Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects

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Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects Yi Zou a, Xinsong Yang a,∗, Rongqiang Tang a, Zunshui Cheng b a Department

b School

of Mathematics, Chongqing Normal University, Chongqing 401331, China of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao 266061, China Received 26 December 2018; received in revised form 7 March 2019; accepted 6 May 2019 Available online xxx

Abstract Usually, it is very difficult to overcome the effects of time delay and impulse simultaneously in finite-time control area. The present paper solves this problem by considering finite-time synchronization (FTS) and control in an array of competitive neural networks (CNNs) with discontinuous activations, proportional delays, as well as impulsive effects. Since the proportional delays are unbounded and the discontinuous activations usually induce uncertain Filippov solutions, new analytical techniques have to be developed. Quantized control scheme is designed to reduce the control cost and save channel resources. By constructing new Lyapunov functionals and comparison system, several sufficient conditions are obtained to ensure that the coupled CNNs achieve synchronization in a finite settling time which is explicitly estimated. Finally, numerical simulations are provided to illustrate the effectiveness of theoretical analysis. © 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Neural networks (NNs) have been widely applied in many fields such as signal and image processing, pattern recognition, combinatorial optimization [1–3]. Coupled NNs, as a special ∗

Corresponding author. E-mail addresses: [email protected] (Y. Zou), [email protected] (X. Yang), [email protected] (R. Tang), [email protected] (Z. Cheng). https://doi.org/10.1016/j.jfranklin.2019.05.017 0016-0032/© 2019 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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dynamical networks, contain several subsystems, which can exhibit more complicated and important behaviors than those of any subsystems. So, investigating dynamical behaviors of coupled NNs is of great significance. Synchronization in an array of coupled NNs is interesting and has been successfully applied to secret communications, information processing, and biological systems [4–7]. Thus, considerable attention has been attracted to synchronization of coupled NNs [8–18]. Roughly speaking, synchronization can be classified into two types: synchronization as time goes infinity including asymptotic synchronization [8–11] and exponential synchronization [13–15] and synchronization in a settling time [16–18]. It is well known that, compared with the former, FTS is optimal and has properties such as better robustness and disturbance rejection [19]. Therefore, FTS and finite-time control techniques have been extensively considered in recent works [20–26]. From the perspective of control theory, synchronization of CNNs is an absolutely special one because this kind of NNs possesses two types of state variables: the short-term memory (STM) and the long-term memory (LTM) [27]. In this model, the STM describes the fast neural activity, whereas the LTM depicts the slow and unsupervised synaptic modifications. In 1996, Meyer–Ba¨ se introduced the so-called CNNs with different time scales in [28], which extend many NN models. Hitherto, fruitful results concerning stability and synchronization of CNNs have been published [29–39]. For example, the authors in [31] study the exponential synchronization of CNNs by designing state feedback control; Yang et al. [33] consider lag synchronization of CNNs with mixed delays and uncertain hybrid perturbations; Based on LMI approach, exponential synchronization of switched stochastic CNNs with mixed delays is studied in [34]. Note that most of existing results on synchronization and stability of CNNs are obtained in the framework that the activation functions are continuous and even Lipschitz continuous. However, in the case that the activation functions have high-slope nonlinear elements, it is unreasonable to model CNNs by differential equations with continuous activation functions. In fact, dynamical neuron systems with discontinuous activation functions have many important applications and are frequently encountered [40]. In the literature, the stability of CNNs with discontinuous activation functions (or discontinuous CNNs (DCNNs)) has been investigated [41–45]. But, as far as the authors know, few published papers consider FTS of DCNNs, which inspires the present study. Time delay is unavoidable in NNs due to finite information exchanging between different units. In [31,33,35], asymptotic synchronization of CNNs with time-varying delay, constant delay, and unbounded distributed delay are studied, respectively. Recently, another kind of delay called proportional delay has aroused interests of researchers [29,30]. However, few results consider synchronization of CNNs with proportional delay. Proportional delay is timevarying and unbounded which is different from the distributed delay as that in [33]. Although FTS of CNNs with continuous activations and unbounded distributed delay has been studied in [38,39], seldom results are reported for FTS of CNNs with proportional delay. It should be noted that the settling time cannot be given for FTS of CNNs with unbounded distributed delay [38]. Therefore, one may wonder whether the settling time can be estimated for FTS of CNNs with proportional delay. This paper will also answer this question. In the evolution of a dynamical system, system’s states may be suddenly changed at certain moments. Usually, a dynamical system with such phenomenon can be modeled by impulsive differential equation [46,48]. Recently, FTS of coupled systems with impulsive effects is investigated in [47–49]. However, the methods utilized in [47–49] are difficult to be extended to FTS of time-delay systems, let alone systems with proportional delay. This is another main inspiration of this paper. Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017

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In addition, quantizing signals before transmission is significant and necessary for communication with limited channel resources [50–52]. Quantized control techniques can not only save channel resources, but also reduce transmitted information and channel blocking. Recently, some interesting results concerning synchronization and quantized control have been published in [15,53–56]. Especially, based on quantized control, FTS of complex networks without delay is considered in [54] by partitioning the impulse intervals and using convex combination technique. Unfortunately, the analytical techniques proposed in [54] are also difficult to be applied to time-delay systems. Different from [54], this paper establishes new methods to study FTS of coupled DCNNs with proportional delay, and the given results can be easily extended to other systems with delays. This paper investigates FTS of coupled DCNNs with both proportional delay and impulsive effects via quantized control. The main contributions are summarized as follows: 1. It is the first time that FTS of CNNs with discontinuous activation functions, proportional delay, and impulsive effects is investigated. Coping with these three factors simultaneously such that the coupled CNNs achieve FTS is not an easy work. 2. Without directly utilizing the error signals, quantized controller is designed, by which the effects of both the uncertainties caused by Filippov solutions and the proportional delay are well overcome. 3. By designing new Lyapunov functionals and combining 1-norm analytical technique and comparison system method, the effect of impulses on the FTS is well dealt with. 4. Sufficient conditions are obtained for the FTS, which essentially improve those given in [38,39,45]. Moreover, the settling time is explicitly estimated. The paper is organized as follows. In Section 2, the model of the DCNNs with proportional delay and impulsive effects is presented. Some useful definitions and assumptions are also given in this part. Section 3 gives some FTS criteria by strict mathematical proofs. Then, simulation examples are presented to illuminate the effectiveness of the theoretical results in Section 4. Finally, conclusions are given in Section 5. Notations: R is the set of real numbers, Rn denotes the set of n × 1 real vectors, and Rn×m denotes the set of n × m matrices;  · ∞ and  · 1 are the ∞-norm and 1-norm of a vector or a matrix. 1n is a column vector with all n elements being 1. |B| is a vector derived by taking absolute values of all elements of a vector B; co[E ] is the closure of the convex hull of the set E ⊂ Rn . N+ is the set of nonnegative integers. 2. Model description and preliminaries Consider a DCNN with proportional delay as follows: ⎧ n )) ⎨STM : ε x˙r (t ) = −c r xr (t ) + j=1 ar j f j (x j (t p + nj=1 br j f j (x j (qt )) + Er l=1 ℘rl (t )wl , r = 1, 2 . . . , n, ⎩ LTM : ℘˙rl (t ) = −hr℘rl (t ) + mr wl fr (xr (t )), l = 1, 2, . . . , p,

(1)

where n denotes the number of the STM states, p denotes the number of the LTM states, xr (t) is the neuron current activity level, ℘rl (t) represents the synaptic efficiency, cr > 0 describes the rate with which the rth neuron will reset its potential to the resting state, f (x(t )) = ( f1 (x1 (t )), f2 (x2 (t )), . . . , fn (xn (t )))T stands for the activation function which is assumed to be discontinuous, arj and brj represent the connection weight and the delayed connection Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017

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weight between the rth and jth neurons, respectively; wl is the external stimulus, hr > 0 and mr represent disposable scaling constants, Er is the strength of the external stimulus, ε is the time scale of STM state. q ∈ (0, 1] is the proportional delay factor. The initial conditions of system (1) are stated as: xr (0) ∈ R and ℘rl (0) ∈ R. Remark 1. Note that the proportional delay in DCNN (1) is (1 − q)t. The proportional delay is a kind of time-varying infinite delay, which is different from the unbounded distributed 0 delay −∞ Ki j (t − s)ds considered in [33,34,38,39]. p Let sr (t ) = l=1 ℘rl (t )wl = ℘rT (t )w, r = 1, 2, . . . , n, s(t ) = (s1 (t ), s2 (t ), . . . , sn (t ))T , and p w2 = l wl2 be a constant. Without loss of generality, the input stimulus vector w is assumed to be normalized with unit magnitude w2 = 1. Then, the DCNN (1) can be rewritten in the following form:  1 1 1 1 STM : x˙(t ) = − Cx (t ) + A f (x (t )) + B f (x (qt )) + E s(t ), (2) ε ε ε ε LTM : s˙(t ) = −H s(t ) + M f (x(t )), where A = (ar j )n×n , x(t ) = (x1 (t ), x2 (t ), . . . , xn (t ))T , C = diag(c1 , c2 , . . . , cn ), H = diag(h1 , h2 , . . . , hn ), B = (br j )n×n , f (x(t )) = ( f1 (x1 (t )), f2 (x2 (t )), . . . , fn (xn (t )))T , M = diag(m1 , m2 , . . . , mn ), E = diag(E1 , E2 , . . . , En ), f (x(qt )) = ( f1 (x1 (qt )), f2 (x2 (qt )), . . . , fn (xn (qt )))T . The following assumption conditions are utilized in our study. (A1 ) For each r = 1, 2, . . . , n, fr (·) : R → R is continuous except on a countable set of isolated points {ρkr }, where both the right and left limits fr+ (ρkr ) and fr− (ρkr ) exist. Moreover, fr (·) has at most a finite number of jump discontinuous in every bounded compact interval R. (A2 ) For each r = 1, 2, . . . , n, 0 ∈ co[ fr (0)], and there exist nonnegative constants κ r and ϱr such that |αr − βr | ≤ κr |u − v| + r holds for ∀u, v ∈ R, where αr ∈ co[ fr (u)], βr ∈ co[ fr (v)] with co[ fr (·)] = [min{ fr− (·), fr+ (·)}, max{ fr− (·), fr+ (·)}]. The Filippov set-valued map given below is needed to define Filippov solution. Definition 1 [57]. The Filippov set-valued map of f(x) at x ∈ Rn is defined as follows:   F[ f (x)] = co[ f (B(x, δ)\ )], δ>0 μ( )=0

where B(x, δ) = {y : y − x ≤ δ}, and μ( ) is the Lebesgue measure of set . By Yang et al. [16,17], the conditions (A1 ) and (A2 ) ensure that the DCNN (2) has at least one Filippov solution in Rn as follows:  1 1 1 1 STM : x˙(t ) = − Cx(t ) + Aα(t ) + Bα(qt ) + E s(t ), (3) ε ε ε ε LTM : s˙(t ) = −H s(t ) + Mα(t ), where α(t ) ∈ F[ f (x)]. Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017

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Consider linearly coupled DCNNs with impulsive effects as follows: ⎧ 1 1 1 1 ⎪ STM : y˙i (t ) = − Cyi (t ) + A f (yi (t )) + B f (yi (qt )) + E Si (t ) ⎪ ⎪ ⎪ ε ε ε ε ⎪ ⎨ + Nl=1 dil yl (t ) + Ri (t ), t = tk , y (t ) = μki ei (tk− ), t = tk , ⎪ ⎪ i k  ⎪ ˙ ⎪ LTM : S (t ) = −H S (t ) + M f (yi (t )) + Nl=1 uil Sl (t ) + Ki (t ), t = tk , ⎪ i i ⎩ − Si (tk ) = νik zi (tk ), t = tk , k ∈ N+ , i = 1, 2, . . . , N,

5

(4)

where yi (t ) = (yi1 (t ), yi2 (t ), . . . , yin (t ))T and Si (t ) = (Si1 (t ), Si2 (t ), . . . , Sin (t ))T are the state vector of the ith node,  = diag(γ1 , γ2 , . . . , γn ) > 0 and = diag(θ1 , θ2 , . . . , θn ) > 0 are the inner coupling matrices, D = (dil )N×N and U = (uil )N×N  are the outer-coupling matrices and satisfy the following condition: dil ≥ 0, i = l and dii = − Nl=1,i =l dil , uil ≥ 0, i = l and uii =  − Nl=1,i =l uil , respectively. The constants μki and νik are impulse gains. {tk , k ∈ N+ } is the impulse sequence satisfying 0 = t0 < · · · < tk < . . . , limk→∞ tk = +∞. ei (t ) = yi (t ) − x(t ), zi (t ) = Si (t ) − s(t ), yi (tk ) = yi (tk ) − yi (tk− ), Si (tk ) = Si (tk ) − Si (tk− ), yi (tk ) = yi (tk+ ) = limt →tk+ yi (t ), yi (tk− ) = limt →tk− yi (t ), Si (tk ) = Si (tk+ ) = limt →tk+ Si (t ), Si (tk− ) = limt →tk− Si (t ), Ki (t) and Ri (t) are controllers to be designed. The Filippov solution of Eq. (4) satisfies the following equations: ⎧ N 1 1 1 1 ⎪ ⎪STM : y˙i (t ) = − Cyi (t )+ Aβi (t )+ Bβi (qt )+ E Si (t ) + l=1 dil yl (t ) + Ri (t ), t = tk , ⎪ ⎨ ε ε ε ε yi (tk ) = μki ei (tk− ), t = tk ,  ⎪ ˙ ⎪ Si (t ) + Mβi (t ) + Nl=1 uil Sl (t ) + Ki (t ), t = tk , ⎪ ⎩LTM : Si (t ) k= −H Si (tk ) = νi zi (tk− ), t = tk , k ∈ N+ , i = 1, 2, . . . , N, (5) where βi (t ) ∈ F[ f (yi )]. Our main aim is to design suitable controllers Ri (t) and Ki (t) such that the synchronization defined below is realized. Definition 2. The coupled DCNNs (4) are said  to be finite-timely synchronizedonto Eq. (2) if there exists a constant T such that limt→T Ni=1 (ei (t ) + zi (t ) ) = 0 and Ni=1 (ei (t ) + zi (t ) ) ≡ 0 for t ≥ T, where T is called settling time and depends on the initial value conditions of systems (2) and (4). Definition 3 [46,47]. An impulse sequence {tk , k ∈ N+ } is said to have average impulsive interval Ta if there exist positive integer N0 and positive constant Ta such that t − t0 t − t0 − N0 ≤ N (t, t0 ) ≤ + N0 , Ta Ta where N(t, t0 ) is the number of impulsive times of the impulse sequence {tk , k ∈ N+ } in (t0 , t). 3. Main results In this section, sufficient conditions are derived to ensure the FTS of the controlled DCNNs (4) through strict mathematical proof. Considering the advantages of quantized control techniques, two simple but effective quantized controllers are designed. Moreover, the settling time is theoretically estimated. Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017

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According to Definition 2, synchronizing Eq. (4) with Eq. (2) is equivalent to synchronizing Eq. (5) with Eq. (3). Hence, subtracting Eq. (3) from Eq. (5) derives error system: ⎧  1 1 1 1 ⎪ STM : e˙i (t ) = − Cei (t )+ Aξi (t )+ Bξi (qt )+ E zi (t ) + Nl=1 dil el (t ) + Ri (t ), t = tk , ⎪ ⎪ ⎨ ε ε ε ε ei (tk ) = (1 + μki )ei (tk− ), t = tk ,  ⎪ ⎪ zi (t ) + Mξi (t ) + Nl=1 uil zl (t ) + Ki (t ), t = tk , ⎪ ⎩LTM : z˙i (t ) = −H zi (tk ) = (1 + νik )zi (tk− ), t = tk , k ∈ N+ , i = 1, 2, . . . , N, (6) where ξi (t ) = βi (t ) − α(t ). A quantizer q(·) is a mapping from R to ℵ, where ℵ = {±w : w = ρ i w0 , 0 < ρ < 1, i = 0, ±1, ±2, . . . } ∪ {0} with sufficient large constant w0 > 0. For ∀v ∈ R, the quantizer q(v) is defined as follows: ⎧ 1 1 ⎪ w
Ri (t ) = −πi q(ei (t )) − ζ sign (q(ei (t ))), (8) Ki (t ) = −i q(zi (t )) − ηsign (q(zi (t ))),

where i = 1, 2, . . . , N, π i ≥ 0, ϖi ≥ 0, ζ > 0, and η > 0 are control gains to be designed, sign(·) is the standard sign function, sign (q(ei (t ))) = (sign (q(ei1 (t ))), . . . , sign (q(ein (t ))))T , sign (q(zi (t ))) = (sign (q(zi1 (t ))), . . . , sign (q(zin (t ))))T . Remark 2. It is worth noting that, compared with the state-based controllers in [38,39,45], the quantized controllers in Eq. (8) need limited information and can avoid channel blocking. Moreover, the quantized controllers in Eq. (8) are simple to be carried out in practice, while the controller in [45] is complex. Therefore, our results improve the corresponding ones in [38,39,45].  1 κ1 Denote 1i = − εc + M1 κ1 + 1ε A1 κ1 + Nl=1,i =l dli  + dii  + Bqε , 2i =  N 1 −h + ε E 1 + uii + l=1,i =l uil , c = min{c j , j = 1, 2, . . . , n}, h = min{h j , j = 1, 2, . . . , n},  = min{γi , i = 1, 2, . . . , n},  = max{γi , i = 1, 2, . . . , n}, = min{θi , i = 1, 2, . . . , n}, = max{θi , i = 1, 2, . . . , n}, κ = (κ1 , κ2 , . . . , κn )T and  = (1 , 2 , . . . , n )T . Our main results are given below. Theorem 1. Suppose that (A1 )–(A2 ) are satisfied and the average impulse interval of the sequence {tk , k ∈ N+} is Ta . If the control gains in Eq. (8) are chosen to satisfy ⎧ ⎨ 1 i , 1i > 0, πi = (1 − δ) (9) ⎩0,  ≤ 0, 1i

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⎧ ⎨ 2i , 2i > 0, i = (1 − δ) ⎩0, 2i ≤ 0,

7

(10)

1 ζ = (|A|∞ + |B|∞ ) + 1 , ε

(11)

η =|M|∞ + 2 ,

(12)

for i = 1, 2, . . . , N, then the coupled DCNNs (4) can be synchronized onto Eq. (2) in a finite settling time. Moreover, the settling time is estimated as (1) T ≤ (2) T ≤

Ta ln μ  Ta ln μ 

μ  ln ( (1 +(2 )1T+a μ22N)T0a−V ) for |μ|  < 1; (0) ln μ  2N0

+2 )Ta ln ( (1 +2 )T(a1− ) for |μ|  > 1 and Ta > μ2N0 V (0) ln μ 

μ 2N0 ln μV  (0) , 1 + 2

where  μ = max{|1 + νik |, |1 + μki |, i = 1, 2, . . . , n, k ∈ N+ },  1 and  2 are positive constants. Proof. Inspired by Yang et al. [10], define the following Lyapunov functional: V (t ) = V1 (t ) + V2 (t ), (13)  N N N t 1 κ1 where V1 (t ) = i=1 ei (t )1 + i=1 zi (t )1 , V2 (t ) = i=1 |1 + μki | Bqε qt ei (s)1 ds. Since the impulse gains μki work only at each specified impulse instant and remain zero during every impulse interval, for t ∈ (tk−1 , tk ), k ∈ N+ the function V(t) can be written as

N B1 κ1 t V (t ) = V1 (t ) + ei (s)1 ds. qε qt i=1 For t = tk , calculating the time derivative of V1 (t) along the trajectory of the error system (6) yields V˙1 (t ) =

N i=1

N  1 1 1 1 1Tn diag(ϑi (t )) − Cei (t ) + Aξi (t ) + Bξi (qt ) + E zi (t ) + dil el (t ) ε ε ε ε l=1

N   i (t ) + − πi q(ei (t )) − ζ ϑ 1T diag(θi (t )) − H zi (t ) + Mξi (t )

(14)

i=1

+

N

 uil zl (t ) − i q(zi (t )) − η θi (t ) ,

l=1

i (t ) = (ϑ i1 (t ), ϑ i2 (t ), . . . , ϑ in (t ))T , θi (t ) = where ϑi (t ) = (ϑi1 (t ), ϑi2 (t ), . . . , ϑin (t ))T , ϑ T  T i j (t ), θ ij (t), and  (θi1 (t ), θi2 (t ), . . . , θin (t )) , θi (t ) = ( θi1 (t ),  θi2 (t ), . . . ,  θin (t )) . ϑij (t), ϑ θi j (t ), j = 1, 2, . . . , n, i = 1, 2, . . . , N are chosen as follows [17]:



sign (ei j (t )), if ei j (t ) = 0,  sign (q(ei j (t ))), if ei j (t ) = 0, ϑi j (t ) = ϑ (t ) = 0, if ei j (t ) = 0, i j 0, if ei j (t ) = 0, Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017

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θi j (t ) =

sign (zi j (t )), if zi j (t ) = 0,  sign (q(zi j (t ))), if zi j (t ) = 0, θi j (t ) = 0, if zi j (t ) = 0, 0, if zi j (t ) = 0.

It follows from (A2 ) that n n   κr |alr ||eir (t )| + r |alr | ≤ A1 κ1 ei (t )1 + |A|∞ . (15) 1Tn diag(ϑi (t ))Aξi (t ) ≤ l=1 r=1

Similar to Eq. (15), one has 1Tn diag(ϑi (t ))Bξi (qt )

≤ B1 κ1 ei (qt )1 + |B|∞

(16)

1Tn diag(θi (t ))Mξi (t ) ≤ M1 κ1 ei (t )1 + |M|∞ .

(17)

and i j (t ) and θi j (t ) =  On the other hand, it is easy to obtain that ϑi j (t ) = ϑ θi j (t ) due to q(ei j (t )) = (1 + ςi j )ei j (t ) and q(zi j (t )) = (1 + ςˆi j )zi j (t ), where ς ij and ςˆi j ∈ [−δ, δ), δ ∈ [0, 1). Then, the following equations hold: i (t ) = −ζ and − 1Tn diag(θi (t ))η −1Tn diag(ϑi (t ))ζ ϑ θi (t ) = −η. (18) Combining Eqs. (14)–(18) leads N  c − + M1 κ1 + V˙1 (t ) ≤ ε i=1

to

 N 1 A1 κ1 + dli  + dii  − (1 − δ)πi ei (t )1 ε l=1,i =l

 N 1 1 + B1 κ1 ei (qt )1 + E 1 − h + uii + uil ε ε l=1,i =l   − (1 − δ)1 zl (t )1 − (1 + 2 ) ,

(19)

where 1 = ζ − 1ε (|A|∞ + |B|∞ ) and 2 = η − |M|∞ . Differentiating V2 (t) reaches that  N  H 1 κ1 1 V˙2 (t ) = ei (t )1 − H 1 κ1 ei (qt )1 . qε ε i=1

(20)

Therefore, it is easy to check from Eqs. (19) and (20) that N   V˙ (t ) ≤ [1i − (1 − δ)πi ]ei (t )1 + [2i − (1 − δ)i ]zi (t )1 − (1 + 2 ).

(21)

i=1

By conditions (9) and (10), one knows V˙ (t ) ≤ −(1 + 2 ).

(22)

When t = tk , it can be obtained from Eq. (13) that V (tk ) =

N

ei (tk )1 +

i=1

=

N

N

zi (tk )1 +

i=1

(1 + μki )ei (tk− )1 +

i=1

≤ μV  (tk− ).

N i=1

N i=1

|1 + μki |

H 1 κ1 qε

(1 + νik )zi (tk− )1 +

N i=1

tk

ei (s)1 ds

qtk

|1 + μki |

H 1 κ1 qε

tk−

qtk−

ei (s)1 ds (23)

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Consider the following comparison system: ⎧

−(1 + 2 ), t = tk , ⎪ ⎪ ˙ )= ⎨υ(t 0, t = tk , + − υ(t ) = μυ(t  ) , k ∈ N ⎪ +, k ⎪ k ⎩ υ(0) = V (0).

9

(24)

Comparing Eqs. (22) and (23) with Eq. (24), one can see that 0 ≤ V (t ) ≤ υ(t ), which implies that if there exists a T > 0 such that υ(t ) ≡ 0 for t > T,  then V(t) ≡ 0 for t > T. With the help of the discussions in [47], V(t) ≡ 0 for t ≥ T means that Ni=1 (ei (t ) + zi (t ) ) ≡ 0 for t ≥ T. Therefore, one only needs to prove that there is a T such that υ(T ) = 0 and υ(t ) ≡ 0 for t > T. By using ordinary differential equation theory, one has from Eq. (24) that, for t ∈ [0, t1 ),

t

t1 v(t ) = v(0) − (1 + 2 )ds, and v(t1− ) = v(0) − (1 + 2 )ds. 0

0

From Eq. (24), one gets v(t1+ ) = μv(t  1− ) = μv(0)  −μ 

t1

(1 + 2 )ds.

0

For t ∈ [t1 , t2 ), one has

t

v(t ) = v(t1 ) − (1 + 2 )ds, and v(t2+ ) = μv(0)  −μ  t1

v(t2+ ) = μv(t  2− ) = μ 2 v(0) − μ 2

t1

(1 + 2 )ds − μ 

0

t2

t1

(1 + 2 )ds −

0

t2

(1 + 2 )ds,

t1

(1 + 2 )ds.

t1

When v(t ) > 0, by mathematical introduce, one has for t ∈ [tk , tk+1 ), ∀k ∈ N+ ,

t1

t2 v(t ) = μ k v(0) − μ k (1 + 2 )ds − μ k−1 (1 + 2 )ds − · · ·

μ −

0 tk

t1

t

(1 + 2 )ds −

tk−1

(1 + 2 )ds,

tk

=μ N (t,0) v(0) − (1 + 2 )

t

μ N (t,s) ds.

(25)

0

Next, let us prove the existence of time instant T. t−s t−s (s,t ) When 0 < μ  < 1, one has μ  Ta +N0 ≤ μ N ≤ μ  Ta −N0 . It can be deduced from Eq. (25) that

t t t−s −N 0 T v(t ) ≤ μ a v(0) − (1 + 2 ) μ  Ta +N0 ds 0 −N0

= (μ 

v(0) − (1 + 2 )μ 

N0

t Ta Ta )μ .  Ta + (1 + 2 ) μN0 ln μ  ln μ 

(26)

In consideration of ln μ  < 0, one has μ −N0 v(0) − (1 + 2 )μ N0 lnTaμ > 0 and (1 + 2 )  μN0 lnTaμ < 0. So, there is T ∈ (0, +∞ ) such that the right hand side of Eq. (26) becomes 0, and T =

Ta ln μ 

μ  ln ( (1 +(2 )1T+a μ22N)T0a−v(0) ). ln μ  2N0

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When μ  > 1, one has μ  Ta −N0 ≤ μ N ≤ μ  Ta +N0 .

t (1 + 2 ) t t−s v(t ) ≤ μ  Ta +N0 v(0) − μ  Ta ds μ  0 t Ta Ta = (μ N0 v(0) − (1 + 2 )μ −N0 μ Ta + (1 + 2 )μ −N0 ) . ln μ  ln μ  (s,t )

t−s

t−s

(27)

ln μ  Since ln μ  > 0 and Ta > μ 1v(0) , one has μ N0 v(0) − (1 + 2 )μ −N0 lnTaμ < 0 and (1 + + 2 2 )  μ−N0 lnTaμ > 0. Therefore, there is T ∈ (0, +∞ ) such that the right hand side of Eq. (27) be2N0

comes 0, and T =

Ta ln μ 

+2 )Ta ln ( (1 +2 )(Ta1− ).  μ2N0 v(0) ln μ 

Remark 3. From the proof of Theorem 1 one can find that the value of μ  plays an important role in the synchronization of the coupled systems. Actually, the impulse plays a positive role for the synchronization when 0 < μ  < 1, while the the impulse hampers the synchronization when μ  > 0. Recently, synchronization of coupled networks with both impulsive effects and time delays was discussed in [7,46], respectively. However, the synchronization in [7,46] can be guaranteed only when time goes to infinity, which is not so optimal as the finite-time synchronization obtained in this paper. Therefore, results of this paper improve the corresponding one in [7,46]. Remark 4. Note that Theorems 1 is obtained without utilizing the finite-time stability theorem in [19], which cannot be directly utilized to study FTS of time-delay systems [17]. By constructing new 1-norm-based Lyapunov functional and using the quantized controllers in Eq. (8), the coupled DCNN (4) with proportional delay and impulsive effects can be finitetimely synchronized onto Eq. (2). The key step in the proof of Theorem 1 is to obtain the inequalities (22) and (23). Note that the analysis methods developed in this paper can be applied to FTS of dynamical systems with or without delays, which are completely different from those used in [18,19,45]. If f(·) in Eqs. (2) and (4) is continuous, then r = 0, r = 1, 2, . . . , n. Namely, f(·) satisfies the following assumption: 2 ) There exist nonnegative constants  (A κr , r = 1, 2, . . . , n such that | f (u)r − f (v)r | ≤ κr |u − v| holds for ∀u, v ∈ R.  Then, by the same analysis as that in the proof of Theorem 1, the following corollary can be easily obtained. 2 ) is satisfied and the average impulsive interval of impulsive Corollary 1. Suppose that (A sequence {tk , k ∈ N+} is Ta . If the control gains ϖi and π i satisfy Eqs. (9) and (10) for i = 1, 2, . . . , N, then the coupled CNNs Eq. (4) can be synchronized onto CNN (2) in a finite settling time. Moreover, the settling time is estimated as 1. T ≤ 2. T ≤

Ta ln μ  Ta ln μ 

μ  ln ( (1 +(2 )1T+a μ22N)T0a−V ) for |μ|  < 1; (0) ln μ  2N0

+2 )Ta ln ( (1 +2 )T(a1− ) for |μ|  > 1 and Ta > μ2N0 V (0) ln μ 

μ 2N0 ln μV  (0) , 1 + 2

where μ  = max{|1 + νik |, |1 + μki |, i = 1, 2, . . . , n}, η and ζ are any positive constants. Remark 5. Corollary 1 is also applicable to FTS of CNNs without impulsive effects. Recently, without considering impulsive effect, that authors in [38] investigated the FTS of Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017

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0.6

3

0.4

2

11

0.2

S2 (t)

x2 (t)

1 0

0

−0.2

−1 −0.4

−2

−0.6

−3 −4 −4

−3

−2

−1

0

x1 (t)

1

2

3

4

−0.8 −2

−1.5

−1

−0.5

0

S1 (t)

0.5

1

1.5

2

Fig. 1. Trajectory of Eq. (28) with initial conditions x(t ) = (1.4, 1.6)T and S(t ) = (1.2, 0.5)T .

CNNs with continuous activation functions and unbounded distributed delay. However, it is reported in [38] that the settling time cannot be estimated. This paper shows that, although the proportional  delay is unbounded, the settling time t N can be explicitly given. The main reason 1 κ1 is that V2 = Ni=1 Hqε  e (s)  ds = 0 if i 1 i=1 ei (s) = 0 for s ∈ [qt, t]. In this sense, qt Corollary 1 essentially extend the results in [38]. Remark 6. it is can be found from Theorem 1 and Corollary 1 that the control gains π i and ϖi , i = 1, 2, . . . , N in Eq. (8) are used to stabilize the error system, while the parameters η and ζ are used to eliminate proportional delay and the effects of uncertain caused by Filippov solutions. Hence, larger values of η and ζ will shorten the settling time.

4. Numerical examples In this section, numerical examples are given to illustrate the effectiveness of the theoretical analysis. In the simulations, the time-step size is taken as 0.001 and the quantizer density is assumed to be ρ = 0.8. A 2-dimensional DCNN with proportional delay is described as follows: 

1 1 1 1 STM : x˙(t ) = − Cx (t ) + A f (x (t )) + B f (x (qt )) + E s(t ), ε ε ε ε LTM : s˙(t ) = −H s(t ) + M f (x(t )),

(28)

where x(t ) = (x1 (t ), x2 (t ))T , S(t ) = (S1 (t ), S2 (t ))T , q = 0.8, ε = 1.2, C = diag(2, 2), 1.0 1.0 −1.5 2.0 E = diag(0.5, 1.5), H = diag(1, 1), M = diag(1, 0.5), A = (−5 .2 3.2 ), B = ( 2.0 3.5 ), T the activation function is f (x(t )) = ( f1 (x1 (t )), f2 (x2 (t ))) with f j (x j (t )) =

tanh (x j (t )) + 0.05, if x j (t ) ≥ 0, j = 1, 2, tanh (x j (t )) − 0.05, if x j (t ) < 0, j = 1, 2. Fig. 1 depicts the trajectory of Eq. (28) with initial values x(0) = (1.4, 1.6)T and S(0) = (1.2, 0.5)T . By simple computation, (A1 ) and (A2 ) are satisfied with κ1 = κ2 = 1, 1 = 2 = 0.1. Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017

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a

ei(t)

1

ei (t), zi (t), 1 ≤ i ≤ 4

zi(t) 0.5

0

−0.5

−1

−1.5

0

2

4

b

6

t

1.5

8

10

1.5 ei1(t)

0.5

0

−0.5

0.5

0

−0.5

−1

−1.5

q(ei2(t))

1

q(ei (t)), 1 ≤ i ≤ 4

1

ei (t), 1 ≤ i ≤ 4

q(ei1(t))

ei2(t)

−1

0

0.05

0.1

0.15

0.2

t

0.25

0.3

0.35

0.4

−1.5

0

0.05

0.1

0.15

0.2

t

0.25

0.3

0.35

0.4

c 2

2 zi1(t)

1.5

q(zi1(t)) 1.5

zi2(t)

1

q(zi (t)), 1 ≤ i ≤ 4

zi (t), 1 ≤ i ≤ 4

1

q(zi2(t))

0.5 0

−0.5

0.5 0

−0.5

−1

−1.5

−1

−1.5

−2

−2

−2.5

−2.5

0

0.05

0.1

0.15

0.2

t

0.25

0.3

0.35

0.4

0

0.05

0.1

0.15

0.2

t

0.25

0.3

0.35

0.4

Fig. 2. (a) The considered network without controller (8). (b) Trajectories ei (t) and q(ei (t)) (i = 1, 2, 3, 4) of the considered network. (c) Trajectories zi (t) and q(zi (t)) (i = 1, 2, 3, 4) of the considered network.

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a

13

6 ei(t) zi(t)

ei (t), zi (t), 1 ≤ i ≤ 4

4

2

0

−2

−4

−6 0

2

4

b

t

6

2.5

q(ei1(t))

2

ei2(t)

1.5

q(ei2(t))

1.5

q(ei (t)), 1 ≤ i ≤ 4

1 0.5 0

−0.5

1 0.5 0

−0.5

−1

−1.5

−1

−1.5

−2

−2

−2.5

−2.5

−3

10

2.5 ei1(t)

2

ei (t), 1 ≤ i ≤ 4

8

0

0.05

0.1

0.15

0.2

t

0.25

0.3

0.35

−3

0.4

0

0.05

0.1

0.15

0.2

t

0.25

0.3

0.35

0.4

c zi1(t)

3

q(zi1(t))

3

zi2(t)

q(zi2(t)) 2

q(zi (t)), 1 ≤ i ≤ 4

zi (t), 1 ≤ i ≤ 4

2 1 0

−1

1 0

−1

−2

−2

−3

−3

−4

−4

−5

0

0.05

0.1

0.15

0.2

t

0.25

0.3

0.35

0.4

−5

0

0.05

0.1

0.15

0.2

t

0.25

0.3

0.35

0.4

Fig. 3. (a) The considered network without controller (8). (b) Trajectories ei (t) and q(ei (t)) (i = 1, 2, 3, 4) of the considered network. (c) Trajectories zi (t) and q(zi (t)) (i = 1, 2, 3, 4) of the considered network.

Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017

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Consider ⎛ −2 ⎜1 D=⎜ ⎝1 1

Eq. (4) with four (28), where  = = diag(2, 1), and ⎞ ⎛ ⎞ 0 1 1 −6 3 1 2 ⎜ −3 1 1⎟ −7 1 2⎟ ⎟ and U = ⎜ 4 ⎟. ⎝2 2 −4 1⎠ 3 −6 1⎠ 1 0 −2 3 1 2 −6

(29)

By simple computation, it is easy to obtain from Eqs. (9)–(12) that 11 = 14 = 1 12.7292, 12 = 13.7292, 13 = 14.7292, 21 = 23 = 24 = 6.25, 22 = 7.25, 1−δ = 1.125, |A|∞ + |B|∞ = 1.39 and |M|∞ = 0.15. The initial values of the coupled network are randomly chosen from (−3, 3) by uniform distribution and get V (0) = 16.2. Case 1: μ  < 1. Take |1 + μki | = |1 + νik | = 0.4, i = 1, 2, 3, 4, k ∈ N+ , and N0 = Ta = 2. Choose gains of the controller (8) as π11 = π14 = 14.3203, π12 = 15.4453, π13 = 16.5703, 21 = 23 = 24 = 7.0313, 22 = 8.1563, and 1 = 2 = 0.5. According to the condition (1) in Theorem 1, the considered network can achieve FTS in the settling time T = 12.3826. Fig. 2(a) shows that the network without control is not synchronized, while Fig. 2(b) and (c) describes the error trajectories of the the considered coupled network with controllers (8), from which one can see that the network with impulsive effect satisfying the condition 1) in Theorem 1 achieves synchronization before T = 12.3826. Case 2: μ  > 1. Taking |1 + μki | = |1 + νik | = 1.4, i = 1, 2, 3, 4, k ∈ N+ , and 1 = 2 = 5, the condition (2) in Theorem 1 implies that Ta > 4.1042. Choose Ta = 4.3, N0 = 3, the control parameters are taking the same as those in Case 1. Then, the considered networks can achieve FTS in T = 4.6488. Fig. 3(a) shows that the network without controllers (8) cannot realize synchronization. Fig. 3(b) and (c) describes the error trajectories of the the considered coupled network with controllers (8), which shows that the network with impulsive effects satisfying the the condition (2) in Theorem 1 achieve synchronization before T = 4.6488. 5. Conclusions In this paper, FTS of coupled DCNNs with both proportional delay and impulsive effects has been investigated. Novel quantized controllers have been designed such that the effects of the uncertainties caused by Filippov solutions and the proportional delay are overcome. By using 1-norm analytical technique and comparison system method and designing novel Lyapunov functional, the effect of impulses on the FTS has been well dealt with. Moreover, the settling time is estimated. Numerical simulations have been provided to illustrate the effectiveness of theoretical analysis. Obviously, if the quantized control is added to only a small fraction of nodes in the network, i.e, quantized pinning control, the control cost can be further reduced. Therefore, FTS of coupled DCNNs with proportional delay via quantized pinning control is our next research topic. Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 61673078 and 61374011, and the Basic and Frontier Research Project of Chongqing under Grant No. cstc2018jcyjAX0369, the Bowang Scholar of Chongqing Normal University. Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017

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Please cite this article as: Y. Zou, X. Yang and R. Tang et al., Finite-time quantized synchronization of coupled discontinuous competitive neural networks with proportional delay and impulsive effects, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2019.05.017