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2.2.Thequadraticfunctionalforalinearsystemwithonedelay
Hence,
Ż
/
dv(xt,t)
dt
dt=lim
t→Ż
v(xt,t)−lim
t→to
v(xt,t)=−v(lim
t→to
(xt,t))=−v(I,to).
to
19
(2.9)
AssumethatthetimederivativeoftheLyapunovfunctional
v
isgivenasaquadratic
form
dv(xt,t)
dt
≡−xT(t,to,I)Wx(t,to,I)
(2.10)
fort≥to,whereW∈Rn×nisapositivedefnitematrix.
Itfollowsfrom(2.7)and(2.10)that
Ż
I=
/
xT(t,to,I)Wx(t,to,I)dt=v(I,to).
to
(2.11)
quadraticform
parametricoptimizationproblem.
Ifoneconstructsapositivedefnitefunctionalsuchthatitstimederivative
computedonthetrajectoryofsystem
itybutalsocalculateavalueofasquareindicatorofquality
Corollary2.10
(2.10)
onecannotonlyinvestigatethesystem
(2.1)
isgivenasanegativedefnite
(2.7)
(2.1)
ofthe
stabil-
2.2.Thequadraticfunctional
foralinearsystemwithonedelay
2.2.1.Mathematicalmodelof
alineartimedelaysystemwithonedelay
Letusconsideralinearsystemwitharetardedtypetimedelaywhosedynamicsis
describedbyafunctional-diferentialequation(FDE)
{
x(to+9)=I(9)
dx(t)
dt
=Ax(t)+Bx(t−r),
(2.12)
t≥to,9∈[−r,o],r≥o,A,B∈Rn×n,x(t)∈Rn,I∈PC([−r,o],Rn).
Thesolutionofthefunctional-diferentialequation
(2.12)
forinitialfunction
I
is
anabsolutelycontinuousfunctiondefnedfor
t≥to
withvaluesin
Rn
andisdenoted
asx(·,to,I).
