Stability of Impulsive Cellular Neural Networks with Time-varying Delays
704
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
Stability of Impulsive Cellular Neural Networks with Time-varying Delays Yuanqiang Chen College, Guizhou Minzu University, Guiyang550025, China Email: [email protected]
Abstract—The problems of exponential stability and exponential convergence rate for a class of impulsive cellular neural networks with time-varying delays are studied. By means of the Lyapunov stability theory and discrete-time Halanay-type inequality technique, stability criteria for ensuring global exponential stability of noimpulsive discrete-time cellular neural networks and impulsive discrete-time cellular neural network with timevarying delays are derived respectively, and the estimated exponential convergence rate is provided as well. Finally, the validity of the obtained results is shown by two numerical examples. Index Terms—Cellular neural networks, stability, Discrete-time Halanay inequality
I. INTRODUCTION
Exponential
stability of CNN and DCNN is known to be an important problem in theory and applications. However, the state of electronic networks is often subject to instantaneous perturbations and experience abrupt changes at certain instants [5]. Since delays and impulses can affect the dynamical behaviors of the system, it is necessary to investigate both delay and impulsive effects on the stability of cellular neural networks. A large number of the criteria on the stability of DCNN without impulse has been derived (see [6-7]).Correspondingly, there is not much work dedicated to investigate the stability of impulsive discrete-time cellular neural networks. Recently, some mathematical models of impulsive cellular neural networks described by measure differential equations or general impulsive differential equations have been formulated ([1],[2],[8-25]). Some interesting results on the stability of impulsive discretetime cellular neural networks without delay or with constant delays have been obtained. However, in practical evolutionary processes of the networks, absolute constant delay may be scarce and delays are frequently varied with time. In this paper, we shall deal with a class of discrete-time cellular neural networks with time-varying delays subject to impulsive perturbations:
In 1988, a novel class of information-processing systems called cellular neural networks (CNN) been proposed [1-3]. Their key features are asynchronous parallel processing and global interaction of network elements. In many evolutionary systems there are two common phenomena: delay effects and impulsive effects. In implementation of electronic networks, for example, delays frequently appear because of the finite switching n speed of amplifiers. On the other hand, the state of + 1) ci xi ( m ) + ∑ aij f j ( x j ( m ) ) xi ( m = electronic networks is often subject to instantaneous j =1 perturbations and experience abrupt change at certain n instants which may be caused by switching phenomenon, + ∑ bij f j x j ( m − τ j ( m ) ) , m ≠ N k , j =1 frequency change or other sudden noise, that is, do (1) exhibit impulsive effects. Even in biological neural ∆xi ( m= ) xi ( m + 1) − xi ( m=) Pi ( xi ( m ) ) , networks, impulsive effects are likely to exist. For m =N k , m ∈ N (1) , k ∈ N ( 0 ) , instance, when a stimulus from the body or the external xi = ( m ) φi ( m ) , m ∈ N ( −τ , 0 ) , i ∈ N (1, n ) , environment is received by receptors the electrical impulses will be conveyed to the neural net and impulsive = where x ( x1 , x2 , , xn ) ∈ R n is the state of system, effects arise naturally in the net. Therefore, neural continuous function f j : R → R denote the output of the network model with delays and impulsive effects should be more accurate to describe the evolutionary process of j th unit and f j ( 0 ) = 0 ， ci , aij , bij are constants, ci , the systems. Since delays and impulses can affect the 0 ≤ ci < 1 , represents the rate with which the i th unit will dynamical behaviors of the system by creating oscillatory reset its potential to the resting state in isolation when and unstable characteristics, it is necessary to investigate disconnected from the network and external inputs, both delay and impulsive effects on the stability of neural aij , bij denote the strength of the j th unit on the i th unit networks. Cellular neural networks and delayed cellular neural networks (DCNN) have been applied to various at time m and m − τ j ( m ) respectively, τ j ( m ) fields such as linear and nonlinear programming, corresponds to the transmission delay along the axon of optimization, pattern recognition, associative memory the j th unit and satisfies 0 ≤ τ j ( t ) ≤ τ ( τ is integer) and and computer vision [4]. Therefore, the study of the
(
© 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.3.704-711
)
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
705
m → ∞ as m − τ ij ( m ) → ∞ , integer N k , k ∈ N ( 0 ) are the moments of impulsive perturbations and satisfy 0 < N 0 < N1 < and k → ∞ as N k → ∞ , continuous function
Pi ( xi ( N k ) ) : R → R
represents
the
abrupt
change of the state at the impulsive moment N k and
Pi ( 0 ) = 0 , φ j : R → R is continuous function,
N ( k ) = {k , k + 1, k + 2,} , N ( k , l ) = {k , k + 1, k + 2, , l} . By utilizing the Lyapunov stability theory and discretetime Halanay-type inequality, we shall obtain global exponential stability criteria and estimated exponential convergence rate for impulsive discrete-time cellular neural networks with time-varying delays. The rest of this paper is organized as follows. In Section II, impulsive cellular neural networks with time-varying delays are introduced and some preliminary lemmas are presented. In Section III, based on the Lyapunov stability theory and discrete-time Halanay-type inequality technique, global exponential stability criteria for ensuring global exponential stability of no-impulsive discrete-time cellular neural networks and impulsive discrete-time cellular neural network with time-varying delays are derived respectively. Moreover, two numerical examples are presented in Sections IV. Section V concludes the paper.
Without loss of generality, we may assume system (1) satisfies the following assumptions. Assumption A: The sequence {N k } of the impulsive time points satisfies N k + 2 < N k +1 . Assumption B: For the impulsive increment function sequence Pi ( xi ( N k ) ) , there exists wik > 0 , such that xi ( t ) , t ∈ N ( 0 ) , the following condition is satisfied,
xi ( t ) + Pi ( xi ( t ) ) ≤ wik xi ( t ) , i ∈ N (1, n ) .
(3)
Assumption C: There exists a constant l j > 0 , such that ∀t1 , t2 ∈ N ( 0 ) , the function f j ( m ) in system (1) is bounded and satisfies the following Lipschitz condition, (4) f j ( t1 ) − f j ( t2 ) ≤ l j t1 − t2 , j ∈ N (1, n ) . III. MAIN RESULTS In this section, we shall investigate the global exponential stability criteria and the estimated exponential convergence rate of impulsive discrete-time cellular neural networks. The following theorem presents the results on the global exponential stability of the noimpulsive system (1). Consider a discrete-time neural network described by
xi ( m = + 1) ci xi ( m ) + ∑ aij f j ( x j ( m ) ) n
II. PRELIMINARIES
j =1
(
)
+ ∑ bij f j x j ( m − τ j ( m ) ) , m ∈ N (1) , n
Let us introduce the following necessary definitions and lemmas. Lemma 1: [26] (Discrete-time Halanay-type Inequality) Suppose that the real numbers sequence {α n }n ≥− h
j =1
(5)
xi = ( m ) φi ( m ) , m ∈ N ( −τ , 0 ) , i ∈ N (1, n ) , The following theorem presents the results on the satisfied global exponential stability of the system (5). ∆α n = −εα n + g ( n, α n , α n −1 , , α n − h ) , n ∈ N (1) , ε ∈ ( 0,1] , Theorem 1: Suppose that Assumption C and the following inequality is satisfied, if there exists a δ ∈ ( 0, ε ) such that (6) c +l 0 and r ∈ ( 0,1) such that
x ( m ) ≤ Kr , ∀m ∈ N (1) , m
Then, the trivial equilibrium point of (5) is globally exponentially stable with the convergence rate λ , which is the smallest root in the interval ( 0,1) of the following equation
λ τ +1 + cλ τ − l =0 . Proof: From the trajectory { x ( m )} , m ∈ N (1) of system (5), we have m −1
= xi ( m ) cim xi ( 0 ) + ∑ cim −1− s ∑ aij f j ( x j ( s ) )
(2)
n
=s 0=j 1
Where r is called the exponential convergence rate. If (2) is satisfied for any initial condition x ( m ) ∈ R n , m ∈ N ( −τ ,0 ) , the
m −1
(
)
+ ∑ cim −1− s ∑ bij f j x j ( s − τ j ( s ) ) , n
= s 0=j 1
trivial equilibrium point is globally exponentially stable for the impulsive discrete-time cellular neural network (1).
m ∈ N (1) , i ∈ N (1, n ) , Then,
© 2013 ACADEMY PUBLISHER
}} .
706
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
xi ( m ) ≤ ci
m −1
xi ( 0 ) + ∑ ci
m
m − s −1
xi ( m = + 1) ci xi ( m ) + ∑ aij f j ( x j ( m ) ), m ∈ N (1) ,
(∑ aij f j ( x j ( s ) )
n
n
j =1
=s 0=j 1
(
xi (= 0 ) x0 , i ∈ N (1, n ) , Corollary1. Suppose that Assumption C and the following inequality is satisfied, c +l
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
Stability of Impulsive Cellular Neural Networks with Time-varying Delays Yuanqiang Chen College, Guizhou Minzu University, Guiyang550025, China Email: [email protected]
Abstract—The problems of exponential stability and exponential convergence rate for a class of impulsive cellular neural networks with time-varying delays are studied. By means of the Lyapunov stability theory and discrete-time Halanay-type inequality technique, stability criteria for ensuring global exponential stability of noimpulsive discrete-time cellular neural networks and impulsive discrete-time cellular neural network with timevarying delays are derived respectively, and the estimated exponential convergence rate is provided as well. Finally, the validity of the obtained results is shown by two numerical examples. Index Terms—Cellular neural networks, stability, Discrete-time Halanay inequality
I. INTRODUCTION
Exponential
stability of CNN and DCNN is known to be an important problem in theory and applications. However, the state of electronic networks is often subject to instantaneous perturbations and experience abrupt changes at certain instants [5]. Since delays and impulses can affect the dynamical behaviors of the system, it is necessary to investigate both delay and impulsive effects on the stability of cellular neural networks. A large number of the criteria on the stability of DCNN without impulse has been derived (see [6-7]).Correspondingly, there is not much work dedicated to investigate the stability of impulsive discrete-time cellular neural networks. Recently, some mathematical models of impulsive cellular neural networks described by measure differential equations or general impulsive differential equations have been formulated ([1],[2],[8-25]). Some interesting results on the stability of impulsive discretetime cellular neural networks without delay or with constant delays have been obtained. However, in practical evolutionary processes of the networks, absolute constant delay may be scarce and delays are frequently varied with time. In this paper, we shall deal with a class of discrete-time cellular neural networks with time-varying delays subject to impulsive perturbations:
In 1988, a novel class of information-processing systems called cellular neural networks (CNN) been proposed [1-3]. Their key features are asynchronous parallel processing and global interaction of network elements. In many evolutionary systems there are two common phenomena: delay effects and impulsive effects. In implementation of electronic networks, for example, delays frequently appear because of the finite switching n speed of amplifiers. On the other hand, the state of + 1) ci xi ( m ) + ∑ aij f j ( x j ( m ) ) xi ( m = electronic networks is often subject to instantaneous j =1 perturbations and experience abrupt change at certain n instants which may be caused by switching phenomenon, + ∑ bij f j x j ( m − τ j ( m ) ) , m ≠ N k , j =1 frequency change or other sudden noise, that is, do (1) exhibit impulsive effects. Even in biological neural ∆xi ( m= ) xi ( m + 1) − xi ( m=) Pi ( xi ( m ) ) , networks, impulsive effects are likely to exist. For m =N k , m ∈ N (1) , k ∈ N ( 0 ) , instance, when a stimulus from the body or the external xi = ( m ) φi ( m ) , m ∈ N ( −τ , 0 ) , i ∈ N (1, n ) , environment is received by receptors the electrical impulses will be conveyed to the neural net and impulsive = where x ( x1 , x2 , , xn ) ∈ R n is the state of system, effects arise naturally in the net. Therefore, neural continuous function f j : R → R denote the output of the network model with delays and impulsive effects should be more accurate to describe the evolutionary process of j th unit and f j ( 0 ) = 0 ， ci , aij , bij are constants, ci , the systems. Since delays and impulses can affect the 0 ≤ ci < 1 , represents the rate with which the i th unit will dynamical behaviors of the system by creating oscillatory reset its potential to the resting state in isolation when and unstable characteristics, it is necessary to investigate disconnected from the network and external inputs, both delay and impulsive effects on the stability of neural aij , bij denote the strength of the j th unit on the i th unit networks. Cellular neural networks and delayed cellular neural networks (DCNN) have been applied to various at time m and m − τ j ( m ) respectively, τ j ( m ) fields such as linear and nonlinear programming, corresponds to the transmission delay along the axon of optimization, pattern recognition, associative memory the j th unit and satisfies 0 ≤ τ j ( t ) ≤ τ ( τ is integer) and and computer vision [4]. Therefore, the study of the
(
© 2013 ACADEMY PUBLISHER doi:10.4304/jnw.8.3.704-711
)
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
705
m → ∞ as m − τ ij ( m ) → ∞ , integer N k , k ∈ N ( 0 ) are the moments of impulsive perturbations and satisfy 0 < N 0 < N1 < and k → ∞ as N k → ∞ , continuous function
Pi ( xi ( N k ) ) : R → R
represents
the
abrupt
change of the state at the impulsive moment N k and
Pi ( 0 ) = 0 , φ j : R → R is continuous function,
N ( k ) = {k , k + 1, k + 2,} , N ( k , l ) = {k , k + 1, k + 2, , l} . By utilizing the Lyapunov stability theory and discretetime Halanay-type inequality, we shall obtain global exponential stability criteria and estimated exponential convergence rate for impulsive discrete-time cellular neural networks with time-varying delays. The rest of this paper is organized as follows. In Section II, impulsive cellular neural networks with time-varying delays are introduced and some preliminary lemmas are presented. In Section III, based on the Lyapunov stability theory and discrete-time Halanay-type inequality technique, global exponential stability criteria for ensuring global exponential stability of no-impulsive discrete-time cellular neural networks and impulsive discrete-time cellular neural network with time-varying delays are derived respectively. Moreover, two numerical examples are presented in Sections IV. Section V concludes the paper.
Without loss of generality, we may assume system (1) satisfies the following assumptions. Assumption A: The sequence {N k } of the impulsive time points satisfies N k + 2 < N k +1 . Assumption B: For the impulsive increment function sequence Pi ( xi ( N k ) ) , there exists wik > 0 , such that xi ( t ) , t ∈ N ( 0 ) , the following condition is satisfied,
xi ( t ) + Pi ( xi ( t ) ) ≤ wik xi ( t ) , i ∈ N (1, n ) .
(3)
Assumption C: There exists a constant l j > 0 , such that ∀t1 , t2 ∈ N ( 0 ) , the function f j ( m ) in system (1) is bounded and satisfies the following Lipschitz condition, (4) f j ( t1 ) − f j ( t2 ) ≤ l j t1 − t2 , j ∈ N (1, n ) . III. MAIN RESULTS In this section, we shall investigate the global exponential stability criteria and the estimated exponential convergence rate of impulsive discrete-time cellular neural networks. The following theorem presents the results on the global exponential stability of the noimpulsive system (1). Consider a discrete-time neural network described by
xi ( m = + 1) ci xi ( m ) + ∑ aij f j ( x j ( m ) ) n
II. PRELIMINARIES
j =1
(
)
+ ∑ bij f j x j ( m − τ j ( m ) ) , m ∈ N (1) , n
Let us introduce the following necessary definitions and lemmas. Lemma 1: [26] (Discrete-time Halanay-type Inequality) Suppose that the real numbers sequence {α n }n ≥− h
j =1
(5)
xi = ( m ) φi ( m ) , m ∈ N ( −τ , 0 ) , i ∈ N (1, n ) , The following theorem presents the results on the satisfied global exponential stability of the system (5). ∆α n = −εα n + g ( n, α n , α n −1 , , α n − h ) , n ∈ N (1) , ε ∈ ( 0,1] , Theorem 1: Suppose that Assumption C and the following inequality is satisfied, if there exists a δ ∈ ( 0, ε ) such that (6) c +l 0 and r ∈ ( 0,1) such that
x ( m ) ≤ Kr , ∀m ∈ N (1) , m
Then, the trivial equilibrium point of (5) is globally exponentially stable with the convergence rate λ , which is the smallest root in the interval ( 0,1) of the following equation
λ τ +1 + cλ τ − l =0 . Proof: From the trajectory { x ( m )} , m ∈ N (1) of system (5), we have m −1
= xi ( m ) cim xi ( 0 ) + ∑ cim −1− s ∑ aij f j ( x j ( s ) )
(2)
n
=s 0=j 1
Where r is called the exponential convergence rate. If (2) is satisfied for any initial condition x ( m ) ∈ R n , m ∈ N ( −τ ,0 ) , the
m −1
(
)
+ ∑ cim −1− s ∑ bij f j x j ( s − τ j ( s ) ) , n
= s 0=j 1
trivial equilibrium point is globally exponentially stable for the impulsive discrete-time cellular neural network (1).
m ∈ N (1) , i ∈ N (1, n ) , Then,
© 2013 ACADEMY PUBLISHER
}} .
706
JOURNAL OF NETWORKS, VOL. 8, NO. 3, MARCH 2013
xi ( m ) ≤ ci
m −1
xi ( 0 ) + ∑ ci
m
m − s −1
xi ( m = + 1) ci xi ( m ) + ∑ aij f j ( x j ( m ) ), m ∈ N (1) ,
(∑ aij f j ( x j ( s ) )
n
n
j =1
=s 0=j 1
(
xi (= 0 ) x0 , i ∈ N (1, n ) , Corollary1. Suppose that Assumption C and the following inequality is satisfied, c +l
