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PureMathematicsnØêÆ,2021,11(8),1559-1569
PublishedOnlineAugust2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.118174
óÀ››eCXêžCž¢[œ ²ä
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ÏLOóÀ››,ïÄCXêžCž¢[œ ²äÓÚ¯K"æ^Lyapunov•¼•
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[œ ²ä§óÀ››§ÓÚ§žCž¢§CXê
SynchronizationofFuzzyCellular
NeuralNetworkswithVariable
CoefficientsandTimeDelays
underImpulseControl
SonghuanZhang
∗
,YangLiu
DepartmentofMathematics,ShanghaiNormalUniversity,Shanghai
∗ÏÕŠö"
©ÙÚ^:Üt‚,4.óÀ››eCXêžCž¢[œ ²äÓÚ[J].nØêÆ,2021,11(8):
1559-1569.DOI:10.12677/pm.2021.118174
Üt‚§4
Received:Jul.18
th
,2021;accepted:Aug.19
th
,2021;published:Aug.26
th
,2021
Abstract
Thispaperinvestigates thesynchronization offuzzycellular neuralnetworkswithvari-
ablecoefficientsandtime-varyingdelaybydesigningaimpulsivecontrol.Bytaking
Lyapunovfunctionalmethodandthematrixinequalitymethod,thelinearmatrixin-
equality conditions are given to ensure the synchronization of the system.Meanwhile,
exponentialsynchronizationconditionsandasymptoticbehaviorofunknownparam-
etersarederived.Finally,asimulationexampleisgiventoverifytheeffectivenessof
theproposedmethod.
Keywords
FuzzyCellularNetworks,ImpulsiveControl, Synchronization, Time-Varying, Variable
Coefficients
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2021.1181741560nØêÆ
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j
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^
j=1
¯η
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(t)
˜
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j
[˜u
j
(t−τ(t),x)]+
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^
j=1
T
ij
v
j
+
n
_
j=1
¯
β
ij
(t)
˜
f
j
[˜u
j
(t−τ(t),x)]
+
n
_
j=1
S
ij
v
j
+ε
i
(t)(˜u
i
(t,x)−u
i
(t,x)),i= 1,2,···,n,t≥0,t6= t
k
,
4˜u
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k
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(t
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k
,x)−˜u
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(t
−
k
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ik
˜u
i
(t
k
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+
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k
,
˜u
i
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˜u
i
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n
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n
(i=1,2,···,n)L«››ìOÊ.òÓÚØ½Â
DOI:10.12677/pm.2021.1181741561nØêÆ
Üt‚§4
•e
i
(t,x) =˜u
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(t,x)−u
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
˙e
i
(t,x) =
m
X
k=1
∂
∂x
k
(D
ik
∂e
i
(t,x)
∂x
k
)−ω
i
e
i
(t,x)+
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j=1
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ij
f
j
(e
j
(t,x))
+
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^
j=1
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ij
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∗
j
[e
j
(t−τ(t),x)]+
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j=1
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j
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j
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i
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i
)˜u
i
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j=1
(
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b
ij
(t)−b
ij
)
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j
(˜u
j
(t,x))
+
n
^
j=1
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ij
(t)−η
ij
)
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j
[˜u
j
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n
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j=1
(
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β
ij
(t)−β
ij
)
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[˜u
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(t−τ(t),x)]
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(t)e
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,
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k
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k
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i
(t
−
k
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ik
e
i
(t
k
,x),i= 1,2,...,n,k∈Z
+
,t= t
k
,
e
i
(t,x) = 0,(t,x) ∈[−τ,+∞]×∂Ω,i= 1,2,...,n,
e
i
(t,x) = ψ
i
(t,x)−φ
i
(t,x),(t,x) ∈[−τ,0]×Ω,i= 1,2,...,n.
(2.3)
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f
j
(e
j
(t,x)) =
˜
f
j
(˜u
j
(t,x))−
˜
f
j
(u
j
(t,x)),
n
V
j=1
η
ij
f
∗
j
[e
j
(t−τ(t),x)] =
n
V
j=1
η
ij
˜
f
j
[˜u
j
(t−τ(t),x)]−
n
V
j=1
η
ij
˜
f
j
[u
j
(t−τ(t),x)],
n
W
j=1
β
ij
f
∗∗
j
[e
j
(t−τ(t),x)] =
n
W
j=1
β
ij
˜
f
j
[˜u
j
(t−τ(t),x)]−
n
W
j=1
β
ij
˜
f
j
[u
j
(t−τ(t),x)].
½Â2.1.éuψ(t,x) =(ψ
1
(t,x),...ψ
n
(t,x))
T
∈C([−τ,0]×R
m
,R
n
)´l[−τ,0]×R
m
R
n
þ
ëY¼êNBanach˜m,½Â
kψ(t,x) k
2
=
v
u
u
t
n
X
i=1
kψ
i
(t,x) k
2
2
,
Ù¥kψ
i
(t,x) k
2
=

Z
Ω
sup
−τ≤t≤0,x∈Ω
|ψ
i
(t,x) |
2
dx

1
2
.
½Â2.2.XJ•3L>1,ε>0¦
k˜u
i
(t,x)−u
i
(t,x) k
2
=ke(t,x) k
2
≤Lkψ−φk
2
e
−εt
,(t,x) ∈[0,+∞)×Ω.
K¡XÚ(2.1)Ú(2.2)´Û•êÓÚ"Ù¥
k˜u
i
(t,x)−u
i
(t,x) k
2
=
Z
Ω
n
X
i=1
|˜u
i
(t,x)−u
i
(t,x) |
2
dx
!
1
2
.
DOI:10.12677/pm.2021.1181741562nØêÆ
Üt‚§4
Ún2.1.eh(x) ∈C
1
(Ω,R),h(x) |
∂Ω
= 0,Ω´k.;8,|x
k
|≤w
k
(k= 1,2,...,m).Kk
Z
Ω
h
2
(x)dx≤w
2
k
Z
Ω

∂h
∂x
k

2
dx.
Ún2.2.([2]).éu∀η
ij
,β
ij
∈R,k
|
n
V
j=1
η
ij
˜
f
j
(u
j
)−
n
V
j=1
η
ij
˜
f
j
(˜u
j
)|≤
n
P
j=1
|η
ij
||
˜
f
j
(u
j
)−
˜
f
j
(˜u
j
)|,
|
n
W
j=1
β
ij
˜
f
j
(u
j
)−
n
W
j=1
β
ij
˜
f
j
(˜u
j
)|≤
n
P
j=1
|β
ij
||
˜
f
j
(u
j
)−
˜
f
j
(˜u
j
)|.
Ún2.3.éu∀x,y∈R
n
Ú½ÝQ∈R
n×n
,k
2x
T
y≤x
T
Qx+y
T
Q
−1
y.
Ún2.4.(SchurÖÚn)éu©¬Ý
X=


X
11
X
12
X
T
12
X
22


>0,
Ù¥X
11
,X
22
´•,e^‡´dµ
1)X
11
>0,X
22
−X
T
12
X
−1
11
X
12
>0,
2)X
22
>0,X
11
−X
12
X
−1
22
X
T
12
>0.
3.̇(J
•(XÚ(2.1)Ú(2.2)ÓÚ,b±e^‡¤áµ
(H
1
).
˜
f
i
(x)´LipschitzëY,•3h
i
>0¦
|
˜
f
i
(x)−
˜
f
i
(y)|≤h
i
|x−y|,
∀x,y∈R,x6= y, i= 1,2,···,n.…h=max
1≤i≤n
(h
2
i
).
(H
2
).τ(t) >0,0 ≤˙τ(t) ≤µ≤1,é?¿t,ùpµ´~ê.
(H
3
).θ
ik
∈RÚθ
ik
∈[0,2],i= 1,2,...,n.k∈Z
+
.
(H
4
).e‚5ÝØª


ΓQ(|η|+|β|)
(|η|+|β|)
T
QE


>0,(3.1)
DOI:10.12677/pm.2021.1181741563nØêÆ
Üt‚§4
Ù¥Q=diag(q
i
),W =diag(ω
i
),B=(b
ij
)
n×n
,|η|=(η
ij
)
n×n
,|β|=(|β
ij
|)
n×n
,H=diag(h
i
),
Γ = QD+QW−λQ−
1
2
(Q|B|H+H|B
T
|Q)−
1
1−µ
H
T
H,λ>0,E´ü Ý.
½n3.1.Äu(H
1
)−(H
4
).b¯ω
i
(t),
¯
b
ij
(t),¯η
ij
(t),
¯
β
ij
(t),ε
i
(t)(i,j= 1,2,···,n)÷v:
˙
¯ω
i
(t)=
1
γ
i
e
λt
e
i
(t,x)˜u
i
(t,x),
˙
¯
b
ij
(t)=−
1
α
ij
e
λt
e
i
(t,x)˜g
j
(˜u
j
(t,x)),
˙
¯η
ij
(t)=−
1
%
ij
e
λt
|e
i
(t,x)˜g
j
[˜u
j
(t−τ(t),x)]|sgn(¯η
ij
(t)−η
ij
),
˙
¯
β
ij
(t)=−
1
σ
ij
e
λt
|e
i
(t,x)˜g
j
[˜u
j
(t−τ(t),x)]|sgn(
¯
β
ij
(t)−β
ij
),
˙ε
i
(t)=−
1
δ
i
e
λt
e
2
i
(t,x).
(3.2)
¿…ε
i
(t
k
)=ε
i
(t
+
k
),¯ω
i
(t
k
)=¯ω
i
(t
+
k
),
¯
b
ij
(t
k
)=
¯
b
ij
(t
+
k
),¯η
ij
(t
k
)=¯η
ij
(t
+
k
),
¯
β
ij
(t
k
)=
¯
β
ij
(t
+
k
).q
i
,ω
i
,
δ
i
,γ
i
,α
ij
,%
ij
,σ
ij
´?¿~ê,KXÚ(2.1)Ú(2.2)ÓÚ,¿…
lim
t→∞
(¯ω
i
(t)) = ω
i
,lim
t→∞
(
¯
b
ij
(t)) = b
ij
,lim
t→∞
(¯η
ij
(t)) = η
ij
,lim
t→∞
(
¯
β
ij
(t)) = β
ij
.(3.3)
y².ELyapunov-krasovskii¼ê
V(t)=
Z
Ω
{
1
2
n
X
i=1
q
i
{e
λt
e
2
i
(t,x)+γ
i
(¯ω
i
(t)−ω
i
)
2
+
n
X
j=1
α
ij
(
¯
b
ij
(t)−b
ij
)
2
+
n
P
j=1
%
ij
(¯η
ij
(t)−η
ij
)
2
+
n
P
j=1
σ
ij
(
¯
β
ij
(t)−β
ij
)
2
+δ
i
ε
2
i
(t)}
+
Z
t
t−τ(t)
1
1−µ
e
λ(s+τ)
|f
T
[e(s,x)]||f[e(s,x)]|ds}dx.
(3.4)
éut≥0,t6= t
k
,k∈Z
+
,OŽ(3.4)ê,
˙
V(t)=
Z
Ω
{
n
X
i=1
q
i
{e
λt
e
i
(t,x)
∂e
i
(t,x)
∂t
+
1
2
λe
λt
e
2
i
(t,x)+γ
i
(¯ω
i
(t)−ω
i
)
˙
¯ω
i
(t)
+
n
P
j=1
α
ij
(
¯
b
ij
(t)−b
ij
)
˙
¯
b
ij
(t)+
n
P
j=1
%
ij
(¯η
ij
(t)−η
ij
)
˙
¯η
ij
(t)+
n
P
j=1
σ
ij
(
¯
β
ij
(t)−β
ij
)
˙
¯
β
ij
(t)
+δ
i
ε
i
(t)˙ε
i
(t)}+
1
1−µ
e
λ(t+τ)
|f
T
[e(t,x)]||f[e(t,x)]|
−
1−˙τ(t)
1−µ
e
λ(t−τ(t)+τ)
|f
T
[e(t−τ(t),x)]||f[e(t−τ(t),x)]|}dx
≤
Z
Ω
{
n
X
i=1
q
i
{e
λt
e
i
(t,x)
m
X
k=1
∂
∂x
k
(D
ik
∂e
i
(t,x)
∂x
k
)−ω
i
e
λt
e
2
i
(t,x)
+
n
P
j=1
b
ij
e
λt
e
i
(t,x)f
j
(e
j
(t,x))+λe
λt
e
2
i
(t,x)+
n
P
j=1
|η
ij
|e
λt
e
i
(t,x)|f
j
[e
j
(t−τ(t),x)]|
+
n
P
j=1
|β
ij
|e
λt
e
i
(t,x)|f
j
[e
j
(t−τ(t),x)]|}+
1
1−µ
e
λ(t+τ)
|f
T
[e(t,x)]||f[e(t,x)]|
−
1−˙τ(t)
1−µ
e
λ(t−τ(t)+τ)
|f
T
[e(t−τ(t),x)]||f[e(t−τ(t),x)]|}dx,
DOI:10.12677/pm.2021.1181741564nØêÆ
Üt‚§4
dÚn2.19>Š^‡
Z
Ω
e
i
(t,x)
m
X
k=1
∂
∂x
k

D
ik
∂e
i
(t,x)
∂x
k

dx
=
Z
Ω
e
i
(t,x)
m
X
k=1
∂
∂x
k

D
ik
∂e
i
(t,x)
∂x
k

dx
=
Z
Ω
e
i
(t,x)∇

D
ik
∂e
i
(t,x)
∂x
k

m
k=1
dx
=

Z
∂Ω
e
i
(t,x)

D
ik
∂e
i
(t,x)
∂x
k

m
k=1
dx−
Z
Ω

D
ik
∂e
i
(t,x)
∂x
k

m
k=1
∇e
i
(t,x)dx

= −
Z
Ω

D
ik
∂e
i
(t,x)
∂x
k

m
k=1
∇e
i
(t,x)dx
= −
m
P
k=1
Z
Ω
D
ik

∂e
i
(t,x)
∂x
k

2
dx
≤−
m
P
k=1
Z
Ω
D
ik
w
2
k
e
2
i
(t,x)dx,
Ù¥∇= (
∂
∂x
1
,
∂
∂x
2
,...,
∂
∂x
m
)
T
´FÝŽf,¿…
(D
ik
∂e
i
(t,x)
∂x
k
)
m
k=1
= (D
i1
∂e
i
(t,x)
∂x
1
,D
i2
∂e
i
(t,x)
∂x
2
,...D
im
∂e
i
(t,x)
∂x
m
).
Šâ(H
1
)ÚÚn2.2,·‚k
˙
V(t)≤
Z
Ω
{−
n
X
i=1
q
i
(
m
X
k=1
D
ik
w
2
k
+ω
i
)e
λt
e
2
i
(t,x)+
n
X
i=1
n
X
j=1
e
λt
q
i
|e
i
(t,x)||b
ij
||f
j
(e
j
(t,x))|
+
n
P
i=1
q
i
λe
λt
e
2
i
(t,x)+
n
P
i=1
n
P
j=1
e
λt
q
i
|e
i
(t,x)||η
ij
||f
j
[e
j
(t−τ(t),x)]|
+
n
P
i=1
n
P
j=1
e
λt
q
i
|e
i
(t,x)||β
ij
||f
j
[e
j
(t−τ(t),x)]|+
1
1−µ
e
λ(t+τ)
|f
T
[e(t,x)]||f[e(t,x)]|
−
1−˙τ(t)
1−µ
e
λ(t−τ(t)+τ)
|f
T
[e(t−τ(t),x)]||f[e(t−τ(t),x)]|}dx
≤
Z
Ω
{−e
λt
e
T
(t,x)(QD+QW)e(t,x)+e
λt
n
X
i=1
n
X
j=1
q
i
|e
i
(t,x)||b
ij
||h
i
||e
j
(t,x)|
+λe
λt
e
T
(t,x)Qe(t,x)+e
λt
|e
T
(t,x)|Q(|η|+|β|)|f[e(t−τ(t),x)]|
−
1−˙τ(t)
1−µ
e
λ(t−τ(t)+τ)
|f
T
[e(t−τ(t),x)]||f[e(t−τ(t),x)]|
+
1
1−µ
e
λ(t+τ)
|f
T
[e(t,x)]||f[e(t,x)]|}dx,
DOI:10.12677/pm.2021.1181741565nØêÆ
Üt‚§4
d(H
2
)ÚÚn2.3,Œ
˙
V(t)≤
Z
Ω
{−e
λt
e
T
(t,x)(QD+QW)e(t,x)+e
λt
|e
T
(t,x)|
1
2
(Q|B|H+H|B
T
|Q)|e(t,x)|
+λe
λt
e
T
(t,x)Qe(t,x)+
1
2
e
λt
|e
T
(t,x)|Q(|η|+|β|)(|η|+|β|)
T
Q
T
|e(t,x)|
+
1
2
e
λt
f
T
[e(t−τ(t),x)]||f[e(t−τ(t),x)]|
−
1−˙τ(t)
1−µ
e
λ(t−τ(t)+τ)
|f
T
[e(t−τ(t),x)]||f[e(t−τ(t),x)]|
+
1
1−µ
e
λ(t+τ)
|f
T
[e(t,x)]||f[e(t,x)]|}dx
≤
Z
Ω
e
λt
{−|e
T
(t,x)|{QD+QW−λQ−
1
2
(Q|B|H+H|B
T
|Q)
−
1
2
Q(|η|+|β|)(|η|+|β|)
T
Q
T
−
1
1−µ
H
T
H}|e(t,x)|}dx
=−Ke
λt
ke
i
(t,x) k
2
2
≤0,
Ù¥K= QD+QW−λQ−
1
2
(Q|B|H+H|B
T
|Q)−
1
2
Q(|η|+|β|)(|η|+|β|)
T
Q
T
−
1
1−µ
H
T
H.ÏL
Ún2.4,K>0.,kV(t)≤−Ke
λt
ke
i
(t,x) k
2
2
≤0.ÏdV(t)≤V(t
+
k−1
)é?¿t∈(t
k−1
,t
k
],
k∈Z
+
,ùpV(0
+
) = V(0).d(2.3)(3.2)(3.4)Ú(H
3
),Υ
V(t
+
k
)=
Z
Ω
{
1
2
n
X
i=1
q
i
{e
λt
+
k
e
2
i
(t
+
k
,x)+γ
i
(¯ω
i
(t
+
k
)−ω
i
)
2
+
n
X
j=1
α
ij
(
¯
b
ij
(t
+
k
)−b
ij
)
2
+
n
P
j=1
%
ij
(¯η
ij
(t
+
k
)−η
ij
)
2
+
n
P
j=1
σ
ij
(
¯
β
ij
(t
+
k
)−β
ij
)
2
+δ
i
ε
2
i
(t
+
k
)}
+
Z
t
+
k
t
+
k
−τ(t
k
)
1
1−µ
e
λ(s+τ)
|f
T
[e(s,x)]||f[e(s,x)]|ds}dx
=
Z
Ω
{
1
2
n
X
i=1
q
i
{e
λt
k
(1−θ
ik
)
2
e
2
i
(t
k
,x)+γ
i
(¯ω
i
(t
k
)−ω
i
)
2
+
n
X
j=1
α
ij
(
¯
b
ij
(t
k
)−b
ij
)
2
+
n
P
j=1
%
ij
(¯η
ij
(t
k
)−η
ij
)
2
+
n
P
j=1
σ
ij
(
¯
β
ij
(t
k
)−β
ij
)
2
+δ
i
ε
2
i
(t
k
)}
+
Z
t
k
t
k
−τ(t
k
)
1
1−µ
e
λ(s+τ)
|f
T
[e(s,x)]||f[e(s,x)]|ds
−
Z
t
+
k
−τ(t
k
)
t
k
−τ(t
k
)
1
1−µ
e
λ(s+τ)
|f
T
[e(s,x)]||f[e(s,x)]|ds
+
Z
t
+
k
t
k
1
1−µ
e
λ(s+τ)
|f
T
[e(s,x)]||f[e(s,x)]|ds}dx
≤V(t
k
).
Ïd,XÚ(2.1)Ú(2.2)´ÓÚ.½n3.1y..
íØ3.1.Äu½n2.1^‡,XJε
i
(0)=0,¯ω
i
(0)=ω
i
,
¯
b
ij
(0)=b
ij
,¯η
ij
(0)=η
ij
,
¯
β
ij
(0)=
β
ij
.KXÚ(2.1)Ú(2.2)Û•êÓÚ.
53.1.θ
ik
≡0,i= 1,2,...,n.k∈Z
+
,XÚ(2.1)Ú(2.2)òz•©z[3,4]¥ÃóÀA
ò´[œ ²äœ¹.
DOI:10.12677/pm.2021.1181741566nØêÆ
Üt‚§4
4.êŠ[
~4.1.3(2.1)ª¥D
ik
= 0§i= 1,2.
(ω
ij
)
2×2
=
10
01
!
, (b
ij
)
2×2
=
2.0−0.1
−5.02.8
!
,
(β
ij
)
2×2
= (η
ij
)
2×2
=
−1.5−0.1
−0.21−1.8
!
,τ(t) =
e
t
1+e
t
,˜g
j
(t) = tanh(t).
Щ^‡Xeµu
1
(t) = −0.01,u
2
(t) = 0.05, ˜u
1
(t) = −0.5,˜u
2
(t) = 0.5,
ε
1
(0) = −0.1, ε
2
(0) = 0.02;b
1
(0) = −0.02, b
2
(0) = 4.46,
η
1
(0) = −0.02, η
2
(0) = 0.01;β
1
(0) = 0.2, β
2
(0) = −0.2.θ
ik
= 0.1.
XÚëêìC5Æ
˙
¯
b
1
(t)=−9.43×(˜u
1
−u
1
)tanh(˜u
1
),
˙
¯
b
2
(t) = −2.173×(˜u
2
−u
2
)tanh(˜u
2
).
˙
¯η
1
(t)=−6.65×sgn(¯η
1
+1.5)|(˜u
1
−u
1
)tanh(˜u
1
(t−τ))|,
˙
¯η
2
(t)=−1.45×sgn(¯η
2
+1.8)|(˜u
2
−u
2
)tanh(˜u
2
(t−τ))|.
˙
¯
β
1
(t)=−9.22×sgn(
¯
β
1
+1.5)|(˜u
1
−u
1
)tanh(˜u
1
(t−τ))|,
˙
¯
β
2
(t)=−2.5×sgn(
¯
β
2
+1.8)|(˜u
2
−u
2
)tanh(˜u
2
(t−τ))|.
˙ε
1
(t)=−0.5×(˜u
1
−u
1
)
2
,˙ε
2
(t) = −0.35×(˜u
2
−u
2
)
2
.
ÏLã1Úã2Œ•,XÚ(2.1)Ú(2.2)ÓÚ,¿…XÚ(2.2)Xê,
¯
b
1
,
¯
b
2
,¯η
1
,¯η
2
,
¯
β
1
,
¯
β
2
,©Oì
Cª•u2.0,2.8,-1.5,-1.8,-1.5,-1.8.
Figure1.Synchronizationerrorse
1
(t)ande
2
(t)
ã1.ØÓÚe
1
(t)Úe
2
(t)
DOI:10.12677/pm.2021.1181741567nØêÆ
Üt‚§4
Figure2.Asymptoticbehaviorofsystem(2.2)parameters
ã2.XÚ(2.2)XêìCã”
Ä7‘8
©óŠdI[g,‰ÆÄ7(No.12071302)|±.
ë•©z
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andSystems,35,1257-1272.https://doi.org/10.1109/31.7600
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https://doi.org/10.1016/j.physleta.2008.04.053
[4]Wang,L.,Ding, W. and Chen,D. (2010) Synchronization Schemes of a Class ofFuzzy Cellular
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https://doi.org/10.1016/j.cnsns.2009.02.013
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Üt‚§4
[6]Ding,W.,Han,M.andLi,M.(2009)ExponentialLagSynchronizationofDelayedFuzzy
CellularNeuralNetworkswithImpulses.PhysicsLettersA,373,832-837.
https://doi.org/10.1016/j.physleta.2008.12.049
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DOI:10.12677/pm.2021.1181741569nØêÆ

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