Stability analysis of quaternion-valued neural networks with both discrete and distributed delays

Applied Mathematics and Computation 343 (2019) 342–353

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Stability analysis of quaternion-valued neural networks with both discrete and distributed delays Zhengwen Tu a,∗, Yongxiang Zhao a, Nan Ding a, Yuming Feng b, Wei Zhang b a

School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404100, China Key Laboratory of Intelligent Information Processing and Control of Chongqing Municipal Institutions of Higher Education, Chongqing Three Gorges University, Wanzhou 404100, China

b

a r t i c l e

i n f o

Keywords: Quaternion-valued neural networks (QVNNs) Stability Distributed delays Linear matrix inequality

a b s t r a c t The existence, uniqueness and stability of the equilibrium of quaternion-valued neural networks (QVNNs) with both discrete and distributed delays are investigated in this paper. The considered model is managed as a single entirety without decomposition. Based on homeomorphic mapping theorem and linear matrix inequality, several sufficient criteria are derived to ascertain the aforementioned QVNNs to be globally asymptotically stable and exponentially stable. Moreover, provided criteria can be verified by the linear matrix inequality (LMI) toolbox in MATLAB. Finally, one simulation example is demonstrated to verify the effectiveness of obtained results. © 2018 Elsevier Inc. All rights reserved.

1. Introduction Neural networks are with various specialty, such as powerful self-learning ability, parallel computing, powerful tolerance ability, etc, and they have been successfully applied in optimal computation, intelligent control, pattern recognition, signal processing and so on. Therefore, the dynamical characteristic analysis of neural networks has attracted extensive attentions [1–8]. Cao et al. have investigated the recurrent neural networks and some sufficient criteria for ascertaining the global asymptotic stability have been provided in [1,2]. The stability problem of Cohen–Grossberg neural networks was discussed detailedly in [6,7]. The synchronization problem was investigated deeply by sampled-data control in [9,10]. The asymptotic stability analysis of cellular neural networks were conducted in [11,12]. Based on impulsive differential inequality, Liu et al. analyzed the stability of impulsive neural networks [13,14]. By employing LMI optimization approach, the stability of neural networks with neutral delay was investigated in [15]. Zhang et al. have discussed the stability and stabilization problem with different control strategies in [16,17]. The stability and synchronization problem for memristor neural networks were investigated by designing intermittent controllers and adaptive controllers [18,19]. Real-valued neural networks (RVNNs) are with certain limitation in dealing with the problem involving multidimensional data. For instance, when encounter the problem of detection of symmetry, RVNNs may become helplessness, while this problem can be resolved ideally with complex-valued neural networks(CVNNs)[20]. Furthermore, it has been proved that a better performance of CVNNs can be achieved than that of RVNNs in handling the problem with complex signals. Therefore, dynamics characteristics of CVNNs have been discussed intensively [20–28]. Based on the assumption that the activation function can be separated into real and imaginary parts, Hopf bifurcation and local asymptotical stability of ∗

Corresponding author. E-mail address: [email protected] (Z. Tu).

https://doi.org/10.1016/j.amc.2018.09.049 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

Z. Tu et al. / Applied Mathematics and Computation 343 (2019) 342–353

343

CVNNs were investigated in [29]. The problem of finite-time stability for fractional-order CVNNs was investigated by employing Cauchy–Schiwartz inequality and Gronwall inequality [30]. Zhang et al. have investigated the stability of fractionalorder CVNNs with leakage and discrete delays by employing contraction mapping principle, and several delay-dependent conditions were obtained [23]. And the Lagrange stability of CVNNs has also been investigated in [31,32]. The μ-stability of delayed CVNNs was discussed with the help of Lyapunov theory and free weighting matrix method in [33]. However, the plural is also with certain limitation in dealing with the problem involving three-dimensional or four-dimensional data. As an extension of complex number, quaternion is with unique superiority in solving the problem involving three-dimensional and four-dimensional data. The spatial rotation can be expressed efficiently and tersely by the quaternion [34], and the phenomenon of gimbal lock can be elegantly avoided. Quaternion-valued neural networks (QVNNs) combine the merits of both quaternion and neural networks. Hence, it is of great value to investigate dynamics characteristics and applications of QVNNs [35–42,44,45]. By virtue of QVNNs, Buchholz and Bihanthe realized the optimum separation of polarized signals [35]. The robust stability of delayed QVNNs with parameter uncertainties was considered via Lyapunov theory and Homeomorphic mapping theory[38]. Based on homeomorphic mapping theory and complex decomposition method, the global asymptotical stability of continuous-time QVNNs and discrete-time QVNNs was considered in [39]. By decomposing QVNNs into two CVNNs, the μ-stability of QVNNs was discussed, and some sufficient conditions were established in the form of LMIs [42]. As is known, the stability of designed neural networks is the first requirement to various applications, such as optimal computation, associative memory, pattern recognition. Hence, it is not only important but also necessary to study stability of QVNNs. However, most existing results concerning stability were obtained by equivalently decomposing QVNNs into two CVNNs or four RVNNs [39–42]. Few results considered the dynamic characteristic of QVNNs by taking QVNNs as an entirety without decomposition, which is exactly our motivations to conduct this research. Time delays are inescapable in neural system due to the limited propagation velocity between different neurons. And dynamical behaviors of neural networks could become more complex owing to the appearance of time delays, and it may lead to performance degradation, such as instability, oscillation, bifurcation and so on. On the other hand, due to the existence of vast parallel pathways with different axon sizes, it is more rational way to introduce continuously distributed delays into neural networks models [46]. And plenty of interesting results have been reported [47–52]. By constructing a new Lyapunov–Krasovskii functionals, the asymptotic stability for neural networks with distributed delay was conducted and some delay-dependent sufficient conditions were established with the help of integral inequalities and delay partitioning technique [47]. Samidurai et al. considered the exponential stability of neutral neural networks with mixed delays and impulsive effects, some sufficient conditions were derived by constructing a new Lyapunov–Krasovskii functional [51]. The exponential stability of impulsive CVNNs with mixed delays were investigated by using analytic technique and M-matrix theory [52]. As far as we know, however, there is few result concerning the dynamics of QVNNs with both discrete and distributed delays. Furthermore, neural network models with mixed delays are closer to practical models and engineering application. The stability analysis of QVNNs with both discrete and distributed delays isis still an open and challenging issue, which is another motivation to conduct this research. In this paper, the asymptotic stability and exponential stability of QVNNs with mixed delays are investigated, and the QVNN is operated as an entirety rather than by decomposing into CVNNs or RVNNs. The main contributions of this paper can be listed as follows: (i) Results concerning the stability of QVNNs with both discrete and distributed delays are rare. This paper is one of the first to do this attempt, and asymptotic stability and exponential stability are also investigated. (ii) The quaternion-valued linear matrix inequality is equivalently translated into real-valued linear matrix inequality, which can be checked easily by LMI tool box in Matlab. (iii) Several freedom matrices are blended into sufficient criteria, which can make our results with less conservativeness. The remainder part of this paper is organized as follows. Model descriptions and preliminaries are exhibited in Section 2. The stability analysis for QVNNs with both discrete and distributed delays is conducted in Section 3. The validity of obtained results is checked by numerical simulations examples in Section 4. Conclusions are made in Section 5. 2. Preliminaries In order to present main results precisely, some preliminaries are presented firstly. x is said to be a quaternion, if x = x(r ) + x(i ) i + x( j ) j + x(k ) k, where x(r ) , x(i ) , x( j ) , x(k ) ∈ R. And x(r) is the real part of x, x(i ) i + x( j ) j + x(k ) k is the imaginary part of x, i, j and k are imaginary units, and the following equalities are satisfied

i2 = j2 = k2 = i jk = −1, i j = − ji = k, jk = −k j = i, ki = −ik = j.

(1)

Different from real number and complex number, the commutative property is not true for quaternion due to Hamilton rule (1). Notations: R represents the one-dimensional real space. C represents the one-dimensional complex space. Cm×n  {(csl )m×n |csl ∈ C, s = 1, 2, . . . , m, l = 1, 2, . . . , n}. Q  {x(r ) + x(i) i + x( j ) j + x(k) k |x(r ) , x(i) , x( j ) , x(k) ∈ R}. x¯ = x(r ) − x(i) i − √ ( j ) ( k ) ( r ) ( i ) ( j ) ( k ) x j − x k is the conjugate of x = x + x i + x j + x k. |x|  xx¯ = (x(r ) )2 + (x(i ) )2 + (x( j ) )2 + (x(k ) )2. The n 1 dimensional quaternion space is denoted by Qn , x  ( np=1 |x p |2 ) 2 , for x = (x1 , x2 , . . . , xn )T ∈ Qn . ϕ   supt0 −τ ≤s≤t0 |ϕ (s )| for ϕ ∈ C ([t0 − τ , t0 ]; Qn ). Qm×n  {(qsl )m×n |qsl ∈ Q, s = 1, 2, . . . , m, l = 1, 2, . . . , n}. The positive definite Hermitian matrix S is denoted by S0, and SCn> (Q )  {S ∈ Qn |S  0}, S≺0 means −S  0.

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Z. Tu et al. / Applied Mathematics and Computation 343 (2019) 342–353

Considering the QVNN with mixed delays:

q˙ (t ) = −Dq(t ) + C f (q(t )) + A f (q(t − τ (t ))) + B



t

t−τ

f (q(s ))ds + U,

(2)

where q(t ) = (q1 (t ), q2 (t ), . . . , qn (t ))T ∈ Qn is the state vector. D = diag{d1 , d2 , . . . , dn } denotes the self-feedback matrix with dl > 0, l = 1, 2, . . . , n. A, B, C ∈ Qn×n are link weights matrices; f( · ) is the activation function; τ (t) and τ are the discrete timevarying delay and distributed delay respectively. τ (t) meets with 0 < τ (t) ≤ τ , τ˙ (t ) ≤ μ. U = (u1 (t ), u2 (t ), . . . , un (t ))T ∈ Qn is the external input vector. The initial condition is given as q(s ) = ϕ (s ), s ∈ [t0 − τ , t0 ]. Just as the definition presented in [53], if qˆ satisfies −Dqˆ + C f (qˆ ) + A f (qˆ ) + τ B f (qˆ ) + U = 0, then qˆ is an equilibrium point of (2). Assumption. (H) There exists positive constant ml such that

| f p (x ) − f p (y )| ≤ ml |x − y|, l = 1, 2, . . . , n. And let  = diag{m1 , m2 , . . . , mn }. Definition 1. [41] The unique equilibrium point Q¯ of QVNN (2) is said to be globally exponentially stable if there exists a positive constant α such that

 Q (t ) − Q¯ ≤ ϕ − Q¯  e−α (t−t0 ) , t ≥ t0 . 

S11 T S12

Lemma 1. [55] Let S =

S12 S22



T , S T ∈ Qn×n with S11 = S11 22 = S22 , then S ≺ 0 is equivalent to one of following conditions:

−1 T (i) S22 ≺0, S11 − S12 S22 S12 ≺ 0; −1 T (ii) S11 ≺0, S22 − S12 S11 S12 ≺ 0.

Lemma 2. Let x, y ∈ Qn , P ∈ SCn> (Q ), then x∗ y + y∗ x ≤ 1ε x∗ P −1 x + ε y∗ P y. Lemma 3. [21] Given a Hermitian matrix Q ∈ Cn×n , then Q≺0 is equivalent to



Q (r ) Q (i )

−Q (i ) Q (r )



≺ 0,

(3)

where Q (r ) = Re(Q ), Q (i ) = Im(Q ). Let q = q(r ) + q(i ) i + q( j ) j + q(k ) k, then q can be uniquely expressed as

q = z1 + z2 j, z1 , z2 ∈ C. In fact, q = (q(r ) + q(i ) i ) + (q( j ) + q(k ) i ) j, and let z1 = q(r ) + q(i ) i, z2 = q( j ) + q(k ) i, then q = z1 + z2 j. Correspondingly, the matrix Q ∈ Qm×n can be uniquely expressed as

Q = Q1 + Q2 j, Q1 , Q2 ∈ Cm×n , where Q1 = Q1(r ) + Q1(i ) i, Q2 = Q2(r ) + Q2(i ) i, Q1(r ) = Q (r ) , Q1(i ) = Q (i ) , Q2(r ) = Q ( j ) , Q2(i ) = Q (k ) . Lemma 4. [54] (1) Let Z = Z1 + Z2 i ∈ Cm×n , Z1 , Z2 ∈ Rm×n , then (i) jZ = Z¯ j; (ii) jZ j = −Z¯ ; (iii) jZ ∗ = Z¯ T j. (2) Let Q = Q1 + Q2 j ∈ Qm×n , Q1 , Q2 ∈ Cm×n , then Q ∗ = Q1∗ − Q2T j. Lemma 5. Given a Hermitian matrix Q ∈ Qn×n , then Q ≺ 0 is equivalent to

⎛

Q (r )

−Q ( j )

−Q (i )

Q (k )

⎞

⎜ Q ( j) ⎜ (i ) ⎝ Q

Q (r )

Q (k )

−Q (k )

Q (r )

Q (i ) ⎟

−Q ( j ) ⎠

−Q (k )

−Q (i )

Q ( j)

Q (r )

⎟ ≺ 0.

Proof. For a quaternion vector q, considering the quadric form as

q∗ Qq = (z1∗ − z2T j )(Q1 + Q2 j )(z1 + z2 j ) = z1∗ Q1 z1 + z1∗ Q1 z2 j + z1∗ Q2 jz1 + z1∗ Q2 jz2 j −(z2T jQ1 z1 + z2T jQ1 z2 j + z2T jQ2 jz1 + z2T jQ2 jz2 j ) Based on Lemma 4, one gets

q∗ Qq = z1∗ Q1 z1 + z1∗ Q1 z2 j + z1∗ Q2 z¯1 j − z1∗ Q2 z¯2 −z2T Q¯ 1 z¯1 j + z2T Q¯ 1 z¯2 + z2T Q¯ 2 z1 + z2T Q¯ 2 z2 j

(4)

Z. Tu et al. / Applied Mathematics and Computation 343 (2019) 342–353

= z1∗ Q1 z1 − z1∗ Q2 z¯2 + z2T Q¯ 1 z¯2 + z2T Q¯ 2 z1



+ z1∗ Q2 z¯1 + z1∗ Q1 z2 − z2T Q¯ 1 z¯1 + z2T Q¯ 2 z2 j

  z1 z¯2

=

∗

 

−Q2 Q¯ 1

Q1 Q¯ 2

z1 z¯2

  

+

345

z¯1 z2

T

Q2 −Q¯ 1

Q1 Q¯ 2

  z¯1 z2

j.

(5)

Since Q is a Hermitian matrix, then Q = Q ∗ , i.e. Q1 + Q2 j = (Q1 + Q2 j )∗ = Q1∗ − Q2T j, which means that Q1∗ = Q1 , −Q2T = Q Q1 Q2 . Hence the matrix ( ¯ 2 ) is a skew-symmetric matrix. In fact, Q¯ 2 −Q 1



Q2 −Q¯ 1

Q1 Q¯ 2





T

=

−Q¯ 1T Q¯ T

Q2T Q1T

−Q2 Q¯ 1

Q1 Q¯ 2



−Q2 Q¯ 1

=

2

Q z Therefore, q∗ Qq = ( 1 )∗ ( ¯ 1 z¯2 Q2







−Q1 −Q¯ 2

Q1(r ) ⎜ Q (r ) ⎜ 2(i) ⎝ Q 1 −Q2(i )

−Q1(i ) Q2(i ) Q1(r ) Q2(r )

and then Q≺0 is equivalent to

Q (r ) ⎜ Q ( j) ⎝ Q (i ) −Q (k )

Q2 −Q¯ 1



Q1 . Q¯ 2

≺ 0.

−Q2(r ) Q1(r ) −Q2(i ) −Q1(i )

⎛



=−

−Q2 z1 )( ) which implies that Q ≺ 0 is equivalent to Q¯ 1 z¯2

Q According to Lemma 3, one can obtain that ( ¯ 1 Q2

⎛



−Q ( j ) Q (r ) −Q (k ) −Q (i )

−Q (i ) Q (k ) Q (r ) Q ( j)

⎞

−Q2 ) ≺ 0 is equivalent to Q¯ 1

Q2(i ) Q1(i ) ⎟ ⎟ ≺ 0, −Q2(r ) ⎠ Q1(r )

⎞

Q (k ) Q (i ) ⎟ ⎠ ≺ 0. −Q ( j ) Q (r )

 Remark 1. The problem of how to resolve linear matrix inequality involving quaternion is well answered by Lemma 5, which will make the criteria involving quaternion can be easily checked by the LMI tool box in Matlab. Lemma 6. [38] If the continuous map H (q ) : Qn → Qn satisfies the following conditions (i) H(q) is a injective on Qn , (ii) lim H (q ) = ∞, q→∞

then H(q) is a homeomorphism of Qn onto itself. 3. Main results Theorem 3.1. If assumption (H) holds, and there exists positive matrix P1 , three positive diagonal matrices W1 , W2 , W3 , such that

⎛

11

⎜C P =⎝ ∗ 1 ∗

A P1 B∗ P1

P1C −W1 0 0

P1 A 0 −W2 0

⎞

P1 B 0 ⎟ ≺ 0, 0 ⎠ −W3

(6)

then the QVNN (2) has a unique equilibrium, where 11 = −P1 D − DP1 + W1  + W2  + τ 2 W3  . Proof. Firstly, the existence and uniqueness of the equilibrium of the QVNN (2) will be discussed. Considering the following continuous map

H (q ) = −Dq + C f (q ) + A f (q ) + τ B f (q ) + U. Now, we shall prove that the continuous map H(q) is an injective under the condition (6). Otherwise, there exist two different quaternion vectors q, q˘, such that H (q ) − H (q˘ ) = 0, i.e.

−D(q − q˘ ) + C ( f (q ) − f (q˘ )) + A( f (q ) − f (q˘ )) + τ B( f (q ) − f (q˘ )) = 0.

(7)

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Z. Tu et al. / Applied Mathematics and Computation 343 (2019) 342–353

Multiplying both sides of Eq. (7) by (q − q˘ )∗ P1 , one can get

− (q − q˘ )∗ P1 D(q − q˘ ) + (q − q˘ )∗ P1C ( f (q ) − f (q˘ )) + (q − q˘ )∗ P1 A( f (q ) − f (q˘ )) + τ (q − q˘ )∗ P1 B( f (q ) − f (q˘ )) = 0.

(8)

Obviously, the following equation is also true since it is just the conjugate transpose of (8),

− (q − q˘ )∗ DP1 (q − q˘ ) + ( f (q ) − f (q˘ ))∗C ∗ P1 (q − q˘ ) + ( f (q ) − f (q˘ ))∗ A∗ P1 (q − q˘ ) + τ ( f (q ) − f (q˘ ))∗ B∗ P1 (q − q˘ ) = 0.

(9)

Combining (8) and (9), one can obtain

− (q − q˘ )∗ P1 D(q − q˘ ) − (q − q˘ )∗ DP1 (q − q˘ ) + (q − q˘ )∗ P1C ( f (q ) − f (q˘ )) + ( f (q ) − f (q˘ ))∗C ∗ P1 (q − q˘ ) + (q − q˘ )∗ P1 A( f (q ) − f (q˘ )) + ( f (q ) − f (q˘ ))∗ A∗ P1 (q − q˘ ) + τ (q − q˘ )∗ P1 B( f (q ) − f (q˘ )) + τ ( f (q ) − f (q˘ ))∗ B∗ P1 (q − q˘ ) = 0.

(10)

Based on Lemma 2, the following inequalities can be obtained

(q − q˘ )∗ P1C ( f (q ) − f (q˘ )) + ( f (q ) − f (q˘ ))∗C ∗ P1 (q − q˘ ) ≤ (q − q˘ )∗ P1CW1−1C ∗ P1 (q − q˘ ) + ( f (q ) − f (q˘ ))∗W1 ( f (q ) − f (q˘ )), (q − q˘ )∗ P1 A( f (q ) − f (q˘ )) + ( f (q ) − f (q˘ ))∗ A∗ P1 (q − q˘ ) ≤ (q − q˘ )∗ P1 AW2−1 A∗ P1 (q − q˘ ) + ( f (q ) − f (q˘ ))∗W2 ( f (q ) − f (q˘ )), τ (q − q˘ )∗ P1 B( f (q ) − f (q˘ )) + τ ( f (q ) − f (q˘ ))∗ B∗ P1 (q − q˘ ) ≤ (q − q˘ )∗ P1 BW3−1 B∗ P1 (q − q˘ ) + ( f (q ) − f (q˘ ))∗W3 ( f (q ) − f (q˘ )),

(11)

where W1 , W2 and W3 are positive diagonal matrices. According to assumption (H), the following inequalities can be derived

( f (q ) − f (q˘ ))∗Wp ( f (q ) − f (q˘ )) ≤ (q − q˘ )∗ Wp (q − q˘ ), p = 1, 2, 3.

(12)

In light of (10)–(12), one can get

 (q − q˘ )∗ − P1 D − DP1 + P1CW1−1C ∗ P1 + P1 AW2−1 A∗ P1  + P1 BW3−1 B∗ P1 + W1  + W2  + τ 2 W3  (q − q˘ ) ≥ 0.

(13)

Since  ≺ 0, the following inequality can be established by using Lemma 1

− P1 D − DP1 + P1CW1−1C ∗ P1 + P1 AW2−1 A∗ P1 + P1 BW3−1 B∗ P1 + W1  + W2  + τ 2 W3  ≺ 0. Hence,

  (q − q˘ )∗ − P1 D − DP1 + P1CW1−1C ∗ P1 + P1 AW2−1 A∗ P1 + P1 BW3−1 B∗ P1 + W1  + W2  + τ 2 W3  (q − q˘ ) < 0

with q = q˘, which contradicts with (13). Therefore, H (q ) = H (q˘ ) for q = q˘, which implies that the continuous map H(q) is an injective. Next, we shall prove that H (q ) → +∞ as q → +∞. For a nonzero quaternion vector q, one has





2q∗ (H (q ) − H (0 )) ≤ q∗ − P1 D − DP1 + P1CW1−1C ∗ P1 + P1 AW2−1 A∗ P1 + P1 BW3−1 B∗ P1 + W1  + W2  + τ 2 W3  q ˇ q∗ q, ≤ −λ

(14)

ˇ is the minimum eigenvalue of P1 D + DP1 − P1CW −1C ∗ P1 − P1 AW −1 A∗ P1 − P1 BW −1 B∗ P1 − W1  − W2  − τ 2 W3  . where λ 1 2 3 Accordingly,

ˇ q2 ≤ 2qP (H (q ) + H (0 ) ), λ i.e.

ˇ q λ ≤ ( H ( q ) + H ( 0 ) ). 2P  Based on the above discussion, one can acquire that the continuous map H(q) is an injective on Qn . In light of Lemma 6, the existence and uniqueness of equilibrium point of QVNN (2) can be guaranteed.  Let qˆ be the equilibrium point of QVNN (2), and it can be translated to the origin with the variable substitution qˆ(t ) = q(t ) − qˆ. Meanwhile, QVNN (2) can be transformed as

qˆ˙ (t ) = −Dqˆ(t ) + Cg(qˆ(t )) + Ag(qˆ (t − τ (t ))) + B



t

t−τ

g(qˆ (s ))ds,

(15)

Z. Tu et al. / Applied Mathematics and Computation 343 (2019) 342–353

347

where g(qˆ(t )) = f (qˆ(t ) + qˆ ) − f (qˆ ). Theorem 3.2. If assumption (H) holds, μ < 1, and there exists positive matrix P1 , three positive diagonal matrices W1 , W2 , W3 , such that

⎛

11

P1C W2 + τ 2W3 − W1 0 0

C ∗ P1  =⎜  ⎝ ∗

A P1 B∗ P1

⎞

P1 A 0 −(1 − μ )W2 0

P1 B 0 ⎟ ≺ 0, 0 ⎠ −W3

(16)

then the QVNN (2) is globally asymptotically stable, where 11 = −P1 D − DP1 + W1  + W2  + τ 2 W3  .  ≺ 0, the following inequality can be derived from (16), Proof. Since 

⎛

11

⎜C ∗ P1 ⎝A∗ P ∗

1

B P1

P1C −W1 0 0

⎞

P1 A 0 −W2 0

P1 B 0 ⎟ ≺ 0, 0 ⎠ −W3

(17)

then conditions in Theorem 3.1 can be satisfied, which means that the existences and uniqueness of equilibrium point can  ≺ 0. be ascertained with  Considering the Lyapunov functional as



V (qˆ(t )) = qˆ∗ (t )P1 qˆ(t ) + +τ





0 −τ

t

t+s

t

t −τ (t )

qˆ∗ (s )Rqˆ(s )ds +



t

t −τ (t )

g∗ (qˆ(s ))W2 g(qˆ (s ))ds

g∗ (qˆ(θ ))W3 g(qˆ (θ ))dθ ds.

(18)

where R = W2  + τ 2 W3  . Calculating the derivative of V (qˆ(t )) along (15), one derives

dV (qˆ(t )) dqˆ∗ (t ) dqˆ(t ) = P1 qˆ(t ) + qˆ∗ (t )P1 + qˆ∗ (t )Rqˆ(t ) − (1 − τ˙ (t ))qˆ∗ (t − τ (t ))Rqˆ(t − τ (t )) dt dt dt +g∗ (qˆ(t ))W2 g(qˆ (t )) − (1 − τ˙ (t ))g∗ (qˆ(t − τ (t )))W2 g(qˆ(t − τ (t )))  0  +τ g∗ (qˆ(t ))W3 g(qˆ (t )) − g∗ (qˆ(t + s ))W3 g(qˆ (t + s )) ds =

−τ

− Dqˆ(t ) + Cg(qˆ(t )) + Ag(qˆ(t − τ (t ))) + B



t

t−τ



g(qˆ(s ))ds ∗ P1 qˆ(t )

+qˆ (t )P1 − Dqˆ(t ) + Cg(qˆ(t )) + Ag(qˆ(t − τ (t ))) + B ∗



t

t−τ

g(qˆ (s ))ds



+qˆ∗ (t )Rqˆ(t ) − (1 − τ˙ (t ))qˆ∗ (t − τ (t ))Rqˆ(t − τ (t )) + g∗ (qˆ(t ))W2 g(qˆ(t )) −(1 − τ˙ (t ))g∗ (qˆ(t − τ (t )))W2 g(qˆ (t − τ (t ))) + τ 2 g∗ (qˆ(t ))W3 g(qˆ(t ))  t −τ g∗ (qˆ(s ))W3 g(qˆ(s ))ds t−τ

≤ qˆ∗ (t )(−DP1 − P1 D + R )qˆ(t ) + g∗ (qˆ(t ))C ∗ P1 qˆ(t ) + qˆ∗ (t )P1Cg(qˆ(t ))  t  +g∗ (qˆ(t − τ (t )))A∗ P1 qˆ(t ) + qˆ∗ (t )P1 Ag(qˆ(t − τ (t ))) + g(qˆ(s ))ds ∗ B∗ P1 qˆ(t ) +qˆ∗ (t )P1 B



t−τ

t

t−τ

g(qˆ (s ))ds − (1 − μ )qˆ∗ (t − τ (t ))Rqˆ(t − τ (t ))

+g∗ (qˆ(t ))W2 g(qˆ (t )) − (1 − μ )g∗ (qˆ(t − τ (t )))W2 g(qˆ(t − τ (t )))  t   t  +τ 2 g∗ (qˆ(t ))W3 g(qˆ (t )) − g(qˆ (s ))ds ∗W3 g(qˆ (s ))ds t−τ

−g∗ (qˆ(t ))W1 g(qˆ (t )) + qˆ∗ (t )W1  qˆ(t )  η (t ). ≤ η∗ (t ) t

t−τ

(19)

where η (t ) = (qˆ(t ), g(qˆ(t )), g(qˆ(t − τ (t ))), t−τ g(qˆ(s ))ds )T .  ≺ 0, dV (qˆ(t )) < 0 for any η(t) = 0. And dV (qˆ(t )) = 0 if and only if η (t ) = 0. On the other hand, V (qˆ(t )) → Due to  dt dt +∞ as qˆ(t ) → +∞. Thus, the equilibrium point of QVNN (2) is globally asymptotically stable.  Generally speaking, compared to asymptotic stability, faster convergence can be attained with the exponential stability. Next, the exponential stability of delayed QVNNs will be discussed.

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Theorem 3.3. If assumption (H) holds, and there exists positive matrix P1 , four positive diagonal matrices W1 , W2 , W3 , W4 , appropriate dimension matrices S1 , S2 , S3 , such that

⎛

ˆ 11 

⎜ C ∗ P1 ⎜ ∗ A P  =⎜  ⎜ ∗ 1 ⎜ B P1 ⎝S∗ − S 2

P1C  22 

P1 A 0  33 

0 0 0 0

1

S3∗ − S1

0 0 0

⎞

S2 − S1∗ 0 0 0  55 

P1 B 0 0 −W3 0 0

S3 − S1∗ ⎟ 0 ⎟ ⎟ 0 ⎟ ≺ 0, 0 ⎟ ∗⎠ −S3 − S2 −S3∗ − S3

−S3∗ − S2

then the QVNN (2) is globally exponentially stable, where  = eατ W + τ 2 eατ W − W ,   = −(1 − μ )W − W ,  22 2 3 1 33 2 4

(20)

 = α P − DP − P D + W  + eατ (W  + τ 2 W ) + S∗ + S ,  11 1 1 1 1 2 3 1 1  = W  − (1 − μ )(W  + τ 2 W ) − S∗ − S .  55 4 2 3 2 2

Proof. By simple calculation, the following inequality can be established from (20),

⎛

11

⎜C ∗ P1 ⎝A∗ P ∗

1

B P1

P1C −W1 0 0

⎞

P1 A 0 −W2 0

P1 B 0 ⎟ ≺ 0, 0 ⎠ −W3

(21)

then conditions in Theorem 3.1 are satisfied, which implies that the existences and uniqueness of equilibrium point can be  ≺ 0. ascertained with  Introducing the Lyapunov functional as

V (qˆ(t )) = eαt qˆ∗ (t )P1 qˆ(t ) + +τ



0 −τ



t

t+s



t

t −τ (t )

eα (s+τ ) qˆ∗ (s )Rqˆ(s )ds +



t

t −τ (t )

eα (s+τ ) g∗ (qˆ(s ))W2 g(qˆ(s ))ds

eα (θ +τ ) g∗ (qˆ(θ ))W3 g(qˆ(θ ))dθ ds,

(22)

where R = W2  + τ 2 W3  . Considering the derivative of V (qˆ(t )) along (15), one can derive

dV (qˆ(t )) = dt



α eαt qˆ∗ (t )P1 qˆ(t ) + eαt − Dqˆ(t ) + Cg(qˆ(t )) + Ag(qˆ(t − τ (t ))) + B

×P1 qˆ(t ) + eαt qˆ∗ (t )P1 − Dqˆ(t ) + Cg(qˆ(t )) + Ag(qˆ (t − τ (t ))) + B



t

t−τ t

t−τ

g(qˆ (s ))ds

g(qˆ (s ))ds

∗



+eα (t+τ ) qˆ∗ (t )Rqˆ(t ) − eα (t+τ −τ (t )) (1 − τ˙ (t ))qˆ∗ (t − τ (t ))Rqˆ(t − τ (t ))

+eαt g∗ (qˆ(t ))W2 g(qˆ (t )) − (1 − τ˙ (t ))eα (t+τ −τ (t )) g∗ (qˆ(t − τ (t )))W2 g(qˆ(t − τ (t )))  t +τ 2 eα (t+τ ) g∗ (qˆ(t ))W3 g(qˆ (t )) − τ eαt g∗ (qˆ(s ))W3 g(qˆ(s ))ds t−τ



≤ eαt qˆ∗ (t )(α P1 − DP1 − P1 D + eατ R )qˆ(t ) + g∗ (qˆ(t ))C ∗ P1 qˆ(t ) + qˆ∗ (t )P1Cg(qˆ(t )) +g∗ (qˆ(t − τ (t )))A∗ P1 qˆ(t ) + qˆ∗ (t )P1 Ag(qˆ(t − τ (t ))) + +qˆ∗ (t )P1 B



t

t−τ



t

t−τ

g(qˆ (s ))ds



B P1 qˆ(t )

∗ ∗

g(qˆ(s ))ds − (1 − μ )qˆ∗ (t − τ (t ))Rqˆ(t − τ (t ))

+eατ g∗ (qˆ(t ))W2 g(qˆ(t )) − (1 − μ )g∗ (qˆ(t − τ (t )))W2 g(qˆ (t − τ (t )))  t   t  +τ 2 eατ g∗ (qˆ(t ))W3 g(qˆ(t )) − g(qˆ(s ))ds ∗W3 g(qˆ(s ))ds . t−τ

t−τ

According to Newton-Leibnitz formula, one gets



qˆ(t ) − qˆ(t − τ (t )) −



t

t −τ (t )

qˆ˙ (s )ds

 ∗

S1 qˆ(t ) + S2 qˆ(t − τ (t )) + S3





t

t −τ (t )

qˆ˙ (s )ds = 0.

Combining (23) and (24), one obtains



dV (qˆ(t )) ≤ eαt qˆ∗ (t )(α P1 − DP1 − P1 D + eατ R )qˆ(t ) + g∗ (qˆ(t ))C ∗ P1 qˆ(t ) + qˆ∗ (t )P1Cg(qˆ(t )) dt  t  +g∗ (qˆ(t − τ (t )))A∗ P1 qˆ(t ) + qˆ∗ (t )P1 Ag(qˆ (t − τ (t ))) + g(qˆ (s ))ds ∗ B∗ P1 qˆ(t ) +qˆ (t )P1 B ∗



t−τ

t

t−τ

g(qˆ (s ))ds − (1 − μ )qˆ (t − τ (t ))Rqˆ(t − τ (t )) ∗

(23)

(24)

Z. Tu et al. / Applied Mathematics and Computation 343 (2019) 342–353

349

+eατ g∗ (qˆ(t ))W2 g(qˆ(t )) − (1 − μ )g∗ (qˆ(t − τ (t )))W2 g(qˆ (t − τ (t ))) + τ eατ g∗ (qˆ(t ))  t   t  ×W3 g(qˆ(t )) − g(qˆ(s ))ds ∗W3 g(qˆ(s ))ds + qˆ∗ (t )(S1 + S1∗ )qˆ(t ) t−τ

+qˆ (t )S2 qˆ(t − τ (t )) + qˆ (t )S3 ∗

∗

t−τ



t

t −τ (t )

qˆ˙ (s )ds − qˆ∗ (t − τ (t ))S1 qˆ(t )

−qˆ∗ (t − τ (t ))(S2 + S2∗ )qˆ(t − τ (t )) − qˆ∗ (t − τ (t ))S3 −qˆ∗ (t )S1∗ qˆ(t − τ (t )) − qˆ∗ (t )S1∗  ×

t

t −τ (t )



−



t

t −τ (t )



−

qˆ˙ (s )ds −

t

t −τ (t )

qˆ˙ (s )ds qˆ˙ (s )ds



 

t

t −τ (t ) ∗

t

t −τ (t )

qˆ˙ (s )ds

(S3 + S3∗ )

∗ ∗ S3 qˆ







∗

t −τ (t )

t

t −τ (t )

qˆ˙ (s )ds

qˆ˙ (s )ds + qˆ∗ (t − τ (t ))S2∗ qˆ(t ) − qˆ∗ (t − τ (t ))S2∗

S1 qˆ(t ) −

t







qˆ˙ (s )ds +

t

t −τ (t )



qˆ˙ (s )ds

t

t −τ (t )



∗

S2 qˆ(t − τ (t ))

qˆ˙ (s )ds



∗ ∗ S3 qˆ

(t )

(t − τ (t )) − g∗ (qˆ(t ))W1 g(qˆ(t )) + qˆ∗ (t )W1  qˆ(t )

−g∗ (qˆ(t − τ (t )))W4 g(qˆ (t − τ (t ))) + qˆ∗ (t − τ (t ))W4  qˆ(t − τ (t ))  ξ (t ), ≤ ξ ∗ (t )

where ξ (t ) = qˆ(t ), g(qˆ(t )), g(qˆ(t − τ (t ))),

t

t−τ

g(qˆ(s ))ds, qˆ(t − τ (t )),

t

ˆ (s )ds t −τ (t ) q ˙



(25)

T.

 ≺ 0, then Since 

dV (qˆ(t )) < 0, dt for qˆ(t )) = 0. Consequently

eαt xT (t )P1 x(t ) ≤ V (t ) ≤ V (t0 ), sup

and then

xT (t )x(t )

≤

t0 −τ ≤s≤t0

(V (s ))

λmin (P1 )

e−αt , which implies that the equilibrium point of system (2) is globally exponentially

stable.  If the QVNN (2) is without distributed delays, then the following QVNN model can be obtained

q˙ (t ) = −Dq(t ) + C f (q(t )) + A f (q(t − τ (t ))) + U,

(26)

which is just the QVNN model discussed in [40–42]. Similar to the proof of Theorem 3.3, the following corollary can be derived. Corollary 3.1. If assumption (H) holds, and there exists positive matrix P1 , four diagonal matrices W1 , W2 , W4 , appropriate dimension matrices S1 , S2 , S3 , such that

⎛

ˆ 11 

⎜ C ∗ P1 ⎜ ∗ ˇ =⎜  ⎜ A∗ P1 ⎜ B P1 ⎝S∗ − S 2

P1C ˇ 22 

1

S3∗ − S1

0 0 0 0

P1 A 0 ˇ 33  0 0 0

S2 − S1∗ 0 0 0 ˇ 55  −S3∗ − S2

S3 − S1∗ 0 0 0 −S3 − S2∗ −S3∗ − S3

⎞

⎟ ⎟ ⎟ ⎟ ≺ 0, ⎟ ⎠

(27)

ˇ 11 = α P1 − DP1 − P1 D + W1  + eατ W2  + S∗ + S1 ,  ˇ 22 = then the QVNN (26) is globally exponentially stable, where  1 ατ ˇ 33 = −(1 − μ )W2 − W4 ,  ˇ 55 = W4  − (1 − μ )W2  − S∗ − S2 . e W2 − W1 ,  2

Remark 2. The stability analysis of QVNNs has been discussed by equivalently decomposing QVNNs into two CVNNs or four RVNNs [39–42]. Different from [39–42], QVNNs are discussed directly, which is more natural and compact than discussed by decomposing QVNNs into two CVNNs or four RVNNs, and sufficient conditions proposal here are concise in form and can be easily checked by Matlab. Remark 3. Criteria proposed here are in the form of linear matrix inequality with quaternion to ascertain the desired performance, which makes that provided criteria can be easily checked by the LMI tool box in Matlab with Lemma 5. Remark 4. Dynamics characteristics of QVNNs with time delays have been discussed intensively. The stability of both continuous and discrete QVNNs with time delay was considered in [39]. The global μ-stability of QVNNs with unbounded time delays is discussed in [40,41]. The boundedness and periodicity of delayed discrete-time QVNNs were investigated [43]. The

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Fig. 4.1. The transient behaviors of x(r) (t), x(i) (t), x(j) (t), x(k) (t) of QVNN (28).

dissipativity of QVNNs with time-varying delay was also studied in [44]. As far as we know, however, there is few results about the QVNNs with both discrete and distributed delays being reported. This paper is one of the first to explore the dynamic of QVNNs discrete and distributed delays. Moreover, some easily checked criteria are derived to guarantee the asymptotic stability and exponential stability respectively. 4. Illustrative examples In this section, one numerical example will be deduced to show the correctness and effectiveness of obtained results. Example 4.1. Considering the QVNN model as follow:

q˙ (t ) = −Dq(t ) + C f (q(t )) + A f (q(t − τ (t ))) + B where

 C=

 A=

 B=



t

t−τ

f (q(s ))ds + U,

(28)



1.3 + 3.2i − 3.5 j + 6.3k 5.8 − 4.1i + 1.5 j − 1.6k 3.8 − 5.1i + 3.5 j − 3.6k

1.6 + 6.5i − 3.7 j − 4.3k 1.2 + 3.1i − 3.5 j + 5.2k 2.1 + 4.1i − 3.6 j + 3.6k

2.6 + 5.5i − 4.9 j − 3.4k 3.6 + 6.5i − 3.9 j − 2.4k , 4.7 + 2.6i − 5.9 j − 3.4k

1.3 − 3.3i + 6.4 j + 1.3k 2.3 + 3.1i − 7.5 j − 1.4k 2.2 + 1.3i − 4.3 j − 2.2k

1.3 + 2.5i + 4.3 j − 2.6k 8.2 + 3.3i + 4.6 j + 1.3k 5.4 + 2.2i − 3.3 j + 4.6k

1.4 + 2.6i + 3.3 j − 1.6k 7.2 + 6.3i + 2.6 j + 5.3k , 4.3 + 7.2i − 2.1 j + 5.0k

3.4 − 1.3i − 3.1 j + 1.4k −1.9 − 4.1i − 3.2 j + 1.6k −2.9 − 4.5i − 3.8 j + 1.7k

−7.1 − 1.1i + 9.2 j − 1.1k 1.2 − 3.3i − 8.4 j − 3.2k 2.2 − 5.3i − 2.4i − 6.2k





−7.1 − 1.1i + 9.2 j − 1.1k 4.1 − 2.1i + 3.2 j − 4.1k , −5.1 − 3.1i + 6.2 j − 2.1k

Z. Tu et al. / Applied Mathematics and Computation 343 (2019) 342–353

351

Fig. 4.2. The transient behaviors of x(r) (t), x(i) (t), x(j) (t), x(k) (t) of QVNN (28) without distributed delay.

D = diag{12, 13, 14}, U = (2 + 1.2i + 2 j + 3k, 1 − 3.2i + j + 2k, 3 − 2.1i + j − 3k )T , ˆ ≺ 0 can be resolved as: 0.45 sin t + 0.55. Let α = 3.0, solutions of LMI 

 P1 =

 W2 =

 W4 =

 S2 =





2.7466 −0.3863 0.5065

−0.3863 3.1734 −1.7906

0.5065 −1.7906 , W1 = 3.8122

661.0115 0 0

0 661.0115 0

0 0 , W3 = 661.0115

31.9775 0 0 24.5393 −1.9068 2.6463

0 31.9775 0 −1.9068 26.2839 −10.7937





0 0 , S1 = 31.9775





1279.4 0 0



227.0783 0 0

3.9182 −1.3851 1.8005

2.6463 −10.7937 , S3 = 31.4923

0 1279.4 0



24.5448 −1.9075 2.6474

τ (t ) =



0 0 , 1279.4

0 227.0783 0

−1.3851 6.9127 −6.1025

f (q(t )) = 0.18 tanh(q(t )),



1.8005 −6.1025 , 9.8330

−1.9075 26.2899 −10.7994



0 0 , 227.0783



2.6474 −10.7994 . 31.5007

Therefore, according to Theorem 3.3, one can get that the QVNN (28) is with a unique equilibrium point and it is globally exponentially stable. And solution trajectories of the QVNN (28) are shown in Fig. 4.1. ˆ ≺ 0 without freedom matrices S1 , S2 , S3 is unsolvRemark 5. If choosing D = diag{3.8, 3.8, 3.8} in Example 4.1, the LMI  able, while it can be solved with freedom matrices S1 , S2 , S3 . From this point, the conservativeness can be reduced by introducing freedom matrices into criterion.

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Z. Tu et al. / Applied Mathematics and Computation 343 (2019) 342–353

Remark 6. If choosing B = 0 in Example 4.1, then the QVNN model (28) degrades into the QVNN model (26), solutions of ˇ ≺ 0 can be resolved as: LMI 



P1 =

 W2 =

 S1 =

 S3 =

4.0760 −1.0063 1.1586

−1.0063 3.2347 −2.1582



1.1586 −2.1582 , W1 = 4.1179





984.7350 0 0

666.0466 0 0

0 666.0466 0

0 0 , W4 = 666.0466

−12.7891 −5.8970 7.1972

−5.8970 −14.3234 −12.4155

7.1972 −12.4155 , S2 = −7.9068

32.5601 −1.2096 1.4513

−1.2096 32.2788 −2.5108









0 984.7350 0



0 0 , 984.7350



18.0856 0 0

0 18.0856 0

0 0 , 18.0856

32.5585 −1.2096 1.4513

−1.2096 32.2777 −2.5108

1.4513 −2.5108 , 33.5298



1.4513 −2.5108 . 33.5309

According to Corollary 3.1, one can get that the QVNN (28) is with a unique equilibrium point and it is globally exponentially stable. And solution trajectories of the QVNN (28) without distributed delay are shown in Fig. 4.2. On the other hand, criteria of [41] cannot be met, and Theorem 3.1 in [41] is deactivated to QVNN (28). From this point, criteria obtained here are with better applications than Theorem 3.1 of [41]. 5. Conclusions The asymptotic stability and exponential stability of QVNNs with both discrete and distributed delays are investigated respectively in this paper. The existence and uniqueness of the equilibrium of QVNNs with mixed delays are also discussed with homeomorphic mapping theorem. QVNNs are discussed directly rather than the method by decomposing them into CVNNs or RVNNs. Some sufficient conditions in the form of LMIs involving quaternion are obtained with Lyapunov functional and inequality technique. Furthermore, the solving problem of quaternion-valued linear matrix inequality is also well answered here. The correctness and validity of our results are verified by a numerical simulation example. The topic on dynamical analysis of QVNNs is meaningful and interesting, and we will consider the stabilization, finite-time stability and fixed-time stability of QVNNs in the near future. Acknowledgments This work was jointly supported by the National Natural Science Foundation of China under Grant No. 11601047, the Chongqing Research Program of Basic Research and Frontier Technology Grant No. cstc2018jcyjAX0588, the Youth Fund of Chongqing Three Gorges University Grant No. 18QN02, Program for Innovation Team Building at Institutions of Higher Education in Chongqing No. CXTDX201601035, Project supported by Program of Chongqing Development and Reform Commission Grant No. 2017[1007], and Key Laboratory of Chongqing Municipal Institutions of Higher Education (Grant No. [2017]3). References [1] J.D. Cao, J. Wang, Global asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 50 (1) (2003) 34–44. [2] X.S. Yang, J.D. Cao, Z.C. Yang, Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM J. Control Optim. 51 (5) (2013) 3486–3510. [3] C.D. Li, X.F. Liao, R. Zhang, Global robust asymptotical stability of multi-delayed interval neural networks: an LMI approach, Phys. Lett. A 328 (6) (2004) 452–462. [4] T.W. Huang, C.D. Li, S.K. Duan, J.A. Starzyk, Robust exponential stability of uncertain delayed neural networks with stochastic perturbation and impulse effects, IEEE Trans. Neural Netw. Learn. Syst. 23 (2012) 866–875. [5] R.X. Li, J.D. Cao, A. Alsaedi, F. Alsaadi, Exponential and fixed-time synchronization of Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms, Appl. Math. Comput. 313 (2017) 37–51. [6] W.L. Lu, T.P. Chen, Dynamical behaviors of Cohen-Grossberg neural networks with discontinuous activation functions, Neural Netw. 18 (3) (2005) 231–242. [7] K. Mathiyalagan, J.H. Pak, R. Sakthivel, S.M. Anthoni, Delay fractioning approach to robust exponential stability of fuzzy Cohen-Grossberg neural networks, Appl. Math. Comput. 230 (230) (2014) 451–463. [8] H.B. Bao, J.H. Park, J.D. Cao, Matrix measure strategies for exponential synchronization and anti-synchronization of memristor-based neural networks with time-varying delays, Appl. Math. Comput. 270 (2015) 543–556. [9] R.M. Zhang, D.Q. Zeng, S.M. Zhong, Novel master-slave synchronization criteria of chaotic Lur’e systems with time delays using sampled-data control, J. Frankl. Inst. 354 (12) (2017) 4930–4954. [10] R.M. Zhang, D.Q. Zeng, J.H. Park, Y.J. Liu, S.M. Zhong, Quantized sampled-data control for synchronization of inertial neural networks with heterogeneous time-varying delays, IEEE Trans. Neural Netw. Learn. Syst. (2018), doi:10.1109/TNNLS.2018.2836339. [11] S. Arik, An analysis of global asymptotic stability of delayed cellular neural networks, IEEE Trans. Neural Netw. 13 (5) (2002) 1239–1242. [12] J.H. Park, A new stability analysis of delayed cellular neural networks, Appl. Math. Comput. 181 (1) (2006) 200–205. [13] B. Wu, Y. Liu, J.Q. Lu, New results on global exponential stability for impulsive cellular neural networks with any bounded time-varying delays, Math. Comput. Modell. 55 (3) (2012) 837–843. [14] Y. Liu, R.J. Yang, J.Q. Lu, B. Wu, X.S. Cai, Stability analysis of high-order hopfield-type neural networks based on a new impulsive differential inequality, Int. J. Appl. Math. Comput. Sci. 23 (1) (2013) 201–211.

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