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26
Chapter2.Quadraticfunctionalsforlinearretardedtypetimedelaysystem
Thesetofadiferentialequation(2.42)takesaform
[
d;(9)
dK(9)
d9]=[−!
d9
kop
T
T
−
kop
T
!
T
][;(9)
K(9)].
Thefundamentalmatrixofsystem(2.57)takesaform
Φ(9)=[coshA9−!
kop
TAsinhA9
TAsinhA9
coshA9+
−
kop
TAsinhA9
TAsinhA9],
!
where
A=
J!2−k2
T
op
2
.
Thesetofalgebraicequations(2.50)takesaform
(2.57)
(2.58)
(2.59)
(
I
−2
TI+K(−r)=−1,
!
I
4
I
I
l
2I
[cosh
kop
T+;(−r)=o,
Ar
2−
!+kop
TA
sinh
Ar
2];(−r)+[−cosh
Ar
2−
!+kop
TA
sinh
Ar
2]K(−r)=o.
(2.60)
Fromanequation
(2.60)
oneobtainsaparameter
I
andtheinitialconditionsofthe
diferentialequation(2.57)
I=
2(Asinh
cosh
Ar
2+
Ar
2+
!+kop
TA
!+kop
T
sinh
cosh
Ar
2
Ar
2)
,
;(−r)=
kop
T(cosh
Asinh
Ar
2+
Ar
2+
!+kop
!+kop
T
TA
cosh
sinh
Ar
2
Ar
2)
,
K(−r)=
−
kop
T(cosh
Asinh
Ar
2+
Ar
2−
!+kop
T
!+kop
TA
cosh
sinh
Ar
2
Ar
2)
.
(2.61)
(2.62)
(2.63)
Havingafundamentalmatrix
(2.58)
andtheinitialconditionsofthediferential
equation(2.57)oneobtains
;(9)=
T(Asinh
Ar
2+
kop
!+kop
T
cosh
Ar
2)[(!+kop
TA
cosh
Ar
2
−sinh
Ar
2)sinhA9
+(!
+kop
TA
sinh
Ar
2
−cosh
Ar
2)coshA9],
(2.64)
