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AdvancesinAppliedMathematics
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,2023,12(6),2643-2649
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.126266
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5
ANovelHybridSystem
JinfeiWei
SchoolofMathematicalSciences,ZhejiangNormalUniversity, Jinhua Zhejiang
Received:May5
th
,2023;accepted:May28
th
,2023;published:Jun.6
th
,2023
Abstract
Impulsivesystems,asatypeofhybridsystem,canbecombinedwithlogicaldynamic
systemstostudysystemswith logicselectiveimpulsiveeffects.Duetothefactthat the
powersystemconsistsofimpulsiveeffectsandlogicaloperations,theyaremoregeneral
andcomplexhybridsystems.Byusingthesemi-tensorproductmethodofmatrices,
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the logicalfunction istransformedinto an equivalent algebraicform, making thestudy
ofsuchimpulsivesystemspossible.Thisarticlemainlyusesthesemitensorproduct
methodofmatricestocombineimpulsivesystemswithlogicaldynamicsystems.
Keywords
ImpulsiveSystem,LogicalDynamicSystem,TheSemiTensorProductofMatrices,
Stability
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2023.1262662646
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).
é
u
p
≥
2,
k
ω
(
t
k
−
1
)
n
(
x
(
t
k
))
p
=
D
p
−
1
[
m,m
]
ω
(
t
k
−
1
)
p
n
x
(
t
−
k
)
p
=
D
p
−
1
[
m,m
]
(
p
−
1
Y
s
=0
W
[
n
p
−
s
m
p
−
s
,n
]
ξ
p
(
t
−
k
)
(8)
½
Â
Γ
1
:=
L
g
W
[
N
2
n,m
]
DG
(0)
L
h
L
g
M
[
r,m
]
.
(9)
p
≥
2,
Γ
p
:=
L
g
W
[
N
2
n
p
,m
]
DG
(0)
L
h
L
g
M
[
r,m
]
D
p
−
1
[
m,m
]
(
p
−
1
Y
s
=0
W
[
n
p
−
s
m
p
−
s
,n
]
)
.
(10)
Γ(
ξ
(
t
−
k
)) :=
∞
X
p
=1
1
p
!
Γ
p
ξ
p
(
t
−
k
)
.
(11)
Ï
d
,
X
Ú
(1)
Œ
±
=
†
¤
±
e
#
X
Ú
µ
˙
x
(
t
) =
Ax
(
t
)
,t
6
=
t
k
,
x
(
t
) =
D
η
(
t
)
x
(
t
−
)
,t
=
t
k
,k
∈
Z
+
,
x
(0) =
x
0
,
(12)
DOI:10.12677/aam.2023.1262662647
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ê
Æ
?
Ð
Ÿ
7
œ
Ù
¥
A
Ú
D
η
(
t
)
´
~
ê
Ý
.
…
½
Â
G
(
x
(
t
−
k
)) := [
D
N
−
1
x
(
t
−
k
)
D
N
−
2
x
(
t
−
k
)
...D
0
x
(
t
−
k
)](13)
Ú
D
:= [
D
N
−
1
D
N
−
2
...D
0
](14)
X
Ú
(12)
Œ
±
¤
˙
ξ
(
t
) =
Aξ
(
t
)
,t
6
=
t
k
,
x
(
t
) = Λ(
t
)
ξ
(
t
−
)
,t
=
t
k
,k
∈
Z
+
,
ξ
(0)
∈
∆
m
nR
n
,
(15)
Ù
¥
Λ :=
L
g
W
[
N
2
n,m
]
DL
h
L
g
M
[
r,m
]
.
4.
o
(
L
·
‚
•
ï
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Ü
6
Ä
X
Ú
½
ö
ó
À
X
Ú
,
3
©
¥
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‚
ò
Ü
6
Ä
X
Ú
Ú
ó
À
X
Ú
ƒ
(
Ü
,
ò
Ü
6
Ä
X
Ú
Ñ
Ñ
Š
•
ó
À
À
J
,
Ï
L
Ý
Œ
Ü
þ
È
•{
,
˜
‡
#.
·
Ü
X
Ú
,
™
5
·
‚
Œ
±
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Ä
T
X
Ú
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5
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5
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©
z
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Ÿ
7
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