1 Upper-Lower Solution Method for Differential Riccati Equations ...

8
aEßGßKb Ä aE FOßG HO, ZaObb. (20)
Therefore, a representation for N O follows from (17) under replacement (20); that is,
NO
œ D X T ab =D€I( B X aT € Z aO b€ _ aOàTbbB.>
, (21)
where _aOàT b is L aT
b defined (5) under replacement (20), namely,
=
>
"
w
X
X
_aOàTbœ aE FOb T€TaE FO b€ aG HOb TaG HObÞ
(22)
Propositions 4 and 5 below reveal some important structural properties of LQaT b.
Proposition 4. Suppose T­P "ß_ a 8
Mß b _ 5‚8
’ is feasible and O ­ P aMß‘
b. Let LQ a T b , Z a O b and
_aOàT b be defined in (5), (19) and (22), respectively. Then
(i) LQ aT b€ a^aTb Ob X e aTba^aTb
Ob
œ Z aO b€ _ aOàTb,
(23)
(ii)
Ú LQaTb Ÿ Z aO b€ _ aOàTb if eaT b !à (24.1)
Û LQaT b Z aO b€ _ aOàTb if eaTb
Ÿ!à (24.2)
Ü LQaTb œ Z a^aTbb€ _ a^aT bàTb.
(24.3)
(24)
Remark 1. From the proof below, it follows that identities (23) and (24.3) hold with ^aT b replaced by
any O­ s ŠaTb. Identity (24.3) has been verified directly in [26, Proof of Thm 7.2, Ch. 6]. Here it
follows from (23) with Oœ^aT
b.
Inequalities (24.1) and (24.2) follow directly from (23) and the
assumptions eaT b ! and eaTb
Ÿ! . So we only need to verify (23), which could be done directly.
Our derivation of (23) is based the fact that both sides of (23) represent the cost N a?
b in (2) with
RœT a> " b and ?œ OB.
Proof of (23). For a=ßDb
­M‚V 8 "ß_ 8
and a given T­P aM ß’ b, consider the cost N a?
b in (2) with
RœT a> " b and ?œ OB. Then N a? b is precisely NO
in (18), which is represented in (21). On the other
hand, by (15) with RœT a> b, Oœ s aT b and ?œ OB, we have
" ^
X X w
X
N œD T ab =D€I( ˜ B ˆ T € LQaT b€ a^aTb Ob e aTba^aTb
Ob
‰ B .>Þ (25)
O
=
>
"
So the two integrals in (25) and (21) are identical for the solution
a=ßDb
­M‚‘ 8 . By Proposition 3, (23) must hold. ¨
B
to equation (18.1) for each