具时变系数和时滞的模糊细胞神经网络的固定时间同步 Fixed-Time Synchronization for Fuzzy Cellular Neural Networks with Time-Varying Coefficients and Delays

doi:10.12677/PM.2021.117154

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PureMathematicsnØêÆ,2021,11(7),1369-1378

PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.117154

äžCXêÚž¢[œ ²ä

½žmÓÚ

444§§§ÜÜÜttt‚‚‚§§§¶¶¶UUU

∗

þ°“‰ŒÆ§ênÆ§þ°

Email:

∗

dingwei@shnu.edu.cn

ÂvFÏµ2021c611F¶¹^FÏµ2021c713F¶uÙFÏµ2021c721F

Á‡

©ïÄ´äkžCXêÚžCž¢[œ ²ä(FCNNs)½žmÓÚ¯K§8

´ÏLEÜ·››ì§¦ü‡äkžCXêÚžCž¢FCNNs3k•žmSÓÚ§¿…

Œ±ÏLUC››ìëê5ýk½XÚÓÚ¤I‡žm"Ì‡•{´EÜ·››ì±

9Lyapunov¼ê§|^½žm-½5nØ9˜ØªE|§äkžCXêÚžCž¢

FCNNs½žmÓÚ˜#¿©^‡"•„‰˜‡êŠ[~fy²©Ì‡(J

k5"

'…c

[œ ²ä§½žmÓÚ§žCXê§žCž¢

Fixed-TimeSynchronizationforFuzzy

CellularNeuralNetworkswith

Time-VaryingCoeﬃcientsandDelays

YangLiu,SonghuanZhang,WeiDing

∗

∗ÏÕŠö"

©ÙÚ^:4,Üt‚,¶U.äžCXêÚž¢[œ ²ä½žmÓÚ[J].nØêÆ,2021,11(7):

1369-1378.

DOI:10.12677/pm.2021.117154

4

DepartmentofMathematics,ShanghaiNormalUniversity,Shanghai

Email:

∗

dingwei@shnu.edu.cn

Received:Jun.11

,2021;accepted:Jul.13

,2021;published:Jul.21

,2021

Abstract

This paper studiesthe ﬁxed-timesynchronization problem offuzzycellularneural net-

works(FCNNs)withtime-varyingcoeﬃcientsandtime-varyingdelays.Thepurpose

istobeabletomaketwoFCNNswithtime-varyingcoeﬃcientsandtime-varyingde-

lays canbesynchronized in aﬁnitetimeby constructing asuitablecontroller, andthe

time required forsystem synchronization can be presetby changingthe parametersof

thecontroller.ThemainmethodistoconstructasuitablecontrollerandLyapunov

function,andusetheﬁxed-timestabilitytheoryandsomeinequalitytechniquesto

obtainsomenewsuﬃcientconditionsforﬁxed-timesynchronizationofFCNNswith

time-varyingcoeﬃcientsandtime-varyingdelays.Finally, anexampleisgiventoprove

thevalidityofthemainresultsofthispaper.

Keywords

FuzzyCellularNeuralNetwork,Fixed-TimeSynchronization,Time-Varying

Coeﬃcient,Time-VaryingDelay

2021byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

3 ²äù˜+•, ²äÓÚ¯K3LA›cpÚå<‚4Œ'5,¿…

²äÓÚ••)õ«ÓÚa..3ùÓÚa.¥,½žm ÓÚÚk •žmÓÚA^•

\2•,Ï•Nõ¢SA^Ñ‡¦XÚU3k•žmS¢yÓÚ.

DOI:10.12677/pm.2021.1171541370nØêÆ

4

[œ ²ä(FCNNs) ´˜«Óž•ÄÜ6ÚÛÜëÏ5š‚5ä(.é

uFCNNs k•žm9½žmÓÚ¯K,®²kNõkdŠ¤J,~X©z[1][2][3],Ù¥

©z[1][2]ïÄÑ´k•žmÓÚ¯K,¤OÓÚžmÑ†XÚÐ©^‡ƒ'.ƒ'u©

z[1][2],©ïÄK´½žmÓÚ¯K,ÓU¢yXÚuk•žmSˆÓÚ,`³3

u¤OÓÚžmÚXÚÐ©^‡vk'X,3¢SA^¥•\•B…2•.

2..£ã

3©¥,·‚ïÄ´XeäkžCXêÚžCž¢FCNNs:











˙x

(t) = −c

(t)x

(t)+

j=1

(t)f

(t))+

j=1

(t)f

(t−τ

(t)))+

j=1

(t)v

(t)

j=1

(t)v

(t)+

j=1

(t)f

(t−τ

(t)))+

j=1

(t)v

(t)

j=1

(t)f

(t−τ

(t)))+I

(t),

(s) = φ

(s),s∈[−τ,0],

(2.1)

Ù¥i,j∈J={1,2,...,n},x

(t)“L1i‡ ²3tžG;c

(t)´P~Ç;f

(·)´-

y¼ê;3a

(t),b

(t)Úd

(t)¥,cü‡´‡"†ƒ,•˜‡´c"†ƒ;

(

) L«Ú(½)$Ž; T

(t) (S

(t)) Úα

(t) (β

(t)) ©O´c"•(•Œ)†Ú

‡"•(•Œ)†ƒ;v

(t) ÚI

(t)©O“LÑ\† ˜;τ

(t) •žCž¢;XÚ(2.1)

ÐŠ•φ

(s)∈C([−τ,0],R

),Ù¥C([−τ,0],R

)´l[−τ,0]NR

ëY¼ê8,¿…

τ=max

1≤j≤n

{τ

(t)}.

þ¡XÚ(2.1)´°ÄXÚ,éA•AXÚXe¤«:











˙y

(t) = −c

(t)y

(t)+

j=1

(t)f

(t))+

j=1

(t)f

(t−τ

(t)))+

j=1

(t)v

(t)

j=1

(t)v

(t)+

j=1

(t)f

(t−τ

(t)))+

j=1

(t)v

(t)

j=1

(t)f

(t−τ

(t)))+I

(t)+u

(t),

(s) = ϕ

(s),s∈[−τ,0],

(2.2)

Ù¥u

(t)•ƒ‡O››ì.

DOI:10.12677/pm.2021.1171541371nØêÆ

4

^XÚ(2.2)~XÚ(2.1) ŒØXÚ•:











˙e

(t) = −c

(t)e

(t)+

j=1

(t)f

(t))+

j=1

(t)f

(t−τ

(t)))

j=1

(t)f

(t−τ

(t)))+

j=1

(t)f

(t−τ

(t)))+u

(t),

(s) = ϕ

(s)−φ

(s),s∈[−τ,0],

(2.3)

Ù¥e

(·) = y

(·)−x

(·),f

(·)) = f

(·))−f

(·)).

½Â2.1.e•3~êT

÷v^‡:(i)lim

t→T

(t)=0,(ii)∀t>T

(t)≡0,i∈J.@oXÚ(2.1)

Ú(2.2)3-½žmT

U½žmÓÚ,Ù¥T

ÚXÚÐ©^‡vk'X.

Ún2.1([4]).éui,j∈J, x

,ζ

,ς

∈R,f

: R→R´ëY,Kk

j=1

)−

j=1

)|≤

j=1

|ζ

||f

)−f

)|,

j=1

)−

j=1

)|≤

j=1

|ς

||f

)−f

)|.

Ún2.2([5]).-0 <µ≤1 <λ,κ

≥0,ι

≥0,i∈J,K

i=1

≥(

i=1

)

i=1

≥n

1−λ

(

i=1

)

Ún2.3 ([6]).e•3˜‡»•Ã.,½¿…ëY¼êV :R

→R

{0}k

V(e(t))≤

−aV

(e(t))−bV

(e(t)),Ù¥0 <µ<1 <λ,a>0,b>0,@oØXÚ(2.3) ½žm-½,KX

Ú(2.1)Ú(2.2) 3-½žmT

ž½žmÓÚ,¿…k

≤T

max

a(λ−1)

b(1−µ)

3.Ì‡(J

©kXeü^b:

).éu∀x,y∈R

,x6= y,•3~êω

>0 ÷v|f

(y)−f

(x)|≤ω

|y−x|,i∈J.

).bc

(t),a

(t),b

(t),d

(t),T

(t),α

(t),S

(t),β

(t),v

(t),I

(t):R→R´ëY¿…k

.¼ê.éuëYk.¼êg(t),kXePÒ:g

−

=inf

t∈R

g(t),g

= sup

t∈R

g(t).

•¦XÚ(2.1)Ú(2.2) U¢y½žmÓÚ,·‚EXe››ì:

(t) = −k

(t)−η

sign(e

(t))|e

(t−τ

(t))|−ρ

sign(e

(t))|e

(t)|

−l

sign(e

(t))|e

(t)|

,(3.1)

DOI:10.12677/pm.2021.1171541372nØêÆ

4

Ù¥k

≥0,η

≥0,ρ

>0,l

>0,0 <µ<1 <λ,i∈J.

½n3.1.3(A

),(A

)Ú››ì(3.1) e,eeØª¤á:











−

j=1

≥0,

−

j=1

(|b

+|α

+|β

)ω

≥0,

(3.2)

@oXÚ(2.1)Ú(2.2) Œ±3½žmT

ÓÚ,¿…

≤T

max

ρn

1−λ

(λ−1)

l(1−µ)

,(3.3)

Ù¥ρ=min

1≤i≤n

{ρ

}>0,l=min

1≤i≤n

}>0,i,j∈J.

y².•ÄXeLyapunov¼ê

V(e(t)) =

i=1

(t)|.(3.4)

(i)e

(t) =0ž,éu››ì(3.1) ku

(t) =0,¿…ØXÚ(2.3) 3ùž´-½.Ïdo´k

Ü·ëê÷v^‡(3.2).•Ò´`,½n3.1˜½´¤á.

(ii)e

(t) 6= 0ž,OŽ(3.4) ê

V(e(t)) =

i=1

sign(e

(t))˙e

(t).

ò(2.3)Ú(3.1) “\?1˜ Œ

V(e(t)) =

i=1

sign(e

(t))[−c

(t)e

(t)+

j=1

(t)f

(t))+

j=1

(t)f

(t−τ

(t)))

j=1

(t)f

(t−τ

(t)))+

j=1

(t)f

(t−τ

(t)))−k

(t)

−η

sign(e

(t))|e

(t−τ

(t))|−ρ

sign(e

(t))|e

(t)|

−l

sign(e

(t))|e

(t)|

]

≤

i=1

−(c

(t)+k

)|e

(t)|+

i=1

j=1

(t)||f

(t))|+

i=1

j=1

(t)||f

(t−τ

(t)))|

i=1

j=1

(t)f

(t−τ

(t)))|+

i=1

j=1

(t)f

(t−τ

(t)))|

−

i=1

(t−τ

(t))|−

i=1

(t)|

−

i=1

(t)|

DOI:10.12677/pm.2021.1171541373nØêÆ

4

ŠâÚn2.1k

V(e(t)) ≤

i=1

−(c

(t)+k

)|e

(t)|+

i=1

j=1

(t)||f

(t))|

i=1

j=1

(t)||f

(t−τ

(t)))|

i=1

j=1

|α

(t)||f

(t−τ

(t)))|

i=1

j=1

|β

(t)||f

(t−τ

(t)))|

−

i=1

(t−τ

(t))|−

i=1

(t)|

−

i=1

(t)|

Šâ(A

)Ú(A

)Œ

V(e(t)) ≤

i=1

−(c

−

)|e

(t)|+

i=1

j=1

(t)|

i=1

j=1

(|b

+|α

+|β

)ω

(t−τ

(t))|

−

i=1

(t−τ

(t))|−

i=1

(t)|

−

i=1

(t)|

i=1

−(c

−

)|e

(t)|+

i=1

j=1

(t)|

i=1

j=1

(|b

+|α

+|β

)ω

(t−τ

(t))|

−

i=1

(t−τ

(t))|−

i=1

(t)|

−

i=1

(t)|

i=1

−(c

−

j=1

)|e

(t)|

−

i=1

[η

−

j=1

(|b

+|α

+|β

)ω

]|e

(t−τ

(t))|

−

i=1

(t)|

−

i=1

(t)|

DOI:10.12677/pm.2021.1171541374nØêÆ

4

d^‡(3.2)9Ún2.2 OŽ

V(e(t)) ≤−

i=1

(t)|

−

i=1

(t)|

≤−ρ

i=1

(t)|

−l

i=1

(t)|

≤−ρn

1−λ

(

i=1

(t)|)

−l(

i=1

(t)|)

= −ρn

1−λ

(e(t))−lV

(e(t)),

Ù¥ρ=min

1≤i≤n

{ρ

}>0,l=min

1≤i≤n

}>0.

ÏLÚn2.3,·‚UXÚ(2.1) Ú(2.2) Œ±3-½žmT

¢y½žmÓÚ,¿…

≤T

max

ρn

1−λ

(λ−1)

l(1−µ)

.y..

íØ3.1.eXÚ(2.1)Ú(2.2) ¥¤kžCXêÑòz•~ê,@o½n3.1 (ØE,¤á.

íØ3.2.eXÚ(2.1), (2.2) 9››ì(3.1) ¥¤kžCž¢t−τ

(t) ÑO†•'~ž¢q

t,Ù

¥q

∈(0,1],K½n3.1 (ØE,¤á.

(−1,1) tanh ¼ê5O†.@o››ì(3.1) ÒC¤

(t) = −k

(t)−η

tanh(ξ

(t))|e

(t−τ

(t))|−ρ

tanh(%

(t))|e

(t)|

−l

tanh(p

(t))|e

(t)|

(3.5)

Ù¥k

≥0,η

≥0,ξ

>0,ρ

>0,%

>0,l

>0,p

>0,0 <µ<1 <λ,i∈J.

4.êŠ[

~4.1.éuXÚ(2.1),(2.2)9››ì(3.5) kXeŠ:

(t)=β

(t)=

1+sinit

,α

(t)=β

(t)=

1+cosit

(t)=

2+3sint

(t)=

2+3cost

(t)=

1+sinit

(t)=b

(t)=

1+cosit

,τ

(t)=

exp(t)

1+exp(t)

(x)=0.5(|x+ 1|−|x−1|),I

(t)=

−4,T

(t) = v

(t) = S

(t) = d

(t) = 1,i,j= 1,2.

,ϕ(s)=(−0.2,0.4)

.››ì(3.5) ¥kk

=3,η

=5,ρ

DOI:10.12677/pm.2021.1171541375nØêÆ

4

1,l

= 3,λ= 1.5,µ= 0.9,ξ

= %

= p

= 2.

ω

= 1, K²OŽŒ(A

), (A

)9^‡(3.2)Ñ÷v.¿d(3.3)Œ-½žmT

≤T

max

≈

6.161.Ïd,Šâ½n3.1Œ•,3››ì(3.5)e,XÚ(2.1) ÚXÚ(2.2)3T

žŒ¢y½žm

ÓÚ.

dMatlab ^‡ ŒXen‡ã”:ã1L²eÃ››ì(3.5) ,ØXÚ(2.3) Ø-½,KXÚ

(2.1)ÚXÚ(2.2) ØU¢yÓÚ(„ã1).ã2KL²3››ì(3.5)e,ØXÚ(2.3) 3½žm

ž-½,@oXÚ(2.1)ÚXÚ(2.2)3T

≤T

max

≈6.161 žU½žmÓÚ(„ã2).

Figure1.Timeevolutionofe

(t)ande

(t)withoutthe

controller(3.5)

ã1.Ã››ì(3.5)že

(t)Úe

(t);,

Figure2.Timeevolutionofe

(t)ande

(t)withthe

controller(3.5)

ã2.k››ì(3.5)že

(t)Úe

(t);,

DOI:10.12677/pm.2021.1171541376nØêÆ

4

54.1.k

= 1,η

= 1,i=1,2.K^‡(3.2) Ø÷v.lã3ŒwÑe^‡(3.2) Ø÷v,=¦3

››ì(3.5)e,XÚ(2.1)ÚXÚ(2.2) •ØU¢y½žmÓÚ§„ã3.

Figure3.Theevolutionofsynchronizationerrore

(t)

ande

(t)thatdon’tsatisfy(3.2)

ã3.Ø÷v^‡(3.2)že

(t)Úe

(t);,

Ä7‘8

ë•©z

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