Journal of the Franklin Institute 338 (2001) 481–495

Stability of stochastic delay neural networks

$

Steve Blythe

a

,Xuerong Mao

a,

*,Xiaoxin Liao

b

a

Department of Statistics and Modelling Science,University of Strathclyde,Glasgow G1 1XH,UK

b

Department of Control,Huazhong University of Science and Technology,Wuhan,

People’s Republic of China

Received 17 March 2000;received in revised form 20 February 2001

Abstract

The authors in their papers (Liao and Mao,Stochast.Anal.Appl.14 (2) (1996a) 165–185;

Neural,Parallel Sci.Comput.4 (2) (1996b) 205–244) initiated the study of stability and

instability of stochastic neural networks and this paper is the continuation of their research in

this area.The main aim of this paper is to discuss almost sure exponential stability for a

stochastic delay neural network dxðtÞ ¼ ½BxðtÞ þAgðx

t

ðtÞÞ dt þsðxðtÞ;gðx

t

ðtÞ;tÞ dwðtÞ.The

techniques used in this paper are diﬀerent from those in their earlier papers.Especially,the

nonnegative semimartingale convergence theorem will play an important role in this paper.

Several examples are also given for illustration.#2001 The Franklin Institute.Published by

Elsevier Science Ltd.All rights reserved.

Keywords:Delay neural network;Brownian motion;Martingale convergence theorem;Lyapunov

exponent;Exponential stability

1.Introduction

Theoretical understanding of neural-network dynamics has advanced greatly in

the past 15 years [1–6].In many networks,time delays cannot be avoided.For

example,in electronic neural networks,time delays will be present due to the ﬁnite

switching speed of ampliﬁers.Marcus and Westervelt [7] proposed,in a similar way

as Hopﬁeld [2],a model for a network with delays as follows:

C

i

’

u

i

ðtÞ ¼

1

R

i

u

i

ðtÞ þ

X

n

j¼1

T

ij

g

j

ðu

j

ðt t

j

ÞÞ;14i4n;ð1:1Þ

Supported by the Royal Society and the EPSRC=BBSRC.

*Corresponding author.Tel.:+44-141-548-3669;fax:+44-141-552-2079.

E-mail addresses:xuerong@stams.strath.ac.uk (X.Mao),liaoxx@public.wh.hb.cn (X.Liao).

0016-0032/01/$20.00#2001 The Franklin Institute.Published by Elsevier Science Ltd.All rights reserved.

PII:S 0 0 1 6 - 0 0 3 2 ( 0 1 ) 0 0 0 1 6 - 3

on t50.The variable u

i

ðtÞ represents the voltage on the input of the ith neuron.Each

neuron is characterized by an input capacitance C

i

,a time delay t

i

and a transfer

function g

i

ðuÞ.The connection matrix element T

ij

has a value þ1=R

ij

when the

noninverting output of the jth neuron is connected to the input of the ith neuron

through a resistance R

ij

,and a value 1=R

ij

when the inverting output of the jth

neuron is connected to the input of the ith neuron through a resistance R

ij

.The

parallel resistance at the input of each neuron is deﬁned R

i

¼ ð

P

n

j¼1

jT

ij

jÞ

1

.The

nonlinear transfer function g

i

ðuÞ is sigmoidal,saturating at 1 with maximumslope

at u ¼ 0.That is,in mathematical terms,g

i

ðuÞ is nondecreasing and

jg

i

ðuÞj41 ^b

i

juj for all 15u51;ð1:2Þ

where b

i

is the slope of g

i

ðuÞ at u ¼ 0 and is supposed to be ﬁnite.By deﬁning

b

i

¼

1

C

i

R

i

;a

ij

¼

T

ij

C

i

;

Eq.(1.1) can be re-written as

’

u

i

ðtÞ ¼ b

i

u

i

ðtÞ þ

X

n

j¼1

a

ij

g

j

ðu

j

ðt t

j

ÞÞ;14i4n ð1:3Þ

or,in matrix form,

’

uðtÞ ¼ BuðtÞ þAgðu

t

ðtÞÞ;t50;ð1:4Þ

where

uðtÞ ¼ ðu

1

ðtÞ;...;u

n

ðtÞÞ

T

;u

t

ðtÞ ¼ ðu

1

ðt t

1

Þ;...;u

n

ðt t

n

ÞÞ

T

;

B ¼ diagðb

1

;...;b

n

Þ;A ¼ ða

ij

Þ

n n

;gðuÞ ¼ ðg

1

ðu

1

Þ;...;g

n

ðu

n

ÞÞ

T

:

Moreover,there is a relationship

b

i

¼

X

n

j¼1

ja

ij

j;14i4n:ð1:5Þ

It is clear that whenever given initial data uðsÞ ¼ xðsÞ for

%

t4s40;Eq.(1.4) has a

unique global solution on t50,where

%

t ¼ max

14i4n

t

i

and x ¼ fxðsÞ:

%

t4s40g is

a Cð½

%

t;0;R

n

Þ-valued function.

Haykin [8] points out that in real nervous systems,synaptic transmission ‘‘...is a

noisy process brought on by random ﬂuctuations from the release of neurotrans-

mitters,and other probabilistic causes’’ (p.309–310).One approach to the

mathematical incorporation of such eﬀects is to use probabilistic threshold models

(e.g [8]);the approach used in the current paper is to view neural networks as

nonlinear dynamical systems with intrinsic noise,that is,to include a representation

of the inherent stochasticity in the neurodynamics.Le Cun et al.[9] describe a

network where noise is injected into the ﬁrst hidden layer (only),and use the result to

obtain error derivatives,thereby avoiding back-propagation.

We therefore suppose that there exists a stochastic perturbation to the neural

network and the stochastically perturbed network with delays is described by a

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495482

stochastic diﬀerential delay equation

dxðtÞ ¼ ½BxðtÞ þAgðx

t

ðtÞÞ dt þsðxðtÞ;x

t

ðtÞ;tÞ dwðtÞ on t50;

xðsÞ ¼ xðsÞ on

%

t4s40:ð1:6Þ

Here wðtÞ ¼ ðw

1

ðtÞ;...;w

m

ðtÞÞ

T

is an m-dimensional Brownian motion deﬁned on a

complete probability space ðO;F;PÞ with a natural ﬁltration fF

t

g

t50

(i.e.F

t

¼

sfwðsÞ:04s4tg),xðtÞ ¼ ðx

1

ðtÞ;...;x

n

ðtÞÞ

T

,x

t

ðtÞ ¼ ðx

1

ðt t

1

Þ;...;x

n

ðt t

n

ÞÞ

T

and

s:R

n

R

n

R

þ

!R

n m

i.e.sðx;y;tÞ ¼ s

ij

ðx;y;tÞÞ

n m

.Assume,throughout

this paper,that sðx;y;tÞ is locally Lipschitz continuous and satisﬁes the linear

growth condition as well.So it is known (cf.[10–12]) that Eq.(1.6) has a unique

global solution on t50,which is denoted by xðt;xÞ.Moreover,assume also that

sð0;0;tÞ 0 for the stability purpose of this paper.So Eq.(1.6) admits an

equilibrium solution xðt;0Þ 0.

Stability of stochastic diﬀerential delay equations have been studied intensively

and the reader is referred,for example,to Arnold [13],Friedman [14],Has’minskii

[15],Kolmanovskii and Myshkis [16] and Mao [10,11].However,stochastic

delay neural networks have their own characteristics and it is desirable to obtain

stability criteria that make full use of these characteristics.It was in this spirit

that the authors in their earlier papers Liao and Mao [17,18] initiated the study

of stability and instability of stochastic neural networks and this paper is

the continuation of their research in this area.The main aim of this paper is to

discuss the almost sure exponential stability of the stochastic delay neural network

(1.6).Liao and Mao [18] mainly discussed the mean square exponential stability

fromwhich,along with an additional condition,the almost sure exponential stability

follows.But this paper investigates the almost sure exponential stability directly

and the criteria obtained here are absolutely new.Moreover,the techniques used

in this paper are diﬀerent from those in the authors’ earlier papers.Especially,

the nonnegative semimartingale convergence theorem will play an important role

in this paper.This paper is organized as follows:The main results of this paper

are developed in Section 2 where several suﬃcient criteria are established for almost

sure exponential stability of the stochastic delay neural network (1.6).By making

use of the special construction of the network,a number of very useful corollaries

are obtained in Section 3.These corollaries are described only in terms of

given system parameters and hence are extremely useful in applications.Finally,a

number of examples will be provided as illustrations of the use of the theorems in

Section 4.

2.Main results

Let C

2;1

ðR

n

R

þ

;R

þ

Þ denote the family of all nonnegative functions Vðx;tÞ on

R

n

R

þ

which are continuously twice diﬀerentiable in x and once diﬀerentiable in t.

For each V 2 C

2;1

ðR

n

R

þ

;R

þ

Þ,deﬁne an operator LV,associated with the

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495 483

stochastic delay neural network (1.6),from R

n

R

n

R

þ

to R by

LVðx;y;tÞ ¼V

t

ðx;tÞ þV

x

ðx;tÞ½Bx þAgðyÞ

þ

1

2

trace½s

T

ðx;y;tÞV

xx

ðx;tÞsðx;y;tÞ;

where

V

t

ðx;tÞ ¼

@Vðx;tÞ

@t

;V

x

ðx;tÞ ¼

@Vðx;tÞ

@x

1

;...;

@Vðx;tÞ

@x

n

;

V

xx

ðx;tÞ ¼

@

2

Vðx;tÞ

@x

i

@x

j

n n

:

Let us stress that LV is deﬁned on R

n

R

n

R

þ

while V on R

n

R

þ

.Let

CðR

n

;R

þ

Þ denote the family of all continuous functions from R

n

to R

þ

,while

Cð½

%

T;0;R

n

Þ,CðR;R

þ

Þ;etc.

Although conditions (1.2) and (1.5) are the characteristics of the delay network

and are of course assumed to hold throughout this paper,they will be mentioned

whenever they are used explicitly.

Theorem 2.1.Assume that there exist a number of functions V 2 C

2;1

ðR

n

R

þ

;R

þ

Þ;

f 2 CðR

n

;R

þ

Þ;f

i

2 CðR;R

þ

Þ ð14i4nÞ and two constants l

1

> l

2

50 such that

LVðx;y;tÞ4l

1

fðxÞ þl

2

X

n

i¼1

f

i

ðy

i

Þ;ðx;y;tÞ 2 R

n

R

n

R

þ

;ð2:1Þ

Vðx;tÞ4fðxÞ;ðx;tÞ 2 R

n

R

þ

ð2:2Þ

and

X

n

i¼1

f

i

ðx

i

Þ4fðxÞ;x 2 R

n

:ð2:3Þ

Then;for every x 2 Cð½t;0;R

n

Þ;the solution of Eq.ð1:6Þ has the property

limsup

t!1

1

t

log ðVðxðt;xÞ;tÞÞ4g a:s:;ð2:4Þ

where g 2 ð0;l

1

l

2

Þ is the unique root of

l

1

¼ g þl

2

e

g

%

t

:ð2:5Þ

ðRecall

%

t ¼ max

14i4n

t

i

.)

The proof of this theorem is based on the following semimartingale convergence

theorem established by Liptser and Shiryayev [19,Theorem 7,p.139].

Lemma 2.2.Let AðtÞ and UðtÞ be two continuous adapted increasing processes on t50

with Að0Þ ¼ Uð0Þ ¼ 0 a.s.Let MðtÞ be a real-valued continuous local martingale with

Mð0Þ ¼ 0 a.s.Let z be a nonnegative F

0

-measurable random variable with Ez51.

Deﬁne

XðtÞ ¼ z þAðtÞ UðtÞ þMðtÞ for t50:

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495484

If XðtÞ is nonnegative;then

lim

t!1

AðtÞ51

lim

t!1

XðtÞ51

\lim

t!1

UðtÞ51

a:s:;

where B D a.s.means PðB\D

c

Þ ¼ 0.In particular;if lim

t!1

AðtÞ51 a.s.;then

for almost all o 2 O

lim

t!1

Xðt;oÞ51 and lim

t!1

Uðt;oÞ51;

that is both XðtÞ and UðtÞ converge to ﬁnite random variables.

Proof.Fix initial data x 2 Cð½t;0;R

n

Þ arbitrarily and write simply xðt;xÞ ¼ xðtÞ.

Deﬁne

Uðx;tÞ ¼ e

gt

Vðx;tÞ for ðx;tÞ 2 R

n

R

þ

;

which is in C

2;1

ðR

n

R

þ

;R

þ

Þ obviously.Using conditions (2.1) and (2.2),one can

compute

LUðx;y;tÞ ¼e

gt

½gVðx;tÞ þLVðx;y;tÞ

4e

gt

ðl

1

gÞfðxÞ þl

2

X

n

i¼1

f

i

ðy

i

Þ

"#

:

The It

#

o formula shows that for any t50

e

gt

VðxðtÞ;tÞ ¼Vðxð0Þ;0Þ þ

Z

t

0

LVðxðsÞ;x

t

ðsÞ;sÞ ds

þ

Z

t

0

e

gs

V

x

ðxðsÞ;sÞsðxðsÞ;x

t

ðsÞ;sÞ dwðsÞ

4Vðxð0Þ;0Þ ðl

1

gÞ

Z

t

0

e

gs

fðxðsÞÞ ds

þl

2

X

n

i¼1

Z

t

0

e

gs

f

i

ðx

i

ðs t

i

ÞÞ ds

þ

Z

t

0

e

gs

V

x

ðxðsÞ;sÞsðxðsÞ;x

t

ðsÞ;sÞ dwðsÞ:ð2:6Þ

On the other hand,it is easy to see that

Z

t

tt

i

e

gs

f

i

ðx

i

ðsÞÞ ds ¼

Z

t

t

i

e

gs

f

i

ðx

i

ðsÞÞ ds

Z

t

0

e

gðst

i

Þ

f

i

ðx

i

ðs t

i

ÞÞ ds

4

Z

t

%

t

e

gs

f

i

ðx

i

ðsÞÞ ds e

g

%

t

Z

t

0

e

gs

f

i

ðx

i

ðs t

i

ÞÞ ds:

This,together with condition (2.3),implies

X

n

i¼1

Z

t

tt

i

e

gs

f

i

ðx

i

ðsÞÞ ds4

Z

t

%

t

e

gs

fðxðsÞÞ ds e

g

%

t

X

n

i¼1

Z

t

0

e

gs

f

i

ðx

i

ðs t

i

ÞÞ ds:

ð2:7Þ

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495 485

It then follows from (2.6) and (2.7) that

e

gt

VðxðtÞ;tÞ þl

2

e

g%t

X

n

i¼1

Z

t

tt

i

e

gs

f

i

ðx

i

ðsÞÞ ds

4Vðxð0Þ;0Þ þ

Z

0

%t

fðxðsÞÞ ds ðl

1

g l

2

e

g

%

t

Þ

Z

t

0

e

gs

fðxðsÞÞ ds

þ

Z

t

0

e

gs

V

x

ðxðsÞ;sÞsðxðsÞ;x

t

ðsÞ;sÞ dwðsÞ:

Making use of (2.5) yields that

e

gt

VðxðtÞ;tÞ þl

2

e

g

%

t

X

n

i¼1

Z

t

tt

i

e

gs

f

i

ðx

i

ðsÞÞ ds4XðtÞ;ð2:8Þ

where

XðtÞ:¼ Vðxð0Þ;0Þ þ

Z

0

%

t

fðxðsÞÞ ds þ

Z

t

0

e

gs

V

x

ðxðsÞ;sÞ sðxðsÞ;x

t

ðsÞ;sÞ dwðsÞ;

which is a nonnegative martingale,and Lemma 2.2 shows

lim

t!1

XðtÞ51 a:s:

It therefore follows from (2.8) that

limsup

t!1

½e

gt

VðxðtÞ;tÞ51 a:s:

which implies

limsup

t!1

1

t

log ðVðxðtÞ;tÞÞ4g a:s:

as required.The proof is complete.&

Theorem 2.3.Let ð1:2Þ hold.Assume that there exist symmetric nonnegative-deﬁnite

matrices C

1

;C

2

and C

3

¼ diagðd

1

;...;d

n

Þ such that

trace½s

T

ðx;y;tÞsðx;y;tÞ4x

T

C

1

x þg

T

ðyÞC

2

gðyÞ þy

T

C

3

y ð2:9Þ

for all ðx;y;tÞ 2 R

n

R

n

R

þ

.Assume also that there exists a positive-deﬁnite

diagonal matrix D ¼ diagðd

1

;...;d

n

Þ such that the symmetric matrix

H ¼

2BþC

1

þC

3

þ

%

D A

A

T

DþC

2

0

@

1

A

is negative-deﬁnite;where

%

D ¼ diagðd

1

b

2

1

;...;d

n

b

2

n

Þ.Let l ¼ l

max

ðHÞ;the biggest

eigenvalue of H so l > 0.Then;for every x 2 Cð½t;0;R

n

Þ;the sample Lyapunov-

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495486

exponent of the solution of Eq.ð1:6Þ can be estimated as

limsup

t!1

1

t

log ðjxðt;xÞjÞ4

g

2

a:s:;ð2:10Þ

where g > 0 is the unique root to the equation

l

1

¼ g þl

1

l

2

e

g%t

ð2:11Þ

with

l

1

¼ min

14i4n

ðl þd

i

þd

i

b

2

i

Þ and l

2

¼ max

14i4n

d

i

þðd

i

lÞb

2

i

l þd

i

þd

i

b

2

i

:ð2:12Þ

In other words;the stochastic delay neural network ð1:6Þ is almost surely exponentially

stable.

Proof.Let Vðx;tÞ ¼ jxj

2

.Then the operator LV has the form

LVðx;y;tÞ ¼ 2x

T

½Bx þAgðyÞ þtrace½s

T

ðx;y;tÞsðx;y;tÞ:

Compute,by the hypotheses,

LVðx;y;tÞ4 2x

T

Bx þx

T

AgðyÞ þg

T

ðyÞA

T

x þx

T

C

1

x þg

T

ðyÞC

2

gðyÞ þy

T

C

3

y

¼x

T

ð2BþC

1

þC

3

þ

%

DÞx þx

T

AgðyÞ þg

T

ðyÞA

T

x

þg

T

ðyÞðDþC

2

ÞgðyÞ x

T

ðC

3

þ

%

DÞx þy

T

C

3

y þg

T

ðyÞDgðyÞ

¼ðx

T

;g

T

ðyÞÞ H

x

gðyÞ

!

x

T

ðC

3

þ

%

DÞx þy

T

C

3

y þg

T

ðyÞDgðyÞ

4 lðjxj

2

þjgðyÞj

2

Þ x

T

ðC

3

þ

%

DÞx þy

T

C

3

y þg

T

ðyÞDgðyÞ

¼

X

n

i¼1

ðl þd

i

þd

i

b

2

i

Þx

2

i

þ

X

n

i¼1

½d

i

y

2

i

þðd

i

lÞg

2

i

ðy

i

Þ:

It is easy to see fromthe construction of H that l4d

i

for all 14i4n.Using (1.2) one

can then derive that

LVðx;y;tÞ4

X

n

i¼1

ðl þd

i

þd

i

b

2

i

Þx

2

i

þ

X

n

i¼1

½d

i

þðd

i

lÞb

2

i

y

2

i

:ð2:13Þ

In order to apply Theorem 2.1,deﬁne f 2 CðR

n

;R

þ

Þ and f

i

2 CðR;R

þ

Þ by

fðxÞ ¼

1

l

1

X

n

i¼1

ðl þd

i

þd

i

b

2

i

Þx

2

i

and f

i

ðy

i

Þ ¼

1

l

1

ðl þd

i

þd

i

b

2

i

Þy

2

i

:

It is obvious that

fðxÞ5jxj

2

¼ Vðx;tÞ and fðxÞ ¼

X

n

i¼1

f

i

ðx

i

Þ:

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495 487

Moreover,

LVðx;y;tÞ4 l

1

fðxÞ þ

X

n

i¼1

d

i

þðd

i

lÞb

2

i

l þd

i

þd

i

b

2

i

l

1

f

i

ðy

i

Þ

4 l

1

fðxÞ þl

1

l

2

X

n

i¼1

f

i

ðy

i

Þ:

By Theorem 2.1,for every x 2 Cð½t;0;R

n

Þ,the solution of Eq.(1.6) has the

property

limsup

t!1

1

t

log ðjxðt;xÞj

2

Þ4g a:s:

and the required assertion (2.10) follows.The proof is complete.&

Theorem 2.4.Let ð1:2Þ hold.Assume that there exist nonnegative numbers m

i

;y

i

and d

i

such that

trace½s

T

ðx;y;tÞsðx;y;tÞ4

X

n

i¼1

½m

i

x

2

i

þy

i

g

2

i

ðy

i

Þ þd

i

y

2

i

ð2:14Þ

for all ðx;y;tÞ 2 R

n

R

n

R

þ

.Assume also that there exists a positive-deﬁnite

diagonal matrix D ¼ diagðd

1

;...;d

n

Þ such that the symmetric matrix

%

H ¼

2B þ

%

D A

A

T

D

!

is negative-deﬁnite;where

%

D is the same as deﬁned in Theorem 2:3;namely

%

D ¼

diagðd

1

b

2

1

;...;d

n

b

2

n

Þ.Let

%

l ¼ l

max

ð

%

HÞ so

%

l > 0.If

ðm

i

þd

i

Þ _y

i

5

%

l;14i4n;ð2:15Þ

then the stochastic delay neural network ð1:6Þ is almost surely exponentially

stable.Moreover;the sample Lyapunov exponent ði.e.the left hand side of ð2:10ÞÞ

can be estimated by ð2:10Þ as long as the l in ð2:12Þ is determined by

l ¼ min

14i4n

½

%

l ðm

i

þd

i

Þ _y

i

:ð2:16Þ

Proof.Set

C

1

¼ diagðm

1

;...;m

n

Þ;C

2

¼ diagðy

1

;...;y

n

Þ;C

3

¼ diagðd

1

;...;d

n

Þ:

Then (2.14) can be written as

trace½s

T

ðx;y;tÞsðx;y;tÞ4x

T

C

1

x þg

T

ðyÞC

2

gðyÞ þy

T

C

3

y:

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495488

In view of Theorem 2.3,it is suﬃcient to verify that the matrix H deﬁned there is

negative-deﬁnite.To do so,for any x;y 2 R

n

,compute

ðx

T

;y

T

ÞH

x

y

!

¼ðx

T

;y

T

Þ

%

H

x

y

!

þðx

T

;y

T

Þ

C

1

þC

3

0

0 C

2

!

x

y

!

4

%

lðjxj

2

þjyj

2

Þ þ

X

n

i¼1

½ðm

i

þd

i

Þx

2

i

þy

i

y

2

i

4lðjxj

2

þjyj

2

Þ;

where l is deﬁned by (2.16) and is positive due to (2.15).The proof is therefore

complete.&

3.Further results

In the previous section several general criteria were obtained for the almost

sure exponential stability of the delay network.The use of these criteria depends

very much on the construction of the Lyapunov function V (Theorem 2.1) or

the choice of positive numbers d

i

(Theorems 2.3 and 2.4).However,it would be

nice and convenient to have some criteria which are only based on the system

parameters e.g.b

i

and b

i

.Moreover,relations (1.2) and (1.5) are both the properties

of the neural network but the criteria established in the previous have only used

property (1.2).In this section we shall also make use of the nice property (1.5) in

order to obtain further results.In particular,the new criteria here will be described

only in terms of the system parameters b

i

;b

i

;etc.given by (1.2),(1.5) and (2.14)

hence these criteria can be veriﬁed easily and should be proven very useful in

applications.

Corollary 3.1.Let ð1:2Þ;ð1:5Þ and ð2:14Þ hold.Assume

b

i

> b

2

i

X

n

j¼1

ja

ji

j for all 14i4n:ð3:1Þ

Let

%

l ¼ min

14i4n

b

i

b

2

i

P

n

j¼1

ja

ji

j

1 þb

2

i

:ð3:2Þ

If

ðm

i

þd

i

Þ _y

i

5

%

l;14i4n;ð3:3Þ

then the stochastic delay neural network ð1:6Þ is almost surely exponentially stable.

Proof.Choose

d

i

¼

b

i

þ

P

n

j¼1

ja

ji

j

1 þb

2

i

for 14i4n

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495 489

and then deﬁne the symmetric matrix

%

H the same as in Theorem 2.4.For any

x;y 2 R

n

,compute

ðx

T

;y

T

Þ

%

H

x

y

!

¼

X

n

i¼1

ð2b

i

þd

i

b

2

i

Þx

2

i

þ2

X

n

i;j¼1

a

ij

x

i

y

j

X

n

i¼1

d

i

y

2

i

4

X

n

i¼1

ð2b

i

þd

i

b

2

i

Þx

2

i

þ

X

n

i;j¼1

ja

ij

jðx

2

i

þy

2

j

Þ

X

n

i¼1

d

i

y

2

i

¼

X

n

i¼1

ðb

i

d

i

b

2

i

Þx

2

i

X

n

i¼1

d

i

X

n

j¼1

ja

ji

j

!

y

2

i

;

where condition (1.5) has been used.But

b

i

d

i

b

2

i

¼ d

i

X

n

j¼1

ja

ji

j ¼

b

i

b

2

i

P

n

j¼1

ja

ji

j

1 þb

2

i

5

%

l:

So

ðx

T

;y

T

ÞH

x

y

!

4

%

lðjxj

2

þjyj

2

Þ;

which implies l

max

ð

%

HÞ4

%

l.Now the conclusion of this corollary follows from

Theorem 2.4.The proof is complete.&

In practice,the networks are often symmetric in the sense ja

ij

j ¼ ja

ji

j.For such

symmetric networks the following result are particularly useful.

Corollary 3.2.Let ð1:2Þ;ð1:5Þ and ð2:14Þ hold.Assume the network is symmetric in the

sense

ja

ij

j ¼ ja

ji

j for all 14i;j4n:ð3:4Þ

Assume

b

i

51 for all 14i4n:ð3:5Þ

Let

%

l ¼ min

14i4n

b

i

ð1 b

2

i

Þ

1 þb

2

i

:ð3:6Þ

If ð3:3Þ holds;then the stochastic delay network ð1:6Þ is almost surely exponentially

stable.

Proof.By (1.5),(3.4) and (3.5),

b

2

i

X

n

j¼1

ja

ji

j5

X

n

j¼1

ja

ij

j ¼ b

i

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495490

for all 14i4n.Also

min

14i4n

b

i

b

2

i

P

n

j¼1

ja

ji

j

1 þb

2

i

¼ min

14i4n

b

i

ð1 b

2

i

Þ

1 þb

2

i

:

Hence the conclusion follows from Corollary 3.1 directly.The proof is complete.

In the sequel we shall denote by jjAjj the operator norm of matrix A,i.e.

jjAjj ¼ supfjAxj:x 2 R

n

;jxj ¼ 1g.

Corollary 3.3.Let ð1:2Þ and ð2:14Þ hold.Assume

2b

i

> jjAjjð1 þb

2

i

Þ for all 14i4n:ð3:7Þ

Let

%

l ¼ min

14i4n

2b

i

1 þb

2

i

jjAjj

!

:ð3:8Þ

If ð3:3Þ holds;then the stochastic delay network ð1:6Þ is almost surely exponentially

stable.

Proof.Choose

d

i

¼

2b

i

1 þb

2

i

for 14i4n

and then deﬁne the symmetric matrix

%

H the same as in Theorem 2.4.For any

x;y 2 R

n

,compute

ðx

T

;y

T

Þ

%

H

x

y

!

¼

X

n

i¼1

ð2b

i

þd

i

b

2

i

Þx

2

i

þ2x

T

Ay

X

n

i¼1

d

i

y

2

i

4

X

n

i¼1

ð2b

i

þd

i

b

2

i

Þx

2

i

þjjAjjðjxj

2

þjyj

2

Þ

X

n

i¼1

d

i

y

2

i

¼

X

n

i¼1

ð2b

i

d

i

b

2

i

jjAjjÞx

2

i

X

n

i¼1

ðd

i

jjAjjÞy

2

i

:

Note

2b

i

d

i

b

2

i

jjAjj ¼ d

i

jjAjj ¼

2b

i

1 þb

2

i

jjAjj5

%

l:

So

ðx

T

;y

T

ÞH

x

y

!

4

%

lðjxj

2

þjyj

2

Þ

which yields l

max

ð

%

HÞ4

%

l.Now the conclusion of this corollary follows from

Theorem 2.4.The proof is complete.&

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495 491

4.Examples

This section provides the examples as illustrations of the use of the theorems.The

networks described here are a little bit artiﬁcial but how the theory of this paper can

be applied is clearly illustrated.

The use of the criteria established in Section 2 depends very much on the

construction of the Lyapunov function V (Theorem 2.1) or the choice of positive

numbers d

i

(Theorems 2.3 and 2.4).The following example illustrates how to choose

positive numbers d

i

so that Theorem 2.3 can be applied.It also illustrates how to

estimate the sample Lyapunov exponent.

Example 4.1.Consider a stochastic delay network

d

x

1

ðtÞ

x

2

ðtÞ

!

¼

4 0

0 2

!

x

1

ðtÞ

x

2

ðtÞ

!

dt þ

2 2

1 1

!

g

1

ðx

1

ðt t

1

ÞÞ

g

2

ðx

2

ðt t

2

ÞÞ

!

dt

þ

0:2x

2

ðt t

2

ÞÞ

0:5x

1

ðt t

1

ÞÞ

!

dwðtÞ;ð4:1Þ

where wðtÞ is a real-valued Brownian motion,t

1

and t

2

are both positive numbers

while

g

i

ðu

i

Þ ¼

1 e

u

i

1 þe

u

i

:

It is easily shown that (1.2) is satisﬁed with b

1

¼ b

2

¼ 1.To apply Theorem2.3,note

in this example that

sðx;y;tÞ ¼ ð0:2y

2

;0:5y

1

Þ

T

whence s

T

ðx;y;tÞsðx;y;tÞ ¼ 0:25y

2

1

þ0:04y

2

2

:

Hence condition (2.9) is satisﬁed with C

1

¼ C

2

¼ 0 and C

3

¼ diagð0:25;0:04Þ.Now,

choose D ¼ diagðd

1

;d

2

Þ such that

2BþC

3

þD ¼ D;namely

8 0

0 4

!

þ

0:25 0

0 0:04

!

¼

2d

1

0

0 2d

2

!

which gives d

1

¼ 3:875 and d

2

¼ 1:98.So the matrix H deﬁned in Theorem 2.3

becomes

H ¼

3:875 0 2 2

0 1:98 1 1

2 1 3:875 0

2 1 0 1:98

0

B

B

B

@

1

C

C

C

A

:

Compute l

max

ðHÞ ¼ 0:19578 which means that H is negative-deﬁnite.By Theorem

2.3,the delay network (4.1) is almost surely exponentially stable.To estimate the

sample Lyapunov exponent,compute,by (2.12),l

1

¼ 2:21578 and l

2

¼ 0:9094 so

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495492

(2.11) becomes

2:21578 ¼ g þ2:015e

g%t

:ð4:2Þ

If both t

1

and t

2

are known,e.g.t

1

¼ t

2

¼ 0:1 then

%

t ¼ 0:1 and ð4:2Þ reduces to

2:21578 ¼ g þ2:015e

0:1g

which has a unique root g ¼ 0:1668.Therefore,Theorem 2.3 shows that the

sample Lyapunov exponent of the solution of Eq.(4.1) should not be greater than

0:0834.

By making use of the special construction of the network,a number of very useful

results are obtained in Section 3.Since these results are described only in terms of

given system parameters,they are very convenient to be used in applications.The

following example illustrates that if the neural network is symmetric how we can

make use of this nice symmetric property to show the stability.

Example 4.2.Consider a three-dimensional stochastic delay neural network

dxðtÞ ¼½BxðtÞ þAgðx

t

ðtÞÞ dt þB

1

xðtÞ dw

1

ðtÞ

þðy

1

sinðx

1

ðt t

1

ÞÞ;y

2

sinðx

2

ðt t

2

ÞÞ;y

3

sinðx

3

ðt t

3

ÞÞÞ

T

dw

2

ðtÞ:ð4:3Þ

Here ðw

1

ðtÞ;w

2

ðtÞÞ is a two-dimensional Brownian motion,B

1

a 3 3 constant

matrices,y

i

’s are real numbers and t

i

’s positive constants while

B ¼ diagð2;3;4Þ;A ¼

0 1 1

1 1 1

1 1 2

0

B

@

1

C

A

;

g

i

ðy

i

Þ ¼ ðb

i

y

i

^1Þ _ð1Þ with b

1

¼ 0:4;b

2

¼ 0:5;b

3

¼ 0:6;

gðyÞ ¼ ðg

1

ðy

1

Þ;g

2

ðy

2

Þ;g

3

ðy

3

ÞÞ

T

:

Clearly (1.2) and (1.5) holds and the network is symmetric so (3.4) is satisﬁed as well.

By (3.6),compute

%

l ¼ min

2ð1 0:4

2

Þ

1 þ0:4

2

;

3ð1 0:5

2

Þ

1 þ0:5

2

;

4ð1 0:6

2

Þ

1 þ0:6

2

¼ 1:448:

On the other hand,note in this example that

sðx;y;tÞ ¼ ðB

1

x;ðy

1

sin y

1

;y

2

sin y

2

;y

3

sin y

3

Þ

T

Þ:

Note also

sin

2

y

i

4ððy

i

^1Þ _ð1ÞÞ

2

4

1

b

2

i

ððb

i

^1Þ _ð1ÞÞ

2

¼

g

2

i

ðy

i

Þ

b

2

i

:

Consequently

trace½s

T

ðx;y;tÞ sðx;y;tÞ4jjB

1

jj

2

jxj

2

þ

y

2

1

0:16

g

2

1

ðy

1

Þ þ

y

2

2

0:25

g

2

2

ðy

2

Þ þ

y

2

1

0:36

g

2

3

ðy

3

Þ:

ð4:4Þ

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495 493

By Corollary 3.2,if

jjB

1

jj

2

51:448;y

2

1

50:23168;y

2

2

50:362;y

2

3

50:52128;ð4:5Þ

then the stochastic delay network (4.3) is almost surely exponentially stable.Of

course,one may estimate trace½s

T

ðx;y;tÞsðx;y;tÞ in a diﬀerent way in order to

obtain an alternative result.For instance,given

jjB

1

jj

2

¼ 0:5;y

1

¼ y

2

¼ y

3

¼ 1 ð4:6Þ

(4.5) is not satisﬁed.However,one can estimate

trace½s

T

ðx;y;tÞsðx;y;tÞ40:5jxj

2

þsin

2

ðy

1

Þ þsin

2

ðy

2

Þ þsin

2

ðy

2

Þ

40:5jxj

2

þ0:8jyj

2

þ0:2½sin

2

ðy

1

Þ þsin

2

ðy

2

Þ þsin

2

ðy

2

Þ

40:5jxj

2

þ0:8jyj

2

þ1:25g

2

1

ðy

1

Þ þ0:8g

2

2

ðy

2

Þ þ0:56g

2

3

ðy

3

Þ:

So (3.3) is satisﬁed and Corollary 3.2 shows that the stochastic delay network (4.3) is

still almost surely exponentially stable under condition (4.6).

The following example illustrates that for a given nonsymmetric network how we

can obtain a robust result on the intensity of noise provided the system parameters

are speciﬁed.

Example 4.3.Consider another three-dimensional stochastic delay neural

network

dxðtÞ ¼ ½BxðtÞ þAgðx

t

ðtÞÞ dt þB

1

gðx

t

ðtÞÞ dwðtÞ;ð4:7Þ

where wðtÞ is a scalar Brownian motion,B

1

a 3 3 constant matrix and

B ¼ diagð3;4;3Þ;A ¼

1 0 2

1 2 1

1 1 1

0

B

@

1

C

A

;

gðyÞ ¼ ðg

1

ðy

1

Þ;g

2

ðy

2

Þ;g

3

ðy

3

ÞÞ

T

;g

i

ðy

i

Þ ¼

e

b

i

y

i

e

b

i

y

i

e

b

i

y

i

þe

b

i

y

i

with b

1

¼ 0:4;b

2

¼ 0:5;b

3

¼ 0:4.Clearly (1.2) and (1.5) are satisﬁed.Note that

sðx;y;tÞ ¼ B

1

gðyÞ and for any e > 0,

s

T

ðx;y;tÞsðx;y;tÞ4jjB

1

jj

2

jgðyÞj

2

4jjB

1

jj

2

½eb

2

1

y

2

1

þeb

2

2

y

2

2

þeb

2

3

y

2

3

þð1 eÞjgðyÞj

2

4jjB

1

jj

2

½0:25ejyj

2

þð1 eÞjgðyÞj

2

:

Letting e ¼ 0:8 gives

s

T

ðx;y;tÞsðx;y;tÞ40:2jjB

1

jj

2

ðjyj

2

þjgðyÞj

2

Þ:

That is,(2.14) holds with m

i

¼ 0 and y

i

¼ d

i

¼ 0:2jjB

1

jj

2

.To apply Corollary 3.1,

compute by (3.2) that

%

l ¼ min

3 0:16 3

1 þ0:16

;

4 0:25 3

1 þ0:25

;

3 0:16 4

1 þ0:16

¼ 2:034:

S.Blythe et al./Journal of the Franklin Institute 338 (2001) 481–495494

Hence,(3.3) becomes

0:2jjB

1

jj

2

52:034;namely jjB

1

jj53:189:ð4:8Þ

Corollary 3.3 therefore concludes that the delay network (4.7) is almost surely

exponentially stable as long as jjB

1

jj53:189.

Acknowledgements

The authors would like to thank the referees for their very helpful suggestions.

They would also like to thank the Royal Society (UK) and the EPSRC=BBSRC for

their ﬁnancial supports.

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