New approaches on stability criteria for neural networks with interval time-varying delays

Applied Mathematics and Computation 218 (2012) 9953–9964

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

New approaches on stability criteria for neural networks with interval time-varying delays O.M. Kwon a, S.M. Lee b, Ju H. Park c,⇑, E.J. Cha d a

School of Electrical Engineering, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 361-763, Republic of Korea School of Electronics Engineering, Daegu University, Gyungsan 712-714, Republic of Korea c Nonlinear Dynamics Group, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea d Department of Biomedical Engineering, School of Medicine, Chungbuk National University, 52 Naesudong-ro, Heungduk-gu, Cheongju 361-763, Republic of Korea b

a r t i c l e

i n f o

Keywords: Neural networks Time-varying delays Stability Lyapunov method

a b s t r a c t This paper concerns the problem of delay-dependent stability criteria for neural networks with interval time-varying delays. First, by constructing a newly augmented Lyapunov– Krasovskii functional and combining with a reciprocally convex combination technique, less conservative stability criterion is established in terms of linear matrix inequalities (LMIs), which will be introduced in Theorem 1. Second, by taking different interval of integral terms of Lyapunov–Krasovskii functional utilized in Theorem 1, further improved stability criterion is proposed in Theorem 2. Third, a novel approach which divides the bounding of activation function into two subinterval are proposed in Theorem 3 to reduce the conservatism of stability criterion. Finally, through two well-known numerical examples used in other literature, it will be shown the proposed stability criteria achieves the improvements over the existing ones and the effectiveness of the proposed idea. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Recently, neural networks have been found successful applications in various fields including image and signal processing, pattern recognition, fault diagnosis, associative memories, fixed-point computations, combinatorial optimization, and other scientific areas (for instance, see [1–4]). Since these applications heavily depend on the dynamic behavior of the equilibrium point, the stability analysis of the equilibrium points of the designed network has been one of important issue. In the implementation of neural networks, time delays frequently occur due to the finite switching speed of amplifies and may cause instability or oscillation of neural networks. Therefore, considerable efforts have been done to asymptotic stability analysis of neural networks with time-delays [5–36]. Especially, delay-dependent stability analysis has been investigated by many researchers [18–36] since it is well known that delay-dependent stability criteria are generally less conservative than delay-independent ones when the sizes of time-delays are small. In the field of delay-dependent stability analysis, an important index for checking the conservatism is to find maximum delay bounds such that the asymptotic stability of the designed network can be guaranteed for any admissible delays less than maximum delay bounds. Thus, how to choose Lyapunov–Krasovskii functional and obtain an upper bound of timederivative of it play key roles to increase the feasible region of stability criteria. Since Li et al. [21] pointed that the stability conditions are hardly improved by using the same Lyapunov–Krasovskii functional, delay-partitioning approach, which was firstly introduced by Gu [38], has been attracted by many researchers. One of the

⇑ Corresponding author. E-mail addresses: [email protected] (O.M. Kwon), [email protected] (S.M. Lee), [email protected] (J.H. Park), [email protected] (E.J. Cha). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.082

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main advantage of this approach is to estimate more tighter upper bounds than the the results without delay-partitioning approach. In this regard, with the idea of delay-partitioning, new exponential stability criterion for neural networks with constant time-delay was investigated in [29]. Hu, Gao and Zheng [30] investigated the asymptotic stability for a class of cellular neural networks with interval time-varying delays by introducing a novel Lyapunov functional which divide the lower bound of the time-varying delay. The stability analysis using delay-partitioning was further investigated for stochastic Hopfield neural networks with constant time-delays and norm-bounded parameter uncertainties [31]. In [32], a piecewise delay method which divides delay interval into two was proposed to reduce the conservatism of delay-dependent stability criteria for neural networks with interval time-varying delays. Recently, the idea of the proposed method [30] was extended to the problem of exponential stability on stochastic neural networks with discrete interval and distributed delays [33]. Very recently, by employing the improved delay-partitioning technique and general convex combination, delay-derivative-dependent stability criteria for delayed neural networks with unbounded distributed delay were proposed in [35] and further improvement results on [35] are in [36]. All of the works [29–36] mentioned focused on the application of delay-partitioning approach. However, as mentioned in [37], the choice of activation functions influence the ability and performance of neural networks. Thus, the bounding properties of activation function is another important factor to determine the feasible region of stability criteria. Motivated by this mentioned above,in this paper, three new delay-dependent stability criteria for neural networks with interval time-varying delays will be proposed by employing different approaches. The contribution of this paper lies in three aspects. 1. Unlike the method of [29–36], no delay-partitioning methods are utilized. Instead, by taking more information of states and activation functions as augmented vectors, an augmented Lyapunov–Krasovskii’s functional is proposed. Then, inspired by the work of [39–42], a sufficient condition such that the considered neural networks are asymptotically stable is derived in terms of linear matrix inequalities (LMIs) which will be presented in Theorem 1 and 2. 2. A novel approach partitioning the bounding of activation function will be proposed for the first time. As a tradeoff between time-consuming and improvement of the feasible region, the bounding of activation function is divided into two subintervals. Through two well-known numerical examples, it will be shown that in spite of no employing delay-partitioning approaches the proposed stability criteria can provide larger delay bounds than the recent results in which delay-partitioning techniques were utilized.Notation. Throughout this brief, Rn denotes n-dimensional Euclidean space, and Rnm is the set of all n  m real matrices. For symmetric matrices X and Y, the notation X > Y (respectively, X P Y) means that the matrix X  Y is positive definite, (respectively, nonnegative). diagf  g denotes the block diagonal matrix. I represents the elements below the main diagonal of a symmetric matrix. The subscript ‘T 0 denotes the transpose of the matrix. 2. Problem statement Consider the following neural networks with interval time-varying delays:

_ yðtÞ ¼ AyðtÞ þ W 0 gðyðtÞÞ þ W 1 gðyðt  hðtÞÞÞ þ b; T

ð1Þ

n

where yðtÞ ¼ ½y1 ðtÞ; . . . ; yn ðtÞ 2 R is the neuron state vector, n denotes the number of neurons in a neural network, gðyðtÞÞ ¼ ½g 1 ðy1 ðtÞÞ; . . . ; g n ðyn ðtÞÞT 2 Rn means the neuron activation function, gðyðt  hðtÞÞÞ ¼ ½g 1 ðy1 ðt  hðtÞÞÞ; . . . ; g n ðyn ðt  hðtÞÞÞT 2 Rn , A ¼ diagfai g 2 Rnn is a positive diagonal matrix, W 0 ¼ ðw0ij Þnn 2 Rnn and W 1 ¼ ðw1ij Þnn 2 Rnn are T the interconnection matrices representing the weight coefficients of the neurons, and b ¼ ½b1 ; b2 ; . . . ; bn  2 Rn represents a constant input vector. The delay, hðtÞ, is a time-varying continuous function satisfying

0 6 hL 6 hðtÞ 6 hU ;

_ hðtÞ 6 hD ;

ð2Þ

where hL and hU are positive scalars and hD is any constant one. The activation functions, g i ðyi ðtÞÞ; i ¼ 1; . . . ; n, are assumed to bounded and hold the following condition:

g i ðuÞ  g i ðv Þ þ 6 ki ; u; v 2 R; uv u – v ; i ¼ 1; . . . ; n; 

k 6

 ki

ð3Þ

þ ki

where and are constant values.  T For simplicity, in stability analysis of the neural networks (1), the equilibrium point y ¼ y1 ; . . . ; yn whose uniqueness  has been reported in [18] is shifted to the origin by utilizing the transformation xðÞ ¼ yðÞ  y , which leads the system (1) to the following form:

_ xðtÞ ¼ AxðtÞ þ W 0 f ðxðtÞÞ þ W 1 f ðxðt  hðtÞÞÞ; T

n

where xðtÞ ¼ ½x1 ðtÞ; . . . ; xn ðtÞ 2 R is the state vector of the transformed system, f ðxðtÞÞ ¼ ½f 1 ðx1 ðtÞÞ; . . . ; f n ðxn ðtÞÞ f j ðxj ðtÞÞ ¼ g j ðxj ðtÞ þ yj Þ  g j ðyj Þ with f j ð0Þ ¼ 0ðj ¼ 1; . . . ; nÞ.

ð4Þ T

and

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O.M. Kwon et al. / Applied Mathematics and Computation 218 (2012) 9953–9964

It should be noted that the activation functions f i ðÞði ¼ 1; . . . ; nÞ satisfy the following condition:

f i ðuÞ  f i ðv Þ þ 6 ki ; u; v 2 R; uv u – v ; i ¼ 1; . . . ; n: 

ki 6

If

ð5Þ

v ¼ 0 in (5), then we have 

ki 6

f i ðuÞ þ 6 ki ; u

8u – 0; i ¼ 1; . . . ; n;

ð6Þ

which is equivalent to

   þ  f i ðuÞ  ki u f i ðuÞ  ki u 6 0;

i ¼ 1; . . . ; n:

ð7Þ

The objective of this paper is to investigate the delay-dependent stability conditions of system (4) which will be conducted in Section 3. Before deriving our main results, we state the following lemmas. Lemma 1. For any constant positive-definite matrix M 2 Rnn and b 6 s 6 a, the following inequalities hold:

ða  bÞ

Z

a

_ P x_ T ðsÞM xðsÞds

b

Z

ða  bÞ2 2

a

b

Z

Z

a

T Z _ M xðsÞds

b

a

_ P x_ T ðuÞMxðuÞduds

Z

s

a

a

 _ xðsÞds ;

ð8Þ

b

T Z a _ M xðuÞduds

Z s

b

a

b

 a _ xðuÞuds :

Z

ð9Þ

s

Proof. According to Jensen’s inequality in [43], one can obtain (8). Moreover, the following inequality holds

ða  sÞ

Z

a

_ P x_ T ðuÞM xðuÞdu

Z

s

a

T Z _ M xðuÞdu

s

a

 _ xðuÞdu :

ð10Þ

s

By Schur Complements [44], Eq. (10) is equivalent to the following

"Ra s

# Ra T _ x_ T ðuÞM xðuÞdu x_ ðuÞdu s P 0: Ra _ xðuÞdu ða  sÞM1 s

ð11Þ

Integration of (11) from b to a yields

"Ra Ra b

_ x_ T ðuÞMxðuÞduds Ra Ra _ xðuÞduds b s s

Ra Ra b

Ra b

s

x_ T ðuÞduds

ða  sÞM 1 ds

# P 0:

ð12Þ

Therefore, the inequality (12) is equivalent to the inequality (9) according to Schur Complements. This complete the proof. h Lemma 2 [45]. Let f 2 Rn ; U ¼ UT 2 Rnn , and B 2 Rmn such that rankðBÞ < n. Then, the following statements are equivalent: (1) fT Uf < 0; Bf ¼ 0; f – 0, (2) ðB? ÞT UB? < 0, where B? is a right orthogonal complement of B.

3. Main results In this section, using augmented Lyapunov–Krasovskii functionals and some new approaches, novel delay-dependent stability criteria for systems (4) will be proposed. For the sake of simplicity of matrix representation, ei ði ¼ 1; . . . ; 17Þ 2 R17nn are defined as block entry matrices. (For example, eT3 ¼ ½0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0). The notations for some matrices are defined as:

" fðtÞ ¼ xT ðtÞ xT ðt  hðtÞÞ xT ðt  hL Þ xT ðt  hU Þ x_ T ðtÞ x_ T ðt  hL Þ x_ T ðt  hU Þ

Z

t

xT ðsÞds

thL

 f T ðxðtÞÞ f T ðxðt  hðtÞÞÞ f T ðxðt  hL ÞÞ f T ðxðt  hU ÞÞ

Z

t

thL

f T ðxðsÞÞds

Z

thL thðtÞ

Z

thL

xT ðsÞds

thðtÞ

f T ðxðsÞÞds

Z

thðtÞ

thU

Z

thðtÞ

thU

xT ðsÞds #

f T ðxðsÞÞds ;

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O.M. Kwon et al. / Applied Mathematics and Computation 218 (2012) 9953–9964









aðtÞ ¼ xT ðtÞ x_ T ðtÞ f T ðxðtÞÞ ; bðtÞ ¼ xT ðtÞ f T ðxðtÞÞ ; P1 ¼ ½ e1 e3 e4 e8 e9 þ e10 e15 e16 þ e17 ; P2 ¼ ½ e5 e6 e7 e1  e3 e3  e4 e11  e13 e13  e14 ; P3 ¼ ½ e1 e5 e11 ; P6 ¼ ½ e1 e11 ;

P4 ¼ ½ e3 e6 e13 ;

P7 ¼ ½ e2 e12 ;

P5 ¼ ½ e4 e7 e14 ;

P8 ¼ ½ e8 e1  e3 e15 ;

P9 ¼ ½ e9 e3  e2 e16 e10 e2  e4 e17 ;

C ¼ ½ A 0 0 0 I 0 0 0 0 0 W 0 W 1 0 0 0 0 0 ; U ¼ e11 KeT5 þ e5 KeT11 þ e1 K p DeT5 þ e5 DK p eT1  e11 DeT5  e5 DeT11 ; 

2

N¼ðh2L =2Þ2 e5 Q 3 eT5 ðhL e1 e8 ÞQ 3 ðhL e1 e8 ÞT þ ðh2U h2L Þ=2 e5 Q 4 eT5 ððhU hL Þe1  e9  e10 ÞQ 4 ððhU  hL Þe1  e9  e10 ÞT ; W ¼ hL e1 Q 5 eT1 þ hL e5 Q 6 eT5 þ ðhU  hL Þe1 Q 7 eT1 þ ðhU  hL Þe5 Q 8 eT5 þ e1 P1 eT1 þ e3 ðP1 þ P 2 ÞeT3 þ e2 ðP2 þ P3 ÞeT2  e4 P 3 eT4 ;  ¼ e1 ðK p þ K m ÞH1 eT11 þ e11 H1 ðK p þ K m ÞeT1  2e11 H1 eT11  2e1 K m H1 K p eT1 þ e2 ðK p þ K m ÞH2 eT12 þ e12 H2 ðK p þ K m ÞeT2  2e12 H2 eT12  2e2 K m H2 K p eT2 þ e3 ðK p þ K m ÞH3 eT13 þ e13 H3 ðK p þ K m ÞeT3  2e13 H3 eT13  2e3 K m H3 K p eT3 þ e4 ðK p þ K m ÞH4 eT14 þ e14 H4 ðK p þ K m ÞeT4  2e14 H4 eT14  2e4 K m H4 K p eT4 ;

R1 ¼ N þ U þ W þ P1 RPT2 þ P2 RPT1 þ P3 N PT3 þ P4 ðN þ MÞPT4  P5 MPT5 þ P6 GPT6  ð1  hD ÞP7 GPT7 " 2

þ hL P3 Q1 PT3  P8 Q1 PT8 þ ðhU  hL Þ2 P3 Q2 PT3  P9

Q2

S

I

Q2

#

PT9 :

ð13Þ

Now, we have the following theorem.





þ

Theorem 1. For given positive scalars hL and hU , any scalar hD , diagonal matrices K m ¼ diagfk1 ; . . . ; kn g and K p ¼ diagfk1 ; þ _ . . . ; kn g, the system (1) is asymptotically stable for 0 6 hL 6 hðtÞ 6 hU and hðtÞ 6 hD if there exist positive diagonal matrices   K ¼ diagfk1 ; . . . ; kn g, D ¼ diagfd1 ; . . . ; dn g, Hi ¼ diagfhi1 ; . . . ; hin gði ¼ 1; . . . ; 4Þ, positive definite matrices R ¼ Rij 77 2 R7n7n ;           M ¼ Mij 33 2 R3n3n , G ¼ Gij 22 2 R2n2n ; Q1 ¼ Q 1;ij 33 2 R3n3n ; Q2 ¼ Q 2;ij 33 2 R3n3n ; N ¼ N ij 33 2 R3n3n ,   Q i ði ¼ 3; . . . ; 8Þ, symmetric matrices Pi ði ¼ 1; . . . ; 3Þ, and any matrix S ¼ Sij 33 2 R3n3n , satisfying the following LMIs:

ðC? ÞT fR1 þ  gC? < 0; " # Q2 S > 0; I Q2 " # Q 5 P1 > 0; I Q6 " # Q 7 P2 > 0; I Q8 " # Q 7 P3 > 0; I Q8

ð14Þ ð15Þ

ð16Þ

ð17Þ

ð18Þ

where R1 ;  , and C are defined in (13), and C? is the right orthogonal complement of C. Proof. For positive diagonal matrices K; D and positive definite matrices R; N , M; G; Q1 ; Q2 ; Q i ði ¼ 3; . . . ; 8Þ let us consider P the following Lyapunov–Krasovskii’s functional candidate V ¼ 6i¼1 V i where

O.M. Kwon et al. / Applied Mathematics and Computation 218 (2012) 9953–9964

9957

3T 2 3 xðtÞ xðtÞ 6 xðt  hL Þ 7 6 xðt  hL Þ 7 7 6 7 6 7 6 7 6 6 xðt  hU Þ 7 6 xðt  hU Þ 7 7 6 R 7 6 R t t 7 6 7 6 6 thL xðsÞds 7 6 thL xðsÞds 7 V1 ¼ 6 R 7 R6 R 7; 7 6 thL 7 6 thL 6 th xðsÞds 7 6 th xðsÞds 7 U U 7 6R 7 6R 7 6 t 7 6 t 7 6 th f ðxðsÞÞds 7 6 th f ðxðsÞÞds L L 5 4 5 4 R thL R thL f ðxðsÞÞds f ðxðsÞÞds thU thU Z xi ðtÞ  n  P þ V2 ¼ 2 ki f i ðsÞ þ di ðki s  f i ðsÞÞ ds ; 2

V3 ¼

Z

0

i¼1 t

T

a ðsÞN aðsÞds þ

thL

Z

V 4 ¼ hL

thL 2

V 5 ¼ ðhL =2Þ V6 ¼

Z

t

thL

Z

Z

aT ðsÞMaðsÞds þ

t

aT ðuÞQ1 aðuÞduds þ ðhU  hL Þ

s

Z

t

thL t

thL

thU

Z

t

Z

t

t

t

u

bðsÞT GbðsÞds;

thðtÞ

Z

thL

thU

Z

s

Z

Z

t

aT ðuÞQ2 aðuÞduds;

s

_ v Þdv duds; þððh2U  h2L Þ=2Þ x_ T ðv ÞQ 3 xð

 T _ x ðuÞQ 5 xðuÞ þ x_ T ðuÞQ 6 xðuÞ duds þ

Z

s

thL

thU

Z

t

Z



thL

thU

Z s

t

Z

t

_ v Þdv duds; x_ T ðv ÞQ 4 xð

u

_ xT ðuÞQ 7 xðuÞ þ x_ T ðuÞQ 8 xðuÞ duds:

ð19Þ

s

The time-derivative of V 1 is calculated as

2

3T

xðtÞ

3 2 _ xðtÞ 6 xðt  hL Þ 7 7 6 7 6 _  hL Þ xðt 7 6 7 6 7 6 xðt  hU Þ 7 6 7 6 7 6 R _  hU Þ xðt 7 6 t xðsÞds 7 6  7 6 7 T T T thL _V 1 ¼ 26 xðtÞ  xðt  hL Þ 7 R6 7 ¼ f ðtÞ P1 RP2 þ P2 RP1 fðtÞ: 6 R 7 6 7 6 thL 6 th xðsÞds 7 6 xðt  hL Þ  xðt  hU Þ 7 U 7 6 7 6R 7 6 7 6 t 6 th f ðxðsÞÞds 7 4 f ðxðtÞÞ  f ðxðt  hL ÞÞ 5 L 5 4 R thL f ðxðt  hL ÞÞ  f ðxðt  hU ÞÞ f ðxðsÞÞds thU

ð20Þ

By calculation of V_ 2 , we have

  T _ þ 2 K p xðtÞ  f ðxðtÞÞ T DxðtÞ _ V_ 2 ¼ 2f ðxðtÞÞKxðtÞ ¼ fT ðtÞUfðtÞ:

ð21Þ

_ With the condition hðtÞ 6 hD , an upper bound of V 3 is obtained as

V_ 3 6 aT ðtÞN aðtÞ  aT ðt  hL ÞN aðt  hL Þ þ aT ðt  hL ÞMaðt  hL Þ  aT ðt  hU ÞMaðt  hU Þ þ bT ðtÞGbðtÞ   ð1hD ÞbT ðthðtÞÞGbðthðtÞÞ¼fT ðtÞ P3 N PT3 þ P4 ðN þ MÞPT4  P5 MPT5 þ P6 GPT6 ð1hD ÞP7 GPT7 fðtÞ: By use of Lemma 1 and Theorem 1 in [40], an estimation of V_ 4 is 2 V_ 4 ¼ hL aT ðtÞQ1 aðtÞ  hL

 ðhU  hL Þ

Z

Z

t

aT ðsÞQ1 aðsÞds þ ðhU  hL Þ2 aT ðtÞQ2 aðtÞ  ðhU  hL Þ

Z

thL

thL

ð22Þ

aT ðsÞQ2 aðsÞds

thðtÞ

thðtÞ

aT ðsÞQ2 aðsÞds thU

2

6 hL aT ðtÞQ1 aðtÞ 

Z

t

T

aðsÞds Q1

thL

t



aðsÞds þ ðhU  hL Þ2 aT ðtÞQ2 aðtÞ

thL

!T !  Z thðtÞ Z thðtÞ hU  hL aðsÞds Q2 aðsÞds  aðsÞds Q2 aðsÞds hU  hðtÞ thðtÞ thðtÞ thU thU Z t T Z t  2 aðsÞds Q1 aðsÞds þ ðhU  hL Þ2 aT ðtÞQ2 aðtÞ 6 hL aT ðtÞQ1 aðtÞ  

hU  hL  hðtÞ  hL

2R

thL thðtÞ

 4 R thðtÞ

 Z

aðsÞds

thL

3T 5

thL



Q2 I

!T

Z Z

thL

!



thL

2 3  R thL S 4 thðtÞ aðsÞds 5 R thðtÞ Q2 aðsÞds thU

aðsÞds th U

Q2 2 ¼ fT ðtÞ hL P3 Q1 PT3  P8 Q1 PT8 þ ðhU  hL Þ2 P3 Q2 PT3  P9 I

 S PT9 fðtÞ: Q2

ð23Þ

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O.M. Kwon et al. / Applied Mathematics and Computation 218 (2012) 9953–9964

By Lemma 2, V_ 5 is bounded as 2 _  ðh2L =2Þ V_ 5 ¼ ðhL =2Þ2 x_ T ðtÞQ 3 xðtÞ

 Z 2 2  ðhU  hL Þ=2

thL

thU

2

_  6 ðhL =2Þ2 x_ T ðtÞQ 3 xðtÞ 2

2

Z

Z

Z

Z

t

thL

t s

2 2 _ _ x_ T ðuÞQ 3 xðuÞduds þ ððhU  hL Þ=2Þ2 x_ T ðtÞQ 4 xðtÞ

t

_ x_ T ðuÞQ 4 xðuÞduds s t

thL

Z

t

T Z _ Q3 xðuÞduds

s

_  þ ððhU  hL Þ=2Þ2 x_ T ðtÞQ 4 xðtÞ

Z

t

thL

Z

Z

thL

thU

!T

t

_ xðuÞduds

t

 _ xðuÞduds

s

Z

Q4

thL

thU

s

Z

!

t

_ xðuÞduds

¼ fT ðtÞNfðtÞ:

ð24Þ

s

Calculation of V_ 6 leads to

V_ 6 ¼ hL xT ðtÞQ 5 xðtÞ  Z



Z

t

_  xT ðsÞQ 5 xðsÞds þ hL x_ T ðtÞQ 6 xðtÞ

thL thL

_  xT ðsÞQ 7 xðsÞds þ ðhU  hL Þx_ T ðtÞQ 8 xðtÞ

thU

Z

Z

t

_ x_ T ðsÞQ 6 xðsÞds þ ðhU  hL ÞxT ðtÞQ 7 xðtÞ

thL thL

_ x_ T ðsÞQ 8 xðsÞds:

ð25Þ

thU

Inspired by the work of [42], the following three zero equality with any symmetric matrices P 1 ; P2 , and P 2 are considered:

0 ¼ xT ðtÞP1 xðtÞ  xT ðt  hL ÞP1 xðt  hL Þ  2

Z

t

_ xT ðsÞP1 xðsÞds;

thL

0 ¼ xT ðt  hL ÞP2 xðt  hL Þ  xT ðt  hðtÞÞP2 xðt  hðtÞÞ  2

Z

thL

_ xT ðsÞP2 xðsÞds;

thðtÞ

0 ¼ xT ðt  hðtÞÞP 3 xðt  hðtÞÞ  xT ðt  hU ÞP3 xðt  hU Þ  2

Z

thðtÞ

_ xT ðsÞP3 xðsÞds:

ð26Þ

thU

With the above three zero equalities, an upper bound of V_ 6 is



 



Z thL

xðsÞ xðsÞ T Q 5 P1 xðsÞ T Q 7 ds _ _ _ I Q 6 xðsÞ I thðtÞ xðsÞ thL xðsÞ 

 

Z thðtÞ

xðsÞ xðsÞ T Q 7 P3 ds:  _ _ xðsÞ I Q 8 xðsÞ thU

V_ 6 6fT ðtÞWfðtÞ

Z

t



P2 Q8



 xðsÞ ds _ xðsÞ ð27Þ

From (7), for any positive diagonal matrices H1 ¼ diagfh11 ; . . . ; h1n g; H2 ¼ diagfh21 ; . . . ; h2n g; H3 ¼ diagfh31 ; . . . ; h3n g, and H4 ¼ diagfh41 ; . . . ; h4n g, the following inequality holds

0 6 2

n P

i¼1

n      P  þ  h1i f i ðxi ðtÞÞ  ki xi ðtÞ f i ðxi ðtÞÞ  ki xi ðtÞ  2 h2i f i ðxi ðt  hðtÞÞÞ  ki xi ðt  hðtÞÞ i¼1

n      P þ  þ  f i ðxi ðt  hðtÞÞÞ  ki xi ðt  hðtÞÞ  2 h3i f i ðxi ðt  hL ÞÞ  ki xi ðt  hL Þ f i ðxi ðt  hL ÞÞ  ki xi ðt  hL Þ i¼1

2

n P

i¼1

    þ h4i f i ðxi ðt  hU ÞÞ  ki xi ðt  hU Þ f i ðxi ðt  hU ÞÞ  ki xi ðt  hU Þ ¼ fT ðtÞ fðtÞ:

ð28Þ

From Eqs. (19)–(28) and by application of S-procedure [44], if Eqs. (16)–(18) hold, then an upper bound of V_ is

V_ 6 fT ðtÞfR1 þ  gfðtÞ;

ð29Þ

where R1 are defined in (13). By Lemma 2, fT ðtÞðR1 þ  ÞfðtÞ < 0 with 0 ¼ CfðtÞ is equivalent to ðC? ÞT ðR1 þ  ÞC? < 0. Therefore, if LMIs (14)–(18) hold, then the neural networks (4) is asymptotically stable. This completes our proof. h

Remark 1. In Theorem 1, the augmented vector fðtÞ has integrating terms of activation function f ðxðtÞÞ which are R thL R thðtÞ Rt f ðxðsÞÞds; thðtÞ f ðxðsÞÞds and thU f ðxðsÞÞds. By taking these integral terms as augmented vector which has not been conthL sidered in other literature, more past history of f ðxðtÞÞ can be utilized, which may lead less conservative results. Remark 2. Recently, the reciprocally convex optimization technique to reduce the conservatism of stability criteria for systems with time-varying delays was proposed in [40]. Motivated by this work, the proposed method of [40] was utilized in Eq. R thL R thðtÞ R thL R thðtÞ (23). However, an augmented vector with thðtÞ xðsÞds; thU xðsÞds, thðtÞ f ðxðsÞÞds, and thU f ðxðsÞÞds was used in Theorem 1, which is different from the method of [40].

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O.M. Kwon et al. / Applied Mathematics and Computation 218 (2012) 9953–9964

R th R t Remark 3. In the proposed Lyapunov–Krasovskii’s functional of Theorem 1, the terms ðhU  hL Þ thUL s aT ðuÞQ2 aðuÞduds, R thL R t R t T R thL R t T 2 2 _ v Þdv duds, and th s ðx ðuÞQ 7 xðuÞ þ x_ T ðuÞQ 8 xðuÞÞduds _ ððhU  hL Þ=2Þ thU s u x_ ðv ÞQ 4 xð were proposed at each V 4 ; V 5 , and U V 6 , respectively. In double integral terms, we have t  hU 6 s 6 t  hL and s 6 u 6 t. Also, in triple integral terms, one can confirm t  hU 6 s 6 t  hL ; s 6 u 6 t, and u 6 v 6 t. Since the parameter s has the integral interval from t  hU to t  hL , it may be effective that the maximum values of u and v are changed as t  hL instead of t. With this regard, in Theorem 2, R th R th R th R th R th R thL R thL T _ v Þdv duds, and th the terms ðhU  hL Þ thUL s L aT ðuÞQ2 aðuÞduds, ððhU  hL Þ2 =2Þ thUL s L u L x_ T ðv ÞQ 4 xð ðx ðuÞ s U _ Q 7 xðuÞ þ x_ T ðuÞQ 8 xðuÞÞduds will proposed. Next, based on the results of Theorem 1 and Remark 3, improved stability criteria for system (1) will be introduced as Theorem 2 by taking different interval of integral terms of V i ði ¼ 4; 5; 6Þ. The notations for some matrices which will be utilized in Theorem 2 are defined as Pa ¼ ½ e3 e6

e13 ;  2 2 2 e N ¼ ðhL =2Þ e5 Q 3 eT5  ðhL e1  e8 ÞQ 3 ðhL e1  e8 ÞT þ ðhU  hL Þ2 =2 e6 Q 4 eT6  ððhU  hL Þe3  e9  e10 ÞQ 4 ððhU  hL Þe3  e9  e10 ÞT ; e ¼ hL e1 Q 5 eT þ hL e5 Q 6 eT þ ðhU  hL Þe3 Q 7 eT þ ðhU  hL Þe6 Q 8 eT þ e1 P1 eT þ e3 ðP1 þ P2 ÞeT þ e2 ðP 2 þ P3 ÞeT  e4 P 3 eT ; W 1 5 3 6 1 3 2 4

e þ P1 RPT þ P2 RPT þ P3 N PT þ P4 ðN þ MÞPT  P5 MPT þ P6 GPT  ð1  hD ÞP7 GPT þ h2 P3 Q1 PT e þUþW R2 ¼ N 2 1 3 4 5 6 7 3 L  P8 Q1 PT8 þ ðhU  hL Þ2 Pa Q2 PTa  P9



Q2 I

 S PT9 : Q2

ð30Þ

and other notations expressed in R2 are the same ones as in Eq. (13). Now, we have the following theorem. 



þ

Theorem 2. For given positive scalars hL and hU , any scalar hD , diagonal matrices K m ¼ diagfk1 ; . . . ; kn g and K p ¼ diagfk1 ; þ _ . . . ; kn g, the system (1) is asymptotically stable for 0 6 hL 6 hðtÞ 6 hU and hðtÞ 6 hD if there exist positive diagonal matrices   K ¼ diagfk1 ; . . . ; kn g, D ¼ diagfd1 ; . . . ; dn g, Hi ¼ diagfhi1 ; . . . ; hin gði ¼ 1; . . . ; 4Þ, positive definite matrices R ¼ Rij 77           2 R7n7n ; N ¼ N ij 33 2 R3n3n , M ¼ M ij 33 2 R3n3n , G ¼ Gij 22 2 R2n2n ; Q1 ¼ Q 1;ij 33 2 R3n3n ; Q2 ¼ Q 2;ij 33   2 R3n3n ; Q i ði ¼ 3; . . . ; 8Þ, symmetric matrices P i ði ¼ 1; . . . ; 3Þ, and any matrix S ¼ Sij 33 2 R3n3n , satisfying the following LMIs:

ðC? ÞT fR2 þ  gC? < 0;

 Q2 S > 0; I Q2

 Q 5 P1 > 0; I Q6

 Q 7 P2 > 0; I Q8

 Q 7 P3 > 0; I Q8

ð31Þ ð32Þ ð33Þ ð34Þ ð35Þ

where R2 was defined in Eq. (30). Proof. For positive diagonal matrices K; D and positive definite matrices R; N , M; G; Q1 ; Q2 ; Q i ði ¼ 3; . . . ; 8Þ let us consider P the following Lyapunov–Krasovskii’s functional candidate V ¼ 6i¼1 V i where

V 4 ¼ hL

Z

Z

t

thL 2

V 5 ¼ ðhL =2Þ V6 ¼

Z

t

thL

Z

Z

t

aT ðuÞQ1 aðuÞduds þ ðhU  hL Þ

s t

Z

thL t

t

Z

thL

thU

Z

s

Z

t

thL

aT ðuÞQ2 aðuÞduds;

s

_ v Þdv duds; þððhU  hL Þ2 =2Þ x_ T ðv ÞQ 3 xð

u

Z

 T _ x ðuÞQ 5 xðuÞ þ x_ T ðuÞQ 6 xðuÞ duds þ

s

thL

thU

Z

thL

Z

thU

Z

thL



thL

s

Z

thL

_ v Þdv duds; x_ T ðv ÞQ 4 xð

u

_ xT ðuÞQ 7 xðuÞ þ x_ T ðuÞQ 8 xðuÞ duds;

ð36Þ

s

and V i ði ¼ 1; 2; 3Þ are the same ones as in Eq. (19). By using the similar method presented at Eq. (23), an upper bound of time-derivative of V_ 4 can be 2 V_ 4 6 hL aT ðtÞQ1 aðtÞ  hL

 ðhU  hL Þ

Z

thL thðtÞ

Z

t

aT ðsÞQ1 aðsÞds þ ðhU  hL Þ2 aT ðt  hL ÞQ2 aðt  hL Þ

thL

aT ðsÞQ2 aðsÞds  ðhU  hL Þ

Z

thðtÞ

aT ðsÞQ2 aðsÞds

thU



Q2 2 6 fT ðtÞ hL P3 Q1 PT3  P8 Q1 PT8 þ ðhU  hL Þ2 Pa Q2 PTa P9 I

S Q2



PT9 fðtÞ:

ð37Þ

9960

O.M. Kwon et al. / Applied Mathematics and Computation 218 (2012) 9953–9964

By calculating V_ 5 , we have 2 _  ðh2L =2Þ V_ 5 ¼ ðhL =2Þ2 x_ T ðtÞQ 3 xðtÞ

 Z  ðhU  hL Þ2 =2

thL

Z

thU 2

_  6 ðhL =2Þ2 x_ T ðtÞQ 3 xðtÞ Z



thL

Z

thU

Z

t

thL

thL

t

_ _  hL Þ x_ T ðuÞQ 3 xðuÞduds þ ððhU  hL Þ2 =2Þ2 x_ T ðt  hL ÞQ 4 xðt

s

_ x_ T ðuÞQ 4 xðuÞduds

s

Z

t

thL

!T

thL

Z

_ xðuÞduds s

Z

t

T Z _ xðuÞduds Q3

s

Q4

t

thL

Z

thL thU

Z

thL

Z

t

 _ _  hL Þ xðuÞduds þ ððhU  hL Þ2 =2Þ2 x_ T ðt  hL ÞQ 4 xðt

s

!

_ xðuÞduds

s

e fðtÞ: ¼ fT ðtÞ N

ð38Þ

From the result of V_ 6 , one can obtain

V_ 6 ¼ hL xT ðtÞQ 5 xðtÞ  

Z

Z

t

_  xT ðsÞQ 5 xðsÞds þ hL x_ T ðtÞQ 6 xðtÞ

thL thL

Z

t

_  hL Þ  xT ðsÞQ 7 xðsÞds þ ðhU  hL Þx_ T ðt  hL ÞQ 8 xðt

thU

_ x_ T ðsÞQ 6 xðsÞds þ ðhU  hL ÞxT ðt  hL ÞQ 7 xðt  hL Þ

thL

Z

thL

_ x_ T ðsÞQ 8 xðsÞds:

ð39Þ

thU

With the obtained results (37)–(39), the other procedure is straightforward from the proof of Theorem 1, so we omit it.

h

Remark 4. Since a delay-partitioning idea was firstly proposed in [38], it is well recognized that delay-partitioning approach can increase the feasible region of stability criteria owing to the fact that this method can obtain more tighter upper bounds obtained by calculating the time-derivative of Lyapunov–Krasovskii functional, which leads to less conservative results. However, when the number of delay-partitioning number increases, the matrix formulation becomes more complex and the computational burden and time-consuming grow bigger. Noticing this fact mentioned above, in Theorem 3, the bounding  þ   þ of activation function ki 6 fi ðuÞ 6 ki will be divided into two subintervals such as ki 6 fi ðuÞ 6 ðki þ ki Þ=2 and u u  þ þ fi ðuÞ ðki þ ki Þ=2 6 u 6 k instead of no using delay-partitioning approach. To the authors’ best knowledge, this approach has not been proposed. Through two numerical examples, it will be shown Theorem 3 significantly improves the feasible region of stability criterion comparing with those of Theorem 2. Finally, based on the results of Theorem 2 and Remark 4 mentioned above, a novel approach for delay-range-dependent stability criterion for system (4) will be introduced. For the sake of simplicity in matrix representation, the notations for some matrices of Theorem 3 are defined as

 a ¼ e1 ðð3K m þ K p Þ=2ÞH1 eT11 þ e11 H1 ðð3K m þ K p Þ=2ÞeT1  2e11 H1 eT11  2e1 K m H1 ððKm þ KpÞ=2ÞeT1 þ e2 ðð3K m þ K p Þ=2ÞH2 eT12 þ e12 H2 ðð3K m þ K p Þ=2ÞeT2  2e12 H2 eT12  2e2 K m H2 ððKm þ KpÞ=2ÞeT2 þ e3 ðð3K m þ K p Þ=2ÞH3 eT13 þ e13 H3 ðð3K m þ K p Þ=2ÞeT3  2e13 H3 eT13  2e3 K m H3 ððKm þ KpÞ=2ÞeT3 þ e4 ðð3K m þ K p Þ=2ÞH4 eT14 þ e14 H4 ðð3K m þ K p Þ=2ÞeT4  2e14 H4 eT14  2e4 K m H4 ððKm þ KpÞ=2ÞeT4 ;  b ¼ e1 ððK m þ 3K p Þ=2ÞH5 eT11 þ e11 H5 ððK m þ 3K p Þ=2ÞeT1  2e11 H5 eT11  2e1 ððKm þ KpÞ=2ÞH5 K p eT1 þ e2 ððK m þ 3K p Þ=2ÞH6 eT12 þ e12 H6 ððK m þ 3K p Þ=2ÞeT2  2e12 H6 eT12  2e2 ððKm þ KpÞ=2ÞH6 K p eT2 þ e3 ððK m þ 3K p Þ=2ÞH7 eT13 þ e13 H7 ððK m þ 3K p Þ=2ÞeT3  2e13 H7 eT13  2e3 ððKm þ KpÞ=2ÞH7 K p eT3 þ e4 ððK m þ 3K p Þ=2ÞH8 eT14 þ e14 H8 ððK m þ 3K p Þ=2ÞeT4  2e14 H8 eT14  2e4 ððKm þ KpÞ=2ÞH8 K p eT4 :

ð40Þ

Now, we have the following theorem. 



þ

þ

Theorem 3. For given positive scalars hL and hU , any scalar hD , diagonal matrices K m ¼ diagfk1 ; . . . ; kn g and K p ¼ diagfk1 ; . . . ; kn g, _ the system (1) is asymptotically stable for 0 6 hL 6 hðtÞ 6 hU and hðtÞ 6 hD if there exist positive diagonal matrices K ¼ diagfk ; . . . ; k g, D ¼ diagfd ; . . . ; d g, H ¼ diagfh ; . . . ; h gði ¼ 1; .. . ; 8Þ, positive definite R¼ n n 1 1 i i1 in         matrices Rij 77 2 R7n7n ; N ¼ N ij 33 2 R3n3n , M ¼ M ij 33 2 R3n3n , G ¼ Gij 22 2 R2n2n ; Q1 ¼ Q 1;ij 33 2 R3n3n ; Q2 ¼     Q 2;ij 33 2 R3n3n ; Q i ði ¼ 3; . . . ; 8Þ, symmetric matrices Pi ði ¼ 1; . . . ; 3Þ, and any matrix S ¼ Sij 33 2 R3n3n , satisfying the following LMIs:

ðC? ÞT fR2 þ  a gC? < 0;

ð41Þ

ðC? ÞT fR2 þ  b gC? < 0;

 Q2 S > 0; I Q2

ð42Þ ð43Þ

O.M. Kwon et al. / Applied Mathematics and Computation 218 (2012) 9953–9964



Q5

P1



I Q7

Q6  P2



I Q7

Q8  P3

I

Q8

9961

 > 0;

ð44Þ

> 0;

ð45Þ

> 0;

ð46Þ

where R2 , is defined in (30),  a , and  b are in Eq. (40). Proof. For positive diagonal matrices K; D and positive definite atrices R; N , M; G; Q1 ; Q2 ; Q i ði ¼ 3; . . . ; 8Þ let us consider the same Lyapunov–Krasovskii functional (36) proposed in Theorem 2.   þ Case I: ki 6 fi ðuÞ u 6 ðki þ ki Þ=2. From (7), for any positive diagonal matrices H1 ¼ diagfh11 ; . . . ; h1n g; H2 ¼ diagfh21 ; . . . ; h2n g; H3 ¼ diagfh31 ; . . . ; h3n g, and H4 ¼ diagfh41 ; . . . ; h4n g, the following inequality holds

0 6 2

n P

     þ h1i fi ðxi ðtÞÞ  ki xi ðtÞ fi ðxi ðtÞÞ  ððki þ ki Þ=2Þxi ðtÞ

i¼1 n P

2

2 2

i¼1 n P i¼1 n P i¼1

     þ h2i fi ðxi ðt  hðtÞÞÞ  ki xi ðt  hðtÞÞ fi ðxi ðt  hðtÞÞÞ  ððki þ ki Þ=2Þxi ðt  hðtÞÞ      þ h3i fi ðxi ðt  hL ÞÞ  ki xi ðt  hL Þ fi ðxi ðt  hL ÞÞ  ððki þ ki Þ=2Þxi ðt  hL Þ      þ h4i fi ðxi ðt  hU ÞÞ  ki xi ðt  hU Þ fi ðxi ðt  hU ÞÞ  ððki þ ki Þ=2Þxi ðt  hU Þ ¼ fT ðtÞ a fðtÞ:

ð47Þ

  þ Then, from the proof of Theorem 1 and 2, when ki 6 fi ðuÞ 6 ðki þ ki Þ=2, an upper bound of V_ can be u

V_ 6 fT ðtÞfR2 þ  a gfðtÞ

ð48Þ

with 0 ¼ CfðtÞ. Therefore, from Lemma 2 and S-procedure [44], if (41), (42), (42), (43, (46) hold, then system (4) is asymp  þ _ totically stable for 0 6 hL 6 hðtÞ 6 hU ; hðtÞ 6 hD , and ki 6 fi ðuÞ 6 ðki þ ki Þ=2. u  þ þ Case II: ðki þ ki Þ=2 6 fi ðuÞ 6 k . i u þ þ Note that the condition ki =2 6 fi ðuÞ u 6 ki is equivalent to

   þ þ  fi ðuÞ  ððki þ ki Þ=2Þu fi ðuÞ  ki u < 0;

i ¼ 1; . . . ; n:

ð49Þ

From (49), for any positive diagonal matrices H5 ¼ diagfh51 ; . . . ; h5n g; H6 ¼ diagfh61 ; . . . ; h6n g; H7 ¼ diagfh71 ; . . . ; h7n g, and H8 ¼ diagfh81 ; . . . ; h8n g, the following inequality holds

0 6 2

n P

    þ þ h5i fi ðxi ðtÞÞ  ððki þ ki Þ=2Þxi ðtÞ fi ðxi ðtÞÞ  ki xi ðtÞ

i¼1 n P

2

2 2

i¼1 n P i¼1 n P i¼1

    þ þ h6i fi ðxi ðt  hðtÞÞÞ  ððki þ ki Þ=2Þxi ðt  hðtÞÞ fi ðxi ðt  hðtÞÞÞ  ki xi ðt  hðtÞÞ     þ þ h7i fi ðxi ðt  hL ÞÞ  ððki þ ki Þ=2Þxi ðt  hL Þ fi ðxi ðt  hL ÞÞ  ki xi ðt  hL Þ     þ þ h8i fi ðxi ðt  hU ÞÞ  ððki þ ki Þ=2Þxi ðt  hU Þ fi ðxi ðt  hU ÞÞ  ki xi ðt  hU Þ ¼ fT ðtÞ b fðtÞ:

ð50Þ

Therefore, from Lemma 2 and S-procedure [44], if (42), (42), (42), (43, (46) hold, then system (4) is asymptotically stable for  þ þ _ 0 6 hL 6 hðtÞ 6 hU ; hðtÞ 6 hD , and ðki þ ki Þ=2 6 fi ðuÞ 6 ki . Thus, the feasibility of (42), (42), (42), (43, (46) means that system u  þ _ (4) is asymptotically stable for 0 6 hL 6 hðtÞ 6 hU ; hðtÞ 6 hD , and ki 6 fi ðuÞ 6 ki . This completes the proof of Theorem 3. h u _ Remark 5. When the information of an upper bound of hðtÞ is unknown or larger than one, Theorem 1, 2 also can provide delay-dependent stability criteria for (1) by letting G ¼ 0. 4. Numerical examples Example 1. Consider the neural networks (4) where

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O.M. Kwon et al. / Applied Mathematics and Computation 218 (2012) 9953–9964

Table 1 Delay bounds hU with hL ¼ 3 and different hD (Example 1). hD

0.1

0.5

0.9

Unknown (or hD P 1)

[30]ðm ¼ 2Þ [30]ðm ¼ 4Þ [36]ðm ¼ 2Þ Theorem 1 Theorem 2 Theorem 3

3.65 3.71 3.78 4.0071 4.0130 4.1967

3.32 3.36 3.45 3.3960 3.4470 3.6246

3.26 3.29 3.39 3.3033 3.3403 3.5961

3.24 3.28 3.38 3.2827 3.3196 3.5952

⁄ m is delay-partitioning number.

2

1:2769

0

0

0

0

0:6231

0

0

0

0

0:9230

0

0

0

0

0:4480

6 6 A¼6 6 4

3 7 7 7; 7 5

3 0:2336 7 6 6 1:6033 0:5988 0:3224 1:2352 7 7; W0 ¼ 6 7 6 4 0:3394 0:0860 0:3824 0:5785 5 2

0:0373

0:1311 2

0:8674

0:4852

0:3351

0:3253

0:9534 0:5015

1:2405 0:5325

6 6 0:0474 W1 ¼ 6 6 4 1:8495

0:9164

0:0360

2:6117

0:3788

0:5179

1:1734

2:0413

3

0:0220

7 0:9816 7 7; 7 0:8428 5 0:2775

K p ¼ diagf0:1137; 0:1279; 0:7994; 0:2368g:

ð51Þ

For this system, when hL ¼ 3, by dividing the lower bound of the time-varying delay, improved delay-dependent stability criterion was proposed in [30]. Very recently, with the consideration of the lower bound of delay derivative and delaypartitioning technique, less conservative results were presented in [36]. Table 1 gives the comparison results on the maximum delay bound allowed via the methods in recent works and our new study. From Table 1, it can be seen that Theorem 2

2

x1(t)

1.5

x (t) 2

x (t) 3

1

x4(t)

x(t)

0.5

0

−0.5

−1

−1.5

−2 0

10

20

30

40

50 time(sec)

60

70

80

Fig. 1. State trajectories of system (51) when hðtÞ ¼ 3 þ 0:59jsinð10tÞj.

90

100

9963

O.M. Kwon et al. / Applied Mathematics and Computation 218 (2012) 9953–9964 Table 2 Delay bounds hU with different hD (Example 2). hD ¼ 0:8 [23] [32] Theorem 1 Theorem 2 Theorem 3

hL ¼ 1

[23] [32] Theorem 1 Theorem 2 Theorem 3

hL ¼ 100

hD ¼ 0:9

Unknown hD (or hD P 1)

2.5967 3.8359 4.8278 4.8278 4.8668

2.0443 2.9234 3.6889 3.6889 3.8047

1.9621 2.7532 3.2975 3.2975 3.6001

101.5946 102.8335 103.7774 103.7776 103.8101

101.0443 101.9234 102.6887 102.6887 102.8047

100.9621 101.7532 102.2975 102.2975 102.6001

provides larger delay bounds than those of Theorem 1, which supports the effectiveness of the proposed idea introduced in Theorem 2. Note that Theorem 1 and 2 give larger delay bounds than those of [30] but fails the improvement of the feasible region when hD is 0.5, 0.9, and unknown comparing with the results of [36]. However, Theorem 3 significantly reduces the conservatism of Theorem 1 and 2 and provides larger delay bounds than the existing ones of [36] in spite of no utilizing delay-partitioning techniques. To confirm the obtained results of Theorem 3 when hðtÞ is unknown, the state trajectories of system (51) when hðtÞ ¼ 3 þ 0:59jsinð10tÞj are shown in Fig. 1. Example 2. Consider the neural networks (4) with the parameters

 1 1 W0 ¼ ; 0 2 1 1

 0:88 1 W1 ¼ ; K p ¼ diagf0:4; 0:8g; 1 1 A¼



2 0

 ;

K m ¼ diagf0; 0g:

ð52Þ

For this system, when hL ¼ 1 and hL ¼ 100, the maximum delay bounds obtained by the methods of [23,32] for various conditions of hD are listed in Table 2. In [32], by dividing delay interval into two and employing different free-weighting matrices at each interval, improved maximum delay bounds were obtained. With the same conditions presented in Table 2, the obtained results by applying Theorem 1, 2 to the above system (52) are also compared with the existing ones of [23,32]. From Table 2, one can see Theorem 1 which does not employ delay partitioning technique provides larger delay bounds than the results of [32]. Even though the results of Theorem 2 do not significantly enhance the feasible region of Theorem 1, Theorem 3 gives larger delay bounds than those of Theorem 2. This supports the effectiveness of the proposed idea in Theorem 3 in reducing the conservatism of stability criteria.

5. Conclusions In this paper, three delay-dependent stability criteria for neural networks with interval time-varying delays have been proposed by the use of Lyapunov method and LMI framework. In Theorem 1, by constructing the augmented Lyapunov– Krasovskii functional and utilizing reciprocal convex optimization approach introduced in [40], less conservative results of stability criterion has been proposed without the use of delay-partitioning techniques. Based on the results of Theorem 1, it was shown that improved feasible region of stability criterion can be obtained by modifying some intervals of integral terms in the proposed Lyapunov–Krasovskii functional. By dividing the bounding of activation functions into two, the further improved stability criterion was proposed in Theorem 3. Through two well-known examples, the improvement of the proposed stability criteria has been successfully verified. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0000479), and by a grant of the Korea Healthcare Technology R & D Project, Ministry of Health & Welfare, Republic of Korea (A100054). References [1] L.O. Chua, L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits Syst. 35 (1988) 1273–1290. [2] A. Cichocki, R. Unbehauen, Neural Networks for Optimization and Signal Processing, Wiley, Hoboken, NJ, 1993. [3] G. Joya, M.A. Atencia, F. Sandoval, Hopfield neural networks for optimization: study of the different dynamics, Neurocomputing 43 (2002) 219–237.

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