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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 8, AUGUST 2013
Exponential Synchronization of Coupled Switched Neural Networks With Mode-Dependent Impulsive Effects Wenbing Zhang, Yang Tang, Member, IEEE, Qingying Miao, and Wei Du
Abstract— This paper investigates the synchronization problem of coupled switched neural networks (SNNs) with modedependent impulsive effects and time delays. The main feature of mode-dependent impulsive effects is that impulsive effects can exist not only at the instants coinciding with mode switching but also at the instants when there is no system switching. The impulses considered here include those that suppress synchronization or enhance synchronization. Based on switching analysis techniques and the comparison principle, the exponential synchronization criteria are derived for coupled delayed SNNs with mode-dependent impulsive effects. Finally, simulations are provided to illustrate the effectiveness of the results. Index Terms— Coupled switched neural networks (SNNs), exponential synchronization, mode-dependent impulsive effects.
I. I NTRODUCTION
S
INCE small-world and scale-free complex networks were proposed in [1] and [2], complex dynamical networks that consist of interacting dynamical entities with an interplay between dynamical states and interaction patterns have received increasing attention from various fields of science and engineering [3]–[5]. A large number of systems in nature can be modeled by complex dynamical networks, e.g., power grids, communication networks, the Internet, the World Wide Web, metabolic systems and food webs, etc. Many interesting behaviors can be observed from complex dynamical networks, e.g., synchronization, consensus, self-organization, and spatiotemporal chaos spiral waves (see [6] and references therein). Synchronization as an important collective behavior of complex dynamical networks has gained much attention in the last two decades [7]–[16].
Manuscript received November 12, 2012; revised January 27, 2013; accepted April 3, 2013. Date of publication May 13, 2013; date of current version June 28, 2013. This work was supported in part by the National Natural Science Foundation of China under Grant 61203235, the China PostDoctoral Science Foundation under Grant 2011M501040, the Alexander von Humboldt Foundation of Germany, the Key Creative Project of the Shanghai Education Community under Grant 13ZZ050, and the Key Foundation Project of Shanghai under Grant 12JC1400400. W. Zhang is with the Department of Mathematics, Yangzhou University, Jiangsu 225002, China (e-mail: [email protected]). Y. Tang is with the Institute of Physics, Humboldt University of Berlin, Berlin 12489, Germany, and also with the Potsdam Institute for Climate Impact Research, Potsdam 14415, Germany (e-mail: [email protected]). Q. Miao is with the School of Continuing Education, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]). W. Du is with the Institute of Textile and Clothing, The Hong Kong Polytechnic University, Hong Kong, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2013.2257842
As a special class of complex networks, linearly coupled neural networks have been a hot topic because they have wide applications in a variety of areas, such as signal processing, pattern recognition, static image processing, associative memory, and combinatorial optimization [17]. On the other hand, time delay is ubiquitous in coupled neural networks because of the finite information transmission and processing speeds among the units. It is well known that time delay may be an important source of oscillation, divergence, and instability in systems. Thus, delayed neural networks have been widely studied [18]–[21]. In [18], the exponential synchronization problem is studied for a class of coupled neural networks with discrete and distributed delays under time-varying sampling. In [19], the robust synchronization problem for an array of coupled stochastic discrete-time neural networks with timevarying delay is studied. In [20], the synchronization problem is studied for a class of Markovian coupled neural networks with nonidentical node delays and random coupling strengths. On the other hand, in real-world systems, because of link failures and the creation of new links that may happen at times, the jumps between different topologies are inevitable [6]. Hence, when modeling real-world dynamical networks, it is desirable to model such networks using model switchings. Recently, synchronization of complex networks with switching topology has become a hot topic [9], [14], [22]. By assuming the switching interval to be less than a given constant, the synchronization problem of switched complex networks is investigated in [22]. In [9], the stability and synchronization problem has been addressed for a class of discrete-time neural networks with mode-dependent mixed time delay and Markovian jump switching. It is also shown in [23] and [24] that an arbitrary switching may destroy the stability of switched systems and that one useful method to investigate the stability of switched systems is to employ the so-called average dwell time approach. By this approach, the synchronization of switched networks under arbitrary switching was investigated [6], [25]–[28]. In [27], based on impulsive control, the synchronization problem for coupled switched neural networks (SNNs) with mixed delays is studied. In [28], the local and global exponential synchronization is investigated for a class of complex dynamical networks with switching topology and time-varying coupling delays. In addition, in real life, many electronic and biological networks are often subject to instantaneous disturbances and experience abrupt changes at certain instants, which may
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ZHANG et al.: EXPONENTIAL SYNCHRONIZATION OF COUPLED SNNs
be caused by frequency change or other sudden noise, i.e., they exhibit impulsive effects [8], [29]. Since impulsive and switching effects can seriously affect the dynamical behaviors of the networks, it is necessary to investigate both impulsive and switching effects on the synchronization of dynamical networks. Recently, impulsive switched systems have found important applications in various fields, such as biological neural networks, bursting rhythm models in pathology, optimal control models in economics, frequency-modulated signal processing systems, and flying-object motion [24], [30]–[32]. As such, impulsive switched systems have been drawing increasing attention (see [31], [33]–[37] and references therein). Although there are many results concerning the stability of impulsive switched systems, the synchronization problem of switched dynamical networks with impulsive effects has received relatively little attention [33], [38]. Moreover, in most existing results regarding impulsive switched systems, it is implicitly assumed that impulsive effects occur at the switching points [31], [33]–[37]. Such an assumption is conservative, as networks may be subjected to instantaneous disturbances at any time instead of only at the switching points. Therefore, it is of great importance to consider a more general case where impulsive effects can exist not only at the instants coinciding with the system switching but also at the instants when there is no system switching. However, to our best knowledge, the synchronization problem of impulsive coupled SNNs with such a relaxed assumption has received little research attention primarily due to their mathematical complexity. Therefore, the main objective of this paper is to shorten such a gap by launching a study on the synchronization problem of coupled SNNs with mode-dependent impulsive effects where the impulsive effects exist not only at the instants coinciding with the system switching but also at instants when there is no system switching. In this paper, we are concerned with the exponential synchronization problem for a class of coupled delayed SNNs with mode-dependent impulsive effects. Based on the average dwell time approach, comparison principle, and the Halanay differential inequality approach, several criteria are presented for the synchronization of coupled delayed SNNs with modedependent impulsive effects. Simulations are given to illustrate the effectiveness of the proposed method. The main contributions of this paper can be highlighted as follows: 1) a new coupled SNN model with impulsive effects is established to consider both mode-dependent impulsive effects and system switching, where the impulsive effects can exist not only at the instants coinciding with the system switching but also at the instants when there is no system switching; 2) by using the average dwell time approach, comparison principle, and the Halanay differential inequality approach, sufficient criteria are derived to ensure that coupled delayed SNNs with modedependent impulsive effects are exponentially synchronized; 3) mode-dependent impulsive effects, time delays, and switchings are considered for modeling the coupled neural networks simultaneously, which renders more practical significance of our current research. Notati ons: Throughout this paper, N and Rn denote, respectively, the set of nonnegative integers and the
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n-dimensional Euclidean space. N+ denotes the set of positive integers. For x ∈ Rn , x T denotes its transpose. The vector √ norm is defined as x = x T x. In denotes n-dimensional identity matrix. For matrix A ∈ Rn×n , A = λmax (A T A), where λmax (·) represents the largest eigenvalue. Moreover, for real symmetric matrices X and Y , the notation X ≤ Y (respectively, X < Y ) means that the matrix X − Y is negative semidefinite (respectively, negative definite). Let PC(m) denote the class of piecewise right continuous function ψ : [t0 − τ, t] → Rm with the norm defined by ψ(t)τ = sup−τ ≤t ≤0 ψ(t + s).
II. M ODEL F ORMULATION AND S OME P RELIMINARIES In this section, some preliminaries are given, including model formulation, lemmas, and definitions. Consider the following coupled SNNs consisting of N linearly coupled identical nodes: x˙i (t) = Cσ (t ) x i (t) + Bσ (t ) f (σ (t), x i (t)) + Dσ (t ) f (σ (t), x i (t − τ (t))) +α
N
ai j,σ (t ) σ (t ) x j (t)
(1)
j =1
where i = 1, 2, . . . , N, x i (t) = [x i1 (t), x i2 (t), . . . , x in (t)]T ∈ Rn is the state vector of the i th node at time t, σ : [0, +∞] → S = {1, 2, . . . , N }, which is represented by σ (t) according to σ (t) = r ∈ S for t ∈ [tk , tk+1 ), is a piecewise constant function, continuous from the right, specifying the index of the active subsystem. t0 is the initial time. For the sake of generality, in this paper we assume that there is no switching or impulsive effects in the initial state. Cr < 0 is a diagonal matrix, and Br and Dr ∈ Rn×n denote the connection weight matrices of the neurons. f : S × Rn → Rn is a continuous map function. r = diag{γ1,r , γ2,r , . . . , γn,r } > 0 is the inner coupling matrix between two connected nodes for all 1 ≤ i, j ≤ N. α > 0 is the coupling strength of the coupled SNNs (1). τ (t) is the time-varying delay satisfying 0 ≤ τ (t) ≤ τ . The configuration coupling matrices (Ar ) N×N are defined as follows: if there is a connection from node j to node i (i = j ), then ai j,r > 0, otherwise, ai j,r = 0, and the diagonal elements are defined as aii,r = −
N
ai j,r , i = 1, 2, . . . , N, r ∈ S.
j =1, j =i
The initial values of (1) are given by x i (t) = ϕi (t), t0 − τ ≤ t ≤ t0 , i = 1, 2, . . . , N where ϕi (t) ∈ C([−τ, 0], Rn ) is the set of continuous functions from [−τ, 0] to Rn . By introducing the mode-dependent impulsive effects into the coupled SNNs (1), one can obtain the following coupled SNNs with mode-dependent impulsive
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 8, AUGUST 2013
⎧ x˙i (t) = Cr x i (t) + Br f (r, x i (t)) ⎪ ⎪ ⎪ ⎪ +Dr f (r, x i (t − τ (t))) ⎪ ⎪ ⎪ ⎨ N +α ai j,r r x j (t), t = tk, k , ⎪ j =1 ⎪ ⎪ − ⎪ ⎪ x i (tk, k ) = μr x i (tk,
), t = tk, k , ⎪ ⎪ ⎩ x (t) = ϕ (t), t − τ ≤k t ≤ t i i 0 0
(2)
+ − ) − x i (tk,
), {tk, k , k ∈ where k ∈ N x i (tk, k ) = x i (tk,
k k N+ } are the impulsive instances satisfying t0 < t0,1 < t0,2 < . . . < t1 ≤ t1,1 < t1,2 < . . . < t2 ≤ . . . tk ≤ tk,1 < tk,2 < . . . < tk+1 . μr represents the corresponding mode-dependent impulsive strength. In this paper, we always assume that x i (t) is right continuous at t = tk, k . Remark 1: The assumption in model (2) does not lose any generality. The reasons are as follows. If there exists some tk , such that there are no impulsive effects in [tk , tk+1 ), then model (2) not only includes the case where the impulsive effects occur between two switching instants but also the case in which the switching effects occur between two impulsive instants. For simplicity, we assume that the impulsive strengths − ). in the same mode are equal, i.e., x i (tk, k ) = μr x i (tk,
k In model (2), the impulsive effects can occur not only at the instants coinciding with the system switching but also at the instants when there is no system switching. However, in most existing results regarding impulsive switched systems, it is assumed that the impulsive effects occur at instants coinciding with mode switching [31], [33]–[37]. Hence, model (2) is more general than most existing results. For the activation function f (r, ·), we have the following assumption: Assumption 1: The activation function f (r, ·) satisfies the following Lipschitz condition:
f (r, x) − f (r, y) ≤ L r x − y
(3)
∀x, y ∈ Rn , where L r are known diagonal positive matrices. The following definitions and lemmas are presented for the derivation of the main results. Definition 1 ([23]): For a switching signal σ (t) and any T > t, let N(T, t) be the switching numbers of σ (t) over the interval [t, T ). If T −t ∀T ≥ t ≥ 0 (4) N(T, t) ≤ N0 + Ta for N0 > 0, Ta > 0, then N0 and Ta are called the chatter bound and the average dwell time, respectively. Definition 2: The coupled SNN (2) with mode-dependent impulsive effects is said to be exponentially synchronized if there exist η > 0, t0 ≥ 0, and M0 > 0 such that for any initial values ϕi (s) x i (t) − x j (t) ≤ M0 e−η(t −t0 )
(5)
holds for all t ≥ t0 , and for any i, j = 1, 2, . . . , N. Then M0 and η are called the decay coefficient and decay rate, respectively. ˆ K ) is the set of matrices with Definition 3 ([39]): T ( R, entries in Rˆ such that the sum of the entries in each row ˆ is equal to K for some K ∈ R.
Lemma 1: Let ⊗ denote the Kronecker product, and A, B, C, and D are matrices with appropriate dimensions. Then the following properties are satisfied: 1) (a A) ⊗ B = A ⊗ (a B), where a is a constant; 2) (A + B) ⊗ C = A ⊗ C + B ⊗ C; 3) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (B D). Lemma 2 ([39]): Let A be an N × N matrix in the set ˆ K ). Then the (N − 1) × (N − 1) matrix H defined by T ( R, H = M A J satisfies M A = H M, where ⎤ ⎡ 1 −1 ⎥ ⎢ 1 −1 ⎥ ⎢ , M =⎢ ⎥ . .. ⎦ ⎣ ⎡
1 ⎢0 ⎢ ⎢ ⎢ J =⎢ ⎢ ⎢ ⎣0 0
1 1 1 1 .. . ... 0 ... 0 0
1 −1 (N−1)×N ⎤ ... 1 . . . 1⎥ ⎥ ⎥ 1⎥ ⎥ 1 1⎥ ⎥ 0 1⎦ . . . 0 N×(N−1)
ˆ in which 1 is the multiplicative identity of R. Lemma 3 ([29]): Suppose p > q ≥ 0, and u(t) satisfies the scalar impulsive differential inequality ⎧ + ⎨ D u(t) < − pu(t) + qsup−τ ≤s≤0 u(t + s), t = tk , t ≥ t0 , (6) u(t + ) ≤ ρk u(tk− ), k ∈ N+ , ⎩ k u(t) = φ(t), t ∈ [t0 − τ, t0 ] where u(t) is continuous at t = tk , t > t0 , u(tk ) = u(tk+ ), and u(tk− ) exists, φ ∈ PC(1). Then ρk )e−λ(t −t0) , t ≥ t0 (7) u(t) ≤ φ(t0 )τ ( t0 0 is the unique solution of the equation λ − p + qeλτ = 0. Remark 2: Lemma 3 extends the famous Halanay differential inequality [40] to impulsive delay differential systems, and it will be used to derive main results here. Lemma 4 ([41]): Let 0 ≤ τi (t) ≤ τ . F(t, u, u¯ 1 , u¯ 2 , m+1 + . . . , u¯ m ) : R × R × · · · × R → R be nondecreasing in u¯ i for each fixed (t, u, u¯ 1 , . . . , u¯ i−1 , u¯ i+1 , . . . , u¯ m ), i = 1, 2, . . . , m, and Ik (u) : R → R be nondecreasing in u. Suppose that + D u(t) ≤ F(t, u(t), u(t − τ1 (t)), . . . , u(t − τm (t))) k ∈ N+ u(tk+ ) ≤ Ik (u(tk− )), and
D + v(t) > F(t, v(t), v(t − τ1 (t)), . . . , v(t − τm (t))) v(tk+ ) ≥ Ik (v(tk− )), k ∈ N+
where the upper-right Dini derivative D + y(t) is defined as D + y(t) = limh→0+ (y(t + h) − y(t))/ h. Then u(t) ≤ v(t), for −τ ≤ t ≤ 0 implies that u(t) ≤ v(t), for t ≥ 0. Lemma 5: Let x, y ∈ Rn , and Q a diagonal positive definite matrix with appropriate dimensions; then the following inequality holds: x T y + y T x ≤ x T Qx + y T Q −1 y.
(8)
ZHANG et al.: EXPONENTIAL SYNCHRONIZATION OF COUPLED SNNs
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Proof: Equation (8) can be obtained directly from the inequality [Qx − y]T Q −1 [Qx − y] ≥ 0. Lemma 6 (Schur Complement [42]): The linear matrix inequality Q(x) S(x) >0 (9) S T (x) R(x) where Q(x) = Q T (x), R(x) = R T (x), is equivalent to either of the following conditions: 1) Q(x) > 0, R(x) − S T (x)Q(x)−1 S(x) > 0 2) R(x) > 0, Q(x) − S T (x)R(x)−1 S(x) > 0. For a clear presentation, here we let x(t) = [x 1T (t), x 2T (t), . . . , x NT (t)]T , Ar = Ar ⊗ r , f(r, x(t)) = [ f T (r, x 1 (t)), . . . , f T (r, x N (t)]T , ϕ(t) = [ϕ1T (t), . . . , ϕ NT (t)]T , CrN = (I N ⊗ Cr ), BrN = (I N ⊗ Br ), DrN = (I N ⊗ Dr ). Then coupled neural networks (2) can be rewritten as ⎧ x(t) ˙ = CrN x(t) + BrN f(r, x(t)) + αAr ⎪ ⎪ ⎨ + DrN f(r, x(t − τ (t))), t = tk, k , − ), t = tk, k , x(tk, k ) = μr x(tk,
⎪ ⎪ k ⎩ x(s) = ϕ(s), −τ ≤ s ≤ 0.
(10)
III. M AIN R ESULTS In this section, the exponential synchronization of coupled SNNs with mode-dependent impulsive effects is investigated. As was shown in [43], there are two types of impulses: one is the desynchronizing impulses (the impulsive effects suppressing synchronization); and the other is the synchronizing impulses (the impulsive effects enhancing synchronization) [31], [44], [45]. In the following, Theorem 1 aims to investigate the exponential synchronization problem of coupled delayed SNNs with desynchronizing impulses. In addition, in Theorem 2 we investigate the synchronization problem of coupled delayed SNNs with synchronizing impulses. The following assumption is necessary before presenting the main results. Assumption 2: i) When the mode-dependent impulses suppress synchronization, we assume inf{tk, k +1 − tk, k } = k ; ii) when the mode-dependent impulses can contribute to the achievement of synchronization, we assume sup{tk, k +1 − tk, k } = k . Theorem 1: Suppose that Assumption 1 and condition i) of Assumption 2 hold and there exist positive definite matrices Pr ∈ Rn(N−1)×n(N−1) , diagonal positive definite matrices Q r ∈ Rn(N−1)×n(N−1) and Rr ∈ Rn(N−1)×n(N−1) , and positive constants ζr , ηr (r ∈ S) such that (H1 ) for r ∈ S, the following inequalities hold: ⎤ ⎡ r Pr BrN−1 Pr DrN−1 0 ⎢(B N−1 )T Pr −Q r 0 0⎥ r ⎥η≥0
(12)
where ζ = minr∈S {ζr }, η = maxr∈S {ηr }; (H3 ) there exist positive constants ρ ≥ 1, λ > 0, satisfying Pr ≤ ρ Pr¯ ∀r, r¯ ∈ S λ−
ln M0 Ta
λ−ζ +
ln β >0 λτ ηe = 0
−
(13) (14) (15)
where M0 = maxr∈S {βr ρ, βr exp((λ − lnβ )τ )}, βr = (μr + 1)2 > 1, β = maxr {βr }, = mink {k }. Then the SNNs (10) with mode-dependent desynchronizing impulses are exponentially synchronized with decay rate ϑ/2, where 0 − lnβ . Proof: See Appendix. ϑ = λ − lnTM a If there are no switching parameters in the model (10), then the following corollary can be obtained directly: Corollary 1: Suppose that Assumption 1 and condition i) of Assumption 2 hold. Let ζ¯ = −λmax (2C N−1 + 2α A˜ + L T L + (B N−1 )T B N−1 + (D N−1 )T D N−1 ) and η¯ = λmax (L T L). Then, the linearly coupled neural networks (10) without switching parameters will be globally exponentially synchronized if ζ¯ > η, ¯ λ−
2 ln |1 + μ| >0
where λ > 0 is the unique solution of the equation λ − ζ¯ + ηe ¯ λτ = 0. Proof: Choose V (t) = x T (t)MT Mx(t). Let Q r = Rr = I in Theorem 1. Then the results can be obtained directly. Remark 3: When there are no switching parameters in the model (10), Theorem 1 can be reduced to the synchronization criterion of coupled DNNs with impulsive disturbances. Recently, in [29], the synchronization problem of coupled neural networks with impulsive disturbances was investigated by using the average impulsive approach. It should be pointed out that the configuration coupling matrix in [29] is assumed to be irreducible. However, Theorem 1 can be applied to the case where the configuration matrix is reducible. Hence, Theorem 1 has a wider application than the results in [29]. If there are no impulsive effects in the model (10), we can get the following synchronization criterion for coupled delayed SNNs. Corollary 2: Suppose that Assumption 1 and condition i) of Assumption 2 hold and there exist positive definite matrices Pr ∈ Rn(N−1)×n(N−1) , diagonal positive definite matrices Q r ∈ Rn(N−1)×n(N−1) and Rr ∈ Rn(N−1)×n(N−1) , and positive constants ζr , ηr (r ∈ S) such that (H1 ) for r ∈ S, the following inequalities hold: η≥0 where ζ = minr∈S {ζr }, η = maxr∈S {ηr };
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(H3 ) there exist positive constants ρ > 1, λ > 0, satisfying Pr ≤ ρ Pr¯ ∀r, r¯ ∈ S, λ−
ln M0 Ta
1.2
> 0,
λ − ζ + ηeλτ = 0
1
where M0 = max{ρ, exp(λτ )}. Then the coupled SNNs (10) without impulses are exponentially synchronized with decay 0 rate (λ − lnTM )/2. a In Theorem 1, based on Lemma 3, the synchronization problem of coupled SNNs with mode-dependent desynchronizing impulses is investigated. Inequalities (11) and (12) imply that all the subsystems of the hybrid networks (10) are synchronized. Therefore, Theorem 1 is only suitable for the case where each subsystem of (10) is self-synchronized. Hence it is desirable to consider the case where the SNNs cannot be self-synchronized. Remark 4: In the following, we will investigate the case where the mode-dependent impulses can contribute to the achievement of synchronization. However, Lemma 3 can only be applied to the case where the impulsive-free coupled neural networks are self-synchronized. Hence, it is not suitable for us to use Lemma 3 to investigate the synchronization problem of coupled SNNs where the impulsive-free SNNs are not synchronized. Fortunately, in [41], the comparison principle has been proposed to deal with the impulsive control problem. Hence, we will use the comparison principle to investigate the synchronization problem of coupled SNNs with modedependent synchronizing impulses. Theorem 2: Suppose that Assumption 1 and condition ii) of Assumption 2 hold and there exist positive definite matrices Pr ∈ Rn(N−1)×n(N−1) , diagonal positive definite matrices Q r ∈ Rn(N−1)×n(N−1) and Rr ∈ Rn(N−1)×n(N−1) , and positive constants ζr , ηr (r ∈ S) such that (H1 ) for r ∈ S, the following inequalities hold: ˜ 0, satisfying Pr ≤ ρ Pr¯ ∀r, r¯ ∈ S ζ˜ + β˜ η˜ + lnβ < 0 ˜ β˜ ηe λ1 + ζ+ ˜ λ1 τ + λ1 −
ln M˜ 0 Ta
ln β
>0
1.4
=0
(17) (18) (19) (20)
where M˜ 0 = β˜ max{ρ, exp(λ1 τ )}, β˜ = maxr∈S {βr−1 }, βr = (μr + 1)2 < 1, ζ˜ = maxr∈S {ζ˜r } and η˜ = maxr∈S {η˜r }, = minr∈S {r }. Then the coupled SNNs (10) with modedependent synchronizing impulses are exponentially synchro˜0 ˜ . nized with decay rate ϑ/2, where ϑ˜ = λ1 − lnTM a Proof: See Appendix. Remark 5: Recently, the synchronization problem of the switched complex networks with impulsive control was investigated in [38], where the impulsive intervals were allowed to be different from the network switching intervals occasionally.
0.8
0.6
0.4
0.2
0
0
Fig. 1.
2
4
6
8
10
Impulsive sequence of Example 1.
However, the results derived in [38] were based on the assumption x i (t − τ )2 ≤ ρi x i (t)2 , i = 1, 2, . . . , N. Clearly, this assumption may bring inconvenience and conservativeness in a practical application. In Theorem (2), this restrictive condition is removed. Hence, Theorem 2 can be applied more easily than the results in [38]. If there are no switching parameters in Theorem 2, then we have the following corollary. Corollary 3: Suppose that Assumption 1 and condition ii) of Assumption 2 hold. Then the coupled neural networks in (10) with synchronizing impulses are exponentially synchronized if the following inequality holds: ζˆ + β −1 ηˆ +
ln β
0.8298. Let Ta = 1. The impulsive signal is simulated in Fig. 1. Figs. 2 and 3 depict the synchronization errors of x 11 − x i1 and x 12 − x i2 , respectively. From them, one can see the exponential synchronization errors x i (t) − x j (t), which indicate that synchronization can be achieved. Example 2: In this example, we will show that the impulsive effects exist only at the instants coinciding with the system switching. Here, we consider (22) with the same parameters
11
31
0.8
i1
(22)
x −x
1
x11−x
x˙ i (t) = Cr x i (t) + Dr f (r, x i (t − τ (t)))) N ai j,r r x j (t). + Br f (r, x i (t)) + α
Synchronization errors of x12 − xi2 , i = 2, 3.
Fig. 3.
0.6
0.4
0.2
0
−0.2
0
2
4
6
8
10
t
Fig. 4.
Synchronization errors of x11 − xi1 , i = 2, 3.
as in Example 1. Note that the impulsive effects exist only at the instants coinciding with the system switching, and thus in (14) is +∞. Let μ1 = μ2 = 0.1. We solve (14) and then Ta > 0.6571. Let Ta = 0.7, and the synchronization errors of x 11 − x i1 and x 12 − x i2 are plotted in Figs. 4 and 5, respectively. From these figures, we can see that, when the impulsive effects exist only at the instants coinciding with the system switching, the coupled delayed SNNs (22) can be synchronized with the mode-dependent desynchronizing impulses. Example 3: In this example, we will show that the modedependent impulses can contribute to the achievement of synchronization. Even the impulse-free coupled SNNs cannot reach synchronization by choosing the appropriate k and μr ; then the coupled neural networks can achieve synchronization with the impulsive effects. Consider the delayed coupled SNN
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can be synchronized with the mode-dependent synchronizing impulses. The simulations presented here verify the theoretical results in this paper very well.
1.2 x12−x22 x −x 12
1
32
V. C ONCLUSION
x12−x
i2
0.8
0.6
0.4
0.2
0
−0.2
0
2
4
6
8
10
t
Fig. 5.
Synchronization errors of x12 − xi2 , i = 2, 3.
In this paper, we investigated the problem of exponential synchronization for a class of coupled SNNs with time delay and mode-dependent impulsive effects. The new concept of mode-dependent impulses was proposed, where the impulsive effects can occur not only at the instants coinciding with the system switching but also at the instants when there is no system switching. By using the Lyapunov method, comparison principle, and the Halanay differential inequality, the synchronization of coupled SNNs with mode-dependent desynchronizing and synchronizing impulsive effects was studied. Further research topics include the extension of our results to more general switched complex dynamical networks with stochastic perturbations, coupling delay, discrete-time switched dynamical networks, and neuronal networks [48].
2
10
A PPENDIX A. Proof of Theorem 1
0
10
Proof: Denote M = M ⊗ In . Construct the switched Lyapunov function in the form V (t) = x T (t)MT Pr Mx(t), t ∈ [tk , tk+1 ), r ∈ S.
log10(err(t))
−2
10
Taking the derivative of V (t) along the trajectory of the coupled SNNs (10), it yields
−4
10
D + V (t) = 2x T (t)MT Pr M[CrN x(t) + BrN f(r, x(t)) + αAr x(t) + DrN f(r, x(t − τ (t)))]
−6
10
t ∈ [tk, k , tk, k +1 ).
−8
10
0
2
4
6
8
10
t
Fig. 6.
(23)
3 2 Synchronization error of (x1i − x j i )2 . i=1
j =1
model (22) with the following parameters: 0.1 0 0.2 0 , C2 = − , C1 = − 0 0.1 0 0.2 ⎡ ⎤ −0.6 0.3 0.3 A1 = A2 = ⎣ 0.3 −0.6 0.3 ⎦ 0.3 0.3 −0.6 and the other parameter are same as in Theorem 1. By simple calculation, it can be seen that Theorem 1 does not have any feasible solutions. However, solving (16) and (17) in Theorem 2, we can get ζ1 = ζ2 = 1.0648, η1 = η2 = 1. Let μ1 = μ2 = −0.3, 1 = 2 = . . . = . Solving (18), we can get < 0.2297. Let = 0.15. We solve (19), and then λ1 = 0.4595. We solve (20), and then Ta = 2.0092.Let Ta = 2.01. Then the 3 2 synchronization error err(t) = 2i=1 j =1 (x 1i − x j i ) with impulses is plotted in Fig. 6. From the error we can see that, by choosing appropriate k and μr , the coupled delayed SNNs
(24)
By the definition of M, the following equalities are clearly true: MBrN = BrN−1 M, MCrN = CrN−1 M, MDrN = DrN−1 M, MAr = A˜ r M
(25)
where A˜ r = A˜ r ⊗ r , A˜ r = M Ar J . Substituting (25) into (24), we have D + V (t) = 2x T (t)MT Pr CrN−1 Mx(t) + 2x T (t)MT Pr BrN−1 Mf(r, x(t)) + 2x T (t)MT Pr DrN−1 Mf(r, x(t − τ (t))) + 2αx T (t)MT Pr A˜ r Mx(t) t ∈ [tk, k , tk, k +1 ).
(26)
In view of Lemma 5, we have 2x T (t)MT Pr BrN−1 Mf(r, x(t))
≤ x T (t)MT Pr BrN−1 Q r−1 (BrN−1 )T Pr Mx(t) + f T (r, x(t))MT Q r Mf(r, x(t)) ≤ x T (t)MT Pr BrN−1 Q r−1 (BrN−1 )T Pr Mx(t) + x T (t)MT L r Q r L r Mx(t)
(27)
ZHANG et al.: EXPONENTIAL SYNCHRONIZATION OF COUPLED SNNs
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By (31), (32), and (36), for t ∈ [tm+1 , tm+2 ),
and
V (t) ≤ β M0 M0m e(−λ+
2x T (t)MT Pr DrN−1 Mf(r, x(t − τ (t))) ≤ x T (t)MT Pr DrN−1 Rr−1 (DrN−1 )T Pr Mx(t)
×e
+x T (t − τ (t))MT L r Rr L r Mx(t − τ (t)).
(28)
It follows from (11), (24)–(28), and Lemma 6 that for t ∈ [tk, k , tk, k +1 ),
=
− τ (t))
When t = tk, k , it follows from the second equation of (10) that (30)
where βr = (μr + 1)2 . Since t ∈ [tk , tk+1 ), σ (t) = r , from Lemma 3, for t ∈ [tk , tk+1 ), k ≥ 1, we have, when tk = tk,1 t−tk k
= V (tk )τ e
(−λ+ lnβr )(t −tk ) k
where ϑ = λ −
ln β )(t −tk )
V (t) ≤ V (tk )τ βr
= βr V (tk )τ e
+1
(31)
k
ln β )(t −tk )
.
(32)
Then, we show by induction, that V (t) ≤ βV (t0 )τ M0k e(−λ+
ln β )(t −t0 )
where M0 = maxr∈S {βr ρ, βr exp((λ − [t0 , t1 ), it follows from Lemma 3 that: V (t) ≤ V (t0 )τ β
t−t0 0
≤ βV (t0 )τ e
e(−λ+
ln β )(t −t0 )
ln M0 Ta
−
ln β
(38)
> 0. Then, we have (39)
where ω = minr∈S {λmin (Pr )}, ν = maxr∈S {λmax (Pr )}. Hence ν x T (t)MT Mx(t) ≤ β Mϕ(s)τ M0N0 e−ϑ(t −t0) . (40) ω From Definition 2 and the definition of M, coupled SNNs (10) with mode-dependent desynchronizing impulses are exponentially synchronized. This completes the proof.
Proof: Consider the same Lyapunov function as Theorem 1. Similar to the proof of Theorem 1, for t ∈ [tk, k , tk, k +1 ), we have D + V (t) ≤ ζ˜ V (t) + ηV ˜ (t − τ (t))
(41)
+ − V (tk,
) = βr V (tk,
), t = tk, k . k k
(42)
and e−λ(t −tk )
(−λ+ lnβr )(t −tk )
≤ βr V (tk )τ e(−λ+
e
(t−t0 ) ln M0 Ta
ωx T (t)MT Mx(t) ≤ νβMϕ(s)τ M0N0 e−ϑ(t −t0)
and when tk < tk,1 t−tk k
(37)
B. Proof of Theorem 2
e−λ(t −tk )
≤ V (tk )τ e(−λ+
.
= β M0N0 V (t0 )τ e−ϑ(t −t0 ) , t ≥ t0
(29)
+ − V (tk,
) = βr V (tk,
) k k
ln β )(t −t0 )
N(t,t0 ) (−λ+ lnβ )(t −t0 )
V (t) ≤ βV (t0 )τ M0
+ ηr V (t − τ (t)) − ζr V (t)
V (t) ≤ V (tk )τ βr
βV (t0 )τ M0m+1 e(−λ+
≤ β M0N0 V (t0 )τ e
≤ −ζ V (t) + ηV (t − τ (t)).
V (t0 )τ
(−λ+ lnβ )(t −tm+1 )
Therefore, by the induction principle, we see that (33) holds for all k ∈ N. Then from Definition 1, we have
D + V (t) ≤ x T (t)MT [Pr CrN−1 + (CrN−1 )T Pr + 2α Pr A˜ r + L r Q r L r + Pr BrN−1 Q r−1 (BrN−1 )T Pr + Pr DrN−1 Rr−1 (DrN−1 )T Pr + ζr Pr ]Mx(t) + x T (t − τ (t))MT (L r Rr L r − ηr Pr )Mx(t
ln β )(tm+1 −t0 )
, t ∈ [tk , tk+1 ) ln β )τ )}.
(33)
When t ∈
+1 −λ(t −t0 ) e
(−λ+ lnβ )(t −t0 )
.
˜ (tk )τ e−λ1 (t −tk ) . V (t) ≤ βV
(43)
For any ε > 0, let v(t) be a unique solution of the impulsive delay system ⎧ ˙ = ζ˜ v(t) + ηv(t ˜ − τ (t)) + ε, t = tk, k , ⎨ v(t) + − (44) ) = β v(t ), v(tk,
r k, k t = tk, k , k ⎩ v(t) = V (t), tk − τ ≤ t ≤ tk . Then it follows from Lemma 4 that:
(34)
Now, assume that (33) holds for k = 0, 1, . . . , m, where m ≥ 1. Then we show that (33) holds for k = m +1. Note that Pr ≤ ρ Pr¯ ∀r, r¯ ∈ S; then we have − ), tm+1 = t(m+1),1 , βρV (tm+1 V (tm+1 ) ≤ (35) − ρV (tm+1 ), tm+1 < t(m+1),1 . Then we can get from (31), (32), and (35) that ln β )τ } ln β ×βV (t0 )τ M0m e(−λ+ )(tm+1−t0 ) .
In the following, we show that for t ∈ [tk , tk+1 ):
V (tm+1 )τ ≤ max{βρ, β exp(λ −
(36)
V (t) ≤ v(t), t ∈ [tk , tk+1 ).
(45)
By the formula for the variation of parameters, one obtain from (44) that for t ∈ [tk , tk+1 ) t W (t, s)[ηv(s ˜ − τ (s)) + ε]ds v(t) = W (t, tk )v(tk ) + tk
(46) where W (t, s), t, s ≥ tk is the Cauchy matrix of the linear system y˙ (t) = ζ˜ v(t), t = tk, k , (47) + − ) = βr v(tk,
), t = tk, k . v(tk,
k k
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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 8, AUGUST 2013
According to the representation of the Cauchy matrix, we can get the following estimation: ˜ βr . (48) W (t, s) ≤ eζ (t −s) s 0, there exists a unique λ1 > 0, h(+∞) = +∞ and h(ς such that ln β + β˜ ηe ˜ λ1 τ = 0. (52) λ1 + ζ˜ +
ν ˜ Mϕ(s)τ β˜ M˜ 0N0 eϑ(t −t0 ) . (61) ω Therefore, the coupled SNNs (10) with mode-dependent synchronizing impulses are exponentially synchronized. This completes the proof. x T (t)MT Mx(t) ≤
ACKNOWLEDGMENT
ln β ).
It is obvious from (18) that
The authors would like to thank the Associate Editor and the anonymous reviewers for their helpful comments.
v(t) ≤ ξ < ξ e−λ1 (t −tk ) +
ε , tk − τ ≤ t ≤ tk . (53) −1 ˜ β δ − η˜
R EFERENCES
Denote δ = −(ζ˜ + β˜ −1 δ − η˜ > 0; hence
Then we claim that
ε
v(t) < ξ e−λ1 (t −tk ) +
, tk ≤ t < tk+1 . β˜ −1 δ − η˜ If it is not true, there exists a t ∗ ∈ [tk , tk+1 ) such that ε v(t ∗ ) ≥ ξ e−λ1 (t −tk ) + β˜ −1 δ − η˜ and v(t) < ξ e−λ1 (t −tk ) +
ε β˜ −1 δ − η˜
From (50), we have ∗
v(t ) ≤ ξ e
−δ(t ∗ −tk )
+
t∗
, t ∈ [tk − τ, t ∗ ).
(54)
(55)
(56)
˜ −δ(t ∗ −s) [ηv(s ˜ − τ (s)) βe
tk
+ ε]ds t∗ ˜ δ(s−tk )
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 8, AUGUST 2013
Exponential Synchronization of Coupled Switched Neural Networks With Mode-Dependent Impulsive Effects Wenbing Zhang, Yang Tang, Member, IEEE, Qingying Miao, and Wei Du
Abstract— This paper investigates the synchronization problem of coupled switched neural networks (SNNs) with modedependent impulsive effects and time delays. The main feature of mode-dependent impulsive effects is that impulsive effects can exist not only at the instants coinciding with mode switching but also at the instants when there is no system switching. The impulses considered here include those that suppress synchronization or enhance synchronization. Based on switching analysis techniques and the comparison principle, the exponential synchronization criteria are derived for coupled delayed SNNs with mode-dependent impulsive effects. Finally, simulations are provided to illustrate the effectiveness of the results. Index Terms— Coupled switched neural networks (SNNs), exponential synchronization, mode-dependent impulsive effects.
I. I NTRODUCTION
S
INCE small-world and scale-free complex networks were proposed in [1] and [2], complex dynamical networks that consist of interacting dynamical entities with an interplay between dynamical states and interaction patterns have received increasing attention from various fields of science and engineering [3]–[5]. A large number of systems in nature can be modeled by complex dynamical networks, e.g., power grids, communication networks, the Internet, the World Wide Web, metabolic systems and food webs, etc. Many interesting behaviors can be observed from complex dynamical networks, e.g., synchronization, consensus, self-organization, and spatiotemporal chaos spiral waves (see [6] and references therein). Synchronization as an important collective behavior of complex dynamical networks has gained much attention in the last two decades [7]–[16].
Manuscript received November 12, 2012; revised January 27, 2013; accepted April 3, 2013. Date of publication May 13, 2013; date of current version June 28, 2013. This work was supported in part by the National Natural Science Foundation of China under Grant 61203235, the China PostDoctoral Science Foundation under Grant 2011M501040, the Alexander von Humboldt Foundation of Germany, the Key Creative Project of the Shanghai Education Community under Grant 13ZZ050, and the Key Foundation Project of Shanghai under Grant 12JC1400400. W. Zhang is with the Department of Mathematics, Yangzhou University, Jiangsu 225002, China (e-mail: [email protected]). Y. Tang is with the Institute of Physics, Humboldt University of Berlin, Berlin 12489, Germany, and also with the Potsdam Institute for Climate Impact Research, Potsdam 14415, Germany (e-mail: [email protected]). Q. Miao is with the School of Continuing Education, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]). W. Du is with the Institute of Textile and Clothing, The Hong Kong Polytechnic University, Hong Kong, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2013.2257842
As a special class of complex networks, linearly coupled neural networks have been a hot topic because they have wide applications in a variety of areas, such as signal processing, pattern recognition, static image processing, associative memory, and combinatorial optimization [17]. On the other hand, time delay is ubiquitous in coupled neural networks because of the finite information transmission and processing speeds among the units. It is well known that time delay may be an important source of oscillation, divergence, and instability in systems. Thus, delayed neural networks have been widely studied [18]–[21]. In [18], the exponential synchronization problem is studied for a class of coupled neural networks with discrete and distributed delays under time-varying sampling. In [19], the robust synchronization problem for an array of coupled stochastic discrete-time neural networks with timevarying delay is studied. In [20], the synchronization problem is studied for a class of Markovian coupled neural networks with nonidentical node delays and random coupling strengths. On the other hand, in real-world systems, because of link failures and the creation of new links that may happen at times, the jumps between different topologies are inevitable [6]. Hence, when modeling real-world dynamical networks, it is desirable to model such networks using model switchings. Recently, synchronization of complex networks with switching topology has become a hot topic [9], [14], [22]. By assuming the switching interval to be less than a given constant, the synchronization problem of switched complex networks is investigated in [22]. In [9], the stability and synchronization problem has been addressed for a class of discrete-time neural networks with mode-dependent mixed time delay and Markovian jump switching. It is also shown in [23] and [24] that an arbitrary switching may destroy the stability of switched systems and that one useful method to investigate the stability of switched systems is to employ the so-called average dwell time approach. By this approach, the synchronization of switched networks under arbitrary switching was investigated [6], [25]–[28]. In [27], based on impulsive control, the synchronization problem for coupled switched neural networks (SNNs) with mixed delays is studied. In [28], the local and global exponential synchronization is investigated for a class of complex dynamical networks with switching topology and time-varying coupling delays. In addition, in real life, many electronic and biological networks are often subject to instantaneous disturbances and experience abrupt changes at certain instants, which may
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ZHANG et al.: EXPONENTIAL SYNCHRONIZATION OF COUPLED SNNs
be caused by frequency change or other sudden noise, i.e., they exhibit impulsive effects [8], [29]. Since impulsive and switching effects can seriously affect the dynamical behaviors of the networks, it is necessary to investigate both impulsive and switching effects on the synchronization of dynamical networks. Recently, impulsive switched systems have found important applications in various fields, such as biological neural networks, bursting rhythm models in pathology, optimal control models in economics, frequency-modulated signal processing systems, and flying-object motion [24], [30]–[32]. As such, impulsive switched systems have been drawing increasing attention (see [31], [33]–[37] and references therein). Although there are many results concerning the stability of impulsive switched systems, the synchronization problem of switched dynamical networks with impulsive effects has received relatively little attention [33], [38]. Moreover, in most existing results regarding impulsive switched systems, it is implicitly assumed that impulsive effects occur at the switching points [31], [33]–[37]. Such an assumption is conservative, as networks may be subjected to instantaneous disturbances at any time instead of only at the switching points. Therefore, it is of great importance to consider a more general case where impulsive effects can exist not only at the instants coinciding with the system switching but also at the instants when there is no system switching. However, to our best knowledge, the synchronization problem of impulsive coupled SNNs with such a relaxed assumption has received little research attention primarily due to their mathematical complexity. Therefore, the main objective of this paper is to shorten such a gap by launching a study on the synchronization problem of coupled SNNs with mode-dependent impulsive effects where the impulsive effects exist not only at the instants coinciding with the system switching but also at instants when there is no system switching. In this paper, we are concerned with the exponential synchronization problem for a class of coupled delayed SNNs with mode-dependent impulsive effects. Based on the average dwell time approach, comparison principle, and the Halanay differential inequality approach, several criteria are presented for the synchronization of coupled delayed SNNs with modedependent impulsive effects. Simulations are given to illustrate the effectiveness of the proposed method. The main contributions of this paper can be highlighted as follows: 1) a new coupled SNN model with impulsive effects is established to consider both mode-dependent impulsive effects and system switching, where the impulsive effects can exist not only at the instants coinciding with the system switching but also at the instants when there is no system switching; 2) by using the average dwell time approach, comparison principle, and the Halanay differential inequality approach, sufficient criteria are derived to ensure that coupled delayed SNNs with modedependent impulsive effects are exponentially synchronized; 3) mode-dependent impulsive effects, time delays, and switchings are considered for modeling the coupled neural networks simultaneously, which renders more practical significance of our current research. Notati ons: Throughout this paper, N and Rn denote, respectively, the set of nonnegative integers and the
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n-dimensional Euclidean space. N+ denotes the set of positive integers. For x ∈ Rn , x T denotes its transpose. The vector √ norm is defined as x = x T x. In denotes n-dimensional identity matrix. For matrix A ∈ Rn×n , A = λmax (A T A), where λmax (·) represents the largest eigenvalue. Moreover, for real symmetric matrices X and Y , the notation X ≤ Y (respectively, X < Y ) means that the matrix X − Y is negative semidefinite (respectively, negative definite). Let PC(m) denote the class of piecewise right continuous function ψ : [t0 − τ, t] → Rm with the norm defined by ψ(t)τ = sup−τ ≤t ≤0 ψ(t + s).
II. M ODEL F ORMULATION AND S OME P RELIMINARIES In this section, some preliminaries are given, including model formulation, lemmas, and definitions. Consider the following coupled SNNs consisting of N linearly coupled identical nodes: x˙i (t) = Cσ (t ) x i (t) + Bσ (t ) f (σ (t), x i (t)) + Dσ (t ) f (σ (t), x i (t − τ (t))) +α
N
ai j,σ (t ) σ (t ) x j (t)
(1)
j =1
where i = 1, 2, . . . , N, x i (t) = [x i1 (t), x i2 (t), . . . , x in (t)]T ∈ Rn is the state vector of the i th node at time t, σ : [0, +∞] → S = {1, 2, . . . , N }, which is represented by σ (t) according to σ (t) = r ∈ S for t ∈ [tk , tk+1 ), is a piecewise constant function, continuous from the right, specifying the index of the active subsystem. t0 is the initial time. For the sake of generality, in this paper we assume that there is no switching or impulsive effects in the initial state. Cr < 0 is a diagonal matrix, and Br and Dr ∈ Rn×n denote the connection weight matrices of the neurons. f : S × Rn → Rn is a continuous map function. r = diag{γ1,r , γ2,r , . . . , γn,r } > 0 is the inner coupling matrix between two connected nodes for all 1 ≤ i, j ≤ N. α > 0 is the coupling strength of the coupled SNNs (1). τ (t) is the time-varying delay satisfying 0 ≤ τ (t) ≤ τ . The configuration coupling matrices (Ar ) N×N are defined as follows: if there is a connection from node j to node i (i = j ), then ai j,r > 0, otherwise, ai j,r = 0, and the diagonal elements are defined as aii,r = −
N
ai j,r , i = 1, 2, . . . , N, r ∈ S.
j =1, j =i
The initial values of (1) are given by x i (t) = ϕi (t), t0 − τ ≤ t ≤ t0 , i = 1, 2, . . . , N where ϕi (t) ∈ C([−τ, 0], Rn ) is the set of continuous functions from [−τ, 0] to Rn . By introducing the mode-dependent impulsive effects into the coupled SNNs (1), one can obtain the following coupled SNNs with mode-dependent impulsive
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effects:
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 8, AUGUST 2013
⎧ x˙i (t) = Cr x i (t) + Br f (r, x i (t)) ⎪ ⎪ ⎪ ⎪ +Dr f (r, x i (t − τ (t))) ⎪ ⎪ ⎪ ⎨ N +α ai j,r r x j (t), t = tk, k , ⎪ j =1 ⎪ ⎪ − ⎪ ⎪ x i (tk, k ) = μr x i (tk,
), t = tk, k , ⎪ ⎪ ⎩ x (t) = ϕ (t), t − τ ≤k t ≤ t i i 0 0
(2)
+ − ) − x i (tk,
), {tk, k , k ∈ where k ∈ N x i (tk, k ) = x i (tk,
k k N+ } are the impulsive instances satisfying t0 < t0,1 < t0,2 < . . . < t1 ≤ t1,1 < t1,2 < . . . < t2 ≤ . . . tk ≤ tk,1 < tk,2 < . . . < tk+1 . μr represents the corresponding mode-dependent impulsive strength. In this paper, we always assume that x i (t) is right continuous at t = tk, k . Remark 1: The assumption in model (2) does not lose any generality. The reasons are as follows. If there exists some tk , such that there are no impulsive effects in [tk , tk+1 ), then model (2) not only includes the case where the impulsive effects occur between two switching instants but also the case in which the switching effects occur between two impulsive instants. For simplicity, we assume that the impulsive strengths − ). in the same mode are equal, i.e., x i (tk, k ) = μr x i (tk,
k In model (2), the impulsive effects can occur not only at the instants coinciding with the system switching but also at the instants when there is no system switching. However, in most existing results regarding impulsive switched systems, it is assumed that the impulsive effects occur at instants coinciding with mode switching [31], [33]–[37]. Hence, model (2) is more general than most existing results. For the activation function f (r, ·), we have the following assumption: Assumption 1: The activation function f (r, ·) satisfies the following Lipschitz condition:
f (r, x) − f (r, y) ≤ L r x − y
(3)
∀x, y ∈ Rn , where L r are known diagonal positive matrices. The following definitions and lemmas are presented for the derivation of the main results. Definition 1 ([23]): For a switching signal σ (t) and any T > t, let N(T, t) be the switching numbers of σ (t) over the interval [t, T ). If T −t ∀T ≥ t ≥ 0 (4) N(T, t) ≤ N0 + Ta for N0 > 0, Ta > 0, then N0 and Ta are called the chatter bound and the average dwell time, respectively. Definition 2: The coupled SNN (2) with mode-dependent impulsive effects is said to be exponentially synchronized if there exist η > 0, t0 ≥ 0, and M0 > 0 such that for any initial values ϕi (s) x i (t) − x j (t) ≤ M0 e−η(t −t0 )
(5)
holds for all t ≥ t0 , and for any i, j = 1, 2, . . . , N. Then M0 and η are called the decay coefficient and decay rate, respectively. ˆ K ) is the set of matrices with Definition 3 ([39]): T ( R, entries in Rˆ such that the sum of the entries in each row ˆ is equal to K for some K ∈ R.
Lemma 1: Let ⊗ denote the Kronecker product, and A, B, C, and D are matrices with appropriate dimensions. Then the following properties are satisfied: 1) (a A) ⊗ B = A ⊗ (a B), where a is a constant; 2) (A + B) ⊗ C = A ⊗ C + B ⊗ C; 3) (A ⊗ B)(C ⊗ D) = (AC) ⊗ (B D). Lemma 2 ([39]): Let A be an N × N matrix in the set ˆ K ). Then the (N − 1) × (N − 1) matrix H defined by T ( R, H = M A J satisfies M A = H M, where ⎤ ⎡ 1 −1 ⎥ ⎢ 1 −1 ⎥ ⎢ , M =⎢ ⎥ . .. ⎦ ⎣ ⎡
1 ⎢0 ⎢ ⎢ ⎢ J =⎢ ⎢ ⎢ ⎣0 0
1 1 1 1 .. . ... 0 ... 0 0
1 −1 (N−1)×N ⎤ ... 1 . . . 1⎥ ⎥ ⎥ 1⎥ ⎥ 1 1⎥ ⎥ 0 1⎦ . . . 0 N×(N−1)
ˆ in which 1 is the multiplicative identity of R. Lemma 3 ([29]): Suppose p > q ≥ 0, and u(t) satisfies the scalar impulsive differential inequality ⎧ + ⎨ D u(t) < − pu(t) + qsup−τ ≤s≤0 u(t + s), t = tk , t ≥ t0 , (6) u(t + ) ≤ ρk u(tk− ), k ∈ N+ , ⎩ k u(t) = φ(t), t ∈ [t0 − τ, t0 ] where u(t) is continuous at t = tk , t > t0 , u(tk ) = u(tk+ ), and u(tk− ) exists, φ ∈ PC(1). Then ρk )e−λ(t −t0) , t ≥ t0 (7) u(t) ≤ φ(t0 )τ ( t0 0 is the unique solution of the equation λ − p + qeλτ = 0. Remark 2: Lemma 3 extends the famous Halanay differential inequality [40] to impulsive delay differential systems, and it will be used to derive main results here. Lemma 4 ([41]): Let 0 ≤ τi (t) ≤ τ . F(t, u, u¯ 1 , u¯ 2 , m+1 + . . . , u¯ m ) : R × R × · · · × R → R be nondecreasing in u¯ i for each fixed (t, u, u¯ 1 , . . . , u¯ i−1 , u¯ i+1 , . . . , u¯ m ), i = 1, 2, . . . , m, and Ik (u) : R → R be nondecreasing in u. Suppose that + D u(t) ≤ F(t, u(t), u(t − τ1 (t)), . . . , u(t − τm (t))) k ∈ N+ u(tk+ ) ≤ Ik (u(tk− )), and
D + v(t) > F(t, v(t), v(t − τ1 (t)), . . . , v(t − τm (t))) v(tk+ ) ≥ Ik (v(tk− )), k ∈ N+
where the upper-right Dini derivative D + y(t) is defined as D + y(t) = limh→0+ (y(t + h) − y(t))/ h. Then u(t) ≤ v(t), for −τ ≤ t ≤ 0 implies that u(t) ≤ v(t), for t ≥ 0. Lemma 5: Let x, y ∈ Rn , and Q a diagonal positive definite matrix with appropriate dimensions; then the following inequality holds: x T y + y T x ≤ x T Qx + y T Q −1 y.
(8)
ZHANG et al.: EXPONENTIAL SYNCHRONIZATION OF COUPLED SNNs
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Proof: Equation (8) can be obtained directly from the inequality [Qx − y]T Q −1 [Qx − y] ≥ 0. Lemma 6 (Schur Complement [42]): The linear matrix inequality Q(x) S(x) >0 (9) S T (x) R(x) where Q(x) = Q T (x), R(x) = R T (x), is equivalent to either of the following conditions: 1) Q(x) > 0, R(x) − S T (x)Q(x)−1 S(x) > 0 2) R(x) > 0, Q(x) − S T (x)R(x)−1 S(x) > 0. For a clear presentation, here we let x(t) = [x 1T (t), x 2T (t), . . . , x NT (t)]T , Ar = Ar ⊗ r , f(r, x(t)) = [ f T (r, x 1 (t)), . . . , f T (r, x N (t)]T , ϕ(t) = [ϕ1T (t), . . . , ϕ NT (t)]T , CrN = (I N ⊗ Cr ), BrN = (I N ⊗ Br ), DrN = (I N ⊗ Dr ). Then coupled neural networks (2) can be rewritten as ⎧ x(t) ˙ = CrN x(t) + BrN f(r, x(t)) + αAr ⎪ ⎪ ⎨ + DrN f(r, x(t − τ (t))), t = tk, k , − ), t = tk, k , x(tk, k ) = μr x(tk,
⎪ ⎪ k ⎩ x(s) = ϕ(s), −τ ≤ s ≤ 0.
(10)
III. M AIN R ESULTS In this section, the exponential synchronization of coupled SNNs with mode-dependent impulsive effects is investigated. As was shown in [43], there are two types of impulses: one is the desynchronizing impulses (the impulsive effects suppressing synchronization); and the other is the synchronizing impulses (the impulsive effects enhancing synchronization) [31], [44], [45]. In the following, Theorem 1 aims to investigate the exponential synchronization problem of coupled delayed SNNs with desynchronizing impulses. In addition, in Theorem 2 we investigate the synchronization problem of coupled delayed SNNs with synchronizing impulses. The following assumption is necessary before presenting the main results. Assumption 2: i) When the mode-dependent impulses suppress synchronization, we assume inf{tk, k +1 − tk, k } = k ; ii) when the mode-dependent impulses can contribute to the achievement of synchronization, we assume sup{tk, k +1 − tk, k } = k . Theorem 1: Suppose that Assumption 1 and condition i) of Assumption 2 hold and there exist positive definite matrices Pr ∈ Rn(N−1)×n(N−1) , diagonal positive definite matrices Q r ∈ Rn(N−1)×n(N−1) and Rr ∈ Rn(N−1)×n(N−1) , and positive constants ζr , ηr (r ∈ S) such that (H1 ) for r ∈ S, the following inequalities hold: ⎤ ⎡ r Pr BrN−1 Pr DrN−1 0 ⎢(B N−1 )T Pr −Q r 0 0⎥ r ⎥η≥0
(12)
where ζ = minr∈S {ζr }, η = maxr∈S {ηr }; (H3 ) there exist positive constants ρ ≥ 1, λ > 0, satisfying Pr ≤ ρ Pr¯ ∀r, r¯ ∈ S λ−
ln M0 Ta
λ−ζ +
ln β >0 λτ ηe = 0
−
(13) (14) (15)
where M0 = maxr∈S {βr ρ, βr exp((λ − lnβ )τ )}, βr = (μr + 1)2 > 1, β = maxr {βr }, = mink {k }. Then the SNNs (10) with mode-dependent desynchronizing impulses are exponentially synchronized with decay rate ϑ/2, where 0 − lnβ . Proof: See Appendix. ϑ = λ − lnTM a If there are no switching parameters in the model (10), then the following corollary can be obtained directly: Corollary 1: Suppose that Assumption 1 and condition i) of Assumption 2 hold. Let ζ¯ = −λmax (2C N−1 + 2α A˜ + L T L + (B N−1 )T B N−1 + (D N−1 )T D N−1 ) and η¯ = λmax (L T L). Then, the linearly coupled neural networks (10) without switching parameters will be globally exponentially synchronized if ζ¯ > η, ¯ λ−
2 ln |1 + μ| >0
where λ > 0 is the unique solution of the equation λ − ζ¯ + ηe ¯ λτ = 0. Proof: Choose V (t) = x T (t)MT Mx(t). Let Q r = Rr = I in Theorem 1. Then the results can be obtained directly. Remark 3: When there are no switching parameters in the model (10), Theorem 1 can be reduced to the synchronization criterion of coupled DNNs with impulsive disturbances. Recently, in [29], the synchronization problem of coupled neural networks with impulsive disturbances was investigated by using the average impulsive approach. It should be pointed out that the configuration coupling matrix in [29] is assumed to be irreducible. However, Theorem 1 can be applied to the case where the configuration matrix is reducible. Hence, Theorem 1 has a wider application than the results in [29]. If there are no impulsive effects in the model (10), we can get the following synchronization criterion for coupled delayed SNNs. Corollary 2: Suppose that Assumption 1 and condition i) of Assumption 2 hold and there exist positive definite matrices Pr ∈ Rn(N−1)×n(N−1) , diagonal positive definite matrices Q r ∈ Rn(N−1)×n(N−1) and Rr ∈ Rn(N−1)×n(N−1) , and positive constants ζr , ηr (r ∈ S) such that (H1 ) for r ∈ S, the following inequalities hold: η≥0 where ζ = minr∈S {ζr }, η = maxr∈S {ηr };
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(H3 ) there exist positive constants ρ > 1, λ > 0, satisfying Pr ≤ ρ Pr¯ ∀r, r¯ ∈ S, λ−
ln M0 Ta
1.2
> 0,
λ − ζ + ηeλτ = 0
1
where M0 = max{ρ, exp(λτ )}. Then the coupled SNNs (10) without impulses are exponentially synchronized with decay 0 rate (λ − lnTM )/2. a In Theorem 1, based on Lemma 3, the synchronization problem of coupled SNNs with mode-dependent desynchronizing impulses is investigated. Inequalities (11) and (12) imply that all the subsystems of the hybrid networks (10) are synchronized. Therefore, Theorem 1 is only suitable for the case where each subsystem of (10) is self-synchronized. Hence it is desirable to consider the case where the SNNs cannot be self-synchronized. Remark 4: In the following, we will investigate the case where the mode-dependent impulses can contribute to the achievement of synchronization. However, Lemma 3 can only be applied to the case where the impulsive-free coupled neural networks are self-synchronized. Hence, it is not suitable for us to use Lemma 3 to investigate the synchronization problem of coupled SNNs where the impulsive-free SNNs are not synchronized. Fortunately, in [41], the comparison principle has been proposed to deal with the impulsive control problem. Hence, we will use the comparison principle to investigate the synchronization problem of coupled SNNs with modedependent synchronizing impulses. Theorem 2: Suppose that Assumption 1 and condition ii) of Assumption 2 hold and there exist positive definite matrices Pr ∈ Rn(N−1)×n(N−1) , diagonal positive definite matrices Q r ∈ Rn(N−1)×n(N−1) and Rr ∈ Rn(N−1)×n(N−1) , and positive constants ζr , ηr (r ∈ S) such that (H1 ) for r ∈ S, the following inequalities hold: ˜ 0, satisfying Pr ≤ ρ Pr¯ ∀r, r¯ ∈ S ζ˜ + β˜ η˜ + lnβ < 0 ˜ β˜ ηe λ1 + ζ+ ˜ λ1 τ + λ1 −
ln M˜ 0 Ta
ln β
>0
1.4
=0
(17) (18) (19) (20)
where M˜ 0 = β˜ max{ρ, exp(λ1 τ )}, β˜ = maxr∈S {βr−1 }, βr = (μr + 1)2 < 1, ζ˜ = maxr∈S {ζ˜r } and η˜ = maxr∈S {η˜r }, = minr∈S {r }. Then the coupled SNNs (10) with modedependent synchronizing impulses are exponentially synchro˜0 ˜ . nized with decay rate ϑ/2, where ϑ˜ = λ1 − lnTM a Proof: See Appendix. Remark 5: Recently, the synchronization problem of the switched complex networks with impulsive control was investigated in [38], where the impulsive intervals were allowed to be different from the network switching intervals occasionally.
0.8
0.6
0.4
0.2
0
0
Fig. 1.
2
4
6
8
10
Impulsive sequence of Example 1.
However, the results derived in [38] were based on the assumption x i (t − τ )2 ≤ ρi x i (t)2 , i = 1, 2, . . . , N. Clearly, this assumption may bring inconvenience and conservativeness in a practical application. In Theorem (2), this restrictive condition is removed. Hence, Theorem 2 can be applied more easily than the results in [38]. If there are no switching parameters in Theorem 2, then we have the following corollary. Corollary 3: Suppose that Assumption 1 and condition ii) of Assumption 2 hold. Then the coupled neural networks in (10) with synchronizing impulses are exponentially synchronized if the following inequality holds: ζˆ + β −1 ηˆ +
ln β
0.8298. Let Ta = 1. The impulsive signal is simulated in Fig. 1. Figs. 2 and 3 depict the synchronization errors of x 11 − x i1 and x 12 − x i2 , respectively. From them, one can see the exponential synchronization errors x i (t) − x j (t), which indicate that synchronization can be achieved. Example 2: In this example, we will show that the impulsive effects exist only at the instants coinciding with the system switching. Here, we consider (22) with the same parameters
11
31
0.8
i1
(22)
x −x
1
x11−x
x˙ i (t) = Cr x i (t) + Dr f (r, x i (t − τ (t)))) N ai j,r r x j (t). + Br f (r, x i (t)) + α
Synchronization errors of x12 − xi2 , i = 2, 3.
Fig. 3.
0.6
0.4
0.2
0
−0.2
0
2
4
6
8
10
t
Fig. 4.
Synchronization errors of x11 − xi1 , i = 2, 3.
as in Example 1. Note that the impulsive effects exist only at the instants coinciding with the system switching, and thus in (14) is +∞. Let μ1 = μ2 = 0.1. We solve (14) and then Ta > 0.6571. Let Ta = 0.7, and the synchronization errors of x 11 − x i1 and x 12 − x i2 are plotted in Figs. 4 and 5, respectively. From these figures, we can see that, when the impulsive effects exist only at the instants coinciding with the system switching, the coupled delayed SNNs (22) can be synchronized with the mode-dependent desynchronizing impulses. Example 3: In this example, we will show that the modedependent impulses can contribute to the achievement of synchronization. Even the impulse-free coupled SNNs cannot reach synchronization by choosing the appropriate k and μr ; then the coupled neural networks can achieve synchronization with the impulsive effects. Consider the delayed coupled SNN
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can be synchronized with the mode-dependent synchronizing impulses. The simulations presented here verify the theoretical results in this paper very well.
1.2 x12−x22 x −x 12
1
32
V. C ONCLUSION
x12−x
i2
0.8
0.6
0.4
0.2
0
−0.2
0
2
4
6
8
10
t
Fig. 5.
Synchronization errors of x12 − xi2 , i = 2, 3.
In this paper, we investigated the problem of exponential synchronization for a class of coupled SNNs with time delay and mode-dependent impulsive effects. The new concept of mode-dependent impulses was proposed, where the impulsive effects can occur not only at the instants coinciding with the system switching but also at the instants when there is no system switching. By using the Lyapunov method, comparison principle, and the Halanay differential inequality, the synchronization of coupled SNNs with mode-dependent desynchronizing and synchronizing impulsive effects was studied. Further research topics include the extension of our results to more general switched complex dynamical networks with stochastic perturbations, coupling delay, discrete-time switched dynamical networks, and neuronal networks [48].
2
10
A PPENDIX A. Proof of Theorem 1
0
10
Proof: Denote M = M ⊗ In . Construct the switched Lyapunov function in the form V (t) = x T (t)MT Pr Mx(t), t ∈ [tk , tk+1 ), r ∈ S.
log10(err(t))
−2
10
Taking the derivative of V (t) along the trajectory of the coupled SNNs (10), it yields
−4
10
D + V (t) = 2x T (t)MT Pr M[CrN x(t) + BrN f(r, x(t)) + αAr x(t) + DrN f(r, x(t − τ (t)))]
−6
10
t ∈ [tk, k , tk, k +1 ).
−8
10
0
2
4
6
8
10
t
Fig. 6.
(23)
3 2 Synchronization error of (x1i − x j i )2 . i=1
j =1
model (22) with the following parameters: 0.1 0 0.2 0 , C2 = − , C1 = − 0 0.1 0 0.2 ⎡ ⎤ −0.6 0.3 0.3 A1 = A2 = ⎣ 0.3 −0.6 0.3 ⎦ 0.3 0.3 −0.6 and the other parameter are same as in Theorem 1. By simple calculation, it can be seen that Theorem 1 does not have any feasible solutions. However, solving (16) and (17) in Theorem 2, we can get ζ1 = ζ2 = 1.0648, η1 = η2 = 1. Let μ1 = μ2 = −0.3, 1 = 2 = . . . = . Solving (18), we can get < 0.2297. Let = 0.15. We solve (19), and then λ1 = 0.4595. We solve (20), and then Ta = 2.0092.Let Ta = 2.01. Then the 3 2 synchronization error err(t) = 2i=1 j =1 (x 1i − x j i ) with impulses is plotted in Fig. 6. From the error we can see that, by choosing appropriate k and μr , the coupled delayed SNNs
(24)
By the definition of M, the following equalities are clearly true: MBrN = BrN−1 M, MCrN = CrN−1 M, MDrN = DrN−1 M, MAr = A˜ r M
(25)
where A˜ r = A˜ r ⊗ r , A˜ r = M Ar J . Substituting (25) into (24), we have D + V (t) = 2x T (t)MT Pr CrN−1 Mx(t) + 2x T (t)MT Pr BrN−1 Mf(r, x(t)) + 2x T (t)MT Pr DrN−1 Mf(r, x(t − τ (t))) + 2αx T (t)MT Pr A˜ r Mx(t) t ∈ [tk, k , tk, k +1 ).
(26)
In view of Lemma 5, we have 2x T (t)MT Pr BrN−1 Mf(r, x(t))
≤ x T (t)MT Pr BrN−1 Q r−1 (BrN−1 )T Pr Mx(t) + f T (r, x(t))MT Q r Mf(r, x(t)) ≤ x T (t)MT Pr BrN−1 Q r−1 (BrN−1 )T Pr Mx(t) + x T (t)MT L r Q r L r Mx(t)
(27)
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By (31), (32), and (36), for t ∈ [tm+1 , tm+2 ),
and
V (t) ≤ β M0 M0m e(−λ+
2x T (t)MT Pr DrN−1 Mf(r, x(t − τ (t))) ≤ x T (t)MT Pr DrN−1 Rr−1 (DrN−1 )T Pr Mx(t)
×e
+x T (t − τ (t))MT L r Rr L r Mx(t − τ (t)).
(28)
It follows from (11), (24)–(28), and Lemma 6 that for t ∈ [tk, k , tk, k +1 ),
=
− τ (t))
When t = tk, k , it follows from the second equation of (10) that (30)
where βr = (μr + 1)2 . Since t ∈ [tk , tk+1 ), σ (t) = r , from Lemma 3, for t ∈ [tk , tk+1 ), k ≥ 1, we have, when tk = tk,1 t−tk k
= V (tk )τ e
(−λ+ lnβr )(t −tk ) k
where ϑ = λ −
ln β )(t −tk )
V (t) ≤ V (tk )τ βr
= βr V (tk )τ e
+1
(31)
k
ln β )(t −tk )
.
(32)
Then, we show by induction, that V (t) ≤ βV (t0 )τ M0k e(−λ+
ln β )(t −t0 )
where M0 = maxr∈S {βr ρ, βr exp((λ − [t0 , t1 ), it follows from Lemma 3 that: V (t) ≤ V (t0 )τ β
t−t0 0
≤ βV (t0 )τ e
e(−λ+
ln β )(t −t0 )
ln M0 Ta
−
ln β
(38)
> 0. Then, we have (39)
where ω = minr∈S {λmin (Pr )}, ν = maxr∈S {λmax (Pr )}. Hence ν x T (t)MT Mx(t) ≤ β Mϕ(s)τ M0N0 e−ϑ(t −t0) . (40) ω From Definition 2 and the definition of M, coupled SNNs (10) with mode-dependent desynchronizing impulses are exponentially synchronized. This completes the proof.
Proof: Consider the same Lyapunov function as Theorem 1. Similar to the proof of Theorem 1, for t ∈ [tk, k , tk, k +1 ), we have D + V (t) ≤ ζ˜ V (t) + ηV ˜ (t − τ (t))
(41)
+ − V (tk,
) = βr V (tk,
), t = tk, k . k k
(42)
and e−λ(t −tk )
(−λ+ lnβr )(t −tk )
≤ βr V (tk )τ e(−λ+
e
(t−t0 ) ln M0 Ta
ωx T (t)MT Mx(t) ≤ νβMϕ(s)τ M0N0 e−ϑ(t −t0)
and when tk < tk,1 t−tk k
(37)
B. Proof of Theorem 2
e−λ(t −tk )
≤ V (tk )τ e(−λ+
.
= β M0N0 V (t0 )τ e−ϑ(t −t0 ) , t ≥ t0
(29)
+ − V (tk,
) = βr V (tk,
) k k
ln β )(t −t0 )
N(t,t0 ) (−λ+ lnβ )(t −t0 )
V (t) ≤ βV (t0 )τ M0
+ ηr V (t − τ (t)) − ζr V (t)
V (t) ≤ V (tk )τ βr
βV (t0 )τ M0m+1 e(−λ+
≤ β M0N0 V (t0 )τ e
≤ −ζ V (t) + ηV (t − τ (t)).
V (t0 )τ
(−λ+ lnβ )(t −tm+1 )
Therefore, by the induction principle, we see that (33) holds for all k ∈ N. Then from Definition 1, we have
D + V (t) ≤ x T (t)MT [Pr CrN−1 + (CrN−1 )T Pr + 2α Pr A˜ r + L r Q r L r + Pr BrN−1 Q r−1 (BrN−1 )T Pr + Pr DrN−1 Rr−1 (DrN−1 )T Pr + ζr Pr ]Mx(t) + x T (t − τ (t))MT (L r Rr L r − ηr Pr )Mx(t
ln β )(tm+1 −t0 )
, t ∈ [tk , tk+1 ) ln β )τ )}.
(33)
When t ∈
+1 −λ(t −t0 ) e
(−λ+ lnβ )(t −t0 )
.
˜ (tk )τ e−λ1 (t −tk ) . V (t) ≤ βV
(43)
For any ε > 0, let v(t) be a unique solution of the impulsive delay system ⎧ ˙ = ζ˜ v(t) + ηv(t ˜ − τ (t)) + ε, t = tk, k , ⎨ v(t) + − (44) ) = β v(t ), v(tk,
r k, k t = tk, k , k ⎩ v(t) = V (t), tk − τ ≤ t ≤ tk . Then it follows from Lemma 4 that:
(34)
Now, assume that (33) holds for k = 0, 1, . . . , m, where m ≥ 1. Then we show that (33) holds for k = m +1. Note that Pr ≤ ρ Pr¯ ∀r, r¯ ∈ S; then we have − ), tm+1 = t(m+1),1 , βρV (tm+1 V (tm+1 ) ≤ (35) − ρV (tm+1 ), tm+1 < t(m+1),1 . Then we can get from (31), (32), and (35) that ln β )τ } ln β ×βV (t0 )τ M0m e(−λ+ )(tm+1−t0 ) .
In the following, we show that for t ∈ [tk , tk+1 ):
V (tm+1 )τ ≤ max{βρ, β exp(λ −
(36)
V (t) ≤ v(t), t ∈ [tk , tk+1 ).
(45)
By the formula for the variation of parameters, one obtain from (44) that for t ∈ [tk , tk+1 ) t W (t, s)[ηv(s ˜ − τ (s)) + ε]ds v(t) = W (t, tk )v(tk ) + tk
(46) where W (t, s), t, s ≥ tk is the Cauchy matrix of the linear system y˙ (t) = ζ˜ v(t), t = tk, k , (47) + − ) = βr v(tk,
), t = tk, k . v(tk,
k k
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According to the representation of the Cauchy matrix, we can get the following estimation: ˜ βr . (48) W (t, s) ≤ eζ (t −s) s 0, there exists a unique λ1 > 0, h(+∞) = +∞ and h(ς such that ln β + β˜ ηe ˜ λ1 τ = 0. (52) λ1 + ζ˜ +
ν ˜ Mϕ(s)τ β˜ M˜ 0N0 eϑ(t −t0 ) . (61) ω Therefore, the coupled SNNs (10) with mode-dependent synchronizing impulses are exponentially synchronized. This completes the proof. x T (t)MT Mx(t) ≤
ACKNOWLEDGMENT
ln β ).
It is obvious from (18) that
The authors would like to thank the Associate Editor and the anonymous reviewers for their helpful comments.
v(t) ≤ ξ < ξ e−λ1 (t −tk ) +
ε , tk − τ ≤ t ≤ tk . (53) −1 ˜ β δ − η˜
R EFERENCES
Denote δ = −(ζ˜ + β˜ −1 δ − η˜ > 0; hence
Then we claim that
ε
v(t) < ξ e−λ1 (t −tk ) +
, tk ≤ t < tk+1 . β˜ −1 δ − η˜ If it is not true, there exists a t ∗ ∈ [tk , tk+1 ) such that ε v(t ∗ ) ≥ ξ e−λ1 (t −tk ) + β˜ −1 δ − η˜ and v(t) < ξ e−λ1 (t −tk ) +
ε β˜ −1 δ − η˜
From (50), we have ∗
v(t ) ≤ ξ e
−δ(t ∗ −tk )
+
t∗
, t ∈ [tk − τ, t ∗ ).
(54)
(55)
(56)
˜ −δ(t ∗ −s) [ηv(s ˜ − τ (s)) βe
tk
+ ε]ds t∗ ˜ δ(s−tk )
