[IJCT-V3I2P19]

International Journal of Computer Techniques -– Volume 3 Issue 2, Mar-Apr 2016 RESEARCH ARTICLE

OPEN ACCESS

Exponential Stabilization of the Coupled Dynamical Neural Networks with Nodes of Different Dimensions and Time Delays Jieyin Mai, Xiaojun Li (Department of Mathematics, Jinan University, Guangzhou Guangdong 510632, China)

----------------------------------------************************----------------------------------

Abstract: A class of coupled neural networks with different internal time-delays and coupling delays is investigated, which consists of nodes of different dimensions. By constructing suitable Lyapunov functions and using the linear matrix inequality, the criteria of exponential stabilization for the coupled dynamical system are established, and formulated in terms of linear matrix inequality. Finally, numerical examples are presented to verify the feasibility and effectiveness of the proposed theoretical results.

Keywords----nodes of different dimensions; coupled dynamical system with time delays; exponential stabilization; linear matrix inequality; Lyapunov function ----------------------------------------************************---------------------------------I.

INTRODUCTION

With the rapid development of science and technology, neural networks are gradually and widely applied to the optimal control, combinatorial optimization, pattern recognition, and other fields.

Due to the finite speeds of transmission and traffic congestion, time delays are common in actual systems, and the delays may be constant, timevarying with parameter or mixed (see [1]).

In the past two decades, stability or stabilization problems have been the hot topics for neural networks with time delays, including isolated neural networks and coupled neural networks (see [2]-[17] and references therein). For instance, the stabilization problem of delayed recurrent neural networks is investigated by a state estimation based approach in [2]. The exponential stability problem for a class of impulsive cellular neural networks with time delay is investigated by Lyapunovfunctions and the method of variation of parameters in[3]. Some studies on exponential stability or stabilization problem are about neural networks with time-varying delays (see [5]-[8]).

Some sufficient conditions to ensure the existence, uniqueness and global exponential stability of the equilibrium point of cellular neural networks with variable delays are derived in[5]. For the problem of exponential stabilization for a class of nonISSN: 2394-2231

autonomous neural networks with mixed discrete and distributed time-varying delays, some new delay-dependent conditions for designing a memoryless state feedback controller which stabilizes the system with an exponential convergence of the resulting closed-loop system are established in [6]. A new class of stochastic CohenGrossberg neural networks with reaction-diffusion and mixed delays is studied in [7]. Since the form of coupling is various, there exist different coupled neural networks, e.g. coupled term with or without time delays, or hybrid both. Some researches on stability and synchronization of different coupled neural networks are investigated (see [9]-[11]).

Some global stability criteria for arrays of linearly coupled delayed neural networks with nonsymmetric coupling are established on the basis of linear matrix inequality method, in which the coupling configuration matrix may be arbitrary matrix with appropriate dimensions in [9]. Since there may be some chaotic behaviours as the dynamics performance of time-delay neural network in certain situations, chaotic neural networks are formed (see [12]-[14] and references therein). Some researches on stability and synchronization of chaotic neural network are investigated in [12]-[14]. Furthermore, sufficient condition for exponential stability of the

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International Journal of Computer Techniques -– Volume 3 Issue 2, Mar-Apr 2016

equilibrium solution of uncontrolled stochastic interval system is also presented in [15]. The decentralized continuous adaptive controller can be designed to make the solutions of the closed-loop system exponentially convergent to a ball, which can be rendered arbitrary small by adjusting design parameters in [16]. Mostof researches on synchronization problem of neural network are also based on linear matrix inequality method and Lyapunovfunctions.

Although there are lots of researches for the exponential stability or stabilization problems of neural networks with time delay, most of them concern with the same dimension of the states. If a network is constructed by nodes with different state dimension, the network will exhibit different dynamical behaviours (see [17]-[19] and references therein). Dimensions of nodes are actually different in many practical situations, so such coupled complex networks need more in-depth study.

Motivated by the above discussion, in this paper we consider the exponential stabilization problem of the coupled dynamical system. The dimensions of states in each isolated network are different, and the internal time-delays are different from the coupling delays. In Sec.3, the criteria of exponential stabilization for the coupled dynamical system are given and proved on basis of the linear matrix inequality and Lyapunovfunction. In Sec.4, two numerical simulation examples are given to demonstrate the effectiveness of the proposed theoretical results. In Sec.5, some concluding remarks are given.

where i = 1, 2,L , N ; xi (t ) = ( xi1 (t ), xi 2 (t ),L , xin (t ))T ∈ ℜ n

i

i

is the state vector of the i th node; A i , Bi ∈ ℜn ×n are constant matrices that representing the feedback matrix without and with time delays respectively; Di = diag (d i1 , di 2 ,L , din ) > 0 is a constant and diagonal i

i

i

matrix; f i (∗) is the activation function; G = ( g ij ) N × N is an outer coupling matrix representing the coupling strength and the topological structure of the neural networks; C ij , Γij ∈ ℜn ×n are inner coupling matrices respectively representing the inner-linking strengths between the cells without and with time delays,when ni ≠ n j , C ij , Γij ∈ ℜn ×n are arbitrary matrices; α i , βi are the strengths of the constant coupling and delayed coupling, respectively; τ i1 ,τ i 2 are the constant and positive delays, τ i1 > 0,τ i 2 > 0 and τ i1 < τ ,τ i 2 < τ ; ui (t ) = (ui1 (t ), ui 2 (t ),L , uin (t ))T ∈ ℜ n is an external input.

The initial condition associated with (1) is given as follows: xij ( s ) = φij ( s ) ∈ C ([ −τ , 0] , ℜ ) ,(3) i

j

i

j

i

i

where τ i = max{τ i1 ,τ i 2 } , τ = max{τ1 ,τ 2 ,L ,τ N } , i = 1, 2,L , N , j = 1, 2,L , ni .

For convenience, the notations are givens as follows: I denotes M × M identity matrix; I n denotes ni × ni i

T

identity matrix; y = y y denotes the norm of y ; λmax (∗) and λmin (∗) respectively denote the maximum and minimum eigenvalue of ∗ ; M = n1 + n2 + L + nN ; X (t ) = ( x1T (t ), x2T (t ),L , xNT (t ))T ， II. SYSTEM DESCRIPTION AND PRELIMINARIES U (t ) = (u1T (t ), u 2T (t ),L , u NT (t ))T ， Consider a coupled dynamical system with nodes F ( X (t )) = ( f1T ( x1 (t )), f 2T ( x2 (t )),L , f NT ( xN (t )))T ， of different dimensions and time delays: F ( X (t − τ 1 )) = ( f1T ( x1 (t − τ 11 )), f 2T ( x2 (t − τ 21 )), dxi (t ) ， T T = − D x (t ) + A f ( x (t )) + B f ( x (t − τ )) dt

i i

i

i

i

i

N

N

j =1

j =1

i

i

+α i ∑ gij Cij x j (t ) + βi ∑ g ij Γij x j (t − τ i 2 ) + ui (t ),

(1)

dxi (t ) = − Di xi (t ) + Ai fi ( xi (t )) + Bi fi ( xi (t − τ i1 )), dt

，

X (t − τ 2 ) = ( x1T (t − τ 12 ), x2T (t − τ 22 ),L , xNT (t − τ N 2 ))T

diag (L) denotes

in which each isolated node network with different dimensions and time delays is considered as:

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L , f N ( xN (t − τ N 1 )))

i1

a block-diagonal matrix; D = diag ( D1 , D2 ,L , DN ) ; A = diag ( A1 , A2 ,L , AN ) ; B = diag ( B1 , B2 ,L , BN ) ; Wi = diag ( wi1 , wi 2 ,L , win ) ; i

(2)

W = diag (W1 , W2 ,L , WN ) ; X * 

Y is Z 

X

Y

defined as a matrix in form of  T .

Z  Y

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In order to study the exponential stabilization of the coupled dynamical system (1), the linear controller isdesigned as U (t ) = − KX (t ) ,i.e., ui (t ) = − K i xi (t ) , (4) where K i = diag ( ki1 , ki 2 ,L , kin ) , K = diag ( K1 , K 2 ,L , K N ) .

Hence, equation (1) can be rewritten as i

dX (t ) = −( D + K ) X (t ) + AF ( X (t )) + BF ( X (t − τ 1 )) (5) dt + H1 X (t ) + H 2 X (t − τ 2 ),

Φ ( E (t )) = (φ1T (e1 (t )), φ2T (e2 (t )),L , φ NT (eN (t )))T = F ( X (t )) − F ( X * ), Φ ( E (t − τ 1 )) = F ( X (t − τ 1 )) − F ( X * ), E (t − τ 2 ) = (e1T (t − τ 12 ), e2T (t − τ 22 ),L , eNT (t − τ N 2 ))T .

Assumption 1.The activation function fi ( xi (t )) = ( fi1 ( xi1 (t )), fi 2 ( xi 2 (t )),L , fini ( xini (t )))T

isLipschitz continuous, i.e., there exist constants wil > 0 , such that fil (ξ1 ) − fil (ξ 2 ) ≤ wil ξ1 − ξ 2

where α1 g12 C12 L α1 g1N C1N   α1 g11C11 α g C α L α 2 g 2 N C2 N  2 g 22 C22 , H1 =  2 21 21   M M O M   α N g N 1CN 1 α N g N 2 CN 2 L α N g NN CNN  β1 g12 Γ12 L β1 g1N Γ1N   β1 g11Γ11 β g Γ L β 2 g 2 N Γ 2 N  β 2 g 22 Γ 22 .

H 2 =  2 21 21   M M O M    β N g N 1Γ N 1 β N g N 2 Γ N 2 L β N g NN Γ NN  is Assume that X * = ( x1*T , x2*T ,L , x*N T )T

holdsfor any different ξ1 , ξ 2 ∈ ℜ , and ξ1 ≠ ξ 2 , where i = 1, 2,L , N , l = 1, 2,L , ni .

Definition 1[13]For given k > 0 , the dynamical system (1) is said to be exponential stabilization, if there exist constant Z > 0 such that the following inequality E (t ) 2 ≤ Z ϕ 2 e − kt holds for all initial conditions eij ( s )(i = 1,L , N ; j = 1,L , ni ) of system (7) and any t ≥ T (sufficiently large T > 0 ), where

the equilibrium point of the coupled dynamical system (1), then itsatisfies

N

ϕ =sup −τ ≤ s ≤ 0

ni

∑∑ e ( s) .

2

ij

i =1 j =1

Lemma 1 [4] For any x, y ∈ ℜ n and positive * dX = −( D + K ) X * + Af ( X * ) + Bf ( X * ) + H1 X * + H 2 X * , (6) definite matrix Q ∈ ℜ n× n , the following matrix dt inequality holds: i.e., −( D + K ) X * + Af ( X * ) + Bf ( X * ) + H1 X * + H 2 X * = 0 , 2 xT y ≤ xT Qx + yT Q −1 y . where xi* = ( xi*1 , xi*2 ,L, xin* )T .

Lemma 2[6]The linear matrix inequality Define the linear coordinate transformation  Q( x) S ( x)   S T ( x) R( x)  > 0 , E (t ) = X (t ) − X * , andthe new dynamical systems can   be described as follows: where QT ( x ) = Q ( x) , RT ( x) = R ( x ) , is equivalent to i

dei (t ) = − Di ei (t ) + Aiφi (ei (t )) + Biφi (ei (t − τ i1 )) dt

and Q ( x) − S ( x ) R −1 ( x ) S T ( x) > 0 .

(7) Lemma 3 [5]If Q , R are real symmetric matrices, N N + α i ∑ gij Cij e j (t ) + βi ∑ g ij Γij e j (t − τ i 2 ) + U i (t ) and Q > 0 , R ≥ 0 , then a positive constant σ exist, j =1 j =1 such that the following inequality holds: That is, −Q + σ R < 0 . dE (t ) = − DE (t ) + AΦ ( E (t )) + BΦ( E (t − τ 1 )) ,(8) dt III. EXPONENTIAL STABILIZATION ANALYSIS + H1 E (t ) + H 2 E (t − τ 2 ) + U (t ) Theorem 1.Under the assumption 1, the coupled where T dynamical system (1) is exponential stabilization,if ei (t ) = (ei1 (t ), ei 2 (t ),L , ein (t )) M × M positive definite diagonal matrices there exist = ( xi1 (t ) − xi*1 , xi 2 (t ) − xi*2 ,L , xin (t ) − xin* )T , P , Q , R and K ,such that the following linear φi (ei (t )) = f i ( xi (t )) − f i ( xi∗ ), U i (t ) = − K i ei (t ), matrix inequalities hold: Ξ > 0 ,where R( x) > 0 ,

i

i

i

φi (ei (t − τ i1 )) = f i ( xi (t − τ i1 )) − f i ( xi∗ ), U (t ) = − KE (t ), E (t ) = (e1T (t ), e2T (t ),L , eNT (t ))T , E (t ) = X (t ) − X * ,

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International Journal of Computer Techniques -– Volume 3 Issue 2, Mar-Apr 2016 ψ *  Ξ = *  *  *

PA PB PH1 Q 0 0 * R 0 * * Q * * *

PH 2  0  0  > 0 ,(9)  0  Q 

N

V&3 (t ) = ∑ eλ (t +τ )φiT (ei (t )) Riφi (ei (t )) i =1 N

−∑ eλ (t −τ i1 +τ )φiT (ei (t − τ i1 )) Riφi (ei (t − τ i1 )) (13) i =1

≤ eλ (t +τ ) ΦT ( E (t )) RΦ ( E (t )) −eλt ΦT ( E (t − τ 1 )) RΦ ( E (t − τ 1 ))

ψ = PD + D T P + K + K T − WQW − WRW − 2Q ， Pi = diag ( pi1 , pi 2 ,L , pini ), P = diag ( P1 , P2 ,L , PN ),

From Lemma 1 and Assumption 1, we obtain 2 E T (t ) PAΦ( E (t ))

Qi = diag ( qi1 , qi 2 ,L , qini ), Q = diag (Q1 , Q2 ,L , QN ),

≤ E T (t ) PAQ −1 AT PE (t ) + ΦT ( E (t ))QΦ ( E (t )) (14)

Ri = diag (ri1 , ri 2 ,L , rini ), R = diag ( R1 , R2 ,L , RN ),

≤ E T (t ) PAQ −1 AT PE (t ) + E T (t )WQWE (t ),

Wi = diag ( wi1 , wi 2 ,L , wini ),W = diag (W1 ,W2 ,L ,WN ).

Moreover, the gain matrix of a desired controller of the form (4) is given by K = P −1 K .

Proof. From Lemmas 2 and 3, we get the following conclusion: there exist a positive constant λ , such that ψ  * Ξ1 =  *  * * 

PH 2   0  0 <0,  0  −Q  * λτ where ψ = −ψ + λ P + (e − 1)(Q + WRW ) . *

PA PB PH1 −Q 0 0 * −R 0 * * −Q * * *

≤ E T (t ) PH 2Q −1 ( H 2 )T PE (t ) + E T (t − τ 2 )QE (t − τ 2 ) 2 E T (t ) PH 1 E (t ) ≤ E T (t ) PH1Q −1 ( H1 )T PE (t ) + E T (t )QE (t ) (11)~(17) into V& (t ) , it yields

+ eλτ WRW + Q + PAQ −1 AT P + PBR −1 BT P −1

(16) (17)

−1

T

(18)

T

+ PH1Q ( H1 ) P + PH 2 Q ( H 2 ) P − 2 PK ]E (t )}

From Lemma 2, Ξ1 < 0 is equivalent to ∆ = λ P − P( D + K ) − ( D + K )T P + WQW + eλτ Q +eλτ WRW + Q + PAQ −1 AT P + PBR −1 BT P

N

V1 (t ) = eλt ∑ eiT (t ) Pe i i (t )

+ PH1Q −1 ( H1 )T P + PH 2 Q −1 ( H 2 )T P

i =1

t

2 E T (t ) PH 2 E (t − τ 2 )

(15)

V& (t ) ≤ eλt {E T (t )[λ P − PD − DT P + WQW + eλτ Q

V (t ) = V1 (t ) + V2 (t ) + V3 (t ) ,(10)

N

≤ E T (t ) PBR −1 B T PE (t ) + ΦT ( E (t − τ 1 )) RΦ ( E (t − τ 1 )),

Substituting

Consider the Lyapunovfunction as

V2 (t ) = ∑ ∫

2 E T (t ) PBΦ ( E (t − τ 1 ))

(19)

<0

eλ ( s +τ ) eiT ( s )Qi ei ( s )ds

It is derived that V& (t ) ≤ 0 . Therefore, we know V (t ) is decreasing from t =0 , and V (t ) ≤ V (0) . t V3 (t ) = ∑ ∫ eλ ( s +τ )φiT (ei ( s )) Riφi (ei ( s ))ds . t −τ From the initial conditions (3) of the coupled i =1 Calculating the time derivatives of V1 (t ) , V2 (t ) and dynamical system (1),we obtain the initial conditions of thenew dynamical system (8) are V3 (t ) along the trajectories of system (7), we have N N eij ( s ) = ϕij ( s ) − xij* ( s ) ∈ C ([ −τ , 0 ] , ℜ ) . λt T λt T & & V1 (t ) = λ e ∑ ei (t ) Pe i i (t ) + 2e ∑ ei (t ) Pe i i (t ) (11) It follows from (10) that i =1 i =1 i =1

t −τ i 2

N

i1

= λ eλt E T (t ) PE (t ) + 2eλt E T (t ) PE& (t )

N

N

V&2 (t ) = ∑ eλ ( t +τ ) eiT (t )Qi ei (t )

i =1

N

+∑ ∫

i =1 N

−∑ e

N

V (0) = e0 ∑ eiT (0) Pe i i (0) + ∑ ∫

λ ( t −τ i 2 +τ ) T

(12)

ei (t − τ i 2 )Qi ei (t − τ i 2 )

i =1

≤ eλ (t +τ ) E T (t )QE (t ) − eλt E T (t − τ 2 )QE (t − τ 2 )

i =1

−τ i 1

i =1

e

φi (ei ( s)) Riφi (ei ( s))ds N

i =1

+∑ ∫ i =1

0

−τ i 1

eλ ( s +τ ) eiT ( s)Qi ei ( s )ds

λ ( s +τ ) T

= E T (0) PE (0) + ∑ ∫ N

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0

0

−τ i 2

0

−τ i 2

(20) eλ ( s +τ ) eiT ( s )Qi ei ( s)ds

eλ ( s +τ )φiT (ei ( s)) Riφi (ei ( s))ds

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International Journal of Computer Techniques -– Volume 3 Issue 2, Mar-Apr 2016 N

∑∫ τ 0

−

i =1

eλ ( s +τ ) eiT ( s )Qi ei ( s )ds i2

N

≤ ∑ ∫ eλ ( s +τ ) eiT ( s )Qi ei ( s )ds = eλτ ∫ eλ s E T ( s )QE ( s )ds (21) i =1

0

0

−τ

−τ

0

≤ eλτ λmax (Q ) ∫ eλ s ds ⋅ ϕ

2

=

−τ

N

∑∫ τ i =1

0

−

eλτ − 1

λ

⋅ λmax (Q) ϕ

2

eλ ( s +τ )φiT (ei ( s )) Riφi (ei ( s ))ds i2

0

≤ e λτ ∫ e λ s E T ( s )WRWE ( s )ds −τ

0

≤ e λτ λmax (WRW ) ∫ eλ s ds ⋅ ϕ

(22)

2

−τ

=

eλτ − 1

λ

⋅ λmax (WRW ) ϕ

2

Substituting (21)~(22) into V (0) , we get 2

V (0) ≤ λmax ( P ) ϕ +

eλτ − 1

λ

⋅ [λmax (Q ) + λmax (WRW )] ϕ

(23)

λτ

= {λmax ( P) +

e −1

⋅ [λmax (Q ) + λmax (WRW )]} ϕ λ 2 e λt ⋅ λmin ( P ) E (t ) ≤ V1 (t ) ≤ V (t )

From

2

2

V (t ) ≤ V (0) ,we have

1

λmin ( P)

2

≤ V (0) .

⋅ {λmax ( P)

e λτ − 1

2

⋅ [λmax (Q ) + λmax (WRW )]} ⋅ ϕ e − λt λ 2 = Z ϕ e − λt +

where

Z=

1

λmin ( P )

0.6  -0.8  , -0.2 

0.9  , 0.3   -0.2 -0.4   0.9 -0.9 0.7  Γ11 =   , Γ12 =  -0.6 0.1 0.3  , 0.9 0.4      -0.8 -0.1 0.8   0.3 0.1  C22 = -0.2 -0.3 -0.9  , Γ 21 =  -0.2 -0.1 , -0.6 0.9 0.5   -0.6 -0.7   0.9 0.1 0.7   0.2 0.6  Γ 22 =  0.4 -0.8 0.6  , G =  ,  0.7 0.1 -0.3 0.8 -0.4  α1 = 0.1 , α 2 = 0.2 , β1 = 0.9 , β 2 = 0.1 ,

τ 11 = 0.5 , τ 12 = 0.7 , τ 21 = 0.6 , τ 22 = 0.4 ,

By calculating, we know that equilibrium points of the two isolated node networks (2) are T

x%1 = 10 −7 * ( 0.0111, −0.0034 )

T

Combining with (23) and (24), we obtain 2

0.3 0.7   -0.6  -0.7 0.3  , C21 =  -0.2  0.7 0.4 -0.1 -0.8   0.4 0.3 , C12 =   0.3   -0.7 0.1

x%2 = 10 −7 * ( 0.0756, −0.0117, 0.1720 )

e λt ⋅ λmin ( P ) E (t )

E (t ) ≤

and

-0.8 B2 = -0.5  -0.3  0.1 C11 =   -0.1

⋅ {λmax ( P) +

e

λτ

−1

λ

, (24) respectively. Therefore, it is easy to derive that wil = 1 , and W = I 5 , for i = 1, 2,L , N , l = 1, 2,L , ni .

By using the MATLAB LMI Control Toolbox, solving the linear matrix inequalities in Theorem 1, (25) it yields the following feasible solutions:

⋅ [λmax (Q ) + λmax (WRW )]} .

P = diag ( 3.7960,3.4276,3.9682, 4.5893,5.0527 )

Q = diag ( 20.2148, 20.3078, 20.4814, 20.5548, 20.2048 ) R = diag ( 20.2106, 20.2726, 20.3884, 20.4373, 20.2040 )

K = diag ( 20.1689,19.9210,19.4580,19.2622, 20.1957 )

That is, the coupled dynamical system (1) with Then, we have nodes of different dimensions and time delays is K = diag ( 5.3132,5.8119, 4.9035, 4.1972,3.9970 ) . exponential stabilization.This completes the proof.

According to Theorem 1, given the initial IV. NUMERICAL EXAMPLES condition: X (0) = (0.5, - 0.2, - 0.3, 0.2, 0.5)T , a coupled dynamical Example 1.Consider system (1) is composed of two isolated node the coupled dynamical system (1) can achieve the networks, in which the parameters are given as exponential stabilization, and the equilibrium point is follows: X ∗ = 10 −10 * (0.0913, −0.0236, 0.1704, −0.0514, 0.3277)T , N = 2 , n1 = 2 , n2 = 3 , f ( xil ) = sin( xil ) , i = 1, 2 , l = 1,..., ni , the simulation results are plotted in Fig. 1. All state 8 0   0.2 -0.1  -0.9 0.8  D1 =   , A1 =  -0.5 0.9  , B1 =  0.7 0.6  , trajectories converge to the equilibrium point, and 0 9       the equilibrium point of the coupled dynamical 3 0 0  0.1 -0.1 -0.2  system (1) is different from that of each isolated D2 = 0 7 0  , A2 =  0.1 0.2 -0.3 , node network (2).     0 0 6 

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 -0.5

0.4

0.9 

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International Journal of Computer Techniques -– Volume 3 Issue 2, Mar-Apr 2016

respectively. Therefore, it is easy to derive that wil = 2 , and W = 2 I 5 ,for i = 1, 2,L , N , l = 1, 2,L , ni .

By using the MATLAB LMI Control Toolbox, solving the linear matrix inequalities Ξ > 0 in Theorem 1, it yields the following feasible solutions:

0.6 x11(t)

x 1 (t)

0.4

x12(t)

0.2 0 -0.2

0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

Q = diag ( 24.8830, 23.2186, 22.5922, 24.6529, 26.5236 )

1

x 2 (t)

0.5

x 21(t)

R = diag ( 24.6942, 22.3220, 21.7158, 26.2661, 29.3649 )

x 22(t)

K = diag ( 44.4094,53.7464, 62.1294, 47.1680,39.8634 )

x 23(t)

Then, we have

0

-0.5

P = diag ( 3.5079, 2.2782, 2.0888, 4.6528,3.3211)

K = diag (12.6598, 23.5912, 29.7447,10.1376,12.0031) .

0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

the initial condition T X (0) = (1, −1, −2, 2,1) ,according to Theorem 1,the Fig. 1.The state trajectories of coupled coupled dynamical system (1) can achieve the dynamical system in Example 1. exponential stabilization and the equilibrium point Example 2. Consider a coupled dynamical system (1) is composed of two isolated node is X ∗ = ( −0.0376, 0.1092, −0.1181, 0.1190, 0.0145)T , networks, in which the parameters are given as the simulation results are plotted in Fig. 2. All state follows: trajectories converge to the equilibrium point, and N = 2 , n1 = 2 , n2 = 3 , f ( xil ) = sin( xil )+1 , the equilibrium point of the coupled dynamical i = 1, 2 , l = 1,..., ni , system (1) is different from that of each isolated  29 0  1 -1  -3 1  D1 =  A = B = node network (2). , 1  , 1  ,

 3 -3 Γ 21 =  2 1  ,  -3 5 

-4 

4

5

1 4 -5 -5 4  1 -1 , B2 =  1 2 4  ,  2 -4 3  1 1   4 -2 -4  = ,  -2 -4 2 

-2   3 -5 -5  , Γ12 =   , 2  -1 -3 1  4 -4 2 1   -2  , C22 =  3 -5 4 , -4 4 4  4  -4 3 -3  0.9 0.7  Γ 22 =  -5 2 3  , G =  ,  0.1 0.4  -2 4 3 

α1 = 0.1 , α 2 = 0.4 , β1 = 0.9 , β 2 = 0.6 , τ 11 = 0.5 , τ12 = 1 , τ 21 = 0.8 , τ 22 = 0.4 ,

1.5

x 1(t)

5  -3 A2 =  1  -2 -1 , C12 -2 

1

x 11(t)

0.5

x 12(t)

0 -0.5 -1 -1.5

x 2(t)

 0 36  33 0 0  D2 =  0 21 0  ,  0 0 34   -5 C11 =  1  -1 Γ11 =   -1 1 C21 = -5  2

Given

0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

3

x 21(t)

2

x 22(t)

1

x 23(t)

5

0 -1 -2 0

0.5

1

1.5

2

2.5 t

3

3.5

4

4.5

5

Fig. 2.The state trajectories of coupled dynamical system in Example 2.

V. CONCLUSION By calculating, we know that equilibrium points In this paper, the exponential stabilization of the two isolated node networks (2) are problem has been investigated for the coupled T neural networks. Thedimensions of states in each x%1 = ( −0.0645, 0.2690 ) isolated network can be different, and dimensions T x%2 = ( −0.3305, 0.3848, −0.0039 ) , of nodes are actually differentin many practical

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International Journal of Computer Techniques -– Volume 3 Issue 2, Mar-Apr 2016

situations, so the derived results will have wider applicability.

The criteria of exponentialstabilization for the coupled dynamical system are established on basis of the linear matrix inequality andLyapunovfunction. The effectiveness of the proposed theoretical results has been demonstrated by two numerical simulation examples. In addition, finite-time control could be an alternative method for the exponentialstabilization of such kind of coupled dynamical system, which deserves our further investigation.

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