New delay-interval-dependent stability criteria for

Cogn Neurodyn DOI 10.1007/s11571-016-9396-y

RESEARCH ARTICLE

New delay-interval-dependent stability criteria for switched Hopfield neural networks of neutral type with successive time-varying delay components R. Manivannan1 • R. Samidurai1 • Jinde Cao2,3

•

Ahmed Alsaedi4

Received: 14 April 2016 / Revised: 23 June 2016 / Accepted: 8 July 2016 Springer Science+Business Media Dordrecht 2016

Abstract This paper deals with the problem of delay-interval-dependent stability criteria for switched Hopfield neural networks of neutral type with successive timevarying delay components. A novel Lyapunov–Krasovskii (L–K) functionals with triple integral terms which involves more information on the state vectors of the neural networks and upper bound of the successive time-varying delays is constructed. By using the famous Jensen’s inequality, Wirtinger double integral inequality, introducing of some zero equations and using the reciprocal convex combination technique and Finsler’s lemma, a novel delayinterval dependent stability criterion is derived in terms of linear matrix inequalities, which can be efficiently solved via standard numerical software. Moreover, it is also assumed that the lower bound of the successive leakage

and discrete time-varying delays is not restricted to be zero. In addition, the obtained condition shows potential advantages over the existing ones since no useful term is ignored throughout the estimate of upper bound of the derivative of L–K functional. Using several examples, it is shown that the proposed stabilization theorem is asymptotically stable. Finally, illustrative examples are presented to demonstrate the effectiveness and usefulness of the proposed approach with a four-tank benchmark real-world problem.

& Jinde Cao [email protected]

Introduction

R. Manivannan [email protected] R. Samidurai [email protected] Ahmed Alsaedi [email protected] 1

Department of Mathematics, Thiruvalluvar University, Vellore, Tamil Nadu 632 115, India

2

Department of Mathematics, and Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 210 096, China

3

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

4

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Keywords Hopfield neural networks Neutral type Leakage delay Interval time-varying delay Lyapunov– Krasovskii functional Four-tank benchmark

Over the past decades, switched neural networks (SNNs) have become a popular research topic that attracts researcher’s attention, various delayed neural networks such as Hopfield NNs, Cohen–Grossberg NNs, cellular NNs and bidirectional associative memory NNs have been extensively investigated. Switched systems are an important class of hybrid dynamical systems which are composed of a family of continuous-time or discrete-time subsystems and a rule that orchestrates the switching among them. Switched systems provide a natural and convenient unified framework for mathematical modeling of many physical phenomena and practical applications, such as autonomous transmission systems, computer disc drivers, room temperature control, power electronics, chaos generators, to name but a few. In recent years, considerable efforts have been focused on the analysis and design of switched systems. In this regard, lots of valuable results in

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the stability analysis and stabilization for linear or nonlinear hybrid and switched systems were established (see Liberzon and Morse 1999; Song et al. 2008; Zong et al. 2008; Hetel et al. 2008 and references therein). Within the last few decades, many researcher’s have well-focused on the dynamic analysis of Hopfield NNs, which was first introduced by Hopfield (1982, 1984), has drawn considerable attention due to their many applications in different areas such as pattern recognition, associative memory and combinatorial optimization. Since, the stability is one of the most important behaviors for the NNs, a great deal of results concerning the asymptotic or exponential stability have been proposed (see e.g., Xu 1995; Cao and Ho 2005; Cao et al. 2007, 2008, 2016; Manivannan et al. 2016; Aouiti et al. 2016; Yang et al. 2006; Zhou et al. 2009 and the references therein). It is well known that time delays are often encountered in NNs which may degrade the system performance and cause oscillation, leading to instability. Therefore, it is of great importance to study the asymptotic or exponential stability of NNs with time delay. Meanwhile, neutral time-delay systems are frequently encountered in many practical situations such as in chemical reactors, water pipes, population ecology, heat exchangers, robots in contact with rigid environments (Zhang and Yu 2010; Niculescu 2001), and so on. A neutral time-delay system contains delays both in its state, and in its derivatives of state. Therefore, many dynamical NNs are described with neutral functional differential equations that include neutral delay differential equations as their special case. These NNs are called neutral type NNs or NNs of neural-type. Since, we know that successive time-varying delay model has a more strapping application background in remote control and control system. For example, we consider a state-feedback networked control, where the physical plant, controller, sensor, and actuator are placed at different places and signals are transmitted from one device to another. Along with the delays, there are two network-induced ones, one from sensor to controller and the other from controller to actuator. Then, the closed loop system will appear with two additive time delays in the state. Thus, in the network transmission settings, the two delays are usually time varying with dissimilar properties. Therefore, it is of substantial importance to study the stability of systems with two additive timevarying delay components. Motivated by the previous discussion, in this paper we are concerned with the problem of stability analysis for SHNNs of neutral type with successive time-varying delay components. In this connection, recently a new form of NNs with two additive time-varying delays has been considered in Zhao et al. (2008), Gao et al. (2008) and Shao and Han (2011). In Lam et al. (2007) and Rakkiyappan et al. (2015a, b), it

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was mentioned that in network controlled system (NCS), if the signal transmitted from one point to another passes through few segments of networks, then successive delays are induced with different properties owing to variable transmission conditions. That is, if the physical plant and the state-feedback controller are given by zðtÞ _ ¼ AzðtÞ þ BuðtÞ and uc ðtÞ ¼ Kxc ðtÞ, then it is appropriate to consider time-delays in the dynamical model as zðtÞ _ ¼ AzðtÞ þ BKzðt h1 ðtÞ h2 ðtÞÞ, where h1 ðtÞ is the time-delay induced from sensor to controller and h2 ðtÞ is the delay induced from controller to the actuator. Therefore, the stability analysis of such system was earlier carried out by adding up all the successive delays into a single delay, that is h1 ðtÞ þ h2 ðtÞ ¼ hðtÞ to develop a sufficient stability condition. Therefore, the problem of stability analysis of NNs with successive time-varying delays in the state has received more and more attention and become more popular in recent years (see Rakkiyappan et al. 2015a, b; Senthilraj et al. 2016; Samidurai and Manivannan 2015; Dharani et al. 2015 and the references therein). Recently, the stability of systems with leakage delays becomes one of the hot topics and it has been studied by many researcher’s in the literature. The research about the leakage delay (or forgetting delay), which has been found in the negative feedback of system, can be traced back to 1992. In Kosko (1992), it was observed that the leakage delay had great impact on the dynamical behavior of the system. Since then, many researcher’s have paid much attention to the systems with leakage delay and some interesting results have been derived. For example, Gopalsamy (1992), considered a population model with leakage delay and found that the leakage delay can destabilize a system. In Gopalsamy (2007), the bidirectional associative memory (BAM) neural networks with constant leakage delays were investigated based on L–K functions and properties of M-matrices. Inspired by Gopalsamy (2007), recently it is essential important to study the stability of delayed NNs with leakage effects have been existing in Samidurai and Manivannan (2015), Sakthivel et al. (2015), Li et al. (2011, 2015), Lakshmanan et al. (2013), Li and Yang (2015), and Balasubramaniam et al. (2012). So far, recently Rakkiyappan et al. (2015a, b), established the exponential synchronization of complex dynamical networks with control packet loss and additive time-varying delays. Currently, Senthilraj et al. (2016), proposed the problem of stability analysis of uncertain neutral type BAM neural networks with two additive timevarying delay components. Very recently, robust passivity analysis for delayed stochastic impulsive NNs with leakage and additive time-varying delays have been established by Samidurai and Manivannan (2015). Very recently,

Cogn Neurodyn

Rakkiyappan et al. (2015a, b), analyzed synchronization for singular complex dynamical networks with Markovian jumping parameters and two additive time-varying delay components. More recently, new stability criteria for switched Hopfield NNs of neutral type with additive timevarying discrete delay components and finitely distributed delay were studied by Dharani et al. (2015). Lakshmanan et al. (2013), stability problem concerned with the BAM neural networks with leakage time delay and probabilistic time-varying delays was studied. Li and Yang (2015) analyzed the leakage delay has significant impacts on the dynamical behavior of genetic regulatory networks (GRNs) and can bring tendency to destabilize systems. Recently, in Li et al. (2015) considered stability problem for a class of impulsive NNs model, which includes simultaneously parameter uncertainties, stochastic disturbances and two additive time-varying delays in the leakage term. Balasubramaniam et al. (2012), deals with the problem of delaydependent global asymptotic stability of uncertain switched Hopfield NNs with discrete interval and distributed timevarying delays and time delay in the leakage term. Very recently, Sakthivel et al. (2015), considered the issue of state estimation for a class of BAM neural networks with leakage term. Fuzzy cellular NNs with timevarying delays in the leakage terms have been extensively studied by Yang (2014), without assuming the boundedness on the activation functions. In Zhang et al. (2010), studied a class of new NNs referred to as switched neutral-type NNs with time-varying delays, which combines switched systems with a class of neutral-type NNs. By using an average dwell time method and new L–K functional to assure the global exponential stability and decay estimation for a class of switched Hopfield NNs of neutral type in Zong et al. (2010). In Li and Cao (2013), proposed the switched exponential state estimation and robust stability for interval neural networks with the average dwell time. Very recently, Li et al. (2014) concerned with a class of nonlinear uncertain switched networks with discrete timevarying delays, based on the strictly complete property of the matrices system and the delay-decomposing approach. In Ahn (2010) first time, proposed the H1 weight learning law to study not only guarantee the asymptotical stability of switched Hopfield NNs, but also reduce the effect of external disturbance to an H1 norm constraint. With the motivation mentioned above, a new delayinterval-dependent stability criterion for SHNNs of neutral type with successive time-varying delay components is proposed in this paper. By fully using the available information about time-delays and activation functions, a novel L–K functional is constructed. Our main goal is to establish the delay-interval-dependent stability criteria, such that the concerned NNs are asymptotically stable.

Make use of new technique to estimate the lower and upper bound information of the time-varying delay and L–K functional with double and triple integral terms, we apply WDII, introducing of some zero equations and using the RCC technique and Finsler’s lemma, new stability criteria for a class of SHNNs of neutral type is obtained in terms of LMIs, which ensures the asymptotic stability. Finally, four numerical examples are given to demonstrate the effectiveness and applicability of our theoretical results. The main contribution of this paper lies in the following aspects: •

•

•

•

A novel L–K functional is introduced which includes more information about successive time-varying delays and slope of the neuron activation function. Such type of L–K functional has not yet been considered in the previous literature on the stability of SHNNs of neutral type with successive time-varying delay components are introduced. Different from others in Dharani et al. (2015), Balasubramaniam et al. (2012), Zong et al. (2010), Li and Cao (2013), Li et al. (2014), Cao et al. (2013) and Ahn (2010); several numerical examples are presented to illustrate the validity of the main results with a realworld simulation. This implies that the results of the present paper are essentially new. Inspired by the works in Kwon et al. (2014a, (2014b), some zero equations which would include more quadratic and integral terms are introduced. These terms are merged with the time derivative of L–K functional and combined with RCC approach, which in turn can enhance the feasibility region of stability criterion. Moreover, WDII Lemma is taken into account to bound the time-derivative of triple integral L–K functionals, this gives more tighter bounding technology to deal with such L–K functionals, this technique has been never used in previous literature for the stability of SHNNs of neutral type.

Notations Throughout this paper, the superscripts T and 1 mean the transpose and the inverse of a matrix respectively. Rn denotes the n-dimensional Euclidean space, Rnm is the set of all n m real matrices. For symmetric matrices P and Q; P [ Q (respectively, P ¼ Q) means that the matrix P Q is positive definite (respectively, non-negative). In ; 0n and 0m ; n stands for n n identity matrix, n n and n m zero matrices, respectively and symmetric term in a symmetric matrix is denoted by ; X ? denotes a basis for the null-space of X. If the Matrices are not explicitly stated, it is assumed to compatible dimensions.

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Problem formulation and preliminaries Consider the following delayed Hopfield neural network model Dharani et al. (2015) of neutral type with successive time-varying delay components and distributed delay as: _ ¼ Dyðt d1 ðtÞ d2 ðtÞÞ þ Af ðyðtÞÞ yðtÞ þ Bf ðyðt h1 ðtÞ h2 ðtÞÞÞ Z t _ rðtÞÞ þ J; þC f ðyðsÞÞds þ Eyðt

ð1Þ

tsðtÞ

yðtÞ ¼ uðtÞ;

t 2 ½r; 0;

where yðtÞ ¼ ½y1 ðtÞ; y2 ðtÞ; . . .; yn ðtÞT 2 Rn is the state vector of the network at time t, n corresponds to the number of neurons, f ðyðtÞÞ ¼ ½f1 ðy1 ðtÞÞ; f2 ðy2 ðtÞÞ; . . .; fn ðyn ðtÞÞT 2 Rn is the neuron activation function. The matrix D ¼diagðd1 ; d2 ; . . .; dn Þ is a diagonal matrix with positive entries di [ 0: A; B; C; E are the connection weight matrix and coefficient matrix, the discretely delayed connection weight matrix, the distributively delayed connection weight matrix and coefficient matrix of the time derivative of the delayed states, respectively. J ¼ ½J1 ; J2 ; . . .; Jn T is the constant external input vector. ui ðtÞði 2 NÞ is a continuous vector-valued initial function on ½ r ; 0; r ¼maxfd1U ; d2U ; h1U ; h2U ; s; rg. d1 ðtÞ; d2 ðtÞ and h1 ðtÞ; h2 ðtÞ are leakage and discrete interval timevarying continuous functions that represent the two delay components in the state respectively, sðtÞ and rðtÞ are denotes the distributive and neutral time delays, and which satisfies the following: 0 d1L d1 ðtÞ d1U ; 0 d2L d2 ðtÞ d2U ; 0 dL dðtÞ dU ; 0 h1L h1 ðtÞ h1U ; 0 h2L h2 ðtÞ h2U ;

d_1 ðtÞ g1 ; d_2 ðtÞ g ;

d1UL ¼ d1U d1L ;

d2UL ¼ d2U d2L ; _ g; dUL ¼ dU dL ; dðtÞ

2

h_1 ðtÞ l1 ; h_2 ðtÞ l2 ;

h1UL ¼ h1U h1L ;

h2UL ¼ h2U h2L ; _ l; hUL ¼ hU hL ; hðtÞ

0 hL hðtÞ hU ; _ sD ; 0 sðtÞ s; sðtÞ

0 rðtÞ r;

_ rD ; rðtÞ ð2Þ

where d1U d1L ; d2U d2L ; dU dL ; h1U h1L ; h2U h2L ; hU hL ; s; r; g1 ; g2 ; l1 ; l2 ; sD and rD are known real constants. Note that d1L ; d2L ; dL ; h1L ; h2L ; hL may not be equal to 0. we denote dðtÞ ¼ d1 ðtÞ þ d2 ðtÞ; d1 ¼ d1L þ d1U ; d2 ¼ d2L þ d2U ; g ¼ g1 þ g2 ;

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hðtÞ ¼ h1 ðtÞ þ h2 ðtÞ;

h1 ¼ h1L þ h2U ; h2 ¼ h2L þ h2U ;

l ¼ l1 þ l2 :

ð3Þ

Remark 2.1 The first term in the right side of (1) variously known as forgetting or leakage term. It is known from the literature on population dynamics [see Gopalsamy (1992)] that time delays in the stabilizing negative feedback terms will have a tendency to destabilize a system. fj ðÞ; j ¼ 1; 2; . . .; n are signal transmission functions. Furthermore, system (1) contains some data about the derivative of the past state to further analysis and model the dynamics for such complex neural responses. Hence system (1) has been referred to as neutral-type system, in which the system has both the state delay and the state derivative with delay, the so-called neutral delay. Throughout this paper, it is assumed that each neuron activation function fj ðÞ in (1) satisfies: Assumption (H) (Liu et al. 2006) For any j 2 f1; 2; . . .; ng; fj ð0Þ ¼ 0 and their exist constants kj and kjþ such that kj

fj ða1 Þ fj ða2 Þ kjþ ; a1 a2

ð4Þ

for all a1 6¼ a2 , where a1 ; a2 2 R: Then by Brouwer’s fixed-point theorem Cao (2000) and Assumption H, it can be proved that there exist at least one equilibrium point for system (1). Let z ¼ ½z1 ; z2 ; . . .; zn T be one equilibrium point of system (1). For convenience we shift z to the origin by making the following transformation: zðÞ ¼ yðÞ y and then system (1) can be rewritten as zðtÞ _ ¼ Dzðt dðtÞÞ þ AgðzðtÞÞ þ Bgðzðt hðtÞÞÞ Z t þC gðzðsÞÞds þ Ezðt _ rðtÞÞ;

ð5Þ

tsðtÞ

zðtÞ ¼ /ðtÞ;

t 2 ½r; 0;

where zðtÞ ¼ ½z1 ðtÞ; z2 ðtÞ; . . .; zn ðtÞT is the state vector of the transformed system, the initial condition /ðtÞ ¼ uðtÞ z ; gðzðtÞÞ ¼ ½g1 ðz1 ðtÞÞ; g2 ðz2 ðtÞÞ; . . .; gn ðzn ðtÞÞT ; gj ðzj ðtÞÞ ¼ fj ðzj ðtÞ þ zj Þ fj ðzj Þ; j ¼ 1; 2; . . .; n: According to Assumption H, function gj ðÞ satisfies the following condition: gj ðaÞ kjþ ; gj ð0Þ ¼ 0; a i ¼ 1; 2; . . .; n:

kj

8a 2 R;

a 6¼ 0; ð6Þ

The switched Hopfield neural network of neutral type with discrete and distributed delays are described as

Cogn Neurodyn

zðtÞ _ ¼ D.ðtÞ zðt dðtÞÞ þ A.ðtÞ gðzðtÞÞ þ B.ðtÞ gðzðt hðtÞÞÞ Z t þ C.ðtÞ gðzðsÞÞds þ E.ðtÞ zðt _ rðtÞÞ; tsðtÞ

zðtÞ ¼ /ðtÞ;

t 2 ½r; 0; ð7Þ

where .ðtÞ is a switching signal which takes its values in the finite set K ¼ f1; 2; . . .; mg: Define the indicator function cðtÞ ¼ ½c1 ðtÞ; c2 ðtÞ; . . .; cn ðtÞT , where

ck ðtÞ ¼

m X

ck ðtÞ½Dk zðt dðtÞÞ þ Ak gðzðtÞÞ þ Bk gðzðt hðtÞÞÞ

k¼1

þ Ck

Z

#

t

gðzðsÞÞds þ Ek zðt _ rðtÞÞ :

ð9Þ

tsðtÞ

As (9) must be satisfied under any switching rules, it folP lows that m k¼1 ck ðtÞ ¼ 1: Next, we present some preliminary lemmas, which are needed in the proof of our main results. Lemma 2.1 (Gu 2000) For any positive definite matrix M 2 Rnn , scalars h2 [ h1 [ 0, vector function w : ½h1 ; h2 ! Rn such that the integrations concerned are well defined, the following inequality holds: Z th1 ðh2 h1 Þ wT ðsÞMwðsÞds

a

s

where

1; when the switched system is described by the kth mode; Dk ; Ak ; Bk ; Ck ; Ek ; 0; otherwise,

and k 2 K: Thus, the model (8) can also be described by zðtÞ _ ¼

inequality holds for all continuously differentiable function in ½a; b ! Rn : Z b Z b T Z Z ðb aÞ2 b b T _ _ M x_ ðuÞM xðuÞduds xðuÞduds 2 a s a s Z b Z b x_T ðuÞduds þ 2HTd MHd :

Z

th2 th1

th2

T Z wðsÞds M

th1

Hd ¼

Z

b a

Z

b

_ xðuÞduds þ

s

ð8Þ

3 ba

Z

b

Z

b

Z

b

_ xðvÞdvduds: a

s

v

Remark 2.2 So far, very recently the WDII is proposed by Park et al. (2015). Employing WDII is sure to get less conservative criteria than applying the Jensen’s inequality. Therefore, this integral inequality takes advantage of the following information from three aspects: the first is to use the information on the state such as x(t), the second is to benefit information on the integral of the state over the Rt Rt period of the delay such as ts xðsÞds or tsðtÞ xðsÞds and the third is to employ the information on the double integral of the state over the period of the delay such as R0 Rt R0 Rt s tþu xðsÞds or sðtÞ tþu xðsÞds: Therefore, which gives the more information about the plant states such as Rt Rt R0 Rt xðtÞ; ts xðsÞds or tsðtÞ xðsÞds and s tþu xðsÞds or R0 Rt sðtÞ tþu xðsÞds: Hence, Lemma 2.3 may provide tighter bound than the Jensen’s inequality.

wðsÞds th2

Lemma 2.2 (Park et al. 2011) Let f1 ; f2 ; . . .; fN : Rm ! R have positive values in an open subset D of Rm . Then, the reciprocally convex combination of fi over D satisfies X1 X X min fi ðtÞ ¼ fi ðtÞ þ max gi;j ðtÞ P gi;j ðtÞ fai jai [ 0; ai ¼1g i ai i i6¼j i

subject to f ðtÞ gi;j : Rm ! R; gj;i ðtÞ,gi;j ðtÞ; i gj;i ðtÞ

gi;j ðtÞ 0 fj ðtÞ

Lemma 2.3 (Park et al. 2015) For a given matrix M [ 0, given scalars a and b satisfying a\b, the following

Lemma 2.4 (Boyd et al. 1994) Let n 2 Rn ; U ¼ UT 2 Rnn such that rank ðBÞ\n. The following statements are equivalent (i) (ii)

nT Un\0; ?T

?

8Bn ¼ 0;

B UB \0; where B complement of B.

n 6¼ 0; ?

is a right orthogonal

Lemma 2.5 (Boyd et al. 1994) For a given matrices A11 ; A12 ; A21 ; A22 with appropriate dimensions, A11 A12 \0, holds if and only if A22 \0; A11 A12 A21 A22 T A1 22 A12 \0.

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Cogn Neurodyn 2

Main results In this section, we will propose a stability criteria for system (9). For the sake of simplicity of matrix and vector representation, ei 2 R56nn ði ¼ 1; 2; . . .; 56Þ are defined as block entry matrices (for example eT4 ¼ ½0n ; 0n ; 0n ; In ; 0n ; . . .. . .. . .; 0n Þ. The other notations are defined as |fflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflffl} 52 times

fðtÞ ¼ zT ðtÞ zT ðt h1L Þ zT ðt h1 ðtÞÞ zT ðt h1U Þ zT ðt h2L Þ zT ðt h2 ðtÞÞ zT ðt h2U Þ zT ðt hL Þ: Z t Z t T T T z ðt hðtÞÞ z ðt hU Þ z ðsÞds zT ðsÞds Z Z Z Z

t

zT ðsÞds

th1L

Z

th1L

th1 ðtÞ

Z

T

z ðsÞds th1U thL

th2 ðtÞ

Z

zT ðsÞds

zT ðsÞdsdu

Z Z

Z Z

h1U

zT ðsÞdsdu

h2 ðtÞ Z t

Z

hU T

zT ðsÞdsdu zT ðsÞdsdu

tþu

h2L

Z

h2 ðtÞ

T

z ðsÞdsdu

Z

hL

Z

hðtÞ

tþu

hðtÞ Z t

t

h2L tþu h1L Z t

tþu

h2U

Z

0

h1 ðtÞ

tþu

h1 ðtÞ Z t

zT ðsÞds

th2U

zT ðsÞds

zT ðsÞdsdu

hL

th2 ðtÞ

thðtÞ

thU

thðtÞ 0 Z t

Z

T

z ðsÞds

h1L tþu Z 0 Z t

Z

zT ðsÞds

th2L

T

t

zT ðsÞdsdu

tþu t

zT ðsÞdsdu

tþu

T

z ðsÞdsdu g ðzðtÞÞ g ðzðt h1U ÞÞ

tþu T

g ðzðt h1 ðtÞÞÞ g ðzðt h2U ÞÞ g ðzðt h2 ðtÞÞÞ gT ðzðt hU ÞÞ gT ðzðt hðtÞÞÞ z_T ðtÞ z_T ðt h1U Þ Z t T T z_ ðt h2U Þ z_ ðt hU Þ gT ðzðsÞÞds tsðtÞ

zT ðt d1 Þ zT ðt d1 ðtÞÞ zT ðt d2 Þ zT ðt d2 ðtÞÞ Z t zT ðt dÞ zT ðt dðtÞÞ zT ðsÞds Z Z

t T

z ðsÞds td2 ðtÞ td1 ðtÞ

tdL

tdðtÞ

t T

z ðsÞds

Z

zT ðsÞds

Z

td2L td2 ðtÞ

zT ðsÞds

Z

td1L

zT ðsÞds

td1 ðtÞ

tdðtÞ

td1U

Z

td1 ðtÞ

Z

zT ðsÞds

Z

td2 ðtÞ

zT ðsÞds

td2U

tdðtÞ

zT ðsÞds z_T ðt rðtÞÞ tdU

Dk

0n . . .. . .0n Bk |fflfflfflfflfflffl{zfflfflfflfflfflffl} 5 times 3

0n . . .. . .0n |fflfflfflfflfflffl{zfflfflfflfflfflffl}

0n . . .. . .0n |fflfflfflfflfflffl{zfflfflfflfflfflffl}

Ck

4 times

0n . . .. . .0n |fflfflfflfflfflffl{zfflfflfflfflfflffl} 5 times

Ek 5;

9 times

P1 ¼½e1 e49 Dk ; P2 ¼ ½e36 e1 Dk þ ð1 gÞe46 Dk ;

P3 ¼ 2ðe29 km e1 ÞK1 eT36 þ 2 kp e1 e29 D1 eT36 þ 2ðe30 km e4 ÞeT37

þ 2 kp e4 e30 D2 eT37 þ 2ðe32 km e7 ÞK3 eT38 þ 2 kp e7 e32 D3 eT38

þ 2ðe34 km e10 ÞK4 eT39 þ 2 kp e10 e34 D4 eT39 ; P4 ¼ e36 ðT1 þ T2 þ T3 ÞeT36 e37 T1 eT37 e38 T2 eT38 e39 T3 eT39 ; P5 ¼ e1 ðP2 þ P3 þ P4 ÞeT1 þ e42 ðð1 g1 ÞP2 þ e46 ðð1 gÞP4 ð1 gÞP5 ð1 gÞP6 ÞeT46 P7 P8 þ ½e1 e29 ½e1 e29 T P9 P7 P8 ð1 l1 Þ½e3 e31 ½e3 e31 T P9 P10 P11 þ ½e1 e29 ½e1 e29 T P12 P10 P11 ð1 l2 Þ½e6 e33 ½e6 e33 T P12 P13 P14 þ ð1 l1 Þ½e3 e31 ½e3 e31 T P15 P13 P14 ð1 lÞ½e9 e35 ½e9 e35 T P15 P16 P17 þ ð1 l2 Þ½e6 e33 ½e6 e33 T P18 P16 P17 ð1 lÞ½e9 e35 ½e9 e35 T ; P18 P6 ¼ e1 ðQ1 þ Q2 þ Q3 þ Q6 þ Q9 ÞeT1 þ e41 ðQ1 þ Q4 ÞeT41

T

T

28 times

þ ð1 g1 ÞP5 ÞeT42 þ e44 ðð1 g2 ÞP3 þ ð1 g2 ÞP6 ÞeT44

th2L

th1 ðtÞ

thL

C ¼ 40n . . .. . .0n Ak |fflfflfflfflfflffl{zfflfflfflfflfflffl}

þ e43 ðQ2 þ Q3 ÞeT43 þ e45 ðQ3 Q4 Q5 ÞeT45 þ e1 ðQ7 þ Q10 ÞeT29 þ e29 ðQ8 þ Q11 ÞeT29 Q6 Q7 Q6 Q7 þ ½e1 e29 ½e1 e29 T ½e4 e30 ½e4 e30 T Q8 Q8 Q9 Q10 Q9 Q10 þ ½e1 e29 ½e1 e29 T ½e7 e32 ½e7 e32 T Q11 Q11 Q12 Q13 þ ½e4 e30 ½e4 e30 T Q14 Q12 Q13 ½e10 e34 ½e10 e34 T Q14 Q15 Q16 þ ½e7 e32 ½e7 e32 T Q17 Q15 Q16 ½e10 e34 ½e10 e34 T ; Q17

P7 ¼ e1 d21L U þ d22L V þ d21L W þ d21UL X þ d22UL Y þ d2UL Z eT1 e47 Ue47 e48 VeT48 e49 WeT49 e50 XeT50 2e50 XeT51 e51 XeT51

#T :

e52 YeT52 2e52 YeT52 e53 YeT53 e54 ZeT54 2e54 ZeT55 e55 ZeT55 ; P8 ¼ e2 h1UL Q1 eT2 þ e3 ðh1UL Q1 þ h1UL Q2 ÞeT3 e4 h1UL Q2 eT4 þ e5 h2UL Q3 eT5 þ e6 ðh2UL Q3 þ h2UL Q4 ÞeT6 e7 h2UL Q4 eT7 þ e8 hUL Q5 eT8 þ e9 ðhUL Q5 þ hUL Q6 ÞeT9 e10 hUL Q6 eT10 ;

123

Cogn Neurodyn

þ h22L V þ h2L W þ h21UL X þ h22UL Y þ h2UL Z ½e1 e36 T ½e11 e1 e2 U ½e11 e1 e2 T e36 h21L U ½e13 e1 e8 T ½e12 e1 e5 V½e12 e1 e5 T ½e13 e1 e8 W X L ½e14 e2 e3 e15 e3 e4 ½e14 e2 e3 e15 e3 e4 T X Y M ½e16 e5 e6 e17 e6 e7 ½e16 e5 e6 e17 e6 e7 T Y Z N ½e18 e8 e9 e19 e9 e10 ½e18 e8 e9 e19 e9 e10 T ; Z ! h41L h42L h4L ðh21U h21L Þ2 ðh22U h22L Þ2 ðh2U h2L Þ2 R1 þ R 2 þ R3 þ R4 þ R5 þ R6 eT36 ¼ e36 4 4 4 4 4 4 3 3 3 6 18 e1 R1 þ R2 þ R3 þ 3R4 þ 3R5 þ 3R6 eT1 þ e1 3R1 eT20 e11 3R1 eT11 þ e11 R1 eT20 e20 2 R1 eT20 2 2 2 h1L h1L 6 18 6 þ e1 3R2 eT21 e12 3R2 eT12 þ e12 R2 eT21 e21 2 R2 eT21 þ e1 3R3 eT22 e13 3R3 eT13 þ e13 R3 eT22 h2L hL h2L 18 6 18 e22 2 R3 eT22 þ e1 3R4 eT23 e14 3R4 eT14 þ e14 R4 eT23 e23 R4 eT23 h1U h1L hL ðh1U h1L Þ2 6 18 þ e1 3R4 eT24 e15 3R4 eT15 þ e15 R4 eT24 e24 R4 eT24 þ e1 3R5 eT25 e16 3R5 eT16 h1U h1L ðh1U h1L Þ2 6 18 6 þ e16 R5 eT25 e25 R eT þ e1 3R5 eT26 e17 3R5 eT17 þ e17 R5 eT26 2 5 25 h2U h2L h h2L 2U ðh2U h2L Þ 18 6 18 e26 R eT þ e1 3R6 eT27 e18 3R6 eT18 þ e18 R6 eT27 e27 R6 eT27 2 5 26 h U hL ðh2U h2L Þ ðhU hL Þ2 6 18 þ e1 3R6 eT28 e19 3R6 eT19 þ e19 R6 eT28 e28 R6 eT28 ; hU hL ðhU hL Þ2

P9 ¼ ½e1

P10

P11 ¼ e29 s2 S1 eT29 e40 S1 eT40 þ e36 S2 eT36 e56 ð1 rD ÞS2 eT56 ; P12 ¼ e36 ðH H T ÞeT36 2e36 HAk eT46 þ 2e36 HBk eT29 þ 2e36 HCk eT35 þ 2e36 HDk eT40 þ 2e36 HEk eT56 ; P13 ¼ e1 G1 R1 eT1 þ 2e1 G1 R2 eT29 e29 G1 eT29 e3 G2 R1 eT3 þ 2e3 G2 R2 eT31 e31 G2 eT31 e4 G3 R1 eT4 þ 2e4 G3 R2 eT30 e30 G3 eT30 e6 G4 R1 eT6 þ 2e6 G4 R2 eT33 e33 G4 eT33 e7 G5 R1 eT7 þ 2e7 G5 R2 eT32 e32 G5 eT32 e9 G6 R1 eT9 þ 2e9 G6 R2 eT35 e35 G6 eT35 e10 G7 R1 eT10 þ 2e10 G7 R2 eT34 e34 G7 eT34 ; N ¼ P1 PPT2 þ P2 PPT1 þ

13 X

Pi ;

i¼3

diag k1þ ; k2þ ; . . .. . .; knþ ;

Km ¼ diag k1 ; k2 ; . . .. . .; kn ; þ þ

k1 þ k1þ k2 þ k2þ kn þ knþ þ ; ; . . .. . .; R1 ¼ diag k1 k1 ; k2 k2 ; . . .. . .; kn kn ; R2 ¼ diag ; 2 2 2 0n F 1 0 n F2 0 n F3 0 n F4 0n ; F2 ¼ ; F3 ¼ ; F4 ¼ ; F5 ¼ F1 ¼ F1 0 n F2 0 n F 3 0n F4 0 n F5

Kp ¼

F5 0n

; F6 ¼

0n

F6

F6

0n

:

123

Cogn Neurodyn

Theorem 3.1 For given positive scalars d1L ; d1U ; d2L ; d2U ; h1L ; h1U ; h2L ; h2U ; d1 ; d2 ; h1 ; h2 ; s; r; g1 ; g2 ; l1 ; l2 ; sD ; rD and diagonal matrices Kp ; Km , then the neural network described by (9) is globally asymptotically stable, for any time-varying delay dðtÞ; hðtÞ; sðtÞ and rðtÞ satisfying (2), if there exist positive definite matrices Pi ði ¼ 1; 2; . . .; 18Þ 2 Rnn ; Ti ði ¼ 1; 2; 3Þ 2 Rnn ; Qi ði ¼ R2n2n ; 1; 2; . . .; 17Þ 2 Rnn U; V; W; X; Y; Z 2 Rnn ; U2 R2n2n ; X2 R2n2n ; Y2 R2n2n ; Z2 R2n2n , R2n2n ; W2 V2 nn nn Ri ði ¼ 1; 2; . . .; 6Þ 2 R ; Si ði ¼ 1; 2Þ 2 R , positive diagonal matrices Dl ¼ diagfkl1 ; kl2 ; . . .; kln g; Kl ¼diag . . .; lln g; H 2 Rnn ; Gi ði ¼ 1; 2; . . .; 7Þ 2 Rnn , fll1 ; ll2 ; any symmetric matrices Fi 2 Rnn ði ¼ 1; 2; . . .; 6Þ, any matrices L; M; N 2 R2n2n such that the following LMIs hold: ðC? ÞT N C? \0; 2 3 X þ F 1 L 4 5 0; X þ F 2 2 3 M Y þ F 3 4 5 0; Y þ F4 2 3 Z þ F 5 N 4 5 0; Z þ F6

ð10Þ

V3 ðzðtÞ; tÞ ¼ V4 ðzðtÞ; tÞ ¼

Z

t

z_T ðsÞT1 zðsÞds _ þ

th1U Z t

ð12Þ

Z

9 X

tdðtÞ

ð14Þ

Z

V1 ðzðtÞ; tÞ ¼

zðtÞ Dk

V2 ðzðtÞ; tÞ ¼ 2

n X i¼1

þ2

"

zi ðtÞ

k1i

" n X

þ

Z

þ2

Z

þ2

i¼1

123

zi ðth1U Þ

k2i 0

Z

zi ðth2U Þ

k3i 0

i¼1

" n X

gi ðsÞ

Z

zi ðthU Þ

k4i 0

ki s

ds þ d1i

Z 0

kiþ s

gi ðsÞ ki s ds þ d2i gi ðsÞ ki s ds þ d3i

gi ðsÞ ki s ds þ d4i

Z

gi ðsÞ ds

T

z ðsÞQ4 zðsÞds þ td t

zðsÞ f ðzðsÞÞ

d1L tþh Z 0 Z t

zi ðth1U Þ

þ dL

Z

0

Z

zi ðth2U Þ

0

Z

zi ðthU Þ

0

kiþ s gi ðsÞ ds

kiþ s gi ðsÞ ds

#

þ d2UL

Z

Z

þ hL

d1U Z d2L

Z

zT ðsÞP6 zðsÞds

td td2

T

z ðsÞQ5 zðsÞds td

zT ðsÞVzðsÞdsdh

tþh Z t

zT ðsÞXzðsÞdsdh zT ðsÞYzðsÞdsdh

d2U tþh dL Z t

zT ðsÞZzðsÞdsdh;

Z

0

tþh t

nT ðsÞUnðsÞdsdh nT ðsÞVnðsÞdsdh

h2L tþh 0 Z t

Z

nT ðsÞWnðsÞdsdh

hL

þ h1UL þ h2UL þ hUL

T

tþh d1L Z t

dU

#

kiþ s gi ðsÞ ds ;

Z

z ðsÞWzðsÞdsdh

dL

þ h2L

#

td2 ðtÞ

d2L tþh 0 Z t T

h1L tþh Z 0 Z t

#

Z

td2 td1

zðsÞds ;

zi ðtÞ

zT ðsÞP3 zðsÞds

zðsÞ Q6 Q7 ds Q8 gðzðsÞÞ th1U Z t zðsÞ zðsÞ T Q9 Q10 þ ds Q11 gðzðsÞÞ th2U gðzðsÞÞ T Z th1U zðsÞ zðsÞ Q12 Q13 þ ds Q14 gðzðsÞÞ gðzðsÞÞ thU T Z th2U zðsÞ zðsÞ Q15 Q16 þ ds; Q17 gðzðsÞÞ gðzðsÞÞ thU Z 0 Z t V6 ðzðtÞ; tÞ ¼ d1L zT ðsÞUzðsÞdsdh Z

V7 ðzðtÞ; tÞ ¼ h1L

!

t tdðtÞ

0

i¼1

" n X

P1 zðtÞ Dk

zðsÞds

Z

tdðtÞ

Z

t

þ dUL !T

z_T ðsÞT3 zðsÞds; _ thU

tdðtÞ

td1

where t

t

zðsÞ zðsÞ T P7 P8 ds gðzðsÞÞ gðzðsÞÞ P th1 ðtÞ 9 T Z t zðsÞ zðsÞ P10 P11 þ ds P12 gðzðsÞÞ th2 ðtÞ gðzðsÞÞ T Z th1 ðtÞ zðsÞ zðsÞ P13 P14 ds þ P15 gðzðsÞÞ gðzðsÞÞ thðtÞ T Z th2 ðtÞ zðsÞ zðsÞ P16 P17 ds; þ gðzðsÞÞ P18 gðzðsÞÞ thðtÞ Z t Z t Z t V5 ðzðtÞ; tÞ ¼ zT ðsÞQ1 zðsÞds þ zT ðsÞQ2 zðsÞds þ zT ðsÞQ3 zðsÞds þ

i¼1

Z

zT ðsÞP5 zðsÞds þ

þ d2L

Vi ðzðtÞ; tÞ;

th2U Z t

Z

zT ðsÞP4 zðsÞds

td1 ðtÞ

þ d1UL

VðzðtÞ; tÞ ¼

z_T ðsÞT2 zðsÞds _ þ

tdðtÞ

ð13Þ

Proof Let us consider the following Lyapunov–Krasoskii functional candidate:

t

td2 ðtÞ

þ

þ

ð11Þ

zT ðsÞP2 zðsÞds þ

td1 ðtÞ Z t

þ

Z

Z

tþh h1L Z t

h1U Z h2L

Z

tþh Z t

h2U tþh hL Z t

nT ðsÞXnðsÞdsdh nT ðsÞYnðsÞdsdh

nT ðsÞZnðsÞdsdh;

hU

tþh

Cogn Neurodyn

V8 ðzðtÞ; tÞ ¼

Z

h21L 2 þ

2 h2L

V9 ðzðtÞ; tÞ ¼ s

Z

2

Z

h2L h 0 Z 0 hL

z_T ðsÞR1 zðsÞdsdudh _

h22U 2

s

þ

Z

þ zT ðt d2 ðtÞÞ½ð1 g2 ÞP3 þ ð1 g2 ÞP6 zðt d2 ðtÞÞ

tþu t

z_T ðsÞR3 zðsÞdsdudh _ Z

0

Z

t


h tþu h1U Z Z Z h22L h2L 0 t

h2U h2L 2 Z 0Z t

þ zT ðt d1 ðtÞÞ½ð1 g1 ÞP2 þ ð1 g1 ÞP5 zðt d1 ðtÞÞ


tþu h1L

Z

V_4 ðzðtÞ; tÞ zT ðtÞ½P2 þ P3 þ P4 zðtÞ

t

Z

h

h21U h21L 2

þ þ

0

h1L h tþu Z Z 0Z t h22L 0

þ þ

Z

0


h2U h tþu hL Z 0 Z t T

Z

z_ ðsÞR6 zðsÞdsdudh; _

hU

h

tþu

gT ðzðsÞÞS1 gðzðsÞÞdsdh

tþh t

z_T ðsÞS2 zðsÞds: _

trðtÞ

nT ðtÞ ¼ colfzðtÞ; zðtÞ _ g: Taking the time derivative of V(z(t), t) along the trajectories of system (9) yields _ VðzðtÞ; tÞ ¼

9 X

V_i ðzðtÞ; tÞ;

fT ðtÞP5 fðtÞ;

ð19Þ

ð15Þ V_5 ðzðtÞ; tÞ ¼ zT ðtÞ½Q1 þ Q2 þ Q3 þ Q6 þ Q9 zðtÞ

i¼1

where V_1 ðzðtÞ; tÞ 2 zðtÞ Dk

þ zT ðt dðtÞÞ½ð1 gÞP4 ð1 gÞP5 ð1 gÞP6 zðt dðtÞÞ zðtÞ zðtÞ T P7 P8 þ P9 gðzðtÞÞ gðzðtÞÞ zðt h1 ðtÞÞ zðt h1 ðtÞÞ T P7 P8 ð1 l1 Þ gðzðt h1 ðtÞÞÞ P9 gðzðt h1 ðtÞÞÞ zðtÞ zðtÞ T P10 P11 þ P12 gðzðtÞÞ gðzðtÞÞ zðt h2 ðtÞÞ T P10 P11 zðt h2 ðtÞÞ ð1 l2 Þ gðzðt h2 ðtÞÞÞ P12 gðzðt h2 ðtÞÞÞ zðt h1 ðtÞÞ T P13 P14 zðt h1 ðtÞÞ þ ð1 l1 Þ gðzðt h1 ðtÞÞÞ P15 gðzðt h1 ðtÞÞÞ T P13 P14 zðt hðtÞÞ zðt hðtÞÞ ð1 lÞ P15 gðzðt hðtÞÞÞ gðzðt hðtÞÞÞ zðt h2 ðtÞÞ T P16 P17 zðt h2 ðtÞÞ þ ð1 l2 Þ gðzðt h2 ðtÞÞÞ P18 gðzðt h2 ðtÞÞÞ zðt hðtÞÞ zðt hðtÞÞ T P16 P17 ð1 lÞ P18 gðzðt hðtÞÞÞ gðzðt hðtÞÞÞ

Z

þ zT ðt d1 Þ½Q1 þ Q4 zðt d1 Þ

!T

t

zðsÞds

þ zT ðt d2 Þ½Q2 þ Q5 zðt d2 Þ þ zT ðt dÞ P1 ðzðtÞ _

½Q3 Q4 Q5 zðt dÞ

tdðtÞ

Dk zðtÞ þ ð1 gÞDk zðt dðtÞÞÞ 2fT ðtÞPT1 P1 P2 fðtÞ; ð16Þ T V_2 ðzðtÞ; tÞ ¼ 2½gðzðtÞÞ km zðtÞT K1 zðtÞ _ þ 2 kp zðtÞ gðzðtÞÞ D1 zðtÞ _ þ 2½gðzðt h1U ÞÞ km zðt h1U ÞT K2 zðt _ h1U Þ T _ h1U Þ þ 2 kp zðt h1U Þ gðzðt h1U ÞÞ D2 zðt _ h2U Þ þ 2½gðzðt h2U ÞÞ km zðt h2U ÞT K3 zðt T þ 2 kp zðt h2U Þ gðzðt h2U ÞÞ D3 zðt _ h1U Þ _ hU Þ þ 2½gðzðt hU ÞÞ km zðt hU ÞT K4 zðt T _ hU Þ þ 2 kp zðt hU Þ gðzðt hU ÞÞ D4 zðt ¼ fT ðtÞP3 fðtÞ;

ð17Þ V_3 ðzðtÞ; tÞ ¼ z_ ðtÞ½T1 þ T2 þ T3 zðtÞ _ T

z_T ðt h1U ÞT1 z_T ðt h1U Þ z_T ðt h2U ÞT2 z_T ðt h2U Þ z_T ðt hU ÞT3 z_T ðt hU Þ ¼ fT ðtÞP4 fðtÞ;

ð18Þ

þ zT ðtÞ½Q7 þ Q10 gðzðtÞÞ þ gT ðzðtÞÞ½Q8 þ Q11 gðzðtÞÞ zðtÞ T Q6 Q7 zðtÞ þ gðzðtÞÞ Q8 gðzðtÞÞ zðt h1U Þ T Q6 Q7 zðt h1U Þ gðzðt h1U ÞÞ Q8 gðzðt h1U ÞÞ T zðtÞ zðtÞ Q9 Q10 þ gðzðtÞÞ Q11 gðzðtÞÞ zðt h2U Þ T Q9 Q10 zðt h2U Þ gðzðt h2U ÞÞ Q11 gðzðt h2U ÞÞ T zðt h1U Þ zðt h1U Þ Q12 Q13 þ gðzðt h1U ÞÞ Q14 gðzðt h1U ÞÞ T zðt hU Þ zðt hU Þ Q12 Q13 gðzðt hU ÞÞ Q14 gðzðt hU ÞÞ T zðt h2U Þ zðt h2U Þ Q15 Q16 þ gðzðt h2U ÞÞ Q17 gðzðt h2U ÞÞ T zðt hU Þ zðt hU Þ Q15 Q16 gðzðt hU ÞÞ Q17 gðzðt hU ÞÞ ¼ fT ðtÞP6 fðtÞ;

ð20Þ

123

Cogn Neurodyn

V_6 ðzðtÞ; tÞ ¼ zT ðtÞ d21L U þ d22L V þ d2L W þ d21UL X þ d22UL Y þ d2UL Z zðtÞ Z t Z t d1L zT ðsÞUzðsÞds d2L zT ðsÞVzðsÞds dL

Z

td1L t

td2L Z td1L

zT ðsÞWzðsÞds d1UL

tdL Z td2L

d2UL

zT ðsÞXzðsÞds

td1U Z tdL

zT ðsÞYzðsÞds dUL

td2U

zT ðsÞZzðsÞds:

tdU

Applying Lemma 2.1, we have V_6 ðzðtÞ; tÞ zT ðtÞ d21L U þ d22L V þ d2L W þ d21UL X þ d22UL Y þ d2UL Z zðtÞ Z t Z t zT ðsÞdsU zðsÞds

td1 ðtÞ Z t

Z

zT ðsÞdsV

zðsÞds

td2 ðtÞ t

Z

td2 ðtÞ t

T

z ðsÞdsW tdðtÞ

Z

td1L

zT ðsÞdsX

Z

zðsÞds zðsÞds

td1 ðtÞ

Z

td1 ðtÞ

td1U Z td2L

zT ðsÞdsX

zT ðsÞdsY

2

Z

zT ðsÞdsY

td2 ðtÞ

zT ðsÞdsY

zT ðsÞdsZ

Z

zT ðsÞds

td2 ðtÞ

2h2UL

td2U tdL

zðsÞds

zT ðsÞdsZ

tdðtÞ tdðtÞ

zT ðsÞdsZ

Z

Z

2hUL

tdðtÞ

tdðtÞ

zðsÞds tdU

ð21Þ Inspired by the ideas in the works of Kwon et al. (2014a, b), following six zero equalities with any symmetric matrices Fi ; i ¼ 1; 2; . . .; 6 are introduced: h 0 ¼h1UL zT ðt h1L ÞF1 zðt h1L Þ zT ðt h1 ðtÞÞ Z th1L i ð22Þ zT ðsÞF1 zðsÞds _ ; F1 zðt h1 ðtÞÞ 2 th1 ðtÞ

ð27Þ

th1 ðtÞ th1 ðtÞ

th1U Z th2L

Z

zT ðsÞF2 zðsÞds _ T

z ðsÞF3 zðsÞ _ 2h2UL

th2 ðtÞ thL

zT ðsÞF5 zðsÞ _ 2hUL

thðtÞ

zT ðsÞds tdU

V_6 ðzðtÞ; tÞ f ðtÞP7 fðtÞ:

123

Z

2h1UL

td2U

tdðtÞ

tdL

tdU T

td2 ðtÞ

zðsÞds

Z

tdðtÞ

2

zðsÞds td1U td2L

Z

ð26Þ

By summing the above six zero equalities given in the Eqs. (22)–(27), it can be obtained Z th1L 0 ¼ fT ðtÞP8 fðtÞ 2h1UL zT ðsÞF1 zðsÞ _

zðsÞds

td2 ðtÞ

Z

0 ¼ hUL zT ðt hL ÞF5 zðt hL Þ zT ðt hðtÞÞ Z thL i zT ðsÞF5 zðsÞds _ ; F5 zðt hðtÞÞ 2

thU

zT ðsÞds

td1 ðtÞ

Z

ð25Þ

th2U

h

td2 ðtÞ

td2L

td2U Z tdL

td1 ðtÞ td1U

Z

Z

td2 ðtÞ

Z

Z

zT ðsÞdsX

ð24Þ

th2 ðtÞ

0 ¼ h2UL zT ðt h2 ðtÞÞF4 zðt h2 ðtÞÞ zT ðt h2U Þ Z th2 ðtÞ i zT ðsÞF4 zðsÞds _ ; F4 zðt h2U Þ 2

0 ¼ hUL zT ðt hðtÞÞF6 zðt hðtÞÞ zT ðt hU Þ Z thðtÞ i F6 zðt hU Þ 2 zT ðsÞF6 zðsÞds _ :

td1L td1 ðtÞ

td1L

th1U

h 0 ¼ h2UL zT ðt h2L ÞF3 zðt h2L Þ zT ðt h2 ðtÞÞ Z th2L i zT ðsÞF3 zðsÞds _ ; F3 zðt h2 ðtÞÞ 2

h

tdðtÞ

Z

ð23Þ

thðtÞ

Z

td1 ðtÞ

2

td1 ðtÞ t

h 0 ¼ h1UL zT ðt h1 ðtÞÞF2 zðt h1 ðtÞÞ zT ðt h1U Þ Z th1 ðtÞ i F2 zðt h1U Þ 2 zT ðsÞF2 zðsÞds _ ;

Z

Z

th2 ðtÞ

zT ðsÞF4 zðsÞds _

th2U thðtÞ

zT ðsÞF6 zðsÞds; _ thU

þ h22L V þ h2L W þ h21UL X V_7 ðzðtÞ; tÞ ¼ nT ðtÞ h21L U Z t nT ðsÞUnðsÞds þ h22UL Y þ h2UL ZnðtÞ h1L th 1L Z t Z t h2L nT ðsÞVnðsÞds hL nT ðsÞWnðsÞds th2L Z th1L

h1UL Z hUL

th1U thL

thL T

n ðsÞXnðsÞds h2UL

Z

th2L

nT ðsÞYnðsÞds

th2U

nT ðsÞZnðsÞds:

thU

ð28Þ Using Lemma 2.1, the following inequalities hold

Cogn Neurodyn

þ h22L V þ h2L W þ h21UL X V_7 ðzðtÞ; tÞ nT ðtÞ h21L U þ h22UL Y þ h2UL Z nðtÞ 3T 2 3 2 Rt 32 R t 11 U 12 U th1L zðsÞds th1L xðsÞds 7 6 7 6 76 4 5 4 5 54 U22 zðtÞ zðt h1L Þ zðtÞ zðt h1L Þ 32 R t 2 Rt 3T 2 3 V11 V12 th2L zðsÞds th2L zðsÞds 76 6 7 6 7 4 54 5 4 5 V22 zðtÞ zðt h2L Þ zðtÞ zðt h2L Þ 32 R t 3T 2 3 2 Rt W11 W12 thL zðsÞds thL zðsÞds 76 7 6 7 6 4 54 5 4 5 22 W zðtÞ zðt hL Þ zðtÞ zðt hL Þ Z th1L Z th2L h1UL nT ðsÞXnðsÞds h2UL nT ðsÞYnðsÞds hUL

th1U thL

Z

th2U

nT ðsÞZnðsÞds:

thU

ð29Þ By considering integral terms in (29) with the equation (28), if the inequalities in (11), (12) and (13) are holds, then by utilizing Lemmas 2.1 and 2.2, it follows that

h1UL

Z

th1L

nT ðsÞXnðsÞds 2h1UL

Z

th1 ðtÞ th1U th1L

¼ h1UL h1UL

Z

th1L

Z

th2L

nT ðsÞYnðsÞds 2h2UL

2h2UL

Z

th2U Z th2L

¼ h2UL h2UL



th2L


zT ðsÞF4 zðsÞds _ nT ðsÞfY þ F 3 gnðsÞds

th2 ðtÞ

Z

th2 ðtÞ

nT ðsÞfY þ F 4 gnðsÞds

th2U

Z th2L h2UL nT ðsÞfY þ F 3 gnðsÞds h2 ðtÞ h2L th2 ðtÞ Z th2 ðtÞ h2UL nT ðsÞfY þ F 4 gnðsÞds h2U h2 ðtÞ th2U 2 R th2L 3T 2 3 Y þ F 3 M th2 ðtÞ nðsÞds 6 7 6 7 4 5 4 5 R th2 ðtÞ Y þ F 4 th2U nðsÞds 3 2 R th2L th2 ðtÞ nðsÞds 7 6 5; 4 R th2 ðtÞ th2U nðsÞds ð31Þ Z

thL

2hUL

Z

¼ hUL

nT ðsÞfX þ F 1 gnðsÞds

Z

th2 ðtÞ th2 ðtÞ

nT ðsÞZnðsÞds 2hUL

Z

thU

hUL

th1 ðtÞ th1 ðtÞ

Z

th2U

hUL

th1 ðtÞ

th1U

2h1UL

Z

h2UL

Z

thL


thðtÞ thðtÞ


thU Z thL

nT ðsÞfZ þ F 5 gnðsÞds

thðtÞ thðtÞ

nT ðsÞfZ þ F 6 gnðsÞds thU

nT ðsÞfX þ F 2 gnðsÞds

th1U

Z th1L h1UL nT ðsÞfX þ F 1 gnðsÞds h1 ðtÞ h1L th1 ðtÞ Z th1 ðtÞ h1UL nT ðsÞfX þ F 2 gnðsÞds h1U h1 ðtÞ th1U 2 R th1L 3T 2 3 X þ F 1 L th1 ðtÞ nðsÞds 6 7 6 7 4 5 4 5 R th1 ðtÞ þ F2 X nðsÞds th1U 3 2 R th1L th1 ðtÞ nðsÞds 7 6 5; 4 R th1 ðtÞ th1U nðsÞds and similarly, we have

Z thL hUL nT ðsÞfZ þ F 5 gnðsÞds hðtÞ hL thðtÞ Z thðtÞ hUL nT ðsÞfZ þ F 6 gnðsÞds hU hðtÞ thU 3T 2 2 R thL 3 Z þ F 5 N thðtÞ nðsÞds 7 6 6 7 4 5 4 5 R thðtÞ Z þ F6 thU nðsÞds 2 R thL 3 thðtÞ nðsÞds 6 7 4 5: R thðtÞ thU nðsÞds

ð30Þ

ð32Þ From (30)–(32), it is concluded that

123

Cogn Neurodyn

V_ 7 ðzðtÞ; tÞ 2h1UL 2h2UL 2hUL

Z

Z

Z

th1L

zT ðsÞF1 zðsÞ _ 2h1UL

th1 ðtÞ th2L

T

z ðsÞF3 zðsÞ _ 2h2UL th2 ðtÞ thL

zT ðsÞF5 zðsÞ _ 2hUL

thðtÞ

Z

Z

Z

th1 ðtÞ


th1U th2 ðtÞ


th2U thðtÞ


thU

þ h22L V þ h2L W þ h21UL X þ h22UL Y þ h2UL Z nðtÞ nT ðtÞ h21L U 3T 2 3 2 Rt 32 R t 11 U 12 U th1L zðsÞds th1L xðsÞds 7 6 7 6 76 4 5 4 5 54 22 U zðtÞ zðt h1L Þ zðtÞ zðt h1L Þ 3T 2 3 2 Rt 32 R t V11 V12 th2L zðsÞds th2L zðsÞds 7 6 7 6 76 4 5 4 5 54 V22 zðtÞ zðt h2L Þ zðtÞ zðt h2L Þ 2 Rt 3T 2 3 32 R t 11 W 12 W thL zðsÞds thL zðsÞds 6 7 6 7 76 4 5 4 5 54 22 W zðtÞ zðt hL Þ zðtÞ zðt hL Þ 2 3T R th1L th1 ðtÞ zðsÞds 3 6 7 2 L 6 zðt h1L Þ zðt h1 ðtÞÞ 7 X þ F 1 6 7 6 7 7 4 6 5 6 7 R th1 ðtÞ 6 7 X þ F2 4 5 zðsÞds th1U

zðt h1 ðtÞÞ zðt h1U Þ 2 3 R th1L th1 ðtÞ zðsÞds 6 7 6 zðt h1L Þ zðt h1 ðtÞÞ 7 6 7 6 7 6 7 R th1 ðtÞ 6 7 4 5 th1U zðsÞds zðt h1 ðtÞÞ zðt h1U Þ 3T 2 R th2L th2 ðtÞ zðsÞds 7 2 6 6 zðt h2L Þ zðt h2 ðtÞÞ 7 Y þ F 3 7 6 6 7 4 6 7 6 R th2 ðtÞ 7 6 5 4 zðsÞds th2U

ð34Þ Applying Lemma 2.3, the integral terms in (34) can be rewritten as Z 0 Z t h2 1L z_T ðsÞR1 zðsÞdsdu _ 2 h1L tþu T Z t Z t h1L zðtÞ zðsÞds R1 h1L zðtÞ zðsÞds

3

M Y þ F 4

7 5

th1L

zðt h2 ðtÞÞ zðt h2U Þ 3 R th2L th2 ðtÞ zðsÞds 6 7 6 zðt h2L Þ zðt h2 ðtÞÞ 7 6 7 6 7 6 7 R th2 ðtÞ 6 7 4 5 th2U zðsÞds

thU

fT ðtÞH1 fðtÞ; N Z þ F 6

3

ð35Þ

7 5

Similarly, we have Z t Z h22L 0 z_T ðsÞR2 zðsÞdsdu _ fT ðtÞH2 fðtÞ; ð36Þ 2 h2L tþu Z Z t h2 0 z_T ðsÞR3 zðsÞdsdu _ fT ðtÞH3 fðtÞ; ð37Þ L 2 hL tþu Z Z h2 h21L h1L t T z_ ðsÞR4 zðsÞdsdu _ fT ðtÞH4 fðtÞ; 1U 2 h1 ðtÞ tþu

zðt hðtÞÞ zðt hU Þ 3 R thL thðtÞ zðsÞds 6 7 6 zðt hL Þ zðt hðtÞÞ 7 6 7 6 7 6 7 R thðtÞ 6 7 4 5 zðsÞds thU 2

zðt hðtÞÞ zðt hU Þ fT ðtÞP9 fðtÞ;

ð33Þ

123

th1L

T Z 0 Z t Z t h1L 3 2 zðtÞ zðsÞds þ zðsÞdsdu R1 h1L h1L tþu 2 th1L Z 0 Z t Z t h1L 3 zðtÞ zðsÞds þ zðsÞdsdu h1L h1L tþu 2 th1L

2

zðt h2 ðtÞÞ zðt h2U Þ 2 3T R thL thðtÞ zðsÞds 6 7 2 6 zðt hL Þ zðt hðtÞÞ 7 Z þ F 5 6 7 6 7 4 6 6 7 R thðtÞ 6 7 4 5 zðsÞds

h4 h4 h4 ðh2 h21L Þ2 R4 V_8 ðzðtÞ; tÞ ¼ z_T ðtÞ 1L R1 þ 2L R2 þ L R3 þ 1U 4 4 4 4 ! ðh22U h22L Þ2 ðh2U h2L Þ2 R5 þ R6 zðtÞ þ _ 4 4 Z 0 Z t Z 0 h2 h2 z_T ðsÞR1 zðsÞdsdu _ 2L 1L 2 h1L tþu 2 h2L Z t 2Z 0 Z t h z_T ðsÞR2 zðsÞdsdu _ L z_T ðsÞ 2 hL tþu tþu Z Z h2 h21L h1L t T _ 1U z_ ðsÞ dsR3 zðsÞdu 2 h1 ðtÞ tþu Z Z h21U h21L h1 ðtÞ t T _ z_ ðsÞ dsR4 zðsÞdsdu 2 h1U tþu Z Z h2 h22L h2L t T _ 2U z_ ðsÞ R4 zðsÞdsdu 2 h2 ðtÞ tþu Z Z h2 h22L h2 ðtÞ t T _ 2U z_ ðsÞ R5 zðsÞdsdu 2 h2U tþu Z Z h2 h2L hL t T _ U z_ ðsÞ R5 zðsÞdsdu 2 hðtÞ tþu Z Z h2 h2L hðtÞ t T _ U z_ ðsÞR6 zðsÞdsdu: _ R6 zðsÞdsdu 2 hU tþu

ð38Þ

Cogn Neurodyn

h2 h21L 1U 2

Z

h1 ðtÞ

Z

h1U

t

z_T ðsÞR4 zðsÞdsdu _ fT ðtÞH5 fðtÞ; tþu

ð39Þ

h22U h22L 2

Z

h2L Z h2 ðtÞ

t

2

h22L

Z

tsðtÞ

þ z_T ðt rðtÞÞðð1 rD ÞT2 Þzðt _ rðtÞÞ:

tþu

h2 ðtÞ

Z

h2U

t

z_T ðsÞR5 zðsÞdsdu _ fT ðtÞH7 fðtÞ; tþu

ð41Þ

h2U

2

h2L

Z

hL

Z

hðtÞ

V9 ðzðtÞ; tÞ s2 gT ðzðtÞÞS1 gðzðtÞÞ Z t s gT ðzðsÞÞS1 gðzðsÞÞds þ z_T ðtÞT2 zðtÞ _

z_T ðsÞR5 zðsÞdsdu _ fT ðtÞH6 fðtÞ; ð40Þ

h22U

V8 ðzðtÞ; tÞ fT ðtÞP10 fðtÞ;

Utilizing Lemma 2.1, we have V9 ðzðtÞ; tÞ gT ðzðtÞÞ s2 S1 gðzðtÞÞ !T Z Z t gðzðsÞÞds ðS1 Þ þ tsðtÞ

t

2

h2L

Z

hðtÞ hU

Z

gðzðsÞÞds

tsðtÞ

_ þ z_T ðtÞS2 zðtÞ

z_T ðsÞR6 zðsÞdsdu _ fT ðtÞH8 fðtÞ;

tþu

þ z_T ðt rðtÞÞðð1 rD ÞS2 Þzðt _ rðtÞÞ;

ð42Þ h2U

!

t

fT ðtÞP11 fðtÞ:

t

z_T ðsÞR6 zðsÞdsdu _ fT ðtÞH9 fðtÞ:

ð44Þ

tþu

ð43Þ where H1 ¼ ½h1L e1 e11 R1 ½h1L e1 e11 T T h1L 3 h1L 3 2 e1 e11 þ e20 R1 e1 e11 þ e20 h1L h1L 2 2 H2 ¼ ½h2L e1 e12 R2 ½h2L e1 e12 T T h2L 3 h2L 3 2 e1 e12 þ e21 R2 e1 e12 þ e21 ; h2L h2L 2 2 H3 ¼ ½hL e1 e13 R3 ½hL e1 e13 T T hL 3 hL 3 2 e1 e13 þ e22 R3 e1 e13 þ e22 ; hL hL 2 2 H4 ¼ ½h1UL e1 e14 R4 ½h1UL e1 e14 T T h1UL 3 h1UL 3 2 e23 R4 e23 ; e1 e14 þ e1 e14 þ h1UL h1UL 2 2 H5 ¼ ½h1UL e1 e15 R4 ½h1UL e1 e15 T T h1UL 3 h1UL 3 e1 e15 þ e1 e15 þ 2 e24 R4 e24 ; h1UL h1UL 2 2 H6 ¼ ½h2UL e1 e16 R5 ½h2UL e1 e16 T T h2UL 3 h2UL 3 e1 e16 þ e1 e16 þ 2 e25 R5 e25 ; h2UL h2UL 2 2 H7 ¼ ½h2UL e1 e17 R5 ½h2UL e1 e17 T T h2UL 3 h2UL 3 e1 e17 þ e1 e17 þ 2 e26 R5 e26 ; h2UL h2UL 2 2 H8 ¼ ½hUL e1 e18 R6 ½hUL e1 e18 T T hUL 3 hUL 3 2 e1 e18 þ e27 R6 e1 e18 þ e27 ; hUL hUL 2 2 H9 ¼ ½hUL e1 e19 R6 ½hUL e1 e19 T T hUL 3 hUL 3 2 e1 e19 þ e28 R6 e1 e19 þ e28 : hUL hUL 2 2

On the other hand, for any matrix H with appropriate dimension, it is true that m h X 0 ¼ 2z_T ðtÞH ck ðtÞ zðtÞ _ Dk zðt dðtÞÞ þ Ak gðzðtÞÞ k¼1

þBk gðzðt hðtÞÞÞ # Z t gðzðsÞÞds þ Ek zðt _ rðtÞÞ ; þCk tsðtÞ T

¼ f ðtÞP12 fðtÞ:

ð45Þ

From (6), the following inequality holds for any positive diagonal matrices Gi ; i ¼ 1; 2; . . .; 7 T z ðtÞðG1 R1 ÞzðtÞþ2zT ðtÞðG1 R2 ÞgðzðtÞÞþgT ðzðtÞÞðG1 ÞgðzðtÞÞ T þ z ðth1 ðtÞÞðG2 R1 Þzðth1 ðtÞÞþ2zT ðth1 ðtÞÞðG2 R2 Þgðzðth1 ðtÞÞÞ þgT ðzðth1 ðtÞÞÞðG2 Þgðzðth1 ðtÞÞÞ T þ z ðth1U ÞðG3 R1 Þzðth1U Þþ2zT ðth1U ÞðG3 R2 Þgðzðth1U ÞÞ þgT ðzðth1U ÞÞðG3 Þgðzðth1U ÞÞ T þ z ðth2 ðtÞÞðG4 R1 Þzðth2 ðtÞÞþ2zT ðth2 ðtÞÞðG4 R2 Þgðzðth2 ðtÞÞÞ þgT ðzðth2 ðtÞÞÞðG4 Þgðzðth2 ðtÞÞÞ T þ z ðth2U ÞðG5 R1 Þzðth2U Þ þ2zT ðth2U ÞðG5 R2 Þgðzðth2U ÞÞþgT ðzðth2U ÞÞðG5 Þgðzðth2U ÞÞ þ zT ðthðtÞÞðG6 R1 ÞzðthðtÞÞþ2zT ðthðtÞÞðG6 R2 ÞgðzðthðtÞÞÞ þgT ðzðthðtÞÞÞðG6 ÞgðzðthðtÞÞÞ T þ z ðthU ÞðG7 R1 ÞzðthU Þ þ2zT ðthU ÞðG7 R2 ÞgðzðthU ÞÞþgT ðzðthU ÞÞðG7 ÞgðzðthU ÞÞ

0¼

¼fT ðtÞP13 fðtÞ:

ð46Þ From Eqs. (16)–(46), by using S-procedure in Boyd et al. (1994), if Eqs. (11)–(13) hold, then an upper bound of _ VðzðtÞ;tÞ can be written as _ VðzðtÞ; tÞ fT ðtÞNfðtÞ:

ð47Þ

From (34)–(43), it gives that

123

Cogn Neurodyn

Based on Lemma 2.4, fT ðtÞ N fðtÞ\0 with C fðtÞ ¼ 0 is equivalent to ðC? ÞT N C? \0: Therefore, if the inequality (10) holds, the equilibrium point of system (9) is asymptotically stable. This completes the proof Remark 3.1 For the case of SHNNs without neutral term, we let Ek ¼ 0 in (9) and the following corollary can be obtained with a proof similar to Theorem 3.1. In this case, network (9) can be rewritten as m h X zðtÞ _ ¼ ck ðtÞ Dk zðt dðtÞÞ þ Ak gðzðtÞÞ þ Bk gðzðt k¼1 Z t i hðtÞÞÞ þ Ck gðzðsÞÞds : tsðtÞ

ð48Þ Corollary 3.1 For given positive scalars d1L ; d1U ; d2L ; d2U ; h1L ; h1U ; h2L ; h2U ; d1 ; d2 ; h1 ; h2 ; s; g1 ; g2 ; l1 ; l2 ; sD , and diagonal matrices Kp ; Km , then the neural network described by (48) is asymptotically stable, for any time-varying delay dðtÞ; hðtÞ and sðtÞ satisfying (2), if there exist positive definite matrices Pi ði ¼ 1; 2; . . .; 18Þ 2 Rnn ; Ti ði ¼ 1; 2; 3Þ 2 Rnn ; Qi ði ¼ 1; 2; . . .; 17Þ 2 nn nn 2n2n R2n2n ; W R U; V; W; X; Y; Z 2 R ; U2 R ; V2 2n2n 2n2n 2n2n 2n2n 2R ;X 2 R ;Y 2 R ; Z2 R ; Ri ði ¼ 1; 2; . . .; 6Þ 2 Rnn ; S1 2 Rnn , positive diagonal matrices Dl ¼ diagfkl1 ; kl2 ; . . .; kln g; Kl ¼ diagfll1 ; ll2 ; . . .; lln g; H 2 Rnn ; Gi ði ¼ 1; 2; . . .; 7Þ 2 Rnn , any symmetric nn matrices Fi 2 R ði ¼ 1; 2; . . .; 6Þ, any matrices L; M; N 2 R2n2n such that the following LMIs hold: T ? ? C N C \0; ð49Þ 2 3 2 3 X þ F 1 Y þ F 3 L M 4 5 0; 4 5 0; X þ F 2 3 Y þ F 4 2 Z þ F 5 N 4 5 0; Z þ F 6 ð50Þ where N is same as defined in Theorem 3.1 with Ek ¼ 0: Proof For the proof, consider the same Lyapunov–Krasovskii functional (10) with S2 ¼ 0 in V9 ðzðtÞ; tÞ: Then by following the same procedure in Theorem 3.1, we obtain N 2

with

S2 ¼ 0:

Then

by

defining

C ¼ 40n . . .. . .0n Ak |fflfflfflfflfflffl{zfflfflfflfflfflffl}

28 times i 0n . . .. . .0n Bk 0n . . .. . .0n Ck 0n . . .. . .0n Dk 0n . . .. . .0n and |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl}

5 times

4 times

9 times

5 times

T

its right orthogonal complement by C we conclude the proof similar to Theorem 3.1. h

123

Remark 3.2 For the case of SHNNs without leakage and neutral term, we let Ek ¼ 0 in (9) and the following corollary can be obtained with a proof similar to Theorem 3.1. In this case, network (9) can be rewritten as m h X zðtÞ _ ¼ ck ðtÞ Dk zðtÞ þ Ak gðzðtÞÞ k¼1

þ Bk gðzðt hðtÞÞÞ þ Ck

Z

i

t

ð51Þ

gðzðsÞÞds : tsðtÞ

Corollary 3.2 For given positive scalars h1L ; h1U ; h2L ; h2U ; h1 ; h2 ; s; l1 ; l2 ; sD , and diagonal matrices Kp ; Km , then the neural network described by (51) is asymptotically stable, for any time-varying delay h(t) and sðtÞ satisfying (2), if there exist positive definite matrices Pi ði ¼ 1; 7; . . .; 18Þ 2 Rnn ; Ti ði ¼ 1; 2; 3Þ 2 Rnn ; Qi ði ¼ R2n2n ; V2 R2n2n ; W2 6; 7; . . .; 17Þ 2 Rnn ; U2 R2n2n ; 2n2n 2n2n 2n2n R R X2 R ; Y2 ; Z2 ,Ri ði ¼ 1; 2; . . .; 6Þ 2 Rnn ; S1 2 Rnn , positive diagonal matrices Dl ¼ diagfkl1 ; kl2 ; . . .; kln g; Kl ¼ diagfll1 ; ll2 ; . . .; lln g; H 2 Rnn ; Gi ði ¼ 1; 2; . . .; 7Þ 2 Rnn , any symmetric matrices Fi 2 Rnn ði ¼ 1; 2; . . .; 6Þ, any matrices L; M; N 2 R2n2n such that the following LMIs hold: T b ? \0; b? N C C ð52Þ 2 3 2 3 X þ F 1 Y þ F 3 L M 4 5 0; 4 5 0; X þ F2 3 Y þ F4 2 Z þ F 5 N 4 5 0; Z þ F6 ð53Þ where N is same as defined in Theorem 3.1 with Ek ¼ 0: Proof For the proof, consider the same Lyapunov–Krasovskii functional (10) with Pi ¼ 0; i ¼ 2; 3; . . .; 6; Qi ; i ¼ 1; 2; . . .; 5; U ¼ V ¼ W ¼ X ¼ Y ¼ Z ¼ 0; S2 ¼ 0 in V4 ðz ðtÞ; tÞ; V5 ðzðtÞ; tÞ; V6 ðzðtÞ; tÞ and V9 ðzðtÞ; tÞ: Then by following the same procedure in Theorem 3.1, we obtain N with Pi ¼ 0; i ¼ 2; 3; . . .; 6; Qi ; i ¼ 1; 2; . . .; 5; U ¼ V ¼ b¼ W ¼ X ¼ Y ¼ Z ¼ 0; S2 ¼ 0: Then by defining C 2 3 4DK 0n . . .. . .0n Ak 0n . . .. . .0n Bk 0n . . .. . .0n Ck 5 and its |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} 27 times

4 times

5 times

T

right orthogonal complement by C we conclude the proof similar to Theorem 3.1. h Remark 3.3 We may also consider the case of SHNNs without leakage, distributed and neutral term, we let dðtÞ ¼ Ck ¼ Ek ¼ 0 in (9) and the following corollary can be obtained with a proof similar to Theorem 3.1. In this case, network (9) can be rewritten as

Cogn Neurodyn

zðtÞ _ ¼

m X

ck ðtÞ½Dk zðtÞ þ Ak gðzðtÞÞ þ Bk gðzðt hðtÞÞÞ:

k¼1

ð54Þ Corollary 3.3 For given positive scalars h1L ; h1U ; h2L ; h2U ; h1 ; h2 ; l1 ; l2 , and diagonal matrices Kp ; Km , then the neural network described by (54) is asymptotically stable, for any time-varying delay h(t) satisfying (2), if there exist positive definite matrices Pi ði ¼ 1; 7; . . .; 18Þ 2 Rnn ; Ti ði ¼ 1; 2; 3Þ 2 Rnn ; Qi ði ¼ 6; 7; nn 2n2n R R2n2n ; W2 R2n2n ; X2 . . .; 17Þ 2 R ; U2 ; V2 2n2n 2n2n 2n2n nn R ; Y2 R ; Z2 R ; Ri ði ¼ 1; 2; . . .; 6Þ 2 R , positive diagonal matrices Dl ¼ diagfkl1 ; kl2 ; . . .; kln g; Kl ¼ diagfll1 ; ll2 ; . . .; lln g; H 2 Rnn ; Gi ði ¼ 1; 2; . . .; 7Þ 2 Rnn , any symmetric matrices Fi 2 Rnn ði ¼ 1; 2; . . .; 6Þ, any matrices L; M; N 2 R2n2n such that the following LMIs hold:

? T W N W? \0; ð55Þ 2 3 2 3 Y þ F 3 X þ F 1 L M 4 5 0; 4 5 0; þ F2 þ F4 X Y 2 3 Z þ F 5 N 4 5 0; Z þ F 6 ð56Þ where N is same as defined in Theorem 3.1 with dðtÞ ¼ Ck ¼ Ek ¼ 0: Proof For the proof, consider the same Lyapunov–Krasovskii functional (10) with Pi ¼ 0; i ¼ 2; 3; . . .; 6; Qi ; i ¼ 1; 2; . . .; 5; U ¼ V ¼ W ¼ X ¼ Y ¼ Z ¼ 0; S1 ¼ S2 ¼ 0 in V4 ðzðtÞ; tÞ; V5 ðzðtÞ; tÞ; V6 ðzðtÞ; tÞ and V9 ðzðtÞ; tÞ: Then by following the same procedure in Theorem 3.1, we obtain N with Pi ¼ 0; i ¼ 2; 3; . . .; 6; Qi ; i ¼ 1; 2; . . .; 5; U ¼ V ¼ W ¼ X ¼ Y ¼ Z ¼ 0; S1 ¼ S2 ¼ 0: Then by defining W ¼ 2 3 4DK 0n . . .. . .0n Ak 0n . . .. . .0n Bk 0n . . .. . .0n 5 and its right |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} |fflfflfflfflfflffl{zfflfflfflfflfflffl} 27 times

5 times

4 times

orthogonal complement by WT we conclude the proof similar to Theorem 3.1. h Remark 3.4 In order to use more information about neuron activation functions, in this paper terms on the slope of neuron activation functions are introduced in the L–K functional to study the stability of addressed NNs. In Shao and Han (2011) have used the term. n Z zi X 2 di gi ðsÞds; where di 0; i ¼ 1; 2; . . .; n i¼1

0

in their L–K functional for the neuron activation function gðzðÞÞ. By utilizing the condition (4) about the slope of the

neuron activation functions into the L–K functional, the term " Z # Z zi ðtÞ n zi ðtÞ X þ k1i ðgi ðsÞ ki sÞds þ d1i ðki s gi ðsÞÞds ; 2 i¼1

0

0

has been introduced in Li et al. (2011). Recently, only few authors have employed delay bounds into the slope of neuron activation functions in the L–K functional, see Kwon et al. (2014a, b). Inspired by these works, in this paper, we consider a new V2 ðzðtÞ; tÞ, which indicates that more information about neuron activations has been used and it has not been considered in any of the previous works that deal with the stability of SHNNs with successive timevarying delay components. Remark 3.5 In order to reduce the conservatism of stability conditions, inspired by the ideas in Kwon et al. (2014b), six zero integral equalities in (22)–(27) are introduced and terms involving these inequalities are merged with Eq. (29) during the calculation of V7 ðzðtÞ; tÞ. After then, reciprocal convex combination technique is utilized in the proof of Theorem 3.1, which can lead to a further improvement of the stability criterion. It is noted that introducing augmented L–K functional and zero integral inequalities and utilizing reciprocal convex combination technique can lead to less conservative results. Remark 3.6 The number of decision variables used in Theorem 3.1 is larger than the previous studies in Rakkiyappan et al. (2015a, b), Senthilraj et al. (2016), and Dharani et al. (2015). Because, the reason is the proposed model consists of an additive interval timedelay components in the state both of discrete delay and leakage delay with newly augmented form of L–K functionals. As we know that, in order to reduce the computational burden the Finsler’s lemma was conducted in the proof of Theorem 3.1, which in turn to reduces the computational burden. As a result, proposed stability criteria gives better results while maintaining lower computational burden. Remark 3.7 It is important to note that very limited works have been done on stability of switched Hopfield NNs of neutral-type with time-varying delays. More particularly, stability analysis of switched Hopfield NNs of neutral-type with successive interval time-varying delay components in the state both of discrete and leakage delay has not been completely studied in previous literature (see e.g., Rakkiyappan et al. 2015a, b; Senthilraj et al. 2016; Dharani et al. 2015). In order to fill such a gap, in this paper we aimed to obtain new stability criteria for switched Hopfield NNs of neutral-

123

Cogn Neurodyn

type with successive interval time-varying delay components in the state both of discrete and leakage delay is proposed. Therefore, the results of the present paper are essentially new. Hence, unfortunately we could not provide any comparison results over existing methods in order to show the improvements. Remark 3.8 It is noted that, very recently Zeng et al. (2015) proposed the free-matrix-based integral inequality and this integral inequality used for handling the double integral L–K functionals, that offers a new tighter information on the upper bounds of time-varying delay and its interval for the time-delay systems. Therefore, we utilizing this integral inequality to deal with such L–K functionals, which turn to reduce the conservatism further. Thus, there is no limit for such improvements on delay bounds of time-delay systems it’s basically depends on choosing good L–K functionals and computing it’s derivative with an newly improved integral inequalities or some other techniques called delay-partitioning approaches and so on. Thus, in the future, the inequality proposed in Zeng et al. (2015) can be used in order to achieve improved results for delayed NNs. Remark 3.9 It is well-known that most of the existing results concerning the stability problem of delayed switched Hopfield NNs of neutral type. However, switched Hopfield NNs of neutral type with successive interval time-varying delay components in the state of discrete delay and leakage delay has not been considered in the previous works. In contrast to the system models in Rakkiyappan et al. (2015a, b), Senthilraj et al. (2016), Dharani et al. (2015); one can find that their results cannot be applicable to system (1). This indicates that the proposed system model and obtained results are essentially new. There is no doubt that studying stability analysis for the systems described in (9), with leakage and discrete interval time-varying delays is sure not only to enhance the dynamic research theory of system model proposed in (9), but also further enrich the foundation of realistic application for the delayed SHNNs, as shown in the following numerical section.

Numerical examples In this section, we provide four numerical examples to demonstrate the effectiveness of our delay-dependent stability criteria. Example 4.1

123

Consider system (9) with n ¼ k ¼ 2 and

0 4:6 0 ; D2 ¼ ; 04:7 04:3 1:1 0:7 0:8 1:1 A1 ¼ ; A2 ¼ ; 0:9 1:2 0:9 0:8 1:2 0:6 ; B1 ¼ 0:8 1 0:6 0:7 0:8 0:9 B2 ¼ ; C1 ¼ ; 0:7 0:6 0:9 0:8 0:6 0:6 0:8 1:0 ; E1 ¼ ; C2 ¼ 0:65 0:6 0:9 0:8 0:9 1:2 : E2 ¼ 0:9 0:9 D1 ¼

5:1

The activation functions are assumed to be gi ðzi Þ ¼ 0:5ðjzi þ 1j jzi 1jÞ; i ¼ 1; 2: It is easy to check that the activation functions are satisfied (6) with Km ¼ diagf0; 0g; Kp ¼ diagf1; 1g. Also let d1L ¼ 0:10; d1U ¼ 0:20; d1 ¼ 0:30; d2L ¼ 0:15; d2U ¼ 0:25; d2 ¼ 0:40; h1L ¼ 0:50; h1U ¼ 1:0; h1 ¼ 1:50; h2L ¼ 0:80; h2U ¼ 1:0; h2 ¼ 1:80; s ¼ 0:30; r ¼ 0:40; g1 ¼ 0:4; g2 ¼ 0:5; l1 ¼ 0:4; l2 ¼ 0:5; sD ¼ 0:5; rD ¼ 0:5: By our Theorem 3.1 and Matlab LMI toolbox, it is found that the equilibrium point of system (9) is asymptotically stable. It can also be verified that the LMIs (10)–(13) are feasible for larger upper delay bounds d1 ; d2 ; h1 ; h2 ; s and r. lt shows that all the conditions stated in Theorem 3.1 have been satisfied and hence system (9) with the above given parameters are asymptotically stable. Example 4.2 Consider the switched Hopfield neural network without neutral term as in (48) with the parameters Dk ; Ak ; Bk ; Ck ðk ¼ 1; 2Þ as defined in Example 4.1. By choosing d1 ðtÞ ¼ 0:1 þ 0:1 cosð0:5tÞ; d2 ðtÞ ¼ 0:2 þ0:2 cos ð0:5tÞ; h1 ðtÞ ¼ 0:6 þ 0:6 sinð0:5tÞ; h2 ðtÞ ¼ 0:7 þ 0:7 sin ð0:5tÞ; sðtÞ ¼ 0:25 þ 0:25 cosð3tÞ, we let d1L ¼ 0:05; d1U ¼ 0:15; d1 ¼ 0:20; d2L ¼ 0:10; d2U ¼ 0:30; d2 ¼ 0:40; h1L ¼ 0:40; h1U ¼ 0:80; h1 ¼ 1:20; h2L ¼ 0:50; h2U ¼ 1:0; h2 ¼ 1:50; s ¼ 0:50 and g1 ¼ 0:2; g2 ¼ 0:3; l1 ¼ 0:4; l2 ¼ 0:5; sD ¼ 0:5: Also letting gi ðzi Þ ¼ 0:5ðjzi þ 1j jzi 1jÞ; i ¼ 1; 2: it can be easily verified that the activation functions holds with Km ¼ diagf0; 0g; Kp ¼ diagf1; 1g. By using Matlab LMI toolbox, it is found that LMI (49) and (50) is feasible. Thus, it can be conclude that the switched NNs (48) is asymptotically stable and the state trajectories of the dynamical system is converges to the zero equilibrium point with an initial state ½0:2; 0:2T , it can be shown in Fig. 1. Suppose, if we take leakage timevarying delay d1 ðtÞ ¼ 0:15 þ 0:15 cosð0:5tÞðd1 0:30Þ;

Cogn Neurodyn 1

0.2 z1 z2

0.15

z1 z2 0.8

0.1

0.6 0.05

z(t)

z(t)

0.4 0

0.2 −0.05

0

−0.1

−0.2

−0.15 −0.2

0

10

20

30

40

50

time

z1 z2

2 1.5

z(t)

1 0.5 0 −0.5 −1 −1.5

10

20

time

2

4

6

8

10

Fig. 3 State trajectory of the system (51) in Example 4.3

2.5

0

0

time


−2

−0.4

30

40

50


d2 ðtÞ ¼ 0:25 þ 0:25 cosð0:5tÞðd2 0:50Þ, it is found that the neural network (48) is actually unstable and the state trajectories of the dynamical system is not converges to the zero equilibrium point, it can be shown in Fig. 2. According to this example, it can be conclude that the leakage delay has a significant effect in the dynamical behaviour of the switched NNs. Remark 4.1 As is well-known that the leakage time delays are unavoidable and their occurrence causes instability or oscillation, it can be verified through different simulation results for different time delays especially for the leakage delay that the oscillation of the dynamics increases when time delays are chosen to be larger, which would obviously affect the stability. Thus, time delays in the leakage term have a great impact on the stability of the considered switched system.

Example 4.3 Consider the switched Hopfield neural network without leakage and neutral term as in (51) with the parameters Ak ; Bk ; Ck ; Dk ðk ¼ 1; 2Þ as defined in Example 4.1. By choosing h1 ðtÞ ¼ 0:8 þ 0:8 sinð0:5tÞ; h2 ðtÞ ¼ 1:2 þ 1:2 sinð0:5tÞ; sðtÞ ¼ 0:5 þ 0:5 cosð3tÞ; we let h1L ¼ 0:50; h1U ¼ 1:10; h1 ¼ 1:60; h2L ¼ 0:70; h2U ¼ 1:70; h2 ¼ 2:40; s ¼ 1:0 and l1 ¼ 0:3; l2 ¼ 0:35; sD ¼ 0:5. Also letting g1 ðzÞ ¼ g2 ðzÞ ¼ 0:5ðjz þ 1j jz 1jÞ, it can be easily verified that the neuron activation function holds with Km ¼ diagf0; 0g; Kp ¼ diagf1; 1g. By using Matlab LMI toolbox, it is found that LMIs in Corollary 3.2 is feasible. Thus, we can conclude that the model (51) is asymptotically stable. The simulation results for the above mentioned delay values also ensure the asymptotic stability of the model (51). Hence, the convergence of the SHNNs (51) is shown in Fig. 3, with an initial state ½0:4; 0:8T . Example 4.4 So far, originally NNs embody the characteristics of real biological neurons that are connected or functionally related in a nervous system. On the other hand, NNs can represent not only biological neurons but also other practical systems namely the quadruple-tank process system can be shown in Fig. 5. The setup consists of four interacting tanks, two water pumps and two valves. The two process inputs are the voltages t1 and t2 supplied to the two pumps. Tank 1 and Tank 2 are placed below Tank 3 and Tank 4 to receive water flow by the action of gravity. Hence as shown in Fig. 4, the quadruple-tank process can be expressed clearly using the neural network model, see for instance, Samidurai and Manivannan (2016), Lee et al. (2013), Huang et al. (2012), Haoussi et al. (2011) and Johansson (2000); proposed the state-space equation of the quadruple-tank process and designed the state feedback controller as follows:

123

Cogn Neurodyn 0.5 z1 z2 z3 z4

0.4 0.3

state z(t)

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4

0

500

1000

1500

2000

2500

3000

times t


3

in this present study transport delays between valves and tanks being additive interval time-varying, it is also taken into account but not exists in previous literature in the following aspects. For simplicity, it was assumed that s1 ¼ 0; s2 ¼ 0 and s3 ¼ hðtÞ ¼ h1 ðtÞ þ h2 ðtÞ (since h1L h1 ðtÞ h1U and h2L h2 ðtÞ h2U ). Here, the control input uðtÞ, means that the amount of water supplied by the pumps. Therefore, it is true that uðtÞ has a threshold value due to the limited area of the hose and the capacity of the pumps. Therefore, it is natural to consider uðtÞ, as a nonlinear function as follows:

7 7 7; 5

uðtÞ ¼ K gð zðtÞÞ; uðt sðtÞÞ

Fig. 4 Schematic representation of the quadruple-tank process. Source: From Johansson (2000)

x_ðtÞ ¼ A0 xðtÞ þ A1 xðt s1 Þ þ B0 uðt s2 Þ þ B1 uðt s3 Þ; ð57Þ where 2

0:0021 0 6 0 0:0021 6 A0 ¼ 6 4 0 0 2

0 60 6 A1 ¼ 6 40 B0 ¼ B1 ¼

0

0 0

0 0

0:0424

0

0

0

0

0:1113c1

0

0

0

0

0:1042c2

0

0

0 0

0:0424

0 0 0 3 0 0:0424 0 0 0 0:0424 7 7 7; 0 0 0 5

0 0

T

0:1765 0:0795 0:1579 0:2288

;

zðtÞ _ ¼ D1 zðtÞ þ A1 gðzðtÞÞ þ B1 gðzðt hðtÞÞÞ; yðtÞ ¼ uðtÞ;

T ;

0:2073 : 0:0772

Generally speaking, the differential equations representing the mass balances in the delayed [transport delay hðtÞ ¼ h1 ðtÞ þ h2 ðtÞ] equations. To derive a more interesting control problem, transport delays can easily be added by delaying the inlet of water to the tanks, so it is the possible approach used to examine in this paper. Moreover,

123

i ¼ 1; . . .; 4:

The quadruple-tank process (57) can be rewritten to the form of system (54) with k ¼ 1, as follows:

0 0:1113ð1 c1 Þ 0:1042ð1 c2 Þ 0 c2 ¼ 0:307; u ¼ KxðtÞ;

c1 ¼ 0:333; 0:1609 K¼ 0:1977

gð zðtÞÞ ¼ ½g1 ðz1 ðtÞÞ; . . .; g4 ðz4 ðtÞÞT ; gi ðzi ðtÞÞ ¼ 0:1ðj zi ðtÞ þ 1 j j zi ðtÞ 1 jÞ;

ð58Þ

where B1 ¼ B1 K; gðÞ ¼ gðÞ. D1 ¼ A0 A1 ; A1 ¼ B0 K; In addition, Km ¼ diagf0; 0; 0; 0g; Kp ¼ diagf0:1; 0:1; 0:1; 0:1g with h1L ¼ 0:60; h1U ¼ 1:20; h1 ¼ 1:80; h2L ¼ 0:80; h1U ¼ 1:50; h2 ¼ 2:30; l1 ¼ l2 ¼ 0:5. Using MATLAB LMI control Toolbox and by solving LMIs in Corollary 3.3, we found that the quadruple-tank process system (58) is asymptotically stable. By choosing h1 ðtÞ ¼ 0:9 þ 0:9 sinð0:5tÞ; h2 ðtÞ ¼ 1:15 þ 1:15 sinð0:5tÞ; l1 ¼ l2 ¼ 0:5 and gi ðzi Þ ¼ 0:1ðj zi þ 1 j j zi 1 jÞ; i ¼ 1; 2; . . .; 4, it can be easily verified that Assumption (H) is

Cogn Neurodyn

holds. Figure 5 shows the state trajectories of the system is converges to zero equilibrium point with an initial state ½0:3; 0:2; 0:5; 0:4, hence it is found that the dynamical behavior of the quadruple-tank process system (58) is asymptotically stable.

Conclusions In this paper, the problem of new delay-interval-dependent stability criteria for SHNNs of neutral type with time delays have been investigated. In order to achieving stability results, some suitable L–K functional under the weaker assumption of neuron activation function divided by states are utilized to enhance the feasible region of proposed stability criteria. By using the famous Jensen’s inequality, WDII Lemma, introducing of some zero equations and combined with RCC technique, a novel delayinterval-dependent stability criterion is derived in terms of linear matrix inequalities (LMIs). Then the feasibility and effectiveness of the developed methods have been shown by interesting numerical simulation examples. The proposed approach is finally demonstrate the numerical simulation of the benchmark problem that takes into account additive time-varying delays, showing the feasibility of the proposed approach on a realistic problem. Therefore, our results have an important significance in theory and design, as well as in applications of neutral type SHNNs with delays in leakage terms. Acknowledgments The work of first two authors was supported by the Department of Science and Technology—Science and Engineering Research Board (DST-SERB), Government of India, New Delhi, for its financial support through the research Project Grant No. SR/ FTP/MS-041/2011.

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