On global exponential stability for impulsive cellular neural networks with time-varying delays

Computers and Mathematics with Applications 59 (2010) 3508–3515

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On global exponential stability for impulsive cellular neural networks with time-varying delays Ivanka M. Stamova a,∗ , Rajcho Ilarionov b a

Bourgas Free University, 8000 Bourgas, Bulgaria

b

Technical University of Gabrovo, 5300 Gabrovo, Bulgaria

article

info

Article history: Received 1 December 2009 Received in revised form 21 March 2010 Accepted 22 March 2010 Keywords: Global exponential stability Impulsive cellular neural networks Delays Lyapunov method

abstract In this paper, the problem of global exponential stability for cellular neural networks (CNNs) with time-varying delays and fixed moments of impulsive effect is studied. A new sufficient condition has been presented ensuring the global exponential stability of the equilibrium points by using piecewise continuous Lyapunov functions and the Razumikhin technique combined with Young’s inequality. The results established here extend those given previously in the literature. Compared with the method of Lyapunov functionals as in most previous studies, our method is simpler and more effective for stability analysis. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Cellular neural networks (CNNs) were introduced by Chua and Yang in 1988 [1,2]. Impressive applications of CNNs have been proposed for various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition and computer vision [3–7]. However, it is necessary to solve some dynamic image processing and pattern recognition problems by using Delayed Cellular Neural Networks (DCNNs). The study of the stability of CNNs and DCNNs is known to be an important problem in theory and applications. On the other hand, the state of electronic networks is often subject to instantaneous perturbations and the networks experience abrupt change at certain instants which may be caused by a switching phenomenon, frequency change or other sudden noise; that is, the networks exhibit impulsive effects [8,9]. For instance, according to Arbib [10] and Haykin [11], when a stimulus from the body or the external environment is received by receptors the electrical impulses will be conveyed to the neural net and impulsive effects arise naturally in the net. Therefore, a neural network model with delay and impulsive effects should be more accurate to describe the evolutionary process of the systems. Since delays and impulses can affect the dynamical behavior of the system, it is necessary to investigate the effects of both delay and impulsive effects on the stability of neural networks. Such a generalization of the DCNN notion should enable us to study different types of classical problems as well as to ‘‘control’’ the solvability of the DCNN (without impulses). A large number of criteria on the stability of DCNNs without impulses have been derived (see, e.g., [12,3–5,13–15]). Correspondingly, there is not much work dedicated to investigating the stability of impulsive neural networks. Recently, Akca et al. [16], Stamov [17], and Stamov and Stamova [18] have formulated some mathematical models of impulsive neural networks described by measure differential equations or general impulsive differential equations. Impulses can make unstable systems stable, so they have been widely used in many fields such as physics, chemistry, biology, population

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (I.M. Stamova).

0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.03.043

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dynamics, and industrial robotics. The abrupt changes in the voltages produced by faulty circuit elements are exemplary of impulse phenomena that can affect the transient behavior of the network. Some stability results for impulsive neural networks with delays have been given; for example, see [19,20,9] and the references therein. In our paper [19], we formulate a neural network model with time-varying delays and impulsive effects. The aim of the present article is to provide some new results on global exponential stability for cellular neural networks with timevarying delays. The main results are obtained by using piecewise continuous Lyapunov functions [8] and the Razumikhin technique [21,9,22,7] combined with Young’s inequality [23]. Examples are given to demonstrate the effectiveness of the results. The results in the earlier references emerge as special cases of the main results given here. 2. Statement of the problem. Preliminary notes and definitions Let Rn denote the n-dimensional Euclidean space, R+ = [0, ∞) and t0 ∈ R+ . We consider the following impulsive CNNs with time-varying delays:

 n n  x˙ (t ) = −c x (t ) + X a f x (t )
+ X b f x (t − τ (t ))
+ I , i i i ij j j ij j j j i j =1 j =1  1x (t ) = x (t + 0) − x (t ) = P (x (t )), k = 1, 2, . . . , i k i k i k i i k

t 6= tk , t > t0 ,

(2.1)

i = 1, 2, . . . , n, where n corresponds to the numbers of units in a neural network; xi (t ) corresponds to the state of the ith unit at time t; fj (xj (t )) denotes the output of the jth unit at time t; aij , bij , Ii , ci are constants: aij denotes the strength of the jth unit on the ith unit at time t, bij denotes the strength of the jth unit on the ith unit at time t − τj (t ), Ii denotes the external bias on the ith unit, where τj (t ) corresponds to the transmission delay along the axon of the jth unit and satisfies 0 ≤ τj (t ) ≤ τ (τ = const ), and ci represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs; and tk , k = 1, 2, . . ., are the moments of impulsive perturbations and satisfy t0 < t1 < t2 < · · · and limk→∞ tk = ∞. The numbers xi (tk ) and xi (tk + 0) are, respectively, the states of the ith unit before and after impulse perturbation at the moment tk ; and Pi (xi (tk )) represents the abrupt change of the state xi (t ) at the impulsive moment tk . Let J ⊂ R be an interval. Define the following classes of functions. PC [J , Rn ] = {σ : J → Rn : σ (t ) is continuous everywhere except at some points tk ∈ J at which σ (tk − 0) and σ (tk + 0) exist and σ (tk − 0) = σ (tk )}. PCB[J , Rn ] = {σ ∈ PC [J , Rn ] : σ (t ) is bounded on J }. Let ϕ ∈ PCB[[−τ , 0], Rn ]. Denote by x(t ) = x(t ; t0 , ϕ), x ∈ Rn , the solution of system (2.1), satisfying the initial condition



x(t ; t0 , ϕ) = ϕ(t − t0 ), t0 − τ ≤ t ≤ t0 , x(t0 + 0; t0 , ϕ) = ϕ(0).

(2.2)

The solution x(t ) = x(t ; t0 , ϕ) = (x1 (t ; t0 , ϕ), . . . , xn (t ; t0 , ϕ))T of problem (2.1) and (2.2) is a piecewise continuous function with points of discontinuity of the first kind tk k = 1, 2, . . ., where it is continuous from the left, i.e. the following relations are valid: xi (tk − 0) = xi (tk ),

k = 1, 2, . . . ,

xi (tk + 0) = xi (tk ) + Pi (xi (tk )),

tk > 0.

In particular, a constant point x∗ ∈ Rn , x∗ = (x∗1 , x∗2 , . . . , x∗n )T , is called an equilibrium point of (2.1), if x∗ = x∗ (t ; t0 , x∗ ) is a solution of (2.1). We introduce the following conditions. H2.1 There exist constants Li > 0 such that

|fi (u) − fi (v)| ≤ Li |u − v| for all u, v ∈ R, u 6= v , i = 1, 2, . . . , n. H2.2 The functions Pik are continuous on R, i = 1, 2, . . . , n, k = 1, 2, . . .. H2.3 t0 < t1 < t2 < · · · tk < tk+1 < · · · and tk → ∞ as k → ∞. H2.4 There exists a unique equilibrium x∗ = col(x∗1 , x∗2 , . . . , x∗n ) of system (2.1) such that ci x∗i =

n X

aij fj (x∗j ) +

j =1

Pik (xi ) = 0, ∗

n X

bij fj x∗j + Ii ,




j =1

i = 1, 2, . . . , n, k = 1, 2, . . . .

Remark 2.1. The problems of the existence and uniqueness of equilibrium states of neural networks without impulses have been investigated by many authors. Efficient sufficient conditions for the existence and uniqueness of an equilibrium of

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systems of type (2.1) are given in [19]. Note that, when a neural network is designed to function as an associative memory, it is required that there exist many stable equilibrium points, whereas in the case of solving optimization problems it is necessary that the designed neural network must have a unique equilibrium point that is globally asymptotically stable. Therefore, it is of great interest to establish conditions that ensure the global asymptotic stability of a unique equilibrium point of a neural network. We introduce the following notations: Gk = (tk−1 , tk ) × Rn , k = 1, 2, . . . ;

G=

∞ [

Gk .

k=1

Let kϕ− x∗ k = supt ∈[t0 −τ ,t0 ] and the equilibrium x∗ ∈ Rn .

hP n

∗ p i=1 |ϕi (t − t0 )− xi |

i1/p

, p ≥ 1, be the distance between the function ϕ ∈ PCB[[−τ , 0], Rn ]

Definition 2.1. The equilibrium x∗ = col(x∗1 , x∗2 , . . . , x∗n ) of system (2.1) is said to be globally exponentially stable if there exists constants α > 0 and M ≥ 1 such that

kx(t ; t0 , ϕ) − x∗ k ≤ M kϕ − x∗ ke−α(t −t0 ) ,

t ≥ t0 .

Later, we shall use piecewise continuous Lyapunov functions V : [t0 , ∞) × Rn → R+ . Definition 2.2. We say that the function V : [t0 , ∞) × Rn → R+ belongs to the class V0 if the following conditions are fulfilled: 1. The function V is continuous in ∪∞ k=1 Gk and V (t , 0) = 0 for t ∈ [t0 , ∞). 2. The function V satisfies locally the Lipschitz condition with respect to x on each of the sets Gk . 3. For each k = 1, 2, . . . there exist the finite limits V (tk − 0, x) = lim V (t , x), t →tk t
V (tk + 0, x) = lim V (t , x). t →tk t >tk

4. For each k = 1, 2, . . . the following equalities are valid: V (tk − 0, x) = V (tk , x). Let V ∈ V0 . For any (t , x) ∈ [tk−1 , tk ) × Rn , k = 1, 2, . . ., the upper right-hand derivative D+ V (t , x(t )) with respect to system (2.1) is defined by D+ V (t , x(t )) = lim inf

1h

i

V (t + h, x(t + h)) − V (t , x(t )) .

h In the proof of the main results we shall use the following lemma. h→0+

Lemma 2.1 (Young’s Inequality [23]). Assume that a > 0, b > 0, p > 1, ab ≤

1 p

ap +

1 q

1 p

+

1 q

= 1; then the inequality

bq

holds. 3. Main results Theorem 3.1. Assume that: 1. Conditions H2.1–H2.4 hold. 2. There exist real constants ζij , ηij γi > 0, i, j = 1, 2, . . . , n and p ≥ 1 such that the following inequalities hold:

( min

1≤i≤n

pci −

n  X γj j =1

γi

p(1−ζji )

Li |aji |

+ (p − 1)Lj |aij |

pζij p−1

+ (p − 1)Lj |bij |

pηij p−1

)

( > max

1≤i≤n

n X γj j =1

γi

) p(1−ηji )

Li |bji |

3. There exist constants σik , 0 < σik < 2, i = 1, 2, . . . , n, k = 1, 2, . . ., such that the following equalities are satisfied: Pik (xi (tk )) = −σik (xi (tk ) − x∗i ). Then the equilibrium x∗ of (2.1) is globally exponentially stable. Proof. Let yi (t ) = xi (t ) − x∗i and define a Lyapunov function V (t , y) =

n X i=1

γi |yi (t )|p .

> 0.

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Then, for t = tk , from condition 3 of Theorem 3.1, we obtain V (tk + 0, y(tk ) + 1y(tk )) =

n X

γi |xi (tk ) − x∗i − σik (xi (tk ) − x∗i )|p =

n X

i=1

<

n X

γi |1 − σik |p |xi (tk ) − x∗i |p

i=1

γi |xi (tk ) − x∗i |p = V (tk , y(tk )),

k = 1, 2, . . . .

(3.1)

i=1

Let t ≥ t0 and t 6= tk , k = 1, 2, . . .. Then, for the upper right-hand derivative D+ V (t , y(t )) of V with respect to system (2.1) we get D+ V (t , y(t )) =

n X

γi p|yi (t )|p−1 sgn(yi (t ))˙yi (t )

i =1

=

n X

" γi p|yi (t )|p−1 sgn(yi (t )) −ci yi (t ) +

n X

i =1




j=1 n

+

aij fj (yj (t ) + x∗j ) − fj (x∗j )

X

bij fj (yj (t − τj (t )) + xj ) − fj (xj ) ∗

∗

#


j =1

≤

n X

" γi p −ci |yi (t )| + p

i =1

=

|aij |Lj |yi (t )|

p−1

|yj (t )| +

j =1

n X

( γi p −ci |yi (t )| + p

i =1

+

n X

n X

n X

# |bij |Lj |yi (t )|

p−1

|yj (t − τj (t ))|

j =1

Lj |yj (t )||aij |

1−ζij

 |aij |

ζij p−1

p−1 |yi (t )|

j =1

n X

Lj |yj (t − τj (t ))||bij |1−ηij

 |bij |

ηij p−1

p−1 ) |yi (t )| .

(3.2)

j =1

Let a = |yj (t )||aij |1−ζij , b =



p−1 ζij |aij | p−1 |yi (t )| ; by Lemma 2.1, we have

 p−1 ζij pζij p−1 1 |yj (t )||aij |1−ζij |aij | p−1 |yi (t )| |aij | p−1 |yi (t )|p . ≤ |aij |p(1−ζij ) |yj (t )|p + p

(3.3)

p

Similarly, using Lemma 2.1 for a = |yj (t − τj (t ))||bij |

p−1  ηij p − 1 |yi (t )| , and b = |bij |

1−ηij

we get

 p−1 ηij pηij 1 p−1 |yj (t − τj (t ))||bij |1−ηij |bij | p−1 |yi (t )| ≤ |bij |p(1−ηij ) |yj (t − τj (t ))|p + |bij | p−1 |yi (t )|p . p

(3.4)

p

Substituting (3.3) and (3.4) into (3.2), we obtain D V (t , y(t )) ≤ +

n X

( γi p −ci |yi (t )|p +

i =1

+

n X p−1 j =1

p

n X 1

p j =1

pζij

Lj |aij |p(1−ζij ) |yj (t )|p

Lj |aij | p−1 |yi (t )| + p

n X 1 j=1

p

Lj |bij |

|yj (t − τj (t ))| + p

n X p−1 j =1

p

pηij

≤ −k1 V (t , y(t )) + k2 sup V (s, y(s)), t −τ ≤s≤t

)

Lj |bij | p−1 |yi (t )|

# n n n pζij pηij X X X γj p(1−ζji ) p − 1 p − 1 + + = γi −pci + (p − 1)Lj |aij | (p − 1)Lj |bij | Li |aji | |yi (t )|p γi i =1 j =1 j =1 j =1 " # n n X X γj p(1−ηji ) + γi Li |bji | |yi (t − τi (t ))|p γ i i =1 j=1 n X

"

p(1−ηij )

p

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where

( k1 = min

pci −

1≤i≤n

k2 = max

n  X γj

γi

j =1

( n X γj

1≤i≤n

γi

j =1

Li |aji |

p(1−ζji )

pζij

pηij

+ (p − 1)Lj |aij | p−1 + (p − 1)Lj |bij | p−1

)

> 0,

) p(1−ηji )

> 0.

Li |bji |

From the above estimate for any y(t ) which satisfy the Razumikhin condition V (s, y(s)) ≤ V (t , y(t )),

t − τ ≤ s ≤ t,

we have D+ V (t , y(t )) ≤ −(k1 − k2 )V (t , y(t )),

t 6= tk , k = 1, 2, . . . .

By virtue of condition 2 of Theorem 3.1 there exits a real number k > 0 such that k1 − k2 ≥ k, and it follows that D+ V (t , y(t )) ≤ −kV (t , y(t )),

t 6= tk , t ≥ t0 .

(3.5)

Then, using (3.1) and (3.5), we get

γmin kx(t ) − x∗ kp ≤ V (t , y(t )) ≤ e−k(t −t0 ) V (t0 + 0, y(t0 + 0)),

t ≥ t0 .

So, 1/p

kx(t ) − x∗ k ≤

γmax 1/p min

γ

k

kϕ − x∗ ke− p (t −t0 ) ,

and this completes the proof of the theorem.

t ≥ t0 , 

Corollary 3.1. If in Theorem 3.1 condition 2 is replaced by the condition

( min

1≤i≤n

ci −

) ( ) n X γj Li |aji | > max Li |bji | > 0, 1≤i≤n γi γi j=1

n  X γj j =1

where γi > 0, i = 1, 2, . . . , n, then the equilibrium x∗ of (2.1) is globally exponentially stable. Proof. Taking p = 1 in theorem above, then we can easily obtain Corollary 3.1.



In the case when p = 2, ζij = 0.5, ηij = 0.5, i, j = 1, 2, . . . , n, we deduce the following corollary of Theorem 3.1. Corollary 3.2. If in Theorem 3.1 condition 2 is replaced by the condition

( min

1≤i≤n

2ci −

n  X γj j =1

γi

Li |aji | + Lj |aij | + |bij |

)


> max

1≤i≤n

( n X γj j =1

γi

) Li |bji |

> 0,

where γi > 0, i = 1, 2, . . . , n, then the equilibrium x∗ of (2.1) is globally exponentially stable. Remark 3.1. If we let γi = 1, i = 1, 2, . . . , n, then Corollaries 1 and 2 correspond to Theorems 1 and 2 in [19], respectively. That is, our theorem includes the main results in [19] as a special cases. Remark 3.2. In some papers (see for example [20] and the references therein), by constructing Lyapunov functions, results on the global exponential stability of impulsive high-order Hopfield-type neural networks with time-varying delays are presented. Such systems include system (2.1) as a special case. In contrast to our results, all of those results require that the impulsive moments depend on the upper bound of the delay function. Thus, the conditions given in [20] are more restrictive and conservative. Remark 3.3. Young’s inequality was first used by Cao [3]. However, the results in [3] only hold for constant delays.

I.M. Stamova, R. Ilarionov / Computers and Mathematics with Applications 59 (2010) 3508–3515

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4. Examples In the following, we will give three examples to show our results. Example 4.1. Consider the impulsive neural differential equations with time-varying delays

( x˙ i (t ) = −ci xi (t ) +

n X

aij fj xj (t ) +




n X

bij fj xj (t − τj (t )) + Ii ,

where n = 2, I1 = 0.22727272, I2 = 0.2424243, c1 = c2 = 1.5, fi (xi ) =

(aij )2×2 = (bij )2×2 =

a12 a22

b11 b21





=

0.1 0.2

1 2

(|xi +1|−|xi −1|) (i = 1, 2), 0 ≤ τi (t ) ≤ τ (τ = 1),

−0.1 , 0.1    1 0.2 b12 = , b22 −0.3 0.2

a11 a21



(4.1)

j=1

j=1



t 6= tk , t > 0,






with

 1, 8181818 − x1 (tk )  x1 (tk + 0) = ,

k = 1, 2, . . . ,

 x2 (tk + 0) =

k = 1, 2, . . . ,

2 1, 0606064 − x2 (tk ) 6

,

(4.2)

where the impulsive moments are such that 0 < t1 < t2 < · · ·, and limk→∞ tk = ∞. Clearly, fi satisfies assumption H2.1 with L1 = L2 = 1. Also, we have that 0 < σ1k =

3 2

< 2,

0 < σ2k =

7 6

< 2,

and therefore condition 3 of Theorem 3.1 is satisfied. One can check that

( min

1≤i≤n

ci −

n X

) |aji | = 1.2 < max

1≤i≤n

j =1

( n X

) |bji | = 1.3,

j=1

and

( min

1≤i≤n

2ci −

n X

) ) ( n X

|bji | = 1.3. |aji | + |aij | + |bij | = max 1≤i≤n

j=1

j =1

Therefore, the main results in [19] do not hold for this example. On the other hand, if we let p = 3, ζij = ηij = 1.5, γi = 1, i, j = 1, 2 in Theorem 3.1, one can verify that

( min

1≤i≤n

3ci −

n X

) ( ) n X

|aji | + 2 |aij | + |bij | = 1.4 > max |bji | = 1.3.

j=1

1≤i≤n

j=1

Hence, the unique equilibrium x∗ = (x∗1 , x∗2 )T = (0, 6060606, 0, 1515152)T

(4.3)

of (4.1), (4.2) is globally exponential stable. If we consider again the system (4.1) but with impulsive perturbations of the form x1 (tk + 0) = 6x1 (tk ) − 3, 030303, k = 1, 2, . . . , 1, 0606064 − x2 (tk ) x2 (tk + 0) = , k = 1, 2, . . . , 6

(

(4.4)

the point (4.3) will be again an equilibrium of (4.1), (4.4) but there is nothing we can say about its exponential stability, because σ1k = −5 < 0. The example shows that (1) our theorem includes the main results in [19] as a special cases; (2) by means of appropriate impulsive perturbations we can control the neural network system’s dynamics.

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Example 4.2. Consider the impulsive neural differential equations with time-varying delays (4.1), where n = 2, I1 = 0.38235294, I2 = 0.02941178, c1 = c2 = 1, fi (xi ) = 12 (|xi + 1| − |xi − 1|) (i = 1, 2), 0 ≤ τi (t ) ≤ τ (τ = 1),

(aij )2×2 = (bij )2×2 =



a11 a21



b11 b21

 −0.1 , 0.1    0.5 −0.1 b12 = , b22 −0.1 0.5

a12 a22



0.1 0.1



=

with

 0, 7352941 + x1 (tk )  x1 (tk + 0) = ,  x2 (tk + 0) =

2 1, 3235295 + 2x2 (tk ) 5

k = 1, 2, . . . , (4.5)

,

k = 1, 2, . . . ,

where the impulsive moments are such that 0 < t1 < t2 < · · ·, and limk→∞ tk = ∞. Clearly, fi satisfies the assumption H2.1 with L1 = L2 = 1. Also we have that 0 < σ1k =

1 2

< 2,

0 < σ2k =

3

< 2.

5

It is easy to verify that

( min

1≤i≤n

ci −

n X j =1

) |aji | = 0.8 > max

( n X

1≤i≤n

) |bji | = 0.6.

j =1

According to Corollary 3.1 for γi = 1, i = 1, 2, the unique equilibrium x∗ = (x∗1 , x∗2 )T = (0, 7352941, 0, 4411765)T of (4.1), (4.5) is globally exponential stable. By a simple computation, we can seen that the matrix A + AT =

(4.6)



0.2 0



0 0.2

is

not negative semidefinite. Thus the condition in [24] does not hold. If we consider again the system (4.1) but with impulsive perturbations of the form

  

x1 (tk + 0) =

0, 7352941 + x1 (tk )

,

2 x2 (tk + 0) = 5x2 (tk ) − 1, 764706,

k = 1, 2, . . . ,

(4.7)

k = 1, 2, . . . ,

the point (4.6) will be again an equilibrium of (4.1), (4.7) but there is nothing we can say about its exponential stability, because σ2k = −4 < 0. 5. Conclusions This paper presents a sufficient condition for global exponential stability of the equilibrium point for a neural network model with time-varying delays and impulsive effects. The results impose constraint conditions on the network parameters of neural system independent of the delay parameters. The results are applicable to more general neuron activation functions than both the usual sigmoid activation functions in Hopfield networks and the piecewise linear function in standard cellular networks. The results are also compared with the previously reported results in the literature, implying that the results obtained in this paper provide one more set of criteria for determining the stability of impulsive neural networks with time delays. The examples considered show that by means of appropriate impulsive perturbations we can control the stability properties of neural networks. References [1] [2] [3] [4] [5] [6]

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