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24
Chapter2.Quadraticfunctionalsforlinearretardedtypetimedelaysystem
UsingtheKroneckerproductthesetofdiferentialequations
(2.42)
canbereshape
totheform
[
d9col;(9)
d9colK(9)]=[I⊗ATBT⊗I
d
d
−I⊗BT
−AT⊗I][col;(9)
colK(9)]
(2.43)
for9∈[−r,o]withinitialconditionscol;(−r)andcolK(−r).
Thesolutionofinitialvalueproblem(2.43)hasaform
[col;(9)
colK(9)]=[Φ11(9+r)Φ12(9+r)
Φ21(9+r)Φ22(9+r)][col;(−r)
colK(−r)],
(2.44)
whereamatrixΦ(9)=[Φ11(9)Φ12(9)
Φ21(9)Φ22(9)]isafundamentalmatrixofsystem(2.43).
Equation(2.44)implies
col;(9)|
|9=−
r
2
=Φ11(
2)col;(−r)+Φ12(
r
2)colK(−r),
r
colK(9)|
|9=−
r
2
=Φ21(
2)col;(−r)+Φ22(
r
2)colK(−r).
r
Equation(2.41)implies
;
T(9)|
|9=−
2
r
=K(9)|
|9=−
2
r
.
(2.45)
(2.46)
(2.47)
Formula
(2.47)
presentsthealgebraiclinearrelationshipbetweeninitialconditions
col;(−r)andcolK(−r).
Equation(2.41)implies
K(−r)=;T(o).
(2.48)
Formula(2.29)takesaform
ATI+IA+
K(−r)+KT(−r)
2
=−I.
Formulas(2.49),(2.30)and(2.47)createthesetofalgebraicequations
(
I
I
ATI+IA+
K(−r)+KT(−r)
2
=−I,
4
I
I
l
2IB−;(−r)=o,
;T(9)|
|9=−
r
2
=K(9)|
|9=−
2
r
.
(2.49)
(2.50)
Thesetofalgebraicequations
(2.50)
allowsfordeterminationofthematrix
I
andthe
initialconditionsofthesetofdiferentialequations(2.43).
Fromequations(2.36)and(2.41)oneattains
f(9)=BT;(−r−9)=BTKT(9)
for9∈[−r,o].
(2.51)
