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22
Chapter2.Quadraticfunctionalsforlinearretardedtypetimedelaysystem
forξ=t+9hence
∂xt(9)
∂t
=
∂xt(9)
∂9
.
Thetimederivativeofthefunctional
(2.22)
onthetrajectoryofsystem
(2.18)
iscom-
puted.Thistimederivativeisdefnedbytheformula
(2.4)
whichforsystem
(2.18)
takesaform
dv(I)
dt
=limsup
h→o
1
h[v(xt
o+h)−v(I)].
(2.24)
Itistakenthefollowingprocedure.Onecomputesthetimederivativeofeachtermof
theright-hand-sideoftheformula
(2.22)
andonesubstitutesinplaceof
dx(t)/dt
and
∂xt(9)/∂tthefollowingterms
dx(t)
dt
=Ax(t)+Bxt(−r),
∂xt(9)
∂t
=
∂xt(9)
∂9
.
Insuchamanneroneattains(see[98])
dv(xt)
dt
=xT(t)[ATI+IA+;
(o)+;T(o)
2
]x(t)
+xT(t)[2IB−;(−r)]xt(−r)
o
+
−r
/
xT(t)[AT;(9)−
d;(9)
d9
+δT(9,o)]xt(9)d9
o
+
/
xT
t(−r)[BT;(9)−δ(−r,9)]x(t+9)d9
−r
o
o
−
−r
/
9
/
xT
t(9)[
∂δ(9,σ)
∂9
+
∂δ(9,σ)
∂σ
]xt(σ)dσd9.
Toachievenegativedefnitenessofthatderivativeweassumethat
dv(xt)
dt
≡−xT(t)x(t).
Fromequations(2.27)and(2.28)weobtainthesetofequations
ATI+IA+
;(o)+;T(o)
2
=−I,
2IB−;(−r)=o,
AT;(9)−
d;(9)
d9
+δT(9,o)=o,
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.30)
(2.31)
