Global asymptotic robust stability of static neural network models with S-type distributed delays

Mathematical and Computer Modelling 44 (2006) 218–222 www.elsevier.com/locate/mcm

Global asymptotic robust stability of static neural network models with S-type distributed delays Min Wang ∗ , Linshan Wang Department of Mathematics, Ocean University of China, Qingdao, Shandong, 266000, PR China Received 10 August 2005; received in revised form 5 January 2006; accepted 9 January 2006

Abstract This paper investigates the global asymptotic robust stability of static neural network models with S-type distributed delays on a finite interval. We present a theorem and a corollary which generalize the results of related literature. c 2006 Elsevier Ltd. All rights reserved.

Keywords: Static neural network model; S-type distributed delays; Robust stability; Lebesgue–Stieltjes integration; Lyapunov functional

1. Introduction On the basis of the difference of basic variables (local field states or neuron states), the mathematical models of recurrent neural networks can be divided into two types—local field neural network models and static neural network models (see Xu et al. [1]). The basic form of the local field model is τ

n X dxi (t) = −xi (t) + wi j g j (x j (t)) + Ii , dt j=1

i = 1, . . . , n,

(1)

where n denotes the number of neurons; wi j is the value of the synaptic connectivity from neuron j to i; gi (·) is the nonlinear activation function of neuron i; Ii is the external input imposed on neuron i [1,2]. With the same notation, the static model can be written as ! n X dyi (t) = −yi (t) + gi wi j y j (t) + Ii , i = 1, . . . , n. (2) τ dt j=1 Local field neural network models have attracted a great deal of attention. Many scholars studied models with and without time delays (see [3–7]). In contrast, little attention has been paid to the static models. In [2], the authors presented a reference model approach for investigating the stability of system (2), but they did not consider the influence of time delays. In [8], the authors investigated the global asymptotic stability of static models with S-type ∗ Corresponding author. Tel.: +86 532 88832713; fax: +86 532 88832713.

E-mail addresses: wang [email protected] (M. Wang), [email protected] (L. Wang). c 2006 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2006.01.013

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M. Wang, L. Wang / Mathematical and Computer Modelling 44 (2006) 218–222

distributed delays on (−∞, 0]. In this paper, we will study the global asymptotic robust stability of the following model with S-type distributed delays on a finite interval: !  n Z 0   dyi (t) = −a (λ)y (t) + f X y j (t + θ )dwi j (θ, λ) + Ii , i i i (3) dt j=1 −r (λ)   yi (σ + θ ) = φi (θ ), t ≥ σ, θ ∈ [−r (λ), 0], i = 1, . . . , n, where λ ∈ Λ ⊂ R is the parameter, σ ∈ R, φi (θ ) ∈ C[−r (λ), 0], i = 1, . . . , n, wi j (θ, λ), i, j = 1, . . . , n, R0 are nondecreasing bounded variation functions on [−r (λ), 0], −r (λ) y j (t + θ )dwi j (θ, λ), i, j = 1, . . . , n, are Lebesgue–Stieltjes integrable. There exist positive constants r , a i , a¯ i , wi∗j , i, j = 1, . . . , n, such that for any λ ∈ Λ, R0 0 < a i ≤ ai (λ) ≤ a¯ i , 0 ≤ r (λ) ≤ r , | −r (λ) dwi j (θ, λ)| ≤ wi∗j < ∞, i, j = 1, . . . , n. Let W + = (wi∗j )n×n , A = diag(a i ), K = diag(ki ), i = 1, . . . , n, Y = {φ (θ ) | φ (θ ) = (φ1 (θ ), φ2 (θ ), . . . , φn (θ )), φi (θ ) ∈ C[−r (λ), 0], i = 1, . . . , n}. Define kφ kmax = max1≤i≤n {max−r˜(λ)≤θ≤0˜ |φi (θ )|} as the maximum norm. Then Y is a Banach space. ˜ σ ∈ R, a solution of system (3) is a vector function y(t) = y(σ, φ , t) = (y1 (t), . . . , yn (t)) For any φ (θ ) ∈ Y and ˜ ˜ ˜ ˜ satisfying (3) for t ≥ σ . 2. Preliminaries Definition 1. Suppose that y(t) = y(σ, φ , t) is a solution of system (3). The equilibrium y∗ of system (3) is said to be ˜ t (σ, φ ) − y∗ kmax < ε. y∗ is stable if for any ε > 0, σ ∈ ˜R, there˜ is a ˜δ = δ(ε, σ ) such that kφ − y∗ kmax ≤ δ implies ky ∗ ∗ ˜ ˜ ˜ (σ,˜φ )(t)−y∗ k = ˜ 0, said to be attractive if there is a neighborhood O(y ) ⊂ Y such that φ ∈ O(y ) implies lim˜t→∞ ky ∗ ˜ ˜ ˜ ˜ ˜ ˜ where k · k denotes the Euclidean norm. y is said to be asymptotically stable if it is both stable and attractive. y∗ is ˜ said to be globally asymptotically stable if˜ it is asymptotically stable and O(y∗ ) = Y. ˜ Definition 2. System (3) is said to be robust stable or globally asymptotically robust stable if its equilibrium y∗ = (y1∗ , y2∗ , . . . , yn∗ ) is stable or globally asymptotically stable for any r (λ) ∈ [0, r ] and ai (λ) ∈ [a i , a¯ i ], i = 1, . . . ,˜n. First, we will prove two lemmas concerning differentiability of Lebesgue–Stieltjes integration. Lemma 1. If f (t, θ ) is continuous on [a,b;c,d], w(θ ) is a nondecreasing bounded variation function on Rd [c, d], c dw(θ ) = w ∗ < ∞, then Z d g(t) = f (t, θ )dw(θ ) c

is continuous on [a,b]. Proof. f (t, θ ) is continuous on [a, b; c, d], so it is uniformly continuous, i.e., ∀ε ≥ 0, ∃δ ≥ 0 such that when |t1 − t2 | ≤ δ and |θ1 − θ2 | ≤ δ, we have | f (t1 , θ1 ) − f (t2 , θ2 )| ≤ ε. Since −| f (t1 , θ ) − f (t2 , θ )| ≤ f (t1 , θ ) − f (t2 , θ ) ≤ | f (t1 , θ ) − f (t2 , θ )|, Z d Z d Z − | f (t1 , θ ) − f (t2 , θ )|dw(θ ) ≤ ( f (t1 , θ ) − f (t2 , θ ))dw(θ ) ≤ c

c

d

| f (t1 , θ ) − f (t2 , θ )|dw(θ ),

c

Rd ( f (t1 , θ ) − f (t2 , θ ))dw(θ )| ≤ c | f (t1 , θ ) − f (t2 , θ )|dw(θ ) ≤ εw ∗ . Z d Z d |g(t1 ) − g(t2 )| = f (t1 , θ )dw(θ ) − f (t2 , θ )dw(θ ) c c Z d = ( f (t1 , θ ) − f (t2 , θ ))dw(θ) ≤ εw ∗ .

we have |

Rd c

c

Thus g(t) is continuous. Rd c



From Lemma 1, we know that when the conditions of Lemma 1 are satisfied, limt→t ∗ limt→t ∗ f (t, θ )dw(θ ).

Rd c

f (t, θ )dw(θ ) =

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Lemma 2. If f (t, θ ) and f t (t, θ) = dtd f (t, θ ) are continuous on [a,b;c,d], w(θ ) is a nondecreasing bounded Rd variation function on [c, d], c dw(θ ) = w ∗ < ∞, then  Z d Z d d f t (t, θ )dw(θ ). f (t, θ )dw(θ ) = dt c c Rd Proof. Let g(t) = c f (t, θ )dw(θ ). From the mean value theorem, Z d Z d f (t + 1t, θ ) − f (t, θ ) g(t + 1t) − g(t) f t (t + ξ 1t, θ )dw(θ ), ξ ∈ [0, 1]. = dw(θ ) = 1t 1t c c From Lemma 1, Z d d g(t + 1t) − g(t) f t (t + ξ 1t, θ )dw(θ ) g(t) = lim = lim 1t→0 1t→0 c dt 1t Z d Z d = lim f t (t + ξ 1t, θ )dw(θ ) = f t (t, θ )dw(θ ).  c

1t→0

c

3. Global asymptotic robust stability Theorem 3. Assume that (T1) | f i (y1 ) − f i (y2 )| ≤ ki |y1 − y2 |, i = 1, . . . , n, (T2) A − K W + is an M-matrix. Then system (3) is globally asymptotically robust stable. Proof. For any λ ∈ Λ, r (λ) ∈ [0, r ], ai (λ) ∈ [a i , a¯ i ], i = 1, . . . , n, like in the proof in Wang and Gao [5], we can prove the existence of equilibrium of (3) by homotopic invariance, topological degree theory and (T2). Now we will prove the global asymptotic robust stability. From (T2), there exist positive constants βi > 0, i = 1, . . . , n [4,5] such that βi ai −

n X

β j ki wi∗j > 0,

j=1

or βi ai −

n X

β j k j w ∗ji > 0.

j=1

Suppose that y∗ = (y1∗ , . . . , yn∗ ) is an equilibrium of system (3), ∀φ ∈ Y, y(t) is a solution of (3), and let z = y − y∗ . ˜ ˜ ˜ ˜ ˜ ˜ Define the Lyapunov functional !!  Z 0 Z t n n X X V (z, t) = βi |z i (t)| + ki |z j (s)|ds dwi j (θ, λ) . −r (λ) t+θ ˜ i=1 j=1 Since 0 ≤ r (λ) ≤ r , let wi j (θ, λ) = wi j (−r (λ), λ) when θ ∈ [−r, −r (λ)), we have that wi j (θ, λ) can be continued R0 as the bounded variation function on [−r, 0] and | −r dwi j (θ, λ)| ≤ wi∗j , i, j = 1, . . . , n. Let u(kz(t)k) = βkz(t)k, !! ˜ ˜  Z 0 Z t n n X X v(kzt kmax ) = β¯ kzt kmax + ki kzt kmax ds dwi j (θ, λ) , −r t+θ ˜ ˜ ˜ i=1 j=1 where β = min1≤i≤n {βi } > 0, β¯ = max1≤i≤n {βi } > 0, kz(t)k is the Euclidean norm, kzt kmax is the maximum norm. ˜ = 0. ˜ Obviously, u(kz(t)k) ≤ V (z, t) ≤ v(kzt kmax ), u(0) = v(0) ˜ ˜ ˜

M. Wang, L. Wang / Mathematical and Computer Modelling 44 (2006) 218–222

as

221

From Lemma 2, the upper right Dini-derivative of V (z, t) along the solution of system (3) can be calculated ˜ ! n n Z 0 X X DV + ≤ βi −ai (λ)|z i (t)| + f i (z j (t + θ ) + y ∗j )dwi j (θ, λ) + Ii −r (λ) i=1 j=1 ! n Z 0 n Z 0 X X ∗ y j dwi j (θ, λ) + Ii + ki |z j (t)|dwi j (θ, λ) − fi j=1 −r (λ) j=1 −r (λ) ! n Z 0 X − ki |z j (t + θ )|dwi j (θ, λ) j=1 −r (λ)

≤

n X

βi −a i |z i (t)| + ki

i=1 n X

0

j=1 −r (λ) n Z X

|z j (t + θ )|dwi j (θ, λ)

0

!

|z j (t + θ )|dwi j (θ, λ) j=1 −r (λ) n n X n X X −a i βi |z i (t)| + β j k j w ∗ji |z i (t)| i=1 i=1 j=1 + ki

wi∗j |z j (t)| − ki

n Z X

j=1

=

=−

n X

a i βi −

i=1

n X

! β j k j w ∗ji |z i (t)|

j=1

≤ −Lkz(t)k, ˜ P where L = min1≤i≤n {a i βi − nj=1 β j k j w ∗ji } > 0. Obviously, Lkz(t)k > 0 when kz(t)k > 0. So the equilibrium y∗ ˜ Because of the˜ arbitrariness of φ , y∗ is globally ˜ of system (3) is asymptotically stable (see [9, Theorem 5.2.1, p. 105]). ˜ ˜ asymptotically stable. Thus, system (3) is globally asymptotically robust stable.  From the sufficient conditions for the M-matrix [4,5], we have: Corollary 4. Assume that system (3) satisfies (T1) and one of the following: (T3) A − K W + is a matrix with strictly diagonal dominance of column (or row), (T4) a i − ki wii∗ >

Pn

j=1, j6=i

Pn

(T5) max1≤i≤n (T6)

∗ j=1 w ji k j

ai ∗ ∗ j=1 (wi j ki +w ji k j )

Pn

2a i

βi

β j ki wi∗j

, βi > 0, i = 1, . . . , n,

< 1,

< 1, i = 1, . . . , n.

Then system (3) is globally asymptotically robust stable.

4. Remark and conclusion

System (3) includes many models as special cases. For example, when  1 1 wi j , θ = 0 wi j (θ, λ) = , r (λ) = r, ai (λ) = , f i (·) = gi (·), 0, −r ≤ θ < 0 τ τ system (3) is model (2). When

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M. Wang, L. Wang / Mathematical and Computer Modelling 44 (2006) 218–222

X m (k)   wi j (λ),     k=0   m X   (k)   wi j (λ),   k=1 m wi j (θ, λ) = X (k)  wi j (λ),      k=2  ···    (m)    wi j (λ), 0,

θ = r0 = 0 r1 ≤ θ < 0 r2 ≤ θ < r1

,

i, j = 1, . . . , n,

··· rm ≤ θ < rm−1 −r ≤ θ < rm

system (3) becomes a static neural network model with discrete time delays: ! m X n X dyi (t) (k) i = 1, . . . , n. = −ai (λ)yi (t) + f i wi j (λ)y j (t − rk ) + Ii , dt k=0 j=1

(4)

When wi j (θ, λ) ∈ C 1 [−r, 0], i, j = 1, . . . , n, the value of the synaptic connectivity from neuron j to i is a continuous function on [−r, 0], which means that time delays influence the network continuously, system (3) belongs to the static model with continuous time delays: ! n Z 0 X dyi (t) i = 1, . . . , n. (5) y j (t + θ )wi0 j (θ, λ)dθ + Ii , = −ai (λ)yi (t) + f i dt j=1 −r As many neural networks are described by static models (see [2]), system (3) is widely representative. The global asymptotic robust stability of neural networks can be widely applied in solving control and optimization problems, and the M-matrix is easy to verify, so the results in this paper are significant and practical in both theory and application. References [1] Z. Xu, H. Qiao, J. Peng, B. Zhang, A comparative study of two modeling approaches in neural networks, Neural Networks 17 (2004) 73–85. [2] H. Qiao, J. Peng, Z. Xu, B. Zhang, A reference model approach to stability analysis of neural networks, IEEE Transactions on Systems, Man, and Cybernetics 33 (6) (2003) 925–936. [3] J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences of the United States of America 81 (10) (1984) 3088–3092. [4] L. Wang, D. Xu, Global asymptotic stability of bidirectional associative memory neural networks with S-type distributed delays, International Journal of Systems Science 33 (11) (2002) 869–877. [5] L. Wang, Y. Gao, On global robust stability for interval Hopfield neural networks with time delay, Annals of Differential Equations 19 (3) (2003) 421–426. [6] H. Zhao, Global exponential stability and periodicity of cellular neural networks with variable delays, Physics Letters A 336 (2005) 331–341. [7] H. Zhao, G. Wang, Existence of periodic oscillatory solution of reaction–diffusion neural networks with delays, Physics Letters A 343 (2005) 372–383. [8] M. Wang, L. Wang, Global asymptotic stability of static neural network models with S-type distributed delays, Journal of Shandong University 40 (Suppl.) (2005) 5–8. [9] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York Inc., 1977.