Novel Existence and Stability Criteria of Periodic Solutions for Impulsive Delayed Neural Networks Via Coefficient Integral Averages

Author’s Accepted Manuscript Novel Existence and Stability Criteria of Periodic Solutions for Impulsive Delayed Neural Networks Via Coefficient Integral Averages

Huamin Wang, Shukai Duan, Tingwen Huang, Chuandong Li, Lidan Wang www.elsevier.com/locate/neucom

PII: DOI: Reference:

S0925-2312(16)30867-0 http://dx.doi.org/10.1016/j.neucom.2016.08.027 NEUCOM17438

To appear in: Neurocomputing Received date: 13 December 2015 Revised date: 11 July 2016 Accepted date: 6 August 2016 Cite this article as: Huamin Wang, Shukai Duan, Tingwen Huang, Chuandong Li and Lidan Wang, Novel Existence and Stability Criteria of Periodic Solutions for Impulsive Delayed Neural Networks Via Coefficient Integral Averages Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.08.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Novel Existence and Stability Criteria of Periodic Solutions for Impulsive Delayed Neural Networks Via Coefficient Integral Averages Huamin Wanga,b Shukai Duana,∗, Tingwen Huangc , Chuandong Lia , Lidan Wanga a

College of Electronic and Information Engineering, Southwest University, Chongqing 400715, PR China b

Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471022, PR China c

Department of Science, Texas A&M University at Qatar, Doha 23874, Qatar

Abstract. In this paper, an impulsive neural network with periodic coefficients and delayed time is firstly given, then the existence, uniqueness and exponential stability problems of periodic solutions for this system are further investigated.

By means of the characteristic

equation of delayed differential equations and impulsive differential equations, an impulsive delayed differential inequality with novel conditions are constructed. It can be used as an impulsive comparison system to deal with the existence and exponential stability problems. By utilizing the impulsive comparison system, Lyapunov functions and the fixed point theorem, we obtain the periodic solution’s existence and exponential stability criteria. Finally, one numerical example and its simulations are given to illustrate the effectiveness of the theoretical results. Key Words and Phrases:

Delayed neural networks, impulsive delayed differential in-

equality, periodic solution, exponential stability, integral average of periodic function.

1

Introduction Since neural networks(NNs) have huge potential applications in different areas, various neural networks,

including Hopfield NNs (HNNs)[1], Cohen-Grossberg NNs (CGNNs)[2], bidirectional associative memory (BAM) NNs (BAMNNs)[3], cellular NNs (CNNs)[4] and memristive NNs(MNNs) [5] etc., have been deeply investigated in the past thirty years. Many scholars attentions were attracted on the investigation of NNs and lots of conclusions were derived [6]-[20] by use different ways. For instance, the authors in [6]-[8] discussed the stable properties of HNNs, the properties of CGNNs were given in [9, 11], and BAMNNs were studied in [10, 12]. In [15]-[17], synchronization and stability problems of MNNs were investigated by means of different methods. In [17]-[20], the authors researched the dynamical properties of complex-valued NNs and complex dynamic networks, respectively. ∗ Corresponding

author, E-mail: [email protected]

1

As is known to all, NNs can be implemented by VLSI circuits. Therefore, their design of NNs may be affected by many actual factors. The dynamical properties are very important for the design of NNs, so it is necessary to further investigate the dynamical properties of NNs. Among all the dynamical behaviors of NNs, synchronization [21]-[24] and stability [25]-[29] are evergreen hot topics for the NN systems. For example, in [22, 24], synchronization of coupled delayed NNs (DNNs) was studied by pinning control technique. The stable problems of NNs were researched by the aid of impulsive control and intermittent control in [25, 27, 28]. Particularly, stability problem is the most fundamental dynamical property of DNNs. In order to control systems to achieve a certain state, some control strategies are usually chosen to stabilize or synchronize these systems. Among all the control strategies, impulsive control, switch control, pinning control and intermittent control are several commonly used control strategies [22, 27, 28], [30]-[32]. In these control strategies, because impulsive control strategy has some special advantages, it usually can be used as one of the most common strategy[31, 33, 34]. In the past decades, impulsive control strategy have been used to investigated the stability of solution for the DNNs by many researchers and numerous stability criteria have been established in the published literatures [8, 13, 28, 29, 35]. Since the periodic oscillations can be presented by NN models with periodic coefficients, the periodic solution’s properties are very important for us to further study the dynamical behaviors of NNs. According the model of NNs, some important conclusions related to periodic coefficient NNs have been derived by many scholars including [7],[9]-[12],[14, 25, 28, 36]. For instance, in [14], the authors firstly given the existence conditions of CNNs without delays and impulsive effects, then the stability conditions of existed periodic solutions were obtained by means of proper LyapunovCKrasovskii functional. The authors in [7, 11] studied the periodic solution’s properties of impulsive NNs without delays by Lyapunov function. In [9, 12, 36], the sufficient conditions of periodic solution’s existence and globally exponentially stability for the DNNs without impulses were established. In reality, the transient of NNs could be simultaneously affected by the abrupt changes and delays. Therefore, it is necessary to research the stable problems of NNs with delayed time and impulsive effects. In recent years, some important dynamical behaviors of periodic solution for the impulsive DNNs with periodic coefficients have been deeply investigated and some criteria have been obtained [25, 28, 35],[37]-[39]. For example, in [25, 28, 35], by using inequality technique and Lyapunov function, the authors firstly studied the existence of periodic solution, then they discussed the exponential stability of impulsive delayed CGNNs with periodic coefficients. In [39], the periodicity and stability of CNNs with delays and impulses were researched. Although there have been many interesting results on the properties of periodic solutions of NN models with periodic coefficients, some novel and nonconservative conditions can be studied for these systems. Recently, according to the properties of periodic functions, a novel form of periodic function named as integral average was presented by Lisena in [8, 11]. Then, the author used the integral averages of periodic coefficient and other theories to research the periodic solution’s properties. Recently, Lisena studied the existence conditions and exponential stability criteria of HNNs and CGNNs without delays by utilizing some operation skills, respectively. In [40], the novel average form of periodic coefficients was given and used to research the asymptotic properties of differential inequalities with delays and periodic coefficient, by which the stability of periodic coefficient DNNs without impulses was studied and some interesting conclusions were obtained in [36]. Integral average of periodic function is an interesting concept for the

2

dynamics systems with periodic coefficient, so it is worthy discussing some deep questions about this definition in the future. To the authors’ knowledge, the impulsive DNNs with periodic coefficients have not been discussed by this integral averages of periodic coefficients at present. Based on the above analysis, we will discuss periodic solution’s existence and stability of DNNs with impulses and periodic coefficients by using novel integral average form of periodic function in this paper. According to the characteristic equation of delayed differential equations [41] and delayed differential inequality, a novel impulsive differential inequalities on the integral average form of periodic function are constructed, which can be seen as an impulsive comparison system. Then, by means of the new impulsive comparison system, Lyapunov functions and the fixed point theorem, we will firstly study the existence conditions of periodic solutions of impulsive DNNs with periodic coefficients. Then, we will research the global exponential stable problems of this system and we will proof several average form stability criteria for this system. Finally, one numerical example and its simulation are provided to illustrate the theoretical results.

2

Preliminaries The notations, definitions and lemmas used in this paper will be given as follows. Suppose R denotes

the real number set, R+ denotes the nonnegative real number set, N denotes the positive integer number set and Rn denotes the space of n-dimensional column vectors x = col(x1 , x2 , · · · , xn ) with the vector norm ·, φτ = sup−τ ≤s≤0 φ(t0 + s) (τ, t0 are nonnegative number, respectively). I is the n order identity matrix. P C([a, b], S) = {φ : [a, b] → S|φ(t) = φ(t+ ), ∀t ∈ [a, b]; φ(t− ) exists for ∀t ∈ [a, b], and φ(t− ) = φ(t) holds for all but at most a finite number of points t ∈ [a, b]}. In this paper, the following impulsive DNN model will be considered: ⎧ n n ⎪ ⎪ ⎨ x˙ i (t) = −di (t)xi (t) + j=1 aij (t)fj (xj (t)) + j=1 bij (t)gj (xj (t − τ )) + Ii (t), −

−

 xi (t)  xi (t) − xi (t ) = Cik xi (t ), ⎪ ⎪ ⎩ x (t + s) = ϕ (t + s), i 0 i 0

t ≥ t0 , t = tk , t = tk , k ∈ N + ,

(2.1)

s ∈ [−τ, 0],

where n denotes the number of units, xi (t) denotes the state variable of the ith unit at time t, di (t) > 0 is the ith unit self-regulates rate, τ > 0 is the delay, aij (t) and bij (t) denote the connective weights of the interconnection neurons. fj (·) and gj (·) are the bounded feedback functions, which satisfy fj (0) = gj (0) = 0. Ii (t) denotes bias, Cik > 0 is the impulsive perturbation at time tk , x(tk ) = x(t+ k ) = limh→0+ x(tk + h), x(t− k ) = limh→0− x(tk + h), k = 1, 2, · · · , and 0 < t1 < t2 < · · · < tk < · · · (tk → ∞ as k → ∞),

ϕ(t0 + s) = (ϕ1 (t0 + s), ϕ2 (t0 + s), . . . , ϕn (t0 + s))T ∈ P C([−τ, 0], Rn ). Definition 2.1 [37] Suppose that the impulsive DNNs (2.1) is T -periodic, and x(t, t0 , ϕ) is a solution of the impulsive delayed NNs (2.1) through (t0 , ϕ). Then, the NNs (2.1) has a T -periodic solution if and only if there exists a ϕ ∈ P C, such that xt0 +T (t0 , ϕ) = ϕ(t0 ). Definition 2.2 [28] Let x∗ (t, t0 , ϕ∗ ) be a T -periodic solution of the impulsive DNNs (2.1) with initial value ϕ∗ ∈ P C, x∗ (t, t0 , ϕ∗ ) is said to be globally exponentially stable if there exists α > 0 and M > 0 such that for any solution x(t, t0 , ϕ) with initial value x(t0 +s) = ϕ(t0 +s) ∈ P C satisfies x(t, t0 , ϕ)−x∗ (t, t0 , ϕ∗ ) < M ϕ − ϕ∗ τ e−α(t−t0 ) , t ≥ t0 . 3

We firstly consider the following delayed differential equation: u(t) ˙ = −A(t)u(t) + B(t)u(t − τ ),

(2.2)

where A(t) > 0 and B(t) ≥ 0 are T -periodic continuous functions and their integral average form are  t +T  t +T denoted by m[A] = T1 t00 A(s)ds and m[B] = T1 t00 B(s)ds, respectively. Lemma 2.1 [40, 41] Let m[A] > m[B], τ = hT, (h ∈ N ), and λ(t) = A(t) − B(t)eτ z , z is the unique solution of z = m[A] − m[B]eτ z . Then, we obtain that u(t) = φ(t0 )e

−

t

t0

λ(s)ds

t ≥ t0 ,

,

is a solution of equation (2.2) with initial values φ(σ) = φ(t0 )e

−

σ

t0

λ(s)ds

, σ ∈ [t0 − τ, t0 ], where λ(t) is a

continuous T -periodic function. The following impulsive differential equation will be considered: ⎧ ⎪ ˙ = −A(t)u(t), t = tk , t ≥ t0 , ⎪ u(t) ⎨ − u(t+ k ) = (1 + Ck )u(tk ), k ∈ N+ , ⎪ ⎪ ⎩ u(t ) = φ(t ), , 0 0

(2.3)

− where u(t+ k ) = u(tk ), u(tk ) exists and 1 + Ck > 1.

Lemma 2.2 Assume u(t) ∈ P C([t0 , ∞], R), we can conclude that 

 − t A(s)ds u(t) = φ(t0 ) (1 + Ck ) e t0

(2.4)

t0
is a solution of (2.3). Proof. Obviously, u(t) = φ(t0 )e

−

t

t0

A(s)ds

holds for any t ∈ [t0 , t1 ). When t = t1 , we have

− − u(t+ 1 ) = (1 + C1 )u(t1 ) = (1 + C1 )φ(t0 )e

 t− 1 t0

A(s)ds

= φ(t0 )(1 + C1 )e−

 t1 t0

A(s)ds

.

Similarly, when t ∈ [t1 , t2 ), we have u(t) = u(t1 )e

−

t

A(s)ds

= φ(t0 )(1 + C1 )e

−

t

A(s)ds

. 

  − tt A(s)ds 0 holds for any t ∈ [t0 , t2 ). Therefore, it can be derived that u(t) = φ(t0 ) t0
t0

Then, by mathematical induction, we can conclude that equality (2.4) holds for any t ∈ [t0 , ∞).2 Lemma 2.3 [40] Under the conditions of Lemma 2.1, let u(t) be a positive solution of D+ u(t) ≤ −A(t)u(t)+ B(t)u(t − τ ) with initial values u(t0 + σ) = φ(t0 + σ) > 0 in [−τ, 0], then we have u(t) ≤ φ(t0 )τ e

−

t

t0

λ(s)ds

where φ(t0 )τ = sup−τ ≤σ≤0 φ(t0 + σ).

4

,

t ≥ t0 ,

Lemma 2.4 Under the conditions of Lemma 2.1, let initial function be positive continuous, 1 + Ck ≥ 1, u(t), φ(t) ∈ P C([t0 − τ, ∞], R), and u(t) be a positive solution of the delayed impulsive differential inequalities:

⎧ + ⎪ ⎪ ⎨ D u(t) ≤ −A(t)u(t) + B(t)u(t − τ ), − u(t+ k ) ≤ (1 + Ck )u(tk ), ⎪ ⎪ ⎩ u(t + σ) = φ(t + σ), 0

0

Then, we have

 u(t) ≤



t = tk , t ≥ t0 , k ∈ N+ , σ ∈ [−τ, 0].

 − t λ(s)ds (1 + Ck ) φ(t0 )τ e t0 .

t0
Proof. Please see the part of Appendix.2 Remark 2.1 In the past decades, impulsive delayed differential inequality have been deeply investigated and applied to research the dynamical behaviors of dynamic systems[28], [33], [40], etc.. For example, the authors investigated the impulsive delayed differential inequality with periodic coefficients by M -matrix in [28]. In [40], the author studied the asymptotic properties for a delayed impulsive differential inequality by the special T -periodic continuous function γ(t) and linear impulsive equation. However, some types of impulsive delayed differential inequality have not been fully investigated. For this lemma, a novel conclusion for the delayed impulsive differential inequality is obtained by using the integral average form of periodic coefficients, which could be used as an impulsive comparison system to research the stability and synchronization problems of delayed impulsive NNs with periodic coefficients. In order to obtain the main conclusions of this article in the next part, we give the following several assumptions: (A1) The activation functions fj (·) and gj (·) are Lipschitz continuous, that is, there exist real numbers Lj > 0, Mj > 0, such that for any u, v ∈ R, j = 1, 2, · · · , n |fj (u) − fj (v)| ≤ Lj |u − v|,

|gj (u) − gj (v)| ≤ Mj |u − v|.

(A2) The expressions di (t), aij (t), bij (t) and Ii (t) of impulsive delayed system (2.1) are T -periodic continuous functions, i.e. di (t + T ) = di (t),

aij (t + T ) = aij (t),

bij (t + T ) = bij (t),

Ii (t + T ) = Ii (t)..

(A3) There exists q ∈ N such that {t1 , t2 , · · · , tq } ⊂ (0, T ],

tk+q = tk + T,

Ci(k+q) = Cik .

(A4) The norm  ·  in Rn is defined by x(t) =

n

|xi (t)|,

i=1

where x(t) = (x1 (t), x2 (t), · · · , xn (t))T (A5) For any t ≥ t0 , the n × n symmetric matrix Δ(t) = (˜ aij (t))n×n is defined by 

−di (t) + Li |aii (t)| + 12 nh=1 Mh |bih (t)|, i = j, a ˜ij (t) =   1 i = j, 2 Lj |aij (t)| + Li |aji (t)| , 5

3

Main Results

Theorem 3.1 Suppose that assumptions (A1)-(A4) are satisfied, and (i) m[A] > m[B] and τ = hT, (h ∈ N ), where 

n |aji (t)| , A(t) = min di (t) − Li 1≤i≤n

(ii) m[λ(t)] −

1 T

q

k=1

B(t) = max

1≤i≤n

j=1



n Mi |bji (t)| ; j=1

log Jk > 0, where Jk = max1≤i≤n (1 + Cik ) ≥ 1, λ(t) is defined as in Lemma 2.1.

Then, the impulsive DNN (2.1) has an unique globally exponentially stable T -periodic solution. Proof. Suppose that x(t) = (x1 (t), x2 (t), · · · , xn (t))T , y(t) = (y1 (t), y2 (t), · · · , yn (t))T are two solutions of the delayed impulsive NNs (2.1) with initial conditions ϕ(t0 +s) = (ϕ1 (t0 +s), ϕ2 (t0 +s), · · · , ϕn (t0 +s))T and ψ(t0 + s) = (ψ1 (t0 + s), ψ2 (t0 + s), · · · , ψn (t0 + s))T (s ∈ [−τ, 0]), respectively. Construct a Lyapunov function V (t) =

n

|xi (t) − yi (t)|.

i=1

For convenience, let zi (t) = xi (t) − yi (t). When t = tk , the Dini derivative of Lyapunov V is D+ V (t) = =

n i=1 n

(x˙ i − y˙ i )sign(xi − yi )

n sign(xi − yi ) − di (t)(xi (t) − yi (t)) + aij (t)(fj (xj (t)) − fj (yj (t)))

i=1

+

j=1

n

 bij (t)(gj (xj (t − τ )) − gj (yj (t − τ )))

j=1

≤

−

n

n  n

di (t)|zi (t)| +

i=1

i=1

|aij (t)|Lj |zj (t)| +

j=1

n  n i=1

|bij (t)|Mj |zj (t − τ )| .

j=1

Exchanging indices of i and j, we have



n  n  n n D+ V (t) ≤ − di (t) − Li Mi |aji (t)| |zi (t)| + |bji (t)| |zi (t − τ )| i=1

≤

−A(t)

j=1 n

|zi (t)| + B(t)

i=1

i=1 n

j=1

|zi (t − τ )| = −A(t)V (t) + B(t)V (t − τ ).

i=1

When t = tk , V (t+ k) =

n

+ |xi (t+ k ) − yi (tk )| =

i=1

≤

Jk

n

− |1 + Cik ||xi (t− k ) − yi (tk )|

i=1

n

− − |xi (t− k ) − yi (tk | = Jk V (tk ).

i=1

By Lemma 2.4, we can get that V (t) ≤ (



Jk )ϕ − ψτ e

t0
6

−

t

t0

λ(s)ds

.

That is x(t) − y(t) =

n

(

Jk )e

−

Jk )ϕ − ψτ e

t

t0

λ(s)ds

= exp



t0
 log Jk −

t0
It is known that



−

t

λ(s)ds

t0

.

t0
i=1





|xi (t) − yi (t)| ≤ (

t

t0



t

t0

λ(s)ds

˜ λ(s)ds = m[λ(t)](t − t0 ) + λ(t),

˜ where λ(t) is a suitable periodic function. Let q(t) =



log Jk −

t0
q t − t0 log Jk , T k=1

By assumption (A3) {t1 , t2 , · · · , tq } ⊂ (0, T ],

tk+q = tk + T,

Ci(k+q) = Cik

and Jk = max1≤i≤n (1 + Cik ), it can be easily checked that Jk+q = Jk and

q(t + T ) =

log Jk −

t0
=



log Jk +

t0
q t + T − t0 log Jk T k=1



log Jk −

t
q q t − t0 log Jk − log Jk = q(t), T k=1

k=1

which implies that q(t) is T -periodic function. As a consequence, ˜ exp x(t) − y(t) ≤ ϕ − ψτ exp(q(t) − λ(t))



 q 1 log Jk − m[λ(t)] (t − t0 ) , T k=1

which can be represented by x(t) − y(t) ≤ ϕ − ψτ Q(t) exp(−μ(t − t0 )), ˜ where Q(t) = exp(q(t) − λ(t)) and μ = m[λ(t)] −

1 T

q

k=1

(3.1)

log Jk .

In the next, by Banach’s fixed point theorem, we can derive the existence of periodic solution. Introduce the Poincar´ e mapping P : P Ct0 → P Ct0 +T by P(ϕ) = xt0 +T (t0 , ϕ), ϕ ∈ P Ct0 , where xt (t0 , ϕ) is the solution of (2.1) through (t0 , ϕ). According to Lemma 2.3 of [37] and the periodicity of {tk , k ∈ Z}, P Ct0 = P Ct0 +T , which means that P maps P Ct0 into itself. Furthermore, by the solution’s existence and uniqueness of the impulsive DNNs (2.1), we can get Pk (ϕ) = xt0 +kT (t0 , ϕ),

k ∈ N.

From (3.1), there exist constants m ∈ N , K > 1 and 0 < ε < 1 such that for all t ∈ [t0 , ∞), |Q(t)| ≤ K and Pm (ϕ) − Pm (ψ) = xt0 +mT (t0 , ϕ) − xt0 +mT (t0 , ψ) ≤ εϕ − ψτ , 7

which implies that Pm is a contraction mapping in Banach space P Ct0 . Consequently, according to Banach’s fixed point theorem, Pm has a fixed point ϕ∗ ∈ P Ct0 such that Pm (ϕ∗ ) = ϕ∗ . Since Pm (P(ϕ∗ )) = P(Pm (ϕ∗ )) = P(ϕ∗ ), P(ϕ∗ ) is also a fixed point of Pm . According to the uniqueness of fixed point, we can obtain P(ϕ∗ ) = ϕ∗ . Denoting by x∗ (t, t0 , ϕ) as the solution of impulsive DNNs (2.1) through (t0 , ϕ∗ ), that is, x∗ (t) = x∗ (t, t0 , ϕ∗ ) is the unique T -periodic solution of impulsive DNN (2.1). From (3.1), it can be obtained that x(t) − x∗ (t) ≤ Kϕ − ϕ∗ τ exp(−μ(t − t0 )),

(3.2)

which means that the T -periodic solution of (2.1) is globally exponentially stable.2 Remark 3.1 In this theorem, based on the integral average form of periodic coefficients, the periodic solution’s existence conditions and exponential stability criteria for the DNNs with impulsive effects and periodic coefficients are obtained by using of Lyapunov functions, impulsive comparison system and Banach’s fixed point theorem. Although the results of [36] had been obtained basing on the integral average form of periodic coefficients, the impulsive perturbance was not considered. In the impulsive DNNs (2.1), if let Cik = 0, i.e. there is no impulsive perturbance, the related DNNs without impulses has an unique globally exponentially stable T -periodic solution. Therefore, the condition (i) of this theorem can ensure the T -periodic solution’s existence and exponential stability of the initial continuous DNN system. When the system is affected by impulses, the condition (ii) is given to maintain the exponential stability of the original stable system. Based on this analysis, let Cik = 0 and B(t) = 0, respectively, we can derive the following two corollaries. Corollary 3.1 [36] Suppose (A1)-(A4) are satisfied, and assume that m[A] > m[B] and τ = hT, (h ∈ N ), where A(t) = min

1≤i≤n



n di (t) − Li |aji (t)| ,

B(t) = max

j=1

1≤i≤n



n Mi |bji (t)| . j=1

Then, the DNN system (2.1) without impulsive effects has an unique globally exponentially stable T -periodic solution. Theorem 3.2 Suppose that assumptions (A1)-(A5) are satisfied, and 

n (i) m[A] > m[B], where A(t) = −2Λ(t), B(t) = max1≤i≤n Mi j=1 |bji (t)| and Λ(t) is the greatest eigenvalue of matrix Δ(t) for a fixed value t ≥ t0 . (ii)

1 1 2 m[λ(t)] − 2T

q

k=1

log Jk > 0, where Jk = max1≤i≤n (1 + Cik )2 and λ(t) is defined as in Lemma 2.1.

Then, the impulsive DNN system (2.1) has an unique globally exponentially stable T -periodic solution. Proof. Let V (t) =

n

i=1

zi2 (t) 2 ,

where zi (t) is defined as in Theorem 3.1. When t = tk , the Dini derivative of V

8

is D+ V (t) =

−

≤

−

n i=1 n

i=1 n 

≤

di (t)zi2 (t) + di (t)zi2 (t) +

n i=1 n

zi (t)

n

aij (t)f˜j (zj (t)) +

j=1

 n

|zi (t)|

i=1



zi (t)

i=1

n

bij (t)˜ gj (zj (t − τ ))

j=1



n n |aij (t)|Lj |zj (t)| + |zi (t)| |bij (t)|Mj |zj (t − τ )|

j=1

i=1 n

− di (t) + Li |aii (t)| zi2 (t) +

j=1

Lj |aij (t)||zi (t)||zj (t)|

i,j=1;i=j

i=1

+

n

n   1 Mj |bij (t)| zi2 (t) + zj2 (t − τ ) , 2 i,j=1

where f˜j (zj (t)) = fj (xj (t)) − fj (yj (t)), g˜j (zj (t − τ )) = gj (xj (t − τ )) − gj (yj (t − τ )). According to assumption (A5), we have +

D V (t) ≤ ≤

n n  1 Mi a ˜ij (t)|zi (t)||zj (t)| + |bji (t)| zi2 (t − τ ) 2 i,j=1 i=1 j=1 n

−A(t)V (t) + B(t)V (t − τ ).

Since Δ(t) = (˜ aij (t))n×n is n × n symmetric matrix, the following inequality can be obtained: n

a ˜ij (t)|zi (t)||zj (t)| ≤ Λ(t)

i,j=1

n

zi2 (t),

i=1

where Λ(t) is the greatest eigenvalue of matrix Δ(t) for a fixed value t ≥ t0 . Therefore, D+ V (t) ≤ Λ(t)

n

zi2 (t) +

i=1

n n  1 Mi |bji (t)| zi2 (t − τ ) ≤ −A(t)V (t) + B(t)V (t − τ ). 2 i=1 j=1

When t = tk , V (t+ k)=

1 − (1 + Cik )2 zi2 (t− k ) ≤ Jk V (tk ). 2 i=1 n

Then, by means of Lemma 2.4, we can get that V (t) ≤ (



Jk )ϕ − ψτ e

−

t

t0

λ(s)ds

.

t0
That is x(t) − y(t) =

n

 |xi (t) − yi (t)| ≤ 2 V (t) ≤ 2

 (

(

−

t

t0

λ(s)ds

t0
i=1



Jk )ϕ − ψτ e

Jk )e

−

t

t0

λ(s)ds

= exp

t0


 log Jk −

t0


t

t0

λ(s)ds

According to Theorem 3.1, let 

t

t0

˜ λ(s)ds = m[λ(t)](t − t0 ) + λ(t),

q(t) =

t0
9

log Jk −

q t − t0 log Jk , T k=1

.

˜ where λ(t) is a suitable periodic function, and q(t) is T -periodic function. Therefore,  x(t) − y(t) ≤ 2 ϕ − ψτ exp





 q ˜

1 q(t) − λ(t) 1 exp log Jk − m[λ(t)] (t − t0 ) , 2 2T 2 k=1

which can be denoted as follows:  x(t) − y(t) ≤ 2 ϕ − ψτ Q(t) exp(−μ(t − t0 )),

 q ˜ q(t)−λ(t) 1 and μ = 12 m[λ(t)] − 2T where Q(t) = exp k=1 log Jk . 2

(3.3)

Then, by means of Banach’s fixed point theorem, repeat the proof process of inequality (3.2), we can obtain that the impulsive DNN system (2.1) has an unique globally exponentially stable T -periodic solution.2 Remark 3.2 In Theorem 3.1 and Theorem 3.2, we have derived the periodic solution’s existence and exponential stability sufficient conditions of the impulsive DNNs by 1-norm and 2-norm, respectively. From these conditions, we can conclude that it is easier to derive the periodic solution’s existence and exponential stability criteria by 1-norm. Furthermore, these results are different from the published results. For instance, the author of [8, 11] only studied the existence and exponential stability of the HNNs and CGNNs without delays by utilizing impulsive differential inequality  u(t) ˙ ≤ −A(t)u(t), t = tk , t ≥ t0 , − u(t+ k ) ≤ (1 + Ck )u(tk ),

k ∈ N+ .

Thus, in Theorem 3.2, if let B(t) = 0, the similar results in [8, 11] can be obtained as follows: Corollary 3.2 Suppose that assumptions (A1)-(A5) are satisfied, and assume that (i) A(t) = −2Λ(t), where Λ(t) is the greatest eigenvalue of matrix Δ(t) for a fixed value t ≥ t0 . (ii)

1 2 m[A(t)]

−

1 2T

q

k=1

log Jk > 0, where Jk = max1≤i≤n (1 + Cik )2 .

Then, the impulsive NN system (2.1) without delays has an unique globally exponentially stable T -periodic solution. Remark 3.3 For Theorem 3.1 and Theorem 3.2, the condition m[A] > m[B] ≥ 0 is necessary but not conservative. In some published papers, A(t) > B(t) ≥ 0 was necessary to investigate the periodic solution’s existence and exponential stability of NNs with impulses and delays. For example, in [28, 37], when the system was one-dimensional, A(t) > B(t) ≥ 0 was needed to get the impulsive differential inequality, which was used to study the periodic solution’s existence and exponential stability. In [38], the authors used A(t) − B(t) ≥ α > 0, B(t) ≥ 0 to obtain the periodic solution’s global exponential stability and existence sufficient conditions of the generalized high-order NNs. According to the expressions of m[A] and m[B], it can be seen that A(t) > B(t) ≥ 0 can be derived from m[A] > m[B] ≥ 0. Furthermore, let A(t) = A > 0 and B(t) = B ≥ 0 be constants, the Lemma 2.4 will be changed into the generalized Halanay form impulsive inequality, which has been deeply investigated and applied in many published papers[33, 39, 34]. Based on the above analysis, the results in this paper are generalized. 10

z (t)

0.5

1

0

−0.5

0

0.5

1

1.5

2

2.5

1.5

2

2.5

(a),t

x (t)

0.5

1

0

−0.5

0

0.5

1 (b),t

x1(t)−z1(t)

2

0

−2

0

0.5

1

1.5

2

2.5 (c),t

3

3.5

4

4.5

5

Figure 1: State trajectories of the state variable x1 (t). (a) is the state trajectories of x1 (t) without impulses; (b) is the state trajectories of x1 (t) with impulses and (c) is the error curves of these two state trajectories.

4

Numerical Example In this section, one numerical example and its simulated images will be given to illustrate the effective-

ness of the obtained theoretical results. Example 4.1. Consider the following three-dimensional impulsive DNNs with periodic coefficients  3 3 x˙ i (t) = −di (t)xi (t) + j=1 aij (t)fj (xj (t)) + j=1 bij (t)gj (xj (t − τ )) + Ii (t), t ≥ 0, t = tk ,  xi (t)  xi (t) − xi (t− ) = Cik xi (t− ),

t = tk , k ∈ N + ,

(4.1)

where d1 (t) = 3 sin(4πt) + 5, d2 (t) = 2 sin(4πt) + 3, d3 (t) = 2 sin(4πt) + 4, I1 (t) = cos(4πt), I2 (t) = sin(4πt), I3 (t) = sin(4πt), a11 = 0.15 + 0.015 sin(4πt), a12 = 0.2 + 0.02 cos(4πt), a13 = 0.3 + 0.03 sin(4πt), a21 = −0.3 + 0.03 sin(4πt), a22 = 0.1 + 0.01 cos(4πt), a23 = −0.1 + 0.01 sin(4πt), a31 = −0.1 + 0.01 sin(4πt), a32 = 0.1 + 0.01 cos(4πt), a33 = 0.2 + 0.02 cos(4πt), b11 = −0.2 + 0.02 cos(4πt), b12 = −0.3 + 0.03 sin(4πt), b13 = 0.2 + 0.02 sin(4πt), b21 = 0.4 + 0.04 cos(4πt), b22 = −0.1 + 0.01 sin(4πt), b23 = 0.1 + 0.01 cos(4πt), b31 = 0.1 + 0.01 cos(4πt), b32 = 0.1 + 0.01 sin(4πt), b33 = −0.2 + 0.02 sin(4πt), f1 (x) = f2 (x) = f3 (x) = tanh(x), g1 (x) = g2 (x) = g3 (x) = tanh(x), τ = 1, tk =

1 k. 4

From (a) of Figure 1, Figure 2 and Figure 3, it can be seen that the system (4.1) without impulsive perturbance has a stable periodic solution. According 

to the parameters defined in Theorem 3.1, we 3 can obtain A(t) = min1≤i≤3 di (t) − Li j=1 |aji (t)| = 2 sin(4πt) + 3 − 4|0.1 + 0.01 cos(4πt)|, B(t) = 11

z2(t)

0.5 0 −0.5

0

0.5

1

1.5

2

2.5

1.5

2

2.5

(a),t

x2(t)

0.5 0 −0.5

1

0.5

0

(b),t x2(t)−z2(t)

2 0 −2

0

0.5

1

1.5

2

2.5 (c),t

3

3.5

4

4.5

5

Figure 2: State trajectories of the state variable x2 (t). (a) is the state trajectories of x2 (t) without impulses; (b) is the state trajectories of x2 (t) with impulses and (c) is the error curves of these two state trajectories. 

 max1≤i≤3 Mi 3j=1 |bji (t)| = | − 0.2 + 0.02 cos(4πt)| + 0.5 + 0.05 cos(4πt), T = 1/2 and q = 2. Then, we can derive that m[A] = 2.600 and m[B] = 0.700, which implies that the first condition m[A] > m[B] of Theorem 3.1 is satisfied. By the definition of λ(t), it can be obtain that m[λ(t)] = 0.8164. According 3 q to the second condition m[λ(t)] − T1 k=1 log Jk > 0 of Theorem 3.1, we can obtain k=1 log Jk < 0.4082 and J1 J2 J3 < 1.5041. Let Cik = 0.14, then, Jk = 1.14, k = 1, 2, 3, then J1 J2 J3 ≈ 1.2998 < 1.5041. Thus, all the conditions of Theorem 3.1 are satisfied, the impulsive DNN system (4.1) has an unique globally exponentially stable T -periodic solution. Although the impulses are unstable, it can be seen that these sufficient conditions concerned with impulsive perturbation in Theorem 3.1 can maintain the exponential stability of original stable system, which can be shown by the state trajectories curves (b) and the error curves (c) of Figure 1, Figure 2 and Figure 3. Remark 4.1 From Figure 1, 2 and 3, we can see that stable system has certain ability to resist disturbance under some special conditions. The simulations of this example are different with the results of [42], which can be seen from the figures of the two papers. In [42], the authors discussed the impulsive stabilization control of neural networks, i.e. an unstable system could be stabilized by impulses. However, in our paper, we mainly discussed the impulsive perturbation, i.e. the stabilization of a stable system could not be destroyed by unstable impulses under some special conditions.

5

Conclusions The periodic solution’s existence, uniqueness and exponential stability of the impulsive DNNs with

periodic coefficients are the evergreen topics for the neural dynamic systems. Recently, a new definition named as the integral average form of periodic function was provided to study impulsive delayed differential inequalities and the NNs’ dynamic behaviors. In this paper, based on the studies of the periodic coefficient’s average form, an impulsive delayed differential inequalities have been obtained by using the characteristic equation of delayed differential equations and impulsive differential equations. Taking advantage of the novel impulsive delayed differential inequalities, we have firstly researched the periodic solution’s existence 12

z (t)

0.5

3

0

−0.5

1

0.5

0

1.5

2

2.5

1.5

2

2.5

(a),t

x (t)

0.5

3

0

−0.5

0

0.5

1 (b),t

3

0

3

x (t)−z (t)

1

−1 −2

1

0.5

0

2.5 (c),t

2

1.5

3

4.5

4

3.5

5

Figure 3: State trajectories of the state variable x3 (t). (a) is the state trajectories of x3 (t) without impulses; (b) is the state trajectories of x3 (t) with impulses and (c) is the error curves of these two state trajectories. problems of impulsive DNNs with periodic coefficients, then we have studied their exponential stability. By utilizing delayed impulsive comparison systems, Lyapunov functions and fixed point theorem, we have obtained several existence conditions and exponential stability criteria through the expressions of 1-norm and 2-norm, respectively. Furthermore, the presented criteria in this paper are different and generalized by comparing them with the published results. In the future, the integral average form of periodic coefficient could be used to investigate others dynamic behaviors of impulsive DNNs, such as synchronization, passivity etc.. Finally, one numerical example and its simulated images have been shown to illustrate the theoretical results’ effectiveness of this paper.

Appendix Proof of Lemma 2.4: Letw(t) = u(t)e

D+ u(t)e get that

t t0

A(s)ds

w(t+ k)

=

+ u(t)A(t)e u(t+ k )e

 t+ k t0

t t0

A(s)ds

A(s)ds

t

t0

A(s)ds

, then, when t = tk , it can be derived that D+ w(t) =

≤ α(t)w(t − τ ) with α(t) = B(t)e

≤ u(t1k )(1 + Ck )e

 t− k t0

A(s)ds

= (1 +

t

t−τ

A(s)ds

. When t = tk , we can

Ck )w(t− k ).

Let x(t) is the solution of the following impulsive differential equation: ⎧ ⎪ ˙ = β(t)x(t), t = tk , t ≥ t0 , ⎪ x(t) ⎨ − x(t+ k ) = (1 + Ck )x(tk ), k ∈ N+ , ⎪ ⎪ ⎩ x(t ) = φ(t ), , 0 0

with β(t) = B(t)eτ z . Then, according to Lemma 2.2, we have 

 t β(s)ds (1 + Ck ) e t0 x(t) = φ(t0 )

(5.1)

(5.2)

t0
Let x(t) = φ(t0 )e

t

t0

β(s)ds

holds for any t ∈ [−τ, 0]. Assume there exists enough great φ(t0 ) such that

13

w(t) < φ(t0 )e

t

t0

β(s)ds

= x(t) holds for any t ∈ [−τ, 0]. Then, we will prove that w(t) ≤ x(t)

(5.3)

If it is not true, then, there exists t∗ ∈ (t0 , t1 ) such that w(t∗ ) = x(t∗ ), w(t) ≤

holds in [t0 , t1 ).

D+ w(t∗ ) > x(t ˙ ∗ ). Therefore, we can get D+ w(t∗ ) ≤ α(t∗ )w(t∗ − τ ) ≤ x(t) holds in [t0 − τ, t∗ ) and  t∗ −τ t0

α(t∗ )x(t∗ − τ ) = α(t∗ )φ(t0 )e ∗

B(t )e

 t∗

t∗ −τ

A(s)ds −

e

 t∗

t∗ −τ

D+ w(t∗ ) ≤ α(t∗ )φ(t0 )e

τz

B(s)e

 t∗ −τ t0

β(s)ds

ds

β(s)ds

. On the other hand, since β(t∗ ) = B(t∗ )eτ z = B(t∗ )e ∗

= α(t )e

−

 t∗

t∗ −τ

= α(t∗ )φ(t0 )e−

β(s)ds  t∗

t∗ −τ

 t∗

t∗ −τ

λ(s)ds

=

, we have β(s)ds

e

 t∗ t0

β(s)ds

= β(t∗ )φ(t0 )e

 t∗ t0

β(s)ds

= x(t ˙ ∗ ),(5.4)

which contradicts D+ w(t∗ ) > x(t ˙ ∗ ). Thus, the inequality (2.7) holds. Furthermore, we have w(t+ k) ≤

− (1 + Ck )w(t− k ) ≤ (1 + Ck )x(tk ) = x(tk ). Therefore, the inequality (2.7) holds in [t0 , t1 ].

Next, we prove that the inequality w(t) ≤ x(t)

(5.5)

holds in [t0 , tk ) for any k ∈ N . Obviously, it is true for k = 1. Then, suppose that it is true for k ≥ 1, we prove that it is also true for k + 1, that is w(t) ≤ x(t)

(5.6)

− holds in [tk , tk+1 ). When t = tk , we have w(tk ) ≤ (1 + Ck )w(t− k ) ≤ (1 + Ck )x(tk ) = x(tk ). Then, by

the similar process of (2.7), (2.10) holds for any t ∈ [tk , tk+1 ). Thus, (2.9) holds in [t0 , tk ) for any k ∈ N . Therefore, let tk → ∞, we have w(t) = u(t)e

t t0

A(s)ds

≤ x(t) for any t ∈ [t0 , ∞).

Based on the above discussion, we can conclude that u(t) = w(t)e

−

t

t0

A(s)ds

≤ x(t)e

−

t

t0

A(s)ds

=

 φ(t0 )

=

 φ(t0 )



  t β(s)ds − tt A(s)ds (1 + Ck ) e t0 e 0

t0


 − t λ(s)ds (1 + Ck ) e t0 .

t0
Therefore, we can obtain  u(t) ≤



 − t λ(s)ds (1 + Ck ) φ(t0 )τ e t0 .

t0
holds for any t ≥ t0 .2

Acknowledgments The work was supported by National Natural Science Foundation of China (Grant Nos. 61372139, 61503175, 61374078, 61571372), Program for New Century Excellent Talents in University (Grant Nos.[2013]47), Fundamental Research Funds for the Central Universities (Grant Nos. XDJK2016A001, XDJK2014A009), High School Key Scientific Research Project of Henan Province (Grant No. 15A120013). This publication was made possible by NPRP grant NPRP 4-1162-1-181 from the Qatar National Research Fund (a member of Qatar Foundation). 14

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17