓPΓ ei (k),
i
(17)
When k , Nl − 1, l ∈ Z , substituting inequalities (17)-(21) into (16) leads to (X M h E{V(k + 1)} ≤ E eTi (k)AT PAei (k) i=1
+ f T (ei (k − dk ))P f (ei (k − dk ))
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AC ≤c
M X
+ 2eTi (k)AT P f (ei (k − dk ))
eTi (k)
i=1
+
j=1
M X
g2i j AT PAei (k) + cM
j=1
M X
M X j=1
eTj (k)ΓPΓe j (k)
j=1
g2i j AT PA + MΓPΓ ei (k)
+ c f T (ei (k − dk ))
(18)
M X j=1
g2i j P f (ei (k − dk ))
ˆ i (k − dk ) + cMeTi (k)ΓPΓei (k) − eTi (k − dk )U Be + 2eTi (k − dk )U Bˇ f (ei (k − dk )) i − f T (ei (k − dk ))U f (ei (k − dk )) o + ρeTi (k)W T Wei (k)
j=1
and M M X X 2 f T (ei (k − dk ))P c gi j Γe j (k) i=1
M X c2 T ei (k) α2i j ΓPΓ + MΓPΓ ei (k) 2 j=1
+ ceTi (k)
M M X X = c eTi (k) g2i j AT PA + MΓPΓ ei (k), i=1
(21)
+
j=1
i=1
u j [ f j (ei j (k − dk )) − b j1 ei j (k − dk )]T
− f T (ei (k − dk ))U f (ei (k − dk )).
= 2eT (k)(I M ⊗ AT P)[c(G ⊗ Γ)e(k)]
= 2ceT (k)(G ⊗ AT PΓ)e(k) M M X X = 2c eTi (k) gi j AT PΓe j (k)
j=1
·[b j2 ei j (k − dk ) − f j (ei j (k − dk ))]
M M X X 2eTi (k)AT P c gi j Γe j (k) i=1
n X
ˆ i (k − dk ) = −eTi (k − dk )U Be T +2e (k − dk )U Bˇ f (ei (k − dk ))
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similarly
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≤
j=1
M X 2
i = 1, 2, ..., M, j = 1, 2, ..., n.
M
= c2 eT (k)(GT G ⊗ ΓPΓ)e(k) M M X X αi j ΓPΓe j (k) = c2 eTi (k) i=1
(20)
[ f j (ei j (k − dk )) − b j1 ei j (k − dk )]T
Let GT G = (αi j ) M×M , according to the property of the kronecker product and Lemma 1, we have M X
(19)
In addition, it follows from inequality (8) that [10, 13]
o
+[κ(k, ei (k))]T P[κ(k, ei (k))] .
M X
eTi (k)ΓPΓei (k),
here we let F(e(k − dk )) = [ f T (e1 (k − dk )), f T (e2 (k − dk )), ..., f T (e M (k − dk ))]T ∈ R Mn , from Assumption 2 and inequality (10) we have that
M X + 2eTi (k)AT P f (ei (k − dk )) + 2eTi (k)AT P c gi j Γe j (k)
M X
M X i=1
j=1
j=1
j=1
i=1
M M X T X + c gi j Γe j (k) P c gi j Γe j (k) j=1
j=1
i=1
M M X X g2i j P f (ei (k − dk )) ≤ c f T (ei (k − dk ))
j=1
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=E
(X M M h c2 M c2 X 2 αi j ΓPΓ + ΓPΓ eTi (k) AT PA + 2 j=1 2 i=1
< De−µN1 M|ϕ|2 = K|ϕ|2 e−µN1 . (ii) Next, we will prove that:
M X i +c g2i j AT PA + 2cMΓPΓ + ρW T W − αP ei (k)
E{V(k)} ≤ K|ϕ|2 e−µN1 , k ∈ [N0 , N1 ).
j=1
+2eTi (k)AT P f (ei (k − dk )) +eT (k − dk )(−U Bˆ − βP)ei (k − dk ) i
If the inequality (29) does not hold, then N1 − N0 ≥ 2, there exists a k ∈ (N0 , N1 ) (noting that N0 = 0), such that E{V(k)} > K|ϕ|2 e−µN1 . Let k∗ = inf{k ∈ (0, N1 ), |E{V(k)} > K|ϕ|2 e−µN1 }, then E{V(k∗ )} > K|ϕ|2 e−µN1 .
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j=1
¯ k∗ − 1], For any k ∈ [−d,
E{V(k)} ≤ K|ϕ|2 e−µN1 .
j=1
> K|ϕ|2 e−µN1
∀dk∗ −1
E{V(k + 1)} < αE{V(k)} + βE{V(k − dk )}.
ED
(In + Bi(Nl −1) )T P(In + Bi(Nl −1) ) ≤ ηl P, l ∈ Z + .
(23)
< αE{V(k∗ − 1)} + βqE{V(k∗ )},
thus
(24)
E{V(k∗ )} <
i=1 −µξ−µ(Nl+1 −Nl )
≤e
E{V(Nl − 1)}, l ∈ Z + .
α E{V(k∗ − 1)} = (1 + µ)E{V(k∗ − 1)}. (32) 1 − βq
Therefore, it follows from inequalities (26) and (32) that
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o ·(In + Bi(Nl −1) )ei (Nl − 1)} (X M o ≤ ηl E {eTi (Nl − 1)Pei (Nl − 1)}
1 E{V(k∗ )} 1+µ eµξ > K|ϕ|2 e−µN1 e−µξ 1+µ > K|ϕ|2 e−µN1 e−µξ
E{V(k∗ − 1)} >
(25)
= DM|ϕ|2 e−µN1 e−µξ
AC
Choose a constant D > 0 such that λmax (P) ≤ De−µN1 e−µξ < De−µN1 < qλmax (P).
≥ λmax (P)M|ϕ|2 .
(26)
From inequality (28), E{V(0)} ≤ λmax (P)M|ϕ|2 .
Then we will prove that E{V(k)} ≤ K|ϕ|2 e−µNl+1 , k ∈ [Nl , Nl+1 ), ∀l ∈ N,
(31)
E{V(k∗ )} < αE{V(k∗ − 1)} + βE{V(k∗ − 1 − dk∗ −1 )}
It follows from inequalities (13) and (24) that (X M {eTi (Nl − 1)(In + Bi(Nl −1) )T P E{V(Nl )} = E i=1
≥ E{V(k∗ − 1 − dk∗ −1 )}, ¯ dk∗ −1 ∈ Z + . ∈ [d, d],
Since k∗ ∈ (0, N1 ), k∗ , Nl , l ∈ Z + , from (23) and (31) we know
M
From (14), when k , Nl − 1, l ∈ Z + , one has
Applying the Lemma 2 to (15) yields
qE{V(k∗ )} > e2µξ K|ϕ|2 e−µN1
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=
(30)
It can be found from inequalities (12) and (26) that
(22)
[eTi (k), eTi (k − dk ), f T (ei (k − dk ))]T , M M X X c2 M 2 Ω1 (i) = AT PA + c2 α2i j ΓPΓ + g2i j AT PA ΓPΓ + c 2 j=1 j=1 +2cMΓPΓ + ρW T W − αP, Ω2 = −U Bˆ − βP, M X Ω3 (i) = P + c g2i j P − U.
where ζ(k)
(29)
If N1 − N0 = 1, obviously the inequality (29) holds.
+2eTi (k − dk )U Bˇ f (ei (k − dk )) M X o + f T (ei (k − dk ))(P + c g2i j P − U) f (ei (k − dk ))
+αE{V(k)} + βE{V(k − dk )} AT P Ω (i) 0 M X T 1 ˇ ∗ Ω U B ζ(k) = E ζ (k) 2 i=1 ∗ ∗ Ω3 (i) +αE{V(k)} + βE{V(k − dk )},
(28)
Let k¯ = sup{k ∈ [0, k∗ −1), |E{V(k)} ≤ λmax (P)M|ϕ|2 }. Then λmax (P)M|ϕ|2 < E{V(k)} ≤ K|ϕ|2 e−µN1 for k¯ + 1 ≤ k ≤ k∗ − 1. From inequalities (26) and (30), for any k ∈ [k¯ + 1, k∗ − 1], we get
(27)
where K = DM. ¯ N0 ], from Lemma 3 and inequality (26) (i) When k ∈ [−d, we obtain
qE{V(k)} > qλmax (P)M|ϕ|2 > De−µN1 M|ϕ|2 = K|ϕ|2 e−µN1
M X E{V(k)} ≤ λmax (P) ||ei (k)||2 ≤ λmax (P)M|ϕ|2
¯ dk−1 ∈ Z + . ≥ E{V(k − 1 − dk−1 )}, ∀dk−1 ∈ [d, d],
(30)
i=1
6
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Therefore for any k ∈ [k¯ + 1, k∗ − 1], since k , Nl , l ∈ Z + , from (23) we have
= eµξ eµξ K|ϕ|2 e−µ(Nr+1 −Nr ) e−µNr ≥ eµξ K|ϕ|2 e−µNr > K|ϕ|2 e−µNr ≥ E{V(kˆ − 1 − dk−1 ˆ )},
E{V(k)} < αE{V(k − 1)} + βE{V(k − 1 − dk−1 )}
¯ dk−1 ∈ [d, d], ∈ Z+. ∀dk−1 ˆ ˆ
< αE{V(k − 1)} + qβE{V(k)}.
Since kˆ ∈ (Nr , Nr+1 ), kˆ , Nl , l ∈ Z + , from inequalities (23) and (38) we get
Thus, for any k ∈ [k¯ + 1, k∗ − 1], E{V(k)} <
α E{V(k − 1)} = (1 + µ)E{V(k − 1)}. (33) 1 − βq
ˆ < αE{V(kˆ − 1)} + βE{V(kˆ − 1 − dk−1 E{V(k)} ˆ )} ˆ < αE{V(kˆ − 1)} + βqE{V(k)}.
ˆ < E{V(k)}
E{V(k∗ )} < (1 + µ)E{V(k∗ − 1)}
It follows from inequalities (26), (39) and
Thus
(32)
∗
< (1 + µ) E{V(k − 2)} < · · · < (1 + µ)
(33)
k∗ −k¯
¯ E{V(k)}
α E{V(kˆ − 1)} = (1 + µ)E{V(kˆ − 1)}. (39) 1 − βq
≤ K|ϕ|2 e−µN1 < E{V(k∗ )},
(iii) We assume that for t = 1, 2, ..., r,
Next, we will prove that 2 −µNr+1
E{V(k)} ≤ K|ϕ| e
ED
E{V(k)} ≤ K|ϕ|2 e−µNt , k ∈ [Nt−1 , Nt ).
, k ∈ [Nr , Nr+1 ).
(34)
Therefore, for any k ∈ [k˜ + 1, kˆ − 1], k ∈ Z + , from inequalities (11), (12) and (37) we have qE{V(k)} > e2µξ K|ϕ|2 e−µNr+1 e−µξ
(35)
= eµξ K|ϕ|2 e−µ(Nr+1 −Nr ) e−µNr ≥ K|ϕ|2 e−µNr
PT
It follows from inequalities (25) and (34) that
¯ dk−1 ∈ Z + . ≥ E{V(k − 1 − dk−1 )}, ∀dk−1 ∈ [d, d],
E{V(Nr )} ≤ e−µξ e−µ(Nr+1 −Nr ) E{V(Nr − 1)}
Since k ∈ [k˜ + 1, kˆ − 1], k , Nl , l ∈ Z + from inequality (23) one has
≤ e−µξ e−µ(Nr+1 −Nr ) K|ϕ|2 e−µNr 2 −µNr+1
K|ϕ| e
CE
=e
−µξ
< K|ϕ|2 e−µNr+1 .
E{V(k)} < αE{V(k − 1)} + βE{V(k − 1 − dk−1 )}
(36)
If Nr+1 − Nr = 1, obviously the inequality (35) holds.
AC
α E{V(k − 1)} = (1 + µ)E{V(k − 1)}, 1 − βq ∀k ∈ [k˜ + 1, kˆ − 1]. (40)
E{V(k)} <
Furthermore, using inequalities (11), (39) and (40) we derive that ˆ < (1 + µ)E{V(kˆ − 1)} E{V(k)}
From d¯ ≤ Nr − Nr−1 , Nr − d¯ ≥ Nr−1 , from (34), we have E{V(k)} ≤ K|ϕ| e
¯ kˆ − 1]. , ∀k ∈ [Nr − d,
< αE{V(k − 1)} + qβE{V(k)},
thus
If the inequality (35) does not hold, then Nr+1 − Nr ≥ 2, there exists a k ∈ (Nr , Nr+1 ) such that E{V(k)} > K|ϕ|2 e−µNr+1 . Let kˆ = inf{k ∈ (Nr , Nr+1 ), |E{V(k)} > K|ϕ|2 e−µNr+1 }, then ˆ > K|ϕ|2 e−µNr+1 . For any k ∈ [Nr , kˆ − 1], E{V(k)} ≤ E{V(k)} 2 −µNr+1 K|ϕ| e .
2 −µNr
1 ˆ E{V(k)} 1+µ eµξ ˆ −µξ = E{V(k)}e 1+µ eµξ > K|ϕ|2 e−µNr+1 e−µξ 1+µ ≥ K|ϕ|2 e−µNr+1 e−µξ .
From inequality (36), we get E{V(Nr )} ≤ K|ϕ|2 e−µNr+1 e−µξ , define k˜ = sup{k ∈ [Nr , kˆ − 1), |E{V(k)} ≤ K|ϕ|2 e−µNr+1 e−µξ }, ˜ ≤ K|ϕ|2 e−µNr+1 e−µξ . For any k ∈ [k˜ + 1, kˆ − 1], then E{V(k)} 2 −µNr+1 K|ϕ| e ≥ E{V(k)} > K|ϕ|2 e−µNr+1 e−µξ .
M
this is a contradiction, thus inequality (29) holds.
AN US
= (1 + µ)ξ K|ϕ|2 e−µξ e−µN1
≥ 1 that
(39)
¯ ≤ (1 + µ)ξ λmax (P)M|ϕ|2 ≤ (1 + µ)ξ E{V(k)} ≤ (1 + µ)ξ DM|ϕ|2 e−µξ e−µN1
eµξ 1+µ
E{V(kˆ − 1)} >
(11)
(26)
CR IP T
¯ ≤ λmax (P)M|ϕ|2 and (1 + On the other hand, using E{V(k)} µ) e ≤ 1, we derive from inequalities (11), (26), (32) and (33) that ξ −µξ
2
(38)
(39)
ˆ ˜ ˜ < (1 + µ)2 E{V(kˆ − 2)} < · · · < (1 + µ)k−k E{V(k)}
(37)
(40)
≤ (1 + µ)ξ K|ϕ|2 e−µNr+1 e−µξ
It follows from inequalities (11) and (12) that
(11)
ˆ > e2µξ K|ϕ|2 e−µNr+1 qE{V(k)}
ˆ ≤ K|ϕ|2 e−µNr+1 < E{V(k)}, 7
ACCEPTED MANUSCRIPT
ρW T W − α exp(−λ)P, Ψ2 = −U Bˆ − β exp(−λ(dk + P 2 1))P, Ψ3 (i) = P + c M g P − U, then the complex j=1 i j network (1) is exponentially synchronized to (3) in mean square.
this is a contradiction, thus the inequality (35) holds. By the principle of mathematical induction, we have proved inequality (27) holds (noting that N0 = 0), which means
Proof. Choose the following Lyapunov function:
E{V(k)} ≤ K|ϕ|2 e−µNl < K|ϕ|2 e−µk , k ∈ [Nl−1 , Nl ), ∀l ∈ Z + .
V(k) = exp(λk)eT (k)(I M ⊗ P)e(k) =
From Lemma 3 ) ( M X kei (k)k2 ≤ E{V(k)}, ∀k ≥ 0, E λmin (P)
CR IP T
From inequalities (2) and (7), for any k , Nl − 1, l ∈ Z + .
thus E
i=1
2
kei (k)k
)
K ≤ |ϕ|2 e−µk , ∀k ≥ 0. λmin (P)
E{V(k + 1)} = E
(41)
(X M i=1
h exp(λ(k + 1)) Aei (k)
M X iT + f (ei (k − dk )) + c gi j Γe j (k) P
This implies that the complex network (1) is exponentially synchronized to (3) in mean square. Remark 1. In Theorem 1, it is worth noting that α + β can be bigger than 1, which means the error network (7) is allowed to be unstable. In order to stabilize the error network (7), we give the inequality (13), that is the reason why Theorem 1 can be used to deal with the exponential synchronization of complex network (1) and (3) in mean square under impulsive stabilization and pinning control. The next theorem considers another situation when the error network (7) is stable and it is still stable under impulsive perturbations. Remark 2. The reason why we choose Kronecker product to deal with the inequalities (17)-(19) is that we want to get the more convenient representations and it may make the results less conservative. Theorem 2. Under Assumptions 1, 2, if for given positive constants τ˜ > 0, λ > 0, η > 1, α > 0, β > 0, α + β < 1, there exist constant ρ > 0, symmetric positive matrix P ∈ Rn×n and positive diagonal matrix U =diag{u1 , u2 , ..., un } such that
AN US
j=1
h
· Aei (k) + f (ei (k − dk )) + c
+ [κ(k, ei (k))]T P[κ(k, ei (k))]
=E
i=1
M
"
−ηP ∗
(43)
ln η < λ, τ˜
(44)
Ψ1 (i) ∗ ∗
AT P U Bˇ < 0, Ψ3 (i) # exp( 12 λ)(In + Bi(Nl −1) )T P ≤ 0, l ∈ Z + , −P 0 Ψ2 ∗
j=1
o
exp(λ(k + 1)) eTi (k)AT PAei (k)
j=1
j=1
+ 2eTi (k)AT P f (ei (k − dk )) M X + 2eTi (k)AT P c gi j Γe j (k) j=1
M X + 2 f T (ei (k − dk ))P c gi j Γe j (k)
(42)
Nl − Nl−1 ≥ τ˜ , l ∈ Z , +
(X M
M X i gi j Γe j (k)
+ f T (ei (k − dk ))P f (ei (k − dk )) M M T X X gi j Γe j (k) + c gi j Γe j (k) P c
ED
PT
AC
CE
P ≤ ρI,
exp(λk)eTi (k)Pei (k),
i=1
where e(k) = [eT1 (k), eT2 (k), ..., eTM (k)]T ∈ R Mn , k ∈ N.
i=1
(X M
M X
j=1
o + [κ(k, ei (k))]T P[κ(k, ei (k))] .
(47)
It can be verified from inequality (8) that [10, 13] 0 ≤ exp(λ(k + 1))
n X u j [ f j (ei j (k − dk )) − b j1 ei j (k − dk )]T j=1
·[b j2 ei j (k − dk ) − f j (ei j (k − dk ))] ˆ i (k − dk ) = exp(λ(k + 1))[−eT (k − dk )U Be
(45)
i
+2eTi (k − dk )U Bˇ f (ei (k − dk ))
− f T (ei (k − dk ))U f (ei (k − dk ))].
(46)
(48)
When k , Nl − 1, l ∈ Z + , substituting inequalities (17)-(20) and (48) into (47) leads to
12 where Bˆ =diag(b11 b12 , b21 b22 , ..., bn1 bn2 ), Bˇ =diag( b11 +b , 2 bn1 +bn2 b21 +b22 T , ..., 2 ), i = 1, 2, ..., M, Ψ1 (i) = A PA+ 2 M M X X c2 M 2 2 c α ΓPΓ + ΓPΓ + c g2i j AT PA + 2cMΓPΓ+ ij 2 2 j=1 j=1
E{V(k + 1)} ≤ E 8
(X M i=1
h exp(λ(k + 1)) eTi (k)AT PAei (k)
ACCEPTED MANUSCRIPT
+ f T (ei (k − dk ))P f (ei (k − dk )) M X 2 α2i j ΓPΓ + MΓPΓ ei (k) + c2 eTi (k)
·(In + Bi(Nl −1) )e(Nl − 1) ≤ E{ηV(Nl − 1)}, l ∈ Z + .
j=1
E{V(k)} ≤ λmax (P)M|ϕ|2 .
j=1
E{V(k)} ≤ λmax (P)M|ϕ|2 , k ∈ [0, N1 ).
If (53) does not hold, there exists a positive integer k∗ , such that k∗ + 1 = inf{k ∈ (0, N1 ), |E{V(k)} > λmax (P)M|ϕ|2 }. E{V(k)} ≤ λmax (P)M|ϕ|2 for −d¯ ≤ k ≤ k∗ . Thus 0 < k∗ + 1 ≤ N1 − 1,
i=1
M X
h exp(λ(k + 1)) eTi (k) AT PA
E{V(k∗ + 1)} < αE{V(k∗ )} + βE{V(k∗ − dk∗ )}
≤ αλmax (P)M|ϕ|2 + βλmax (P)M|ϕ|2
< λmax (P)M|ϕ|2 ,
+ 2eTi (k)AT P f (ei (k − dk )) + 2eTi (k − dk )U Bˇ f (ei (k − dk )) +eT (k − dk )(−U Bˆ − exp(−λ(dk + 1))βP)ei (k − dk ) i
M X o g2i j P − U) f (ei (k − dk ))
which contradicts with (54), thus (53) holds. (iii) Next we will prove that
M
+ f T (ei (k − dk ))(P + c
j=1
ED
+αE{V(k)} + βE{V(k − dk )} Ψ1 (i) 0 AT P M X T ˇ ζ(k) ∗ Ψ U B exp(λ(k + 1))ζ (k) = E 2 i=1 ∗ ∗ Ψ3 (i) +αE{V(k)} + βE{V(k − dk )},
E{V(k)} ≤ ηλmax (P)M|ϕ|2 , k ∈ [N1 , N2 ).
E{V(N1 )} ≤ E{ηV(N1 − 1)}
(49)
≤ ηλmax (P)M|ϕ|2 .
If (55) does not hold, there exists a positive integer k , such 0 that k + 1 = inf{k ∈ (N1 , N2 ), |E{V(k)} > ηλmax (P)M|ϕ|2 }. 0 0 E{V(k)} ≤ ηλmax (P)M|ϕ|2 for −d¯ ≤ k ≤ k . Thus N1 < k +1 < N2 ,
PT
CE
AC
Ψ2 = −U Bˆ − β exp(−λ(dk + 1))P, Ψ3 (i) = P + c
0
E{V(k + 1)} > ηλmax (P)M|ϕ|2 . Since k , Nl − 1, l ∈ Z + , from (50) one has 0
< ηλmax (P)M|ϕ|2 ,
(50)
which contradicts with (57), thus (55) holds. (iv) Suppose for t = 1, 2, ..., r, we have
exp(λ)(In + Bi(Nl −1) )T P(In + Bi(Nl −1) ) ≤ ηP, l ∈ Z + .
E{V(k)} ≤ ηt−1 λmax (P)M|ϕ|2 , k ∈ [Nt−1 , Nt ).
Thus
i=1
0
≤ αηλmax (P)M|ϕ|2 + βηλmax (P)M|ϕ|2
Applying the Lemma 2 to (46) yields
(X M
0
E{V(k + 1)} < αE{V(k )} + βE{V(k − dk0 )}
j=1
Thus, when k , Nl − 1, l ∈ Z + , from (45) we obtain
E{V(Nl )} = E
(57)
0
M X g2i j P − U.
E{V(k + 1)} < αE{V(k)} + βE{V(k − dk )}.
(56) 0
M X g2i j AT PA + 2cMΓPΓ + ρW T W − α exp(−λ)P, j=1
(55)
In fact, by inequalities (51) and (53), we have
where ζ(k) = [eTi (k), eTi (k − dk ), f T (ei (k − dk ))]T , M X c2 M 2 Ψ1 (i) = AT PA + c2 α2i j ΓPΓ + ΓPΓ 2 j=1 +c
(54)
Since k∗ , Nl − 1, l ∈ Z + , from (50) one has
M X c2 M + c2 α2i j ΓPΓ + ΓPΓ + c g2i j AT PA 2 j=1 j=1 i +2cMΓPΓ + ρW T W − α exp(−λ)P ei (k) 2
E{V(k∗ + 1)} > λmax (P)M|ϕ|2 .
AN US
(X M
(53)
CR IP T
− dk )
o i − f T (ei (k − dk ))U f (ei (k − dk )) + ρeTi (k)W T Wei (k)
=E
(52)
(ii) From (i) and α + β < 1, we will prove
j=1
ˆ i (k + cMeTi (k)ΓPΓei (k) − eTi (k − dk )U Be + 2eTi (k − dk )U Bˇ f (ei (k − dk ))
(51)
¯ N0 ] (noting that N0 = 0), k ≤ 0, from (i) For any k ∈ [−d, Lemma 3 we have
+ 2eTi (k)AT P f (ei (k − dk )) M X + ceTi (k) g2i j AT PA + MΓPΓ ei (k)
M X + c f T (ei (k − dk )) g2i j P f (ei (k − dk ))
We will prove that exp(λNl )eT (Nl − 1)(In + Bi(Nl −1) )T P
E{V(k)} ≤ ηr λmax (P)M|ϕ|2 , k ∈ [Nr , Nr+1 ). 9
(58)
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4. SIMULATION EXAMPLES
E{V(Nr )} ≤ E{ηV(Nr − 1)} r
2
≤ η λmax (P)M|ϕ| .
If (58) does not hold, there exists a positive integer k∗∗ , such that k∗∗ + 1 = inf{k ∈ (Nr , Nr+1 ), |E{V(k)} > ηr λmax (P)M|ϕ|2 }. E{V(k)} ≤ ηr λmax (P)M|ϕ|2 for −d¯ ≤ k ≤ k∗∗ . Thus Nr < k∗∗ + 1 < Nr+1 , E{V(k∗∗ + 1)} > ηr λmax (P)M|ϕ|2 .
(59)
To illustrate the effectiveness and superiority of the given results, two examples are given in this section. Example 1. Consider the synchronization of stochastic impulsive discrete-time delayed networks (1) and (3) via pinning control with the following data: "
# " # 0 −0.8 0 , Bi(Nl −1) = , l ∈ Z+, 1.01 0 −0.8 " # " # 0.1 0 0.1 0 Γ= , W= , 0 0.1 0 0.01 −0.2 0.1 0.1 −0.2 0.1 , H = 0.1 0.1 0.1 −0.2
1.02 A= 0.1
CR IP T
Moreover, it follows from inequality (51) that
Since k∗∗ , Nl − 1, l ∈ Z + , from (50) one has E{V(k∗∗ + 1)} < αE{V(k∗∗ )} + βE{V(k∗∗ − dk∗∗ )}
≤ αηr λmax (P)M|ϕ|2 + βηr λmax (P)M|ϕ|2 < ηr λmax (P)M|ϕ|2 ,
E{V(k)} ≤ ηl λmax (P)M|ϕ|2 , k ∈ [Nl , Nl+1 ), l ∈ N.
AN US
which contradicts with (59), thus (58) holds. Therefore, by the principle of mathematical induction, we have proved that
(60)
From (43) and noted that N0 = 0, for k ∈ [Nl , Nl+1 ), l ∈ N, one obtains k ≥ Nl ≥ (Nl − Nl−1 ) + (Nl−1 − Nl−2 ) + · · · + (N1 − N0 ) ≥ l˜τ. (61)
E{V(k)} ≤ ηl λmax (P)M|ϕ|2 k
ED
≤ η τ˜ λmax (P)M|ϕ|2
= exp( τk˜ ln η)λmax (P)M|ϕ|2 ,
∀k ∈ [Nl , Nl+1 ), l ∈ N.
(62)
PT 2
exp(λk)kei (k)k
i=1
CE
E λmin (P)
M X
)
≤ E{V(k)}, ∀k ≥ 0.
1200
states
AC i=1
x2(1,k) x2(2,k)
600
x3(1,k)
400
x3(2,k)
200
s(1,k) s(2,k)
0
Thus, from (44) one obtain
(X M
x1(2,k)
800
) M X k E λmin (P) exp(λk)kei (k)k2 ≤ exp( ln η)λmax (P)M|ϕ|2 . τ ˜ i=1
E
x1(1,k)
1000
Therefore from inequality (62) for k ∈ [Nl , Nl+1 ), l ∈ N,
(
# 0 . 0.005
ρ = 18.4845, q = 8.3333, therefore, the complex network (1) is exponentially synchronized to (3) in mean square.
Using Lemma 3, the following inequality holds: (
It is easy to check that " # " 0 0 0.005 Bˆ = , Bˇ = 0 0 0
Let α = 1.1, β = 0.01, µ = 0.2, ηl = 0.2, l ∈ Z + . By using LMI toolbox in Matlab, one of the feasible solutions of Theorem 1 is " # " # 15.8391 −2.4451 408.0318 0 P= , U= , −2.4451 1.1751 0 78.9192
M
From (60) and (61) we have
where f˜(xi (k)) = tanh(0.01xi (k)), xi (k) ∈ R2 , i = 1, 2, 3, c = 0.01, the number of coupled identical neural networks is M = 3, the impulsive interval is 0 < Nl − Nl−1 ≤ ξ = 3, l ∈ Z + , p1 = 10, the pinning control u1 (k) = −10Γ(xi (k) − s(k)), ui (k) = 0 for i = 2, 3 and dk = mod(k, 3), k ∈ N.
−200
) λmax (P)M|ϕ|2 ln η kei (k)k2 ≤ exp(−(λ − )k), ∀k ≥ 0. λmin (P) τ˜
−400 −600 −800
Therefore, the complex network (1) is exponentially synchronized to (3) in mean square, this completes the proof. Remark 3. In Theorem 2, α + β < 1 implies that the error network (7) is stable, and η > 1 means that the impulses may have enlarging effects on the states of the networks.
10
0
10
20
30
40
50
k
Figure 1: The state trajectory of the network (1) and (3) in Example 1.
The simulation results for Example 1 are shown in Figure 1 and Figure 2. Figure 1 gives the state trajectory of the
ACCEPTED MANUSCRIPT
20 x (1,k) 1
x1(2,k) x2(1,k)
15
x2(2,k) x3(1,k) x3(2,k)
10
s(1,k) s(2,k)
states
network (1) and (3), which shows network (1) and (3) are not synchronized. The matrix Bi(Nl −1) , l ∈ Z + we give in Example 1 implies impulses do contribute to the synchronization of network (1) and (3). Figure 2 shows the state trajectory of the error network (7) in Example 1 with impulsive control and pinning control, which confirms that the error network (7) with impulsive control and pinning control is exponentially stable. The simulation results have confirmed that the designed controllers perform very well.
5
25
−5
e1(1,k)
0
e1(2,k)
20
e2(1,k) e2(2,k)
15
states
5
10
15 k
20
25
30
Figure 3: The state trajectory of the network (1) and (3) in Example 2.
e3(1,k) 10
CR IP T
0
e (2,k) 3
5 2 0
AN US
0
−5
−2
−10
−4
0
10
20
30
40
50
k
states
−15
Figure 2: The state trajectory of the error network (7) in Example 1.
Example 2. Consider the synchronization of stochastic impulsive discrete-time delayed networks (1) and (3) via pinning control with the following data: " # " # 0.1 −0.3 0.02 0 A= , Bi(Nl −1) = , l ∈ Z+, 0.02 0.1 0 0.02 " # " # 1 0 0.05 0 Γ= , W= , 0 1 0 0.05 −0.2 0.1 0.1 −0.2 0.1 , H = 0.1 0.1 0.1 −0.2
−6 −8
e1(1,k)
−10
e1(2,k)
−12
e2(2,k)
e2(1,k) e3(1,k)
M
−14
ED
−16
e3(2,k) 0
5
10
15 k
20
25
30
Figure 4: The state trajectory of the error network (7) in Example 2.
AC
CE
PT
The simulation results for Example 2 are shown in Figure 3 and Figure 4. Figure 3 gives the state trajectory of the network (1) and (3), which shows network (1) and (3) are synchronized. Figure 4 shows the state trajectory of error network (7). where f˜(xi (k)) = tanh(0.1xi (k)), xi (k) ∈ R2 , i = 1, 2, 3, c = Remark 4. Inspired by some enlightening works (see, e.g., 0.01, the number of coupled identical neural networks is [10–12, 24, 37] and the references therein), it is worthwhile to M = 3, the impulsive interval is 10 = τ˜ ≤ Nl − Nl−1 , l ∈ Z + , investigate synchronization of stochastic impulsive discretep1 = 0.1, the pinning control u1 (k) = −0.1Γ(xi (k) − s(k)), time networks with distributed delays via pinning control in the future. ui (k) = 0 for i = 2, 3 and dk = mod(k, 3), k ∈ N. Remark 5. Examples 1 and 2 cannot be studied by the results in [16–18, 22–24], for instance, the pinning control, time It is easy to check that delay, stochastic effects are not considered in [16, 17], time " # " # 0 0 0.05 0 delay is not considered in [18], time delay, stochastic effects ˆ ˇ B= , B= . 0 0 0 0.05 are not considered in [22], the pinning control and stochastic Let α = 0.5, β = 0.4, λ = 1, η = 3. By using LMI toolbox effects are not considered in [23, 24]. in Matlab, one of the feasible solutions of Theorem 2 is " # " # 5. Conclusion 4.5229 0.8696 17.3033 0 P= , U= , 0.8696 13.2864 0 24.7363 In this paper, we have investigated the synchronization ρ = 21.9660, therefore, the complex network (1) is exponen- problem for a class of stochastic impulsive discrete-time tially synchronized to (3) in mean square. delayed networks via pinning control. By using Lyapunov functions together with Razumikhin technique, some results 11
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are presented. The obtained results show that impulses do contribute to the synchronization of stochastic impulsive discrete-time delayed networks with the objective state. The obtained results also show that even impulses have enlarging effects on the states of the networks, synchronization of the stochastic impulsive discrete-time delayed networks with the objective state can still be achieved under certain conditions. Finally two simulation examples have been given to illustrate the effectiveness and superiority of the obtained results. A possible direction for future work is to obtain some synchronization criteria for stochastic impulsive discrete-time networks with distributed delays via pinning control.
[8] P. Li, J. Cao, Z. Wang, Robust impulsive synchronization of coupled delayed neural networks with uncertainties, Physica A Stat. Mech. its Appl. 373 (2007) 261272. [9] J. Liang, Z. Wang, Y. Liu, X. Liu, Global synchronization control of general delayed discrete-time networks with stochastic coupling and disturbances, IEEE Trans. Syst. Man Cybern. Part B 38 (4) (2008) 1073-1083.
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Acknowledgements The authors would like to thank the associate editor and the anonymous reviewers for their constructive comments and suggestions that improved the quality of the paper.
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Yu Lin Yu Lin received the B.Sc. degree from the School of Mathematical Sciences, Xiamen University, Xiamen, China, in 2015. Now, she is working toward the P.H.D. degree in the School of Mathematical Sciences, Tongji University. Her current research interest is impulsive discrete-time delayed system.
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Yu Zhang Yu Zhang received the B.Sc. degree from the Department of Applied Mathematics, Northeastern University, Shenyang, China, in 2002, and the Ph.D. degree from the Department of Mathematics, Tongji University, Shanghai, China, in 2007, respectively. She was a Research Assistant at the City University of Hong Kong, Hong Kong, China, from July 2005 to October 2005. Since July 2007, she has been with the School ofMathematical Sciences, Tongji University, Shanghai, China. She is currently an Associate Professor at Tongji University. She is the author or coauthor of more than 30 research articles.
[32] Y. Zhang, J. Sun, G. Feng, Impulsive control of discrete systems with time delay, IEEE Trans. Autom. Control 54 (4) (2009) 830–834. [33] Y. Zhang, Robust exponential stability of uncertain impulsive neural networks with time-varying delays and delayed impulses, Neurocomputing 74 (17) (2011) 3268–3276. 13
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Her research interests include stability theory, discrete-time stochastic systems and impulsive control systems.
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Accepted Manuscript
Synchronization of stochastic impulsive discrete-time delayed networks via pinning control Yu Lin, Yu Zhang PII: DOI: Reference:
S0925-2312(18)30077-8 10.1016/j.neucom.2018.01.052 NEUCOM 19254
To appear in:
Neurocomputing
Received date: Revised date: Accepted date:
16 June 2017 6 November 2017 19 January 2018
Please cite this article as: Yu Lin, Yu Zhang, Synchronization of stochastic impulsive discrete-time delayed networks via pinning control, Neurocomputing (2018), doi: 10.1016/j.neucom.2018.01.052
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Highlights • Pinning synchronization of stochastic impulsive discrete delay networks is studied; • Using the Razumikhin technique, synchronization criteria of networks are achieved; • Impulses do contribute to the synchronization of the networks;
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• Synchronization of the networks can be achieved even with enlarging impulses.
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Synchronization of stochastic impulsive discrete-time delayed networks via pinning control✩ Yu Lina , Yu Zhanga,∗ of Mathematical Sciences, Tongji University, 200092, China
Abstract
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a School
This paper is concerned with the synchronization of stochastic impulsive discrete-time delayed networks via pinning control. By using Lyapunov functions together with Razumikhin technique, some sufficient conditions about exponential synchronization of stochastic impulsive discrete-time delayed networks in mean square are developed. Two numerical examples are given to illustrate the effectiveness and superiority of the obtained results.
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Keywords: Delay, Pinning control, Impulse, Discrete-time. 1. Introduction
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causing instability of networks (see, e.g., [3–5, 8–13, 15, 23– 26, 29–37] and the references therein), therefore, it is necIn our realistic world, many real systems can be considered essary to consider dynamical networks with time delay. At as complex systems, such as the Internet, epidemic spread- the same time, the states of real networks are often subject ing networks, the World Wide Web, neural networks and so- to instantaneous perturbations or experience abrupt change at cial networks. A typical complex network is consisted of a certain moments, which can be called impulsive effects, imlarge number of interconnected dynamical nodes, which has pulsive effects can be caused by switching phenomena, frethe property of “scale-free” or “small-world” . Complex net- quency changes or other sudden noises, i.e., impulsive effects works have become a focus of research and have received exist in networks (see, e.g., [2, 4, 7, 8, 15–19, 22–34] and the more and more attention from various fields of science and references therein). Thus, it is necessary to study the discreteengineering. Complex networks tend to exhibit complex and time delayed networks with impulsive effects. interesting dynamical behaviors including synchronization, In the past few years, there have been some literature on the consensus, flocking etc. synchronization of impulsive discrete-time complex networks Synchronization is one of the most interesting and impor- (see, e.g., [16–18, 22–24] and the references therein). For tant collective behaviors in complex dynamical networks, it example, in [16], Zhang etc. studied the impulsive synchrohas attracted special attention of researchers in different fields nization problem of a continuous network composed with the chaotic Chen oscillators and a discrete-time network consist(see, e.g., [1–9, 12–27] and the references therein). There are various types of traditional and new control ing of H´enon map. [17] obtained some sufficient conditions methods have been successfully used to achieve network syn- for asymptotic H-synchronization between the drive system chronization, including impulsive control, adaptive control, and response system by the asymptotic stability criteria of and pinning control etc. Moreover, it is too difficult to con- systems with impulsive effects. In [22], Wu etc. studied the trol all nodes in a large-scale complex network, pinning con- synchronization problem of discrete-time network by impultrol can achieve synchronization of complex dynamical net- sive and pinning control. In [23], Li etc. investigated the exworks by pinning a small fraction of the network nodes, thus ponential synchronization of discrete-time complex networks it is economic and effective to achieve the synchronization with time-varying delay and obtained two synchronization of networks via pinning control. Lately the pinning control criteria by using the average impulsive interval. In [24], Li has stimulated many interesting for the synchronization of etc. obtained several synchronization criteria by using the dynamical networks (see, e.g., [18–20, 22, 25–27] and the Razumikhin technique and the discrete Gronwall inequality. Unfortunately, so far, the synchronization problem for references therein). It is well known that time delay shows as a vital character- stochastic impulsive discrete-time delayed complex networks istic of networks, and becomes one of the main sources for via pinning control has not been investigated yet. Motivated by the above discussions, the purpose of this paper is to investigate the synchronization of stochastic impul∗ Corresponding author at: School of Mathematical Sciences, Tongji Unisive discrete-time delayed networks via pinning control. The versity, 200092, China main contributions of this paper are highlighted as follows: Email address: [email protected], [email protected], Fax:+86-21-65981985. ( Yu Zhang) (1) it is the first time that synchronization of stochastic impulPreprint submitted to Neurocomputing
January 30, 2018
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hii = −
M X
j=1, j,i
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dk ))]T ∈ Rn is the activation function of the networks; c ∈ R (c , 0) is a constant; dk ∈ Z + corresponds to the discrete¯ where d, d¯ are known time delay which satisfies d ≤ dk ≤ d, positive integers and represent the lower and upper delays, respectively; N0 = 0, Nl ∈ Z + for l ∈ Z + , N1 < N2 < · · · < Nl < · · · , Nl → ∞ as l → ∞; κ(k, xi (k)− s(k)) ∈ Rn is the stochastic intensity; Γ ∈ Rn×n represents the inner coupling matrix between the subsystems; matrix H = (hi j ) M×M is the coupling configuration matrix representing the coupling strength and the topological structure of the networks which satisfies
sive discrete-time delayed complex network via pinning control is investigated; (2) based on Lyapunov stability theory, by designing pinning control and using Lyapunov functions together with Razumikhin technique, the synchronization of stochastic impulsive discrete-time delayed complex networks is achieved; (3) the obtained results show that impulses do contribute to the synchronization of stochastic discrete-time delayed networks with the objective state. The obtained results also show that even impulses have enlarging effects on the states of the networks, synchronization of the stochastic impulsive discrete-time delayed networks with the objective state can still be achieved under certain conditions. The rest of this paper is organized as follows. In Section 2, some basic definitions and notations are introduced. In Section 3, some criteria for the stochastic impulsive discretetime delay networks synchronize with the objective state are presented. In Section 4, two numerical examples are given to demonstrate the effectiveness and superiority of the proposed results. Finally, conclusions are drawn in Section 5. Notations: In this paper, Rn and Rn×m denote, respectively, the n-dimensional Euclidean space and the set of all n × m real matrices; In is the identity matrix of order n; the notation X ≥ Y (respectively, X > Y ), where X and Y are symmetric matrices, means that X − Y is positive semi-definite (respectively, positive definite); AT represents the transpose of A; the symmetric terms below the main diagonal of a symmetric matrix are denoted by ∗; λmax (A) and λmin (A) denote the maximum and minimum eigenvalue of A, respectively; for matrices A ∈ Rm×n and B ∈ R p×q , their Kronecker product is a matrix in Rmp×nq denoted by A ⊗ B; E{x(k)} stands for the expectation of the stochastic variable x(k); kx(k)k describes the Euclidean norm of a vector x(k); diag{· · · } stands for a block-diagonal matrix; R denotes the set of real numbers; Z + denotes the set of positive integer numbers; N denotes the set of natural numbers, i.e., N = {0, 1, 2, . . .}.
hi j , hi j = h ji , hi j ≥ 0(i , j), i, j = 1, 2, ..., M.
ω(k) is a one-dimensional Gaussian white noise sequence satisfying
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E{ω(k)} = 0, E{ω2 (k)} = 1, E{ω(i)ω( j)} = 0(i , j).
s(k) is the state vector of the isolate node which satisfies s(k + 1) = As(k) + f˜(s(k − dk )),
(3)
s(k) = [s1 (k), s2 (k), ..., sn (k)]T ∈ Rn . We hope that the complex network (1) can be synchronized to (3).
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In this paper, we have the following assumptions: Assumption 1[10, 13]. Each activation function f˜j (t), t ∈ R( j = 1, 2, ..., n) is continuous and bounded, and there exist constants b j1 , b j2 such that b j1 ≤
f˜j (a) − f˜j (b) ≤ b j2 , j = 1, 2, ..., n, a−b
(4)
where a, b ∈ R, and a , b. Assumption 2[9]. The stochastic intensity κ(k, y) satisfies the following inequality
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2. Preliminaries
(2)
κT (k, y)κ(k, y) ≤ yT W T Wy, i = 1, 2, ..., M, k ∈ N,
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Consider the following stochastic impulsive discrete-time delay networks consisting of M coupled identical neural networks in a complete probability space (Ω, F, P) , where each node of the networks is an n-dimensional system. M X ˜ x (k + 1) = Ax (k) + f (x (k − d )) + c hi j Γx j (k) i i i k j=1 + κ(k, xi (k) − s(k))ω(k) + ui (k), k , Nl − 1; (1) x (k + 1) = xi (k) + Bik (xi (k) − s(k)) i + (A − In )s(k) + f˜(s(k − dk )), k = Nl − 1; x (τ) = ϕ (τ) + s(τ), τ ∈ J , {−d, ¯ −d¯ + 1, ..., 0}, i i
(5)
where y ∈ Rn , W is a matrix with appropriate dimension.
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The pinning controllers in the complex network (1) are designed as follows: ( −pi Γ(xi (k) − s(k)), i = 1, 2, ..., m, ui (k) = (6) 0, i = m + 1, ..., M, where pi > 0 is the feedback gain and m is the number of pinning controllers. Let ei (k) = xi (k) − s(k), noting that
where k ∈ N, xi (k) = [xi1 (k), xi2 (k), . . . , xin (k)]T ∈ Rn is the state vector of the i th node, i = 1, 2, .., M; A ∈ Rn×n is a constant matrix; Bi(Nl −1) ∈ Rn×n is a constant matrix, l ∈ Z + ; f˜(xi (k − dk )) = [ f˜1 (xi1 (k − dk )), f˜2 (xi2 (k − dk )), ..., f˜n (xin (k −
c
M X j=1
3
M X
hi j Γs(k) = 0, thus,
j=1
hi j Γx j (k) − pi Γ(xi (k) − s(k))
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=c
M X j=1
where Q = QT , R = RT , is equivalent to either of the following conditions: 1) Q < 0, R − S T Q−1 S < 0, 2) R < 0, Q − S R−1 S T < 0. Lemma 3 [38]. Let Φ ∈ Rn×n be a positive definite matrix and M ∈ Rn×n be a symmetric matrix, then for any x ∈ Rn , the following inequality holds :
hi j Γ(x j (k) − s(k)) − pi Γ(xi (k) − s(k))
= c[hi1 Γe1 (k) + hi2 Γe2 (k) + · · · + hii Γei (k) + · · · + hiM Γe M (k)] − pi Γei (k)
= c[hi1 Γe1 (k) + hi2 Γe2 (k) + · · · + (hii −
where G = (gi j ) M×M , ( h − gi j = ii hi j ,
M X
λmin (Φ−1 M)xT Φx ≤ xT Mx ≤ λmax (Φ−1 M)xT Φx.
gi j Γe j (k),
j=1
3. Main Results
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+ · · · + hiM Γe M (k)] = c
pi )Γei (k) c
In this section, we will investigate the synchronization of stochastic impulsive discrete-time delayed networks (1) and (3) in mean square. Theorem 1. Under Assumptions 1, 2, if for given constants 0 < ηl < 1, l ∈ Z + , α > 0, β > 0, there exist constants q > 0, ξ > 0, ρ > 0, symmetric positive matrix P ∈ Rn×n and positive diagonal matrix U =diag{u1 , u2 , ..., un } such that
pi c ,i
= 1, 2, ..., m, otherwise.
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Therefore, from inequalities (1), (3) and (6), we get the error system as follows M X e (k + 1) = Ae (k) + f (e (k − d )) + c gi j Γe j (k) i i i k j=1 (7) + κ(k, ei (k))ω(k), k , Nl − 1; ei (k + 1) − ei (k) = Bik ei (k), k = Nl − 1; e (τ) = ϕ (τ), τ ∈ J, i
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In this paper, the following definition and Lemmas are useful for the derivation of the main results. Definition 1. The complex network (1) is said to be exponentially synchronized to (3) in mean square, if there exist constants θ > 0, M0 > 0 such that (X M
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i=1
) kei (k)k2 ≤ M0 |ϕ|e−θ(k−k0 )
T
α − 1 > 0), (1 − βq)
(12)
Ω1 (i) ∗ ∗
(13)
AT P U Bˇ < 0, Ω3 (i) # (In + Bi(Nl −1) )T P ≤ 0, l ∈ Z + , −P
−ηl P ∗
0 Ω2 ∗
(14)
(15)
M X ρW T W − αP, Ω2 = −U Bˆ − βP, Ω3 (i) = P + c g2i j P − U, j=1
(9)
then the complex network (1) is exponentially synchronized to (3) in mean square.
holds for all k ≥ k0 , where |ϕ| = maxτ∈J {kϕi (τ)k, i = 1, 2, ..., M}. Lemma 1 [12]. Let X and Y be any n-dimensional real vectors, and let P be an n × n positive semidefinite matrix. Then, the following matrix inequality holds: T
(11)
12 , where Bˆ =diag(b11 b12 , b21 b22 , ..., bn1 bn2 ), Bˇ =diag( b11 +b 2 bn1 +bn2 b21 +b22 T , ..., ), i = 1, 2, ..., M, Ω (i) = A PA+ 1 2 2 M M 2 X X c M 2 c α2i j ΓPΓ + ΓPΓ + c g2i j AT PA + 2cMΓPΓ+ 2 2 j=1 j=1
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where a ∈ R and a , 0.
d¯ ≤ Nl+1 − Nl ≤ ξ, l ∈ N,
ln ηl < −µξ − µ(Nl+1 − Nl ), l ∈ Z + ,
It can be verified from Assumption 1 that functions f j (t), t ∈ R( j = 1, 2, ..., n) satisfy the following condition: f j (a) ≤ b j2 , j = 1, 2, ..., n, a
(10)
q > e2µξ , (µ =
where ei (k) = [ei1 (k), ei2 (k), ..., ein (k)]T ∈ Rn , f (ei (k − dk )) = f˜(xi (k − dk )) − f˜(s(k − dk )) = [ f1T (ei1 (k − dk )), f2T (ei2 (k − dk )), ..., fnT (ein (k − dk ))]T ∈ Rn , i = 1, 2, .., M.
b j1 ≤
P ≤ ρI,
Proof. Choose the following Lyapunov function: V(k) = eT (k)(I M ⊗ P)e(k) =
M X
eTi (k)Pei (k),
i=1
where e(k) = [eT1 (k), eT2 (k), ..., eTM (k)]T ∈ R Mn , k ∈ N.
T
2X PY ≤ X PX + Y PY.
From inequalities (2) and (7), for any k , Nl − 1, l ∈ Z + ,
Lemma 2 (Schur Complement). The linear matrix inequality " # Q S < 0, ST R
E{V(k + 1)} = E
(X M h i=1
4
Aei (k) + f (ei (k − dk )) + c
M X j=1
gi j Γe j (k)
iT
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= 2F T (e(k − dk ))(I M ⊗ P)(c(G ⊗ Γ)e(k))
= 2cF T (e(k − dk ))(G ⊗ PΓ)e(k)) M M X X gi j PΓe j (k) = 2c f T (ei (k − dk ))
M X i h ·P Aei (k) + f (ei (k − dk )) + c gi j Γe j (k)
+[κ(k, ei (k))]T P[κ(k, ei (k))]
o
j=1
(X M h =E eTi (k)AT PAei (k) + f T (ei (k − dk ))P f (ei (k − dk )) i=1
+cM
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[κ(k, ei (k))]T P[κ(k, ei (k))] ≤ ρeTi (k)W T Wei (k).
M i X +2 f T (ei (k − dk ))P c gi j Γe j (k)
(16)
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c
j=1
i=1
T gi j Γe j (k) P c T
gi j Γe j (k)
j=1
= [c(G ⊗ Γ)e(k)] (I M ⊗ P)[c(G ⊗ Γ)e(k)]
· [b j2 ei j (k − dk ) − f j (ei j (k − dk ))] ≥ 0,
M X
c 2
eTi (k)
j=1
i=1
Therefore
0≤
α2i j ΓPΓ + MΓPΓ ei (k),
i
(17)
When k , Nl − 1, l ∈ Z , substituting inequalities (17)-(21) into (16) leads to (X M h E{V(k + 1)} ≤ E eTi (k)AT PAei (k) i=1
+ f T (ei (k − dk ))P f (ei (k − dk ))
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AC ≤c
M X
+ 2eTi (k)AT P f (ei (k − dk ))
eTi (k)
i=1
+
j=1
M X
g2i j AT PAei (k) + cM
j=1
M X
M X j=1
eTj (k)ΓPΓe j (k)
j=1
g2i j AT PA + MΓPΓ ei (k)
+ c f T (ei (k − dk ))
(18)
M X j=1
g2i j P f (ei (k − dk ))
ˆ i (k − dk ) + cMeTi (k)ΓPΓei (k) − eTi (k − dk )U Be + 2eTi (k − dk )U Bˇ f (ei (k − dk )) i − f T (ei (k − dk ))U f (ei (k − dk )) o + ρeTi (k)W T Wei (k)
j=1
and M M X X 2 f T (ei (k − dk ))P c gi j Γe j (k) i=1
M X c2 T ei (k) α2i j ΓPΓ + MΓPΓ ei (k) 2 j=1
+ ceTi (k)
M M X X = c eTi (k) g2i j AT PA + MΓPΓ ei (k), i=1
(21)
+
j=1
i=1
u j [ f j (ei j (k − dk )) − b j1 ei j (k − dk )]T
− f T (ei (k − dk ))U f (ei (k − dk )).
= 2eT (k)(I M ⊗ AT P)[c(G ⊗ Γ)e(k)]
= 2ceT (k)(G ⊗ AT PΓ)e(k) M M X X = 2c eTi (k) gi j AT PΓe j (k)
j=1
·[b j2 ei j (k − dk ) − f j (ei j (k − dk ))]
M M X X 2eTi (k)AT P c gi j Γe j (k) i=1
n X
ˆ i (k − dk ) = −eTi (k − dk )U Be T +2e (k − dk )U Bˇ f (ei (k − dk ))
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similarly
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≤
j=1
M X 2
i = 1, 2, ..., M, j = 1, 2, ..., n.
M
= c2 eT (k)(GT G ⊗ ΓPΓ)e(k) M M X X αi j ΓPΓe j (k) = c2 eTi (k) i=1
(20)
[ f j (ei j (k − dk )) − b j1 ei j (k − dk )]T
Let GT G = (αi j ) M×M , according to the property of the kronecker product and Lemma 1, we have M X
(19)
In addition, it follows from inequality (8) that [10, 13]
o
+[κ(k, ei (k))]T P[κ(k, ei (k))] .
M X
eTi (k)ΓPΓei (k),
here we let F(e(k − dk )) = [ f T (e1 (k − dk )), f T (e2 (k − dk )), ..., f T (e M (k − dk ))]T ∈ R Mn , from Assumption 2 and inequality (10) we have that
M X + 2eTi (k)AT P f (ei (k − dk )) + 2eTi (k)AT P c gi j Γe j (k)
M X
M X i=1
j=1
j=1
j=1
i=1
M M X T X + c gi j Γe j (k) P c gi j Γe j (k) j=1
j=1
i=1
M M X X g2i j P f (ei (k − dk )) ≤ c f T (ei (k − dk ))
j=1
5
ACCEPTED MANUSCRIPT
=E
(X M M h c2 M c2 X 2 αi j ΓPΓ + ΓPΓ eTi (k) AT PA + 2 j=1 2 i=1
< De−µN1 M|ϕ|2 = K|ϕ|2 e−µN1 . (ii) Next, we will prove that:
M X i +c g2i j AT PA + 2cMΓPΓ + ρW T W − αP ei (k)
E{V(k)} ≤ K|ϕ|2 e−µN1 , k ∈ [N0 , N1 ).
j=1
+2eTi (k)AT P f (ei (k − dk )) +eT (k − dk )(−U Bˆ − βP)ei (k − dk ) i
If the inequality (29) does not hold, then N1 − N0 ≥ 2, there exists a k ∈ (N0 , N1 ) (noting that N0 = 0), such that E{V(k)} > K|ϕ|2 e−µN1 . Let k∗ = inf{k ∈ (0, N1 ), |E{V(k)} > K|ϕ|2 e−µN1 }, then E{V(k∗ )} > K|ϕ|2 e−µN1 .
CR IP T
j=1
¯ k∗ − 1], For any k ∈ [−d,
E{V(k)} ≤ K|ϕ|2 e−µN1 .
j=1
> K|ϕ|2 e−µN1
∀dk∗ −1
E{V(k + 1)} < αE{V(k)} + βE{V(k − dk )}.
ED
(In + Bi(Nl −1) )T P(In + Bi(Nl −1) ) ≤ ηl P, l ∈ Z + .
(23)
< αE{V(k∗ − 1)} + βqE{V(k∗ )},
thus
(24)
E{V(k∗ )} <
i=1 −µξ−µ(Nl+1 −Nl )
≤e
E{V(Nl − 1)}, l ∈ Z + .
α E{V(k∗ − 1)} = (1 + µ)E{V(k∗ − 1)}. (32) 1 − βq
Therefore, it follows from inequalities (26) and (32) that
PT
CE
o ·(In + Bi(Nl −1) )ei (Nl − 1)} (X M o ≤ ηl E {eTi (Nl − 1)Pei (Nl − 1)}
1 E{V(k∗ )} 1+µ eµξ > K|ϕ|2 e−µN1 e−µξ 1+µ > K|ϕ|2 e−µN1 e−µξ
E{V(k∗ − 1)} >
(25)
= DM|ϕ|2 e−µN1 e−µξ
AC
Choose a constant D > 0 such that λmax (P) ≤ De−µN1 e−µξ < De−µN1 < qλmax (P).
≥ λmax (P)M|ϕ|2 .
(26)
From inequality (28), E{V(0)} ≤ λmax (P)M|ϕ|2 .
Then we will prove that E{V(k)} ≤ K|ϕ|2 e−µNl+1 , k ∈ [Nl , Nl+1 ), ∀l ∈ N,
(31)
E{V(k∗ )} < αE{V(k∗ − 1)} + βE{V(k∗ − 1 − dk∗ −1 )}
It follows from inequalities (13) and (24) that (X M {eTi (Nl − 1)(In + Bi(Nl −1) )T P E{V(Nl )} = E i=1
≥ E{V(k∗ − 1 − dk∗ −1 )}, ¯ dk∗ −1 ∈ Z + . ∈ [d, d],
Since k∗ ∈ (0, N1 ), k∗ , Nl , l ∈ Z + , from (23) and (31) we know
M
From (14), when k , Nl − 1, l ∈ Z + , one has
Applying the Lemma 2 to (15) yields
qE{V(k∗ )} > e2µξ K|ϕ|2 e−µN1
AN US
=
(30)
It can be found from inequalities (12) and (26) that
(22)
[eTi (k), eTi (k − dk ), f T (ei (k − dk ))]T , M M X X c2 M 2 Ω1 (i) = AT PA + c2 α2i j ΓPΓ + g2i j AT PA ΓPΓ + c 2 j=1 j=1 +2cMΓPΓ + ρW T W − αP, Ω2 = −U Bˆ − βP, M X Ω3 (i) = P + c g2i j P − U.
where ζ(k)
(29)
If N1 − N0 = 1, obviously the inequality (29) holds.
+2eTi (k − dk )U Bˇ f (ei (k − dk )) M X o + f T (ei (k − dk ))(P + c g2i j P − U) f (ei (k − dk ))
+αE{V(k)} + βE{V(k − dk )} AT P Ω (i) 0 M X T 1 ˇ ∗ Ω U B ζ(k) = E ζ (k) 2 i=1 ∗ ∗ Ω3 (i) +αE{V(k)} + βE{V(k − dk )},
(28)
Let k¯ = sup{k ∈ [0, k∗ −1), |E{V(k)} ≤ λmax (P)M|ϕ|2 }. Then λmax (P)M|ϕ|2 < E{V(k)} ≤ K|ϕ|2 e−µN1 for k¯ + 1 ≤ k ≤ k∗ − 1. From inequalities (26) and (30), for any k ∈ [k¯ + 1, k∗ − 1], we get
(27)
where K = DM. ¯ N0 ], from Lemma 3 and inequality (26) (i) When k ∈ [−d, we obtain
qE{V(k)} > qλmax (P)M|ϕ|2 > De−µN1 M|ϕ|2 = K|ϕ|2 e−µN1
M X E{V(k)} ≤ λmax (P) ||ei (k)||2 ≤ λmax (P)M|ϕ|2
¯ dk−1 ∈ Z + . ≥ E{V(k − 1 − dk−1 )}, ∀dk−1 ∈ [d, d],
(30)
i=1
6
ACCEPTED MANUSCRIPT
Therefore for any k ∈ [k¯ + 1, k∗ − 1], since k , Nl , l ∈ Z + , from (23) we have
= eµξ eµξ K|ϕ|2 e−µ(Nr+1 −Nr ) e−µNr ≥ eµξ K|ϕ|2 e−µNr > K|ϕ|2 e−µNr ≥ E{V(kˆ − 1 − dk−1 ˆ )},
E{V(k)} < αE{V(k − 1)} + βE{V(k − 1 − dk−1 )}
¯ dk−1 ∈ [d, d], ∈ Z+. ∀dk−1 ˆ ˆ
< αE{V(k − 1)} + qβE{V(k)}.
Since kˆ ∈ (Nr , Nr+1 ), kˆ , Nl , l ∈ Z + , from inequalities (23) and (38) we get
Thus, for any k ∈ [k¯ + 1, k∗ − 1], E{V(k)} <
α E{V(k − 1)} = (1 + µ)E{V(k − 1)}. (33) 1 − βq
ˆ < αE{V(kˆ − 1)} + βE{V(kˆ − 1 − dk−1 E{V(k)} ˆ )} ˆ < αE{V(kˆ − 1)} + βqE{V(k)}.
ˆ < E{V(k)}
E{V(k∗ )} < (1 + µ)E{V(k∗ − 1)}
It follows from inequalities (26), (39) and
Thus
(32)
∗
< (1 + µ) E{V(k − 2)} < · · · < (1 + µ)
(33)
k∗ −k¯
¯ E{V(k)}
α E{V(kˆ − 1)} = (1 + µ)E{V(kˆ − 1)}. (39) 1 − βq
≤ K|ϕ|2 e−µN1 < E{V(k∗ )},
(iii) We assume that for t = 1, 2, ..., r,
Next, we will prove that 2 −µNr+1
E{V(k)} ≤ K|ϕ| e
ED
E{V(k)} ≤ K|ϕ|2 e−µNt , k ∈ [Nt−1 , Nt ).
, k ∈ [Nr , Nr+1 ).
(34)
Therefore, for any k ∈ [k˜ + 1, kˆ − 1], k ∈ Z + , from inequalities (11), (12) and (37) we have qE{V(k)} > e2µξ K|ϕ|2 e−µNr+1 e−µξ
(35)
= eµξ K|ϕ|2 e−µ(Nr+1 −Nr ) e−µNr ≥ K|ϕ|2 e−µNr
PT
It follows from inequalities (25) and (34) that
¯ dk−1 ∈ Z + . ≥ E{V(k − 1 − dk−1 )}, ∀dk−1 ∈ [d, d],
E{V(Nr )} ≤ e−µξ e−µ(Nr+1 −Nr ) E{V(Nr − 1)}
Since k ∈ [k˜ + 1, kˆ − 1], k , Nl , l ∈ Z + from inequality (23) one has
≤ e−µξ e−µ(Nr+1 −Nr ) K|ϕ|2 e−µNr 2 −µNr+1
K|ϕ| e
CE
=e
−µξ
< K|ϕ|2 e−µNr+1 .
E{V(k)} < αE{V(k − 1)} + βE{V(k − 1 − dk−1 )}
(36)
If Nr+1 − Nr = 1, obviously the inequality (35) holds.
AC
α E{V(k − 1)} = (1 + µ)E{V(k − 1)}, 1 − βq ∀k ∈ [k˜ + 1, kˆ − 1]. (40)
E{V(k)} <
Furthermore, using inequalities (11), (39) and (40) we derive that ˆ < (1 + µ)E{V(kˆ − 1)} E{V(k)}
From d¯ ≤ Nr − Nr−1 , Nr − d¯ ≥ Nr−1 , from (34), we have E{V(k)} ≤ K|ϕ| e
¯ kˆ − 1]. , ∀k ∈ [Nr − d,
< αE{V(k − 1)} + qβE{V(k)},
thus
If the inequality (35) does not hold, then Nr+1 − Nr ≥ 2, there exists a k ∈ (Nr , Nr+1 ) such that E{V(k)} > K|ϕ|2 e−µNr+1 . Let kˆ = inf{k ∈ (Nr , Nr+1 ), |E{V(k)} > K|ϕ|2 e−µNr+1 }, then ˆ > K|ϕ|2 e−µNr+1 . For any k ∈ [Nr , kˆ − 1], E{V(k)} ≤ E{V(k)} 2 −µNr+1 K|ϕ| e .
2 −µNr
1 ˆ E{V(k)} 1+µ eµξ ˆ −µξ = E{V(k)}e 1+µ eµξ > K|ϕ|2 e−µNr+1 e−µξ 1+µ ≥ K|ϕ|2 e−µNr+1 e−µξ .
From inequality (36), we get E{V(Nr )} ≤ K|ϕ|2 e−µNr+1 e−µξ , define k˜ = sup{k ∈ [Nr , kˆ − 1), |E{V(k)} ≤ K|ϕ|2 e−µNr+1 e−µξ }, ˜ ≤ K|ϕ|2 e−µNr+1 e−µξ . For any k ∈ [k˜ + 1, kˆ − 1], then E{V(k)} 2 −µNr+1 K|ϕ| e ≥ E{V(k)} > K|ϕ|2 e−µNr+1 e−µξ .
M
this is a contradiction, thus inequality (29) holds.
AN US
= (1 + µ)ξ K|ϕ|2 e−µξ e−µN1
≥ 1 that
(39)
¯ ≤ (1 + µ)ξ λmax (P)M|ϕ|2 ≤ (1 + µ)ξ E{V(k)} ≤ (1 + µ)ξ DM|ϕ|2 e−µξ e−µN1
eµξ 1+µ
E{V(kˆ − 1)} >
(11)
(26)
CR IP T
¯ ≤ λmax (P)M|ϕ|2 and (1 + On the other hand, using E{V(k)} µ) e ≤ 1, we derive from inequalities (11), (26), (32) and (33) that ξ −µξ
2
(38)
(39)
ˆ ˜ ˜ < (1 + µ)2 E{V(kˆ − 2)} < · · · < (1 + µ)k−k E{V(k)}
(37)
(40)
≤ (1 + µ)ξ K|ϕ|2 e−µNr+1 e−µξ
It follows from inequalities (11) and (12) that
(11)
ˆ > e2µξ K|ϕ|2 e−µNr+1 qE{V(k)}
ˆ ≤ K|ϕ|2 e−µNr+1 < E{V(k)}, 7
ACCEPTED MANUSCRIPT
ρW T W − α exp(−λ)P, Ψ2 = −U Bˆ − β exp(−λ(dk + P 2 1))P, Ψ3 (i) = P + c M g P − U, then the complex j=1 i j network (1) is exponentially synchronized to (3) in mean square.
this is a contradiction, thus the inequality (35) holds. By the principle of mathematical induction, we have proved inequality (27) holds (noting that N0 = 0), which means
Proof. Choose the following Lyapunov function:
E{V(k)} ≤ K|ϕ|2 e−µNl < K|ϕ|2 e−µk , k ∈ [Nl−1 , Nl ), ∀l ∈ Z + .
V(k) = exp(λk)eT (k)(I M ⊗ P)e(k) =
From Lemma 3 ) ( M X kei (k)k2 ≤ E{V(k)}, ∀k ≥ 0, E λmin (P)
CR IP T
From inequalities (2) and (7), for any k , Nl − 1, l ∈ Z + .
thus E
i=1
2
kei (k)k
)
K ≤ |ϕ|2 e−µk , ∀k ≥ 0. λmin (P)
E{V(k + 1)} = E
(41)
(X M i=1
h exp(λ(k + 1)) Aei (k)
M X iT + f (ei (k − dk )) + c gi j Γe j (k) P
This implies that the complex network (1) is exponentially synchronized to (3) in mean square. Remark 1. In Theorem 1, it is worth noting that α + β can be bigger than 1, which means the error network (7) is allowed to be unstable. In order to stabilize the error network (7), we give the inequality (13), that is the reason why Theorem 1 can be used to deal with the exponential synchronization of complex network (1) and (3) in mean square under impulsive stabilization and pinning control. The next theorem considers another situation when the error network (7) is stable and it is still stable under impulsive perturbations. Remark 2. The reason why we choose Kronecker product to deal with the inequalities (17)-(19) is that we want to get the more convenient representations and it may make the results less conservative. Theorem 2. Under Assumptions 1, 2, if for given positive constants τ˜ > 0, λ > 0, η > 1, α > 0, β > 0, α + β < 1, there exist constant ρ > 0, symmetric positive matrix P ∈ Rn×n and positive diagonal matrix U =diag{u1 , u2 , ..., un } such that
AN US
j=1
h
· Aei (k) + f (ei (k − dk )) + c
+ [κ(k, ei (k))]T P[κ(k, ei (k))]
=E
i=1
M
"
−ηP ∗
(43)
ln η < λ, τ˜
(44)
Ψ1 (i) ∗ ∗
AT P U Bˇ < 0, Ψ3 (i) # exp( 12 λ)(In + Bi(Nl −1) )T P ≤ 0, l ∈ Z + , −P 0 Ψ2 ∗
j=1
o
exp(λ(k + 1)) eTi (k)AT PAei (k)
j=1
j=1
+ 2eTi (k)AT P f (ei (k − dk )) M X + 2eTi (k)AT P c gi j Γe j (k) j=1
M X + 2 f T (ei (k − dk ))P c gi j Γe j (k)
(42)
Nl − Nl−1 ≥ τ˜ , l ∈ Z , +
(X M
M X i gi j Γe j (k)
+ f T (ei (k − dk ))P f (ei (k − dk )) M M T X X gi j Γe j (k) + c gi j Γe j (k) P c
ED
PT
AC
CE
P ≤ ρI,
exp(λk)eTi (k)Pei (k),
i=1
where e(k) = [eT1 (k), eT2 (k), ..., eTM (k)]T ∈ R Mn , k ∈ N.
i=1
(X M
M X
j=1
o + [κ(k, ei (k))]T P[κ(k, ei (k))] .
(47)
It can be verified from inequality (8) that [10, 13] 0 ≤ exp(λ(k + 1))
n X u j [ f j (ei j (k − dk )) − b j1 ei j (k − dk )]T j=1
·[b j2 ei j (k − dk ) − f j (ei j (k − dk ))] ˆ i (k − dk ) = exp(λ(k + 1))[−eT (k − dk )U Be
(45)
i
+2eTi (k − dk )U Bˇ f (ei (k − dk ))
− f T (ei (k − dk ))U f (ei (k − dk ))].
(46)
(48)
When k , Nl − 1, l ∈ Z + , substituting inequalities (17)-(20) and (48) into (47) leads to
12 where Bˆ =diag(b11 b12 , b21 b22 , ..., bn1 bn2 ), Bˇ =diag( b11 +b , 2 bn1 +bn2 b21 +b22 T , ..., 2 ), i = 1, 2, ..., M, Ψ1 (i) = A PA+ 2 M M X X c2 M 2 2 c α ΓPΓ + ΓPΓ + c g2i j AT PA + 2cMΓPΓ+ ij 2 2 j=1 j=1
E{V(k + 1)} ≤ E 8
(X M i=1
h exp(λ(k + 1)) eTi (k)AT PAei (k)
ACCEPTED MANUSCRIPT
+ f T (ei (k − dk ))P f (ei (k − dk )) M X 2 α2i j ΓPΓ + MΓPΓ ei (k) + c2 eTi (k)
·(In + Bi(Nl −1) )e(Nl − 1) ≤ E{ηV(Nl − 1)}, l ∈ Z + .
j=1
E{V(k)} ≤ λmax (P)M|ϕ|2 .
j=1
E{V(k)} ≤ λmax (P)M|ϕ|2 , k ∈ [0, N1 ).
If (53) does not hold, there exists a positive integer k∗ , such that k∗ + 1 = inf{k ∈ (0, N1 ), |E{V(k)} > λmax (P)M|ϕ|2 }. E{V(k)} ≤ λmax (P)M|ϕ|2 for −d¯ ≤ k ≤ k∗ . Thus 0 < k∗ + 1 ≤ N1 − 1,
i=1
M X
h exp(λ(k + 1)) eTi (k) AT PA
E{V(k∗ + 1)} < αE{V(k∗ )} + βE{V(k∗ − dk∗ )}
≤ αλmax (P)M|ϕ|2 + βλmax (P)M|ϕ|2
< λmax (P)M|ϕ|2 ,
+ 2eTi (k)AT P f (ei (k − dk )) + 2eTi (k − dk )U Bˇ f (ei (k − dk )) +eT (k − dk )(−U Bˆ − exp(−λ(dk + 1))βP)ei (k − dk ) i
M X o g2i j P − U) f (ei (k − dk ))
which contradicts with (54), thus (53) holds. (iii) Next we will prove that
M
+ f T (ei (k − dk ))(P + c
j=1
ED
+αE{V(k)} + βE{V(k − dk )} Ψ1 (i) 0 AT P M X T ˇ ζ(k) ∗ Ψ U B exp(λ(k + 1))ζ (k) = E 2 i=1 ∗ ∗ Ψ3 (i) +αE{V(k)} + βE{V(k − dk )},
E{V(k)} ≤ ηλmax (P)M|ϕ|2 , k ∈ [N1 , N2 ).
E{V(N1 )} ≤ E{ηV(N1 − 1)}
(49)
≤ ηλmax (P)M|ϕ|2 .
If (55) does not hold, there exists a positive integer k , such 0 that k + 1 = inf{k ∈ (N1 , N2 ), |E{V(k)} > ηλmax (P)M|ϕ|2 }. 0 0 E{V(k)} ≤ ηλmax (P)M|ϕ|2 for −d¯ ≤ k ≤ k . Thus N1 < k +1 < N2 ,
PT
CE
AC
Ψ2 = −U Bˆ − β exp(−λ(dk + 1))P, Ψ3 (i) = P + c
0
E{V(k + 1)} > ηλmax (P)M|ϕ|2 . Since k , Nl − 1, l ∈ Z + , from (50) one has 0
< ηλmax (P)M|ϕ|2 ,
(50)
which contradicts with (57), thus (55) holds. (iv) Suppose for t = 1, 2, ..., r, we have
exp(λ)(In + Bi(Nl −1) )T P(In + Bi(Nl −1) ) ≤ ηP, l ∈ Z + .
E{V(k)} ≤ ηt−1 λmax (P)M|ϕ|2 , k ∈ [Nt−1 , Nt ).
Thus
i=1
0
≤ αηλmax (P)M|ϕ|2 + βηλmax (P)M|ϕ|2
Applying the Lemma 2 to (46) yields
(X M
0
E{V(k + 1)} < αE{V(k )} + βE{V(k − dk0 )}
j=1
Thus, when k , Nl − 1, l ∈ Z + , from (45) we obtain
E{V(Nl )} = E
(57)
0
M X g2i j P − U.
E{V(k + 1)} < αE{V(k)} + βE{V(k − dk )}.
(56) 0
M X g2i j AT PA + 2cMΓPΓ + ρW T W − α exp(−λ)P, j=1
(55)
In fact, by inequalities (51) and (53), we have
where ζ(k) = [eTi (k), eTi (k − dk ), f T (ei (k − dk ))]T , M X c2 M 2 Ψ1 (i) = AT PA + c2 α2i j ΓPΓ + ΓPΓ 2 j=1 +c
(54)
Since k∗ , Nl − 1, l ∈ Z + , from (50) one has
M X c2 M + c2 α2i j ΓPΓ + ΓPΓ + c g2i j AT PA 2 j=1 j=1 i +2cMΓPΓ + ρW T W − α exp(−λ)P ei (k) 2
E{V(k∗ + 1)} > λmax (P)M|ϕ|2 .
AN US
(X M
(53)
CR IP T
− dk )
o i − f T (ei (k − dk ))U f (ei (k − dk )) + ρeTi (k)W T Wei (k)
=E
(52)
(ii) From (i) and α + β < 1, we will prove
j=1
ˆ i (k + cMeTi (k)ΓPΓei (k) − eTi (k − dk )U Be + 2eTi (k − dk )U Bˇ f (ei (k − dk ))
(51)
¯ N0 ] (noting that N0 = 0), k ≤ 0, from (i) For any k ∈ [−d, Lemma 3 we have
+ 2eTi (k)AT P f (ei (k − dk )) M X + ceTi (k) g2i j AT PA + MΓPΓ ei (k)
M X + c f T (ei (k − dk )) g2i j P f (ei (k − dk ))
We will prove that exp(λNl )eT (Nl − 1)(In + Bi(Nl −1) )T P
E{V(k)} ≤ ηr λmax (P)M|ϕ|2 , k ∈ [Nr , Nr+1 ). 9
(58)
ACCEPTED MANUSCRIPT
4. SIMULATION EXAMPLES
E{V(Nr )} ≤ E{ηV(Nr − 1)} r
2
≤ η λmax (P)M|ϕ| .
If (58) does not hold, there exists a positive integer k∗∗ , such that k∗∗ + 1 = inf{k ∈ (Nr , Nr+1 ), |E{V(k)} > ηr λmax (P)M|ϕ|2 }. E{V(k)} ≤ ηr λmax (P)M|ϕ|2 for −d¯ ≤ k ≤ k∗∗ . Thus Nr < k∗∗ + 1 < Nr+1 , E{V(k∗∗ + 1)} > ηr λmax (P)M|ϕ|2 .
(59)
To illustrate the effectiveness and superiority of the given results, two examples are given in this section. Example 1. Consider the synchronization of stochastic impulsive discrete-time delayed networks (1) and (3) via pinning control with the following data: "
# " # 0 −0.8 0 , Bi(Nl −1) = , l ∈ Z+, 1.01 0 −0.8 " # " # 0.1 0 0.1 0 Γ= , W= , 0 0.1 0 0.01 −0.2 0.1 0.1 −0.2 0.1 , H = 0.1 0.1 0.1 −0.2
1.02 A= 0.1
CR IP T
Moreover, it follows from inequality (51) that
Since k∗∗ , Nl − 1, l ∈ Z + , from (50) one has E{V(k∗∗ + 1)} < αE{V(k∗∗ )} + βE{V(k∗∗ − dk∗∗ )}
≤ αηr λmax (P)M|ϕ|2 + βηr λmax (P)M|ϕ|2 < ηr λmax (P)M|ϕ|2 ,
E{V(k)} ≤ ηl λmax (P)M|ϕ|2 , k ∈ [Nl , Nl+1 ), l ∈ N.
AN US
which contradicts with (59), thus (58) holds. Therefore, by the principle of mathematical induction, we have proved that
(60)
From (43) and noted that N0 = 0, for k ∈ [Nl , Nl+1 ), l ∈ N, one obtains k ≥ Nl ≥ (Nl − Nl−1 ) + (Nl−1 − Nl−2 ) + · · · + (N1 − N0 ) ≥ l˜τ. (61)
E{V(k)} ≤ ηl λmax (P)M|ϕ|2 k
ED
≤ η τ˜ λmax (P)M|ϕ|2
= exp( τk˜ ln η)λmax (P)M|ϕ|2 ,
∀k ∈ [Nl , Nl+1 ), l ∈ N.
(62)
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exp(λk)kei (k)k
i=1
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E λmin (P)
M X
)
≤ E{V(k)}, ∀k ≥ 0.
1200
states
AC i=1
x2(1,k) x2(2,k)
600
x3(1,k)
400
x3(2,k)
200
s(1,k) s(2,k)
0
Thus, from (44) one obtain
(X M
x1(2,k)
800
) M X k E λmin (P) exp(λk)kei (k)k2 ≤ exp( ln η)λmax (P)M|ϕ|2 . τ ˜ i=1
E
x1(1,k)
1000
Therefore from inequality (62) for k ∈ [Nl , Nl+1 ), l ∈ N,
(
# 0 . 0.005
ρ = 18.4845, q = 8.3333, therefore, the complex network (1) is exponentially synchronized to (3) in mean square.
Using Lemma 3, the following inequality holds: (
It is easy to check that " # " 0 0 0.005 Bˆ = , Bˇ = 0 0 0
Let α = 1.1, β = 0.01, µ = 0.2, ηl = 0.2, l ∈ Z + . By using LMI toolbox in Matlab, one of the feasible solutions of Theorem 1 is " # " # 15.8391 −2.4451 408.0318 0 P= , U= , −2.4451 1.1751 0 78.9192
M
From (60) and (61) we have
where f˜(xi (k)) = tanh(0.01xi (k)), xi (k) ∈ R2 , i = 1, 2, 3, c = 0.01, the number of coupled identical neural networks is M = 3, the impulsive interval is 0 < Nl − Nl−1 ≤ ξ = 3, l ∈ Z + , p1 = 10, the pinning control u1 (k) = −10Γ(xi (k) − s(k)), ui (k) = 0 for i = 2, 3 and dk = mod(k, 3), k ∈ N.
−200
) λmax (P)M|ϕ|2 ln η kei (k)k2 ≤ exp(−(λ − )k), ∀k ≥ 0. λmin (P) τ˜
−400 −600 −800
Therefore, the complex network (1) is exponentially synchronized to (3) in mean square, this completes the proof. Remark 3. In Theorem 2, α + β < 1 implies that the error network (7) is stable, and η > 1 means that the impulses may have enlarging effects on the states of the networks.
10
0
10
20
30
40
50
k
Figure 1: The state trajectory of the network (1) and (3) in Example 1.
The simulation results for Example 1 are shown in Figure 1 and Figure 2. Figure 1 gives the state trajectory of the
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x1(2,k) x2(1,k)
15
x2(2,k) x3(1,k) x3(2,k)
10
s(1,k) s(2,k)
states
network (1) and (3), which shows network (1) and (3) are not synchronized. The matrix Bi(Nl −1) , l ∈ Z + we give in Example 1 implies impulses do contribute to the synchronization of network (1) and (3). Figure 2 shows the state trajectory of the error network (7) in Example 1 with impulsive control and pinning control, which confirms that the error network (7) with impulsive control and pinning control is exponentially stable. The simulation results have confirmed that the designed controllers perform very well.
5
25
−5
e1(1,k)
0
e1(2,k)
20
e2(1,k) e2(2,k)
15
states
5
10
15 k
20
25
30
Figure 3: The state trajectory of the network (1) and (3) in Example 2.
e3(1,k) 10
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e (2,k) 3
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0
−5
−2
−10
−4
0
10
20
30
40
50
k
states
−15
Figure 2: The state trajectory of the error network (7) in Example 1.
Example 2. Consider the synchronization of stochastic impulsive discrete-time delayed networks (1) and (3) via pinning control with the following data: " # " # 0.1 −0.3 0.02 0 A= , Bi(Nl −1) = , l ∈ Z+, 0.02 0.1 0 0.02 " # " # 1 0 0.05 0 Γ= , W= , 0 1 0 0.05 −0.2 0.1 0.1 −0.2 0.1 , H = 0.1 0.1 0.1 −0.2
−6 −8
e1(1,k)
−10
e1(2,k)
−12
e2(2,k)
e2(1,k) e3(1,k)
M
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−16
e3(2,k) 0
5
10
15 k
20
25
30
Figure 4: The state trajectory of the error network (7) in Example 2.
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The simulation results for Example 2 are shown in Figure 3 and Figure 4. Figure 3 gives the state trajectory of the network (1) and (3), which shows network (1) and (3) are synchronized. Figure 4 shows the state trajectory of error network (7). where f˜(xi (k)) = tanh(0.1xi (k)), xi (k) ∈ R2 , i = 1, 2, 3, c = Remark 4. Inspired by some enlightening works (see, e.g., 0.01, the number of coupled identical neural networks is [10–12, 24, 37] and the references therein), it is worthwhile to M = 3, the impulsive interval is 10 = τ˜ ≤ Nl − Nl−1 , l ∈ Z + , investigate synchronization of stochastic impulsive discretep1 = 0.1, the pinning control u1 (k) = −0.1Γ(xi (k) − s(k)), time networks with distributed delays via pinning control in the future. ui (k) = 0 for i = 2, 3 and dk = mod(k, 3), k ∈ N. Remark 5. Examples 1 and 2 cannot be studied by the results in [16–18, 22–24], for instance, the pinning control, time It is easy to check that delay, stochastic effects are not considered in [16, 17], time " # " # 0 0 0.05 0 delay is not considered in [18], time delay, stochastic effects ˆ ˇ B= , B= . 0 0 0 0.05 are not considered in [22], the pinning control and stochastic Let α = 0.5, β = 0.4, λ = 1, η = 3. By using LMI toolbox effects are not considered in [23, 24]. in Matlab, one of the feasible solutions of Theorem 2 is " # " # 5. Conclusion 4.5229 0.8696 17.3033 0 P= , U= , 0.8696 13.2864 0 24.7363 In this paper, we have investigated the synchronization ρ = 21.9660, therefore, the complex network (1) is exponen- problem for a class of stochastic impulsive discrete-time tially synchronized to (3) in mean square. delayed networks via pinning control. By using Lyapunov functions together with Razumikhin technique, some results 11
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are presented. The obtained results show that impulses do contribute to the synchronization of stochastic impulsive discrete-time delayed networks with the objective state. The obtained results also show that even impulses have enlarging effects on the states of the networks, synchronization of the stochastic impulsive discrete-time delayed networks with the objective state can still be achieved under certain conditions. Finally two simulation examples have been given to illustrate the effectiveness and superiority of the obtained results. A possible direction for future work is to obtain some synchronization criteria for stochastic impulsive discrete-time networks with distributed delays via pinning control.
[8] P. Li, J. Cao, Z. Wang, Robust impulsive synchronization of coupled delayed neural networks with uncertainties, Physica A Stat. Mech. its Appl. 373 (2007) 261272. [9] J. Liang, Z. Wang, Y. Liu, X. Liu, Global synchronization control of general delayed discrete-time networks with stochastic coupling and disturbances, IEEE Trans. Syst. Man Cybern. Part B 38 (4) (2008) 1073-1083.
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[10] Y. Liu, Z. Wang, X. Liu, Global exponential stability of generalized recurrent neural networks with discrete and distributed delays, Neural Networks 19 (5) (2006) 667-675. [11] Y. Liu, Z. Wang, X. Liu, Asymptotic stability for neural networks with mixed time-delays: the discrete-time case, Neural Networks 22 (1) (2009) 67–74.
Acknowledgements The authors would like to thank the associate editor and the anonymous reviewers for their constructive comments and suggestions that improved the quality of the paper.
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[12] Y. Liu, Z. Wang, J. Liang, X. Liu, Synchronization and state estimation for discrete-time complex networks with distributed delays, IEEE Trans. Syst. Man Cybern. Part B 38 (5) (2008) 1314–1325. [13] Y. Liu, Z. Wang, J. Liang, X. Liu, Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays, IEEE Trans. Neural Networks 20 (7) (2009) 1102-1116.
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Yu Lin Yu Lin received the B.Sc. degree from the School of Mathematical Sciences, Xiamen University, Xiamen, China, in 2015. Now, she is working toward the P.H.D. degree in the School of Mathematical Sciences, Tongji University. Her current research interest is impulsive discrete-time delayed system.
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[29] J. Zhou, Q. Wu, L. Xiang, Pinning complex delayed dynamical networks by a single impulsive controller, IEEE Trans. Circ. Syst. I Regul. Pap. 58 (12) (2011) 2882– 2893.
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[30] J. Lu, Z. Wang, J. Cao, D.W.C. Ho, J. Kurths, Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay, Int. J. Bifurcation Chaos 22 (07) (2012) 1250176. [31] H. Xu, X. Liu, K. L. Teo, A LMI approach to stability analysis and synthesis of impulsive switched systems with time delays, Nonlinear Anal. Hybrid Syst. 2 (1) (2008) 38–50.
Yu Zhang Yu Zhang received the B.Sc. degree from the Department of Applied Mathematics, Northeastern University, Shenyang, China, in 2002, and the Ph.D. degree from the Department of Mathematics, Tongji University, Shanghai, China, in 2007, respectively. She was a Research Assistant at the City University of Hong Kong, Hong Kong, China, from July 2005 to October 2005. Since July 2007, she has been with the School ofMathematical Sciences, Tongji University, Shanghai, China. She is currently an Associate Professor at Tongji University. She is the author or coauthor of more than 30 research articles.
[32] Y. Zhang, J. Sun, G. Feng, Impulsive control of discrete systems with time delay, IEEE Trans. Autom. Control 54 (4) (2009) 830–834. [33] Y. Zhang, Robust exponential stability of uncertain impulsive neural networks with time-varying delays and delayed impulses, Neurocomputing 74 (17) (2011) 3268–3276. 13
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Her research interests include stability theory, discrete-time stochastic systems and impulsive control systems.
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