868

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 6, JUNE 2013

Impulsive Control for Existence, Uniqueness, and Global Stability of Periodic Solutions of Recurrent Neural Networks With Discrete and Continuously Distributed Delays Xiaodi Li and Shiji Song Abstract— In this paper, a class of recurrent neural networks with discrete and continuously distributed delays is considered. Sufficient conditions for the existence, uniqueness, and global exponential stability of a periodic solution are obtained by using contraction mapping theorem and stability theory on impulsive functional differential equations. The proposed method, which differs from the existing results in the literature, shows that network models may admit a periodic solution which is globally exponentially stable via proper impulsive control strategies even if it is originally unstable or divergent. Two numerical examples and their computer simulations are offered to show the effectiveness of our new results. Index Terms— Contraction mapping theorem, delays, existence, impulsive control, periodic solution, recurrent neural networks, stability theory, uniqueness.

I. I NTRODUCTION

R

ECURRENT neural networks (RNNs), especially Hopfield neural networks [1], cellular neural networks (CNNs) [2], [3], Cohen–Grossberg neural networks (CGNNs) [4], bidirectional associative memory (BAM) neural networks [5], and networks with time delays [6]–[9] have been thoroughly investigated in recent years because of their potential applications in the areas of signal and image processing, associative memories and pattern classification, parallel computation, and optimization problems ([10]–[12] and references therein). In the design of RNNs, the dynamical properties of networks, such as the existence-uniqueness and global asymptotic stability or global exponential stability of equilibrium points of the networks, play important roles. For example, in solving optimization problems, the neural network must be designed to have one unique and globally stable equilibrium

Manuscript received October 27, 2011; revised October 20, 2012; accepted December 16, 2012. Date of publication March 25, 2013; date of current version April 5, 2013. This work was supported by the National Natural Science Foundation of China under Grant 61273233, Grant 60834004, and Grant 11226136, the Research Foundation for the Doctoral Program of Higher Education under Grant 20090002110035 and Grant 20120002110035, the Project of Shandong Province Higher Educational Science and Technology Program (J12LI04), and the Research Fund for Excellent Young and Middleaged Scientists of Shandong Province BS2012DX039. X. Li is with the School of Mathematical Sciences, Shandong Normal University, Ji’nan, 250014, China (e-mail: [email protected]). S. Song is with the Department of Automation, Tsinghua University, Beijing 100084, 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.2012.2236352

point [10], [13]. In the analysis of parallel computation, to increase the rate of convergence to the equilibrium point of the networks and reduce the neural computing time, it is necessary to ensure a desired exponential convergence rate of the networks’ trajectories, starting from arbitrary initial states to the equilibrium point which corresponds to the optimal solution [14]; so there is a strong motivation to study the global exponential stability of equilibrium points for neural networks [15]–[20]. Although there are many important results dealing with the static attractors (i.e., equilibrium points) of various RNNs, experimental and theoretical studies have revealed that periodic attractors in RNNs are also a very interesting dynamic behavior, as many biological and cognitive activities (e.g., heartbeat, respiration, mastication, locomotion, and memorization) exhibit periodicity [21], [22]. Persistent oscillation, such as limit cycles (periodic orbits), represents a common feature of neural firing patterns produced by the dynamic interplay between cellular and synaptic mechanisms. Moreover, the periodic solution problem of RNNs has found many applications such as associative memories [23], pattern recognition [24], machine learning [25], robot motion control [26], etc. Hence, investigation of periodic solution on RNNs is indispensable for practical design and engineering applications of network models. In addition, as we know, an equilibrium point can be viewed as a special periodic attractor of neural networks with an arbitrary period. In this sense, the analysis of the periodic attractor of neural networks may be considered to be more general than that of the equilibrium point. Recently, the existence, uniqueness, and global stability of periodic solution of RNNs with/without time delays have received great attention, and many interesting results on this topic have been reported in the literature (e.g., [27]–[29] and references therein). As a kind of dynamic system, RNNs are generally classified into two groups: continuous and discrete. Recently, a new system type has been introduced as dynamic systems with impulses, which displays some kind of dynamic behavior in both continuous and discrete characteristics. These include, e.g., many evolutionary processes, particularly some biological systems such as biological neural networks and bursting rhythm models in pathology [30], [31]. Other examples include optimal control models in economics, frequency-modulated signal processing systems, and flying

2162-237X/$31.00 © 2013 IEEE

LI AND SONG: RECURRENT NEURAL NETWORKS WITH DISCRETE AND CONTINUOUSLY DISTRIBUTED DELAYS

object motion [32]. In recent years, research on periodic solution of RNNs with impulsive effects has attracted considerable interest (see [33]–[47]). However, almost all these results have a common feature in that the corresponding continuous system (i.e., without impulsive effects) is originally considered even exponentially stable, and then they consider the robust stability under some impulsive perturbations. It is known that, in theory and practice, impulsive control has been widely used to stabilize some unstable systems and synchronize some chaotic systems (see [48]–[54]). For example, Yang and Xu [48] investigated the robust stability and impulsive stabilization of CGNNs with time-varying delays. In [49], Li studied the exponential stability of Cohen–Grossberg-type BAM neural networks with time-varying delays via impulsive control. In [52], Zhang and Sun studied the robust synchronization of coupled delayed RNNs under general impulsive control. To our knowledge, however, there are very few results on the existence, uniqueness, and global stability of periodic solution of RNNs with/without time delays via impulsive control. As we know, the main idea of impulsive control of RNNs is to introduce the impulsive effects into topological structure of the networks and then change the states of systems. The main idea of impulsive synchronization is to achieve the synchronization between the derived system and the response system via the impulsive control, while impulsive stationary oscillations are devoted to the investigation of existence of global periodic attractor of RNNs via the impulsive control. Its main idea is to first study the existence of periodic solution from impulsive control point of view, and then to investigate its attractiveness. That is, perhaps a system does not admit periodic solution originally, but it may admit one which is unique and globally attractive under the impulsive control. More recently, in [55] some new criteria were derived to ensure the global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays. An interesting problem is whether we can apply those criteria [55] to the existence, uniqueness, and global stability of periodic solution of RNNs with time delays via impulsive control. Motivated by the above discussion, in this paper we consider the existence, uniqueness, and global stability of periodic solutions for a class of RNNs with discrete and continuously distributed delays based on the established criteria in [52]. By using contraction mapping theorem, some new sufficient conditions ensuring the existence, uniqueness, and global stability of periodic solutions are obtained. The novelty of the obtained result is that the network models may be originally unstable or divergent, but they will admit a periodic solution which is globally stable via the impulsive control strategies that are developed in this paper. The rest of this paper is organized as follows. In Section II, we introduce some necessary notations, definitions, and preliminaries which will be used later. In Section III, several sufficient conditions are derived for the existence, uniqueness, and global stability of periodic solution for the considered RNNs. In Section IV, two numerical examples and their computer simulations are offered to show the effectiveness of our new results. Finally, we draw conclusions in Section V.

869

II. P RELIMINARIES Notations: Let R denote the set of real numbers, R+ the set Rn of positive numbers, Z+ the set of positive integers, and n the n-dimensional real spaces with the norm x = i=1 |x i | for any x ∈ Rn . α ∨β denotes the maximum value of α and β. For any interval J ⊆ R, set V ⊆ Rk (1 ≤ k ≤ n), PC(J, V ) = {ϕ : J → V is continuous everywhere except at finite number of points t, at which ϕ(t + ), ϕ(t − ) exist and ϕ(t + ) = ϕ(t)} and PCB(J, V ) = {ϕ ∈ PC(J, V ) : ϕ is bounded }. In particular, let PCB+ = PCB([0, +∞), R), PCB− = PCB((−∞, 0], R), and for each ϕ ∈ PCB+ (PCB− ), the norm is defined by ϕ B = sups≥0(s≤0) ϕ(s). For any t ≥ t0 ≥ 0 > α ≥ −∞, let F(t, x(s)), where s ∈ [t + α, t] or F(t, x(·)), be a Volterratype functional and its values are in Rn and are determined by t ≥ t0 and the values of x(s), s ∈ [t + α, t]. In the case when α = −∞, the interval [t + α, t] is understood to be replaced by (−∞, t]. Given a continuous bounded function h, which is defined on  ⊂ R, we set . h I = inf h(s), s∈

. h S = sup h(s). s∈

In this paper, we consider the following neural networks with discrete and distributed delays: ⎧ n  ⎪ ⎪ ⎪ x ˙ (t) = −c (t)x (t) + ai j (t) f j (x j (t)) ⎪ i i i ⎪ ⎪ ⎪ j =1 ⎪ ⎪ ⎪ n ⎪  ⎪ ⎪ ⎪ + bi j (t) f j (x j (t − τ (t))) ⎪ ⎨ j =1 (1)  ∞ n ⎪  ⎪ ⎪ ⎪ wi j (t) K i j (s) f j (x j (t − s))ds + ⎪ ⎪ ⎪ 0 ⎪ j =1 ⎪ ⎪ ⎪ ⎪ + Ii (t), t ∈ [tk−1 , tk ), ⎪ ⎪ ⎪ ⎩x (t + s) = φ (s), s ∈ (−∞, 0], i ∈  i 0 i subject to the impulses x i (tk ) = Jik (tk , x i (tk− )), k ∈ Z+ ,

i ∈

(2)

where  = {1, 2, . . . , n}, φi ∈ PCB− ; the impulse times tk satisfy 0 ≤ t0 < t1 < · · · < tk < · · · , limk→+∞ tk = +∞; x i corresponds to the state of the i th unit at time t; n ≥ 2 is the number of units in a neural network; ci denotes the neural self-inhibition at time t; ai j , bi j , and wi j are the connection weights; f j represents the neuron input–output activation; Ii denotes the external inputs from outside the neural networks at time t; τ (t) is the discrete transmission time-varying delay satisfying 0 ≤ τ (t) ≤ τ, where τ is a real constant; K i j is the delay kernel function and Jik denotes the impulsive perturbations of the i th neuron at time tk . Throughout this paper, we make the following assumptions. (H1 ) ci (t), ai j (t), bi j (t), wi j (t), τ (t), Ii (t), and Jik (t, •) are all continuously periodic functions defined on t ∈ [0, ∞) with common period ω > 0, i, j ∈ . (H2 ) The delay kernels K i j , i, j ∈ , are some real-valued nonnegative continuous function defined in [0, ∞)

870

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 6, JUNE 2013

satisfying K i j (t) ≤ K(t) for any i, j ∈ , t ∈ R+ and  ∞ K(s)eηs ds < ∞ 0

in which η is a positive constant. (H3 ) For given ω > 0, there exists an integer q ∈ Z+ such that tk + ω = tk+q and Ji(k+q) (t, •) = Jik (t, •), k ∈ Z+ , i ∈ , t ∈ R+ . (H4 ) There exist some positive constants ρik such that |Jik (t, u) − Jik (t, v)| ≤ ρik |u − v|, u, v ∈ R, t ∈ R+ , i ∈ , k ∈ Z+ . (H5 ) The neuron activation functions f j (·), j ∈ , are bounded and satisfy σ j− ≤

f j (u) − f j (v) ≤ σ j+ , u−v

j ∈

for any u, v ∈ R, u = v, where σ j− , σ j+ , j ∈  are some real constants and they may be positive, zero, or negative. Definition 1: A map x : R+ → Rn is said to be an ωperiodic solution of (1) and (2) if: 1) x(t) is a piecewise continuous map with first-class discontinuity points and satisfies (1) and (2); 2) x(t) satisfies x(t + ω) = x(t), t = tk and x(tk + ω+ ) = x(tk+ ), k ∈ Z+ . Definition 2: The function V : [t0 + α, ∞) × R → R+ belongs to class ν0 if: 1) V is continuous on each of the sets [tk−1 , tk ) × R and lim(t,ϕ1 )→(t −,ϕ2 ) V (t, ϕ1 ) = V (tk− , ϕ2 ) exists; k 2) V (t, x) is locally Lipschitzian in x and V (t, 0) ≡ 0. Lemma 1: PCB+ ( or PCB− ) is a Banach space. Lemma 2: Assume that assumptions (H1) − −(H5) hold. Then system (1), (2) has an ω-periodic solution if there exists a φ = (φ1 , φ2 , . . . , φn )T , φi ∈ PCB− such that x i (t0 + ω, t0 , φi ) = φi (0), i ∈ , where (x 1 (t, t0 , φ1 ), . . . , x n (t, t0 , φn ))T is a solution of equation (1) and (2) through (t0 , φ). Proof: Let (x 1 (t, t0 , φ1 ), . . . , x n (t, t0 , φn ))T be a solution of system (1), (2) satisfying x i (t0 + ω, t0 , φi ) = φi (0), i ∈ , φi ∈ PCB− . Then one may define an ω-periodic version of x i as follows: ⎧ ⎪ ⎨φi (t − t0 ), t ∈ (−∞, t0 ] x i (t) = x i (t), t ∈ [t0 , t0 + ω) ⎪ ⎩ x i (t − nω), t ∈ [t0 + nω, t0 + (n + 1)ω), n ∈ Z+ . It is clear that x i (t) is ω-periodic on [t0 , ∞). Let Ii (t, x t (·)) = −ci (t)x i (t) +

n 

ai j (t) f j (x j (t))

j =1

+ +

n  j =1 n  j =1

bi j (t) f j (x j (t − τ (t))) 

∞

wi j (t)

K i j (s) f j (x j (t − s))ds

0

+Ii (t), t ≥ t0 , i ∈ .

Then it can be deduced from Assumption (H1) that Ii (t + ω, x t +ω (·)) = Ii (t, x t +ω (·)), i ∈ . Next one may prove that x i (t) is a solution of (1) and (2). In fact, for any t ∈ [t0 +nω, t0 +(n +1)ω), when t = tk , k ∈ Z+ , it holds that x˙ i (t) = x˙i (t − nω) = Ii (t − nω, x t −nω (·)) = Ii (t, x t −nω (·)) = Ii (t, x t (·)), i ∈ . And when t = tk , k ∈ Z+ , it holds that x i (tk ) = x i (tk − nω) = x i (tk−nq )

− = Ji(k−nq) (tk−nq , x i (tk−nq ))

= Jik (tk − nω, x i (tk − nω− )) x i (tk− )). = Jik (tk , Hence, ( x1, . . . , x n )T is an ω-periodic solution of (1) and (2) through (σ, φ). By the existence-uniqueness of solutions of (1) and (2), we know that x i = x i . This completes the proof of Lemma 2. Remark 1: The idea of Lemma 2 is inspired by [34]. But the condition x t0 +ω (t0 , φ) = φ developed in [34] is invalid in this paper due to the infinite delay effects, which can be further seen in the following section. In addition, we point out that the necessary condition developed in [34, Lemma 2] is false. In fact, if (1) in [34] has a T -periodic solution x(t, σ, φ), then it follows that x σ +T (σ, φ)(0) = x(σ + T, σ, φ) = x(σ, σ, φ) = φ(0), which implies that x σ +T (σ, φ) = φ for s = 0. However, x σ +T (σ, φ)(s) = x(σ + T + s, σ, φ) = φ(s) for s ∈ [−τ, 0). Consider the following impulsive functional differential equations: ⎧

⎪ ⎨x (t) = F(t, x(·)), t ≥ t0 , t = tk (3) x|t =tk = x(tk ) − x(tk− ) = I¯k (tk , x(tk− )), k ∈ Z+ ⎪ ⎩ x t0 = ϕ(s), α ≤ s ≤ 0. Some detailed information can be found in [55] and [32]; here we omit it. Lemma 3 [55]: Assume that there exist a function V (t, x) ∈ ν0 , constants p > 0, q > 1, c1 > 0, c2 > 0, m > 0, and γ > 0 such that: 1) c1 xm ≤ V (t, x) ≤ c2 xm , (t, x) ∈ [0, ∞) × Rn ; 2) For any t0 ∈ R+ and ψ ∈ PC([α, 0], Rn ), if eγ θ V (t + θ, ψ(θ )) ≤ q V (t, ψ(0)), θ ∈ [α, 0], t = tk , then D + V (t, ψ(0)) ≤ pV (t, ψ(0)); 3) For all (tk , ψ) ∈ R+ × PC([α, 0], Rn ), V (tk , ψ(0) + I¯k (tk , ψ)) ≤ 1/q V (tk− , ψ(0)); 4) supk∈Z+ {tk − tk−1 } < ln q/ p. The trivial solution of (4) is globally exponentially stable and, moreover, for any solution x(t, t0 , ϕ) of (4), it holds that V (t, x(t)) ≤ qc2 ||ϕ||mB e−ε(t −t0) ,

t ≥ t0

where ε is a positive constant. Remark 2: It should be noted that Lemma 3 can be applied to systems with finite or/and infinite delays since α ∈ [−∞, 0]. Based on this point, we shall establish some

LI AND SONG: RECURRENT NEURAL NETWORKS WITH DISCRETE AND CONTINUOUSLY DISTRIBUTED DELAYS

criteria guaranteeing the existence, uniqueness, and global exponential stability of periodic solution for (1) and (2) in Section III.

871

Assumptions (H2 ) and (H5) that D+ V =

n 

z˙ i sgn (z i )

i=1

III. M AIN R ESULTS Theorem 1: Assume that assumptions (H1)–(H5) hold. If there exist constants A > 1, γ ∈ (0, η], B > 0 such that − min ciI i∈

+

n  i=1

+A

max |aiSj |L j j ∈

n  i=1

|σ j+ |

+Ae

j ∈

n  i=1

 max |wiSj |L j

γτ

∞

+

K(s)eγ s ds ≤ B

i∈,k∈Z+

ρik ≤

+

where G ik (tk , z i (tk− )) = Jik (tk , x i (tk− )) − Jik (tk , yi (tk− )), g j (z j (·)) = f j (x j (·)) − f j (y j (·)). Obviously, it yields from Assumptions (H4) and (H5) that

|g j (z j (·))| ≤ L j |z j (·)|. (6)

Choose  = [0, ω] and consider the following Lyapunov function: n 



n 

+

n  n 

Obviously, V ∈ ν0 . Calculating the upper right derivative of function V, it can be deduced from (6) and (7), and

 |wiSj |L j

− min ciI i∈

×

n 

n 

∞

+

+ ≤



+ +

n 

|z i (t)| +

i=1

i=1

 max |aiSj |L j j ∈

|z j (t)|

n  i=1 n 

 max |biSj |L j j ∈ max |wiSj |L j

n 

i=1

n  i=1

n 

+

|z j (t − τ (t))|

j =1



j ∈

i=1

− min ciI i∈

K(s)|z j (t − s)|ds

0

j =1



|aiSj |L j |z j (t)|

|biSj |L j |z j (t − τ (t))|

i=1 j =1

≤

n n   i=1 j =1

i=1

+

K i j (s)|g j (z j (t − s))|ds

0

ciI |z i (t)| +

n n  

∞

K(s)

0

n  i=1

n 

|z j (t − s)|ds

j =1



max |aiSj |L j j ∈

V (t, z(t))



max |biSj |L j j ∈ max |wiSj |L j j ∈

V (t, z(t − τ (t))) 

∞

K(s)V (t, z(t − s))ds.

0

When eγ θ V (t + θ, z(t + θ )) ≤ A V (t, z(t)), θ ∈ (−∞, 0], t ∈ [tk−1 , tk ), from (5) we get D+ V ≤



− min ciI + i∈

+A eγ τ +A

n  i=1

n  i=1

n 

max |aiSj |L j j ∈

max |biSj |L j

i=1

|z i |.

i=1

∞

|wi j (t)|

i=1 j =1

≤−

|ai j (t)||g j (z j (t))|

|bi j (t)||g j (z j (t − τ (t)))|

i=1 j =1

Proof: Let x = x(t, t0 , φ) and y = y(t, t0 , ϕ) be two arbitrary solutions of system (1), (2) with initial values (t0 , φ) and (t0 , ϕ), respectively, where x = (x 1 , . . . , x n )T ∈ Rn , y = (y1 , . . . , yn )T ∈ Rn and φ, ϕ ∈ PCB− . Let z = x − y; then from (1) and (2) we get a new auxiliary system ⎧ n  ⎪ ⎪ ⎪ z ˙ (t) = −c (t)z (t) + ai j (t)g j (z j (t)) i i i ⎪ ⎪ ⎪ ⎪ j =1 ⎪ ⎪ n ⎪ ⎪  ⎪ ⎪ ⎪ + bi j (t)g j (z j (t − τ (t))) ⎪ ⎪ ⎪ ⎨ j =1  ∞ n  (5) ⎪ w (t) K i j (s)g j (z j (t − s))ds + ⎪ i j ⎪ ⎪ 0 ⎪ ⎪ j =1 ⎪ ⎪ ⎪ ⎪ t ∈ [tk−1 , tk ) ⎪ ⎪ ⎪ ⎪ ⎪ z i (t0 + s) = φi − ϕi , s ∈ (−∞, 0], i ∈  ⎪ ⎪ ⎩ z i (tk ) = G ik (tk , z i (tk− )), k ∈ Z+ , i ∈ 

V (t, z) = z =

n n  

0

ln A ⎪ ⎪ . ⎩ sup {tk − tk−1 } < B k∈Z+

n  n  i=1 j =1

n n  

(4)

1 A

|G ik (tk , z i (tk− ))| ≤ ρik |z i (tk− )|,

ci (t)|z i (t)| +

i=1 j =1

|σ j− |,

sup

n  i=1

max |biSj |L j j ∈

where L j = ∨ j ∈ , then (1) admits a periodic solution which is globally exponentially stable under the following impulsive control: ⎧ ⎪ ⎪ ⎨

≤−

j ∈

max |wiSj |L j j ∈

≤ BV (t, z(t)).



∞

 K(s)eγ s ds V (t, z(t))

0

(7)

872

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 6, JUNE 2013

On the other hand, when t = tk , it follows from assumption (H4) that V (tk , z(tk )) = = ≤ ≤

n  i=1 n  i=1 n 

u(t, t0 , φi ) − v(t, t0 , ϕi ) = u(t + kω, t0 , φi ) − v(t + kω, t0 , ϕi ) ≤ A φ  − ϕ   B e−ε(t +kω−t0 ) → 0 as k → ∞

|x i (tk ) − yi (tk )| |Jik (x i (tk− )) − Jik (yi (tk− ))|

which implies that u(t) ≡ v(t), t ≥ 0. Hence, u(t) is the unique ω-periodic solution of (1) and (2) and all other solutions converge exponentially to it, i.e., system (1) admits a unique periodic solution which is globally exponentially stable via impulsive control (2). The proof is therefore complete.  Corollary 1: Assume that assumptions (H1)–(H5) hold and sup ρik < 1. If there exists a constant B > 0 such that

ρik |x i (tk− ) − yi (tk− )|

i=1 sup

i∈,k∈Z+

 ρik V (tk , z(tk− ))

1 V (tk , z(tk− )). ≤ A

(8)

By (8) and (9), and using Lemma 3, we know that there exists a constant ε > 0 such that V (t, z) ≤ A φ − ϕ B e

−ε(t −t0 )

,

v = (v 1 , . . . , v n )T , where v i = v i (t, t0 , ϕi ), ϕi ∈ PCB− . Then by the proof of (10), we can deduce that for t ≥ 0

i∈,k∈Z+

− min ciI + i∈

max |aiSj |L j j ∈

i=1

 1 eγ τ max |biSj |L j j ∈ sup ρik n

+

t ≥ t0 .

n 

i=1

i∈,k∈Z+

That is x − y ≤ A φ − ϕ B e

−ε(t −t0 )

,

t ≥ t0 .

(9)

Then choose a T ≥ t0 such that 1 x − y ≤ φ − ϕ B , 2

t ≥ T.

0

Obviously, operator F maps the set PCB− into itself (here, PCB− is understood to be n-dimensional). By simple induction, it can be deduced that ⎧ ⎨1 ˆ φ, s < 0 k ˆ F φ = 2k ⎩u(t + kω, t , φ), ˆ s = 0, k ∈ Z . 0

0

1 sup ρik

i∈,k∈Z+

i=1

 max |wiSj |L j j ∈

∞

K(s)eγ s ds ≤ B

0

where L j = |σ j+ | ∨ |σ j− |, j ∈ , then (1) admits a unique periodic solution which is globally exponentially stable under the following impulsive control:

ˆ of (1) and (2) through (t0 , φ), ˆ one For any solution u(t, t0 , φ) may define an operator ⎧ ⎨1 ˆ φ, s < 0 F φˆ = 2 ⎩u(t + ω, t , φ), ˆ s = 0. 0

+

n 

+

Choose k large enough such that 1 1 sup φˆ − ϕ ˆ B ˆ + φˆ − ϕ 3 s 1 and B > 0 such that − min ciI + i∈

n  i=1

+A

j ∈

n  i=1

max |aiSj |L j + A 

max |wiSj |L j j ∈

∞

n  i=1

max |biSj |L j j ∈

K(s)eηs ds < B

0

where L j = |σ j+ | ∨ |σ j− |, j ∈ , then (1) admits a unique periodic solution which is globally exponentially stable under the following impulsive control: ⎧ 1 ⎪ ⎪ ⎨ sup ρik ≤ A i∈,k∈Z+

ln A ⎪ ⎪ . ⎩ sup {tk − tk−1 } < B k∈Z+

−1 Remark 2: Let A = supi∈,k∈Z+ ρik , then we can obtain Corollary 1 easily. Note that condition (5) can be replaced by

− min ciI + i∈

+A

n 

i=1 n 

max |aiSj |L j + A eγ τ j ∈

max |wiSj |L j

i=1

j ∈



∞

n  i=1

max |biSj |L j j ∈

K(s)eηs ds ≤ B.

0

Then, in view of the arbitrariness of the positive constant γ , we can obtain Corollary 2. In addition, the impulsive condition

LI AND SONG: RECURRENT NEURAL NETWORKS WITH DISCRETE AND CONTINUOUSLY DISTRIBUTED DELAYS

where ρ > 1, μ > 0 are two given constants. Choose A = ρ and q = ω/μ; then by Theorem 1, the following result can be derived. Corollary 3: Assume that Assumptions (H1), (H2 ), and (H5) hold. If there exist constants γ ∈ (0, η], B > 0 such that n n   − min ciI + max |aiSj |L j + ρeγ τ max |biSj |L j

10 5 0

x(t)

−5 −10 −15

−25

x1

Nonimpulsive effects and ω=2

−20

0

50

100 t

i∈ n 

x2

150

200

10

x 2 (t)

0 −10 −20 −30 10 5

200 150

0 x 1 (t)

100 −10

50 0

t

(b) 10 5

i=1

j ∈

max |wiSj |L j j ∈



∞

i=1

j ∈

K(s)eγ s ds ≤ B

0

Remark 3: Here we should point out that, in this paper, some strict restrictions on the impulsive intervals (i.e., tk − tk−1 ) are imposed in order to study the periodic problem of considered neural networks. As we know, larger impulsive intervals can further reduce the control cost when using impulsive control method. Thus how to improve the control strategies with larger impulsive intervals will be our future important research task. IV. N UMERICAL E XAMPLES

−5

In this section, we will give two examples and their computer simulations to show the effectiveness of the development method. Example 1: Consider the following neural networks with discrete delays: ⎧  2π  ⎪ x ˙ t x 1 (t) (t) = − 0.15 + 0.05 sin ⎪ 1 ⎪ ⎪ ω ⎪ ⎪ 2  ⎪  ⎪ 2π ⎪ ⎪ 0.3 − 0.01 cos (t + j ) f j (x j (t)) + ⎪ ⎪ ω ⎪ ⎪ j =1 ⎪ ⎪ ⎪ 2  ⎪   ⎪ 2π ⎪ ⎪ (t + j ) f j (x j (t − 1)) + 0.1 + 0.04 sin ⎪ ⎪ ⎪ ω ⎪ j =1 ⎪ ⎪ ⎨ 2π t, t = tk + cos ω ⎪  ⎪ 2π  ⎪ ⎪ ⎪ t x 2 (t) x˙2 (t) = − 0.2 + 0.1 sin ⎪ ⎪ ω   ⎪  ⎪ 2 ⎪ 2π ⎪ (t + j ) f j (x j (t)) 0.17 + 0.02 sin + ⎪ j =1 ω ⎪ ⎪ ⎪ 2 ⎪   ⎪ 2π ⎪ ⎪ 0.1 + 0.01 sin (t + j ) f j (x j (t − 1)) + ⎪ ⎪ ⎪ ω ⎪ j =1 ⎪ ⎪ ⎪ ⎪ 2π ⎩ t, t = tk + sin ω subject to impulses

x(t)

0

−10 −15

−25

x1

Nonimpulsive effects and ω=8

−20

0

50

100 t

x2

150

200

(c)

10 0 x 2 (t)

+ρ

i=1

where L j = |σ j+ | ∨ |σ j− |, j ∈ , then system (1) admits a unique periodic solution which is globally exponentially stable under the following impulsive control: ⎧ ⎪ ⎨ μ < ln ρ B ω ⎪ ⎩ ∈ Z+ . μ

(a)

−5

873

−10 −20 −30 10 5 0 x 1 (t)

−5 −10

50 0

100

150

200

t

(d) Fig. 1. control. control. control. control.

(a) State trajectories of system (11) with ω = 2 without (b) Phase portraits of system (11) with ω = 2 without (c) State trajectories of system (11) with ω = 8 without (d) Phase portraits of system (11) with ω = 8 without

(2) is simplified by 1 x i (tk ) = x i (tk− ), k ∈ Z+ , i ∈ , tk = μk ρ

impulsive impulsive impulsive impulsive

x i (tk ) =

1 x i (tk− ), k ∈ Z+ , i = 1, 2 ρ

where f 1 = f 2 = tanh(x), ω > 0, and ρ > 1 are some real constants.

874

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 6, JUNE 2013

10

3 x1

ω=2, ρ=1.5, μ=0.5

2

5

x2

0 1

x(t)

x(t)

−5 0

−10 −1

−15

−2 −3

0

a 10 t

5

15

−25

20

x1

Nonimpulsive effects and ω=6

−20

0

50

100 t

(a)

x2 150

200

(a)

10 5

0 x 2 (t)

x2(t)

0 −5

−10 −20

−10 −15 10

−30 10 20

5

200

5

15

150

10

0

x1(t)

−5

5

x 1 (t)

t

0

100

0 −5

50 0

20

3 x1

ω=8, ρ=1.5, μ=0.5

x2

2

10 0 x(t)

x(t)

1 0

−10

−1

−20

x2

−30 0

5

10

15 t

20

25

0

30

50

100

150 t

200

250

300

(c)

(c)

5

20

0

0 x 2 (t)

x2(t)

x1

Nonimpulsive effects and ω=35

−2 −3

t

(b)

(b)

−5

−20 −10

−40 10

−15 10

5

30

5 x1(t)

300 0

20

0

10 −5

0

x 1 (t)

t

200 −5

100 −10

0

(d)

(d) Fig. 2. (a) State trajectory of system (11) with ω = 2, ρ = 1.5, μ = 0.5. (b) Phase portraits of system (11) with ω = 2, ρ = 1.5, μ = 0.5. (c) State trajectory of system (11) with ω = 8, ρ = 1.5, μ = 0.5. (d) Phase portraits of system (11) with ω = 8, ρ = 1.5, μ = 0.5.

Property 1: Equation (11) admits a unique periodic solution which is globally exponentially stable if there exists a constant q ∈ Z+ such that tk + ω = tk+q and max {tk+1 − tk }
0 small enough such that max {tk+1 − tk }
0 such that ⎧ ln ρ ⎪ ⎨μ < 0.4 + 0.25ρ ω ⎪ ⎩ ∈ Z+ . μ Remark 4: Corollary 4 can be obtained directly via Corollary 3, and here we omit the proof. In the simulations, one may observe that system (11) with ω = 2 or 4 has no periodic solution which is globally exponentially stable when there is no impulsive effect, which can be seen from Fig. 1(a)–(d). However, via the impulsive control strategies that we have established, (11) may admit a unique periodic solution which is globally exponentially stable. For instance, when ω = 2 or 4, one may choose ρ = 1.5 and μ = 0.5 such that the conditions in Corollary 4 hold and so the periodic solution can be guaranteed. The corresponding numerical simulations are shown in Fig. 2(a)–(d). Those simulations results match our development method in this paper perfectly. Remark 5: It should be noted that most of existing results such as [27]–[29] and [33]–[47] on periodic problem of delayed neural networks with/without impulses cannot be applied to equation (11) since it is originally unstable. But the existence-uniqueness and global stability can be effectively guaranteed via the impulsive control strategies developed in this paper. Example 2: Consider the following neural networks with distributed delays: ⎧  2π  ⎪ ⎪ x ˙ t x 1 (t) (t) = − 0.3 + 0.1 cos 1 ⎪ ⎪ ω ⎪ ⎪ 2 ⎪   ⎪ 2π ⎪ ⎪ ⎪ 0.29 − 0.1 cos (t + 1 + j ) f j (x j (t − 10)) + ⎪ ⎪ ω ⎪ ⎪ j =1 ⎪ ⎪ ⎪ 2   ⎪  ⎪ 2π ⎪ ⎪ 0.2 − 0.08 cos (t + 1 + j ) + ⎪ ⎪ ω ⎪ ⎪ ⎪  j =1 ⎪ ∞ ⎪ ⎪ ⎪ ⎪ e−s f j (x j (t − s)) · ⎪ ⎪ ⎪ 0 ⎨ 2π t, t = tk + sin ⎪  ω ⎪ ⎪ 2π  ⎪ ⎪ t x 2 (t) x ˙ (t) = − 0.3 − 0.1 sin ⎪ 2 ⎪ ω ⎪ ⎪ 2 ⎪  ⎪  2π ⎪ ⎪ ⎪ 0.1 + 0.01 sin (t + 2 + j ) f j (x j (t − 10)) + ⎪ ⎪ ω ⎪ ⎪ j =1 ⎪ ⎪ ⎪ ⎪ 2   ⎪  ⎪ 2π ⎪ ⎪ 0.2 + 0.02 sin (t + 2 + j ) + ⎪ ⎪ ω ⎪ ⎪ j =1 ⎪  ⎪ ∞ ⎪ 2π ⎪ ⎪ ⎩ t, t = tk e−s f j (x j (t − s)) + cos (10) · ω 0

x2

1

x(t)

j ∈

x1

ω=6, ρ=2, μ=0.3

1.5

0 −2 0.5

20 15

0 x1(t)

−0.5 −1

5

10 t

0

(b) 3 x1

ω=35, ρ=2, μ=0.35

2

x2

1 x(t)

= 0.4 +

i=1

2

0 −1 −2 −3

0

10

20

30

40 t

50

60

70

80

(c)

5 0 x2(t)

i∈

875

−5 −10 −15 10 5

60

x (t) 1

80 40

0 −5

20 0

t

(d) Fig. 4. (a) State trajectory of system (13) with ω = 6, ρ = 2, μ = 0.3. (b) Phase portraits of system (13) with ω = 6, ρ = 2, μ = 0.3. (c) State trajectory of system (13) with ω = 35, ρ = 2, μ = 0.35. (d) Phase portraits of system (13) with ω = 35, ρ = 2, μ = 0.35.

subject to impulses 1 x i (tk− ), k ∈ Z+ , i = 1, 2 ρ where f1 = f 2 = 0.5(|x + 1| − |x − 1|), ω > 0, and ρ > 1 are some real constants. x i (tk ) =

876

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 6, JUNE 2013

Property 2: Equation (13) admits a unique periodic solution which is globally exponentially stable if there exists a constant q ∈ Z+ such that tk + ω = tk+q and max {tk+1 − tk }
0 small enough such that ln ρ max {tk+1 − tk } < . 1 k∈Z+ −0.2 + 0.5ρe10γ + 0.5ρ 1−γ Note that τ = 10, L j = 1, ai j = 0, K = e−s , and let A = ρ; then we can compute that − min ciI + A eγ τ i∈

+A

n 

max |biSj |L j

i=1

j ∈



max |wiSj |L j

∞

K(s)eγ s ds

j ∈

0

−0.2 + 0.5ρe10γ

+ 0.5ρ

i=1

=

n 

1 . 1−γ

Let B = −0.2+0.5ρe10γ +0.5ρ1/1 − γ ; then by Theorem 1, we can obtain the above result.  Corollary 5: System (13) admits a unique periodic solution which is globally exponentially stable with tk = μk, k ∈ Z+ if there exists a constant μ > 0 such that ⎧ ln ρ ⎪ ⎨μ < ρ − 0.2 ω ⎪ ⎩ ∈ Z+ . μ Remark 6: In particular, if we take ω = 6 or 35, then Fig. 3(a)–(d) tells us that system (13) has no periodic solution which is globally exponentially stable when there is no impulsive effect. In this case, via impulsive control, one may choose ρ = 2, μ = 0.3 when ω = 2 and ρ = 2, μ = 0.35 when ω = 35 such that the conditions in Corollary 2 hold and so the existence-uniqueness and global stability of system (13) can be guaranteed, which are shown Fig. 4(a)–(d). Remark 7: In the simulations, we choose the time step h = 0.01 and initial values φ = (2m, −3m)T in Examples 1 and 2 , where m = 1, . . . , 4. V. C ONCLUSION This paper presented some analytical results on existenceuniqueness and global exponential stability of periodic solution for RNNs with discrete and continuously distributed delays. By using contraction mapping theorem and stability theory on impulsive functional differential equations, some sufficient conditions ensuring the existence-uniqueness and global exponential stability of periodic solution for neural networks have been presented. Our method shows that network models may admit a unique solution which is globally stable via proper impulsive control strategies even if it is originally unstable or divergent. Two numerical examples and their computer simulations have been given to show the effectiveness of our proposed method. In addition, the ideas employed in this paper can be developed to study other delayed systems.

R EFERENCES [1] J. Hopfield, “Neurons with graded response have collective computational properties like those of two-stage neurons,” in Proc. Nat. Academy Sci. United States Amer., vol. 81, no. 10, pp. 3088–3092, May. 1984. [2] L. Chua and L. Yang, “Cellular networks: Theory,” IEEE Trans. Circuits Syst., vol. 35, no. 10, pp. 1257–1272, Oct. 1988. [3] L. Chua and L. Yang, “Cellular neural networks: Applications,” IEEE Trans. Circuits Syst., vol. 35, no. 10, pp. 1273–1290, Oct. 1988. [4] M. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,” IEEE Trans. Syst., Man, Cybern., vol. 13, no. 5, pp. 815–826, Feb. 1983. [5] B. Kosko, “Bi-directional associative memories,” IEEE Trans. Syst., Man, Cybern., vol. 18, no. 1, pp. 49–60, Jan.–Feb. 1988. [6] S. Arik and Z. Orman, “Global stability analysis of Cohen-Grossberg neural networks with time varying delays,” Phys. Lett. A, vol. 341, nos. 5–6, pp. 410–421, Jun. 2005. [7] J. Cao, “A set of stability criteria for delayed cellular neural networks,” IEEE Trans. Circuits Syst. I, vol. 48, no. 4, pp. 494–498, Apr. 2001. [8] Z. Zeng and J. Wang, “Global exponential stability of recurrent neural networks with time-varying delays in the presence of strong external stimuli,” Neural Netw., vol. 19, no. 10, pp. 1528–1537, Dec. 2006. [9] K. Gopalsamy and X. He, “Delay-independent stability in bi-directional associative memory networks,” IEEE Trans. Neural Netw., vol. 5, no. 6, pp. 998–1002, Nov. 1994. [10] J. Hopfield and D. Tank, “Computing with neural circuits: A model,” Science, vol. 233, no. 4764, pp. 625–633, Aug. 1986. [11] M. Hirsch, “Convergent activation dynamics in continuous time networks,” Neural Netw., vol. 2, no. 5, pp. 331–349, Mar. 1989. [12] P. Baldi and A. Atiya, “How delays affect neural dynamics and learning,” IEEE Trans. Neural Netw., vol. 5, no. 4, pp. 612–621, Jul. 1994. [13] S. Arik and V. Tavsanoglu, “Equilibrium analysis of delayed CNNs,” IEEE Trans. Circuits Syst. I, vol. 45, no. 2, pp. 168–171, Feb. 1998. [14] P. Driessche and X. Zou, “Global attractivity in delayed hopfield neural networks model,” SIAM J. Appl. Math., vol. 58, no. 6, pp. 1878–1890, Dec. 1998. [15] Y. Xia, J. Cao, and S. Cheng, “Global exponential stability of delayed cellular neural networks with impulses, Neurocomputing, vol. 70, nos. 13–15, pp. 2495–2501, Aug. 2007. [16] Q. Song and J. Cao, “Stability analysis of Cohen-Grossberg neural network with both time-varying and continuously distributed delays,” J. Comput. Appl. Math., vol. 197, no. 1, pp. 188–203, Dec. 2006. [17] Q. Song, “Exponential stability of recurrent neural networks with both time-varying delays and general activation functions via LMI approach,” Neurocomputing, vol. 71, nos. 13–15, pp. 2823–2830, Aug. 2008. [18] Q. Zhu, C. Huang, and X. Yang, “Exponential stability for stochastic jumping BAM neural networks with time-varying and distributed delays,” Nonlinear Anal., Hybrid Syst., vol. 5, no. 1, pp. 52–77, Feb. 2011. [19] Q. Zhu and J. Cao, “Stability analysis of markovian jump stochastic BAM Neural networks with impulse control and mixed time delays,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 3, pp. 467–479, Mar. 2012. [20] R. Rakkiyappan and P. Balasubramaniam, “On exponential stability results for fuzzy impulsive neural networks,” Fuzzy Sets Syst., vol. 161, no. 13, pp. 1823–1835, Jul. 2010. [21] M. Arbib, Branins, Machines, and Mathematics. New York: SpringerVerlag, 1987. [22] B. Chen and J. Wang, “Global exponential periodicity and global exponential stability of a class of recurrent neural networks with various activation functions,” Neural Netw., vol. 20, no. 10, pp. 1067–1080, Dec. 2007. [23] T. Nishikawa, Y. Lai, and F. Hoppensteadt, “Capacity of oscillatory associative-memory networks with error-free retrieval,” Phys. Rev. Lett., vol. 92, no. 10, pp. 108101-1–108101-4, Mar. 2004. [24] D. Wang, “Emergent synchrony in locally coupled neural oscillators,” IEEE Trans. Neural Netw., vol. 6, no. 4, pp. 941–948, Jul. 1995. [25] A. Ruiz, D. Owens, and S. Townley, “Existence, learning, and replication of periodic motion in recurrent neural networks,” IEEE Trans. Neural Netw., vol. 9, no. 4, pp. 651–661, Jul. 1998. [26] H. Jin and M. Zacksenhouse, “Oscillatory neural networks for robotic yo-yo control,” IEEE Trans. Neural Netw., vol. 14, no. 2, pp. 317–325, Mar. 2003. [27] J. Cao and L. Wang, “Periodic oscillatory solution of bidirectional associative memory networks with delays,” Phys. Rev. E, vol. 61, no. 2, pp. 1825–1828, Feb. 2000.

LI AND SONG: RECURRENT NEURAL NETWORKS WITH DISCRETE AND CONTINUOUSLY DISTRIBUTED DELAYS

[28] Y. Xia, J. Cao, and M. Lin, “New results on the existence and uniqueness of almost periodic solution for BAM neural networks with continuously distributed delays, Chaos, Solitons Fractals, vol. 31, no. 4, pp. 928–936, Feb. 2007. [29] J. Zhou, Z. Liu, and G. Chen, “Dynamics of periodic delayed neural networks,” Neural Netw., vol. 17, no. 1, pp. 87–101, Jan. 2004. [30] V. Lakshmikantham, D. Bainov, and P. Simeonov, Theory of Impulsive Differential Equations. Singapore: World Scientific, 1989. [31] K. Gopalsamy, “Stability of artificial neural networks with impulses,” Appl. Math. Comput., vol. 154, no. 3, pp. 783–813, Jul. 2004. [32] X. Fu, B. Yan, and Y. Liu, Introduction of Impulsive Differential Systems. Beijing, China: Science Press, 2005. [33] X. Yang, X. Liao, D. Evans, and Y. Tang, “Existence and stability of periodic solution in impulsive Hopfield neural networks with finite distributed delays,” Phys. Lett. A, vol. 343, nos. 1–3, pp. 108–116, Aug. 2005. [34] Z. Yang and D. Xu, “Existence and exponential stability of periodic solution for impulsive delay differential equations and applications,” Nonlinear Anal., vol. 64, no. 1, pp. 130–145, Jan. 2006. [35] J. Lu, J. Kurths, J. Cao, N. Mahdavi, and C. Huang, “Synchronization control for nonlinear stochastic dynamical networks: Pinning impulsive strategy,” IEEE Trans. Neural Netw. Learn. Syst., vol. 23, no. 2, pp. 285–292, Feb. 2012. [36] X. Yang, X. Cui, and Y. Long, “Existence and global exponential stability of periodic solution of a cellular neural networks difference equation with delays and impulses, “ Neural Netw., vol. 22, no. 7, pp. 970–976, Sep. 2009. [37] X. Li and J. Shen, “LMI approach for stationary oscillation of interval neural networks with discrete and distributed time varying delays under impulsive perturbations,” IEEE Trans. Neural Netw., vol. 21, no. 10, pp. 1555–1563, Oct. 2010. [38] W. Allegretto, D. Papini, and M. Forti, “Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks,” IEEE Trans. Neural Netw., vol. 21, no. 7, pp. 1110–1125, Sep. 2010. [39] J. Sun, “Stationary oscillation for chaotic shunting inhibitory cellular neural networks with impulses,” Chaos, Interdiscipl. J. Nonlinear Sci., vol. 17, no. 4, pp. 043123-1–043123-5, Dec. 2007. [40] Y. Li, L. Zhao, and T. Zhang, “Global exponential stability and existence of periodic solution of impulsive Cohen-Grossberg neural networks with distributed delays on time scales,” Neural Process. Lett., vol. 33, no. 1, pp. 61–81, 2011. [41] E. Kaslik and S. Sivasundaram, “Multiple periodic solutions in impulsive hybrid neural networks with delays,” Appl. Math. Comput., vol. 217, no. 10, pp. 4890–4899, Jan. 2011. [42] L. Zhao and P. Liu, “Existence and exponential stability of periodic solutions of impulsive high order Hopfield neural network both with discrete and distributed delays on time scales,” Int. J. Pure Appl. Math. vol. 61, no. 1, pp. 95–120, 2010. [43] J. Sun and H. Lin, “Stationary oscillation of an impulsive delayed system and its application to chaotic neural networks,” Chaos, Interdiscipl. J. Nonlinear Sci., vol. 18, no. 3, pp. 033127-1–033127-, Jan. 2008. [44] W. Allegretto, D. Papini, and M. Forti, “Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks,” IEEE Trans. Neural Netw., vol. 21, no. 7, pp. 1110–1125, Jul. 2010. [45] Y. Zhang and Q. Wang, “Stationary oscillation for high-order Hopfield neural networks with time delays and impulses,” J. Comput. Appl. Math., vol. 231, no. 1, pp. 473–477, Sep. 2009. [46] Y. Zhang, “Stationary oscillation for nonautonomous bidirectional associative memory neural networks with impulse,” Chaos, Solitons Fractals, vol. 41, no. 4, pp. 1760–1763, Aug. 2009.

877

[47] Y. Zhang, “Stationary oscillation for cellular neural networks with time delays and impulses,” Math. Comput. Simulat., vol. 79, no. 10, pp. 3174–3178, Jun. 2009. [48] Z. Yang and D. Xu, “Impulsive effects on stability of Cohen-Grossberg neural networks with variable delays,” Appl. Math. Comput., vol. 177, no.1, pp. 63–78, Jun. 2006. [49] X. Li, “Global exponential stability of Cohen-Grossberg-type BAM neural networks with time-varying delays via impulsive control,” Neurocomputing, vol. 73, nos.1–3, pp. 525–530, Dec. 2009. [50] Q. Wang and X. Liu, “Exponential stability of impulsive cellular neural networks with time delay via Lyapunov functionals,” Appl. Math. Comput., vol. 194, no. 1, pp. 186–198, Dec. 2007. [51] C. Li, L. Chen, and K. Aihara, “ Impulsive control of stochastic systems with applications in chaos control, chaos synchronization, and neural networks,” Chaos , vol. 18, no. 2, pp. 023132-1–023132-11, Jun. 2008. [52] Y. Zhang and J. Sun, “Robust synchronization of coupled delayed neural networks under general impulsive control,” Chaos Solitons Fractals, vol. 41, no. 3, pp. 1476–1480, Aug. 2009. [53] X. Yang, J. Cao, and J. Lu, “Stochastic synchronization of complex networks with nonidentical nodes via hybrid adaptive and impulsive control,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 59, no. 2, pp. 371–384, Feb. 2012. [54] X. Yang, C. Huang, and Q. Zhu, “Synchronization of switched neural networks with mixed delays via impulsive control,” Chaos Solitons Fractals, vol. 44, no. 10, pp. 817–826, Oct. 2011. [55] X. Li, “New results on global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays,” Nonlinear Anal., Real World Appl., vol. 11, no. 5, pp. 4194–4201, Oct. 2010.

Xiaodi Li was born in Shandong province, China, in 1982. He received the B.S. and M.S. degrees from Shandong Normal University, Jinan, China, in 2005 and 2008, respectively, and the Ph.D. degree from Xiamen University, Xiamen, China, in 2011, all in applied mathematics. He is currently a Professor with the Department of Mathematics, Shandong Normal University. He has authored or co-authored more than 30 research papers. His current research interests include stability theory, artificial neural networks, impulsive functional differential systems, stochastic functional differential systems, and applied mathematics.

Shiji Song was born in 1965. He received the Ph.D. degree from the Department of Mathematics, Harbin Institute of Technology, Harbin, China, in 1996. He is a Professor with the Department of Automation, Tsinghua University, Beijing, China. His current research and teaching interests include system identification, stochastic optimization, stochastic neural networks, and fuzzy systems.