Synchronization of switched neural networks with ... - Semantic Scholar

Chaos, Solitons & Fractals 44 (2011) 817–826

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Synchronization of switched neural networks with mixed delays via impulsive control Xinsong Yang a, Chuangxia Huang b,⇑, Quanxin Zhu c a

Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China Department of Mathematics, College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410114, China c Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, China b

a r t i c l e

i n f o

Article history: Received 16 October 2010 Accepted 15 June 2011 Available online 3 September 2011

a b s t r a c t This paper concerns the problem of global exponential synchronization for a class of switched neural networks with time-varying delays and unbounded distributed delays via impulsive control method. By using Lyapunov stability theory, new synchronization criterion is derived. In our synchronization criterion, the switching law can be arbitrary and the concept of average impulsive interval is utilized such that the obtained synchronization criterion is less conservative than those based on maximum of impulsive intervals. Numerical simulations are given to show the effectiveness and less conservativeness of the theoretical results. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Synchronization control of chaotic systems has attracted much attention from researchers of many fields thanks to its potential applications such as secure communication, information processing, etc. [1,2]. Among a lot of synchronization phenomena, drive-response synchronization is one of the most important research topics. The principle of drive-response synchronization is this: a chaotic system, called the driver (or master), generates a signal (or transmitted information) sent over a channel to a responder (or slave), which uses this signal to synchronize itself with the driver. In other words, in the drive-response (or master–slave) systems, the response (or slave) system is influenced by the behavior of the drive (or master) system, but the drive (or master) system is independent of the response (or slave) one. On the other hand, it is reported that delayed neural networks can exhibit chaotic behavior if the network’s parameters and delays are appropriately chosen [3–13]. Therefore, many results concerning

⇑ Corresponding author. Tel./fax: +86 731 85258787. E-mail addresses: [email protected] (X. Yang), cxiahuang@126. com (C. Huang), [email protected] (Q. Zhu). 0960-0779/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2011.06.006

synchronization problem of chaotic delayed neural networks have been appeared in the literature [3–13]. It is well-known that control method has great influence on the realization of chaos synchronization. To date, many effective control methods have been developed, such as state feedback control [3–7], adaptive control [8,9], intermittent control [10,11], impulsive control [12–18], etc. State feedback control and adaptive control are continuous state feedback control methods, which demand the states of both drive and response dynamics are known incessantly. From the viewpoint of engineering applications, the control cost of continuous feedback control is expensive. In order to overcome this drawback, in [10,11], intermittent state feedback control was adopted to synchronize coupled neural networks with constant time-delay, which reduce the control cost and the amount of the transmitted information to some extent. It is worth noting that both intermittent control and impulsive control are discontinuous control techniques. The difference between them is that intermittent control is activated during some intervals and does not work during the other intervals while impulsive control is activated only at some isolated points. Obviously, the control cost and the amount of the transmitted information can be further reduced if the response system can be synchronized with derive

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X. Yang et al. / Chaos, Solitons & Fractals 44 (2011) 817–826

system under impulsive control. Recently, authors of [15] studied the global asymptotical synchronization of nondelayed chaotic neural networks by output feedback impulsive control. Authors of [12,13] studied synchronization of neural networks with constant time-delay via impulsive control. However, to the best of our knowledge, result on synchronization of neural networks with distributed delay via impulsive control has not yet been reported, especially on neural networks with unbounded distributed delays. The main reason may be the difficulty in coping with the distributed delay under impulsive control. Since neural networks usually have a spatial extent due to the presence of an amount of parallel pathways with a variety of axon size and lengths [3–7], it is more practical to model them by introducing distributed delays. Bearing in mind that in most cases the time-delay is not a constant and distributed delay is indispensable, in this paper, we shall investigate synchronization of neural networks with both time-varying delay and unbounded distributed delay (mixed delays) via impulsive control. Note that the cost of impulsive control is strongly related to impulsive interval, which is the most important characteristic of impulsive synchronization technique. As far as impulsive controllers with equal control gain are concerned, the larger the impulsive intervals are, the lower the control cost is expended. However, most of existing criteria of impulsive synchronization are conservative since maximum of impulsive intervals is used to derive their main results (see for instance [12–25]), which decreases the impulse distances. In [26], Lu et al. studied the synchronization coupled impulsive dynamical networks by proposing a concept named average impulsive interval. With this concept, the impulse distances in [26] can be increased. Unfortunately, authors of [26] did not consider time delay, and the results cannot be directly extended to the case with delay. Hence, one of this paper’s aims is to modify the concept of average impulsive interval. After that, by using the modified concept we study the synchronization problem of neural networks with mixed delays. On the other hand, as an important class of hybrid systems, switched systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models. A switched system is composed of several dynamical subsystems and a switching law that specifies the active subsystem at each switching instant of time. Recently, there has been increasing interest in the theory of analysis of switched systems since they have numerous applications in communication, control of mechanical systems, automotive industry, aircraft and air traffic control, electric power systems [27–36] and many other fields. A primary application of derive-response synchronization is secure communication. When the switching laws are unknown, it is difficult for an intruder to synchronize with driving switched system [27,28], and hence, the security of communication is improved. In [27,28,30], asymptotic synchronization of switched systems with arbitrary switching law is investigated by using feedback control. Stability of hybrid impulsive and switching systems was studied in [35,36]. But authors of [35,36] did not consider time-delay, let alone unbounded distributed delay. Moreover, the coincidence of impulsive

instants and switching instants in [35,36] leave the limit of their results in real applications. The purpose of this paper is to shorten the gaps mentioned above. Namely, exponential synchronization of switched neural networks with mixed delays is considered via impulsive control. New synchronization criterion is obtained, in which the switching law can be arbitrary and the maximum of impulsive intervals are increased. Therefore, our synchronization technique is flexible and the control cost will be reduced drastically. Numerical simulations are given to show the effectiveness and less conservativeness of the theoretical results. The rest of this paper is organized as follows. In Section 2, the considered model of switched neural networks with mixed delays is presented. Some necessary assumptions, definitions and lemmas are also given in this section. In Section 3, synchronization for the proposed model is studied. Then, in Section 4, one example with simulations is presented to show the effectiveness and less conservativeness of the theoretical results. Finally, Section 5 provides some conclusions and prospects of future works. Notations: In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. Nþ denote the set of positive integers. In denotes the n  n identity matrix. Rn and Rnn denote the n-dimensional and n  n-dimensional real spaces equipped with Euclidean norm kk. For vector x 2 Rn ; kxk2 ¼ xT x, where T denotes transposition. For a matrix A 2 Rnn ; kAk ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmax ðAT AÞ, where kmax() means the largest eigenvalue. A > 0 means A is a symmetric and positive definite matrix. K = {1, 2, . . . , n}. 2. Preliminaries A general neural network model with mixed delays is described as follows:

_ xðtÞ ¼ CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt  sðtÞÞÞ Z t pðt  sÞf ðxðsÞds þ IðtÞ; þD

ð1Þ

1

where xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xn ðtÞÞT 2 Rn represents the state vector of the neural network at time t; n corresponds to the number of neurons; f(x(t)) = (f1(x1(t)), . . . , fn(xn(t)))T is the neuron activation function; C = diag{c1, c2, . . . , cn} is a diagonal matrix with ci > 0, i 2 K; A = (aij)nn, B = (bij)nn and D = (dij)nn are the connection weight matrix, time-delayed weight matrix and the distributively time-delayed weight matrix, respectively; IðtÞ ¼ ðI1 ðtÞ; I2 ðtÞ; . . . ; In ðtÞÞT 2 Rn is an external input vector; s(t) denotes the time-varying delay; p() is a scalar function describing the delay kernel. By introducing switching signal into the system (1) and taking a set of neural networks as the individual subsystems, the switched system can be obtained, which is described as:

_ xðtÞ ¼ C r xðtÞ þ Ar fr ðxðtÞÞ þ Br fr ðxðt  sr ðtÞÞÞ Z t pr ðt  sÞfr ðxðsÞÞds þ Ir ðtÞ; þ Dr 1

ð2Þ

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X. Yang et al. / Chaos, Solitons & Fractals 44 (2011) 817–826

A switching signal is simply a piecewise constant signal taking values on index set. In model (2), the rðtÞ : ½0; þ1Þ ! M ¼ f1; 2; . . . ; mg is a switching signal. Then {Cr, Ar, Br, Dr, fr(), sr(), pr(), Ir(t)} is a family of matrices, activation functions, time-varying delays, delay kernels, and external input vectors parametrized by index set M. In this paper, we assume that the switching law of r is known priori to the receiver, and its instantaneous value is available in real time. Let the switched system (2) the driving system. We construct a controlled response system with the same switching law as that in system (2), which is presented by

_ yðtÞ ¼ C r yðtÞ þ Ar fr ðyðtÞÞ þ Br fr ðyðt  sr ðtÞÞÞ Z t pr ðt  sÞfr ðyðsÞÞds þ Ir ðtÞ þ U r ; þ Dr

ð3Þ

where yðtÞ ¼ ðy1 ðtÞ; y2 ðtÞ; . . . ; yn ðtÞÞT 2 Rn is the state vector of the response neural network at time t, Ur are the control inputs. In the following, the aim is to design impulsive controllers such that the response system (3) can globally exponentially synchronize with the driving system (2). Let z(t) = y(t)  x(t). If limt?+1z(t) = 0, then (3) is said to be synchronized with (2) under the controllers Ur. The impulsive controllers Ur are designed as

Ur ¼

k

Er zðtÞdðt  t k Þ;

k 2 Nþ ;

ð4Þ

k¼1

where the impulsive sequence ft k ; k 2 Nþ g satisfies 0 = t0 < t1 < t2 <    < tk1 < tk <   , and limk!þ1 t k ¼ þ1; Ekr 2 Rnn ; dðÞ is the Dirac impulsive function, that is

dðt  t k Þ ¼



m X



pl ðtÞ C l xðtÞ þ Al fl ðxðtÞÞ þ Bl fl ðxðt  sl ðtÞÞÞ

l¼1

þ Dl

Z

t

 pl ðt  sÞfl ðxðsÞÞds þ Il ðtÞ ;

ð7Þ

1

Accordingly, the hybrid impulsive and switched system (3) becomes:

8 m P > _ > yðtÞ ¼ pl ðtÞ½C l yðtÞ þ Al fl ðyðtÞÞ þ Bl fl ðyðt  sl ðtÞÞÞ > > > l¼1 > < i Rt þDl 1 pl ðt  sÞfl ðyðsÞÞ ds þ Il ðtÞ ; t – t k ; > > > m > P > > : Dyðt k Þ ¼ pl ðtÞEkl zðtk Þ; t ¼ tk ; k 2 Nþ : l¼1

1

1 X

_ xðtÞ ¼

1; t ¼ t k ; k 2 Nþ ; 0; t – t k :

With the impulsive controller (4), the controlled network (3) turns out to the following hybrid impulsive and switched system:

8 _ yðtÞ ¼ C r yðtÞ þ Ar fr ðyðtÞÞ þ Br fr ðyðt  sr ðtÞÞÞ > < Rt þDr 1 pr ðt  sÞfr ðyðsÞÞds þ Ir ðtÞ; t – t k ; > : Dyðt k Þ ¼ yðt k Þ  yðt k Þ ¼ Ekr zðt k Þ; t ¼ t k ; k 2 Nþ ; ð5Þ   þ where yðt k Þ ¼ yðt þ k Þ ¼ limt!t k yðtÞ; yðt k Þ ¼ limt!t k yðtÞ. Define an indicator function p(t) = (p1(t), p2(t), . . . , pm(t))T [27], where

8 1; when the switching system is described by > > > < the lth mode pl ðtÞ ¼ > fC ; A l l ; Bl ; Dl ; fl ðÞ; sl ðÞ; pl ðÞ; I l ðtÞg; > > : 0; otherwise; ð6Þ Pm with l = 1, 2, . . . , m. It follows from (6) that l¼1 pl ðtÞ ¼ 1 under any switching law and the switching time instants have nothing to do with impulsive time instants. With the indicator function (6), the switched system (2) can be written as

ð8Þ Subtracting (7) from (8) yields the following hybrid impulsive and switched error dynamical system:

8 m P > > z_ ðtÞ ¼ pl ðtÞ½C l zðtÞ þ Al g l ðzðtÞÞ þ Bl g l ðzðt  sl ðtÞÞÞ > > > l¼1 > < i Rt þDl 1 pl ðt  sÞg l ðzðsÞÞ ds ; t – tk ; > > > m > P > > : zðtk Þ ¼ pl ðtÞðIn þ Ekl Þzðt k Þ; t ¼ tk ; k 2 Nþ ; l¼1

ð9Þ where gl(z(t)) = fl(y(t))  fl(x(t)). The initial condition associated with the error system (9) is given as z(s) = /(s), s 2 (1, 0]. In order to achieve our main results, we make the following assumptions: (H1) There exist constants sl > 0 such that 0 < sl ðtÞ 6 sl ; l 2 M; t 2 R. (H2) The delay kernels pl : [0, +1) ? [0, +1) are real-valued non-negative continuous functions and there R þ1 exist positive numbers al such that 0 pl ðsÞds ¼ al ; l 2 M: ~f i ; i 2 K (H3) There exist constant such that l jfli ðxÞ  fli ðyÞj 6 ~f il jx  yj; 8x; y 2 R; x – y. Definition 1. The zero solution of hybrid impulsive and switched system (9) is said to be globally exponentially stable if there exist constants k > 0, M > 0, such that for any initial values kz(t)k 6 Mekt hold for t P 0. According to assumption (H3), the hybrid impulsive and switched system (9) admits zero solution z(t) = 0. From Definition 1, if the zero solution is globally exponentially stable, then the global exponential synchronization between switched systems (2) and (3) is achieved under the impulsive controllers (4). Similar to the definition of average impulsive interval in [26], we give the following definition. Definition 2. An impulsive sequence ft k ; k 2 Nþ g is said to have average impulsive interval Ta if there exist positive integer f and positive constant Ta such that

T t T t  f 6 NðT; tÞ 6 f þ ; Ta Ta

8T P t P 0;

ð10Þ

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X. Yang et al. / Chaos, Solitons & Fractals 44 (2011) 817–826

where N(T, t) denotes the number of impulsive times of the impulsive sequence ftk ; k 2 Nþ g on the interval (t, T), the constant f is called the ‘‘elasticity number’’ of the impulsive sequence, which implies that, on the time interval (t, T), the practical number of impulsive times N(T, t) may be more or less than Tt by f. Ta Lemma 1 ([37]). Given any real matrices Q1, Q2, Q3 of appropriate dimensions and a scalar e > 0, if Q 3 ¼ Q T3 > 0, then the following inequality holds:

1 Q T1 Q 2 þ Q T2 Q 1 6 eQ T1 Q 3 Q 1 þ Q T2 Q 1 3 Q 2:

e

Lemma 2. Suppose p(t) is a non-negative bounded scalar R þ1 function defined on [0, +1) and 0 pðuÞdu ¼ a. For any connn stant matrix D 2 R ; D > 0, and vector function x : ð1; t ! Rn for t P 0, one has

a

Z

t

!T Z xðsÞ ds D

thðtÞ

t

thðtÞ

xðsÞds 6 hðtÞ

Z

t

xT ðsÞDxðsÞds

thðtÞ

ð13Þ provided that the integrals are all well defined, where h ¼ maxfhðtÞ; t 2 Rg. Remark 1. Corollary 1 is equivalent to Jensen’s inequality, which is an effective tool for studying problems such as stability or synchronization. However, Jensen’s inequality only applicable to bounded delay. In this paper, based on the obtained Lemma 2, we shall investigate synchronization between (2) and (3), where the distributed delay is unbounded. Notice that Lemma 2 is the extension of Jensen’s inequality, results of this paper are also applicable to switched neural networks with time-varying delay and bounded distributed delay.

t

pðt  sÞxT ðsÞDxðsÞds

1

Z

t

P

T Z pðt  sÞxðsÞ ds D

1

t

pðt  sÞxðsÞds

ð11Þ

1

1 ; . . . ; u  j1 ; u  jþ1 ; . . . ;  i for each fixed ðt; u; u creasing in u Rt  m1 ; 1  m ðsÞdsÞ; j ¼ 1; 2; . . . ; m, and Ik ðuÞ : R ! R u pðt  sÞu

Proof. It follows from D > 0 that D1 > 0. In view of Schur complement lemma [38], the following inequality holds:

D1

xðsÞ

!

xT ðsÞ xT ðsÞDxðsÞ

P 0:

Note that p(u) P 0 on [0, +1) implies p(t  s) P 0 on (1, t] about s. Pre-multiplying the above inequality with p(t  s) and integrating from 1 to t, we have

Rt

aD1 Rt

pðt  sÞxT ðsÞds

1

1 Rt 1

pðt  sÞxðsÞds

!

pðt  sÞxT ðsÞDxðsÞds

P 0: ð12Þ

Again, by virtue of Schur complement lemma, the inequality (12) is equivalent to t

pðt  sÞxT ðsÞDxðsÞ ds  1



Z

1

a

Z

Lemma 3. Let 0 6 si(t) 6 si, i = 1, 2, . . . , m  1, for some non-negative constants si, p(t) is a non-negative 1 ; . . . ; bounded scalar function defined on ½0; þ1Þ; Fðt; u; u Rt  m1 ; 1  m ðsÞdsÞ : Rþ  R      R ! R be nondeu pðt  sÞu |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} mþ1

provided the integrals are all well defined.

Z

Z

T

t

pðt  sÞxðsÞ ds

D

1

t

pðt  sÞxðsÞds P 0:

be nondecreasing in u. Suppose that u(t), v(t) satisfy

8 þ > < D RuðtÞ 6 Fðt; uðtÞ; uðt  s1 ðtÞÞ; . . . ; uðt  sm1 ðtÞÞ; t pðt  sÞuðsÞdsÞ; t P 0; > : 1 uðt k Þ 6 Ik ðuðt k ÞÞ; k 2 Nþ ; 8 þ > < D Rv ðtÞ > Fðt; v ðtÞ; v ðt  s1 ðtÞÞ; . . . ; v ðt  sm1 ðtÞÞ; t pðt  sÞv ðsÞdsÞ; t P 0; > : 1 v ðtk Þ P Ik ðv ðtk ÞÞ; k 2 Nþ ; where the upper-right Dini derivative D+y(t) is defined as Dþ yðtÞ ¼ limh!0þ yðtþhÞyðtÞ , where h ? 0+ means that h aph proaches zero from the right-hand side. Then u(t) 6 v(t) for t 6 0 implies u(t) 6 v(t) for t P 0. By the same procedure of the proof for Lemma 3.2 in [39], one can easily get Lemma 3. We omit its proof here.

1

3. Main results Transposing the terms of the above inequality produces (11). This completes the proof. h

for any scalar hðtÞ > 0; t 2 R, then the following corollary can be derived directly from Lemma 2.

In this section, synchronization criterion with three free positive constants is given first by making use of the concept of average impulsive interval. Then optimal synchronization criterion is also obtained by utilizing extreme value theory. Detail methods including how to determine elasticity number of some special impulsive sequences and how to design average impulsive interval are also given through remarks.

Corollary 1. For any constant matrix D 2 Rnn ; DT ¼ D > 0, scalar h(t) > 0 and vector function x : ½h; t ! Rn for t P 0, one has

Theorem 1. Assume (H1)–(H3) hold, and the impulsive sequence ftk ; k 2 Nþ g satisfies (10) with elasticity number f and average impulsive interval Ta. Furthermore, if there exist

As a special case, if

pðsÞ ¼



0; s > hðtÞ; 1; 0 6 s 6 hðtÞ;

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X. Yang et al. / Chaos, Solitons & Fractals 44 (2011) 817–826

positive constants nl, ql, .l such that for each l 2 M the following inequalities hold:

kIn þ

Ekl k2

6 bl < 1;

k 2 Nþ ;

m X

nh

i

zT ðtÞzðtÞ pl ðtÞ 2cl þ n1 þ nl kAl k2 F 2l þ q1 þ .1 l l l

l¼1

þ ql kBl k2 F 2l zT ðt  sl ðtÞÞzðt  sl ðtÞÞ þ al .l F 2l kDl k2 Z t  pl ðt  sÞzT ðsÞzðsÞ ds

ð14Þ

 2cl þ n1 þ nl kAl k2 F 2l þ q1 þ .1 þ l l l f

_ VðtÞ 6

ln bl Ta

1

f

þ bl ql kBl k2 F 2l þ bl a2l .l F 2l kDl k2 < 0;

ð15Þ

¼

m X

nh

i

VðtÞ pl ðtÞ 2cl þ n1 þ nl kAl k2 F 2l þ q1 þ .1 l l l

l¼1

where C l ¼ diagfc1l ; c2l ; . . . ; cnl g; cl ¼ minfcil ; i 2 Kg; F l ¼ i ~ maxff l ; i 2 Kg, then the zero solution of the hybrid impulsive and switched error dynamical system (9) with any switching law is globally exponentially stable.

T

VðtÞ ¼ z ðtÞzðtÞ:

ð16Þ

_ VðtÞ ¼ 2zT ðtÞ

m X

solution

of

(9)

for

Z



pl ðtÞ C l zðtÞ þ Al g l ðzðtÞÞ þ Bl g l ðzðt  sl ðtÞÞÞ

¼



pl ðtÞ 2zT ðtÞC l zðtÞ þ 2zT ðtÞAl g l ðzðtÞÞ

l¼1

þ 2zT ðtÞBl g l ðzðt  sl ðtÞÞÞ þ 2zT ðtÞDl  Z t  pl ðt  sÞg l ðzðsÞÞ ds :

¼

m X

pl ðtÞkIn þ Ekl k2 zT ðtk Þzðtk Þ

6

m X

pl ðtÞbl Vðtk Þ:

nl g Tl ðzðtÞÞATl Al g l ðzðtÞÞ

2z ðtÞAl g l ðzðtÞÞ 6

T n1 l z ðtÞzðtÞ

þ

6

T n1 l z ðtÞzðtÞ

þ nl kAl k2 F 2l zT ðtÞzðtÞ

ð18Þ

ð23Þ

v ðtÞ P VðtÞ P 0;

t P 0:

þ al .l F 2l kDl k2

2zT ðtÞBl g l ðzðt  sl ðtÞÞÞ ð19Þ

For positive constants .l ; l 2 M, it is derived from (H3), Lemmas 1 and 2 that Z t T 2zT ðtÞDl pl ðt  sÞg l ðzðsÞÞds 6 .1 l z ðtÞzðtÞ 1

t

pðt  sÞg l ðzðsÞÞds

T

1

T 6 .1 l z ðtÞzðtÞ þ al .l kDl k

2

Z

Z

Z

0

 pl ðs  uÞv ðuÞdu þ  ds ;

s

ð25Þ

1

8 h i m P > > _ yðtÞ; þ nl kAl k2 F 2l þ q1 þ .1 ¼ pl ðtÞ 2cl þ n1 > yðtÞ l l l < l¼1 m P

> > > : yðt k Þ ¼

l¼1

pðt  sÞg l ðzðsÞÞds

t – tk ;

pl ðtÞbl yðtk Þ; t ¼ tk ; k 2 Nþ :

According to the representation of the Cauchy matrix [40], we get the following estimation 1

t

P l ðt; sÞ ¼ e½2cl þnl

þnl kAl k2 F 2l þq1 þ.1 ðtsÞ Nðt;sÞ l l bl ;

t P s P 0; l 2 M: ð26Þ

1

It follows from (10), (14) and (26) that

t

1

2 2 T 6 .1 l z ðtÞzðtÞ þ al .l F l kDl k

DTl Dl

Z

where Pl(t, s), t, s P 0, is the Cauchy matrix of linear system

T T T 6 q1 l z ðtÞzðtÞ þ ql g l ðzðt  sl ðtÞÞÞBl Bl gðzðt  sl ðtÞÞÞ 2 2 T T 6 q1 l z ðtÞzðtÞ þ ql kBl k F l z ðt  sl ðtÞÞzðt  sl ðtÞÞ:

ð24Þ

By the formula for the variation of parameters, one obtains from (23) that  Z t m h X v ðtÞ ¼ pl ðtÞ Pl ðt; 0Þv ð0Þ þ Pl ðt; sÞ ql kBl k2 F 2l v ðs  sl ðsÞÞ l¼1

and

þ .l

t – tk ;

It follows from Lemma 3 that

For any positive constants nl and ql ; l 2 M, it is obtained from Lemma 1 and (H3) that

Z

ð22Þ

l¼1

ð17Þ

1

T

pl ðtÞzT ðtk Þzðtk Þ

l¼1

8 i nh m P 2 > > v_ ðtÞ ¼ pl ðtÞ 2cl þ n1 þ nl kAl k F 2l þ q1 þ .1 v ðtÞ > l l l > > l¼1 > > o > > 2 2 2 Rt 2 < þql kBl k F l v ðt  sl ðtÞÞ þ al .l F l kDl k 1 pl ðt  sÞv ðsÞds þ  ; > m P >  > > > > v ðt k Þ ¼ l¼1 pl ðtÞbl v ðtk Þ; t ¼ tk ; k 2 Nþ ; > > > : v ðsÞ ¼ k/ðsÞk2 ; s 2 ð1; 0:

1 m X

m X

For any  > 0, let v(t) be the unique solution of the following impulsive delay system:

 pl ðt  sÞg l ðzðsÞÞ ds

t

Vðt k Þ ¼

l¼1

l¼1

þ Dl

ð21Þ

1

On the other hand, when t ¼ t k ; k 2 Nþ , it is obtained from the second equation of (9) and (14) that

Proof. Consider the following Lyapunov function:

Differentiating V(t) along the t 2 ½t k1 ; t k Þ; k 2 Nþ obtains that

þ ql kBl k2 F 2l Vðt  sl ðtÞÞ þ al .l F 2l kDl k2 Z t  pl ðt  sÞVðsÞds :

pðt  sÞg Tl ðzðsÞÞg l ðzðsÞÞds

1

Pl ðt; sÞ 6 e½2cl þnl

t

pðt  sÞzT ðsÞzðsÞds:

ð20Þ

1

f

1

¼ bl e½2cl þnl f

Substituting (18)–(20) into (17) produces the following inequality:

þnl kAl k2 F 2l þq1 þ.1 ðtsÞ l l

¼ bl edl ðtsÞ ;

tsf

blT a

þnl kAl k2 F 2l þq1 þ.1 ðtsÞ l l

ts

e T a ln bl

t P s P 0; l 2 M;

where dl ¼  2cl þ n1 þ nl kAl k2 F 2l þ q1 þ .1 l l l þ

ð27Þ ln bl Þ Ta

> 0.

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X. Yang et al. / Chaos, Solitons & Fractals 44 (2011) 817–826

It follows from (25) and (27) that  Z t m h X dl ðtsÞ v ðtÞ 6 pl ðtÞ cl edl t þ bf ql kBl k2 F 2l v ðs  sl ðsÞÞ l e 0 l¼1  Z s 2 þ al .l F 2l kDl k pl ðs  uÞv ðuÞdu þ  ds : ð28Þ where cl ¼

1 f bl maxs60 k/ðsÞk2 ,

ql kBl k2 F 2l f f bl ðdl bl

þ

l 2 M. Now define

f

By simple computation, one can get that

 ql kBl k2 F 2l  a2l .l F 2l kDl k2 Þ

.l a2l kDl k2 F 2l

f

hl ðmÞ ¼ m  dl þ bl ql kBl k2 F 2l emsl þ bl al .l F 2l kDl k2 Z þ1  pl ðsÞems ds; l 2 M:

¼

f

 ql kBl k2 F 2l  a2l .l F 2l kDl k2 Þ

f f bl ðdl bl

dl dl bl  ql kBl k2 F 2l  a2l .l F 2l kDl k2 f

;

þ bl



l 2 M:

ð34Þ

0

Notice (H1), one derives from (29), (33) and (34) that

It can be derived from (H2) and (15) that Z

hl ð0Þ ¼ dl þ

f bl

q

2 2 l kBl k F l

þ

f bl

f

2 2 l F l kDl k

þ1

al .

pl ðsÞds

0

f

6 dl þ bl ql kBl k2 F 2l þ bl a2l .l F 2l kDl k2 < 0;

v ðt



Þ
0, .l > 0 as follows: 2 2 1 hl ðnl ; ql ; .l Þ ¼ 2cl þ n1 þ .1 l þ nl kAl k F l þ ql l

þ

ln bl f f þ bl ql kBl k2 F 2l þ bl a2l .l F 2l kDl k2 : Ta

In order to make (15) holds, we only need to find points  l Þ such that hl(wl) takes the minimum value  l; . wl ¼ ð nl ; q on (0, +1)  (0, +1)  (0, +1) and hl ðwl Þ < 0; l 2 M. By simple computation, one has q

2 l

@hl @nl

¼

wl ¼

þ

@hl @ ql



f bl kBl k2 F 2l ; @@h.l l

¼

@hl @ .l

1 kAl kF l

¼ .

2 l

þ

@hl @nl f bl

¼ n2 þ kAl k2 F 2l ; l

@hl @ ql

2 2 2 l F l kDl k .

a

¼

Let

¼ 0, one gets a unique stagnation point  b0:5f ; a Fl kD k . Furthermore, the Hesse matrix

b0:5f ; kBl kF l l

l l

l

of hl(nl, ql, .l) at wl is

0

2ðkAl kF lÞ3 B B B0 Hhl ðwl Þ ¼ B B B @ 0

0 2 0



kBl kF l b0:5f l

3

1

0

C C C 0 C > 0; C  3 C A l kDl k 2 albF0:5f l

Therefore, wl is the minimum value point of hl(nl, ql, .l) on (0, +1)  (0, +1)  (0, +1). Let hl(wl) < 0, we derive (37). This completes the proof. h Remark 2. Based on impulsive control technique, novel synchronization criterion is developed for switched neural networks model with time-varying delays and unbounded distributed delays. To the best of our knowledge, no published result consider synchronization of switched system with or without delay via impulsive control. In our model, the switching law can be arbitrary, the time-delay can be constant or time-varying, specially, the time-delay may be discontinuous or even indifferentiable. However, in order to guarantee exponential stability, switches are required not to happen too frequently in [31,32], timedelays in [6–8,10,12] are constant delays or time-varying delays which are required continuously differentiable and their derivative is simultaneously demanded to be finite or not larger than 1. Therefore, results of this paper are new. Remark 3. Impulsive control can reduce control cost and amount of the transmitted information when synchronizing drive-response systems. In this paper, we modify the concept of average impulsive interval [26] with elasticity number and extend it to delayed systems, which increases maximum of impulsive intervals and further reduce control cost and amount of the transmitted information. The obtained synchronization criterion is less conservative than those in [12–25], which are derived by using the maximum of impulsive intervals. Remark 4. In this paper, the ‘‘average impulsive interval’’ Ta with ‘‘elasticity number’’ f of impulsive sequence is utilized to derive synchronization criterion. Since max{tk  tk1} P Ta, k 2 Nþ , our synchronization criterion increases the impulse distances than those obtained by

using maximum of impulsive intervals. For impulsive control scheme, large impulse distance can lead to the reduction of the control cost and this is the most important characteristic of impulsive synchronization technique. Therefore, results of this paper are less conservative than those in which maximum of impulsive intervals are used to derive the synchronization criteria (see [12,13,18–25]). It should be pointed out that determining the ‘‘elasticity number’’ accurately is not an easy work since the various uncertainties of impulsive sequence in practice. It is obvious that the smaller the f is, the easier the synchronization criterion (37) will be satisfied. For some simple impulsive sequence, the ‘‘elasticity number’’ can be estimated. For example, the ‘‘elasticity number’’ of impulsive sequences with equal impulsive intervals is 1. In [26], Lu et al. constructed a specific impulsive sequence as

1 ¼ f; 2; . . . ; ðN0  1ÞðÞ; N0 T a ; N0 T a þ ; N0 T a þ 2; . . . ; N0 T a þ ðN0  1ÞðÞ; 2N0 T a ; . . .g;

ð38Þ

where  and Ta are positive numbers satisfying  < Ta, and N0 is a positive integer. By analysis we can obtain that the ‘‘elasticity number’’ and average impulsive interval of h i  and Ta, the impulsive sequence are f ¼ N 0  ðN0T1Þ a respectively, where [] denote the integer function. Specially, when  = Ta, the constructed impulsive sequence is an impulsive sequence with equal impulsive intervals  and f = 1, when 0 < ðN0T1Þ 6 1, then f = N0  1. Moreover, a

the maximum of impulsive supk2Nþ ft k  tk1 g ¼ N 0 ðT a  Þ þ .

intervals

is

4. Examples and simulations In this section, we provide one example with simulations to show the effectiveness and feasibility of our theoretical results obtained above. The less conservativeness of our synchronization criterion than those based on the maximum of impulsive intervals is further discussed in details. Consider a switched neural network with two switched subsystems as follows:

_ xðtÞ ¼ C r xðtÞ þ Ar fr ðxðtÞÞ þ Br fr ðxðt  sr ðtÞÞÞ Z t pr ðt  sÞfr ðxðsÞ ds þ Ir ðtÞ; þ Dr

ð39Þ

1

where xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞÞT ; r : ½0; þ1Þ ! M ¼ f1; 2g; p1 ðmÞ ¼ e0:5m ; p2 ðmÞ ¼ e0:2m ; f1 ðxÞ ¼ f2 ðxÞ ¼ ðtanhðx1 Þ; tanhðx2 ÞÞT ; s1 ðtÞ ¼ 1; s2 ðtÞ ¼ 1:5  j sin tj,       1 0 3 0:3 1:4 1 C1 ¼ ; A1 ¼ ; B1 ¼ ; 0 1:2 6 5 0:4 8     1:2 1 1 ; I1 ðtÞ ¼ ; D1 ¼ 2:8 1 1:2

     0:7 0 2 0:3 1:4 1 ; A2 ¼ ; B2 ¼ ; 0 1 5 4:5 0:3 6     1:2 1 2:3 ; I2 ðtÞ ¼ : D2 ¼ 2:8 1:2 0:4 C2 ¼



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X. Yang et al. / Chaos, Solitons & Fractals 44 (2011) 817–826 15

10

10

5

5

0

0

−5

−5

−10

−10 −3

−2

−1

0

1

2

3

4

−15 −2

−1

0

1

2

3

4

5

Fig. 1. Trajectories of subsystems 1 (left) and 2 (right).

Obviously, the assumptions (H1)–(H3) are satisfied with s1 ¼ 1; s2 ¼ 1:5; a1 ¼ 2; a2 ¼ 5; ~f i1 ¼ ~f i2 ¼ 1; i ¼ 1; 2. Hence, F1 = F2 = 1. Fig. 1 describes trajectories of the two subsystems with the same initial valve x(t) = (0.7, 0.8)T, t 2 [5, 0], x(t) = 0 for t 2 (1, 5). Consider the following hybrid impulsive and switched response system:

_ yðtÞ ¼ C r yðtÞ þ Ar fr ðyðtÞÞ þ Br fr ðyðt  sr ðtÞÞÞ Z t pr ðt  sÞfr ðyðsÞÞds þ Ir ðtÞ þ U r ; þ Dr

c1 þ kA1 kF 1 þ

where impulsive Ur controllers are defined as those in (4). Now we construct impulsive controllers. Taking Ek1 ¼ 0:4I2 and Ek2 ¼ 0:65I2 ; k 2 Nþ , it is obtained that b1 = 0.36, b2 = 0.1225. Choose N0 = 2,  = 0.001, Ta = 0.014 in (38), then, by Remark 4, f = 1 for the constructed impulsive sequence. By operating on computer with MATLAB, we get

F1 0:5f b1

ðkB1 k þ a1 kD1 kÞ þ

ln b1 2T a

¼ 4:8245 < 0;

F2 0:5f

b2

ð41Þ

¼ 2:8132 < 0:

ðkB2 k þ a2 kD2 kÞ þ

F1 0:5f b1

ðkB1 k þ a1 kD1 kÞ þ

ln b2 2T a ð42Þ

By virtue of Theorem 2, the switched systems (39) and (40) with arbitrary switching law can realize global exponential synchronization under the impulsive controllers (4). Denote switching sequence fT k ; k 2 Nþ g, where 0 = T0 < T1 < T2 <    < Tk1 < Tk <   , and limk?+1Tk = +1. When t 2 ½T 2k2 ; T 2k1 Þ; k 2 Nþ , the subsystem 1 is active, and the subsystem 2 act for t 2 ½T 2k1 ; T 2k Þ; k 2 Nþ . In the simulations, the switching intervals are taken as ½T 2k2 ; T 2k1 Þ ¼ d1 ; ½T 2k1 ; T 2k Þ ¼ d2 ; k 2 Nþ , where d1 and d2 are two positive constants. The initial condition of system (40) is y(t) = (1.2, 0.5)T, t 2 [5, 0], y(t) = 0 for t 2 (1, 5). Figs. 2–4 show the time responses of synchronization

ln b1 2  0:027

¼ 12:7436 > 0;

ð43Þ

and

c2 þ kA2 kF 2 þ

F2 0:5f

b2

ðkB2 k þ a2 kD2 kÞ þ

¼ 33:2918 > 0:

and

c2 þ kA2 kF 2 þ

Remark 5. In this paper, average impulsive interval instead of the maximum of impulsive intervals is utilized to get less conservative results. In fact, according to Remark 4, the maximum of the impulsive intervals for the constructed impulsive sequence is 0.027. If Ta in (41) and (42) is replaced by 0.027, we get

ð40Þ

1

c1 þ kA1 kF 1 þ

errors between (39) and (40) with arbitrary constants d1 and d2.

ln b2 2  0:027 ð44Þ

Hence, synchronization criteria obtained by using maximum of impulsive intervals fails to judge whether the switched neural networks (39) and (40) can be synchronized under the impulsive controllers. Therefore, results of this paper are less conservative.

5. Numerical conclusions In this paper, the global exponential synchronization of switched neural networks with time-varying delay and unbounded distributed delay has been studied by using impulsive control. Demanding on the time-varying is least and the switching law is arbitrary. By utilizing the concept of average impulsive interval with elasticity number, novel synchronization criterion is derived. Our synchronization criterion can increases the impulse distances, which can drastically reduce control cost and the amount of the transmitted information in practical applications. Numerical simulations confirm the effectiveness and feasibility of the theoretical results.

825

X. Yang et al. / Chaos, Solitons & Fractals 44 (2011) 817–826

unavoidable. How to synchronize switched system with mixed delays under such kind of uncertain perturbations via impulsive control is our future research topic.

Of course the considered model is idealized, since no external perturbation is included. In practice, uncertain perturbations, especially those do not satisfy Brownian motion’s perturbations [41], are

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5

0

2

4

6

8

10

12

14

16

18

20

−0.5

0

2

4

6

8

10

12

14

16

18

20

Fig. 2. Time responses synchronization errors e1(t) (left) and e2(t) (right) between (39) and (40) with d1 = 0.5 and d2 = 1.5.

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5

0

2

4

6

8

10

12

14

16

18

20

−0.5

0

2

4

6

8

10

12

14

16

18

20

18

20

Fig. 3. Time responses synchronization errors e1(t) (left) and e2(t) (right) between (39) and (40) with d1 = d2 = 1.

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 0

2

4

6

8

10

12

14

16

18

20

−0.5

0

2

4

6

8

10

12

14

16

Fig. 4. Time responses synchronization errors e1(t) (left) and e2(t) (right) between (39) and (40) with d1 = d2 = 0.5.

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X. Yang et al. / Chaos, Solitons & Fractals 44 (2011) 817–826

Acknowledgments This work was jointly supported by the Scientific Research Fund of Yunnan Province under Grant No. 2010ZC150, the National Natural Science Foundation of China under Grant No. 11101053 and 10801056, the Key Project of Chinese Ministry of Education (211118), the Foundation of Chinese Society for Electrical Engineering (2008), the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities (11FEFM05), the Excellent Youth Foundation of Educational Committee of Hunan Provincial (10B002). References [1] Li C, Liao X, Wong K. Chaotic lag synchronization of coupled timedelayed systems and its applications in secure communication. Physica D 2004;194(3–4):187–202. [2] Xie Q, Chen G, Bollt E. Hybrid chaos synchronization and its application in information processing. Math Comput Model 2002;35(1–2):145–63. [3] Li X, Bohner M. Exponential synchronization of chaotic neural networks with mixed delays and impulsive effects via output coupling with delay feedback. Math Comput Model 2010;52:643–53. [4] Song Q. Design of controller on synchronization of chaotic neural networks with mixed time-varying delays. Neurocomputing 2009;72:3288–95. [5] Li T, Fei SM, Zhang KJ. Synchronization control of recurrent neural networks with distributed delays. Physica A 2008;387(4):982–96. [6] Li T, Fei SM, Guo YQ. Synchronization control of chaotic neural networks with time-varying and distributed delays. Nonlinear Anal 2009;71:2372–84. [7] Sheng L, Yang H. Exponential synchronization of a class of neural networks with mixed time-varying delays and impulsive effects. Neurocomputing 2008;71:3666–74. [8] Cao J, Wang Z, Sun Y. Synchronization in an array linearly stochastically coupled neural networks with time delays. Physica A 2007;385:718–28. [9] Yang X, Cao J, Long Y, Rui W. Adaptive lag synchronization for competitive neural networks with mixed delays and uncertain hybrid perturbations. IEEE Trans Neural Networks 2010;21(10):1656–67. [10] Huang T, Li C, Liu X. Synchronization of chaotic systems with delay using intermittent linear state feedback. Chaos 2008;18:033122. [11] Yang X, Cao J. Stochastic synchronization of coupled neural networks with intermittent control. Phys Lett A 2009;373:3259–72. [12] Li P, Cao J, Wang Z. Robust impulsive synchronization of coupled delayed neural networks with uncertainties. Physica A 2007;373:261–72. [13] Hu C, Jiang H, Teng Z. Impulsive control and synchronization for delayed neural networks with reaction-diffusion terms. IEEE Trans Neural Networks 2010;21(1):67–81. [14] Wu Q, Zhou J, Xiang L, Liu Z. Impulsive control and synchronization of chaotic HindmarshCRose models for neuronal activity. Chaos Solitons Fractals 2009;41:2706–15. [15] Lu JG, Chen G. Global asymptotical synchronization of chaotic neural networks by output feedback impulsive control: an LMI approach. Chaos Solitons Fractals 2009;41:2293–300. [16] Chen D, Sun J, Huang C. Impulsive control and synchronization of general chaotic system. Chaos Solitons Fractals 2006;28:213–8. [17] Zhang X, Liao X, Li C. Impulsive control, complete and lag synchronization of unified chaotic system with continuous periodic switch. Chaos Solitons Fractals 2005;26:845–54.

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