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20
Chapter2.Quadraticfunctionalsforlinearretardedtypetimedelaysystem
OnecanobtainasolutionofFDE
(2.12)
usingastepmethod.Thestepmethod
isabasicmethodforsolvingFDEwithalumpeddelay.Asolutionisfoundonsuc-
cessiveintervals,oneafteranother,bysolvinganordinaryequationwithoutdelayin
eachinterval.
Fort∈[to,to+r]theequation(2.12)takesaform
{
x(to)=I(o).
dx(t)
dt
=Ax(t)+BI(t−r),
Thesolutionofequation(2.13)isgivenbyaterm
t
x(t)=eA(t−to)
I(o)+
/
eA(t−ξ)BI(ξ−r)dξ,
to
ψ(t)=x(t),
x(to+r)=x1.
Fort∈[to+r,to+2r]theequation(2.12)takesaform
{
x(to+r)=x1
dx(t)
dt
=Ax(t)+Bψ(t−r),
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
andsoon.Bymeansofthisprocedureonecanconstructthesolutioninanyfnite
interval.
Onecanwritetheequation(2.12)intheform
{
xt
dx(t)
dt
o=I
=Ax(t)+Bxt(−r),
(2.18)
for
t≥to
,where
xt∈PC([−r,o],Rn)
isashiftedrestrictionof
x(·,to,I)
tothe
segment[t−r,t).
Thereholdsarelationship
xt
o(·,to,I)=I,
(2.19)
wherext
o(·,to,I)isashiftedrestrictionofx(·,to,I)toaninterval[to−r,to].
Thetheoremsofexistence,continuousdependenceanduniquenessofsolutionsof
equation
(2.18)
aregiveninthearticlebyV.Kharitonov[72].Thecontrollabilityofthe
systemswithtimedelayispresentedintheworkbyJ.Klamka[83].
Inaparametricoptimizationproblemwillbeusedanintegralquadraticperfor-
manceindexofquality
I=
to
/
Ż
xT(t)x(t)dt.
(2.20)
