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2.2.Thequadraticfunctionalforalinearsystemwithonedelay
BT;(9)−δ(−r,9)=o,
∂δ(9,σ)
∂9
+
∂δ(9,σ)
∂σ
=o
for9∈[−r,o],σ∈[−r,o].
Thesolutionofadiferentialequation(2.33)isgivenintheform
δ(9,σ)=f(9−σ),
wheref∈C1([−r,r],Rn×n).
Fromequations(2.34)and(2.32)oneobtains
δ(−r,9)=f(−r−9)=BT;(9),
f(9)=BT;(−r−9),
δT(9,o)=fT(9)=;T(−r−9)B.
Afterputting(2.37)into(2.31)oneattainsaformula
d;(9)
d9
=AT;(9)+;T(−r−9)B
for9∈[−r,o].
Thederivativeofthefunction;(−9−r)withrespectto9iscalculated
d;(−r−9)
d9
=
d;(ξ)
dξ
dξ
d9
=−
d;(ξ)
dξ
=−AT;(ξ)−;T(−r−ξ)B=−AT;(−r−9)−;T(9)B,
where
ξ=−r−9.
Thesetofthediferentialequationsisobtained
{
d;(9)
d;(−r−9)
d9
d9
=AT;(9)+;T(−r−9)B,
=−AT;(−r−9)−;T(9)B
for9∈[−r,o].
Anewfunctionisintroduced
K(9)=;T(−9−r)
for9∈[−r,o].
Thesetofthediferentialequations(2.40)takesaform
{
d;(9)
dK(9)
d9
d9
=AT;(9)+K(9)B,
=−K(9)A−BT;(9)
for9∈[−r,o]withinitialconditions;(−r)andK(−r).
23
(2.32)
(2.33)
(2.34)
(2.35)
(2.36)
(2.37)
(2.38)
(2.39)
(2.40)
(2.41)
(2.42)
