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Exponential stability in Hopfield-type neural networks with impulses
Chaos, Solitons and Fractals 32 (2007) 456–467 www.elsevier.com/locate/chaos
Exponential stability in Hopfield-type neural networks with impulses Sannay Mohamad
*
Department of Mathematics, Faculty of Science, Universiti Brunei Darussalam, Jalan Tunku Link BE1410, Brunei Darussalam Accepted 8 June 2006
Communicated by Prof. M.S. El Naschie
Abstract This paper demonstrates that there is an exponentially stable unique equilibrium state in a Hopfield-type neural network that is subject to quite large impulses that are not too frequent. The activation functions are assumed to be globally Lipschitz continuous and unbounded. The analysis exploits an homeomorphic mapping and an appropriate Lyapunov function, and also either a geometric–arithmetic mean inequality or a Young inequality, to derive a family of easily verifiable sufficient conditions for convergence to the unique globally stable equilibrium state. These sufficiency conditions, in the norm kÆkp where p P 1, include those governing the network parameters and the impulse magnitude and frequency. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Impulses are ubiquitous in both biological and artificial neural networks. For example, peaceful sleep may be suddenly interrupted, perhaps by a nightmare. However, even if there are abrupt changes in the neural state, a person may resume a deeper sleep provided the interruption is not too great nor too frequent. In the case of an artificial network for signal processing, faulty elements can produce sudden changes in the state voltages and thereby affect the normal transient behaviour in processing signals or information, and robust system design is important. Neural networks perceived as either continuous or discrete dynamical systems have been studied extensively, but the mathematical modelling of dynamical systems with impulses is a quite recent development [1,2,14–16,24–26,31]. In this paper, we demonstrate the exponential stability of a unique equilibrium state in a Hopfield-type neural network [17] consisting of m processing units, subjected to impulsive state displacements. Phenomena such as those mentioned above may be interpreted as large impulses affecting otherwise normal transient behaviour, but conditions for a neural network to resist impulses of significant magnitude do not appear to have been found before.
*
Tel.: +673 2 463001; fax: +673 2 461502. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.035
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
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2. Mathematical model The network is described by m X dxi ðtÞ bij fj ðxj ðtÞÞ þ ci ; ¼ ai xi ðtÞ þ dt j¼1
t > t0 ; t 6¼ tk ;
and Dxi jt¼tk ¼ xi ðtþ k Þ xi ðt k Þ ¼ I k ðxi ðt k ÞÞ;
subject to xðt0 Þ ¼ x0 2 Rm ;
ð2:1Þ k ¼ 1; 2; 3; . . .
1 Here xi ðtþ k Þ xi ðt k þ 0Þ, xi ðt k Þ xi ðt k 0Þ for i 2 I ¼ f1; 2; . . . ; mg and the sequence of times ft k gk¼1 satisfies
t0 < t 1 < t 2 < < tk ! 1
as k ! 1
and Dtk ¼ tk tk1 P h
for k = 1, 2, 3, . . ., where the value h > 0 denotes the minimum time of interval between successive impulses. A sufficiently large value of h ensures that impulses do not occur too frequently (see later), but h ! 1 means that the network (2.1) becomes impulse free. We refer to Gopalsamy [14] for earlier discussion of this model formulation and its application. The vector solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xm ðtÞÞT 2 Rm of (2.1) has components xi(t) piece-wise continuous on (t0, b) for some b > t0, such that xðtþ k Þ and xðt k Þ exist and x(t) is differentiable on the open intervals (tk1, tk) (t0, b). Further, we assume that x(t) is right continuous with xðtk Þ ¼ xðtþ k Þ; the functions I k : R ! R that characterize the impulsive jumps are continuous; the neural parameters ai, bij, ci satisfy ai > 0; bij ; ci 2 R; and the activation functions fj : R ! R with fj(0) = 0 may be unbounded and continuous but satisfy jfj ðuÞ fj ðvÞj 6 Lj ju vj for all u; v; 2 R; jfj ðuÞj ! 1 as juj ! 1;
ð2:2Þ
where Lj > 0 for j 2 I denotes a Lipschitz constant. Sufficiency conditions on the neural parameters and the impulses have previously been obtained to guarantee the asymptotic convergence towards a unique equilibrium state of the network (2.1), associated with the norms kÆk1, kÆk2 and kÆk1 [14]. However, the asymptotic stability of this equilibrium state is guaranteed only if the magnitudes of the impulses are neither large nor frequent. Although that is of course consistent with the view that a dynamical system tends to become unstable when subjected to sufficiently frequent impulses of large magnitude [4,22,30], in this paper a family of easily verifiable sufficient conditions on the neural parameters and the impulses is found to more generally guarantee the exponential convergence of the neural states towards the unique equilibrium state, and in any norm kÆkp where p P 1. The results obtained by applying a geometric–arithmetic mean inequality and a Young inequality to an appropriate Lyapunov function significantly enhance the earlier work, both with and without impulses.
3. Existence and uniqueness theorems To provide for an associative memory [8,29,34,37], the network architecture is designed to not only store as many equilibrium states (or memories) as possible, but also to retrieve the relevant stored memory produced by a given external stimulus c = (c1, c2, . . . , cm)T. Thus if a neural network is intended to solve an optimization problem, the circuit design of the network should ensure that all neural states approach a unique equilibrium state of the network [6,13,21,23,32]. The association of an equilibrium state with an external input vector c = (c1, c2, . . . , cm)T avoids the possibility of convergence towards some local minimum that is a spurious equilibrium state, and not the optimal solution of the optimization problem. To ensure that a unique equilibrium state exists, it has been customary to impose restrictions on the neural parameters and the activation functions of the network, such assuming they are bounded and Lipschitz continuous [11] – i.e. gj ð0Þ ¼ 0;
jgj ðuÞj 6 M j ;
jgj ðuÞ gj ðvÞj 6 Lj ju vj
for some positive constants Mj, Lj and any u; v 2 R. Stronger requirements have been adopted [3,9,10,12,18,20,27, 35,36,39], where the activation functions are assumed to be bounded, continuous, monotonic and differentiable – i.e. gj 2 C 1 ðRÞ, g0j ðuÞ > 0 for u 2 R, g0j ð0Þ ¼ supu2R g0j ðuÞ > 0, gj(0) = 0 and gj(u) ! ±1 as u ! ±1. Recent applications, in which the activation functions are either linear and piece-wise continuous or Gaussian, have indicated a number of advantages for removing the differentiability and monotonicity requirements [19,29,34,37]. And in applying a neural network to solve a certain class of optimization problems, Forti et al. [13] have used activation functions of exponentialtype and diode-like, which are unbounded to suit the unbounded constraint requirement of the problems considered.
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S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
One should not overlook the possibility that unbounded activation functions may result in the nonexistence of an equilibrium state of the network, and in any case it is usually important to establish the existence of a unique equilibrium state. In this Section, sufficiency conditions are established for the existence of a unique equilibrium state of the impulsive network (2.1) assuming the functions fj(Æ) satisfy (2.2). Let us denote an equilibrium state of the network (2.1) by the constant vector x ¼ ðx1 ; x2 ; . . . ; xm ÞT 2 Rm , where each xi is governed by the algebraic system 0 ¼ ai xi þ
m X
bij fj ðxj Þ þ ci ;
i 2 I:
ð3:1Þ
j¼1
Here it is assumed that the impulsive jumps Ik(Æ) satisfy I k ðxi Þ ¼ 0 for i 2 I; k 2 Zþ . Gopalsamy [14] used a contraction mapping principle to derive sufficient conditions for the existence of a unique equilibrium state in the impulsive network (2.1), under the usual norms kÆk1, kÆk2 and kÆk1. A result due to Forti et al. [13] and Zhang is used here to prove there is a unique solution of the system (3.1) under the general P[38] m p 1=p , where p P 1 and u ¼ ðu1 ; u2 ; . . . ; um ÞT 2 Rm . Recall that a map h : Rm ! Rm Euclidean norm kukp ¼ i¼1 jui j is a homeomorphism (Rm onto itself) if h 2 C0 is one-to-one and onto, and the inverse map h1 2 C0. The result is that the map h 2 C0 is a homeomorphism on Rm if it is injective on Rm and satisfies limkuk!1kh(u)k ! 1. However, whereas Forti et al. [13] and Zhang [38] employed matrices with negative principal minors to obtain their results under the norm kÆk2, we use certain inequalities of nonnegative real numbers as mentioned earlier. The first invoked in this Section is the geometric–arithmetic mean inequality [5] – namely that, if n1, n2, . . . , np denote p nonnegative real numbers, then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi np þ np þ þ np 1 2 p p ðnp1 np2 npp Þ 6 p and hence n1 n2 np 6
np1 þ np2 þ þ npp ; p
ð3:2Þ
where p P 1 denotes an integer. A particular form of (3.2), namely fp1 1 f2 6
p1 p 1 p f þ f p 1 p 2
for p ¼ 1; 2; 3; . . .
has previously been used to investigate the exponential stability characteristics of a unique equilibrium state of a delayed cellular neural network [28]. Theorem 3.1. Let p be a positive integer, ai > 0; bij ; ci 2 R, and the activation functions fj(Æ) satisfy (2.2). Suppose the condition m m 1X 1X aj pr pr pr ðjbij jpd1;ij Lj 1;ij þ þ jbij jpdp1;ij Lj p1;ij Þ þ jbji jpdp;ji Li p;ji ð3:3Þ ai > p j¼1 p j¼1 ai for real numbers that satisfy Pp i 2 I is satisfied, Pp where the ai are positive numbers, and dl,ij, rl,ij, for l = 1, 2, . . . , p denote * l¼1 dl;ij ¼ 1 and l¼1 rl;ij ¼ 1, respectively. Then there exists a unique equilibrium state x of the impulsive network (2.1). Proof. Consider a map hðuÞ ¼ ðh1 ðuÞ; h2 ðuÞ; . . . ; hm ðuÞÞT 2 C 0 ðRm ; Rm Þ, where m X bij fj ðuj Þ þ ci hi ðuÞ ¼ ai ui þ
ð3:4Þ
j¼1
for ui 2 R, i 2 I. This map h is a homeomorphism on Rm if it injective on Rm and satisfies kh(u)kp ! 1 as kukp ! 1. The injective part of the map h, namely h(u) = h(v) implies u = v for any u; v 2 Rm , is shown as follows. Since ai ui þ
m X
bij fj ðuj Þ þ ci ¼ ai vi þ
j¼1
j¼1
and consequently ai jui vi j 6
m X
m X j¼1
jbij jLj juj vj j;
bij fj ðvj Þ þ ci
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
459
under the assumption (2.2). Now m X
pai ai jui vi jp 6
i¼1
m X i¼1
¼
m X
m X
pai
( ai
jbij jLj jui vi jp1 juj vj j
j¼1 m X
i¼1
) pðjbij j
dl;ij
r Lj l;ij jui
dp1;ij
vi jÞ ðjbij j
r Lj p1;ij jui
vi jÞðjbij j
dp;ij
r Lj p;ij juj
vj jÞ ;
ð3:5Þ
j¼1
where p denotes a positive i are positive real numbers, and dl,ij, rl,ij for l ¼ 1; 2; . . . ; p; i; j 2 I denote real P integer, the aP numbers that satisfy pl¼1 dl;ij ¼ 1 and pl¼1 rl;ij ¼ 1. By applying the geometric–arithmetic mean inequality (3.2) to (3.5), we obtain ( ) m m m X X X p pd1;ij pr1;ij p pdp1;ij prp1;ij p pdp;ij prp;ij p pai ai jui vi j 6 ai ðjbij j Lj jui vi j Þ þ þ ðjbij j Lj jui vi j Þ þ ðjbij j Lj jui vi j Þ i¼1
i¼1
¼
m X
j¼1
( pai
i¼1
) m m 1X 1X aj pdpl;ij prp1;ij pdp1;ij prp1;ij pdp;ji prp;ji jui vi jp ðjbij j Lj þ þ jbij j Lj Þþ jbji j Lj p i¼1 p j¼1 ai
implying that p
m X
ai jui vi jp 6 0;
i¼1
where the number * defined by ( ) m m 1X 1X aj pd1;ij pr1;ij pdp1;ij prp1;ij pdp;ji prp;ji ¼ min ai ðjbij j Lj þ þ jbij j Lj Þ jbji j Li i2I p j¼1 p j¼1 ai
ð3:6Þ
is positive in accordance with the condition (3.3). Consequently, ui = vi for i 2 I (i.e. u = v), hence the map h is injective on Rm . To demonstrate the property kh(u)kp ! 1 as kukp ! 1, we consider the map ^ hðuÞ ¼ hðuÞ hð0Þ – i.e. m X ^hi ðuÞ ¼ hi ðuÞ hi ð0Þ ¼ ai ui þ bij fj ðuj Þ for ui 2 R; i 2 I: j¼1
Then m X
pai jui j
p1
sgnðui Þ^hi ðuÞ ¼
m X
i¼1
i¼1
6
m X
( pai jui j
p1
sgnðui Þ ai ui þ
m X
) bij fj ðuj Þ
j¼1
( p
ai pai jui j þ
i¼1
m X
p1
pjbij jLj jui j
) juj j :
j¼1
Applying the geometric–arithmetic mean inequality (3.2) as before, ( ) m m m X X 1X aj pdp1;ij prp1;ij pdp;ji prp;ji 1 ;ij jui jp pai jui jp1 sgnðui Þh^i ðun Þ 6 pai ai jbij jpd1 ;ij Lpr þ þ jb j L þ jb j L ij ji j j j p a i i¼1 i¼1 j¼1 6 p
m X
ai jui jp :
i¼1
We have p minfai g i2I
m X i¼1
jui jp 6
m X
pai jui jp1 sgnðui Þ^hi ðuÞ 6 p maxfai g
i¼1
which then gives m m X b X jui jp 6 jui jp1 j^hi ðuÞj; i¼1 i¼1
i2I
m X i¼1
hi ðuÞj jui jp1 ^
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S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
where b ¼ maxi2I fai g=mini2I fai g P 1. Applying a Ho¨lder inequality to the above, !1=q !1=p m m m X X b X p ðp1Þq p ^ jui j 6 jui j jhi ðuÞj ; i¼1 i¼1 i¼1 where 1p þ 1q ¼ 1 which in turn yields b ^ khðuÞkp : It is now clear that k^ hðuÞkp ! 1 as kukp ! 1 for any positive integer p, and it follows that kh(u)kp ! 1 as kukp ! 1. Thus for every input c = (c1, c2, . . . , cm)T, under the sufficient condition (3.3) the map h is a homeomorphism on Rm , and hence it has a unique fixed point x*. This fixed point is the unique solution of the algebraic system (3.1) defining the unique equilibrium state of the impulsive network (2.1). The proof is now complete. h kukp 6
Theorem 3.1 can be extended to any real number p > 1, by invoking a Young inequality [5] previously used to establish the exponential stability of a unique equilibrium state in a delayed cellular neural network with bounded activations and without impulses [7]. Thus if n1, n2 denote nonnegative real numbers, then n1 n2 6
nr1 ns2 þ r s
ð3:7Þ
for any real numbers r, s satisfying r > 1 and 1r þ 1s ¼ 1. Theorem 3.2. Let p > 1 be a real number, ai > 0; bij ; ci 2 R, and the activation functions fj(Æ) satisfy (2.2). Suppose the condition ai >
m m p p rij p1 X 1X aj ð1r Þp jbij jdij p1 Lj p1 þ jbji jð1dji Þp Lj ji p j¼1 p j¼1 ai
ð3:8Þ
for i 2 I is satisfied, where the ai are positive numbers and dij, rij denote real numbers. Then there exists a unique equilibrium state x* of the impulsive network (2.1). The proof of Theorem 3.2 is only slightly different to that of Theorem 3.1. Thus rewriting system (3.5) and then applyp ing the Young inequality (3.7) with r ¼ p1 and s = p yields m X
pai ai jui vi jp 6
i¼1
m X
pai
i¼1
¼
m X
6
jbij jLj jui vi jp1 juj vj j
j¼1
pai
i¼1 m X
m X
m X r ð1r Þ ðjbij jdij Lj ij jui vi jp1 Þðjbij jð1dij Þ Lj ij juj vj jÞ j¼1
( pai
i¼1
) m m p X X p p1 1 dij p1 rij p1 p ð1dij Þp ð1rij Þp p ðjbij j ðjbij j Lj jui vi j Þ þ Lj juj vj j Þ : p p j¼1 j¼1
The rest of the proof readily follows as before. We observe that the condition (3.8) is included in the condition (3.3) for integral values of p > 1. For instance, by letting d1;ij ¼ d2;ij ¼ ¼ dp1;ij ¼ dij ;
dp;ij ¼ dij ;
r1;ij ¼ r2;ij ¼ ¼ rp1;ij ¼ rij ;
rp;ij ¼ rij ;
in the condition (3.3) and using p X i¼1
dl;ij ¼ ðp 1Þdij þ dij ¼ 1;
p X
rl;ij ¼ ðp 1Þrij þ rij ¼ 1;
i¼1
one obtains the condition (3.8). For this reason, both theorems are mutually beneficial in the sense that Theorem 3.1 includes Theorem 3.2 for integral values of p while Theorem 3.2 extends Theorem 3.1 for non-integral values of p > 1.
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
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4. Exponential stability theorems As discussed earlier, the existence and stability of a unique equilibrium state is usually a requirement in the design of artificial neural networks for various applications, particularly when there are destabilizing agents such as impulses. However, even if the unique stable state exists, impulses may affect the convergence speed of the network, which in turn can downgrade the performance of the network in applications that demand fast computation in real-time mode. Thus exponential stability is usually desirable for an impulsive network, and sufficient conditions for the global exponential stability of the unique equilibrium state x* of the impulsive network (2.1) are obtained in this section. For analytical convenience, let y i ðtÞ ¼ xi ðtÞ xi ;
gj ðy j ðtÞÞ ¼ fj ðy j ðtÞ þ xj Þ fj ðxj Þ;
y i ðt k Þ þ I k ðy i ðt k Þ þ xi Þ ¼ Gk ðy i ðt k ÞÞ
so that (2.1) can be written as m X dy i ðtÞ ¼ ai y i ðtÞ þ bij gi ðy j ðtÞÞ; dt j¼1
t 6¼ tk ; t > t0 ; ð4:1Þ
yðt0 Þ ¼ x0 x 2 Rm ; y i ðtþ k Þ ¼ Gk ðy i ðt k ÞÞ;
k ¼ 1; 2; 3; . . . ;
where the sequence of times ftk g1 k¼1 satisfies t0 < t1 < < tk ! 1 as k ! 1 and
Dtk ¼ tk tk1 P h > 0 for all k 2 Zþ ;
the functions gi(Æ) inherit the properties of fi(Æ) – namely gi ð0Þ ¼ 0;
jgj ðuÞj 6 Lj juj
for all u 2 R;
jgj ðuÞj ! 1
as juj ! 1;
ð4:2Þ þ
and the functions Gk : R ! R are assumed to be continuous with Gk(0) = 0 for k 2 Z . In particular, let us consider Gk(Æ) of the form y i ðtþ k Þ ¼ Gk ðy i ðt k ÞÞ ¼ ck y i ðt k Þ;
k ¼ 1; 2; 3; . . . ;
ð4:3Þ
þ
where ck 5 0 for k 2 Z . Given the relationship between the two networks (2.1) and (4.1), it suffices to establish the global exponential stability of the trivial equilibrium state y* = 0 of the impulsive network (4.1). The case ck0 ¼ 0 for some k 0 2 Zþ means that the solution y(t) is mapped to the equilibrium state y* = 0 at t ¼ tk0 – i.e. the stability of y* = 0 is achieved immediately at that particular instant t ¼ tk0 . Other authors have considered impulses of the form (4.3) in stability investigations of impulsive neural networks. One category [1,2,15,16,24–26] is restricted to impulsive jumps with small magnitudes 0 < jckj < 1 independent of the inter-impulse intervals Dtk = tk tk1. A second category [14,33] includes some linkage between the magnitudes jckj and the intervals Dtk, and this is the case here. Theorem 4.1. Let p be a positive integer, ai > 0; bij ; ci 2 R, and the activation functions gj(Æ) satisfy (4.2). Suppose the condition (3.3), namely, m m 1X 1X aj pr pr pr ðjbij jpd1;ij Lj 1;ij þ þ jbij jpdp1;ij Lj p1;ij Þ þ jbji jpdp;ji Li p;ji ai > p j¼1 p i¼1 ai for i 2 I holds, and the impulsive jumps characterized by Gk(Æ) of the form (4.3) have magnitudes jckj satisfying 0 < jck j 6 eh
for k ¼ 1; 2; 3; . . . ;
ð4:4Þ
where the number > 0 is associated with the condition (3.3) and the number h > 0 defines Dtk P h. Then the equilibrium state y* = 0 of the impulsive network (4.1) is unique and globally exponentially stable in the sense that there exists a positive number l for which !1=p m X pffiffiffi p jy i ðtÞj < p belðtt0 Þ kyðt0 Þkp for t > t0 ; t 6¼ tk ; ð4:5Þ i¼1 m X
!1=p p jy i ðtþ k Þj
6
ffiffiffi lhk p p be kyðt0 Þkp
i¼1
where b ¼ maxi2I fai g=mini2I fai g P 1.
for k ¼ 1; 2; 3; . . . ;
ð4:6Þ
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S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
Proof. The existence of a unique equilibrium state y* = 0 of the impulsive network (4.1) follows from Theorem 3.1. Then from (4.1) and (4.2) that m m X X dþ jy i ðtÞj dy i ðtÞ ¼ sgnðy i ðtÞÞ ¼ ai y i ðtÞsgnðy i ðtÞÞ þ bij gj ðy i ðtÞÞsgnðy i ðtÞÞ 6 ai jy i ðtÞj þ jbij jLj jy j ðtÞj dt dt j¼1 j¼1
for i 2 I, t > t0, t 5 tk, where d+/dt denotes the upper right derivative. Consequently, from condition (3.3) there exists a positive number l for which the number ( ) m 1X aj pd1;ij pr1;ij pdp1;ij prp1;ij pdp;ji prp;ji ¼ min ðai lÞ jbij j Lj þ þ jbij j Lj þ jbji j Li i21 P j¼1 ai satisfies 0 < < *, where * is associated with the above when l = 0. Let us now define nonnegative functions wi(Æ) by wi ðtÞ ¼ elt jy i ðtÞj
for i 2 I; t > t0 ; t 6¼ tk
ð4:7Þ
so that m X dþ wi ðtÞ dþ jy i ðtÞj ¼ lelt jy i ðtÞj þ elt 6 ðai lÞwi ðtÞ þ jbij jLj wj ðtÞ dt dt j¼1
ð4:8Þ
for i 2 I, t > t0, t 5 tk. Associated with the solution wi(Æ) of (4.8), we define a positive definite Lyapunov function V(Æ) by m X V ðtÞ ¼ V ðwðtÞÞ ¼ ai wpi ðtÞ for t > t0 ; t 6¼ tk ð4:9Þ i¼1
where ai > 0 and p denotes a positive integer. Applying the geometric–arithmetic mean inequality (3.2) gives the rate of change ( ) m m m þ X X dþ V ðtÞ X p1 d wi ðtÞ p p1 ¼ 6 ai pwi ai pðai lÞwi ðtÞ þ pjbij jLj wi ðtÞwj ðtÞ dt dt i¼1 i¼1 j¼1 ( ) m m h i X X d1;ij r1;ij dp1;ij rp1;ij dp;ij rp;ij p ¼ ai pðai lÞwi ðtÞ þ pðjbij j Lj wi ðtÞÞ ðjbij j Lj wi ðtÞÞðjbij j Lj wj ðtÞÞ i¼1
6
m X
j¼1
( ai pðai
lÞwpi ðtÞ
þ
i¼1
m h X
jbij j
pd1;ij
pr Lj 1;ij wpi ðtÞ
þ þ jbij j
pdp1;ij
pr Lj p1;ij wpi ðtÞ
þ jbij j
pdp;ij
pr Lj p;ij wpj ðtÞ
) i
j¼1
so that
( ) m m X dþ V ðtÞ 1X aj pd1;ij pr1;ij pdp1;ij prp1;ij pdp;ji prp;ji 6 wpi ðtÞ pai ðai lÞ jbij j Lj þ þ jbij j Lj þ jbji j Lj dt p a i i¼1 j¼1 6 pV ðtÞ for t > t0 ; t 6¼ tk ;
ð4:10Þ
It follows that the derivative d+V(t)/dt is negative definite, which means V(t) < V(t0) for t > t0, t 5 tk. Then from (4.7) and (4.9), m X
ai elpt jy i ðtÞjp <
i¼1
m X
ai elpt0 jy i ðt0 Þjp
i¼1
which yields m X
!1=p p
jy i ðtÞj
<
ffiffiffi lðtt Þ p p 0 be kyðt0 Þkp
for t > t0 ; t 6¼ tk ;
i¼1
where b ¼ maxi2I fai g=mini2I fai g P 1. From (4.10) and the positivity of V(t) for t 2 (tk1, tk), k = 1, 2, 3, . . . one has that Z tk Z tk Z V ðt Þ Z t k dV k ds 6 p 6 p dt ) ds: V V ðtþ tþ tk1 Þ s tk1 k1 k1
ð4:11Þ
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
463
In the following, Z V ðt Þ Z V ðtþ Þ Z tk Z V ðtþ Þ Z V ðtþ Þ k k k k ds ds ds ds ¼ þ 6 p ds þ þ Þ Þ s s s s V ðtþ V ðt V ðt V ðt Þ Þ t k1 k k k1 k1 which means þ þ V ðtþ V ðtk Þ V ðtk Þ k Þ ln 6 pðtk tk1 Þ þ ln 6 ph þ ln þ V ðtk1 Þ V ðtk Þ V ðt k Þ for k = 1, 2, 3, . . . Now from (4.3) and (4.9), V ðtþ k Þ ¼
m X
þ
p ai epltk jy i ðtþ k Þj ¼
i¼1
m X
p p ai epltk jck jp jy i ðt k Þj ¼ jck j V ðtk Þ
i¼1
for k = 1, 2, 3, . . . Substituting this to the above and applying the condition (4.4), we obtain V ðtþ k Þ ln 6 ph þ ln jck jp 6 ph þ p lnðeh Þ V ðtþ Þ k1 þ which means V ðtþ k Þ 6 V ðtk1 Þ for k = 1, 2, 3, . . . It then follows from (4.7) and (4.9) that m X
p lpðtk tk 1Þ ai jy i ðtþ k Þj 6 e
i¼1
m X
p lph ai jy i ðtþ k1 Þj 6 e
i¼1
6 e2lph
m X
m X
p ai jy i ðtþ k1 Þj
i¼1
p ai jy i ðtþ k2 Þj
i¼1
.. . 6 eklph
m X
p ai jy i ðtþ 0 Þj
i¼1
which yields m X
!1=p p jy i ðtþ k Þj
6
ffiffiffi lhk p p be kyðt0 Þkp
for k ¼ 1; 2; 3; . . . ;
ð4:12Þ
i¼1
* where yðtþ 0 Þ ¼ yðt0 Þ. Thus the global exponential stability of the equilibrium state y = 0 of the impulsive network (4.1) for t > 0 follows from (4.11) and (4.12). The proof of Theorem 4.1 is now complete. h
The proof of a related theorem uses the Young inequality (3.7) in place of the geometric–arithmetic mean inequality (3.2), but otherwise is very similar to the proof of Theorem 3.1. For instance, on applying the Young inequality in calculating the rate of change d+V(t)/dt, one obtains ( ) m m m p X p dþ V ðtÞ p1 X 1X aj dij p1 rij p1 ð1dji Þp ð1rji Þp 6 pai ðai lÞ jbij j Lj jbji j Lj wpi ðtÞ 6 pV ðtÞ dt p j¼1 p i¼1 ai i¼1 for t > t0 ; t 6¼ tk ; where the number > 0 is associated with the condition (3.8) in the following sense: ( ) m m p p p1 X 1X aj dij p1 rij p1 ð1dji Þp ð1rji Þp ¼ min ðai lÞ jbij j Lj jbji j Lj : i2I p j¼1 p i¼1 ai The statement of this related theorem follows, but further details of its proof can be omitted. Theorem 4.2. Let p > 1 be a real number, ai > 0; bij ; ci 2 R, and the activation functions gj(Æ) satisfy (4.2). Suppose the condition (3.8), namely, ai >
m m p p rij p1 X 1X aj ð1r Þp jbij jdij p1 Lj p1 þ jbji jð1dji Þp Li ji p j¼1 p j¼1 ai
464
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
for i 2 I is satisfied, and the impulsive jumps Gk(Æ) of the form (4.3) have magnitudes jckj satisfying (4.4), in which the positive number is associated with the condition (3.8). Then the equilibrium state y* = 0 of the impulsive network (4.1) is unique and globally exponentially stable in the sense of (4.5) and (4.6).
5. Illustrative examples Consider the impulsive neural network given by dy 1 ðtÞ ¼ 5y 1 ðtÞ þ 0:5y 1 ðtÞ þ 0:1 tanhðy 2 ðtÞÞ; dt dy 2 ðtÞ ¼ 5y 2 ðtÞ þ 0:2jy 1 ðtÞj þ 0:4y 2 ðtÞ; dt
ð5:1Þ
for t > 0, t 5 tk = 1, 2, 3, . . . with initial states given by (y1(0), y2(0)) = (50, 45) and (y1(0) ,y2(0)) = (30, 40). The impulsive jumps are characterized by y i ðtþ k Þ ¼ ck y i ðt k Þ for t = tk = 1, 2, 3, . . . with ck = c = 54. For the network (5.1), let us apply the stability conditions associated with the norm kÆk1, consisting of the row-dominance a1 > jb11j + jb12j and a2 > jb21j + jb22j. The derivation of the conditions is given in the next section. One observes that ¼ minfa1 ðjb11 j þ jb12 jÞ; a2 ðjb21 j þ jb22 jÞg ¼ minf5 ð0:5 þ 0:1Þ; 5 ð0:2 þ 0:4Þg ¼ 4:4: Let us pick the number = 4 < * in order to illustrate the usefulness of the condition (4.4), in which case Dtk = h = 1. The magnitudes of the impulsive jumps then satisfy jckj 6 eh = 54.5. In accordance with the earlier theorems, the equilibrium state y* = 0 of the network (5.1) is unique and globally exponentially stable. The exponential convergence dynamics of the network (5.1) with and without impulses are shown in Fig. 1. Consider again the impulsive network (5.1) for t > 0, t 5 tk = 2, 4, 6, . . . subjected to impulsive jumps characterized by y i ðtþ k Þ ¼ ck y i ðt k Þ for tk = 2, 4, 6, . . .with ck = c = 2900. The inter-impulse intervals are Dtk = h = 2. With the above conditions associated with the norm kÆk1, one has = 4.4. On choosing the same number = 4 < *, the magnitudes of the impulsive jumps satisfy jck j 6 eh ¼ 2981, and as before the equilibrium state y* = 0 of the network (5.1) is unique and globally exponentially stable. The exponential convergence dynamics of the network (5.1) with and without impulses are shown in Fig. 2.
Neuron 1 without impulses
30 20
20
10
10
0
0
–10
–10
–20
–20
–30
0
30
1
2
3
4
5
Neuron 1 with impulses
–30
20
10
10
0
0
–10
–10
–20
–20 2
4
6
8
0
30
20
–30 0
Neuron 2 without impulses
30
10
–30 0
1
2
3
4
5
Neuron 2 with impulses
2
4
6
8
10
Fig. 1. Exponential convergence of neural network (5.1) with and without impulses. The impulsive jumps are characterized by ck = c = 54 at times tk = 1, 2, 3, . . . and Dtk = 1.
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467 Neuron 1 without impulses
Neuron 2 without impulses
30
30
20
20
10
10
0
0
–10
–10
–20
–20
–30
–30 0
2
4
6
8
10
0
Neuron 1 with impulses
2
4
6
8
10
Neuron 2 with impulses
30
30
20
20
10
10
0
0
–10
–10
–20
–20
–30
465
–30 0
2
4
6
8
10
0
2
4
6
8
10
Fig. 2. Exponential convergence of neural network (5.1) with and without impulses. The impulsive jumps are characteristics by ck = c = 2900 at times tk = 2, 4, 6, . . . and Dtk = 2.
6. Concluding remarks A set of sufficient conditions for a Hopfleld-type neural network to always converge exponentially towards a unique equilibrium state has been found – cf. (3.3) and (3.8) in particular. The convergence persists even under the influence of impulsive jumps of the form Gk(u) = cku for u 2 R provided the magnitude and the frequency of the impulsive jumps satisfy 0 < jckj 6 eh and Dtk P h > 0 respectively, where > 0 is defined under either (3.3) or (3.8). Note that (4.4)–(4.6) provide the information linking the conditions (3.3) and (3.8) via the positive constants , l and h. The number l > 0 determines the convergence rate of the network, and the constant > 0 defines how large the impulse magnitudes can be. The important role of the constant h > 0 in determining the size of the impulse magnitudes and influencing the convergence of the network is evident in (4.4) and (4.6), respectively. This is consistent with the work of Gopalsamy [14] and Xu et al. [33] in their stability investigations of the impulsive network (4.1) (or equivalently (2.1)). The severe restriction 0 < jckj < 1 of the impulse magnitudes considered in [1,2,15,16,24–26] is merely a special case of the criterion 0 < jckj 6 eh obtained in this paper. The generalization to unbounded activation functions under kÆkp and the sufficiency conditions (3.3) and (3.8) are the other notable developments here. Obviously, the asymptotic stability results obtained previously for the impulsive network (4.1) under the norms kÆk1, kÆk2 and kÆk1 are included. Note that the general Euclidean norm for a vector u ¼ ðu1 ; u2 ; . . . ; um Þ 2 Rm satisfies kÆkp ! kÆk1 as p ! 1, so the relevant condition can be extracted from either the condition (3.3) or the condition (3.8) by appropriately defining the non-network parameters dij, rij. Let the parameters in the condition (3.8) be dij = (p 1)/p and rij = (p 1)/p, so that ai >
m m p1 X 1X aj jbij jLj þ jbji jLi ; p j¼1 p j¼1 ai
Then if p ! 1, one has ai >
m X j¼1
jbij jLj ;
i ¼ 1; 2 . . . ; m
i ¼ 1; 2; . . . ; m:
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as the condition under which the network (4.1) converges exponentially in the norm kÆk1. The condition is consistent with that obtained by Gopalsamy [14]. As another example, set p = 1 in the condition (3.3) such that ai >
m X aj jbji jLi ; ai j¼1
i ¼ 1; 2 . . . ; m;
guarantees the exponential convergence of the impulsive network (4.1) in the norm kÆk1. This condition is an improvement on those obtained previously [1,14,26]. It is also evident that the condition (3.3) with p = 2 improves the sufficiency condition found in [14] under the norm kÆk2.
Acknowledgement The author thanks Professor Roger Hosking of the Department of Mathematics, Universiti Brunei Darussalam for reading the manuscript and suggesting on ways to improve the exposition of the manuscript.
References [1] Akc¸a H, Alassar R, Covachev V, Covacheva Z, Al-Zahrani E. Continuous-time additive Hopfield-type neural networks with impulses. J Math Anal Appl 2004;290:436–51. [2] Akc¸a H, Alassar R, Covachev V, Covacheva Z. Discrete counterparts of continuous-time additive Hopfield-type neural networks with impulses. Dynam Syst Appl 2004;13:77–92. [3] Arik S. Global asymptotic stability of a class of dynamical neural networks. IEEE Trans Circ Syst – I: Fundam Theory Appl 2000;47:568–71. [4] Bainov DD, Simeonov PS. Stability theory of differential equations with impulse effects: theory and applications. Chichester: Ellis Horwood; 1989. [5] Beckenbach EF, Bellman R. Inequalities. New York: Springer-Verlag; 1965. [6] Bouzerdoum A, Pattison TR. Neural network for quadratic optimization with bound constraints. IEEE Trans Neural Networks 1993;4:293–304. [7] Cao J. New results concerning exponential stability and periodic solutions of delayed cellular neural networks. Phys Lett A 2003;307:136–47. [8] Cao J, Tao Q. Estimation of the domain of attraction and the convergence rate of a Hopfield associative memory and an application. J Comput Syst Sci 2000;60:179–86. [9] Chen T, Amari SI. Stability of asymmetric Hopfield networks. IEEE Trans Neural Networks 2001;12:159–63. [10] Cohen MA, Grossberg S. Absolute stability of global pattern formation and parallel memory storage by competitive neural networks. IEEE Trans Syst Man Cybern 1983;13:815–26. [11] van den Driessche P, Zou X. Global attractivity in delayed Hopfield neural network models. SIAM J Appl Math 1998;58:1878–90. [12] Fang Y, Kincaid TG. Stability analysis of dynamical neural networks. IEEE Trans Neural Networks 1996;7:996–1006. [13] Forti M, Tesi A. New conditions for global stability of neural networks with application to linear and quadratic programming problems. IEEE Trans Circuits Syst 1995;42:354–66. [14] Gopalsamy K. Stability of artificial neural networks with impulses. Appl Math Comput 2004;154:783–813. [15] Guan ZH, Lam J, Chen G. On impulsive autoassociative neural networks. Neural Networks 2000;13:63–9. [16] Guan ZH, Chen G. On delayed impulsive Hopfield neural networks. Neural Networks 1999;12:273–80. [17] Hopfield JJ. Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci 1984;81:3088–92. [18] Hu S, Wang J. Global stability of a class of continuous-time recurrent neural networks. IEEE Trans Circ Syst – I: Fundam Theory Appl 2002;49:1334–47. [19] Inoue JI. Retrieval phase diagrams of non-monotonic Hopfield networks. J Phys A: Math Gen 1996;29:4815–26. [20] Juang J. Stability analysis of Hopfield type neural networks. IEEE Trans Neural Networks 1999;10:1366–74. [21] Kennedy MP, Chua LO. Neural networks for nonlinear programming. IEEE Trans Circuits Syst 1988;35:554–62. [22] Lakshmikantham V, Bainov DD, Simeonov PS. Theory of impulsive differential equations. Series in modern applied mathematics, vol. 6. Singapore: World Scientific; 1989. [23] Lee BW, Sheu BJ. Modified Hopfield neural networks for retrieving the optimal solution. IEEE Trans Neural Networks 1991;2:137–42. [24] Li Y, Xing W, Lu L. Existence and global exponential stability of periodic solution of a class of neural networks with impulses. Chaos, Solitons & Fractals 2006;27:437–45. [25] Li Y. Global exponential stability of BAM neural networks with delays and impulses. Chaos, Solitons & Fractals 2005;24:279–85. [26] Li Y, Lu L. Global exponential stability and existence of periodic solution of Hopfield-type neural networks with impulses. Phys Lett A 2004;333:62–71.
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[27] Liu X, Chen T. A new result on the global convergence of Hopfield neural networks. IEEE Trans Circ Syst – I: Fundam Theory Appl 2002;49:1514–6. [28] Mohamad S. Global exponential stability in DCNNs with distributed delays and unbounded activations. J Comput Appl Math, in press. [29] Morita M. Associative memory with nonmonotone dynamics. Neural Networks 1993;6:115–26. [30] Samoilenko AM, Perestyuk NA. Impulsive differential equations. Singapore: World Scientific; 1995. [31] Stamov GT. Impulsive cellular neural networks and almost periodicity. Proc Jpn Acad 2004;80:198–203. [32] Sudharsanan SI, Sundareshan MK. Exponential stability and a systematic synthesis of a neural network for quadratic minimization. Neural Networks 1991;4:599–613. [33] Xu D, Yang Z. Impulsive delay differential inequality and stability of neural networks. J Math Anal Appl 2005;305:107–20. [34] Yanai H, Amari S. Auto-associative memory with two-stage dynamics of non-monotonic neurons. IEEE Trans Neural Networks 1996;7:803–15. [35] Yang H, Dillon TS. Exponential stability and oscillation of Hopfield graded response neural network. IEEE Trans Neural Networks 1994;5:719–29. [36] Ye H, Michel AN, Wang K. Global stability and local stability of Hopfield neural networks with delays. Phys Rev E 1994;50:4206–13. [37] Yoshizawa S, Morita M, Amari S. Capacity of associative memory using a non-monotonic neuron networks. Neural Networks 1993;6:167–76. [38] Zhang J. Absolute stability analysis in cellular neural networks with variable delays and unbounded delay. Comput Math Appl 2004;47:183–94. [39] Zhang J, Jin X. Global stability analysis in delayed Hopfield neural networks models. Neural Networks 2000;13:745–53.
Exponential stability in Hopfield-type neural networks with impulses Sannay Mohamad
*
Department of Mathematics, Faculty of Science, Universiti Brunei Darussalam, Jalan Tunku Link BE1410, Brunei Darussalam Accepted 8 June 2006
Communicated by Prof. M.S. El Naschie
Abstract This paper demonstrates that there is an exponentially stable unique equilibrium state in a Hopfield-type neural network that is subject to quite large impulses that are not too frequent. The activation functions are assumed to be globally Lipschitz continuous and unbounded. The analysis exploits an homeomorphic mapping and an appropriate Lyapunov function, and also either a geometric–arithmetic mean inequality or a Young inequality, to derive a family of easily verifiable sufficient conditions for convergence to the unique globally stable equilibrium state. These sufficiency conditions, in the norm kÆkp where p P 1, include those governing the network parameters and the impulse magnitude and frequency. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Impulses are ubiquitous in both biological and artificial neural networks. For example, peaceful sleep may be suddenly interrupted, perhaps by a nightmare. However, even if there are abrupt changes in the neural state, a person may resume a deeper sleep provided the interruption is not too great nor too frequent. In the case of an artificial network for signal processing, faulty elements can produce sudden changes in the state voltages and thereby affect the normal transient behaviour in processing signals or information, and robust system design is important. Neural networks perceived as either continuous or discrete dynamical systems have been studied extensively, but the mathematical modelling of dynamical systems with impulses is a quite recent development [1,2,14–16,24–26,31]. In this paper, we demonstrate the exponential stability of a unique equilibrium state in a Hopfield-type neural network [17] consisting of m processing units, subjected to impulsive state displacements. Phenomena such as those mentioned above may be interpreted as large impulses affecting otherwise normal transient behaviour, but conditions for a neural network to resist impulses of significant magnitude do not appear to have been found before.
*
Tel.: +673 2 463001; fax: +673 2 461502. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.035
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
457
2. Mathematical model The network is described by m X dxi ðtÞ bij fj ðxj ðtÞÞ þ ci ; ¼ ai xi ðtÞ þ dt j¼1
t > t0 ; t 6¼ tk ;
and Dxi jt¼tk ¼ xi ðtþ k Þ xi ðt k Þ ¼ I k ðxi ðt k ÞÞ;
subject to xðt0 Þ ¼ x0 2 Rm ;
ð2:1Þ k ¼ 1; 2; 3; . . .
1 Here xi ðtþ k Þ xi ðt k þ 0Þ, xi ðt k Þ xi ðt k 0Þ for i 2 I ¼ f1; 2; . . . ; mg and the sequence of times ft k gk¼1 satisfies
t0 < t 1 < t 2 < < tk ! 1
as k ! 1
and Dtk ¼ tk tk1 P h
for k = 1, 2, 3, . . ., where the value h > 0 denotes the minimum time of interval between successive impulses. A sufficiently large value of h ensures that impulses do not occur too frequently (see later), but h ! 1 means that the network (2.1) becomes impulse free. We refer to Gopalsamy [14] for earlier discussion of this model formulation and its application. The vector solution xðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; . . . ; xm ðtÞÞT 2 Rm of (2.1) has components xi(t) piece-wise continuous on (t0, b) for some b > t0, such that xðtþ k Þ and xðt k Þ exist and x(t) is differentiable on the open intervals (tk1, tk) (t0, b). Further, we assume that x(t) is right continuous with xðtk Þ ¼ xðtþ k Þ; the functions I k : R ! R that characterize the impulsive jumps are continuous; the neural parameters ai, bij, ci satisfy ai > 0; bij ; ci 2 R; and the activation functions fj : R ! R with fj(0) = 0 may be unbounded and continuous but satisfy jfj ðuÞ fj ðvÞj 6 Lj ju vj for all u; v; 2 R; jfj ðuÞj ! 1 as juj ! 1;
ð2:2Þ
where Lj > 0 for j 2 I denotes a Lipschitz constant. Sufficiency conditions on the neural parameters and the impulses have previously been obtained to guarantee the asymptotic convergence towards a unique equilibrium state of the network (2.1), associated with the norms kÆk1, kÆk2 and kÆk1 [14]. However, the asymptotic stability of this equilibrium state is guaranteed only if the magnitudes of the impulses are neither large nor frequent. Although that is of course consistent with the view that a dynamical system tends to become unstable when subjected to sufficiently frequent impulses of large magnitude [4,22,30], in this paper a family of easily verifiable sufficient conditions on the neural parameters and the impulses is found to more generally guarantee the exponential convergence of the neural states towards the unique equilibrium state, and in any norm kÆkp where p P 1. The results obtained by applying a geometric–arithmetic mean inequality and a Young inequality to an appropriate Lyapunov function significantly enhance the earlier work, both with and without impulses.
3. Existence and uniqueness theorems To provide for an associative memory [8,29,34,37], the network architecture is designed to not only store as many equilibrium states (or memories) as possible, but also to retrieve the relevant stored memory produced by a given external stimulus c = (c1, c2, . . . , cm)T. Thus if a neural network is intended to solve an optimization problem, the circuit design of the network should ensure that all neural states approach a unique equilibrium state of the network [6,13,21,23,32]. The association of an equilibrium state with an external input vector c = (c1, c2, . . . , cm)T avoids the possibility of convergence towards some local minimum that is a spurious equilibrium state, and not the optimal solution of the optimization problem. To ensure that a unique equilibrium state exists, it has been customary to impose restrictions on the neural parameters and the activation functions of the network, such assuming they are bounded and Lipschitz continuous [11] – i.e. gj ð0Þ ¼ 0;
jgj ðuÞj 6 M j ;
jgj ðuÞ gj ðvÞj 6 Lj ju vj
for some positive constants Mj, Lj and any u; v 2 R. Stronger requirements have been adopted [3,9,10,12,18,20,27, 35,36,39], where the activation functions are assumed to be bounded, continuous, monotonic and differentiable – i.e. gj 2 C 1 ðRÞ, g0j ðuÞ > 0 for u 2 R, g0j ð0Þ ¼ supu2R g0j ðuÞ > 0, gj(0) = 0 and gj(u) ! ±1 as u ! ±1. Recent applications, in which the activation functions are either linear and piece-wise continuous or Gaussian, have indicated a number of advantages for removing the differentiability and monotonicity requirements [19,29,34,37]. And in applying a neural network to solve a certain class of optimization problems, Forti et al. [13] have used activation functions of exponentialtype and diode-like, which are unbounded to suit the unbounded constraint requirement of the problems considered.
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One should not overlook the possibility that unbounded activation functions may result in the nonexistence of an equilibrium state of the network, and in any case it is usually important to establish the existence of a unique equilibrium state. In this Section, sufficiency conditions are established for the existence of a unique equilibrium state of the impulsive network (2.1) assuming the functions fj(Æ) satisfy (2.2). Let us denote an equilibrium state of the network (2.1) by the constant vector x ¼ ðx1 ; x2 ; . . . ; xm ÞT 2 Rm , where each xi is governed by the algebraic system 0 ¼ ai xi þ
m X
bij fj ðxj Þ þ ci ;
i 2 I:
ð3:1Þ
j¼1
Here it is assumed that the impulsive jumps Ik(Æ) satisfy I k ðxi Þ ¼ 0 for i 2 I; k 2 Zþ . Gopalsamy [14] used a contraction mapping principle to derive sufficient conditions for the existence of a unique equilibrium state in the impulsive network (2.1), under the usual norms kÆk1, kÆk2 and kÆk1. A result due to Forti et al. [13] and Zhang is used here to prove there is a unique solution of the system (3.1) under the general P[38] m p 1=p , where p P 1 and u ¼ ðu1 ; u2 ; . . . ; um ÞT 2 Rm . Recall that a map h : Rm ! Rm Euclidean norm kukp ¼ i¼1 jui j is a homeomorphism (Rm onto itself) if h 2 C0 is one-to-one and onto, and the inverse map h1 2 C0. The result is that the map h 2 C0 is a homeomorphism on Rm if it is injective on Rm and satisfies limkuk!1kh(u)k ! 1. However, whereas Forti et al. [13] and Zhang [38] employed matrices with negative principal minors to obtain their results under the norm kÆk2, we use certain inequalities of nonnegative real numbers as mentioned earlier. The first invoked in this Section is the geometric–arithmetic mean inequality [5] – namely that, if n1, n2, . . . , np denote p nonnegative real numbers, then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi np þ np þ þ np 1 2 p p ðnp1 np2 npp Þ 6 p and hence n1 n2 np 6
np1 þ np2 þ þ npp ; p
ð3:2Þ
where p P 1 denotes an integer. A particular form of (3.2), namely fp1 1 f2 6
p1 p 1 p f þ f p 1 p 2
for p ¼ 1; 2; 3; . . .
has previously been used to investigate the exponential stability characteristics of a unique equilibrium state of a delayed cellular neural network [28]. Theorem 3.1. Let p be a positive integer, ai > 0; bij ; ci 2 R, and the activation functions fj(Æ) satisfy (2.2). Suppose the condition m m 1X 1X aj pr pr pr ðjbij jpd1;ij Lj 1;ij þ þ jbij jpdp1;ij Lj p1;ij Þ þ jbji jpdp;ji Li p;ji ð3:3Þ ai > p j¼1 p j¼1 ai for real numbers that satisfy Pp i 2 I is satisfied, Pp where the ai are positive numbers, and dl,ij, rl,ij, for l = 1, 2, . . . , p denote * l¼1 dl;ij ¼ 1 and l¼1 rl;ij ¼ 1, respectively. Then there exists a unique equilibrium state x of the impulsive network (2.1). Proof. Consider a map hðuÞ ¼ ðh1 ðuÞ; h2 ðuÞ; . . . ; hm ðuÞÞT 2 C 0 ðRm ; Rm Þ, where m X bij fj ðuj Þ þ ci hi ðuÞ ¼ ai ui þ
ð3:4Þ
j¼1
for ui 2 R, i 2 I. This map h is a homeomorphism on Rm if it injective on Rm and satisfies kh(u)kp ! 1 as kukp ! 1. The injective part of the map h, namely h(u) = h(v) implies u = v for any u; v 2 Rm , is shown as follows. Since ai ui þ
m X
bij fj ðuj Þ þ ci ¼ ai vi þ
j¼1
j¼1
and consequently ai jui vi j 6
m X
m X j¼1
jbij jLj juj vj j;
bij fj ðvj Þ þ ci
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
459
under the assumption (2.2). Now m X
pai ai jui vi jp 6
i¼1
m X i¼1
¼
m X
m X
pai
( ai
jbij jLj jui vi jp1 juj vj j
j¼1 m X
i¼1
) pðjbij j
dl;ij
r Lj l;ij jui
dp1;ij
vi jÞ ðjbij j
r Lj p1;ij jui
vi jÞðjbij j
dp;ij
r Lj p;ij juj
vj jÞ ;
ð3:5Þ
j¼1
where p denotes a positive i are positive real numbers, and dl,ij, rl,ij for l ¼ 1; 2; . . . ; p; i; j 2 I denote real P integer, the aP numbers that satisfy pl¼1 dl;ij ¼ 1 and pl¼1 rl;ij ¼ 1. By applying the geometric–arithmetic mean inequality (3.2) to (3.5), we obtain ( ) m m m X X X p pd1;ij pr1;ij p pdp1;ij prp1;ij p pdp;ij prp;ij p pai ai jui vi j 6 ai ðjbij j Lj jui vi j Þ þ þ ðjbij j Lj jui vi j Þ þ ðjbij j Lj jui vi j Þ i¼1
i¼1
¼
m X
j¼1
( pai
i¼1
) m m 1X 1X aj pdpl;ij prp1;ij pdp1;ij prp1;ij pdp;ji prp;ji jui vi jp ðjbij j Lj þ þ jbij j Lj Þþ jbji j Lj p i¼1 p j¼1 ai
implying that p
m X
ai jui vi jp 6 0;
i¼1
where the number * defined by ( ) m m 1X 1X aj pd1;ij pr1;ij pdp1;ij prp1;ij pdp;ji prp;ji ¼ min ai ðjbij j Lj þ þ jbij j Lj Þ jbji j Li i2I p j¼1 p j¼1 ai
ð3:6Þ
is positive in accordance with the condition (3.3). Consequently, ui = vi for i 2 I (i.e. u = v), hence the map h is injective on Rm . To demonstrate the property kh(u)kp ! 1 as kukp ! 1, we consider the map ^ hðuÞ ¼ hðuÞ hð0Þ – i.e. m X ^hi ðuÞ ¼ hi ðuÞ hi ð0Þ ¼ ai ui þ bij fj ðuj Þ for ui 2 R; i 2 I: j¼1
Then m X
pai jui j
p1
sgnðui Þ^hi ðuÞ ¼
m X
i¼1
i¼1
6
m X
( pai jui j
p1
sgnðui Þ ai ui þ
m X
) bij fj ðuj Þ
j¼1
( p
ai pai jui j þ
i¼1
m X
p1
pjbij jLj jui j
) juj j :
j¼1
Applying the geometric–arithmetic mean inequality (3.2) as before, ( ) m m m X X 1X aj pdp1;ij prp1;ij pdp;ji prp;ji 1 ;ij jui jp pai jui jp1 sgnðui Þh^i ðun Þ 6 pai ai jbij jpd1 ;ij Lpr þ þ jb j L þ jb j L ij ji j j j p a i i¼1 i¼1 j¼1 6 p
m X
ai jui jp :
i¼1
We have p minfai g i2I
m X i¼1
jui jp 6
m X
pai jui jp1 sgnðui Þ^hi ðuÞ 6 p maxfai g
i¼1
which then gives m m X b X jui jp 6 jui jp1 j^hi ðuÞj; i¼1 i¼1
i2I
m X i¼1
hi ðuÞj jui jp1 ^
460
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
where b ¼ maxi2I fai g=mini2I fai g P 1. Applying a Ho¨lder inequality to the above, !1=q !1=p m m m X X b X p ðp1Þq p ^ jui j 6 jui j jhi ðuÞj ; i¼1 i¼1 i¼1 where 1p þ 1q ¼ 1 which in turn yields b ^ khðuÞkp : It is now clear that k^ hðuÞkp ! 1 as kukp ! 1 for any positive integer p, and it follows that kh(u)kp ! 1 as kukp ! 1. Thus for every input c = (c1, c2, . . . , cm)T, under the sufficient condition (3.3) the map h is a homeomorphism on Rm , and hence it has a unique fixed point x*. This fixed point is the unique solution of the algebraic system (3.1) defining the unique equilibrium state of the impulsive network (2.1). The proof is now complete. h kukp 6
Theorem 3.1 can be extended to any real number p > 1, by invoking a Young inequality [5] previously used to establish the exponential stability of a unique equilibrium state in a delayed cellular neural network with bounded activations and without impulses [7]. Thus if n1, n2 denote nonnegative real numbers, then n1 n2 6
nr1 ns2 þ r s
ð3:7Þ
for any real numbers r, s satisfying r > 1 and 1r þ 1s ¼ 1. Theorem 3.2. Let p > 1 be a real number, ai > 0; bij ; ci 2 R, and the activation functions fj(Æ) satisfy (2.2). Suppose the condition ai >
m m p p rij p1 X 1X aj ð1r Þp jbij jdij p1 Lj p1 þ jbji jð1dji Þp Lj ji p j¼1 p j¼1 ai
ð3:8Þ
for i 2 I is satisfied, where the ai are positive numbers and dij, rij denote real numbers. Then there exists a unique equilibrium state x* of the impulsive network (2.1). The proof of Theorem 3.2 is only slightly different to that of Theorem 3.1. Thus rewriting system (3.5) and then applyp ing the Young inequality (3.7) with r ¼ p1 and s = p yields m X
pai ai jui vi jp 6
i¼1
m X
pai
i¼1
¼
m X
6
jbij jLj jui vi jp1 juj vj j
j¼1
pai
i¼1 m X
m X
m X r ð1r Þ ðjbij jdij Lj ij jui vi jp1 Þðjbij jð1dij Þ Lj ij juj vj jÞ j¼1
( pai
i¼1
) m m p X X p p1 1 dij p1 rij p1 p ð1dij Þp ð1rij Þp p ðjbij j ðjbij j Lj jui vi j Þ þ Lj juj vj j Þ : p p j¼1 j¼1
The rest of the proof readily follows as before. We observe that the condition (3.8) is included in the condition (3.3) for integral values of p > 1. For instance, by letting d1;ij ¼ d2;ij ¼ ¼ dp1;ij ¼ dij ;
dp;ij ¼ dij ;
r1;ij ¼ r2;ij ¼ ¼ rp1;ij ¼ rij ;
rp;ij ¼ rij ;
in the condition (3.3) and using p X i¼1
dl;ij ¼ ðp 1Þdij þ dij ¼ 1;
p X
rl;ij ¼ ðp 1Þrij þ rij ¼ 1;
i¼1
one obtains the condition (3.8). For this reason, both theorems are mutually beneficial in the sense that Theorem 3.1 includes Theorem 3.2 for integral values of p while Theorem 3.2 extends Theorem 3.1 for non-integral values of p > 1.
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
461
4. Exponential stability theorems As discussed earlier, the existence and stability of a unique equilibrium state is usually a requirement in the design of artificial neural networks for various applications, particularly when there are destabilizing agents such as impulses. However, even if the unique stable state exists, impulses may affect the convergence speed of the network, which in turn can downgrade the performance of the network in applications that demand fast computation in real-time mode. Thus exponential stability is usually desirable for an impulsive network, and sufficient conditions for the global exponential stability of the unique equilibrium state x* of the impulsive network (2.1) are obtained in this section. For analytical convenience, let y i ðtÞ ¼ xi ðtÞ xi ;
gj ðy j ðtÞÞ ¼ fj ðy j ðtÞ þ xj Þ fj ðxj Þ;
y i ðt k Þ þ I k ðy i ðt k Þ þ xi Þ ¼ Gk ðy i ðt k ÞÞ
so that (2.1) can be written as m X dy i ðtÞ ¼ ai y i ðtÞ þ bij gi ðy j ðtÞÞ; dt j¼1
t 6¼ tk ; t > t0 ; ð4:1Þ
yðt0 Þ ¼ x0 x 2 Rm ; y i ðtþ k Þ ¼ Gk ðy i ðt k ÞÞ;
k ¼ 1; 2; 3; . . . ;
where the sequence of times ftk g1 k¼1 satisfies t0 < t1 < < tk ! 1 as k ! 1 and
Dtk ¼ tk tk1 P h > 0 for all k 2 Zþ ;
the functions gi(Æ) inherit the properties of fi(Æ) – namely gi ð0Þ ¼ 0;
jgj ðuÞj 6 Lj juj
for all u 2 R;
jgj ðuÞj ! 1
as juj ! 1;
ð4:2Þ þ
and the functions Gk : R ! R are assumed to be continuous with Gk(0) = 0 for k 2 Z . In particular, let us consider Gk(Æ) of the form y i ðtþ k Þ ¼ Gk ðy i ðt k ÞÞ ¼ ck y i ðt k Þ;
k ¼ 1; 2; 3; . . . ;
ð4:3Þ
þ
where ck 5 0 for k 2 Z . Given the relationship between the two networks (2.1) and (4.1), it suffices to establish the global exponential stability of the trivial equilibrium state y* = 0 of the impulsive network (4.1). The case ck0 ¼ 0 for some k 0 2 Zþ means that the solution y(t) is mapped to the equilibrium state y* = 0 at t ¼ tk0 – i.e. the stability of y* = 0 is achieved immediately at that particular instant t ¼ tk0 . Other authors have considered impulses of the form (4.3) in stability investigations of impulsive neural networks. One category [1,2,15,16,24–26] is restricted to impulsive jumps with small magnitudes 0 < jckj < 1 independent of the inter-impulse intervals Dtk = tk tk1. A second category [14,33] includes some linkage between the magnitudes jckj and the intervals Dtk, and this is the case here. Theorem 4.1. Let p be a positive integer, ai > 0; bij ; ci 2 R, and the activation functions gj(Æ) satisfy (4.2). Suppose the condition (3.3), namely, m m 1X 1X aj pr pr pr ðjbij jpd1;ij Lj 1;ij þ þ jbij jpdp1;ij Lj p1;ij Þ þ jbji jpdp;ji Li p;ji ai > p j¼1 p i¼1 ai for i 2 I holds, and the impulsive jumps characterized by Gk(Æ) of the form (4.3) have magnitudes jckj satisfying 0 < jck j 6 eh
for k ¼ 1; 2; 3; . . . ;
ð4:4Þ
where the number > 0 is associated with the condition (3.3) and the number h > 0 defines Dtk P h. Then the equilibrium state y* = 0 of the impulsive network (4.1) is unique and globally exponentially stable in the sense that there exists a positive number l for which !1=p m X pffiffiffi p jy i ðtÞj < p belðtt0 Þ kyðt0 Þkp for t > t0 ; t 6¼ tk ; ð4:5Þ i¼1 m X
!1=p p jy i ðtþ k Þj
6
ffiffiffi lhk p p be kyðt0 Þkp
i¼1
where b ¼ maxi2I fai g=mini2I fai g P 1.
for k ¼ 1; 2; 3; . . . ;
ð4:6Þ
462
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
Proof. The existence of a unique equilibrium state y* = 0 of the impulsive network (4.1) follows from Theorem 3.1. Then from (4.1) and (4.2) that m m X X dþ jy i ðtÞj dy i ðtÞ ¼ sgnðy i ðtÞÞ ¼ ai y i ðtÞsgnðy i ðtÞÞ þ bij gj ðy i ðtÞÞsgnðy i ðtÞÞ 6 ai jy i ðtÞj þ jbij jLj jy j ðtÞj dt dt j¼1 j¼1
for i 2 I, t > t0, t 5 tk, where d+/dt denotes the upper right derivative. Consequently, from condition (3.3) there exists a positive number l for which the number ( ) m 1X aj pd1;ij pr1;ij pdp1;ij prp1;ij pdp;ji prp;ji ¼ min ðai lÞ jbij j Lj þ þ jbij j Lj þ jbji j Li i21 P j¼1 ai satisfies 0 < < *, where * is associated with the above when l = 0. Let us now define nonnegative functions wi(Æ) by wi ðtÞ ¼ elt jy i ðtÞj
for i 2 I; t > t0 ; t 6¼ tk
ð4:7Þ
so that m X dþ wi ðtÞ dþ jy i ðtÞj ¼ lelt jy i ðtÞj þ elt 6 ðai lÞwi ðtÞ þ jbij jLj wj ðtÞ dt dt j¼1
ð4:8Þ
for i 2 I, t > t0, t 5 tk. Associated with the solution wi(Æ) of (4.8), we define a positive definite Lyapunov function V(Æ) by m X V ðtÞ ¼ V ðwðtÞÞ ¼ ai wpi ðtÞ for t > t0 ; t 6¼ tk ð4:9Þ i¼1
where ai > 0 and p denotes a positive integer. Applying the geometric–arithmetic mean inequality (3.2) gives the rate of change ( ) m m m þ X X dþ V ðtÞ X p1 d wi ðtÞ p p1 ¼ 6 ai pwi ai pðai lÞwi ðtÞ þ pjbij jLj wi ðtÞwj ðtÞ dt dt i¼1 i¼1 j¼1 ( ) m m h i X X d1;ij r1;ij dp1;ij rp1;ij dp;ij rp;ij p ¼ ai pðai lÞwi ðtÞ þ pðjbij j Lj wi ðtÞÞ ðjbij j Lj wi ðtÞÞðjbij j Lj wj ðtÞÞ i¼1
6
m X
j¼1
( ai pðai
lÞwpi ðtÞ
þ
i¼1
m h X
jbij j
pd1;ij
pr Lj 1;ij wpi ðtÞ
þ þ jbij j
pdp1;ij
pr Lj p1;ij wpi ðtÞ
þ jbij j
pdp;ij
pr Lj p;ij wpj ðtÞ
) i
j¼1
so that
( ) m m X dþ V ðtÞ 1X aj pd1;ij pr1;ij pdp1;ij prp1;ij pdp;ji prp;ji 6 wpi ðtÞ pai ðai lÞ jbij j Lj þ þ jbij j Lj þ jbji j Lj dt p a i i¼1 j¼1 6 pV ðtÞ for t > t0 ; t 6¼ tk ;
ð4:10Þ
It follows that the derivative d+V(t)/dt is negative definite, which means V(t) < V(t0) for t > t0, t 5 tk. Then from (4.7) and (4.9), m X
ai elpt jy i ðtÞjp <
i¼1
m X
ai elpt0 jy i ðt0 Þjp
i¼1
which yields m X
!1=p p
jy i ðtÞj
<
ffiffiffi lðtt Þ p p 0 be kyðt0 Þkp
for t > t0 ; t 6¼ tk ;
i¼1
where b ¼ maxi2I fai g=mini2I fai g P 1. From (4.10) and the positivity of V(t) for t 2 (tk1, tk), k = 1, 2, 3, . . . one has that Z tk Z tk Z V ðt Þ Z t k dV k ds 6 p 6 p dt ) ds: V V ðtþ tþ tk1 Þ s tk1 k1 k1
ð4:11Þ
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
463
In the following, Z V ðt Þ Z V ðtþ Þ Z tk Z V ðtþ Þ Z V ðtþ Þ k k k k ds ds ds ds ¼ þ 6 p ds þ þ Þ Þ s s s s V ðtþ V ðt V ðt V ðt Þ Þ t k1 k k k1 k1 which means þ þ V ðtþ V ðtk Þ V ðtk Þ k Þ ln 6 pðtk tk1 Þ þ ln 6 ph þ ln þ V ðtk1 Þ V ðtk Þ V ðt k Þ for k = 1, 2, 3, . . . Now from (4.3) and (4.9), V ðtþ k Þ ¼
m X
þ
p ai epltk jy i ðtþ k Þj ¼
i¼1
m X
p p ai epltk jck jp jy i ðt k Þj ¼ jck j V ðtk Þ
i¼1
for k = 1, 2, 3, . . . Substituting this to the above and applying the condition (4.4), we obtain V ðtþ k Þ ln 6 ph þ ln jck jp 6 ph þ p lnðeh Þ V ðtþ Þ k1 þ which means V ðtþ k Þ 6 V ðtk1 Þ for k = 1, 2, 3, . . . It then follows from (4.7) and (4.9) that m X
p lpðtk tk 1Þ ai jy i ðtþ k Þj 6 e
i¼1
m X
p lph ai jy i ðtþ k1 Þj 6 e
i¼1
6 e2lph
m X
m X
p ai jy i ðtþ k1 Þj
i¼1
p ai jy i ðtþ k2 Þj
i¼1
.. . 6 eklph
m X
p ai jy i ðtþ 0 Þj
i¼1
which yields m X
!1=p p jy i ðtþ k Þj
6
ffiffiffi lhk p p be kyðt0 Þkp
for k ¼ 1; 2; 3; . . . ;
ð4:12Þ
i¼1
* where yðtþ 0 Þ ¼ yðt0 Þ. Thus the global exponential stability of the equilibrium state y = 0 of the impulsive network (4.1) for t > 0 follows from (4.11) and (4.12). The proof of Theorem 4.1 is now complete. h
The proof of a related theorem uses the Young inequality (3.7) in place of the geometric–arithmetic mean inequality (3.2), but otherwise is very similar to the proof of Theorem 3.1. For instance, on applying the Young inequality in calculating the rate of change d+V(t)/dt, one obtains ( ) m m m p X p dþ V ðtÞ p1 X 1X aj dij p1 rij p1 ð1dji Þp ð1rji Þp 6 pai ðai lÞ jbij j Lj jbji j Lj wpi ðtÞ 6 pV ðtÞ dt p j¼1 p i¼1 ai i¼1 for t > t0 ; t 6¼ tk ; where the number > 0 is associated with the condition (3.8) in the following sense: ( ) m m p p p1 X 1X aj dij p1 rij p1 ð1dji Þp ð1rji Þp ¼ min ðai lÞ jbij j Lj jbji j Lj : i2I p j¼1 p i¼1 ai The statement of this related theorem follows, but further details of its proof can be omitted. Theorem 4.2. Let p > 1 be a real number, ai > 0; bij ; ci 2 R, and the activation functions gj(Æ) satisfy (4.2). Suppose the condition (3.8), namely, ai >
m m p p rij p1 X 1X aj ð1r Þp jbij jdij p1 Lj p1 þ jbji jð1dji Þp Li ji p j¼1 p j¼1 ai
464
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
for i 2 I is satisfied, and the impulsive jumps Gk(Æ) of the form (4.3) have magnitudes jckj satisfying (4.4), in which the positive number is associated with the condition (3.8). Then the equilibrium state y* = 0 of the impulsive network (4.1) is unique and globally exponentially stable in the sense of (4.5) and (4.6).
5. Illustrative examples Consider the impulsive neural network given by dy 1 ðtÞ ¼ 5y 1 ðtÞ þ 0:5y 1 ðtÞ þ 0:1 tanhðy 2 ðtÞÞ; dt dy 2 ðtÞ ¼ 5y 2 ðtÞ þ 0:2jy 1 ðtÞj þ 0:4y 2 ðtÞ; dt
ð5:1Þ
for t > 0, t 5 tk = 1, 2, 3, . . . with initial states given by (y1(0), y2(0)) = (50, 45) and (y1(0) ,y2(0)) = (30, 40). The impulsive jumps are characterized by y i ðtþ k Þ ¼ ck y i ðt k Þ for t = tk = 1, 2, 3, . . . with ck = c = 54. For the network (5.1), let us apply the stability conditions associated with the norm kÆk1, consisting of the row-dominance a1 > jb11j + jb12j and a2 > jb21j + jb22j. The derivation of the conditions is given in the next section. One observes that ¼ minfa1 ðjb11 j þ jb12 jÞ; a2 ðjb21 j þ jb22 jÞg ¼ minf5 ð0:5 þ 0:1Þ; 5 ð0:2 þ 0:4Þg ¼ 4:4: Let us pick the number = 4 < * in order to illustrate the usefulness of the condition (4.4), in which case Dtk = h = 1. The magnitudes of the impulsive jumps then satisfy jckj 6 eh = 54.5. In accordance with the earlier theorems, the equilibrium state y* = 0 of the network (5.1) is unique and globally exponentially stable. The exponential convergence dynamics of the network (5.1) with and without impulses are shown in Fig. 1. Consider again the impulsive network (5.1) for t > 0, t 5 tk = 2, 4, 6, . . . subjected to impulsive jumps characterized by y i ðtþ k Þ ¼ ck y i ðt k Þ for tk = 2, 4, 6, . . .with ck = c = 2900. The inter-impulse intervals are Dtk = h = 2. With the above conditions associated with the norm kÆk1, one has = 4.4. On choosing the same number = 4 < *, the magnitudes of the impulsive jumps satisfy jck j 6 eh ¼ 2981, and as before the equilibrium state y* = 0 of the network (5.1) is unique and globally exponentially stable. The exponential convergence dynamics of the network (5.1) with and without impulses are shown in Fig. 2.
Neuron 1 without impulses
30 20
20
10
10
0
0
–10
–10
–20
–20
–30
0
30
1
2
3
4
5
Neuron 1 with impulses
–30
20
10
10
0
0
–10
–10
–20
–20 2
4
6
8
0
30
20
–30 0
Neuron 2 without impulses
30
10
–30 0
1
2
3
4
5
Neuron 2 with impulses
2
4
6
8
10
Fig. 1. Exponential convergence of neural network (5.1) with and without impulses. The impulsive jumps are characterized by ck = c = 54 at times tk = 1, 2, 3, . . . and Dtk = 1.
S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467 Neuron 1 without impulses
Neuron 2 without impulses
30
30
20
20
10
10
0
0
–10
–10
–20
–20
–30
–30 0
2
4
6
8
10
0
Neuron 1 with impulses
2
4
6
8
10
Neuron 2 with impulses
30
30
20
20
10
10
0
0
–10
–10
–20
–20
–30
465
–30 0
2
4
6
8
10
0
2
4
6
8
10
Fig. 2. Exponential convergence of neural network (5.1) with and without impulses. The impulsive jumps are characteristics by ck = c = 2900 at times tk = 2, 4, 6, . . . and Dtk = 2.
6. Concluding remarks A set of sufficient conditions for a Hopfleld-type neural network to always converge exponentially towards a unique equilibrium state has been found – cf. (3.3) and (3.8) in particular. The convergence persists even under the influence of impulsive jumps of the form Gk(u) = cku for u 2 R provided the magnitude and the frequency of the impulsive jumps satisfy 0 < jckj 6 eh and Dtk P h > 0 respectively, where > 0 is defined under either (3.3) or (3.8). Note that (4.4)–(4.6) provide the information linking the conditions (3.3) and (3.8) via the positive constants , l and h. The number l > 0 determines the convergence rate of the network, and the constant > 0 defines how large the impulse magnitudes can be. The important role of the constant h > 0 in determining the size of the impulse magnitudes and influencing the convergence of the network is evident in (4.4) and (4.6), respectively. This is consistent with the work of Gopalsamy [14] and Xu et al. [33] in their stability investigations of the impulsive network (4.1) (or equivalently (2.1)). The severe restriction 0 < jckj < 1 of the impulse magnitudes considered in [1,2,15,16,24–26] is merely a special case of the criterion 0 < jckj 6 eh obtained in this paper. The generalization to unbounded activation functions under kÆkp and the sufficiency conditions (3.3) and (3.8) are the other notable developments here. Obviously, the asymptotic stability results obtained previously for the impulsive network (4.1) under the norms kÆk1, kÆk2 and kÆk1 are included. Note that the general Euclidean norm for a vector u ¼ ðu1 ; u2 ; . . . ; um Þ 2 Rm satisfies kÆkp ! kÆk1 as p ! 1, so the relevant condition can be extracted from either the condition (3.3) or the condition (3.8) by appropriately defining the non-network parameters dij, rij. Let the parameters in the condition (3.8) be dij = (p 1)/p and rij = (p 1)/p, so that ai >
m m p1 X 1X aj jbij jLj þ jbji jLi ; p j¼1 p j¼1 ai
Then if p ! 1, one has ai >
m X j¼1
jbij jLj ;
i ¼ 1; 2 . . . ; m
i ¼ 1; 2; . . . ; m:
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S. Mohamad / Chaos, Solitons and Fractals 32 (2007) 456–467
as the condition under which the network (4.1) converges exponentially in the norm kÆk1. The condition is consistent with that obtained by Gopalsamy [14]. As another example, set p = 1 in the condition (3.3) such that ai >
m X aj jbji jLi ; ai j¼1
i ¼ 1; 2 . . . ; m;
guarantees the exponential convergence of the impulsive network (4.1) in the norm kÆk1. This condition is an improvement on those obtained previously [1,14,26]. It is also evident that the condition (3.3) with p = 2 improves the sufficiency condition found in [14] under the norm kÆk2.
Acknowledgement The author thanks Professor Roger Hosking of the Department of Mathematics, Universiti Brunei Darussalam for reading the manuscript and suggesting on ways to improve the exposition of the manuscript.
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