Neural Process Lett (2010) 31:105–127 DOI 10.1007/s11063-009-9128-y
Globally Exponential Stability for Delayed Neural Networks Under Impulsive Control Cheng Hu · Haijun Jiang · Zhidong Teng
Published online: 19 January 2010 © Springer Science+Business Media, LLC. 2010
Abstract In this paper, the dynamic behaviors of a class of neural networks with timevarying delays are investigated. Some less weak sufficient conditions based on p-norm and ∞-norm are obtained to guarantee the existence, uniqueness of the equilibrium point for the addressed neural networks without impulsive control by applying homeomorphism theory. And then, by utilizing inequality technique, Lyapunov functional method and the analysis method, some new and useful criteria of the globally exponential stability with respect to the equilibrium point under impulsive control we assumed are derived based on p-norm and ∞-norm, respectively. Finally, an example with simulation is given to show the effectiveness of the obtained results. Keywords Neural networks · Impulsive control · Time-varying delays · Exponential stability Mathematics Subject Classification (2000)
34D23 · 34K45 · 37N35 · 92B20
1 Introduction In recent years, artificial neural networks have been widely studied due to their extensive applications in many fields such as image processing, associative memories, classification of patterns, quadratic optimization and so on [1]. In implementation of neural networks, time delays especially time-varying delays are unavoidably encountered in the signal transmission among the neurons due to the finite switching speed of neurons and amplifiers, which will affect the stability of the neural system and may lead to some complex dynamic behaviors such as instability, chaos, oscillation or other performance of the neural network. Hence, the control of neural networks with time-varying delays is of both theoretical and practical
C. Hu · H. Jiang (B) · Z. Teng College of Mathematics and System Sciences, Xinjiang University, 830046 Urumqi, Xinjiang, People’s Republic of China e-mail:
[email protected];
[email protected]
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importance. In view of the significance of the control for delayed neural networks, in recent years, the study of neural networks with delays have been extensively investigated by many researchers, for instance, see [1–29]. Impulsive control strategies, as an important control means, in the past several years, have been widely used to stabilize and synchronize nonlinear dynamical systems. From the control point of view, impulsive control, based on the theory of impulsive dynamic systems, is an effective method in the sense that it allows stabilization of a complex system by using only small control impulses, even though the complex system behaviors may follow unpredictable patterns. In view of those merits, up to now, many results on impulsive neural network have been investigated (see [2,22,23,25,30–33]). For instance, in [2], some criteria of the global exponentially stability for cellular neural networks with time-varying delays and fixed moments of impulsive effect were obtained by employing Lyapunov functions and the Razumikhin technique. Some sufficient conditions of the uniform stability of the equilibrium point for the impulsive Hopfield-type neural networks systems with time delays were given by using Lyapunov functions and analysis technique in [25]. And in [33], some criteria ensuring the existence, uniqueness and globally exponential stability of the equilibrium point for impulsive BAM neural networks with time-varying delays were derived by employing the delay differential inequality with impulsive initial conditions and M-matrix theory. However, to the best of our knowledge, most previous results have been restricted to linear impulsive controllers (for instance, see [2,22,23,31]), few authors have considered the nonlinear impulsive controllers on the study of the globally exponential stability of a neural network. Motivated by above discussion, the main purpose of this paper lies in the following two aspects. Firstly, before adding the impulsive control, some less weak sufficient conditions based on p-norm and ∞-norm are obtained to guarantee the existence, uniqueness of the equilibrium point for the originally addressed neural networks by applying homeomorphism theory. And then, some new and useful criteria of the globally exponential stability with respect to the equilibrium point under impulsive control we assumed are also derived based on p-norm and ∞-norm by utilizing inequality technique, Lyapunov functional method and the analysis method. In this letter, the impulsive controllers we assumed can be nonlinear and even dependent of the states of all neurons, and our theorems do not require the activation functions to be differentiable, bounded or monotone nondecreasing, which have been required in [2,3,7,15,17]. In addition, our results are less weak than many previous results (such as [1,2,26]). This paper is organized as follows. In Sect. 2, model description and preliminaries are given. Some criteria are obtained in Sect. 3 to ensure the existence and uniqueness of the equilibrium for the addressed neural networks. In Sect. 4, some sufficient conditions are derived to guarantee the globally exponential stability of the equilibrium under the impulsive controllers we assumed. In Sect. 5, the effectiveness and feasibility of the developed methods have been shown by an example.
2 Preliminary In this paper, we consider a class of neural networks with time-varying delays described by the following form:
x˙i (t) = −ci (xi (t)) +
n j=1
123
ai j f j (x j (t)) +
n j=1
bi j g j (x j (t − τ j (t))) + Ii ,
(1a)
Globally Exponential Stability for Delayed Neural Networks
107
for all i ∈ I = {1, 2, . . . , n}, where n corresponds to the number of units in a neural network; x(t) = (x1 (t), x2 (t), . . . , xn (t))T and xi (t) corresponds to the state of the ith unit at time t; f j (·) and g j (·) denote the activation functions of the jth neuron; ai j and bi j denote the constant connection weight and the constant delayed connection weight of the jth neuron on the ith neuron, respectively; τ j (t) is the time-varying transmission delay and satisfies 0 ≤ τ j (t) ≤ τ j ; ci (·) represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and external inputs; Ii denotes the external inputs on the ith neuron. System (1a) is supplemented with initial value given by: xi (t0 + s) = φi (s), s ∈ [−τ, 0], i ∈ I ,
(1b)
T where τ =n maxi∈I {τi } > 0, φ(s) = (φ1 (s), φ2 (s), . . . , φn (s)) ∈ C and C = [−τ, 0], R denotes the Banach space of all continuous functions mapping [−τ, 0] into R n with p-norm ( p ≥ 1 is a positive integer) or ∞-norm defined by the following forms, respectively:
φ p =
sup
n
s∈[−τ,0]
1
p
|φi (s)|
p
, φ∞ =
i=1
sup s∈[−τ,0]
max |φi (s)| .
1≤i≤n
Throughout this paper, for system (1), we always assume that: (H1 ) Functions ci (·) : R → R are continuous and monotone increasing, that is, there exist real numbers ci > 0 such that ci (u) − ci (v) ≥ ci for all u, v ∈ R, u = v, i ∈ I . u−v (H2 ) Functions f j and g j are Lipschitz-continuous on R with Lipschitz constants L j > 0 and N j > 0, respectively. That is | f j (u) − f j (v)| ≤ L j |u − v| and |g j (u) − g j (v)| ≤ N j |u − v| for all u, v ∈ R and j ∈ I . (H3 ) Functions τ j (t) ∈ C[R + , R + ] and satisfy σ j = inf {1 − τ˙ j (t)} > 0 for j ∈ I . t∈R +
In the following, in order to study the dynamical behaviors of system (1), we introduce the following control system of system (1): x˙i (t) = −ci (xi (t)) +
n j=1
+
∞
ai j f j (x j (t)) +
n
bi j g j (x j (t − τ j (t)))
j=1
δ(t − tk )[ pik (x(t)) − xi (t)] + Ii , i ∈ I ,
(2)
k=1
where k ∈ Z + = {1, 2, . . .}, δ(t) is the Dirac delta function; the time sequence {tk } satisfy 0 < t1 < t2 < · · · < tk < tk+1 . . . , limk→∞ tk = ∞; functions pik (x) = pik (x1 , x2 , . . . , xn ) ∈ [R n , R] denote the external control inputs satisfying the following conditions:
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(H14 ) There exist nonnegative matrixes Pk = ( pikj )n×n such that | pik (u 1 , u 2 , . . . , u n ) − pik (v1 , v2 , . . . , vn )| p ≤
n
pikj |u j − v j | p
j=1
for any (u 1 , u 2 , . . . , u n )T ∈ R n , (v1 , v2 , . . . , vn )T ∈ R n , i ∈ I and k ∈ Z + , where p ≥ 1 is a integer. Integrating from tk − h to tk + h both side of system (2), we have ⎡ t k +h n ⎣−ci (xi (t)) + xi (tk + h) − xi (tk − h) = ai j f j (x j (t)) j=1
tk −h
+
n
bi j g j (x j (t − τ j (t))) +
j=1
∞
⎤
δ(t − tk )[ pik (x(t)) − xi (t)] + Ii ⎦ dt, i ∈ I ,
k=1
where h > 0 is sufficiently small. As h → 0+ , by applying the properties of the Dirac delta function, we have xi (tk+ ) − xi (tk− ) = pik (x(tk )) − xi (tk ).
(3)
In this paper, we assume that x(t) = (x1 (t), x2 (t), . . . , xn (t))T is left continuous at tk (k ∈ Z + ), i.e., xi (tk ) = limt→t − xi (t). In this case, it follows from (3) that k
xi (tk+ ) = pik (x(tk ))
for i ∈ I and k ∈ Z + .
(4)
Thus, from the definition of the Dirac delta function and (4), system (2) can be rewritten as the following form: ⎧ n n ⎨ x˙ (t) = −c (x (t)) + a f (x (t)) + b g (x (t − τ (t))) + I , t = t , i i i ij j j ij j j j i k (5) j=1 j=1 ⎩ xi (tk+ ) = pik (x(tk )), i ∈ I , k ∈ Z + . Remark 1 It is evident that system (5) is equivalent to system (2), system (5) is said to the impulsive control system of (1) and (4) is said to impulsive controllers or impulsive control roles of system (1). Furthermore, if the control is invalid, that is, xi (tk ) ≡ pik (x(tk )) for any i ∈ I and k ∈ Z + , system (5) is reduced to system (1). Remark 2 It is worth noting that the impulsive functions may be nonlinear and even dependent of the states of all neurons in this paper. If impulsive functions pik (x) = pik (xi ) for all i ∈ I and k ∈ Z + , assumption (H41 ) is reduced to the following form: ¯ 4 ) There exist nonnegative diagnose matrixes P k = diag( p¯ 1k , . . . , p¯ nk ) such that (H 1
| pk (x) − pk (y)| ≤ P k |x − y| for all x, y ∈ R n and k ∈ Z + , 1
where pk (x) = ( p1k (x1 ), p2k (x2 ), . . . , pnk (xn ))T and p¯ ik = ( piik ) p . Such function and assumption have been required in [30,32,33]. From this point, our conditions for impulsive functions are more general.
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In the following, for further study, we first give the following definitions and lemmas. Assume that R n be the space of n-dimensional real column vectors. For any x = (x1 , x2 , . . . , xn )T ∈ R n , x denotes a vector norm defined by x p =
n
1
p
|xi |
, x∞ = max |xi |,
p
i∈I
i=1
where p ≥ 1 is a positive integer. Definition 1 A constant vector x ∗ = (x1∗ , x2∗ , . . . , xn∗ )T is said to be an equilibrium point of system (1) if x ∗ satisfies the following equality: ci (xi∗ ) =
n
ai j f j (x ∗j ) +
j=1
n
bi j g j (x ∗j ) + Ii , i ∈ I .
j=1
Definition 2 The equilibrium x ∗ = (x1∗ , x2∗ , . . . , xn∗ )T of system (1) is said to be globally exponentially stable under impulsive controllers (4), if there exist λ > 0 and M ≥ 1 such that x(t) − x ∗ ≤ φ − x ∗ Me−λ(t−t0 ) for t ≥ t0 ≥ 0, where x(t) is an any solution of system (5) with initial value φ ∈ C. Definition 3 (see [32]) A map H : R n → R n is said to a homeomorphism of R n onto itself, if H ∈ C 0 is one-to-one onto itself and the inverse map H −1 ∈ C 0 . Lemma 1 (see [32]) If H : R n → R n is a continuous function and satisfies the following conditions: (1) H (x) is injective on R n , that is, H (x) = H (y) for all x = y. (2) H (x) → ∞ as x → ∞. Then H (x) is homeomorphism of R n .
3 Existence and Uniqueness of Equilibrium Point In this section, we will present some sufficient conditions for the existence and uniqueness of the equilibrium point for system (1). For convenience, we denote that λi = pci −
p−1 n n µj pβ pδ pβ |a ji | pα p, ji L i p, ji , |ai j | pα,i j L j ,i j + |bi j | pγ,i j N j ,i j − µi j=1 =1
ηi =
n µj j=1
µi
j=1
pδ p, ji
|b ji | pγ p, ji Ni
, µ = max{µi }, µ = min{µi }, i∈I
i∈I
where µ real numbers i be positive constant p numbers, nonnegative p p α,i j , β,i j , γ,i j and δ,i j p satisfy =1 α,i j = 1, =1 β,i j = 1, =1 γ,i j = 1 and =1 δ,i j = 1, respectively. The following theorem is provided to guarantee the existence and uniqueness of equilibrium point of system (1).
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Theorem 1 Under the assumptions (H1 ) and (H2 ), system (1) exists a unique equilibrium x ∗ if the following condition is also satisfied: (H5 ) λi > ηi for any i ∈ I . Proof Defining a map H (x) = (h 1 (x), h 2 (x), . . . , h n (x))T ∈ C 0 (R n , R n ), where h i (x) = −ci (xi ) +
n
ai j f j (x j ) +
j=1
n
bi j g j (x j ) + Ii , for xi ∈ R and i ∈ I .
(6)
j=1
In the following, we will prove that H (x) is a homeomorphism by applying Lemma 1. Firstly, we claim that H (x) is an injective map on R n . Otherwise, there exist x T , y T ∈ R n and x T = y T such that H (x) = H (y), then ci (xi ) − ci (yi ) =
n
ai j ( f j (x j ) − f j (y j )) +
j=1
n
bi j (g j (x j ) − g j (y j )), i ∈ I .
j=1
It follows from (H1 ) and (H2 ) that ci |xi − yi | ≤
n
|ai j |L j |x j − y j | +
j=1
n
|bi j |N j |x j − y j |, i ∈ I .
(7)
j=1
From (7), we have n
pµi ci |xi − yi | p
i=1 n
≤
⎡ pµi |xi − yi |
p−1 ⎣
i=1
=
n n
n
|ai j |L j |x j − y j | +
j=1
pµi |ai j |
p
=1 α,i j
n
⎤ |bi j |N j |x j − y j |⎦
j=1 p
Lj
=1
β,i j
|xi − yi | p−1 |x j − y j |
i=1 j=1
+
n n
pµi |bi j |
p
=1 γ,i j
p
Nj
=1 δ,i j
|xi − yi | p−1 |x j − y j |
i=1 j=1
=
n n
β
β
µi p(|ai j |α1,i j L j 1,i j |xi − yi |) × (|ai j |α2,i j L j 2,i j |xi − yi |)
i=1 j=1 β
β
× · · · × (|ai j |α p−1,i j L j p−1,i j |xi − yi |) × (|ai j |α p,i j L j p,i j |x j − y j |) +
n n
δ
δ
µi p(|bi j |γ1,i j N j 1,i j |xi − yi |) × (|bi j |γ2,i j N j 2,i j |xi − yi |)
i=1 j=1 δ
δ
× · · · × (|bi j |γ p−1,i j N j p−1,i j |xi − yi |) × (|bi j |γ p,i j N j p,i j |x j − y j |). Applying the fact that p
p
p
a1 + a2 + · · · + a p ≥ pa1 a2 · · · a p ,
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Globally Exponential Stability for Delayed Neural Networks
111
where ai ≥ 0 and i = 1, 2, · · · , p, we have n
pµi ci |xi − yi | p
i=1 n n
≤
⎡ µi ⎣
n n
⎡ µi ⎣
i=1 j=1
=
n n
+µ j
|ai j |
⎡
p−1
− yi | + |ai j | p
pα p,i j
pβ L j p,i j |x j
− yj|
⎣µi ⎝
p−1
p⎦
⎤ |bi j | pγ,i j
pδ N j ,i j |xi
|ai j |
pβ L j ,i j
− yi | p + |bi j | pγ p,i j
=1
⎛
pα,i j
+
=1
i=1 j=1
⎤ pβ L j ,i j |xi
pα,i j
=1
i=1 j=1
+
p−1
p−1
+ |b ji | pγ p, ji
− y j |p⎦
⎞ |bi j |
=1
pβ |a ji | pα p, ji L i p, ji
pδ N j p,i j |x j
pδ Ni p, ji
pγ,i j
pδ N j ,i j ⎠
⎤ ⎦ |xi − yi | p .
It follows that n
µi ωi |xi − yi | p ≤ 0,
(8)
i=1
where ωi = λi − ηi for any i ∈ I . From (H5 ), we have wi > 0, then it follows from (8) that xi = yi for all i ∈ I . Therefore, x T = y T , which leads to a contradiction with our assumption. Next, we prove that H (x) p → ∞ as x p → ∞. From (H1 ), (H2 ) and (H5 ), we have n
sgn(xi ) pµi (h i (x) − h i (0))|xi | p−1
i=1
≤−
n
pµi ci |xi | p +
i=1
+
n n
pµi |ai j |L j |x j ||xi | p−1
i=1 j=1
n n
pµi |bi j |N j |x j ||xi | p−1
i=1 j=1
≤
n
⎡
⎣− pµi ci + µi
i=1
+
n
p−1 n pβ pδ |ai j | pα,i j L j ,i j + |bi j | pγ,i j N j ,i j j=1 =1
⎤ pβ pδ p, ji p, ji ⎦ |xi | p µ j |a ji | pα p, ji L i + b ji | pγ p, ji Ni
j=1 p
≤ −ωx p ,
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C. Hu et al.
where ω = min {µi ωi } > 0. Then, we have 1≤i≤n
p
x p ≤ −
n 1 sgn(xi ) pµi (h i (x) − h i (0))|xi | p−1 ω
pµ ω
≤
i=1 n
|h i (x) − h i (0)||xi | p−1 .
(9)
i=1
If p = 1, inequality (9) implies that x1 ≤
µ µ H (x) − H (0)1 ≤ (H (x)1 + H (0)1 ). ω ω
If p > 1,using the Hölder inequality, we obtain 1 n 1 n q p pµ ( p−1)q p ≤ |xi | |h i (x) − h i (0)| ω
p x p
i=1
n pµ = |xi | p ω
1− 1 p
i=1
i=1
n
1
p
|h i (x) − h i (0)| p
,
i=1
which leads to x p ≤
pµ pµ H (x) − H (0) p ≤ (H (x) p + H (0) p ), ω ω
where p > 1 and q > 1, and satisfy 1p + q1 = 1. From what have been discussed above, we see that H (x) p → ∞ as x p → ∞. By Lemma 1, H (x) is a homeomorphism on R n , which implies system (1) has a unique equilibrium.
Corollary 1 Under the assumptions (H1 ) and (H2 ), system (1) has a unique equilibrium if one of the following conditions is also satisfied: (a) For p > 1, the following inequality holds: pci > ( p − 1)
n
p
|ai j | p−1
αi j
p
L jp−1
βi j
+ ( p − 1)
j=1
+
n µj j=1
µi
n
p
|bi j | p−1
γi j
p
N jp−1
δi j
j=1 p(1−β ji )
|a ji | p(1−α ji ) L i
+
n µj j=1
µi
p(1−δ ji )
|b ji | p(1−γ ji ) Ni
,
(b) 2ci >
n j=1
+
n j=1
123
2βi j
|ai j |2αi j L j
+
n
2δi j
|bi j |2γi j N j
j=1
µj µj 2(1−β ji ) 2(1−δ ji ) |a ji |2(1−α ji ) L i + |b ji |2(1−γ ji ) Ni , µi µi n
j=1
Globally Exponential Stability for Delayed Neural Networks
113
(c) pci > ( p − 1)
n
|ai j |L j + ( p − 1)
j=1
n
|bi j |N j +
j=1
n µj j=1
µi
|a ji |L i +
n µj j=1
µi
|b ji |Ni ,
(d) ci >
n µj
µi
j=1
n µj
|a ji |L i +
µi
j=1
|b ji |Ni ,
(e) 2ci >
n
|ai j |L j +
n
j=1
|bi j |N j +
j=1
n µj j=1
µi
|a ji |L i +
n µj j=1
µi
|b ji |Ni ,
for all i ∈ I , where µi be positive constant numbers, real numbers αi j , βi j , γi j and δi j satisfy 0 ≤ αi j ≤ 1, 0 ≤ βi j ≤ 1, 0 ≤ γi j ≤ 1 and 0 ≤ δi j ≤ 1. Proof (a) In Theorem 1, when p > 1, choosing α,i j = δ
αi j p−1 , β,i j
=
βi j p−1 , γ,i j
=
γi j p−1
ij and δ,i j = p−1 for = 1, 2, . . . , p − 1, then the result is obtained from (H5 ). (b) The result can be directly obtained by choosing p = 2 in Theorem 1. (c) Taking α,i j = β,i j = γ,i j = δ,i j = 1p for = 1, 2, . . . , p and i, j ∈ I in Theorem 1, we obtain this statement. (d) and (e) follow directly from (c) when p = 1 and p = 2, respectively.
Theorem 1 is given to ensure the existence and uniqueness of the equilibrium of system (1) based on p-norm. The following statements are provided to guarantee the existence and uniqueness of the equilibrium of system (1) based on ∞-norm. Theorem 2 Under the assumptions (H1 ) and (H2 ), system (1) has a unique equilibrium if the following condition is also satisfied: (H5 ) ci >
n
|ai j |L j +
j=1
n
|bi j |N j , i ∈ I .
j=1
Proof Similar to Theorem 1, to complete the proof, it suffices to show that H (x) is a homeomorphism on R n , where H (x) is defined in (6). Firstly, we claim that H (x) is an injective map on R n . In fact, if there exist x T , y T ∈ R n and x T = y T such that H (x) = H (y), then ci (xi ) − ci (yi ) =
n
ai j ( f j (x j ) − f j (y j )) +
n
j=1
bi j (g j (x j ) − g j (y j )), i ∈ I .
j=1
It is evident that there exists ∈ I such that x − y∞ = maxi∈I |xi − yi | = |x − y |. It follows from (H1 ) and (H2 ) that c |x − y | ≤
n
|aj |L j |x j − y j | +
j=1
⎛ ≤⎝
n j=1
n j=1
|aj |L j +
n
|bj |N j |x j − y j | ⎞
|bj |N j ⎠ |x − y |,
j=1
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C. Hu et al.
which implies that ⎛ ⎝c −
n j=1
|aj |L j −
n
⎞ |bj |N j ⎠ x − y∞ ≤ 0,
j=1
this means that x T = y T , which leads to a contradiction with our assumption. So, H (x) is an injective map on R n , On the other hand, similarly, there exists ∈ I such that x∞ = |x |, then sgn(x )(h (x) − h (0)) ≤ −c |x | +
n
|aj |L j |x j | +
j=1
≤ (−c +
n
|aj |L j +
j=1
n
|bj |N j |x j |
j=1 n
|bj |N j )x∞
j=1
= −ω x∞ , which implies that ω x∞ ≤ −sgn(x )(h (x) − h (0)) ≤ H (x) − H (0)∞ ≤ H (x)∞ + H (0)∞ . Therefore, H (x)∞ → ∞ as x∞ → ∞. From Lemma 1, we know that H (x) is a homeomorphism on R n , which implies system (1) has a unique equilibrium.
Remark 3 Condition (d) in Corollary 1 guarantee the existence of equilibrium point of system (1) when 1-norm is defined; (b) and (e) in Corollary 1 are given based on 2-norm. On the other hand, it is known that p-norm is equivalent to ∞-norm when p → ∞. Correspondingly, in this case, condition (c) in Corollary 1 is reduced to (H5 ), which guarantee the existence of the equilibrium of system (1) when ∞-norm is introduced. Remark 4 When bi j = 0 for all i, j ∈ I in Theorem 1, we obtain the condition of Theorem 3.1 in [32]; when µi = 1 and L i = Ni for any i ∈ I in (d) and (H5 ), the conditions of Theorems 2.1 and 2.2 in [18] are obtained, respectively.
4 Exponential Stability of Equilibrium Point In this section, we will establish some sufficient conditions ensuring the globally exponential stability of the equilibrium point x ∗ of system (1) under impulsive control roles (4). It follows from Sect. 3 that Theorems 1 and 2 ensure the existence and uniqueness of the equilibrium point x ∗ of system (1) based on p-norm and ∞-norm, respectively. Under those conditions, in order to stabilize x ∗ , the appropriate control functions pik (x) can be chosen, which satisfy the following condition: (H24 ) pik (x1∗ , x2∗ , . . . , xn∗ ) = xi∗ , i ∈ I , k ∈ Z + .
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Globally Exponential Stability for Delayed Neural Networks
115
Remark 5 Evidently, there always exist the impulsive control functions pik (x) satisfying (H42 ) in applications. For example, the following controllers may be chosen: pik (x(tk )) = xi (tk ) − rik (xi (tk ) − xi∗ ), i ∈ I and k ∈ Z + ,
(4 )
where 0 < rik < 2, i.e., x(tk ) = −rik (xi (tk ) − xi∗ ), which has been given in [2,22,23,31] and satisfy (H42 ). In addition, under the assumption (H42 ), if a constant vector x ∗ is an equilibrium point of system (1), then x ∗ is also an equilibrium of system (5). In order to obtain our main results, the following assumptions are necessary. (H6 ) λi > σηii for any i ∈ I . (H7 ) There exists a constant α ≥ 0 such that
where ρk = maxi∈I {1, nj=1 of the following equation:
ln ρk ≤ α < ε∗ , k ∈ Z + , tk − tk−1 µj µi
p kji }, ε ∗ = mini∈I {εi∗ }, εi∗ is the unique positive solution
λi − εi −
ηi εi τi e = 0, i ∈ I . σi
The following results are established to ensure the globally exponential stability of the equilibrium of system (1) under impulsive controllers (4) based on p-norm. Theorem 3 Assume that (H1 )–(H4 ) and (H7 ) hold, the equilibrium point x ∗ of system (1) is globally exponentially stable under the impulsive controllers (4) if one of the following two conditions is also satisfied: (1) When 0 < σi ≤ 1, (H6 ) holds, that is, λi > σηii for any i ∈ I ; (2) When σi ≥ 1, (H5 ) holds, that is, λi > ηi for any i ∈ I . Proof If 0 < σi ≤ 1, it follows from condition (1) that λi > σηii , which implies λi > ηi . Then, from Theorem 1, system (1) exists a unique equilibrium point x ∗ . On the other hand, if σi ≥ 1, from condition (2), we have λi > ηi , which guarantee the existence and uniqueness of the equilibrium point x ∗ of system (1) and lead to the conclusion that λi > σηii . Hence, regardless of cases, we always have λi > σηii . Let ηi G i (εi ) = λi − εi − eεi τi , σi where εi ≥ 0 for i ∈ I . It is easy that τi ηi εi τi e < 0, σi ηi G i (0) = λi − > 0, i ∈ I . σi
G i (εi ) = −1 −
On the other hand, G i (εi ) is continuous on [0, ∞) and G i (εi ) → −∞ as εi → ∞, then there exists a positive number εi∗ such that G i (εi∗ ) ≥ 0 and G i (εi ) > 0 for εi ∈ (0, εi∗ ). Denoting ε ∗ = mini∈I {εi∗ }, then G i (ε ∗ ) = λi − ε ∗ −
ηi ε∗ τi e ≥ 0, i ∈ I . σi
123
116
C. Hu et al.
Transform x ∗ to the origin by using the transformation yi (t) = xi (t) − x ∗ for i ∈ I . Then system (5) can be rewritten as the following from: ⎧ n n ⎨ y˙ (t) = −c˜ (y (t)) + a f˜ (y (t)) + b g˜ (y (t − τ (t))) t = t , i i i ij j j ij j j j k (10) j=1 j=1 ⎩ + + yi (tk ) = p˜ ik (y(tk )) i ∈ I , k ∈ Z , where c˜i (yi (t)) = ci (yi (t)+xi∗ )−ci (xi∗ ), f˜j (y j (t)) = f j (y j (t)+x ∗j )− f j (x ∗j ), g˜ j (y j (t)) = g j (y j (t) + x ∗j ) − g j (x ∗j ) and p˜ ik (y(tk )) = pik (y(tk ) + x ∗ ) − pik (x ∗ ). From (H1 ), (H2 ) and (10), when t = tk , we have D + |yi (t)| ≤ −ci |yi (t)| +
n
|ai j |L j |y j (t)| +
n
j=1
|bi j |N j |y j (t − τ j (t))|.
(11)
j=1
Now we define: ∗
u i (t) = eε (t−t0 ) µi |yi (t)| p for i ∈ I , n n ∗ u i (t) = eε (t−t0 ) µi |yi (t)| p . u(t) = i=1
(12) (13)
i=1
For t = tk , k ∈ Z + , calculating the upper right derivative of u(t) along the solution of system (10), from (11) and (13), we obtain D + u(t) =
n ∗ ε ∗ u i (t) + pµi eε (t−t0 ) |yi (t)| p−1 D + |yi (t)| i=1
≤
⎧ n ⎨ i=1
⎩
⎡ ε ∗ u i (t) + µi eε
∗ (t−t ) 0
⎣− pci |yi (t)| p +
n
p|ai j |L j |yi (t)| p−1 |y j (t)|
j=1
⎤⎫ n ⎬ p|bi j |N j |yi (t)| p−1 |y j (t − τ j (t))|⎦ + ⎭ j=1 ⎧ ⎡ p−1 n ⎨ n ∗ pβ ≤ |ai j | pα,i j L j ,i j |yi (t)| p ε ∗ u i (t) + µi eε (t−t0 ) ⎣− pci |yi (t)| p + ⎩ j=1 =1
i=1
+
n
pβ p,i j
|ai j | pα p,i j L j
|y j (t)| p +
n
|bi j | pγ p,i j
pδ N j p,i j |y j (t
j=1
⎧⎡
≤
n ⎨ i=1
⎩
pδ,i j
|bi j | pγ,i j N j
|yi (t)| p
j=1 =1
j=1
+
p−1 n
⎣ε − pci + ∗
p−1 n
⎤⎫ ⎬ − τ j (t))| p ⎦ ⎭ ⎤
(|ai j |
pα,i j
pβ L j ,i j
j=1 =1
+ |bi j |
pγ,i j
pδ N j ,i j )⎦
u i (t)
⎫ n n ⎬ µi µ ∗ pβ pδ i |ai j | pα p,i j L j p,i j u j (t) + |bi j | pγ p,i j N j p,i j eε τ j u j (t − τ j (t)) + ⎭ µj µj j=1
123
j=1
Globally Exponential Stability for Delayed Neural Networks
=
117
n ∗ (ε ∗ − λi )u i (t) + ηi eε τi u i (t − τi (t)) . i=1
Now let us define
V (t) = u(t) +
n
ηi e
t
ε∗ τi
i=1
t−τi (t)
u i (s) 1 − τ˙i (ψi−1 (s))
ds
(14)
for t ≥ t0 , where ψi−1 (s) is the inverse function of ψi (s) = s − τi (s). Calculating the upper right derivatives of V (t) along the solutions of system (10), we obtain D + V (t) ≤
n ∗ (ε ∗ − λi )u i (t) + ηi eε τi u i (t − τi (t)) i=1
+ =−
n
ηi e
ε∗ τi
i=1 n
u i (t) − u i (t − τi (t)) σi
(λi − ε ∗ −
i=1
ηi ε∗ τi e )u i (t) σi
≤0 for t ≥ t0 and t = tk , k ∈ Z + . It follows that + ) for t ∈ (tk−1 , tk ], k ∈ Z + , u(t) ≤ V (t) ≤ V (tk−1
(15)
where V (t0+ ) = V (t0 ). On the other hand, from (10), (H41 ) and (H7 ), we have u(tk+ ) = eε
∗ (t −t ) k 0
n
µi |yi (tk+ )| p
i=1
≤e
ε∗ (tk −t0 )
n i=1
= eε
∗ (t −t ) k 0
µi
n
pikj |y j (tk )| p
j=1
n n µj
p k µi |yi (tk )| p µi ji i=1 j=1 ⎧ ⎫ n n ⎨ µj k ⎬ ∗ ≤ max p ji µi eε (tk −t0 ) |yi (tk )| p ⎭ 1≤i≤n ⎩ µi j=1
i=1
≤ ρk u(tk )
123
118
C. Hu et al.
for k ∈ Z + , this together with (14), we have V (tk+ ) = u(tk+ ) +
n
+
ηi e
i=1
≤ ρk u(tk ) + ρk
tk
ε∗ τi
u i (s)
tk+ −τi (tk+ ) n
ηi e
tk
ε∗ τi
i=1
1 − τ˙i (ψi−1 (s))
tk −τi (tk )
ds
u i (s) 1 − τ˙i (ψi−1 (s))
ds
= ρk V (tk ).
(16)
From (15) and (16), we have + ) ≤ ρk−1 V (tk−1 ) ≤ ρ0 ρ1 . . . ρk−1 V (t0 ) u(t) ≤ V (t) ≤ V (tk−1
(17)
for t ∈ (tk−1 , tk ], k ∈ Z + , where ρ0 = 1. In view of (H7 ), we obtain ρk ≤ eα(tk −tk−1 ) , k ∈ Z + . It follows from (17) that u(t) ≤ eα(t1 −t0 ) · · · eα(tk−1 −tk−2 ) V (t0 ) ≤ eα(t−t0 ) V (t0 ) for t ∈ (tk−1 , tk ], k ∈ Z + . It follows that n
1 −(ε∗ −α)(t−t0 ) e V (t0 ) µ ⎡ ⎤ t0 n 1 −(ε∗ −α)(t−t0 ) ⎣ ηi ε∗ τi ≤ e u(t0 ) + max{ e } u i (s)ds ⎦ i∈I σi µ
|yi (t)| p ≤
i=1
t0 −τ i=1
n 1 ηi τ ε∗ τi ∗ ≤ e−(ε −α)(t−t0 ) 1 + max{ e } sup u i (s) i∈I σi µ t0 −τ ≤s≤t0 i=1
≤ Me−(ε
∗ −α)(t−t ) 0
for t ≥ t0 , where M=
sup
n
t0 −τ ≤s≤t0
|yi (s)| p
i=1
µ ηi τ ε∗ τi e } ≥ 1, 1 + max{ i∈I σi µ
(18)
which means that 1
x(t) − x ∗ p ≤ M p φ − x ∗ p e
−ε
∗ −α p (t−t0 )
for t ≥ t0 .
Therefore, the equilibrium point of system (1) is globally exponentially stable under the ∗ impulsive controllers (4) and the exponential convergence rate is ε −α p . the proof of Theorem 3 is completed.
Similar to Corollary 1, the following results are directly gained from Theorem 3.
123
Globally Exponential Stability for Delayed Neural Networks
119
Corollary 2 Under the assumptions (H1 )–(H4 ) and (H7 ), system (1) exists a unique equilibrium point x ∗ and x ∗ is globally exponentially stable under the impulsive controllers (4) if 0 < σi ≤ 1 and one of the following conditions is also satisfied: (a) For p > 1, the following inequality holds: pci − ( p − 1)
n
|ai j |
p p−1 αi j
p p−1 βi j
Lj
+ |bi j |
p p−1 γi j
p p−1 δi j
!
Nj
j=1
−
n µj j=1
µi
p(1−β ji )
|a ji | p(1−α ji ) L i
>
n 1 µj p(1−δ ji ) |b ji | p(1−γ ji ) Ni ; σi µi j=1
(b) 2ci −
n
2βi j
|ai j |2αi j L j
2δi j
+ |bi j |2γi j N j
−
j=1
>
1 σi
n µj j=1
n j=1
µi
2(1−β ji )
|a ji |2(1−α ji ) L i
µj 2(1−δ ji ) |b ji |2(1−γ ji ) Ni ; µi
(c) pci − ( p − 1)
n " j=1
n n # µj 1 µj |ai j |L j + |bi j |N j − |a ji |L i > |b ji |Ni ; µi σi µi j=1
j=1
(d) ci −
n µj j=1
µi
|a ji |L i >
n 1 µj |b ji |Ni ; σi µi j=1
(e) 2ci −
n
|ai j |L j −
j=1
n
|bi j |N j −
j=1
n µj j=1
µi
|a ji |L i >
n 1 µj |b ji |Ni , σi µi j=1
for i ∈ I , where µi be positive constant numbers, real numbers αi j , βi j , γi j and δi j satisfy 0 ≤ αi j ≤ 1, 0 ≤ βi j ≤ 1, 0 ≤ γi j ≤ 1 and 0 ≤ δi j ≤ 1. Remark 6 If 0 < σi ≤ 1, it follows from (a) in Corollary 2 that pci − ( p − 1)
n
p
|ai j | p−1
αi j
p
L jp−1
βi j
p
+ |bi j | p−1
γi j
p
N jp−1
δi j
!
j=1
−
n j=1
µj µj p(1−β ji ) p(1−δ ji ) |a ji | p(1−α ji ) L i > |b ji | p(1−γ ji ) Ni , µi µi n
j=1
123
120
C. Hu et al.
this is less conservative than the following inequality: ⎛ ! n p p p p βi j δi j α γ min ⎝ pci − ( p − 1) |ai j | p−1 i j L jp−1 + |bi j | p−1 i j N jp−1 1≤i≤n
−
j=1
n µj j=1
µi
⎞
p(1−β ji ) ⎠ |a ji | p(1−α ji ) L i
⎛ ⎞ n µj p(1−δ ) ji ⎠ > max ⎝ , |b ji | p(1−γ ji ) Ni 1≤i≤n µi j=1
which has been required in Theorem 2 in [1] and in Theorem in [26]. Remark 7 If the time-varying delays τ j (t) are reduced to constant delays τ j , it is evident that σ j = 1 for all j ∈ I , which implies that (H3 ) holds. In addition, condition (H6 ) is equivalent to (H5 ) in this case. If the time-varying delays τ j (t) reduce to constant delays τ j for all j ∈ I , the following results are easily obtained from Theorems 1, 3 and Remark 7. Corollary 3 Under assumptions (H1 ), (H2 ) and (H5 ), system (1) exists a unique equilibrium x ∗ . Furthermore, if (H4 ) and (H7 ) also hold, then x ∗ is globally exponentially stable under the impulsive controllers (4). Theorem 1 is established to guarantee the equilibrium point x ∗ of system (1) is globally exponentially stable under the impulsive controllers (4) based on p-norm. The following results are provided to guarantee the globally exponential stability of system (1) under the impulsive controllers (4) when ∞-norm is introduced. Firstly, we introduce the following condition: (H41 ) There exist nonnegative matrixes Pk = ( pikj )n×n such that | pik (u 1 , u 2 , . . . , u n ) − pik (v1 , v2 , . . . , vn )| ≤
n
pikj |u j − v j |
j=1
for all (u 1 , u 2 , . . . , u n )T ∈ R n , (v1 , v2 , . . . , vn )T ∈ R n , i ∈ I and k ∈ Z + .
Theorem 4 Under the assumptions (H1 ), (H2 ), (H41 ), (H42 ) and (H5 ), the equilibrium point of system (1) is globally exponentially stable under the impulsive controllers (4) if the following condition is also satisfied: (H7 ) There exists a constant α ≥ 0 such that ln ρk ≤ α < ε∗ , k ∈ Z + , tk − tk−1
where ρk = maxi∈I {1, nj=1 pikj }, ε ∗ = mini∈I {εi∗ } and εi∗ is the unique positive root of the equation εi = λi − ηi eεi τ . Proof Firstly, the existence of the equilibrium point x ∗ of system (1) can be obtained from Theorem 2. Without loss of generality, we denote that λi = ci −
n j=1
123
|ai j |L j , ηi =
n j=1
|bi j |N j for i ∈ I .
Globally Exponential Stability for Delayed Neural Networks
121
Then, from (H5 ), we have λi > ηi for i ∈ I . ∗
Similar to Theorem 3, we know that there exists εi∗ ∈ (0, +∞) such that εi∗ = λi − ηi eεi τ and εi − λi + ηi eεi τ < 0 on εi ∈ [0, εi∗ ) for any i ∈ I . Denoting ε ∗ = mini∈I {εi∗ }, then ε ∗ − λi + ηi eε
∗τ
≤ 0, i ∈ I ,
which implies that ε ∗ − λi + ηi < 0, i ∈ I . For t = tk , from (11), we have D + |yi (t)| ≤ −ci |yi (t)| +
n
|ai j |L j |y j (t)| +
j=1
n
|bi j |N j |y j (t − τ j (t))|.
j=1
Let M=
sup
{max |y j (s)|},
(19)
t0 −τ ≤s≤t0 j∈I
and S(t) = Me−ε
∗ (t−t ) 0
for t ≥ t0 − τ.
(20)
It is evident that |yi (t)| ≤ M = S(t)eε
∗ (t−t ) 0
≤ (t) < hS(t)
(21)
for t ∈ [t0 − τ, t0 ] and i ∈ I , where h > 1 is a real number. In the following, we will prove that |yi (t)| < hS(t) for t ∈ (t0 , t1 ], i ∈ I . Otherwise, there exist some l ∈ I and
t∗
(22)
∈ (t0 , t1 ] such that
|yl (t ∗ )| = hS(t ∗ )
(23)
|yi (t ∗ )| ≤ hS(t ∗ ) and |yi (t)| < hS(t) for t ∈ (t0 , t ∗ ).
(24)
and for any i ∈ I , In view of (21) and (24), for any i ∈ I , |yi (t)| < hS(t) for t ∈ [t0 − τ, t ∗ ).
(25)
If τ j (t ∗ ) = 0 for any j ∈ I , then D + (|yl (t ∗ )| − hS(t ∗ )) ≤ −cl |yl (t ∗ )| +
n
|al j |L j |y j (t ∗ )|
j=1
+
n
|bl j |N j |y j (t ∗ )| + ε ∗ hS(t ∗ )
j=1
≤ (−λl + ηl )hS(t ∗ ) + ε ∗ hS(t ∗ ) < −ε ∗ hS(t ∗ ) + ε ∗ hS(t ∗ ) = 0.
123
122
C. Hu et al.
If there exist some r ∈ I such that τr (t ∗ ) > 0, then from (25), we have |yr (t ∗ − τr (t ∗ ))| < hS(t ∗ − τr (t ∗ )), this together with (24), we have D + (|yl (t ∗ )| − hS(t ∗ )) ≤ −cl |yl (t ∗ )| +
n
|al j |L j |y j (t ∗ )|
j=1
+
n
|bl j |N j |y j (t ∗ − τ j (t ∗ ))| + ε ∗ hS(t ∗ )
j=1
< −cl |yl (t ∗ )| +
n
|al j |L j |y j (t ∗ )|
j=1
+
n
|bl j |N j hS(t ∗ − τ j (t ∗ )) + ε ∗ hS(t ∗ )
j=1 ∗
≤ (−λl + ηl eε τ )hS(t ∗ ) + ε ∗ hS(t ∗ ) ≤ −ε ∗ hS(t ∗ ) + ε ∗ hS(t ∗ ) = 0. From what have been discussed above, we obtain D + (|yl (t ∗ )| − hS(t ∗ )) < 0. Then there exists θ > 0 such that D + (|yl (t)| − hS(t)) < 0 for t ∈ (t ∗ − θ, t ∗ ). Integrating above inequality from t to t ∗ , we have |yl (t)| > hS(t) for t ∈ (t ∗ − θ, t ∗ ), which leads to a contradiction with (25), then the inequality (22) holds. Moreover, from (H41 ) and the definition of ρ1 , we have |yi (t1+ )| =
n j=1
pi1j |y j (t1 )| <
n
pi1j hS(t1 ) ≤ ρ1 hS(t1 ).
j=1
In the following, we will use the mathematical induction to prove that |yi (t)| < ρ0 ρ1 . . . ρk−1 hS(t) for t ∈ (tk−1 , tk ], k ∈ Z + ,
(26)
where ρ0 = 1. As for k = 1, from (22), we know that inequality (26) holds. Assume that inequality (26) holds for all k = 1, 2, . . . , m, that is |yi (t)| < ρ0 ρ1 . . . ρk−1 hS(t) for t ∈ (tk−1 , tk ], k = 1, 2, . . . , m. It follows that |yi (tm+ )| =
n j=1
123
pimj |y j (tm )| < ρ0 ρ1 . . . ρm−1 ρm hS(tm ).
(27)
Globally Exponential Stability for Delayed Neural Networks
123
Similar to (22), we have |yi (t)| < ρ0 ρ1 . . . ρm−1 ρm hS(t) for t ∈ (tm , tm+1 ], i ∈ I . Then by the mathematical induction, we see that (26) holds. Let h → 1, we have |yi (t)| ≤ ρ0 ρ1 . . . ρk−1 S(t) for t ∈ (tk−1 , tk ], k ∈ Z + . In view of
(H7 ),
(28)
we obtain ρk ≤ eα(tk −tk−1 ) , k ∈ Z + .
It follows from (28) that |yi (t)| ≤ eα(t1 −t0 ) eα(t2 −t1 ) · · · eα(tk−1 −tk−2 ) Me−ε ≤ Me−(ε
∗ (t−t ) 0
∗ −α)(t−t ) 0
for t ∈ (tk−1 , tk ], k ∈ Z + , i ∈ I , which implies that x(t) − x ∗ ∞ ≤
{max |y j (s)|}e−ε
sup
∗ (t−t ) 0
t0 −τ ≤s≤t0 j∈I
= φ − x ∗ ∞ e−ε
∗ (t−t ) 0
for t ≥ t0 . The proof is completed.
Remark 8 If impulsive controllers (4) reduce to (4 ), it is evident that condition (H41 ) (or (H41 )) and (H42 ) are satisfied and Pk = E for i ∈ I and k ∈ Z + , where E denotes the identity matrix. Furthermore, it is easy to see that ρk = 1 for k ∈ Z + in Theorems 3 and 4 when µi = 1 for all i ∈ I . Choosing α = 0 in H7 (or H7 ), then H7 (or H7 ) holds in this case. Hence, under the impulsive controllers (4 ), both H4 and H7 (or H7 ) hold. From Theorems 3, 4 and Remark 8, we have the following results. Corollary 4 System (1) exists a unique equilibrium point x ∗ and x ∗ is globally exponentially stable under the impulsive controllers (4 ) if one of the following statements hold: (1) When 0 < σi ≤ 1, H1 − H3 and H6 hold; (2) When σi ≥ 1, H1 − H3 and H5 hold; (3) The assumptions H1 , H2 and H5 hold. Remark 9 In Corollary 4, taking µi = 1, gi (·) = f i (·), 0 < σi ≤ 1 and α,i j = β,i j = γ,i j = δ,i j = 1p for = 1, 2, . . . , p and i, j ∈ I in (H6 ). In this case, for p = 1, it follows from (H6 ) that ci −
n j=1
|a ji |L i >
n
|b ji |L i .
(29)
j=1
Evidently, (29) is less weak than the following inequality: ⎛ ⎞ ⎛ ⎞ n n min ⎝ci − |a ji |L i ⎠ > max ⎝ |b ji |L i ⎠ , 1≤i≤n
j=1
1≤i≤n
j=1
which was asked in Theorem 3.1 in [2]. Similarly, for p = 2, from (H6 ), we obtain 2ci −
n n (|ai j |L j + |bi j |L j + |a ji |L i ) > |b ji |L i , j=1
(30)
j=1
123
124
C. Hu et al.
this is less weak than ⎛
⎛ ⎞ ⎞ n n min ⎝2ci − (|ai j |L j + |bi j |L j + |a ji |L i )⎠ > max ⎝ |b ji |L i ⎠ ,
1≤i≤n
1≤i≤n
j=1
j=1
which was required in Theorem 3.2 in [2].
5 An Example In this section, we will give an example to show the conditions given in the previous sections are less weak than those given in some earlier literatures, such as [2,28]. Consider the following neural networks with variable delays: x˙i (t) = −ci xi (t) +
2
ai j f j (x j (t)) +
j=1
2
bi j g j (x j (t − τ j (t))) + Ii , i = 1, 2, (31)
j=1
where f j (x) = g j (x) = 21 (|x +1|−|x −1|), and c1 = c2 = 2, a11 = 1, a12 = −0.2, a21 = 0.6, a22 = −0.4, b11 = −0.2, b12 = 0.2, b21 = −0.1, b22 = 0.8, I1 = 2, I2 = 4, τ1 (t) = τ2 (t) = e−t . In the following, we introduce the following impulsive controllers: x1 (tk+ ) = 0.098 cos (x1 (tk ) − 1.4) − 0.04x2 (tk ) + 1.4, k ∈ Z + ,
x2 (tk+ ) = −0.05x1 (tk ) + 0.06 sin (x2 (tk ) − 2.45) + 2.52, k ∈ Z + .
(32)
We can verify that (1.4, 2.45)T is an equilibrium of system (31) and (H1 )–(H4 ) are satisfied, and L 1 = L 2 = N1 = N2 = 1, σ1 = σ2 = 1. Choosing µi = 1 for any i ∈ I , for convenience, and we only consider the following three cases: p = 1, p = 2 and p = ∞. For p = 1, we have c1 −
2
|a j1 |L 1 = 0.4 >
j=1
c2 −
2
2
|b j1 |L 1 = 0.3,
j=1
|a j2 |L 2 = 1.2 >
j=1
2
|b j2 |L 2 = 1,
j=1
and Pk =
0.098 0.05
0.04 0.06
! .
For p = 2, we obtain 2c1 −
2 2 (|a1 j |L j + |b1 j |L j + |a j1 |L 1 ) = 0.8 > |b j1 |L 1 = 0.3, j=1
2c2 −
123
j=1
2
2
j=1
j=1
(|a2 j |L j + |b2 j |L j + |a j2 |L 2 ) = 1.7 >
|b j2 |L 2 = 1,
Globally Exponential Stability for Delayed Neural Networks
125
and Pk =
0.392 0.2
0.16 0.24
! .
Then, (H6 ) holds. For p = ∞, we get c1 −
2
|a1 j |L j = 0.8 >
j=1
c2 −
2
2
|b1 j |N j = 0.3,
j=1
|a2 j |L j = 1 >
j=1
2
|b2 j |N j = 0.9,
j=1
and Pk =
0.098 0.05
0.04 0.06
! .
Therefore, (H5 ) holds. On the other hand, from Theorem 3, we have ⎧ ⎫ 2 ⎨ ⎬ p kji = 1, k ∈ Z + , ρk = max 1, ⎭ 1≤i≤2 ⎩ j=1
for p = 1 and p = 2, and it follows from Theorem 4 that ⎧ ⎫ 2 ⎨ ⎬ pikj = 1 ρk = max 1, ⎭ 1≤i≤2 ⎩ j=1
for p = ∞. Therefore, we can take α = 0 and tk − tk−1 = 0.08 in Theorems 3 and 4 for all k ∈ Z + . In this case, from the positivity of ε ∗ , we know that (H7 ) and (H7 ) hold, respectively. Then, all conditions of Theorems 3 and 4 in this paper are satisfied. So (1.4, 2.45)T is globally exponentially stable under impulsive controllers (32). Under above conditions, The simulation results are shown in Fig. 1. From above discussion, it is easy to see that ⎛ ⎞ ⎛ ⎞ 2 2 min ⎝ci − |a ji |L i ⎠ = 0.4 < max ⎝ |b ji |L i ⎠ = 1 1≤i≤2
and
j=1
1≤i≤n
j=1
⎛ ⎞ ⎞ 2 2 min ⎝2ci − (|ai j |L j + |bi j |L j + |a ji |L i )⎠ = 0.8 < max ⎝ |b ji |L i ⎠ = 1,
1≤i≤2
⎛
j=1
1≤i≤2
j=1
which implies that the conditions in [2,28] do not hold for this example. So, our results are less weak than some previous results.
123
126
C. Hu et al. 3.5
x1 x2
3 2.5
x
2 1.5 1 0.5 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t Fig. 1 Stability for system (31) under impulsive controllers (32)
6 Conclusion In this paper, the existence, uniqueness of the equilibrium point of a class of neural network with time-varying delays and without impulsive control are first discussed. And then, by adding appropriate impulsive control input, the globally exponential stability of the equilibrium point is ensured. Those results are derived based on p-norm and ∞-norm, respectively, which extend many previous results, such as [1,2,18,26,32]. In addition, the impulsive controllers in this paper can be nonlinear and even dependent of the states of all neurons, which remove the restriction that the impulsive functions are linear. Finally, the effectiveness and feasibility of the developed methods have been shown by numerical simulation. Acknowledgments This work was supported by the National Natural Science Foundation of P.R. China (60764003), the Major Project of The Ministry of Education of P.R. China (207130) and the Funded by Scientific Research Program of the Higher Education Institution of Xinjiang (XJEDU2007G01 and XJEDU2006I05).
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