Existence and exponential stability of almost periodic solutions for cellular neural networks with time-varying delays

Physics Letters A 341 (2005) 135–144 www.elsevier.com/locate/pla

Existence and exponential stability of almost periodic solutions for cellular neural networks with time-varying delays ✩ Bingwen Liu a,∗ , Lihong Huang b a Department of Mathematics, Hunan University of Arts and Science, Changde, Hunan 415000, PR China b College of Mathematics and Econometrics, Hunan University, Changsha 410082, PR China

Received 8 December 2004; received in revised form 19 April 2005; accepted 20 April 2005 Available online 29 April 2005 Communicated by A.P. Fordy

Abstract In this Letter cellular neural networks with time-varying delays are considered. Sufficient conditions for the existence and exponential stability of the almost periodic solutions are established by using the fixed point theorem and differential inequality techniques. The results of this Letter are new and they complement previously known results.  2005 Elsevier B.V. All rights reserved. MSC: 34C25; 34K13; 34K25 Keywords: Cellular neural networks; Almost periodic solution; Exponential stability; Fixed point theorem; Time-varying delays

1. Introduction Consider the following model for the delayed cellular neural networks (DCNNs) xi (t) = −ci xi (t) +

n
j =1

n    
 aij (t)gj xj (t) + bij (t)gj xj t − τij (t) + Ii (t),

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

(1.1)

j =1

in which n corresponds to the number of units in a neural network, xi (t) corresponds to the state vector of the ith unit at the time t, ci represents the rate with which the ith unit will reset its potential to the resting state in isolation ✩ This work was supported by the NNSF (10371034) of China, the Doctor Program Foundation of the Ministry of Education of China (20010532002) and Key Project of Chinese Ministry of Education ([2002]78). * Corresponding author. E-mail address: [email protected] (B. Liu).

0375-9601/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.04.052

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B. Liu, L. Huang / Physics Letters A 341 (2005) 135–144

when disconnected from the network and external inputs, gj (xj (t)) denotes the output of the j th unit at the time t, aij (t) denotes the strength of the j th unit on the ith unit at time t, bij (t) denotes the strength of the j th unit on the ith unit at time t − τij (t), τij (t)
0 corresponds to the transmission delay of the ith unit along the axon of the j th unit at the time t, and Ii (t) denotes the external bias on the ith unit at the time t. It is well known that the DCNNs have been successfully applied to signal and image processing, pattern recognition and optimization. Hence, they have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of periodic solutions of system (1.1) in the literature. We refer the reader to [1–7] and the references cited therein. However, there exist few results on the existence and exponential stability of the almost periodic solutions of system (1.1). The main purpose of this Letter is to give the conditions for the existence and exponential stability of the almost periodic solutions for system (1.1). By applying fixed point theorem and differential inequality technique, we derive some new sufficient conditions ensuring the existence, uniqueness and exponential stability of the almost periodic solution, which are new and complement previously known results. Throughout this Letter, it will be assumed that ci > 0, τ = max1
i,j
n {maxt∈[0,ω] τij (t)}, and Ii , aij , bij , τij : R → R are almost periodic functions, where ci and τ
0 are constants, and i, j = 1, 2, . . . , n. Then, we can choose constants aij , bij and Ii such that       supbij (t) = bij , supIi (t) = Ii , i, j = 1, 2, . . . , n. supaij (t) = aij , (1.2) t∈R

t∈R

t∈R

We also assume that the following condition (H0 ) holds. (H0 ) for each j ∈ {1, 2, . . . , n}, gj : R → R is Lipschitz with Lipschitz constant Lj , i.e.,   gj (uj ) − gj (vj )  Lj |uj − vj |, for all uj , vj ∈ R. For convenience, we introduce some notations. We will use x = (x1 , x2 , . . . , xn )T ∈ R n to denote a column vector, in which the symbol T denotes the transpose of a vector. We let |x| denote the absolute-value vector given by |x| = (|x1 |, |x2 |, . . . , |xn |)T , and define x = max1
i
n |xi |. For matrix A = (aij )n×n , AT denotes the transpose of A, A−1 denotes the inverse of A, |A| denotes the absolute-value matrix given by |A| = (|aij |)n×n , and ρ(A) denotes the spectral radius of A. A matrix or vector A
0 means that all entries of A are greater than or equal to zero. A > 0 can be defined similarly. For matrices or vectors A and B, A
B (respectively A > B) means that A − B
0 (respectively A − B > 0). Let D = diag(c1 , c2 , . . . , cn ), I = (I1 , I2 , . . . , In ),

A = (aij )n×n ,

B = (bij )n×n ,

L = diag(L1 , L2 , . . . , Ln ),

E = A + B.

For V (t) ∈ C((a, +∞), R), let D − V (t) = lim sup h→0−

V (t + h) − V (t) , h

D− V (t) = lim inf h→0−

V (t + h) − V (t) , h

∀t ∈ (a, +∞).

As usual, we introduce the phase space C([−τ, 0]; R n ) as a Banach space of continuous mappings from [−τ, 0] to R n equipped with the supremum norm defined by   ϕ = max sup ϕi (t) 1
i
n −τ
t
0

for all ϕ = (ϕ1 (t), ϕ2 (t), . . . , ϕn (t))T ∈ C([−τ, 0]; R n ). The initial conditions associated with system (1.1) are of the form xi (s) = ϕi (s),

s ∈ [−τ, 0], i = 1, 2, . . . , n,

where ϕ = (ϕ1 (t), ϕ2 (t), . . . , ϕn (t))T ∈ C([−τ, 0]; R n ).

(1.3)

B. Liu, L. Huang / Physics Letters A 341 (2005) 135–144

137

Definition 1 (see [13,14]). Let u(t) : R → R n be continuous in t. u(t) is said to be almost periodic on R if, for any ε > 0, the set T (u, ε) = {δ: |u(t + δ) − u(t)| < ε, ∀t ∈ R} is relatively dense, i.e., for ∀ε > 0, it is possible to find a real number l = l(ε) > 0, for any interval with length l(ε), there exists a number δ = δ(ε) in this interval such that |u(t + δ) − u(t)| < ε, for ∀t ∈ R. Definition 2. Let Z ∗ (t) = (x1∗ (t), x2∗ (t), . . . , xn∗ (t))T be an almost periodic solution of system (1.1) with initial value ϕ ∗ = (ϕ1∗ (t), ϕ2∗ (t), . . . , ϕn∗ (t))T ∈ C([−τ, 0]; R n ). If there exist constants α > 0 and M > 1 such that for every solution Z(t) = (x1 (t), x2 (t), . . . , xn (t))T of system (1.1) with any initial value ϕ ∈ C([−τ, 0]; R n ),   xi (t) − x ∗ (t)  Mϕ − ϕ ∗ e−αt , i

∀t > 0, i = 1, 2, . . . , n.

Then Z ∗ (t) is said to be globally exponentially stable. Definition 3. A real n × n matrix K = (kij ) is said to be an M-matrix if kij  0, i, j = 1, 2, . . . , n, i = j , and K −1
0. The remaining part of this Letter is organized as follows. In Section 2, we shall derive new sufficient conditions for checking the existence of almost periodic solutions. In Section 3, we present some new sufficient conditions for the uniqueness and exponential stability of the almost periodic solution of (1.1). In Section 4, we shall give some examples and remarks to illustrate our results obtained in previous sections.

2. Existence and uniqueness of almost periodic solutions The following lemma will be very essential to prove our main results of this Letter. Lemma 2.1 (see [12,16]). Let A
0 be an n × n matrix and ρ(A) < 1, then (En − A)−1
0, where En denotes the identity matrix of size n. Theorem 2.1. Let condition (H0 ) holds and ρ(D −1 EL) < 1. Then, there exists exactly one almost periodic solution of system (1.1). Proof. Let    T  X = φ  φ = φ1 (t), φ2 (t), . . . , φn (t) , where φi : R → R is a continuous almost periodic function, i = 1, 2, . . . , n. Then, X is a Banach space with the norm defined by φX = supt∈R max1
i
n |φi (t)|. To proceed further, we need to introduce an auxiliary equation xi (t) = −ci xi (t) +

n
j =1

n    
 aij (t)gj φj (t) + bij (t)gj φj t − τij (t) + Ii (t),

(2.1)

j =1

where i = 1, 2, . . . , n, φ(t) = (φ1 (t), φ2 (t), . . . , φn (t))T ∈ X. Then, notice that τij (t), aij (t), bij (t) and Ii (t) are almost periodic functions, according to [13,14], we know that the auxiliary equation (2.1) has exactly one almost periodic solution

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B. Liu, L. Huang / Physics Letters A 341 (2005) 135–144

 φ T φ x φ (t) = x1 (t), x2 (t), . . . , xnφ (t)  t

n n
   
 −c1 (t−s) = e a1j (s)gj φj (s) + b1j (s)gj φj s − τ1j (s) + I1 (s) ds, . . . , j =1

−∞ t



e−cn (t−s)

n


j =1

T n 
    anj (s)gj φj (s) + bnj (s)gj φj s − τnj (s) + In (s) ds .

j =1

−∞

(2.2)

j =1

Define a mapping Φ : X → X by setting   Φ φ(t) = x φ (t), for all φ ∈ X. Let φ, ψ ∈ X. Then, by (H0 ), we have      Φ φ(t) − Φ ψ(t)           T   =  Φ φ(t) − Φ ψ(t) 1 , . . . ,  Φ φ(t) − Φ ψ(t) n    t n n 
    
     −c1 (t−s) =  e a1j (s) gj φj (s) − gj ψj (s) + b1j (s) gj φj s − τ1j (s)  j =1

−∞

j =1

 t

 n  
          −cn (t−s) ds , . . . ,  e anj (s) gj φj (s) − gj ψj (s) − gj ψj s − τ1j (s)   j =1

−∞

+

n
j =1

T          bnj (s) gj φj s − τnj (s) − gj ψj s − τnj (s) ds  

 t 

e −∞ t

e

−c1 (t−s)



n 
          a1j Lj φj (s) − ψj (s) + b1j Lj φj s − τ1j (s) − ψj s − τ1j (s) ds, . . . ,

j =1

−cn (t−s)

n


j =1

T n 
      anj Lj φj (s) − ψj (s) + bnj Lj φj s − τnj (s) − ψj s − τnj (s)  ds

j =1

−∞



n


n


c1−1 (a1j

j =1

j =1

n
      + b1j )Lj sup φj (t) − ψj (t) , . . . , cn−1 (anj + bnj )Lj supφj (t) − ψj (t) t∈R

j =1

T ,

t∈R

which implies that             T sup Φ φ(t) − Φ ψ(t) 1 , . . . , sup Φ φ(t) − Φ ψ(t) n  t∈R





t∈R

n


c1−1 (a1j

j =1

n
    + b1j )Lj supφj (t) − ψj (t), . . . , cn−1 (anj + bnj )Lj supφj (t) − ψj (t) t∈R

j =1

T

t∈R

  T    F supφ1 (t) − ψ1 (t), . . . , supφn (t) − ψn (t) t∈R

t∈R

      T = F sup φ(t) − ψ(t) 1 , . . . , sup φ(t) − ψ(t) n  ,

t∈R

t∈R

(2.3)

B. Liu, L. Huang / Physics Letters A 341 (2005) 135–144

139

where F = D −1 EL. Let m be a positive integer. Then, from (2.3), we get             T sup Φ m φ(t) − Φ m ψ(t) 1 , . . . , sup Φ m φ(t) − Φ m ψ(t) n  t∈R

t∈R

                T = sup Φ Φ m−1 φ(t) − Φ Φ m−1 ψ(t) 1 , . . . , sup Φ Φ m−1 φ(t) − Φ Φ m−1 ψ(t) n 

t∈R

t∈R



            T  F sup Φ m−1 φ(t) − Φ m−1 ψ(t) 1 , . . . , sup Φ m−1 φ(t) − Φ m−1 ψ(t) n  t∈R

t∈R

.. .

      T  F m sup φ(t) − ψ(t) 1 , . . . , sup φ(t) − ψ(t) n  = Fm



t∈R

t∈R

  T   supφ1 (t) − ψ1 (t), . . . , supφn (t) − ψn (t) . t∈R

(2.4)

t∈R

Since ρ(F ) < 1, we obtain lim F m = 0,

m→+∞

which implies that there exist a positive integer N and a positive constant r < 1 such that  N F N = D −1 EL = (hij )n×n

and

n


hij  r,

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

(2.5)

j =1

In view of (2.4) and (2.5), we have n     N          
 Φ φ(t) − Φ N ψ(t)   sup Φ N φ(t) − Φ N ψ(t)   hij supφj (t) − ψj (t) i i t∈R

   sup max φj (t) − ψj (t) t∈R 1
j
n

j =1

n
j =1

t∈R

  hij  r φ(t) − ψ(t)X ,

for all t ∈ R, i = 1, 2, . . . , n. It follows that  N            Φ φ(t) − Φ N ψ(t)  = sup max  Φ N φ(t) − Φ N ψ(t)   r φ(t) − ψ(t) . X i X t∈R 1
i
n

(2.6)

This implies that the mapping Φ N : X → X is a contraction mapping. By the fixed point theorem of Banach space, Φ possesses a unique fixed point Z ∗ in X such that ΦZ ∗ = Z ∗ . We know from (2.2) that Z ∗ satisfies system (1.1), and therefore, it is the unique almost periodic solution of system (1.1). The proof of Theorem 2.1 is now complete. 2

3. Exponential stability of the almost periodic solution In this section, we establish some results for the exponential stability of the almost periodic solution of (1.1). Theorem 3.1. Suppose that all the conditions of Theorem 2.1 hold. Then system (1.1) has exactly one almost periodic solution Z ∗ (t). Moreover, Z ∗ (t) is globally exponentially stable.

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B. Liu, L. Huang / Physics Letters A 341 (2005) 135–144

Proof. Since ρ(D −1 EL) < 1, it follows from Theorem 3.1 that system (1.1) has a unique almost periodic solution Z ∗ (t) = (x1∗ (t), x2∗ (t), . . . , xn∗ (t))T . Let Z(t) = (x1 (t), x2 (t), . . . , xn (t))T be an arbitrary solution of system (1.1) and define y(t) = Z(t) − Z ∗ (t). Then, set       fj t, yj (t) = gj yj (t) + xj∗ (t) − gj xj∗ (t) , j = 1, 2, . . . , n, we get yi (t) = −ci yi (t) +

n


n    
 aij (t)fj t, yj (t) + bij (t)fj t, yj t − τij (t) ,

j =1

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

(3.1)

j =1

Thus, for i = 1, 2, . . . , n, we have n n  
        
 aij (t)fj t, yj (t)  + bij (t)fj t, yj t − τij (t)  D − yi (t)  −ci yi (t) + j =1

   −ci yi (t) +

j =1

n


(aij + bij )pj

j =1

n  
  sup yj (s) = −ci yi (t) + (aij + bij )pj yj (t),

t−τ
s
t

(3.2)

j =1

where yj (t) = supt−τ
s
t |yj (s)|. Again from ρ(D −1 EL) < 1, it follows from Lemma 2.1 that En − D −1 EL is an M-matrix, we obtain that there exist a constant σ > 0 and a vector ξ = (ξ1 , ξ2 , . . . , ξn )T > (0, 0, . . . , 0)T such that   En − D −1 EL ξ > (σ, σ, . . . , σ )T . Therefore, ξi −

n


dij ξj = ξi −

j =1

n


ci−1 (aij + bij )pj ξj > σ,

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

j =1

which implies that −ci ξi +

n
(aij + bij )pj ξj < −ci σ,

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

(3.3)

j =1

We can choose a positive constant α < 1 such that

n
ατ (aij + bij )pj ξj e αξi + −ci ξi + < 0,

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

(3.4)

j =1

We can choose a constant β > 1 such that βξi e−αt > 1, For ∀ε > 0, let Zi (t) = βξi



for all t ∈ [−τ, 0], i = 1, 2, . . . , n.

n
j =1

(3.5)

yj (0) + ε e−αt ,

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

(3.6)

B. Liu, L. Huang / Physics Letters A 341 (2005) 135–144

141

From (3.4) and (3.6), we obtain



n

n n


−αt ατ D− Zi (t) = −αβξi yj (0) + ε e > −ci ξi + (aij + bij )pj ξj e yj (0) + ε e−αt β j =1

= −ci βξi

n




j =1



yj (0) + ε e

−αt

j =1

+

n


(aij + bij )pj ξj β

j =1



j =1 n




yj (0) + ε e



−αt ατ

e

j =1

n
(aij + bij )pj Zj (t), = −ci Zi (t) +

(3.7)

j =1

where Zj (t) = supt−τ
s
t Zj (s). In view of (3.5) and (3.6), for i = 1, 2, . . . , n, we have

n n

  yj (0) + ε e−αt > yj (0) + ε > yi (t), for all t ∈ [−τ, 0]. Zi (t) = βξi j =1

We claim that   yi (t) < Zi (t),

(3.8)

j =1

for all t > 0, i = 1, 2, . . . , n.

(3.9)

Contrarily, there must exist i ∈ {1, 2, . . . , n} and ti > 0 such that     yi (ti ) = Zi (ti ) and yj (t) < Zj (t), for all t ∈ [−τ, ti ), j = 1, 2, . . . , n,

(3.10)

which implies that     yi (ti ) − Zi (ti ) = 0 and yj (t) − Zj (t) < 0,

(3.11)

for all t ∈ [−τ, ti ), j = 1, 2, . . . , n.

It follows that    [|yi (ti + h)| − Zi (ti + h)] − [|yi (ti )| − Zi (ti ))] 0  D − yi (ti ) − Zi (ti ) = lim sup h h→0−   |yi (ti + h)| − |yi (ti )| Zi (ti + h) − Zi (ti ) − lim inf = D − yi (ti ) − D− Zi (ti ).  lim sup − h h h→0 h→0−

(3.12)

From (3.2), (3.7) and (3.10), we obtain D



−

n n
 
          yi (ti )  −ci yi (ti ) + (aij + bij )pj yj (ti ) = −ci Zi (ti ) + (aij + bij )pj yj (ti ) j =1

j =1

n
 −ci Zi (ti ) + (aij + bij )pj Zj (ti ) < D− Zi (ti ),

(3.13)

j =1

which contradicts (3.12). Hence, (3.9) holds. Letting ε → 0+ and M = n max1
i
n {βξi + 1}, we have from (3.6) and (3.9) that n
    xi (t) − x ∗ (t) = yi (t)  βξi yj (0)e−αt  βξi nϕ − ϕ ∗ e−αt  Mϕ − ϕ ∗ e−αt , i j =1

for all t > 0. This completes the proof.

2

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

142

B. Liu, L. Huang / Physics Letters A 341 (2005) 135–144

Remark 3.1. Most of the existing results on almost periodic solutions for (1.1) have been obtained under the assumption that the row norm of matrix D −1 EL is less than 1. Therefore, the related results in [8] are direct corollaries of Theorem 3.1 of this Letter. Moreover, according to Theorem 3.1, when Ii , aij , bij : R → R are ωperiodic functions with ω > 0, for i, j = 1, 2, . . . , n, we can also obtain the existence and global exponential stability of ω-periodic solutions for system (1.1). Hence, the related results in [4,7] are also our special cases. Corollary 3.1. Let condition (H0 ) holds. Suppose that En − D −1 EL is an M-matrix. Then system (1.1) has exactly one almost periodic solution Z ∗ (t). Moreover, Z ∗ (t) is globally exponentially stable. Proof. Notice that En − D −1 EL is an M-matrix, it follows that there exists a vector η = (η1 , η2 , . . . , ηn )T > 0 such that   En − D −1 EL η > 0,

(3.14)

that is, −ci ηi +

n
(aij + bij )Lj ηj < 0,

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

(3.15)

j =1

For any matrix norm  ·  and nonsingular matrix B, AB = B −1 AB also defines a matrix norm. Let B = diag(η1 , η2 , . . . , ηn ). Then (3.15) implies that the row norm of matrix B −1 (D −1 EL)B is less than 1. Therefore, ρ(D −1 EL) < 1. Corollary 3.1 follows immediately from Theorem 3.1. 2

4. Two examples In this section, we give an example to demonstrate the results obtained in previous sections. Example 4.1. Consider the following DCNNs with delays:   1 x (t) = −x1 (t) + 14 (sin t)g1 (x1 (t)) + 36 (cos t)g2 (x2 (t)) + 14 (sin t)g1 (x1 (t − sin2 t))    1 1 2 + (cos t)g (x (t − 2 sin t)) + I (t), 36

2

2

1

 x  (t) = −x2 (t) + (sin 2t)g1 (x1 (t)) + 13 (cos 4t)g2 (x2 (t)) + (sin 2t)g1 (x1 (t − 3 cos2 t))   2 + 16 (cos 4t)g2 (x2 (t − 4 sin2 t)) + I2 (t),

(4.1)

where g1 (x) = g2 (x) = x, and I1 (t) and I2 (t) are almost periodic functions. Observe that c1 = c2 = L1 = L2 = 1, and 1 1  . D −1 EL = D −1 (A + B)L = 2 18 2 12 So, by easy computation, we can see that ρ(D −1 EL) = ρ(D −1 (A + B)L) = 56 < 1. Thus, from Theorem 3.1, Eq. (4.1) has exactly one almost periodic solution, which is globally exponentially stable. Remark 4.1. Eq. (4.1) is a very simple form of DCNNs equations. One can observe that D −1 (A + B)L1 = 52 , where  · 1 is the row norm of matrix. Therefore, all the results in [1–12] and the references therein cannot be applicable to Eq. (4.1). This implies that the results of this Letter are essentially new.

B. Liu, L. Huang / Physics Letters A 341 (2005) 135–144

Example 4.2. Consider the following CNNs  1  x1 (t) = −x1 (t) + 14 (sin t)g1 (x1 (t)) + 36 (cos t)g2 (x2 (t)) + 14 (sin t)g1 (x1 (t − cos2 t))   1  2  + 36 (cos t)g2 (x2 (t − sin t)) + cos t,  x2 (t) = −x2 (t) + (sin 2t)g1 (x1 (t)) + 14 (cos 4t)g2 (x2 (t)) + (sin 2t)g1 (x1 (t − sin2 t))    + 14 (cos 4t)g2 (x2 (t − cos2 t)) + 2 sin t,     x3 (t) = −x3 (t) + 49 (cos 4t)g3 (x3 (t)) + 49 (sin 2t)g3 (x3 (t − cos2 t)) + cos2 t,

143

(4.2)

where g1 (x) = g2 (x) = g3 (x) = 12 (|x + 1| − |x − 1|). 1 Notice that c1 = c2 = c3 = p1 = p2 = p3 = 1, and a11 = b11 = 14 , a12 = b12 = 36 , a13 = b13 = 0, a21 = b21 = 1 4 1, a22 = b22 = 4 , a23 = b23 = a31 = b31 = a32 = b32 = 0, a33 = b33 = 9 . Then, we have 1 1  0 2 18   −1 D = ci (aij + bij )pj 3×3 =  2 12 0  , 0 0 89 and 8 < 1. 9 Thus, from Theorem 3.1, Eq. (4.2) has exactly one 2π -periodic solution, and the 2π -periodic solution of system (4.2) is globally exponentially stable. ρ(D) =

Remark 4.2. From (4.2), we can easily verify that the results in [1–12,15] and the references cited therein cannot be applicable to system (4.2).

5. Conclusion In this Letter, cellular neural networks with time-varying delays have been studied. Some sufficient conditions for the existence and exponential stability of the almost periodic solutions have been established. These obtained results are new and they complement previously known results. Moreover, two examples are given to illustrate the effectiveness of the new results.

Acknowledgements The authors would like to express their sincere appreciation to the reviewer for his/her helpful comments in improving the presentation and quality of the Letter.

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