Global asymptotic stability of hybrid bidirectional associative memory neural networks with time delays

Physics Letters A 351 (2006) 85–91 www.elsevier.com/locate/pla

Global asymptotic stability of hybrid bidirectional associative memory neural networks with time delays Sabri Arik Istanbul University, Department of Computer Engineering, 34320 Avcilar, Istanbul, Turkey Received 14 October 2005; accepted 19 October 2005 Available online 26 October 2005 Communicated by A.R. Bishop

Abstract This Letter presents a sufficient condition for the existence, uniqueness and global asymptotic stability of the equilibrium point for bidirectional associative memory (BAM) neural networks with distributed time delays. The results impose constraint conditions on the network parameters of neural system independently of the delay parameter, and they are applicable to all bounded continuous non-monotonic neuron activation functions. The results are also compared with the previous results derived in the literature. © 2005 Elsevier B.V. All rights reserved.

1. Introduction In recent years, neural networks have been widely used in various applications such as designing associative memories and solving optimisation problems. In the case where the neural network is designed to solve optimisation problems, it is required that the neural network must possess a unique equilibrium point that is globally asymptotically stable. Therefore, it is of great interest to establish conditions that ensure this type of stability for neural networks. In the literature, many research papers have focused on the equilibria and stability properties of neural networks, presenting various sufficient conditions for the uniqueness and global asymptotic stability of the equilibrium point of different classes of neural networks [1–5]. On the other hand, the delayed version of neural networks have also proved to be important for solving some classes of motion-related optimisation problems. Some results concerning the dynamical behavior of neural networks with delay have been published in [6–15]. Particularly, the stability of the class of bidirectional associative memory (BAM) neural networks with time delays has been given much attention. Some useful results on the uniqueness and global asymptotic stability of the equilibrium point for BAM neural networks with delays can be found in [16–31]. In this Letter, we will present a new sufficient condition for the uniqueness and global asymptotic stability of the equilibrium point for BAM neural networks with distributed time delays. 2. Neural network model Dynamical behavior of a hybrid BAM neural network with constant time delays is described by the following set of differential equations [30,31]: u˙ i (t) = −ai ui (t) +

m 

m      τ wij gj zj (t) + wij gj zj (t − τij ) + Ii ,

j =1

j =1

E-mail address: [email protected] (S. Arik). 0375-9601/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.10.059

∀i

86

S. Arik / Physics Letters A 351 (2006) 85–91

z˙ j (t) = −bj zj (t) +

n 

n      vj i gi ui (t) + vjτ i gi ui (t − σj i ) + Jj ,

i=1

∀j

(1)

i=1

τ , v and v τ are synaptic where ai and bj denote the neuron charging time constants and passive decay rates, respectively; wij , wij ji ji connection strengths; gi and gj represent the activation functions of the neurons and the propagational signal functions, respectively; and Ii and Jj are the exogenous inputs. The activation functions are assumed to satisfy the following conditions:

(H1) There exist positive constants αi , i = 1, 2, . . . , n and βj , j = 1, 2, . . . , m such that   gi (ξ1 ) − gi (ξ2 )  αi |ξ1 − ξ2 |, ξ1 = ξ2   gj (ζ1 ) − gj (ζ2 )  βj |ζ1 − ζ2 |, ζ1 = ζ2 for all ξ1 , ξ2 , ζ1 , ζ2 ∈ R. (H2) There exist positive constants Mi , i = 1, 2, . . . , n and Lj , j = 1, 2, . . . , m such that |gi (u)|  Mi and |gj (z)|  Lj for all u, z ∈ R. Note that this assumption requires that the functions be bounded. 3. Existence, uniqueness and stability of the equilibrium point In this section, we present a theorem which states the conditions that guarantee the existence, uniqueness and global asymptotic stability of the equilibrium point of system (1). Under assumption (H2), neural network defined by (1) always has an equilibrium point [1]. Therefore, what needs to prove is the uniqueness and global asymptotic stability of the equilibrium point. In order to simplify the proof of the uniqueness and global asymptotic stability of the equilibrium point, we will shift the equilibrium point of system (1) to the origin. Using the transformation xi (·) = ui (·) − u∗i ,

i = 1, 2, . . . , n

yj (·) = zj (·) − zj∗ ,

j = 1, 2, . . . , m

we can put system (1) into the following form: x˙i (t) = −ai xi (t) +

m 

m      τ wij sj yj (t) + wij sj yj (t − τij ) ,

j =1 n 

y˙j (t) = −bj yj (t) +

  vj i si xi (t) +

i=1

j =1 n 

  vjτ i si xi (t − σj i ) ,

(2)

i=1

where

      si xi (·) = gi xi (·) + u∗i − gi u∗i ,       sj yi (·) = gj yj (·) + zj∗ − gj zj∗ ,

i = 1, 2, . . . , n j = 1, 2, . . . , m.

It can easily be verified that the functions si and sj satisfy the assumptions on gi and gj , i.e., the functions si and sj have the following properties:   si (xi )  αi |xi |, and si (0) = 0, i = 1, 2, . . . , n   sj (yj )  βj |yj |, and sj (0) = 0, j = 1, 2, . . . , m. Note that the equilibrium point of system (1) is unique and globally asymptotically stable, if the origin of system (2) is a unique and globally asymptotically stable equilibrium point. We are now ready to present the following result: Theorem 1. Let A = diag(a1 , . . . , an ), B = diag(b1 , . . . , bm ), α = diag(α1 , . . . , αn ), β = diag(β1 , . . . , βm ), W = (wij )n×m , τ ) τ τ V = (vj i )m×n , W τ = (wij n×m , V = (vj i )m×n . Then, the origin of system (2) is the unique equilibrium point and it is globally asymptotically stable if the network parameters of the system satisfy the following conditions: m   2   n2 αi2 vjτ i > 0, δi = mai2 − 1 − n2 αi2 λM V T B 2 V −

i = 1, 2, . . . , n

j =1 n   τ 2   Ωj = nbj2 − 1 − m2 βj2 μM W T A2 W − m2 βj2 wij > 0,

j = 1, 2, . . . , m

i=1

where λM (V T B 2 V ) and μM (W T A2 W ) are the maximum eigenvalues of the matrices V T B 2 V and W T A2 W , respectively.

S. Arik / Physics Letters A 351 (2006) 85–91

87

∗ )T be the equilibrium point of system (2). Then, we have the following equiProof. Let x ∗ = (x1∗ , . . . , xn∗ )T and y ∗ = (y1∗ , . . . , ym librium equation:

−ai xi∗ + −bj yj∗ +

m 

m      τ wij sj yj∗ + wij sj yj∗ = 0,

j =1 n 

  vj i si xi∗ +

i=1

i = 1, 2, . . . , n

j =1 n 

  vjτ i si xi∗ = 0,

j = 1, 2, . . . , m.

(3)

i=1

Multiplying both sides of the first equation in (3) by 2mai xi∗ and the second equation in (3) by 2nbj yj∗ results in m 

−2mai2 xi∗ 2 +

m      τ ∗ 2mai wij xi∗ sj yj∗ + 2mai wij xi sj yj∗ = 0,

j =1

−2nbj2 yj∗ 2 +

n 

i = 1, 2, . . . , n

j =1



 ∗

2nbj vj i yj∗ si xi +

i=1

n 

  2nbj vjτ i yj∗ si xi∗ = 0,

j = 1, 2, . . . , m

(4)

i=1

implying that −

n 

2mai2 xi∗ 2 +

m n  m       τ ∗ 2mai wij xi∗ sj yj∗ + 2mai wij xi sj yj∗ − 2nbj2 yj∗ 2

i=1 j =1

i=1

+

m n  

n m  

i=1 j =1

  2nbj vj i yj∗ si xi∗ +

j =1 i=1

n m  

j =1

  2nbj vjτ i yj∗ si xi∗ = 0

j =1 i=1

which is equivalent to the following −

n 

mai2 xi∗ 2 +

m n  

i=1 j =1 n m  

i=1

−

m 

nbj2 yj∗ 2 +

j =1

+

m n      2 ∗2   τ ∗ −ai xi + 2mai wij 2mai wij xi∗ sj yj∗ + xi sj yj∗

2

m

i=1 j =1 n 

=−



   τ 2 2 ∗ wij sj yj −

j =1 i=1 m n  

mai2 xi∗ 2 +

n m  

i=1

j =1 i=1

m 

m n  

nbj2 yj∗ 2 +

−

m

   τ 2 2 ∗ wij sj yj +

mai2 xi∗ 2 +

n

2



2   vjτ i si2 xi∗ −

n m  

n m  

 2   n2 vjτ i si2 xi∗

j =1 i=1

  2nbj vj i yj∗ si xi∗

j =1 i=1 m n  

  2mai wij xi∗ sj yj∗

i=1 j =1 n m   

 2 bj yj∗ − nvjτ i si xi∗

j =1 i=1 n m  

i=1

j =1 i=1

m 

m n  

nbj2 yj∗ 2 +

n m   j =1 i=1

 2   n2 vjτ i si2 xi∗ +

 2 τ ai xi∗ − mwij sj yj∗ −

j =1



i=1 j =1

i=1 j =1

−

2

 τ 2 2  ∗  m2 wij sj yj +

m n   

n 

 2 ∗2   −bj yj + 2nbj vjτ i yj∗ si xi∗

i=1 j =1

j =1

−

  2nbj vj i yj∗ si xi∗ +

j =1 i=1

m n  

−

i=1 j =1 n m  

n m   2      n2 vjτ i si2 xi∗ + 2nbj vj i yj∗ si xi∗ j =1 i=1 m n  

 τ 2 2  ∗  m2 wij sj yj +

i=1 j =1

  2mai wij xi∗ sj yj∗ .

i=1 j =1

We also have m n   i=1 j =1

    2mai xi∗ wij sj yj∗ = 2mx ∗ T AW S y ∗    T  ∗      x − mAW S y ∗ + x ∗ T x ∗ + m2 S T y ∗ W T A2 W S y ∗ = − x ∗ − mAW S y ∗      x ∗ T x ∗ + m2 S T y ∗ W T A2 W S y ∗ ,

(5)

88

S. Arik / Physics Letters A 351 (2006) 85–91 n m  

    2nbj yj∗ vj i si xi∗ = 2my ∗ T BV S x ∗

j =1 i=1

   T  ∗      y − nBV S x ∗ + y ∗ T y ∗ + n2 S T x ∗ V T B 2 V S x ∗ = − y ∗ − nBV S x ∗      y ∗ T y ∗ + n2 S T x ∗ V T B 2 V S x ∗ .

Hence, (5) can be written as : −

n 

mai2 xi∗ 2 +

−

 2       n2 vjτ i si2 xi∗ + x ∗ T x ∗ + m2 S T y ∗ W T A2 W S y ∗

j =1 i=1 m n  

i=1 m 

n m  

nbj2 yj∗ 2 +

j =1

 τ 2 2  ∗      m2 wij sj yj + y ∗ T y ∗ + n2 S T x ∗ V T B 2 V S x ∗  0

i=1 j =1

from which it follows that  n m   T 2     2 2 2 2 2 τ 2 n αi v j i −mai + 1 + n αi λM V B V + xi∗ 2 i=1

+

j =1



m 

−nbj2

+ 1 + m2 βj2 μM





W A W + T

2

j =1

n 

m2 βj2



 τ 2 wij

yj∗ 2 0

i=1

or equivalently −

n 

δi xi∗ 2 −

m 

Ωj yj∗ 2 0.

(6)

j =1

i=1

On the other hand, since δi > 0 for i = 1, . . . , n and Ωj > 0 for j = 1, . . . , m, for x ∗ = (x1∗ , . . . , xn∗ )T = 0 or y ∗ = ∗ )T = 0, we can write (y1∗ , . . . , ym −

n  i=1

δi xi∗ 2 −

m 

Ωj yj∗ 2 < 0.

(7)

j =1

The contradiction between (6) and (7) directly implies that, under the conditions of Theorem 1, system (2) cannot have an equilibrium point at which x ∗ = 0 or y ∗ = 0. In other words, x ∗ = y ∗ = 0 is the only equilibrium point. Thus, the proof of the uniqueness of the equilibrium point is completed. It will be now shown that the conditions of Theorem 1 also imply the global asymptotic stability of the origin of (2). Let us employe the following positive definite Lyapunov functional: m n m n     τ 2    mai xi2 (t) + nbj yj2 (t) + m2 wij V x(t), y(t) = i=1

j =1

i=1 j =1

t sj2

n m    2  yj (η) dη + n2 vjτ i



t−τij

j =1 i=1

t

  si2 xi (ξ ) dξ.

t−σj i

The derivative of V (x(t), y(t)) along the trajectories of the system is obtained as: m n m n     τ 2 2      2mai xi (t)x˙i (t) + 2nbj yj (t)y˙j (t) + m2 wij sj yj (t) V˙ x(t), y(t) = i=1 j =1 n n m m   τ 2 2   2     2  τ 2 2    − m2 wij sj yj (t − τij ) + n vj i si xi (t) − n2 vjτ i si2 xi (t i=1 j =1 j =1 i=1 j =1 i=1 m m n n n        τ =− 2mai2 xi2 (t) + 2mai xi (t)wij sj yj (t) + 2mai xi (t)wij sj yj (t − τij ) i=1 i=1 j =1 i=1 j =1 n n m m m        − 2nbj2 yj2 (t) + 2nbj yj (t)vj i si xi (t) + 2nbj yj (t)vjτ i si xi (t − σj i ) j =1 j =1 i=1 j =1 i=1 m m n n    2     2  τ 2   τ + m2 wij sj2 yj (t) − m wij yj (t − τij ) i=1 j =1 i=1 j =1 i=1

m n  

j =1

 − σj i )

S. Arik / Physics Letters A 351 (2006) 85–91

+

m  n 

m  n     2   2  n2 vjτ i si2 xi (t) − n2 vjτ i si2 xi (t − σj i )

j =1 i=1

=−

n 

−

j =1 i=1

mai2 xi2 (t) +

i=1 j =1

m 

m  n 

nbj2 yj2 (t) +

j =1

+

n  m 

i=1

n  m  2 2     τ −ai xi (t) + 2mai xi (t)wij 2mai xi (t)wij sj yj (t) + sj yj (t − τij )



i=1 j =1

  2nbj yj (t)vj i si xi (t) +

j =1 i=1

n  m 

m  n 

m  n 

  2 2  −bj yj (t) + 2nbj yj (t)vjτ i si xi (t − σj i )

j =1 i=1

 τ 2 2   m2 wij sj yj (t) −

i=1 j =1

+

89

n  m 

 τ 2 2   m2 wij sj yj (t − τij )

i=1 j =1 m  n  2   2     n2 vjτ i si2 xi (t) − n2 vjτ i si2 xi (t − σj i ) .

j =1 i=1

j =1 i=1

We can write the following:    τ 2 2    2  τ τ −ai2 xi2 (t) + 2mai xi (t)wij sj yj (t − τij ) = − −ai xi (t) + mwij sj yj (t − τij ) + m2 wij sj yj (t − τij ) ,    2    2  −bj2 yj2 (t) + 2nbj yj (t)vjτ i si xi (t − σj i ) = − bj yj (t) + nvjτ i si xi (t − σj i ) + n2 vjτ i si2 xi (t − σj i ) . In the light of above equations, V˙ (x(t), y(t)) can be written as: n n  n  m m     2  τ 2 2       m wij sj yj (t − τij ) V˙ x(t), y(t)  − mai2 xi2 (t) + 2mai xi (t)wij sj yj (t) +

−

i=1

i=1 j =1

m 

m  n 

nbj2 yj2 (t) +

j =1

+

n  m 

i=1 j =1

  2nbj yj (t)vj i si xi (t) +

j =1 i=1

+

 τ 2   m2 wij sj yj (t) −

n 

 2   n2 vjτ i si2 xi (t) −

+

n  m 

 τ 2 2   m2 wij sj yj (t − τij )

m  n 

 2   n2 vjτ i si2 xi (t − σj i )

j =1 i=1

mai2 xi2 (t) +

i=1

n  m  i=1 j =1

j =1 i=1

=−

 2  τ 2 2   n vj i si xi (t − σj i )

j =1 i=1

i=1 j =1 m  n 

m  n 

m  n 

m n  m   2   τ 2     n2 vjτ i si2 xi (t) − nbj2 yj2 (t) + m2 wij sj yj (t)

j =1 i=1

  2mai xi (t)wij sj yj (t) +

i=1 j =1

j =1 m  n 

i=1 j =1

  2nbj yj (t)vj i si xi (t) .

j =1 i=1

We also have n  m 

    2mai xi (t)wij sj yj (t) = 2mx T (t)AW S y(t)

i=1 j =1

m  n 

  T    = − x(t) − mAW S y(t) x(t) − mAW S y(t) + x T (t)x(t)     + m2 S T y(t) W T A2 W S y(t)      x T (t)x(t) + m2 S T y(t) W T A2 W S y(t) ,     2nbj yj (t)vj i si xi (t) = 2my T (t)BV S x(t)

j =1 i=1

Hence, we can write

  T        = − y(t) − nBV S x(t) y(t) − nBV S x(t) + y T (t)y(t) + n2 S T x(t) V T B 2 V S x(t)      y T (t)y(t) + n2 S T x(t) V T B 2 V S x(t) .

90

S. Arik / Physics Letters A 351 (2006) 85–91 n n m          2    V˙ x(t), y(t)  − mai2 xi2 (t) + n2 vjτ i si2 xi (t) + x T (t)x(t) + n2 S T x(t) V T B 2 V S x(t) j =1 i=1

i=1

−

m 

nbj2 yj2 (t) +

j =1

−

n 

 τ 2 2       m2 wij sj yj (t) + m2 S T y(t) W T A2 W S y(t) + y T (t)y(t)

i=1 j =1

mai2 xi2 (t) +

n m  

n n  2        n2 vjτ i si2 xi (t) + xi2 (t) + n2 λM V T B 2 V si2 xi (t)

j =1 i=1

i=1

−

m n  

m 

nbj2 yj2 (t) +

j =1

m n  

i=1

i=1

m m  τ 2 2        m2 wij sj yj (t) + yj2 (t) + m2 μM W T A2 W sj2 yj (t) .

i=1 j =1

j =1

j =1

Since, sj2 (yj )  βj2 yj2 , j = 1, 2, . . . , m and si2 (xi )  αi2 xi2 , i = 1, 2, . . . , n, we obtain n n m  n n     2     mai2 xi2 (t) + n2 vjτ i αi2 xi2 (t) + xi2 (t) + n2 λM V T B 2 V αi2 xi2 (t) V˙ x(t), y(t)  − j =1 i=1

i=1

−

m 

nbj2 yj2 (t) +

j =1



=

n 

m n  

i=1

m m    τ 2 2 2  m2 wij βj yj (t) + yj2 (t) + m2 μM W T A2 W βj2 yj2 (t)

i=1 j =1

j =1

  −mai2 + 1 + n2 αi2 λM V T B 2 V +

+

m 

−nbj2

+ 1 + m2 βj2 μM





W A W + T

2

j =1

=−

n 

j =1



 2 2 n2 αi2 vjτ i xi (t)

j =1

i=1

 m 

i=1

n 

m2 βj2



 τ 2 wij

yj2 (t)

i=1

δi xi2 (t) −

m 

Ωj yj2 (t).

j =1

i=1

Since δi > 0 for i = 1, . . . , n and Ωj > 0 for j = 1, . . . , m, it follows that V˙ (x(t), y(t)) < 0 for x(t) = 0 or y(t) = 0. Hence, by the standard Lyapunov-type theorem in functional differential equations we can conclude that the origin of system (2) is globally asymptotically stable. 2 We should point out here that the results of [16–29] are obtained for the pure delayed bidirectional associative memory neural networks in which case the parameters wij = 0 and vj i = 0 for i = 1, 2, . . . , n and j = 1, 2, . . . , m. Therefore, we will compare our result with only the result given in [30], which studies the stability of hybrid BAM neural networks. The result of [30] is stated in the following: Theorem 2 [30]. The origin of system (2) is the unique equilibrium point and it is globally exponentially stable if the network parameters of the system satisfy the following conditions: δˆi =

m  

 τ   βj < a i , |wij | + wij

i = 1, 2, . . . , n

j =1

Ωˆ j =

n  

  |vj i | + vjτ i  αi < bj , j = 1, 2, . . . , m.

i=1

In order to show that the condition we have obtained in Theorem 1 provides a new sufficient criterion for determining the equilibrium and stability properties of system (2), we consider the example where the network parameters are given as follows: ⎡

c ⎢ c W =V =⎣ −c −c

c −c −c c

c c c c

⎤ c −c ⎥ ⎦, c −c

⎡

c ⎢c τ τ W =V =⎣ c c

c c c c

c c c c

⎤ c c⎥ ⎦, c c

⎡

1 ⎢0 A=B =α=β =⎣ 0 0

0 1 0 0

0 0 1 0

⎤ 0 0⎥ ⎦ 0 1

S. Arik / Physics Letters A 351 (2006) 85–91

where c is a real number. The matrices W T A2 W ⎡ 2 4c 0 0 2 0 4c 0 ⎢ W T A2 W = V T B 2 V = ⎣ 0 0 4c2 0 0 0

91

and V T B 2 V are equivalent and have the following diagonal form: ⎤ 0 0 ⎥ ⎦ 0 2 4c

from which it follows that λM (V T B 2 V ) = μM (W T A2 W ) = 4c2 . Thus, under the conditions of Theorem 1, we have δ1 = δ2 = δ3 = δ4 = Ω1 = Ω2 = Ω3 = Ω4 = −3 + 128c2 . Hence, for this example, the sufficient condition for the uniqueness and stability of the equilibrium point is that c2 < 3/128. On the other hand, applying the result of Theorem 2 to this example yields the condition |c| < 1/8, which is equivalent to c2 < 2/128. Thus, it can be concluded that if c2  2/128, the condition of Theorem 2 is not satisfied, whereas the condition of Theorem 1 still holds for 2/128  c2 < 3/128. 4. Conclusions The main contribution of this Letter is the result that ensures the existence, uniqueness and global asymptotic stability of hybrid bidirectional associative memory neural networks with distributed time delays. The stability result establishes a relationship between the network parameters of neural network model independently of the delay parameters. A comparison between our result and that of [24] has also been made, which proved that the result of Theorem 1 holds for different choices of network parameters than those considered in [30]. Acknowledgements This work was supported in part by Scientific and Technical Research Council of Turkey, under Project 104E024 and in part by the Research Fund of Istanbul University under Project BYP-739/07072005. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

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