A Linear Matrix Inequality Approach to Global Synchronization of Multi

CHINESE JOURNAL OF PHYSICS

VOL. 50, NO. 1

February 2012

A Linear Matrix Inequality Approach to Global Synchronization of Multi-Delay Hopfield Neural Networks with Parameter Perturbations Guoliang Cai, Haijian Shao, and Qin Yao Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu 212013, China (Received May 20, 2011) In this paper, a successful linear matrix inequality (LMI) approach is used to analyze multi-time-delay Hopfield neural networks with parameter perturbations by constructing an appropriate Lyapunov-Krasovskii functional. Two novel delay-dependent criteria for asymptotic stability of the multi-time-delay Hopfield neural networks are given by the Lyapunov method combined with the LMI framework. This comprehensive discussion of the situation also includes extensive applications. PACS numbers: 07.05.Mh, 05.45.Gg, 05.45.Pq, 05.45.Xt

I. INTRODUCTION

The HNNs (Hopfield neural networks) are a typical type of recurrent neural network, which has strong associative memory and optimized computing power, so the dynamic properties of the model have received extensive attention [1–3]. Research and applications of the HNNs have penetrated into various fields and achieved rich results [4–6]. HNNs were first proposed with an electrical circuit implementation as a Hopfield-type neural network [7, 8]. HNNs have been extensively studied in the past few years and have found many applications in different areas, such as pattern recognition, associative memory, and combinatorial optimization. Many essential features of these networks, such as the qualitative properties of stability, oscillation, and convergence issues have been investigated by several authors [9, 10]. HNNs with time delays (DHNNs) have been extensively investigated over the years, and various sufficient conditions for the stability of the equilibrium point of this class of neural networks have been presented in [8, 11–15]. Among the sufficient conditions, there are delay-independent and delay-dependent conditions. In general, the delay-independent result is simple and straightforward, while a delay-dependent result is more complex but less restrictive and conservative. So far, most of these existing results for the global asymptotic stability of the DHNNs are independent of delay parameters. However, in some applications, system delays are frequently fixed and their bounds are known. In addition, several famous neural networks are special cases of DHNNs, such as cellular neural networks (CNN) [16, 17], bi-directional associative memory (BAM) neural networks, and so on. For multi-time-delay HNNs (MDHNNs), due to parameter perturbations, external disturbances, and modeling errors, the impact of interconnecting structure, introduced delay, and other uncertain factors may cause shock and system instability. Therefore, multitime-delay, the asymmetric interconnection structure, and the parameter perturbation to the dynamic analysis of HNNs have an important impact. In this paper, we consider a class

http://PSROC.phys.ntu.edu.tw/cjp

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c 2012 THE PHYSICAL SOCIETY ⃝ OF THE REPUBLIC OF CHINA

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of MDHNNs with parameter perturbations described by a nonlinear delay differential equation of neutral type. Attention is focused on the derivation of global asymptotic stability criteria for such a class of delayed neutral networks. By constructing a suitable Lyapunov function, two novel conditions for the stability of MDHNNs are given in terms of a linear matrix inequality. The advantage of the proposed approach is that the resulting stability criteria can be used efficiently via existing numerical convex optimization algorithms, such as the interior-point algorithms for solving LMIs [17, 18]. The paper is organized as follows: In Section II, the system description and preliminaries are stated and some definitions and lemmas are listed. Using Lyapunov stability theory, an LMI approach to global synchronization of MDHNNs with parameter perturbations is designed in Section III. We give an example in Section IV. In Section V, we give the conclusion of this paper.

II. SYSTEM PRELIMINARIES AND DESCRIPTION

Consider the following model MDHNNs with parameter perturbations x(t) ˙ = −(C + ∆C)x(t) + (T0 + ∆T0 )S(x(t)) +

K ∑

(Tk + ∆Tk )S(x(t − τk )) + I,

(1)

k=1

where ∆C = diag [∆C1 , ∆C2 , . . . , ∆Cn ] ∈ Rn×n , ∆TK ∈ Rn×n are norm-bounded timevarying matrices of uncertain items, x = (x1 , x2 , . . . , xn )T ∈ Rn is the state variable that the neurons are associated with; C = diag [c1 , c2 , . . . , cn ], ci > 0 (i = 1, 2, . . . , n) is a diagonal matrix which indicates the feedback of the neurons; T0 ∈ Rn×n is not associated with the delay of the interconnect τK (k = 1, 2, . . . , K), TK ∈ Rn×n is associated with the delay of the interconnect τK (k = 1, 2, . . . , K), τK is the Kth delay, and 0 < τ1 < · · · < τK < +∞; I = (I1 , I2 , . . . , In ) ∈ Rn is a constant input vector; S(x) = [s1 (x1 ), s2 (x2 ), . . . sn (xn )]T , Si (xi ) is the activation and is a bounded function, and si (0) = 0. The initial function of the system is x(s) = φ(s), where s ∈ [−τK , 0], φ ∈ C ([−τK , 0] , Rn ). Assumption 1. The neuron activation functions Si (·) is Lipschitz continuous and is monotonically nondecreasing. Specifically, there exist constants li > 0 such that 0≤

Si (a) − Si (b) ≤ li , a−b

(2)

for any a, b ∈ R, a ̸= b, i = 1, 2, . . . , n. Assumption 2. Parameter perturbations ∆C and ∆Tk are norm-bounded in system (1), ∆C and ∆TK are norm-bounded, and satisfy the following matching conditions: [∆C, ∆T0 , . . . , ∆CTK ] = HF [A, B, . . . , BK ] ,

(k = 1, 2, . . . , K) (3)

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where F is the parameter uncertainty, represented by an unknown matrix, and satisfies F T F ≤ I; A, B0 , . . . , BK are the uncertainties that have the appropriate dimension of the constant matrix. Assume that x∗ = (x∗1 , x∗2 , . . . , x∗n )T is an equilibrium point of Eq. (1), then we will shift the equilibrium point x∗ to the origin. The transformation x ˜i (·) = xi (·) − x∗i puts system (1) into the following form: ˜ x(t)) + x ˜˙ (t) = −(C + ∆C)˜ x(t) + (T0 + ∆T0 )S(˜

K ∑

˜ x(t − τk )). (Tk + ∆Tk )S(˜

(4)

˜ (Tk + ∆Tk )S(y(t − τk )),

(5)

k=1

Let y(t) be shorthand for x ˜(t), then we have ˜ y(t) ˙ = −(C + ∆C)y(t) + (T0 + ∆T0 )S(y(t)) +

K ∑ k=1

where y(t) is the state vector of the transformed system, [ ]T ˜ S(y(t)) = S˜1 (y1 (t)), S˜2 (y2 (t)), . . . , S˜n (yn (t)) , S˜i (yi (t)) = Si (yi (t) + x∗i ) − Si (x∗i ), with S˜i (0) = 0, Si (i = 1, 2, . . . , n) satisfying Assumption 1. The following facts and lemma will be used for deriving the main result. Fact 1. (Schur complement) Given constant symmetric matrices Σ1 , Σ2 , Σ3 where Σ1 = ΣT1 and 0 < Σ2 = ΣT2 , then Σ1 + ΣT3 Σ−1 2 Σ3 < 0 if and only if ( ) ( ) Σ1 ΣT3 −Σ2 Σ3 < 0 or < 0. Σ3 −Σ2 ΣT3 Σ1 Fact 2. For any real vectors a, b and any matrix Q > 0, with appropriate dimensions, it follows that 1 1 aT b ≤ aT Qa + bT Q−1 b. 2 2 Lemma 1. [18] Let U , V , W and M be real matrices of appropriate dimensions with M satisfying M = M T , Then M + U V W + W T V T U T < 0. For all V T V ≤ 1, if and only if there exists a scalar ε > 0, such that M + ε−1 U U T + εW T W < 0. Lemma 2. [19] For a positive matrix Q > 0, any matrices G, F1 , F2 , F3 , F4 , F5 , F6 , and scalar τK ≥ 0 (k = 1, 2, . . . , K), the following inequality holds: K ∫ t K K ∑ ∑ ∑ T T ˜ − y˙ (s)Qy(s)ds ˙ ≤ ςk (t)F ςk (t) + τk ςkT (t)F T Q−1 ςk (t), k=1

t−τk

k=1

k=1

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where F = [F1 , F2 , F3 , F4 , F5 , F6 ],  0 0 F1T 0 0 T ∗ 0 F 0 0 2   ∗ ∗ F3T + F3 F4 F5 F˜ =  ∗ ∗ ∗ 0 0  ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ (∫ t T T T ςk (t) = [y (t), y (t − τk ),

0 0 F6 0 0 0

    ,   

)T y(s)ds ˙ , y˙ T (t), y˙ T (t − τk ), S˜T (y(t − τk ))].

t−τk

Proof. Utilizing Fact 2, we have K ∫ t K (∫ ∑ ∑ T − y˙ (s)Qy(s)ds ˙ ≤2 k=1

t−τk

(F ςk (t) +

K ∫ ∑ k=1



k=1 t

t

)T y˙ (s)ds T

t−τk

ςkT (t)F T Q−1 F ςk (t)ds

t−τk



0 0   K K ∑ ∑ I T   ≤ 2 ςk (t)   F ςk (t) + τk ςkT (t)F T Q−1 F ςk (t) 0  k=1 k=1 0 0 ≤

K ∑

ςkT (t)F˜ ςk (t) +

K ∑

τk ςkT (t)F T Q−1 F ςk (t).

k=1

k=1

III. MAIN RESULT

In this section, we derive two new delay-dependent criteria for the asymptotic stability of the system (5) using the Lyapunov method combined with the linear matrix inequality framework. Before presenting our main result, by utilizing the following zero equation, KGy(t) −

K ∑ k=1

Gy(t − τk ) −

K ∑ k=1

∫

t

G

y(s)ds ˙ = 0, t−τk

(6)

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for a matrix G ∈ Rn×n of appropriate dimension, we can represent the system (5) as y(t) ˙ = − (C + ∆C − KG)y(t) −

K ∑

˜ Gy(t − τk ) + (T0 + ∆T0 )S(y(t))

k=1

+

K ∑

˜ (Tk + ∆Tk )S(y(t − τk )) −

k=1

K ∑ k=1

∫

(7) t

G

y(s)ds. ˙ t−τk

Then we have the following theorem. Theorem 1: For given τK ≥ 0, (k = 1, 2, . . . , K), L = diag{l1 , l2 , . . . , ln } the equilibrium point of Eq. (5) is globally asymptotically stable if there exist positive definite matrices P , U , Z, Q, and any matrices Mi , Fi (i = 1, 2, . . . , 6), satisfying the following LMI:  ¯  ¯ k12 Υk13 M4T M5T M6T F1T Υk11 Υ  ∗ Υk22 Υk23 −M4T −M5T −M6T F2T     ∗ ∗ Υk33 F4 F5 F6 F3T     ∗ ∗ ∗ τk Q + R 0 0 F4T    < 0, T  ∗ ∗ ∗ ∗ −R 0 F5     ∗ ∗ ∗ ∗ ∗ −Z F6T  ∗ ∗ ∗ ∗ ∗ ∗ −τk Q

(8)

where ¯ k11 = −P (C − KG) − (C − KG)T P + 2L T0 + U + L2 Z + M1 + M T Υ 1 K2 L 1 2L HF B0 − P HH T P + εAT A + HF Bk , + 2 K ε K 2G L 2G T ¯ k12 = − Υ + Tk − M1 + M2 , Υk13 = − + F1T − M1 + M3T , K K K Υk22 = −U − M2 − M2T , Υk23 = −M2 − M3T + F2T , Υk33 = F3 + F3T − M3 − M3T . Proof. Consider the following Lyapunov function V (t) = V1 (t) + V2 (t) + V3 (t) + V4 (t) + V5 (t),

(9)

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where V1 (t) = y T (t)P y(t), K ∫ t ∑ V2 (t) = y T (s)U y(s)ds,

V3 (t) =

V4 (t) =

k=1 t−τk K ∫ t ∑ k=1 t−τk K ∫ t ∑

∫

t

y˙ T (u)Qy(u)duds, ˙ s

y˙ T (s)Ry(s)ds, ˙

t−τk

V5 (t) =

k=1 K ∫ t ∑

˜ S˜T (y(s))Z S(y(s))ds.

t−τk

k=1

Then we can get V˙ 1 (t) = y T (t)(−P (C + ∆C − KG) − (C + ∆C − KG)T P )y(t) T

+ 2y (t)(−

K ∑

˜ Gy(t − τk ) + (T0 + ∆T0 )S(y(t))

k=1

+

K ∑

˜ (Tk + ∆Tk )S(y(t − τk )) −

k=1

V˙ 2 (t) = V˙ 3 (t) = V˙ 4 (t) =

K ∑ k=1 K ∑ k=1 K ∑

V˙ 5 (t) =

∫

y T (t)U y(t) −

K ∑

y(s)ds), ˙

τk y˙ T (t)Qy(t) ˙ −

k=1 K ∑ T

(10)

t−τk

y T (t − τk )U y(t − τk ),

k=1 K ∫ t ∑

y˙ T (t)Ry(t) ˙ −

t

G

k=1

y˙ T (s)Qy(s)ds, ˙

(11)

(12)

t−τk

y˙ (t − τk )Ry(t ˙ − τk ),

(13)

k=1

k=1 K ∑

K ∑

˜ S˜T (y(t))Z S(y(t)) −

k=1

K ∑

˜ S˜T (y(t − τk ))Z S(y(t − τk )),

k=1

(14) By utilizing Lemma 2 we obtain K ∫ t K K ∑ ∑ ∑ − y˙ T (s)Qy(s)ds ˙ ≤ ςkT (t)F˜ ςk (t) + τk ςkT (t)F T Q−1 F ςk (t). k=1

t−τk

k=1

k=1

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Thus, we can get a new upper bound of V 3 (t) as follows: V˙ 3 (t) ≤

K ∑

τk y˙ T (t)Qy(t)+ ˙

k=1

K ∑

ςkT (t)F˜ ςk (t)+

k=1

K ∑

τk ςkT (t)F T Q−1 ςk (t). (15)

k=1

Then we can get that V˙ (t) ≤ y T (t)(−P (C + ∆C − KG) − (C + ∆C − KG)T P )y(t)) +

T

2y (t)(−

K ∑

˜ Gy(t − τk + (T0 + ∆T0 )S(y(t))

k=1

+ − +

+

K ∑ k=1 K ∑ k=1 K ∑ k=1 K ∑

˜ (Tk + ∆Tk )S(y(t − τk )) −

K ∑

∫ G

k=1

y T (t − τk )U y(t − τk ) +

K ∑

y(s)ds ˙ t−τk

τk y˙ T (t)Qy(t) ˙ +

ςkT (t)F˜ ςk (t) +

k=1

k=1

K ∑

τk ςkT (t)F T Q−1 F ςk (t)

k=1

y˙ T (t)Ry(t) ˙ −

˜ S˜T (y(t))Z S(y(t)) −

y T (t)U y(t)

k=1

k=1 K ∑

K ∑

K ∑

y˙ T (t − τk )Ry(t ˙ − τk )

k=1 K ∑

˜ S˜T (y(t − τk ))Z S(y(t − τk ))

k=1

+

y T (t)(−P (C + ∆C − KG) − (C + ∆C − KG)T P )y(t) K [ ∑ 1 2 T ˜ y (t)(−Gy(t − τk ) + (T0 + ∆T0 )S(y(t)) K K k=1 ∫ t ˜ y(s)ds) ˙ (Tk + ∆Tk )S(y(t − τk )) − G

+

y (t)U y(t) − y (t − τk )U y(t − τk ) + ςkT (t)F˜ ςk (t)

+

τk ςkT (t)F T Q−1 F ςk (t)

+

τk y˙ T (t)Qy(t) ˙ + y˙ T (t)Ry(t) ˙ − y˙ T (t − τk )Ry(t ˙ − τk ) ] T T ˜ ˜ ˜ ˜ S (y(t))Z S(y(t)) − S (y(t − τk ))Z S(y(t − τk )) .

= +

t−τk

+

T

T

As a tool for deriving a less conservative stability criterion, we add the following one zero equation with a matrix M i (i =1,2,. . . ,6) to be chosen as [ (∫ )T 2

y T (t)M1 + y T (t − τk )M2 +

t

M3 + y˙ T (t)M4 + y˙ T (t − τk )M5

y(s)ds ˙ t−τk

+ S˜T (y(t − τk ))M6 ] × [y(t) − y(t − τk ) −

∫

t

] y(s)ds ˙ = 0.

t−τk

(16)

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The equality (16) can be represented as K ∑

ςkT (t)Ξ1 ςk (t) = 0,

k=1

where

[

(∫

ςkT (t) = y T (t), y T (t − τk ),

t

] )T y(s)ds ˙ , y˙ T (t), y˙ T (t − τk ), S˜T (y(t − τk )) ,

t−τk

 M1 + M1T −M1 + M2T −M1 + M3T M4T M5T M6T  ∗ −M2 − M2T −M2 − M3T −M4T −M5T −M6T     ∗ ∗ −M3 − M3T −M4T −M5T −M6T  (17)  . Ξ1 =  ∗ ∗ ∗ 0 0 0     ∗ ∗ ∗ ∗ 0 0  ∗ ∗ ∗ ∗ ∗ 0 

Let us note that ˜ ˜ S˜T (y(t))Z S(y(t)) ≤ y T (t)LZy(t), 2y T (t)P (T0 + ∆T0 )S(y(t)) ≤ 2y T (t)P (T0 + ∆T0 )Ly(t). ∑ T Then we can get V˙ (t) ≤ K k=1 ςk (t)Πk ςk (t), where ΠK = γK + τ KF T Q−1 F,

(18)



Υk

 Υk11 Υk12 Υk13 M4T M5T M6T  ∗ Υk22 Υk23 −M4T −M5T −M6T     ∗ ∗ Υk33 F4 F5 F6   , =   ∗ ∗ ∗ τ Q + R 0 0 k    ∗ ∗ ∗ ∗ −R 0  ∗ ∗ ∗ ∗ ∗ −Z

Υk11 = −P (C + ∆C − KG) − (C + ∆C − KG)T P +

Υk12 Υk13 Υk23

2L K2

(T0 + ∆T0 ) + U + L2 Z + M1 + M1T , 2G L = − + (Tk + ∆Tk ) − M1 + M2T , K K 2G = − + F1T − M1 + M3T , Υk22 = −U − M2 − M2T , K = −M2 − M3T + F2T , Υk33 = F3 + F3T − M3 − M3T .

We take γK =

∑ K

+ΩK ,

k = 1, 2, . . . , K

(19)

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A LMI APPROACH TO GLOBAL . . .

where

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

Σk =

Σk11 = + Σk12 =

Ωk =

 Σk11 Σk12 Υk13 M4T M5T M6T  ∗ Υk22 Υk23 −M4T −M5T −M6T     ∗ ∗ Υk33 F4 F5 F6   ,  ∗ ∗ ∗ τk Q + R 0 0     ∗ ∗ ∗ ∗ −R 0  ∗ ∗ ∗ ∗ ∗ −Z [∆C, ∆T0 , · · · ∆TK ] = HF [A, B0 , · · · , BK ], 2L −P (C − KG) − (C − KG)T P + 2 T0 + U + L2 Z + M1 K 2L M1T + 2 HF B0 , K 2G L − + Tk − M1 + M2T , K K   L ∆Tk 0 0 0 0 Ωk11 K  ∗ 0 0 0 0 0    ∗  ∗ 0 0 0 0  ,  ∗ ∗ ∗ 0 0 0    ∗ ∗ ∗ ∗ 0 0 ∗

∗

∗ ∗ ∗ 0

Ωk11 = −P ∆C − (∆C) P, T

Ωk11 = −P HF A − AT F T H T P. If we want to get the following equation: Ωk = Ω1k + Ω2k + Ω3k < 0, where



Ω1k

Ω2k

 PH  0     0    F [ −A 0 0 0 0 0 ], =    0   0  0   −AT  0     0  T  F [ H T P 0 0 0 0 0 ],  =    0   0  0

(20)

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59



Ω3k

Ωk

 H  0     0  L  =   0  F [ K Bk 0 0 0 0 0 ],    0  0 = Ω1k + Ω2k + nI6n2 ×6n2 + Ω3k − nI6n2 ×6n2 < 0.

By Lemma 1, we can get that ∀F T F ≤ I if and only if there is a constant ε > 0, diag {nIn×n , nIn×n , nIn×n , nIn×n , nIn×n , nIn×n } + ε−1 diag{−P HH T P, 0, 0, 0, 0, 0} + εdiag{AT A, 0, 0, 0, 0, 0} = diag{nIn×n − ε−1 − P HH T P + εAT A, nIn×n , nIn×n , nIn×n , nIn×n , nIn×n } < 0. (21) Taking a proper value of the constant n, we can get Ω3K − nI6n×6n < 0 clearly, L diag{−nIn×n + HF Bk , −nIn×n , −nIn×n , −nIn×n , −nIn×n , −nIn×n } < 0. (22) K By combinging (17), (18), (19), (20), and (21), we can get  ¯  ¯ k12 Υk13 M4T M5T M6T Υk11 Υ  ∗ Υk22 Υk23 −M4T −M5T −M6T     ∗ ∗ Υk33 F4 F5 F6  ¯   + τk F T Q−1 F < 0, Υk =  (23) ∗ ∗ τk Q + R 0 0   ∗   ∗ ∗ ∗ ∗ −R 0  ∗ ∗ ∗ ∗ ∗ −Z where ¯ k11 = −P (C − KG) − (C − KG)T P + 2L T0 + U + L2 Z + M1 + M1T Υ K2 L 2L 1 + HF B0 − P HH T P + εAT A + HF Bk , K2 ε K ¯ k12 = − 2G + L Tk − M1 + M T , Υk13 = − 2G + F T − M1 + M T , Υ 2 1 3 K K K T T T Υk22 = −U − M2 − M2 , Υk23 = −M2 − M3 + F2 , Υk33 = F3 + F3T − M3 − M3T . If the following LMI (24) is satisfied   ¯ ¯ k12 Υk13 Υk11 Υ M4T M5T M6T F1T  ∗ Υk22 Υk23 −M4T −M5T −M6T F2T     ∗ ∗ Υk33 F4 F5 F6 F3T     ∗ ∗ ∗ τk Q + R 0 0 F4T   < 0.  T  ∗ ∗ ∗ ∗ −R 0 F5     ∗ ∗ ∗ ∗ ∗ −Z F6T  ∗ ∗ ∗ ∗ ∗ ∗ −τk Q

(24)

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we can get Πk = Υk + τk F T Q−1 F < 0. Then we can get V˙ (t) ≤

K ∑

ςkT (t)Πk ςk (t) ≤ 0.

k=1

Thus we complete the proof. Theorem 2: For given τK ≥ 0 (k = 1, 2, . . . , K), L = diag{l1 , l2 , . . . , ln } the equilibrium point of Eq. (5) is globally asymptotically stable if there exist positive definite matrices P , U , Z, Q, and any matrices Mi , Fi (i = 1, 2, . . . , 6), satisfying the following LMI:   ¯1 0 0 0 0 0 Υ  ∗ Υ ¯2 0 0 0 0       ∗ ∗ ... 0 0 0    (25)  ∗ ∗ ∗ Υ ¯ k 0 0  < 0,      ∗ ∗ ∗ ∗ ... 0  ¯K ∗ ∗ ∗ ∗ ∗ Υ where

¯k Υ

¯ k11 Υ

¯ k12 Υ Υk22

 ¯  ¯ k12 Υk13 Υk11 Υ M4T M5T M6T  ∗ Υk22 Υk23 −M4T −M5T −M6T     ∗ ∗ Υk33 F4 F5 F6    + τk F T Q−1 F, =   ∗ ∗ ∗ τ Q + R 0 0 k    ∗ ∗ ∗ ∗ −R 0  ∗ ∗ ∗ ∗ ∗ −Z 2L = −P (C − KG) − (C − KG)T P + 2 T0 + U + L2 Z + M1 + M1T K 2L 1 L + HF B0 − P HH T P + εAT A + HF Bk , 2 K ε K 2G L 2G T = − + Tk − M1 + M2 , Υk13 = − + F1T − M1 + M3T , K K K = −U − M2 − M2T , Υk23 = −M2 − M3T + F2T ,

Υk33 = F3 + F3T − M3 − M3T . Proof. If we let ς T (t)ΛΥ ς(t) ≤ 0,

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61

[ ] T (t) , where ς T (t) = ς1T (t), ς2T (t), . . . , ςkT (t), . . . , ςK   ¯1 0 0 0 0 0 Υ  ∗ Υ ¯2 0 0 0 0      . .  ∗ ∗ . 0 0 0   , ΛΥ =   ¯  ∗ ∗ ∗ Υk 0 0     ∗ ∗ ∗ ∗ ... 0  ¯K ∗ ∗ ∗ ∗ ∗ Υ we can easily get V˙ (t) ≤

K ∑

ςkT (t)Πk ςk (t) ≤ 0.

k=1

The rest of the proof process of Theorem 2 is similar with that of Theorem 1, so here we have omitted it.

IV. AN EXAMPLE

We will through the following numerical simulation of the system (1) verify the validity of the Theorem 1. At this point, only consider the case there are two time-delays, the system model is x(t) ˙ = −(C + ∆C)x(t) + (T0 + ∆T0 )S(x(t)) +

2 ∑

(Tk + ∆Tk )S(x(t − τk )) + I,

k=1

where

[ C=

] 7.9 0 , 0 6.9

[ T0 =

] 3.1 1.9 , 1.0 1.5

[ T1 =

] 1.1 2.0 , 0.9 0.5

[ T2 =

S(x(t)) = [tanh(0.2x1 (t)) , tanh(0.4x2 (t)) ]T . From the linearization of S(x(t)) at the origin, we can get [ ] 0.2 0 Seq = , [∆C ∆T0 ∆T1 ∆T2 ] = HF [A B0 B1 B2 ], 0 0.4 [ ] [ ] [ ] 0.1 0 0.2 0.3 0.2 −0.3 where A = , B0 = , B1 = , 0 0.5 0.1 0.1 0.1 0.1 [ ] [ ] [ ] 0.2 −0.03 1.0 0.3 sin t 0 B2 = , H= , F (t) = . 0.01 0.01 0.2 0 0 sin t

] 6.9 2.0 , 3.0 5.0

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A LMI APPROACH TO GLOBAL . . .

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From an application of the Matlab software in the LMI toolbox to solve the corresponding linear matrix inequalities it can be learned that there are four definite matricies: [ ] [ ] [ ] [ ] 0.27 −0.01 1.103 0 1.101 0 1.103 0 P = , R= , Z= , Q= , −0.01 0.35 0 1.07 0 1.069 0 1.060 and a normal number ε = 1.165. Based on Lyapunov stability, the equilibrium point x = 0 of system (1) is globally stable if LMI (8) holds, where time-delay τK (k = 1, 2) are arbitrary bounded. Set the system’s initial function Φ(s) = [0.8, −0.4]T , F (t) = I2. The numerical simulation is shown in Fig. 1.

FIG. 1: The system state diagram at (a) τ1 = 0.3s, τ2 = 1.1s, (b) τ1 = 0.6s, τ2 = 1.0s.

V. CONLUSION

In this paper, the synchronization problem has been studied for MDHNNs with parameter perturbations. Based on the LMI technique, two convenient solutions can be obtained. Furthermore, the methods we have given in this paper are based on criteria given in the form of LMIs, so one can use the interior-point method in the Matlab environment. The results of a lot of existing papers given by the system convergence criterion are algebraic inequalities, so there is no systematic way to verify the correctness of these inequalities and the procedures in these documents are difficult to apply in practice. Therefore, the results we have given in this paper are more general.

Acknowledgements This work was supported by the National Natural Science Foundations of China (Grant Nos. 70571030 and 90610031), the Society Science Foundation from the Ministry of Education of China (Grant Nos. 12YJAZH002, 08JA790057), and the Advanced Talents’ Foundation and Student’s Foundation of Jiangsu University (Grant Nos. 07JDG054, 07A075 and 10A147).

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