Exponential stability of artificial neural networks with distributed delays and large impulses

Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888 www.elsevier.com/locate/nonrwa

Exponential stability of artificial neural networks with distributed delays and large impulses Sannay Mohamada,∗ , K. Gopalsamyb , Haydar Akçac a Department of Mathematics, Faculty of Science, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam b School of Informatics and Engineering, Flinders University, G. P. O. Box 2100, Adelaide SA5001, Australia c Department of Mathematical Sciences, Faculty of Sciences, United Arab Emirates University, Al Ain, United Arab Emirates

Received 6 September 2006; accepted 5 January 2007

Abstract This paper illustrates that there is a globally exponentially stable unique equilibrium state in an artificial neural network that is subject to delays distributed over unbounded intervals, and also to large impulses that are not too frequent. The activation functions, which may be unbounded, nondifferentiable and/or nonmonotonic, are assumed to be globally Lipschitz continuous. The stability analysis exploits the method of Lyapunov functions and the technique of Halanay inequalities to derive a family of easily verifiable sufficient conditions for convergence to the unique equilibrium state. The sufficiency conditions, in the norm either  · p where p  1 or  · ∞ , include those that govern the network parameters and the impulse magnitude and frequency. 䉷 2007 Elsevier Ltd. All rights reserved. MSC: 34A37; 92B20 Keywords: Neural networks; Distributed delays; Impulsive state displacements; Lyapunov functions; Halanay inequalities; Exponential stability

1. Introduction An artificial neural network is a computational paradigm of a biological neural network in terms of processing signals or information. Its architecture consists of arrays of simple and elementary processing units where each unit processes several input signals and produces a single output signal which is then received by other units connected to it possibly including itself. Although the initial intent of artificial neural networks was to explore and reproduce human information processing tasks such as speech, vision, and knowledge processing, they have demonstrated to date their superior capability in tackling a variety of complex problems that are related to systems control, data compression, system identification, optimizations, classifications, pattern recognition, and function approximations. While an artificial neural network has been known insofar for its transient processing behaviour, its circuit design has never been disentangled from destabilizing factors such as delays and impulses. The finite switching speed of amplifiers within the units’ individual circuits is one of the possible factors that can cause delays in the transmission of ∗ Corresponding author. Tel.: +673 2 463001; fax: +673 2 461502.

E-mail addresses: [email protected] (S. Mohamad), gopal@infoeng.flinders.edu.au (K. Gopalsamy), [email protected] (H. Akça). 1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.01.011

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

873

signals. The abrupt changes in the voltages produced by faulty circuit elements are exemplary of impulse phenomena that can affect the transient behaviour of the network. In this article, we demonstrate the exponential stability of a unique equilibrium state in an artificial neural network consisting of m processing units whose processing architectures contain delays assumed to be distributed over unbounded intervals, and neural states that are subject to impulsive state displacements at fixed instants of time. Although the convergence dynamics of impulsive neural networks with delays have been considered recently [2,4,6,10,12,13,16–21], the sufficiency conditions obtained in this article to resist the destabilizing effects of unbounded distributed delays and impulses of significant magnitude do not appear to have been found before. 2. Model description The network is described by  dxi (t) = −ai xi (t) + bij fj dt m



∞

 Kij (s)xj (t − s) ds + ci ,

t > 0, t  = tk ,

(2.1a)

0

j =1

subject to xi (tk ) = xi (tk+ ) − xi (tk− ) = Ik (xi (tk− )), k ∈ N xi (s) = i (s), s ∈ (−∞, 0], i ∈ I = {1, 2, . . . , m},

and

(2.1b) (2.1c)

in which the sequence of times {tk }k∈N satisfies 0 = t0 < t1 < t2 < · · · < tk → ∞

and

tk = tk − tk−1 

for k ∈ N,

where the value  > 0 denotes the minimum time of interval between successive impulses. A sufficiently large value of  ensures that impulses do not occur too frequently, but  → ∞ means that the network (2.1) becomes impulse free. The vector solution x(t) = (x1 , x2 , . . . , xm )T ∈ Rm of (2.1) has components xi (t) piecewise continuous on (0, ) for some  > 0, such that the limits xi (tk+ ) = limt→t + xi (t) and xi (tk− ) = limt→t − xi (t) for i ∈ I, k ∈ N exist and k k x(t) is differentiable on the open intervals (tk−1 , tk ) ⊂ (0, ). We assume in (2.1) that the functions Ik : R → R that characterize the impulsive jumps are continuous; the initial function i : (−∞, 0] → R is piecewise continuous and bounded in the sense of either  m  1/p   p sup |i (s)| < ∞ where p 1, or p = −∞


∞ = max i∈I

i=1

sup

−∞
 |i (s)| < ∞;

and the self-regulating parameter ai has a positive value, the synaptic connection weights bij are real constants, and the external biases ci denote real numbers. We assume further in (2.1) that the delay kernels Kij : [0, ∞) → R are piecewise continuous and satisfy  ∞ |Kij (s)|K(s) for all i, j ∈ I, s ∈ [0, ∞) and K(s)e0 s ds < ∞, (2.2) 0

in which K(·) corresponds to some nonnegative function defined on [0, ∞) and the constant 0 denotes some positive number (see Wang et al. [17] for a recent application of these delay kernels); and also, the activation function fj : R → R is globally Lipschitz continuous with a Lipschitz constant Lj > 0—namely |fj (u) − fj (v)| Lj |u − v| for all j ∈ I, u, v ∈ R.

(2.3)

3. Existence and uniqueness theorems In this section, a family of sufficient conditions that govern the network parameters and the activation functions is established for the existence of a unique equilibrium state of the network (2.1).

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S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

∗ )T ∈ Rm , We denote an equilibrium state of the impulsive network (2.1) by the constant vector x∗ = (x1∗ , x2∗ , . . . , xm ∗ where the components xi are governed by the algebraic system

0 = −ai xi∗ +

m 



∞

bij fj 0

j =1

 Kij (s)xj∗ ds + ci ,

i ∈ I.

(3.1)

Here, it is assumed that the impulse functions Ik (·) satisfy Ik (xi∗ ) = 0 for all i ∈ I, k ∈ N. Theorem 3.1. Let ai > 0, bij , ci ∈ R, the delay kernels Kij (·) satisfy (2.2), and the activation functions fj (·) satisfy (2.3). Suppose the condition

min{ai } > max i∈I

i∈I

⎧ m ⎨ 

j

⎩

j =1

i



∞

|bj i |Li

K(s) ds

0

⎫ ⎬ (3.2)

⎭

is satisfied, where the i denote positive numbers. Then there exists a unique equilibrium state x∗ of the impulsive network (2.1). Proof. Let yi∗ = i ai xi∗ in (3.1) so that yi∗ = i

m 



 Kij (s)(yj∗ /j aj ) ds + i ci ,

∞

bij fj 0

j =1

i ∈ I.

(3.3)

We consider a mapping G(u) = (G1 (u), G2 (u), . . . , Gm (u))T ∈ Rm , where u = (u1 , . . . , um )T ∈ Rm and Gi (u) = i

m 



∞

bij fj

 Kij (s)(uj /j aj ) ds + i ci ,

i ∈ I.

(3.4)

0

j =1

On applying the hypotheses, G(u) − G(v)1 =

m 

|Gi (u) − Gi (v)|

i=1     ∞  ∞   m     m  i bij fj Kij (s)(uj /j aj ) ds − fj Kij (s)(vj /j aj ) ds  =  0 0  i=1  j =1     m m ∞    ∞   i |bij |Lj  Kij (s)(uj /j aj ) ds − Kij (s)(vj /j aj ) ds  i=1 m 

j =1 m 

0

  ∞  uj − vj   ds  i |bij |Lj |Kij (s)|  j a j  0 i=1 j =1 ⎛ ⎞  ∞ m m  1 ⎝ j  |bj i |Li K(s) ds ⎠ |ui − vi | ai i 0 i=1 m 

=

i=1

j =1

|ui − vi |,



0

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

where the number =

875

∞ j =1 (j /i )|bj i |Li 0 K(s) ds}

m

maxi∈I {

mini∈I {ai }

satisfies 0 <  < 1 by virtue of the condition (3.2). Thus, G(u) − G(v)1 < u − v1 for any two vectors u, v ∈ Rm implying that the mapping G : Rm → Rm is a global contraction on Rm endowed with the norm  · 1 . Hence, there is a unique fixed point y∗ ∈ Rm that satisfies G(y∗ ) = y∗ (i.e., Gi (y∗ ) = yi∗ for i ∈ I) which defines the unique solution of the algebraic system (3.3). The existence of a unique equilibrium state x∗ of the impulsive network (2.1), therefore, follows from xi∗ = yi∗ /i ai , i ∈ I. The proof is now complete.  In the following theorem, we use a result due to Forti and Tesi [5] in order to prove there is a unique solution of the p 1/p , where p > 1 and u = (u , u , . . . , u )T ∈ Rm . system (3.1) under the general Euclidean norm up = ( m 1 2 m i=1 |ui | ) m m m Recall that a map h : R → R is a homeomorphism (R onto itself) if h ∈ C 0 is injective (i.e., one-to-one and onto), and the inverse map h−1 ∈ C 0 . The result is that the map h ∈ C 0 is a homeomorphism on Rm if it is injective on Rm and satisfies limu→∞ h(u) → ∞. In addition, we also invoke a Young inequality of two real numbers—namely that if 1 , 2 denote nonnegative real numbers, then 1 2 

r1 s + 2 r s

for any real numbers r, s satisfying r > 1 and 1/r + 1/s = 1. One may refer to Cao [3] for the application of this inequality in the stability investigation of a delayed cellular neural network. Theorem 3.2. Let p > 1, ai > 0, bij , ci ∈ R, the delay kernels Kij (·) satisfy (2.2), and the activation functions fj (·) satisfy (2.3). Suppose the condition ⎧ ⎫  ∞ m ⎨ ⎬  p−1

p/p−1 min ai − |bij | ij p/p−1 Lj ij K(s) ds ⎭ p i∈I ⎩ 0 j =1 ⎫ ⎧  ∞ m ⎬ ⎨1  j (1− )p |bj i |(1− j i )p Li j i K(s) ds (3.5) > max ⎭ i i∈I ⎩ p 0 j =1

is satisfied, where the i denote positive numbers, and ij , ij denote real numbers. Then there exists a unique equilibrium state x∗ of the impulsive network (2.1). Proof. Consider a map H(u) = (H1 (u), H2 (u), . . . , Hm (u))T ∈ C 0 (Rm , Rm ) endowed with the norm  · p for p > 1, where  ∞  m  bij fj Kij (s)uj ds + ci , i ∈ I. (3.6) Hi (u) = −ai ui + 0

j =1

The map H ∈ C 0 is a homeomorphism on Rm if it is injective on Rm and satisfies H(u)p → ∞ as up → ∞. We demonstrate the injective part, namely H(u) = H(v) implies u = v for any u, v ∈ Rm , as follows. We have −ai ui +

m 



∞

bij fj

 Kij (s)uj ds + ci = −ai vi +

0

j =1

m 

 bij fj

j =1

and consequently ai |ui − vi | 

m  j =1



∞

|bij |Lj 0

 K(s)|uj − vj | ds ,

∞

i∈I

0

 Kij (s)vj ds + ci

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S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

under the hypotheses. Further estimation through the use of the Young inequality leads to m 

i ai |ui − vi |p 

i=1

=

m  i=1 m  i=1 m 

i i

m 



∞

|bij |Lj

j =1 m  ∞  0 j =1  m ∞ 

 K(s)|uj − vj ||ui − vi |p−1 ds

0

K(s)[|bij |

ij

Lj ij |ui

− vi |

p−1

(1− ) ][|bij |(1− ij ) Lj ij |uj

 − vj |] ds



p−1

p/p−1 |ui − vi |p |bij | ij p/p−1 Lj ij p 0 i=1 j =1   1 p (1− ij )p (1− ij )p Lj |uj − vj | ds + |bij | p ⎛ ⎞  ∞ m m   p − 1

p/p−1 = i ⎝ |bij | ij p/p−1 Lj ij K(s) ds ⎠ |ui − vi |p p 0 i=1 j =1 ⎛ ⎞  ∞ m m   j 1 (1−

)p + i ⎝ |bj i |(1− j i )p Li j i K(s) ds ⎠ |ui − vi |p p i 0 

i

K(s)

j =1

i=1

which gives (1 − 2 )

m 

i |ui − vi |p 0,

(3.7)

i=1

where the positive constants ⎫ ⎧  ∞ m ⎬ ⎨  p−1

p/p−1 1 = min ai − |bij | ij p/p−1 Lj ij K(s) ds ⎭ p i∈I ⎩ 0 j =1 ⎧ ⎫  ∞ m ⎨1  ⎬ j (1− )p 2 = max |bj i |(1− j i )p Li j i K(s) ds ⎭ i i∈I ⎩ p 0

and

j =1

satisfy 1 > 2 by virtue of the condition (3.5). It follows, therefore, from (3.7) that ui = vi for i ∈ I (i.e., u = v). Hence, the map H ∈ C 0 is injective on Rm . ˆ Next, we consider the map H(u) = H(u) − H(0)—i.e. Hˆ i (u) = Hi (u) − Hi (0) = −ai ui +

m  j =1





∞

Kij (s)uj ds

bij fj 0

ˆ for ui ∈ R, i ∈ I. It is enough to show H(u) p → ∞ as up → ∞ in order to demonstrate the property H(u)p → ∞ as up → ∞. We have m 

i |ui |p−1 sgn(ui )Hˆ i (u) =

i=1

m  i=1



m  i=1

⎡ i |ui |p−1 sgn(ui ) ⎣−ai ui + ⎡ ⎣−ai i |ui |p + i

m 

 bij fj 0

j =1 m  j =1

∞



∞

|bij |Lj 0

⎤



Kij (s)uj ds ⎦

⎤  K(s)|uj ||ui |p−1 ds ⎦

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

877

⎛

⎞  ∞ m  p−1 p/p−1

 −ai i |ui |p + i ⎝ |bij | ij p/p−1 Lj ij K(s) ds ⎠ |ui |p p 0 i=1 i=1 j =1 ⎛ ⎞  ∞ m m   j 1 (1−

)p j i )p (1− ji + i ⎝ |bj i | Li K(s) ds ⎠ |ui |p p i 0 m 

m 

j =1

i=1

 − (1 − 2 )

m 

i |ui |p

i=1

that yields (1 − 2 )

m 

i |ui |p  −

i=1

m 

i |ui |p−1 sgn(ui )Hˆ i (u) 

i=1

m 

i |ui |p−1 |Hˆ i (u)|.

i=1

On applying a Hölder inequality (1 − 2 ) min {i } i∈I

m 

 |ui |p  max {i }

i=1

i∈I

m 

1/q  |ui |(p−1)q

i=1

m 

1/p |Hˆ i (u)|p

,

i=1

where 1/p + 1/q = 1. It then follows that  m 

1/p |ui |

p

i=1

1/p  m  maxi∈I {i } p  |Hˆ i (u)| (1 − 2 ) mini∈I {i } i=1

ˆ and from which we assert H(u) p → ∞ as up → ∞. We conclude that the map H ∈ C 0 is a homeomorphism on Rm , and this guarantees the existence of a unique solution x∗ ∈ Rm of the algebraic system (3.1) which defines the unique equilibrium state of the impulsive network (2.1). The proof is now complete.  The following theorem gives sufficiency conditions for the existence of a unique equilibrium state x∗ of the impulsive network (2.1) under the Euclidean norm  · ∞ . Theorem 3.3. Let ai > 0, bij , ci ∈ R, the delay kernels Kij (·) satisfy (2.2), and the activation functions fj (·) satisfy (2.3). Suppose the condition ⎧ ⎫  ∞ m ⎨ ⎬  min ai − |bij |Lj K(s) ds > 0 (3.8) ⎭ i∈I ⎩ 0 j =1

is satisfied. Then the impulsive network (2.1) has a unique equilibrium state x∗ . Proof. We note from the condition (3.8) that ai >

m 



∞

|bij |Lj

K(s) ds

for each i ∈ I

0

j =1

and hence, there is a number ⎧ ⎫  ∞ m ⎨1  ⎬ = max |bij |Lj K(s) ds ⎭ i∈I ⎩ ai 0 j =1

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S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

which satisfies 0 < < 1. Now, let us consider a mapping g(u) = (g1 (u), g2 (u), . . . , gm (u))T ∈ Rm , where u = (u1 , u2 , . . . , um )T ∈ Rm and   ∞ m ci 1  Kij (s)uj ds + , i ∈ I. (3.9) gi (u) = bij fj a ai i 0 j =1

On applying the hypotheses

⎧ ⎫   ∞  ∞  ⎬ m ⎨ 1   g(u) − g(v)∞ = max  bij fj Kij (s)uj ds − fj Kij (s)vj ds  i∈I ⎩ ai 0 0 ⎭ j =1 ⎧ ⎫  ∞ ⎬ m ⎨1      max |bij |Lj  Kij (s)(uj − vj ) ds  ⎭ i∈I ⎩ ai 0 j =1 ⎫ ⎧  ∞  m ⎬ ⎨1   max |bij |Lj K(s) ds |uj − vj | ⎭ i∈I ⎩ ai 0 j =1 ⎧ ⎫  ∞ m ⎨1  ⎬  max |bij |Lj K(s) ds · max{|ui − vi |} ⎭ i∈I i∈I ⎩ ai 0 j =1

= u − v∞ .

(3.10)

Since u and v correspond to any vectors in R and the constant satisfies 0 < < 1, we therefore assert from (3.10) that the mapping g : Rm → Rm is a global contraction on Rm endowed with the norm  · ∞ . Thus, there is a unique fixed point x∗ ∈ Rm satisfying g(x∗ ) = x∗ (i.e., gi (x∗ ) = xi∗ for i ∈ I). This point defines the unique equilibrium state of the impulsive network (2.1). The proof is now complete.  m

4. Exponential stability theorems The existence and stability of a unique equilibrium state is usually a requirement in the design of artificial neural networks for various applications, particularly when there are destabilizing agents such as delays and impulses. However, even if the unique stable state exists, these agents may affect the convergence speed of the network, which in turn can downgrade the performance of the network in applications that demand fast computation in real-time mode. Thus, exponential stability is usually desirable for an impulsive network, and sufficient conditions for the global exponential stability of the unique equilibrium state x∗ of the impulsive network (2.1) are obtained in this section. We consider the impulsive network (2.1) in which the impulsive state displacements characterized by Ik : R → R at fixed instants of time t = tk , k ∈ N are defined by xi (tk+ ) − xi (tk− ) = Ik (xi (tk− )) = −dik (xi (tk− ) − xi∗ ),

i ∈ I, k ∈ N,

(4.1)

where dik denote real numbers. This type of impulses has been considered previously in stability investigations of impulsive delayed neural networks [1,2,9–11,13,17,19,21]. For convenience in our analysis, we let yi (t) = xi (t) − xi∗ ,

gj (yj (t)) = fj (yj (t) + xj∗ ) − fj (xj∗ )

(4.2)

so that (2.1) can be rewritten as  dyi (t) bij gj = −ai yi (t) + dt m

j =1



∞

 Kij (s)yj (t − s) ds ,

t > 0, t  = tk

(4.3a)

0

subject to yi (tk+ ) = ik yi (tk− ), k ∈ N and yi (s) = i (s) = i (s) − xi∗ , s ∈ (−∞, 0],

(4.3b) (4.3c)

where ik = 1 − dik and the activation functions gj (·), inheriting the properties of fj (·), satisfy gj (0) = 0,

|gj (u)|Lj |u| for all u ∈ R.

(4.4)

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

879

Given the relationship between the two networks (2.1) and (4.3), it suffices to establish the global exponential stability of the trivial equilibrium state y∗ = 0 of the impulsive network (4.3). To establish the stability of an impulsive neural network, it has been customary to seek for conditions that govern the network parameters and the impulse magnitude so that both parts—i.e. the continuous-time and the discrete-time equations, of the impulsive network become convergent. Particularly, the requirement |yi (tk+ )| < |yi (tk− )| imposed on the discrete-time equation (4.3b) as a result of assuming 0 < | ik | < 1 (or equivalently, 0 < dik < 2) has appeared in the previous studies [1,2,9–11,13,17,19,21]. Although the results provide some insights on the exponential convergence dynamics of impulsive neural networks, their practical use for potential applications of the networks might be limited. In this section, we intend to avoid the severe restriction |yi (tk+ )| < |yi (tk− )| while at the same time achieving the exponential stability of the network (4.3). This turns out to be a novel intention as it leads to allowing the impulse magnitude to be as large as possible through an appropriate linkage between the convergence conditions of the continuous-time equation (4.3a) and the inter-impulse interval tk = tk − tk−1 . The similar linkage has also been observed by a number of researchers recently [4,7,12,14,18,21]. To achieve our results, we employ the method of Lyapunov functions combined with the technique of Halanay inequalities. Some useful results concerning these inequalities are given below. The interested reader may also refer to [8,15] for a number of results concerning continuous-time and discrete-time Halanay inequalities. Lemma 4.1. Let  and  denote positive constants, and v(t) be a nonnegative function defined for all t ∈ R which satisfies the differential inequality 

+

∞

D v(t)  − v(t) + 

K(s)v(t − s) ds

for t > t0 ,

(4.5)

0

in which the function K : [0, ∞) → [0, ∞) is piecewise continuous and number 0 . If  >

∞ 0

K(s)e0 s ds < ∞ for some positive

∞

K(s) ds,

(4.6)

0

then there is a number  satisfying 0 <  < 0 for which v(t)v(t0 )e−(t−t0 )

for all t > t0 ,

(4.7)

where v(t0 ) = sup−∞
∞

F (v) = v −  + 

K(s)evs ds

0

for v ∈ [0, 0 ].

Here we assume, without loss of generality, that F (0 ) > 0. We have F (0) < 0 by virtue of the condition (4.6), and ∞ also F (v) = 1 +  0 K(s)sevs ds > 0 which means that F (v) increases monotonically for v ∈ [0, 0 ]. Consequently, there is a number  ∈ (0, 0 ) that satisfies the equation 

∞

F () =  −  + 

K(s)es ds = 0.

(4.8)

0

Now, let us define w(t) = v(t)e(t−t0 )

for t ∈ R.

(4.9)

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S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

We note from (4.5) and (4.9) that w(t) is continuous for t > t0 and w(t) v(t0 ) for t t0 . By continuity, there is a t ∗ > t0 for which w(t)v(t0 ) for all t ∈ (t0 , t ∗ ). Suppose this is not true. Then, there is a t1 ∈ (t0 , t ∗ ) such that w(t)v(t0 ) for t0 < t < t1 ,

w(t1 ) = v(t0 )

D + w(t1 ) > 0.

and

This together with (4.5) and (4.9) give 0 < D + w(t1 ) = D + v(t1 )e(t1 −t0 ) + v(t1 )e(t1 −t0 )  ∞  − w(t1 ) +  K(s)es w(t1 − s) ds + w(t1 ) 0    ∞

( − )w(t1 ) + 

K(s)es ds

0

sup

−∞
w(r)

F ()v(t0 ), where F () is given by (4.8). Consequently, we have the contradiction 0 < D + w(t1 ) 0. Therefore, the claim w(t) v(t0 ) for all t ∈ (t0 , t ∗ ) is valid. Since the number t ∗ > t0 is arbitrary, we then have from (4.9) that v(t) v(t0 )e−(t−t0 ) for all t > t0 . The proof is complete.  Lemma 4.2. Let i , ij > 0 for i, j ∈ I={1, 2, . . . , m} and K : [0, ∞) → [0, ∞) be a piecewise continuous function ∞ such that 0 K(s)e0 s ds < ∞ for some number 0 > 0. Assume that vi (t) for i ∈ I correspond to nonnegative functions defined for t ∈ R and they satisfy +

D vi (t) − i vi (t) +

m  j =1

 ij

∞

K(s)vj (t − s) ds,

i ∈ I, t > t0

(4.10)

0

subject to vi (t)v i (t0 ) =

sup

−∞
vi (r) < ∞

for i ∈ I, t t0 .

If i >

m 

 ij

j =1

∞

K(s) ds

for i ∈ I,

(4.11)

0

then there is a number  satisfying 0 <  < 0 for which vi (t)v(t0 )∞ e−(t−t0 )

for all i ∈ I, t > t0 ,

(4.12)

where v(t0 )∞ = maxi∈I {v i (t0 )} < ∞. Proof. Let i be the positive solution of the equation i − i +

m 

 ij

j =1

∞

K(s)ei s ds = 0,

i ∈ I.

0

The existence of i ∈ (0, 0 ) follows by the similar arguments given previously in Lemma 4.1. By setting  = mini∈I {i }, we then have  ∈ (0, 0 ) satisfying  − i +

m  j =1

 ij

∞ 0

K(s)es ds 0

for all i ∈ I.

(4.13)

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

881

Now, let us define wi (t) = vi (t)e(t−t0 )

for i ∈ I, t ∈ R.

(4.14)

It follows from the continuity of wi (t) for t > t0 and also wi (t) v(t0 )∞ for all i ∈ I, t t0 , that there is a number t ∗ > t0 for which wi (t)v(t0 )∞ for all i ∈ I, t ∈ (t0 , t ∗ ). Suppose this is not the case in the sense that there is an index i = k ∈ I and a t1 ∈ (t0 , t ∗ ) such that wi (t)v(t0 )∞

for i ∈ I, i  = k, t ∈ (t0 , t ∗ ),

wk (t)v(t0 )∞

for t ∈ (t0 , t1 ), wk (t1 ) = v(t0 )∞

and

D + wk (t1 ) > 0.

This together with (4.10) and (4.14) give 0 < D + wk (t1 ) = D + vk (t1 )e(t1 −t0 ) + vk (t1 )e(t1 −t0 )  − k vk (t1 )e(t1 −t0 ) +

m 

 kj

m 

 kj

j =1

( − k )wk (t1 ) +

m 

∞

 ⎝ − k +

m 



 kj

kj

j =1

∞ 0

K(s)es wj (t1 − s) ds

0 ∞

K(s)e 0

j =1

⎛

K(s)vj (t1 − s)e(t1 −t0 ) ds + vk (t1 )e(t1 −t0 )

0

j =1

= ( − k )wk (t1 ) +

∞

s

 ds

 sup

−∞
wj (r)

⎞ K(s)es ds ⎠ v(t0 )∞ .

The application of (4.13) to the above leads to the contradiction 0 < D + wk (t1 ) 0. Thus, the claim wi (t) v(t0 )∞ for all i ∈ I, t ∈ (t0 , t ∗ ) is valid. Since the number t ∗ > t0 is arbitrary, we have from (4.14) that vi (t) v(t0 )∞ e−(t−t0 ) for all i ∈ I, t > t0 . The proof is now complete.  Theorem 4.1. Let ai > 0, bij , ci ∈ R, the delay kernels Kij (·) satisfy (2.2), and the activation functions fj (·) satisfy (2.3). Suppose the condition (3.2)—namely min{ai } > max i∈I

i∈I

⎧ m ⎨ 

j

⎩

j =1

i



∞

|bj i |Li 0

K(s) ds

⎫ ⎬ ⎭

is satisfied, and there exist positive constants  and  satisfying 0  <  < 0 for which k etk ,

k ∈ N,

(4.15)

where k = max{1, | k |} and | k | = maxi∈I {| ik |}. Then the equilibrium state y∗ = 0 of the impulsive network (4.3) is unique and globally exponentially stable in the sense of m 

|yi (t)|1 e−(−)t

for 0 < t < tk , k ∈ N,

i=1

where  1 and 1 = sup−∞
m

i=1 | i (s)|

< ∞.

(4.16)

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S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

Proof. It follows from Theorem 3.1 that the equilibrium state y∗ = 0 of the network (4.3) exists uniquely. On applying the hypotheses, D + |yi (t)| =

dyi (t) sgn(yi (t)) dt

= − ai yi (t) sgn(yi (t)) +

m 



 − ai |yi (t)| +

 Kij (s)yj (t − s) ds sgn(yi (t))

0

j =1



m 

∞

bij gj ∞

|bij |Lj

K(s)|yj (t − s)| ds

(4.17)

0

j =1

for i ∈ I, t > 0, t  = tk . Consider a Lyapunov function given by V (t) = V (y(t)) =

m 

i |yi (t)|

for t ∈ R,

(4.18)

i=1

and we denote V (t) =

 sup

−∞
V (s) =

sup

m 

−∞
 i |yi (s)|

for t 0.

i=1

Its derivative along the solution of (4.17) is estimated as follows: D + V (t) =

m 

i D + |yi (t)|

i=1



m 

−ai i |yi (t)| +

m 

i=1

= −

i

ai i |yi (t)| +

i=1

m  i=1



∞

 − 1 V (t) + 2



∞

|bij |Lj

K(s)|yj (t − s)| ds

0

j =1

i=1

m 

m 

⎛ ⎞  ∞ m  j ⎝ ⎠ i |bj i |Li K(s)|yi (t − s)| ds i 0 j =1

K(s)V (t − s) ds,

t > 0, t  = tk ,

(4.19)

0

in which the positive numbers 1 = min{ai }, i∈I

satisfy 1 > 2 (4.19),

∞ 0

2 = max i∈I

⎧ m ⎨ 

j

⎩

j =1

i

|bj i |Li

⎫ ⎬ ⎭

K(s) ds in accordance with the condition (3.2). On applying the hypotheses and Lemma 4.1 to

V (t)V (0)e−t

for 0 < t < t1 ,

where the number  ∈ (0, 0 ) satisfies the equation  ∞ K(s)es ds = 0. F () =  − 1 + 2 0

On applying (4.3b) to (4.18) at t = t1 , V (t1+ ) =

m  i=1

i |yi (t1+ )|

m  i=1

i | i1 ||yi (t1− )| 1 V (t1− ) V (0) 1 e−t1 .

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

883

By repeating the analysis to (4.19) for t ∈ (t1 , t2 ), V (t) V (t1+ )e−(t−t1 ) V (0) 1 e−t

for t1 < t < t2

and V (t2+ )V (0) 1 2 e−t2 which is then extended by (4.19) through Lemma 4.1 to give V (t)V (t2+ )e−(t−t2 ) V (0) 1 2 e−t

for t2 < t < t3 .

Inductively, one derives that + )V (0) 0 1 2 · · · k−1 e−tk−1 V (tk−1

V (t)V (0) 0 1 2 · · · k−1 e

−t

for k ∈ N,

for tk−1 < t < tk , k ∈ N,

where t0 = 0 and 0 = 1, which in turn can be estimated further into V (t) V¯ (0) 0 1 2 · · · k−1 e−t

for 0 < t < tk , k ∈ N.

On applying the condition (4.15)—namely k etk where 0  <  < 0 , V (t) V¯ (0)[e(t1 −t0 ) e(t2 −t1 ) · · · e(tk−1 −tk−2 ) ][e−(t−tk−1 ) e−(tk−1 −tk−2 ) · · · e−(t2 −t1 ) e−(t1 −t0 ) ]  V¯ (0)e−(−)(t1 −t0 ) e−(−)(t2 −t1 ) · · · e−(−)(tk−1 −tk−2 ) e−(−)(t−tk−1 ) which is equivalent to m 

|yi (t)|1 e−(−)t

for 0 < t < tk , k ∈ N,

i=1

where  = maxi∈I {i }/mini∈I {i } 1. Since  −  > 0, we must have y(t) converges exponentially towards the equilibrium state y∗ = 0 of the impulsive network (4.3) in the norm  · 1 as t → ∞ (i.e., tk → ∞ as k → ∞). The proof is now complete.  Theorem 4.2. Let p > 1 be a real number, ai > 0, bij , ci ∈ R, the delay kernels Kij (·) satisfy (2.2), and the activation functions fj (·) satisfy (2.3). Suppose the condition (3.5)—namely ⎫ ⎧  ∞ m ⎬ ⎨  p−1

p/p−1 min ai − |bij | ij p/p−1 Lj ij K(s) ds ⎭ p i∈I ⎩ 0 j =1 ⎧ ⎫  ∞ m ⎨1  ⎬ j (1− )p > max |bj i |(1− j i )p Li j i K(s) ds ⎭ i i∈I ⎩ p 0 j =1

is satisfied, and the impulse magnitude k , k ∈ N satisfies (4.15). Then the equilibrium state y∗ = 0 of the impulsive network (4.3) is unique and globally exponentially stable in the sense of  m 1/p  ! p |yi (t)|  p p e−(−)t for 0 < t < tk , k ∈ N, (4.20) i=1

m

where  1, 0  <  < 0 , and p = [sup−∞
i=1 | i (s)|

p )]1/p

< ∞.

Proof. It is clear from the hypotheses and Theorem 3.2 that the equilibrium state y∗ = 0 of the impulsive network (4.3) exists uniquely. To demonstrate its exponential stability via (4.20), we proceed by defining a Lyapunov function

884

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

of the form V (t) = V (y(t)) =

m 

i |yi (t)|p

for p > 1, t ∈ R.

(4.21)

i=1

We denote V (t) =

 sup

−∞
V (s) =

sup

−∞
m 

 i |yi (s)|

for t 0.

p

i=1

The derivative D + V (t) along the solution of (4.17) is estimated as follows: +

D V (t) =

m 

i p|yi (t)|p−1 D + |yi (t)|

i=1



m 

−ai i p|yi (t)|p +

i=1

 −p

m 

i p

i=1 m 

ai i |yi (t)| + p p

m  i=1

i=1

m 



j =1 m  ∞ 

i

j =1 0

K(s)|yi (t)|p−1 |yj (t − s)| ds

0

 K(s)

 1 (1− ij )p (1− ij )p p + |bij | Lj |yj (t − s)| ds p m  = −p ai i |yi (t)|p i=1

∞

|bij |Lj

p−1

p/p−1 |yi (t)|p |bij | ij p/p−1 Lj ij p

⎛

⎞  ∞ m  p − 1

p/p−1 ij +p i ⎝ |bij | ij p/p−1 Lj K(s) ds ⎠ |yi (t)|p p 0 i=1 j =1 ⎛ ⎞  ∞ m m   j 1 (1− )p +p i ⎝ |bj i |(1− j i )p Li j i ⎠ K(s)|yi (t − s)|p ds p i 0 i=1 j =1  ∞ K(s)V (t − s) ds, t > 0, t  = tk ,  − p1 V (t) + p2 m 

(4.22)

0

where the positive numbers ⎧ ⎫  ∞ m ⎨ ⎬  p−1

p/p−1 1 = min ai − |bij | ij p/p−1 Lj ij K(s) ds , ⎭ p i∈I ⎩ 0 j =1 ⎧ ⎫ m ⎨1  ⎬ j (1− )p 2 = max |bj i |(1− j i )p Li j i ⎭ i i∈I ⎩ p j =1

∞ satisfy 1 > 2 0 K(s) ds by virtue of the condition (3.5). The inequality (4.22) is analogous to the Halanay-type inequality (4.5) of Lemma 4.1. Thus, V (t)V (0)e−t

for 0 < t < t1 ,

where  = p with  ∈ (0, 0 ) satisfying the equation  ∞ F () =  − 1 + 2 K(s)es ds = 0. 0

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

885

On applying (4.3b) to V (t) at t = t1 , V (t1+ ) =

m 

i |yi (t1+ )|p | 1 |p

i=1

m 

i |yi (t1− )|p  1 V (t1− ) p

i=1

which then yields V (t1+ )V (0) 1 e−t1 . p

By following the similar inductive arguments as before, we derive that + )V (0) 0 1 2 · · · k−1 e−tk−1 V (tk−1 p p p

p p p V (t)V (0) 0 1 2

p

p · · · k−1 e−t

for k ∈ N,

for tk−1 < t < tk , k ∈ N,

where t0 = 0 and 0 = 1, which can be estimated further into V (t)V (0) 0 1 2 · · · k−1 e−t p p p

p

for 0 < t < tk , k ∈ N.

The application of (4.15) to the above yields V (t)V (0)[ep(t1 −t0 ) ep(t2 −t1 ) · · · ep(tk−1 −tk−2 ) ][e−p(t−tk−1 ) e−p(tk−1 −tk−2 ) · · · e−p(t2 −t1 ) e−p(t1 −t0 ) ] V (0)e−p(−)(t1 −t0 ) e−p(−)(t2 −t1 ) · · · e−p(−)(tk−1 −tk−2 ) e−p(−)(t−tk−1 ) which is equivalent to m 

|yi (t)|p (p )p e−p(−)t

for 0 < t < tk , k ∈ N,

i=1

where  = maxi∈I {i }/mini∈I {i } 1. Hence,  m 1/p  ! p |yi (t)|  p p e−(−)t for 0 < t < tk , k ∈ N. i=1

Since the numbers  and  satisfy 0  <  < 0 , the vector solution y(t) converges to the equilibrium state y∗ = 0 of the impulsive network (4.3) exponentially in the norm  · p as t → ∞ (i.e., tk → ∞ as k → ∞). This completes the proof.  Theorem 4.3. Let ai > 0, bij , ci ∈ R, the delay kernels Kij (·) satisfy (2.2), and the activation functions fj (·) satisfy (2.3). Suppose the condition (3.8)—namely ⎧ ⎫  ∞ m ⎨ ⎬  |bij |Lj K(s) ds > 0 min ai − ⎭ i∈I ⎩ 0 j =1

is satisfied, and the impulse magnitude k , k ∈ N satisfies (4.15). Then the equilibrium state y∗ = 0 of the impulsive network (4.3) is unique and globally exponentially stable in the sense of |yi (t)|∞ e−(−)t

for all i ∈ I, 0 < t < tk , k ∈ N,

(4.23)

where 0  <  < 0 and ∞ = maxi∈I {sup−∞
for all i ∈ I, 0 < t < t1 ,

886

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

where  ∈ (0, 0 ) satisfies the inequality  − ai +

m 



∞

|bij |Lj

K(s)es ds 0

for all i ∈ I.

0

j =1

On applying (4.3b) for t = t1 , |yi (t1+ )|| 1 ||yi (t1− )|∞ 1 e−t1

for all i ∈ I

which is then extended by (4.17) and Lemma 4.2 to the interval (t1 , t2 ) to give 

|yi (t)| max i∈I

sup

−∞
∞ 1 e−t

|yi (s)| e−(t−t1 )

for all i ∈ I, t1 < t < t2 .

By following the similar inductive process as before, one derives that + |yi (tk−1 )|∞ 0 1 2 · · · k−1 e−tk−1

|yi (t)|∞ 0 1 2 · · · k−1 e

−t

for all i ∈ I, k ∈ N,

for all i ∈ I, tk−1 < t < tk , k ∈ N,

which can be further estimated as |yi (t)|∞ 0 1 2 · · · k−1 e−t

for all i ∈ I, 0 < t < tk , k ∈ N.

The application of the condition (4.15) to the above eventually leads to |yi (t)|∞ e−(−)t

for all i ∈ I, 0 < t < tk , k ∈ N,

in which 0  <  < 0 . This asserts the exponential stability of the equilibrium state y∗ = 0 of the network (4.3) under the norm  · ∞ . The proof is complete.  5. Illustrative examples We provide several computer simulations in the attempt to illustrate the exponential convergence of a neural network consisting of unbounded activations given by  ∞   ∞ dy1 (t) K11 (s)y1 (t − s) ds + 0.4 K12 (s)y2 (t − s) ds, = −5y1 (t) + 0.1 tanh dt 0 0    ∞ ∞ dy2 (t) = −5y2 (t) + 0.3 K21 (s)y1 (t − s) ds + 0.2 tanh K22 (s)y2 (t − s) ds , (5.1) dt 0 0 for t > 0, t  = tk , k ∈ N that is subject to impulses of the form yi (tk+ ) = k yi (tk− ) in which | k | = for all k ∈ N. For convenience in our computer simulations, the delay kernels are assumed as follows: K11 (s) = e−0.5s , K12 (s) = e−0.6s , K21 (s) = e−0.7s , K22 (s) = e−0.8s for s ∈ [0, 20], and Kij (s) = 0 for i, j = 1, 2, s > 20. One can check that these kernels satisfy the property (2.2) wherein K(s) = e−0.5s for s ∈ [0, 20], K(s) = 0 for s > 20, and 0 < Kij (s) K(s) for all i, j = 1, 2, s 0. While the network parameters in (5.1) satisfy a number of the sufficiency conditions depicted by (3.2), (3.5), and (3.8) under appropriate choices of the constants i > 0, ij , ij ∈ R and the number p 1, we have chosen to work with the condition (3.8) in order to demonstrate the effect of the impulses towards the convergence of (5.1). Here, we choose  = 0.4 and see that  20  − ai + (bi1 + bi2 ) e(−0.5)s ds 0 for i = 1, 2. 0

The exponential convergence of the network (5.1) subject to impulses yi (tk+ ) = yi (tk− ) at times tk = 2, 4, 6, . . ., where = 2 < etk with  = 0.39 and tk = 2, is shown in Fig. 1. Despite being under the impulsive influence of larger

S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

Neuron 1 without impulse effect

4

1

0

0

-2

-1

-4

0

5

10

15

Neuron 1 with impulse effects

4

-2

1

0

0

-2

-1 0

5

10

0

5

15

-2

10

15

Neuron 2 with impulse effects

2

2

-4

Neuron 2 without impulse effect

2

2

887

0

5

10

15

Fig. 1. Exponential convergence of (5.1) without and with impulses characterized by yi (tk+ ) = 2yi (tk− ) at times tk = 2, 4, 6, . . ..

Neuron 1 without impulse effect

1 0.5

0.5

0

0

-0.5

-0.5

-1

0

10

20

30

Neuron 1 with impulse effects

2

-1

0.5

0

0

-1

-0.5 0

10

20

0

30

-1

10

20

30

Neuron 2 with impulse effects

1

1

-2

Neuron 2 without impulse effect

1

0

10

20

30

Fig. 2. Exponential convergence of (5.1) without and with impulses characterized by yi (tk+ ) = 49yi (tk− ) at times tk = 10, 20, 30, . . ..

magnitude such as = 49 < etk where  = 0.39 and tk = 10, the exponential convergence of the network still persists as shown in Fig. 2 below. Observe in each example that the discrete part of the impulsive network (5.1) satisfies |yi (tk− )| < |yi (tk+ )| for all i = 1, 2, k ∈ N. 6. Concluding remarks We have obtained a family of sufficiency conditions governing the network parameters—cf. (3.2), (3.5) and (3.8), and the magnitude of the impulses (i.e. ¯ k etk ) in order to resist instabilities which may have been caused by unbounded

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S. Mohamad et al. / Nonlinear Analysis: Real World Applications 9 (2008) 872 – 888

distributed delays and large impulses, while achieving the exponential convergence towards a unique equilibrium state. These sufficiency conditions do provide notable contributions to the circuit design and application of the impulsive network for solving real-life problems that demand fast convergence. For instance, the presence of the non-network parameters (i.e. i > 0, ij , ij ∈ R) and the real number p 1 in the conditions does provide a much wider selection of values for designing the circuit of a convergent impulsive network. At the same time, by allowing the magnitude of the impulses to be as large as possible—i.e. exponentially proportional to the size of the inter-impulse interval tk , we have successfully eliminated the serious limitation of the impulsive network that has an absolutely convergent requirement within the discrete part of the network [1,2,9–11,13,17,19]. The application of our results has certainly stretched the functionality of the impulsive network a little further for solving real-life problems. References [1] H. Akça, R. Alassar, V. Covachev, Z. Covacheva, Discrete counterparts of continuous-time additive Hopfield-type neural networks with impulses, Dyn. Syst. Appl. 13 (2004) 77–92. [2] H. Akça, R. Alassar, V. Covachev, Z. Covacheva, E. Al-Zahrani, Continuous-time additive Hopfield-type neural networks with impulses, J. Math. Anal. Appl. 290 (2004) 436–451. [3] J. Cao, New results concerning exponential stability and periodic solutions of delayed cellular neural networks, Phys. Lett. A 307 (2-3) (2003) 136–147. [4] Z. Chen, J. Ruan, Global stability analysis of impulsive Cohen–Grossberg neural networks with delay, Phys. Lett. A 345 (2005) 101–111. [5] M. Forti, A. Tesi, New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Trans. Circuits Syst. I: Fund. Theor. Appl. 42 (1995) 354–366. [6] Z.H. Guan, G. Chen, On delayed impulsive Hopfield neural networks, Neural Networks 12 (1999) 273–280. [7] K. Gopalsamy, Stability of artificial neural networks with impulses, Appl. Math. Comput. 154 (2004) 783–813. [8] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Press, The Netherlands, 1992. [9] Y. Li, W. Xing, L. Lu, Existence and global exponential stability of periodic solution of a class of neural networks with impulses, Chaos Solitons Fractals 27 (2006) 437–445. [10] Y. Li, Global exponential stability of BAM neural networks with delays and impulses, Chaos Solitons Fractals 24 (2005) 279–285. [11] Y. Li, L. Lu, Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses, Phys. Lett. A 333 (2004) 62–71. [12] X. Liu, K.L. Teo, B. Xu, Exponential stability of impulsive high-order Hopfield-type neural networks with time-varying delays, IEEE Trans. Neural Networks 16 (2005) 1329–1339. [13] X.Y. Lou, B.T. Cui, Global asymptotic stability of delay BAM neural networks with impulses, Chaos Solitons Fractals 29 (2006) 1023–1031. [14] S. Mohamad, Exponential stability in Hopfield-type neural networks with impulses, Chaos Solitons Fractals 32 (2007) 456–467. [15] S. Mohamad, K. Gopalsamy, Continuous and discrete Halanay-type inequalities, Bull. Austral. Math. Soc. 61 (2000) 371–385. [16] G.T. Stamov, Impulsive cellular neural networks and almost periodicity, Proc. Jpn. Acad. 80 (2004) 198–203. [17] Y. Wang, W. Xiong, Q. Zhou, B. Xiao, Y. Yu, Global exponential stability of cellular neural networks with continuously distributed delays and impulses, Phys. Lett. A 350 (2006) 89–95. [18] D. Xu, Z. Yang, Impulsive delay differential inequality and stability of neural networks, J. Math. Anal. Appl. 305 (2005) 107–120. [19] X. Yang, X. Liao, D.J. Evans, Y. Tang, Existence and stability of periodic solution in impulsive Hopfield neural networks with finite distributed delays, Phys. Lett. A 343 (2005) 108–116. [20] Z. Yang, D. Xu, Existence and exponential stability of periodic solution for impulsive delay differential equations and applications, Nonlinear Anal. 64 (2006) 130–145. [21] Y. Zhang, J. Sun, Stability of impulsive neural networks with time delays, Phys. Lett. A 348 (2005) 44–50.