Neurocomputing 72(2008)436–444
stochastic
fuzzy cellular
neural networks
with delays $
Ling Chen a,b ,Hongyong Zhao a,?
a
Department of Mathematics,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,PR China
b
Department of Basic Science,Jinling Institute of Technology,Nanjing 210001,PR China
Received 23April 2007;received in revised form 16November 2007;accepted 10December 2007
Communicated by J.Cao Available online 12February 2008
Abstract
In this paper,stochastic fuzzy cellular neural networks with delays are considered.By constructing suitable Lyapunov functionals and using stochastic analysis we give a family of suf?cient conditions ensuring the almost sure exponential stability of the networks.These results obtained are helpful to design stability of networks when stochastic noise is taken into consideration.r 2008Elsevier B.V.All rights reserved.
Keywords:Almost sure exponential stability;Stochastic;Lyapunov functional;Fuzzy cellular neural networks
1.Introduction
In the past few decades,neural networks such as cellular neural networks,Cohen–Grossberg neural networks,and Cohen–Grossberg-type bidirectional associative memory neural networks have drawn much attention,and many important results have been reported,see [7,14,12,27,3,20,5,28,17,6,19,4,1,2,18,23,15,8]for some recent publications.The another fundamental neural networks,Yang et al.in [25]have introduced a new class of neural networks,namely fuzzy cellular neural networks (FCNNs).This class of networks integrate fuzzy logic into the structure of traditional cellular neural networks and maintain local connectedness among cells.Unlike the previous cellular neural network structures,FCNNs have fuzzy logic between their template input and/or output besides the sum of product operation.Studies [22,24,26,9]have shown the potential of FCNNs in image processing and pattern recognition.Such applications heavily depend on the
dynamical behaviors.Thus,the analysis of the dynamical behaviors such as stability is a necessary step for practical design of FCNNs.Recently,many scienti?c and technical works have been joining the study ?elds with great interest,and various interesting results for FCNN models have been obtained,see e.g.[23,15,8]and references therein.Most FCNN models proposed and discussed in existing literature are deterministic.However,a real system is usually affected by external perturbations which in many cases are of great uncertainty and hence may be treated as random,as pointed out by Haykin [11]that in real nervous systems,the synaptic transmission is a noisy process brought on by random ?uctuations from the release of neurotransmitters and other probabilistic causes.Under the effect of the noise,the trajectory of system becomes a stochastic process.There are various kinds stability concepts to describe limiting beha-viors of stochastic processes,see,for example [10].The almost sure exponential stability is the most useful because it is closer to the real situation during computation than other forms of convergence (see [21,13]for the detailed discus-sions).Therefore,it is of great signi?cance to study the almost sure exponential stability for stochastic FCNN models.To the best our knowledge,few authors investigate the almost sure exponential stability for stochastic FCNNs with delays [29],which is still open.
https://www.doczj.com/doc/0419351523.html,/locate/neucom
0925-2312/$-see front matter r 2008Elsevier B.V.All rights reserved.doi:10.1016/j.neucom.2007.12.005
$
This research was supported by the Grant of ‘‘Qing-Lan Engineering’’Project of Jiangsu Province,and the Science Foundation of Nanjing University of Aeronautics and Astronautics.
?Corresponding author.Tel.:+862584489977.
E-mail addresses:zhaohy@https://www.doczj.com/doc/0419351523.html, ,zhaohym@https://www.doczj.com/doc/0419351523.html, (H.Zhao).
Based on the above discussion,our objective in this paper is to study stochastic FCNNs with delays,and give a family of suf?cient conditions ensuring the almost sure exponential stability by constructing suitable Lyapunov functionals and applying stochastic analysis.It is easy to apply these conditions to the real networks.2.Preliminary
R n and C ?X ;Y denote the n -dimensional Euclidean space and a continuous mapping set from the topological space X to the topological space Y ,respectively.Especially,C 9C ??àt ;0 ;R n ,where t 40.
Consider the following stochastic FCNNs with delays:
d x i et T?àc i x i et TtP n j ?1a ij f j ex j et TTtP n j ?1b ij u j tI i "tV n j ?1a ij f j ex j et àt TTtW n j ?1b ij f j ex j et àt TTtV n j ?1T ij u j tW
n j ?1H ij u j #
d t tP n j ?1s ij ex j et T;x j et àt TTd o j et T;t X 0;x i et T?f i et T;àt p t p 0;8
>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:
(1)where i ?1;...;n .a ij ;b ij ;T ij and H ij are elements of fuzzy feedback MIN template,fuzzy feedback MAX template,fuzzy feedforward MIN template and fuzzy feedforward MAX template,respectively.a ij and b ij are elements of feedback template and feedforward template.V and W denote the fuzzy AND and fuzzy OR operation,respec-tively.x i ,u i and I i denote state,input and bias of the i th neuron,respectively.c i 40is the neuron ?ring rate.f j eáTis the activation function.t represents transmission delay.s eá;áT?es ij eá;áTTn ?n is the diffusion coef?cient matrix and o eáT?eo 1eáT;...;o n eáTTT is an n -dimensional Brownian motion de?ned on a complete probability space (O ;F ;P )with a natural ?ltration f F t g t X 0(i.e.F t ?s f o es T:0p s p t g ).f i et Tis the initial function
where f i et T2L 2F 0e?t ;0 ;R n T,here L 2F 0e?t ;0 ;R n
Tdenotes the family of all C -valued random processes x es Tsuch that x es Tis F 0-measurable and R 0
àt E k x es Tk 2ds o 1.Assume,throughout this paper,that f j eáTand s ij eá;áTare locally Lipschitz continuous and satisfy the linear growth condi-tion as well.So it is known that Eq.(1)has a unique global solution on t X 0,which is denoted by x et T,where x et T?ex 1et T;...;x n et TTT .
Assume that the nonlinear functions f j eáTand s ij eá;áTsatisfy the following condition:
(H1)There exist positive constants L j and Z ij such that
j f j eu Tàf j ev Tj p L j j u àv j ,
j s ij eu ;u Tàs ij ev ;v Tj 2
p Z ij ej u àv j 2
tj u àv j 2
T,for any u ;v ;u ;v 2R ,i ;j ?1;...;n .
We ?rst give the following lemmas that are useful in deriving Lemma 1[16]).Let increasing processes on t X 0with A e0T?U e0T?0, a.s .Let M et Tbe a real-valued continuous local martingale with M e0T?0,a.s.Let z be a nonnegative F 0-measurable random variable with E z o 1.De?ne
X et T?z tA et TàU et TtM et Tfor t X 0.If X et Tis nonnegative ,then
lim t !1
A et To 1n
o &lim t !1
X et To 1n o \lim t !1
U et To 1n o
a :s :;
where B &D a.s.means P eB \D c T?0.In particular ,if lim t !1A et To 1a.s.,then for almost all o 2O
lim t !1
X et To 1and
lim t !1
U et To 1,
that is both X et Tand U et Tconverge to ?nite random variables.
Lemma 2(Yang and Yang [23]).Suppose x ?ex 1;...;x n TT and y ?ey 1;...;y n TT are two states of system (1),then we have (1)
^n j ?1
a ij f j ex j Tà
^n j ?1a ij f j ey j T
p X n j ?1
L j j a ij jj x j ày j j ,(2)
_n j ?1
b ij f j ex j Tà
_n j ?1b ij f j ey j T
p X n j ?1
L j j b ij jj x j ày j j ,where L j is given as (H1).
Lemma 3(Chen and Liao [8]).Suppose x ?ex 1;...;x n TT
and y ?ey 1;...;y n TT are two states of system (1),then we have (1)
^n j ?1a ij f j ex j Tà
^n j ?1a ij f j ey j T
p max 1p j p n
f L j j a ij jj x j ày j j
g ,
(2)
_n j ?1
b ij f j ex j Tà
_n j ?1b ij f j ey j T
p max 1p j p n
f L j j b ij jj x j ày j j
g ,
where L j is given as (H1).
L.Chen,H.Zhao /Neurocomputing 72(2008)436–444
437
Throughout the paper,we suppose that (H2)There are a set of positive
constants d 1;...;d n ,such
that
àd i c i t
X n j ?1
j a ji j d j L i t
X n j ?1
j a ji j d j L i t
X n j ?1
j b ji j d j L i o 0,
i ?1;...;n .
(H3)There are a set of positive constants p 1;...;p n ,such
that
àc i t
X n j ?1j a ij j L j p à1j p i tmax 1p j p n
f L j j a ij j p à1
j g p i
tmax 1p j p n
f L j j b ij j p à1
j g p i o 0;i ?1;...;n .
(H4)There are a set of positive constants q 1;...;q n ,such that
à2q i c i tX n j ?1
j a ij j q i L j t
X n j ?1
j a ji j q j L i t
X n j ?1
j a ij j q i L j
t
X n j ?1
j a ji j q j L i tX n j ?1
j b ij j q i L j tX n j ?1
j b ji j q j L i
t2
X n j ?1
Z ji q j o 0;
i ?1;...;n .
(H5)There are a set of positive constants w 1;...;w n ,such that
à2c i t
X n j ?1
j a ij j L j t
X n j ?1
j a ij j L j w à1
j w i tmax 1p j p n
f L j j a ij jg
tmax 1p j p n
f L j j a ij j w à1
j g w i tmax 1p j p n f L j j b ij jg
tmax
1p j p n
f L j j b ij j w à1
j g w i
t2
X n j ?1
Z ij w à1
j w i o 0,
i ?1;...;n .
For any x ?ex 1;...;x n TT 2R n ,we de?ne the vector
norm k ák 1,k ák 2,k ák 1,k ák ,respectively,by k x k 1?
P n i ?1d i j x i j ,k x k 2?????????????????????????P n
i ?1q i j x i j 2
q ,k x k 1?max i f p i j x i jg ,
k x k ??????????????????????????max i f w i j x i j 2p g .
For any f et T?ef 1et T;...;f n et TTT 2L 2F 0e?àt ;0 ;R n
T,we de?ne k f k 1?sup àt p s p 0
k f es Tk 1;k f k 2?sup àt p s p 0
k f es Tk 2;
k f k 1?sup àt p s p 0k f es Tk 1;k f k ?sup àt p s p 0
k f es Tk ,where
k f es Tk 1?P n i ?1
d i j f i es Tj ;k f es Tk 2?????????????????????????
P
n i ?1
q i j f i es Tj 2s ;k f es Tk 1?
max i
f p i j f i es Tj
g ;k f es Tk ?
???????????????????????????????
max i
f w i j f i es Tj 2
g q .
3.Main results
For the deterministic system
d x i et T?àc i x i et TtP n j ?1ea ij f j ex j et TTtb ij u j TtI i "tV n j ?1a ij f j
ex j
et àt TTtW n j ?1
b ij f j ex j
et àt TTt
V n j ?1
T ij u j t
W n j ?1
H ij u j #
d t ;t X 0;
x i et T?f i et T;
àt p t p 0;
8
>>>>>>>>>>>><>>>>>>>>>>>>:
(2)
we have the following results.
Theorem 1.If (H1)holds.Assume furthermore that one of
(H2)–(H5)with Z ij ?0holds.Then system (2)has a unique
equilibrium point x ??ex ?1;...;x ?n TT
.Proof.The proof is similar to that of [20,28].So we omit it here.&
In the paper,we assume that
(H6)s ij ex ?j ;x ?
j T?0;i ;j ?1;...;n .
Thus,system (1)admits an equilibrium point x ??ex ?1;...;x ?n TT .Let y i et T?x i et Tàx ?i ;j i et T?f i et Tàx ?
i ,y et T?ey 1et T;...;y n et TTT ,j et T?ej 1et T;...;j n et TTT ,then system (1)becomes
d y i et T?àc i y i et TtP n j ?1a ij ef j ey j et Ttx ?j Tàf j ex ?j TT"tV n j ?1a ij f j ey j et àt Ttx ?j TàV n j ?1a ij f j ex ?j TtW n j ?1b ij f j ey j et àt Ttx ?j TàV
n j ?1b ij f j ex ?j T#
d t tP n j ?1s ij ey j et Ttx ?j ;y j et àt Ttx ?j Td o j et T;t X 0;y i et T?j i et T;àt p t p 0:
8
>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:
(3)
Clearly,the equilibrium point x ?of (1)is almost surely exponentially stable if and only if the equilibrium point O of system (3)is almost surely exponentially stable.Thus in the following,we only consider the almost sure exponential stability of the equilibrium point O for system (3).
Theorem 2.Suppose that (H1),(H2),and (H6)hold.Then system (3)has an equilibrium point O which is almost surely exponentially stable .
L.Chen,H.Zhao /Neurocomputing 72(2008)436–444
438
Proof.It follows from(H2)that there exists a suf?ciently small constant0o l o min1p i p n f c i g such that
d ielàc iTt
X n
j?1j a ji j d j L ite lt
X n
j?1
j a ji j d j L i
te lt
X n
j?1
j b ji j d j L i p0;i?1;...;n.(4)
Taking VeyetT;tT?e l t P n
i?1
d i j y ietTj,and applying Ito?’s
formula to VeyetT;tT,we have
VeyetT;tTp Veye0T;0TtZ t
l e l s
X n
i?1
d i j y iesTj d s
t
Z t
0e l s
X n
i?1
d iàc i j y iesTj
"
t
X n
j?1a ij?f jey jesTtx?
j
Tàf jex?
j
T
t^n
j?1a ij f jey jesàtTtx?
j
Tà
^n
j?1
a ij f jex?
j
T
t_n
j?1b ij f jey jesàtTtx?
j
Tà
^n
j?1
b ij f jex?
j
T
#
d stMevT,
where
MevT?
Z t
0e l s
X n
i?1
d i sgney iesTT
X n
j?1
s ijey jesTtx?
j
;y jesàtTtx?
j
Td o jesT
By using Lemma2,we obtain
VeyetT;tTp k j k1t
Z t
0l e l s
X n
i?1
d i j y iesTj d s
t
Z t
0e l s
X n
i?1
d iàc i j y iesTjt
X n
j?1
j a ij j L j j y jesTj "
t
X n
j?1
j a ij j L j j y jesàtTj
t
X n
j?1j b ij j L j j y jesàtTj
#
d stMevT
?k j k1t
Z t
0l e l s
X n
i?1
d i j y iesTj d s
t
Z t
0e l s
X n
i?1
àd i c i j y iesTjt
X n
j?1
j a ji j L i d j j y iesTj "
t
X n
j?1
j a ji j L i d j j y iesàtTj
t
X n
j?1j b ji j L i d j j y iesàtTj
#
d stMevT.
Note that
Z t
tàt
e l s j y iesTj d s?
Z t
àt
e l s j y iesTj d s
à
Z t
e lesàtTj y iesàtTj d s.
So
Z t
e l s j y iesàtTj d s?e lt
Z t
àt
e l s j y iesTj d s
àe lt
Z t
tàt
e l s j y iesTj d s,
that is
Z t
e l s j y iesàtTj d s p e lt
Z t
àt
e l s j y iesTj d s.
Following from(4)we have
VeyetT;tTp k j k1t
Z0
àt
e l s e lt
X n
i?1
X n
j?1
j a ji j L i d j j y iesTj d s
t
Z0
àt
e l s e lt
X n
i?1
X n
j?1
j b ji j L i d j j y iesTj d stMevT.
(5)
It is obvious that the right-hand side of(5)is a non-
negative semi-martingale.From Lemma1,it can be easily
seen that
lim sup
t!1
VeyetT;tTot1;P F a:s.
Since
VeyetT;tT?e l t
X n
i?1
d i j y ietTj?
e l t k yetTk1,
we obtain
lim sup
t!1
1
t
logek yetTk1Tpàl;P F a:s.
According to[26,9],the equilibrium point O of(3)is almost
surely exponentially stable.This completes the proof.&
Theorem3.Assume that(H1),(H3),and(H6)hold,then
system(3)has an equilibrium point O which is almost surely
exponentially stable.
Proof.From(H3),there exists a suf?ciently small constant
0o l o min1p i p n f c i g such that
làc it
X n
j?1
j a ij j L j j pà1
j
p ite lt max
1p j p n
f L j j a ij j pà1
j
g p i
te lt max
1p j p n
f L j j b ij j pà1
j
g p i p0;i?1;...;n.
L.Chen,H.Zhao/Neurocomputing72(2008)436–444439
The Lyapunov functional is de?ned as V ey et T;t T?e l t max 1p i p n f p i j y i et Tjg .Suppose p k j y k et Tj ?max 1p i p n
f p i j y i et Tj
g ,where k 2f 1;...;n g .Applying Ito
?’s formula to V ey et T;t T,we have V ey et T;t T
p V ey e0T;0Tt
Z
t
0l e l s p k j y k es Tj d s
t
Z
t
e l s
p k àc k j y k es Tj
"tX n j ?1a kj ?f j ey j es Ttx ?j Tàf j ex ?j T
t^n j ?1
a kj f j ey j es àt Ttx ?j Tà^n j ?1
a kj f j ex ?j T
t_n j ?1
b kj f j ey j es àt Ttx ?j Tà_n j ?1
b kj f j ex ?j T
#d s tZ
t 0
e l s
p k sgn ey k es TT
X n j ?1
s kj ey j es Ttx ?j ;y j es àt Ttx ?
j Td o j es T.
By using Lemma 3,we obtain V ey et T;t Tp k j k 1t
Z
t
0l e l s p k j y k es Tj d s
tZ t 0
e l s
p k àc k j y k es Tj tX n j ?1
j a kj j L j j y j es Tj
"tmax 1p j p n
f L j j a kj j p à1j
g p k j y k es àt Tj tmax 1p j p n
f L j j b kj j p à1
j g p k j y k es àt Tj #
d s t
Z
t
e l s p k sgn ey k es TT
X n j ?1
s kj ey j es Ttx ?j ;y j es àt Ttx ?
j Td o j es T.
Similar to the discussion of Theorem 2,we have V ey et T;t Tp k j k 1t
Z
0àt
e l s e lt p 2k
?max 1p j p n
f L j j a kj j p à1
j gj y k es Tj d s
t
Z
0àt
e l s e lt p 2k max 1p j p n
f L j j b kj j p à1
j gj y k es Tj d s
t
Z
t
e l s
p k sgn ey k es TT
X n j ?1
s kj ey j es Tt
x ?j ;y j es
àt Tt
x ?j Td o j es T.
From Lemma 1,we obtain
lim sup t !11
t
log ek y et Tk 1Tp àl :P F a :s.According to [26,9],the equilibrium point O of (3)is almost surely exponentially stable.We complete the proof.&
Theorem 4.Suppose that (H1),(H4),and (H6)hold.Then system (3)has an equilibrium point O which is almost surely exponentially stable.
Proof.It follows from (H4)that there exists a suf?ciently small constant 0o l o 2min 1p i p n f c i g such that l q i à2q i c i tX n j ?1
j a ij j q i L j t
X n j ?1
j a ji j q j L i
tX n j ?1j a ij j q i L j te lt X n j ?1
j a ji j q j L i t
X n j ?1
j b ij j q i L j te lt
X n j ?1
j b ji j q j L i t
X n j ?1
Z ji q j
te
lt
X n j ?1
Z ji q j p 0;
i ?1;...;n .
Taking V ey et T;t T?e l t P n i ?1q i j y i et Tj 2
,and applying Ito
?’s formula to V ey et T;t T,we have
V ey et T;t Tp V ey e0T;0TtZ t 0
l e l s
X
n i ?1
q i j y i es Tj 2d s
t
Z
t
2e l s
X n i ?1
q i j y i es Tj àc i j y i es Tj
"
tX n
j ?1a ij ?f j ey j
es Ttx ?j Tàf j ex ?j T
t^n j ?1
a ij f j ey j es àt Ttx ?j Tà^n j ?1
a ij f j ex ?j T
t_n j ?1
b ij f j ey j es àt Ttx ?j Tà_n j ?1
b ij f j ex ?j T
#d s tZ t 0
2e l s
X n i ?1
q i y i es TX
n j ?1
s ij ey j es T
tx ?j ;y j es àt Ttx ?j Td o j es T
t
Z
t
e
l s
X n i ?1
q i
X n j ?1
j s ij ey j es T
tx ?j ;y j es àt Ttx ?j Tj 2
d s .
By using Lemma 2and inequality 2ab p a 2tb 2,we obtain
V ey et T;t Tp k j k 22tZ t 0
l e l s
X n i ?1
q i j y i es Tj 2d s
tZ
t
e
l s
X n i ?1
q i à2c i j y i es Tj 2
t
X n j ?1
j a ij j L j j y i es Tj 2
"
tX n j ?1j a ij j L j j y j es Tj 2
t
X n j ?1
j a ij j L j j y i es Tj 2
t
X n j ?1
j a ij j L j j y j es àt Tj 2
t
X n j ?1
j b ij j L j j y i es Tj 2
L.Chen,H.Zhao /Neurocomputing 72(2008)436–444
440
t
X n j ?1
j b ij j L j j y j es àt Tj 2#
d s tZ
t 0
e
l s X n i ?1
q i
X n j ?1
Z ij ej y j es Tj 2tj y j es àt Tj 2Td s
t
Z
t
2e
l s
X n i ?1
q i y i es TX n j ?1
s ij ey j es T
tx ?j ;y j es àt Ttx ?j Td o j es T.
Similar to the discussion of Theorem 2,we have
V ey et T;t Tp k j k 2
2tZ 0àt X n i ?1
q i X n j ?1
e l s e lt ?ej a ij j
tj b ij jTL j tZ ij j y j es Tj 2d s
tZ t 0
2e l s
X n i ?1
q i y i es TX
n j ?1
s ij ey j es T
tx ?j ;y j es àt Ttx ?j Td o j es T.
From Lemma 1,we obtain lim sup t !11t
log ek y et Tk 2Tp àl
2;
P F a :s.
According to [26,9],the equilibrium point O of (3)is almost
surely exponentially stable.This completes the proof.&Theorem 5.Assume that (H1),(H5),and (H6)hold ,then system (3)has an equilibrium point O which is almost surely exponentially stable .
Proof.From (H5),there exists a suf?ciently small constant 0o l o 2min 1p i p n f c i g such that l à2c i t
X n j ?1j a ij j L j t
X n j ?1
j a ij j L j w à1
j w i tmax 1p j p n
f L j j a ij jg
te lt max 1p j p n
f L j j a ij j w à1
j g w i tmax 1p j p n f L j j b ij jg
te lt
max 1p j p n f L j j b ij j w à1
j g w i t
X n j ?1
Z ij w à1
j w i te
lt
X n j ?1
Z ij w à1
j w i p 0;
i ?1;...;n .
The Lyapunov functional is de?ned as V ey et T;t T?
e l t
max 1p i p n f w i j y i et Tj 2g .Suppose w k j y k et Tj 2?max 1p i p n
f w i j y i et Tj 2
g ,where k 2f 1;...;n g .Applying Ito
?’s formula to V ey et T;t T,we have
V ey et T;t Tp V ey e0T;0Tt
Z
t
l e l s w k j y k es Tj 2
d s
tZ
t
02e l s
w k j y k es Tj àc k j y k es Tj
"
tX n j ?1a kj f j ey j es Ttx ?j TàX n j ?1
a kj f j ex ?j T
t^n j ?1
a kj f j ey j es àt Ttx ?j Tà^n j ?1
a kj f j ex ?j T
t_n j ?1b kj f j ey j es àt Ttx ?j Tà_n j ?1b kj f j ex ?j T #d s tZ t 0
2e l s
w k y k es TX
n j ?1
s kj ey j es T
tx ?j ;y j es àt Ttx ?j Td o j es T
t
Z
t
e l s
w k
X n j ?1
s 2kj ey j es Ttx ?j ;y j es àt Ttx ?
j Td s .
By using inequality 2ab p a 2tb 2and Lemma 3,we obtain V ey et T;t Tp k j k 2
t
Z
t
0l e l s w k j y k es Tj 2d s
tZ
t 0e l s
w k à2c k j y k es Tj 2
t
X n j ?1
j a kj j L j j y k es Tj 2
"
tX n j ?1j a kj j L j j y j es Tj 2tmax 1p j p n
f L j j a kj jgj y k es Tj 2
tmax 1p j p n f L j j a kj j w à1j g w k j y k es àt Tj
2
tmax 1p j p n f L j j b kj jgj y k es Tj 2
tmax 1p j p n
f L j j b kj j w à1j
g w k j y k es àt Tj 2#
d s tZ
t
e l s
w k
X n j ?1
Z kj ej y j es Tj 2tj y j es àt Tj 2Td s
t
Z
t
2e l s
w k y k es TX n j ?1
s kj ey j es T
tx ?j ;y j es àt Ttx ?
j Td o j es T.
Similar to the discussion of Theorem 2,we have
V ey et T;t Tp k j k 2
t
Z
àt
e l s e lt
?max 1p j p n
f L j j a kj j w à1j
g w 2k j y k es Tj 2
d s
tZ
àt e l s e lt max 1p j p n
f L j j b kj j w à1j
g w 2k j y k es Tj 2
d s
t
Z 0àt
e l s e lt
w k
X n j ?1
Z kj j y j es Tj 2d s
L.Chen,H.Zhao /Neurocomputing 72(2008)436–444
441
t
Z t
02e l s w k y kesT
X n
j?1
s kjey jesT
tx?
j ;y jesàtTtx?
j
Td o jesT.
From Lemma1,we obtain
lim sup t!11
t
logek yetTkTpà
l
2
;P F a:s.
According to[26,9],the equilibrium point O of(3)is almost surely exponentially stable.We complete the proof.&
When d i?p i?q i?w i?1ei?1;...;nTin Theorems 1–5,we can obtain the following result.
Corollary1.System(3)has an equilibrium point O which is almost surely exponentially stable if the nonlinear functions f jeáTand s ijeá;áTei;j?1;...;nTsatisfy hypothesis(H1). Assume furthermore that one of the following conditions holds:
(A1)
àc it
X n
j?1j a ji j L it
X n
j?1
j a ji j L i
t
X n
j?1
j b ji j L i o0;i?1;...;n. (A2)
àc it
X n
j?1j a ij j L jtmax
1p j p n
f L j j a ij jg
tmax
1p j p n
f L j j b ij j
g o0;i?1;...;n. (A3)
à2c it
X n
j?1j a ij j L jt
X n
j?1
j a ji j L it
X n
j?1
j a ij j L j
t
X n
j?1j a ji j L it
X n
j?1
j b ij j L jt
X n
j?1
j b ji j L i
t2
X n
j?1
Z ji o0;i?1;...;n. (A4)
àc it
X n
j?1j a ij j L jtmax
1p j p n
f L j j a ij jg
tmax
1p j p n f L j j b ij jgt
X n
j?1
Z ij o0;i?1;...;n.
4.Examples
Example 1.Consider the following stochastic FCNNs
with delays:
d x1etT?àc1x1etTt
P2
j?1
ea1j f jex jetTTtb1j u jTtI1
"
t
V2
j?1
a1j f jex jetàtTTt
W2
j?1
b1j f jex jetàtTT
t
V2
j?1
T1j u jt
W2
j?1
H1j u j
#
d t
t
P2
j?1
s1jex jetT;x jetàtTTd o jetT
d x2etT?àc2x2etTt
P2
j?1
ea2j f jex jetTTtb2j u jTtI2
"
t
V2
j?1
a2j f jex jetàtTTt
W2
j?1
b2j f jex jetàtTT
t
V2
j?1
T2j u jt
W2
j?1
H2j u j
#
d t
t
P2
j?1
s2jex jetT;x jetàtTTd o jetT:
8
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>><
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>:
(6)
Let f1ex1T?1
2
ej x1t1jtj x1à1jT,f2ex2T?x2,s ijex jetT;
x jetàtTT?
??
5
p
10
x jetT;i;j?1;2.Obviously,L1?L2?1,
Z ij?1
20
;i;j?1;2.Choose c1?3;c2?2;a11?a12?
1
2
;a21?1
6
;a22?1
2
;a ij?1
4
ei;j?1;2T;b11?b12?b22?1
4
;
b21?1
3
:It can easily see that condition(A3)is satis?ed.
Thus,system(6)has an equilibrium point which is almost
surely exponentially stable.However,
àc2t
X2
j?1
j a j2j L2t
X2
j?1
j a j2j L2t
X2
j?1
j b j2j L2?0,
that is condition(A1)does not hold.
On the other hand,we choose f1ex1T?x1,f2ex2T?
eàx2à1,s11ex1etT;x1etàtTT?2
???
2
p
x1etT,s21ex1etT;x1etàtTT
?
???
7
p
x1etT,s12ex2etT;x2etàtTT?s22ex2etT;x2etàtTT?
x2etT.Obviously,L1?L2?1,Z11?8;Z21?7;Z12?
Z22? 1.Taking c1?36;c2?26;a11?a21?5;a12?8;
a22?6;a11?3;a12?1
4
;a21?9;a22?2;b11?2;b12?1
4
;
b21?10;b22?1.By simple calculation,we obtain
à2c1t
X2
j?1
j a1j j L jt
X n
j?1
j a j1j L1t
X2
j?1
j a1j j L j
t
X2
j?1
j a j1j L1t
X2
j?1
j b1j j L jt
X2
j?1
j b j1j L1
t2
X2
j?1
Z j1?10:540.
L.Chen,H.Zhao/Neurocomputing72(2008)436–444
442
This shows that condition (A3)does not hold.However,it can be easily veri?ed that condition (A1)is satis?ed.
Example 2.For system (6),choose f 1ex 1T?sin x 1,f 2ex 2T?x 2,s ij ex j et T;x j et àt TT?????
10
p x j et T;i ;j ?1;2.Ob-viously,L 1?L 2?1,Z ij ?1
;i ;j ?1;2.Let c 1?1;c 2?
20;a 11?a 21?a 22?120;a 12?1
10;a ij ?14ei ;j ?1;2T;b 11?b 12?b 22?14;b 21?1
3:It can easily see that condition (A4)is satis?ed.Thus,system (6)has an equilibrium point which is almost surely exponentially stable.However,à2c 1tX 2j ?1
j a 1j j L j t
X 2j ?1
j a j 1j L 1t
X 2j ?1
j a 1j j L j
t
X 2j ?1
j a j 1j L 1tX 2j ?1
j b 1j j L j tX 2j ?1
j b j 1j L 1
t2
X 2j ?1
Z j 1?
44
60
40,that is condition (A3)does not hold.
On the other hand,we choose f 1ex 1T?12ej x 1t1j tj x 1à1jT,f 2ex 2T?x 2,s 11ex 1et T;x 1et àt TT?x 1et T;s 21ex 1et T;x 1et àt TT????3p x 1et T;s 22ex 2et T;x 2et àt TT?x 2et T;s 12ex 2et T;x 2et àt TT???2p 2x 2et T.Obviously,L 1?L 2?1,Z 11?Z 22?1;Z 21?3;Z 12?12.Taking c 1?22;c 2?26;a 11
?4;a 21?1;a 22?2;a 12?1
4;a 11?3;a 12?14;a 21?9;a 22?2;b 11?2;b 12?1
4;b 21?5;b 22?1.By simple calculation,we obtain àc 2t
X 2j ?1
j a 2j j L j tmax 1p j p 2f L j j a 2j jg
tmax 1p j p 2
f L j j b 2j j
g t
X 2j ?1
Z 2j ?440.
This shows that condition (A4)does not hold.However,it
can be easily veri?ed that condition (A3)is satis?ed.5.Conclusions
In this paper,stochastic FCNNs model with delays has been investigated.We have derived a family of suf?cient conditions ensuring the almost sure exponential stability for the networks by constructing suitable Lyapunov functionals and applying stochastic analysis.Moreover,two examples are given to demonstrate the advantages of our method.References
[1]J.Cao,A set of stability criteria for delayed cellular neural networks,
IEEE Trans.Circuits Syst.I 48(2001)494–498.
[2]J.Cao,Global stability conditions for delayed CNNs,IEEE Trans.
Circuits Syst.I 48(2001)1330–1333.
[3]J.Cao,H.Li,L.Han,Novel results concerning global robust stability
of delayed neural networks,Nonlinear Anal.Real World Appl.7(2006)458–469.
[4]J.Cao,Q.Song,Stability in Cohen–Grossberg-type bidirectional
associative memory neural networks with time-varying delays,Nonlinearity 19(2006)1601–1617.
[5]T.Chen,S.Amari,New theorems on global convergence of some
dynamical systems,Neural Networks 14(2001)251–255.
[6]T.Chen,L.Rong,Delay-independent stability analysis of
Cohen–Grossberg neural networks,Phys.Lett.A 317(2003)436–449.
[7]T.Chen,L.Wang,Power-rate global stability of dynamical systems
with unbounded time-varying delays,IEEE Trans.Circuits Syst.II 54(2007)705–709.
[8]Y.Chen,X.Liao,Novel exponential stability criteria for fuzzy
cellular neural networks with time-varying delays,in:Lecture Notes in Computer Science,vol.3173,Springer,Berlin,2004,pp.120–125.
[9]T.Feuring,J.Buckley,W.Lippe,A.Tenhagen,Stability analysis of
neural net controllers using fuzzy neural networks,Fuzzy Sets and Systems 101(1999)303–313.
[10]R.Hasminskii,Stochastic stability of differential equations,
D.Louvish,Thans.,Swierczkowski,ED,1980.
[11]S.Haykin,Neural Networks,Prentice-Hall,New Jersey,1994.
[12]H.Jiang,J.Cao,Global exponential stability of periodic neural
networks with time-varying delays,Neurocomputing 70(2006)343–350.
[13]X.Liao,X.Mao,Exponential stability and instability of stochastic
neural networks,Stochast.Anal.Appl.14(1996)165–185.
[14]Q.Liu,J.Cao,Improved global exponential stability criteria of
cellular neural networks with time-varying delays,https://www.doczj.com/doc/0419351523.html,put.Modelling 43(2006)423–432.
[15]Y.Liu,W.Tang,Exponential stability of fuzzy cellular neural
networks with constant and time-varying delays,Phys.Lett.A 323(2004)224–233.
[16]X.Mao,Stochastic Differential Equations and Applications,
Horwood,New York,1997.
[17]J.Park,A new stability analysis of delayed cellular neural networks,
https://www.doczj.com/doc/0419351523.html,put.181(2006)200–205.
[18]Q.Song,J.Cao,Impulsive effects on stability of fuzzy Cohen–
Grossberg neural networks with time-varying delays,IEEE Trans.Syst.Man Cybern.Part B Cybern.37(2007)733–741.
[19]Y.Sun,J.Cao,p th moment exponential stability of stochastic
recurrent neural networks with time-varying delays,Nonlinear Anal.Real World Appl.8(2007)1171–1185.
[20]D.Xu,H.Zhao,H.Zhu,Global dynamics of Hop?eld neural
networks involving variable delays,Comput.Math.Appl.42(2001)39–45.
[21]H.Yang,T.Dillon,Exponential stability and oscillation of Hop?eld
graded response neural network,IEEE Trans.Neural Networks 5(1994)719–729.
[22]T.Yang,C.Yang,L.Yang,The differences between cellular neural
network based and fuzzy cellular neural network based mathematical morphological operations,Int.J.Circuit Theory Appl.26(1998)13–25.
[23]T.Yang,L.Yang,The global stability of fuzzy cellular neural
network,IEEE Trans.Circuits Syst.I 43(1996)880–883.
[24]T.Yang,L.Yang,M.Yang,X.Yang,H.Yang,Linguistic ?ow in
fuzzy discrete-time cellular neural networks and its stability,IEEE Trans.Circuits Syst.I 45(1998)869–878.
[25]T.Yang,L.Yang,C.Wu,L.Chua,Fuzzy cellular neural networks:
theory,in:Proceedings of IEEE International Workshop on Cellular Neural Networks and Application,1996,pp.181–186.
[26]Y.Yang,X.Xu,W.Zhang,Design neural networks based fuzzy
logic,Fuzzy Sets and Systems 114(2000)325–328.
[27]K.Yuan,J.Cao,J.Deng,Exponential stability and periodic
solutions of fuzzy cellular neural networks with time-varying delays,Neurocomputing 69(2006)1619–1627.
[28]H.Zhao,J.Cao,New conditions for global exponential stability of
cellular neural networks with delays,Neural Networks 18(2005)1332–1340.
L.Chen,H.Zhao /Neurocomputing 72(2008)436–444
443
[29]H.Zhao,N.Ding,L.Chen,Almost sure exponential stability of
stochastic fuzzy cellular neural networks with delays,Chaos,Solitons and Fractals,in press,doi:10.1016/j.chaos.2007.09.044
.
Ling Chen is now pursuing the M.S.degree in the Department of Mathematics at Nanjing Univer-sity of Aeronautics and Astronautics,Nanjing,China.
From August 2000until now,she is with the Department of Basic Science at Jinling Institute of Technology,Nanjing,China.Her research interests include neural networks and stability
theory.
Hongyong Zhao received the Ph.D.from Sichuan University,Chendu,China,and the Post-Doctoral Fellow in the Department of Mathematics at Nanjin University,Nanjin,China.
He was with Department of Mathematics at Nanjing University of Aeronautics and Astronau-tics,Nanjin,China.He is currently a Professor of Nanjin University of Aeronautics and Astronau-tics,Nanjin,China.He is also the author or coauthor of more than 60journal papers.His
research interests include nonlinear dynamic systems,neural networks,control theory.
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