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

6
We start with an interpretation of upper and lower solutions to (1) with
Using the notation of the LQR problem (2), we have
w
C œ! , that is,
T € L Q aTb œ!à Ta> b œ R.
(11)
"
Theorem 2. Suppose T ­P
"ß_ "
8
ac=ß> d ß’ b is feasible.
(i) If T is a lower solution to (11) with eaTb !, then N a? b D X T ab =Dfor all ?­ h c=ß>
" d.
(ii) If T is an upper solution to (11) with eaT b Ÿ !, then N a? b Ÿ D X T ab =D for all ?­ h c=ß>
" d.
X
(iii) If T is a solution to (11) with eaTb !(
eaT b Ÿ !), then D T ab =Dis the minimum (maximum,
respectively) value of J a? b over hc=> ß " d, which occurs at ?œ OB s , where O­ s ŠaTb
and B satisfies
.B œ E FOs‘ B.>€ G HOs‘ B.[ßB= ab œDÞ
(12)
"ß_ 8
Proof. Suppose T­P ac=ß> d ß ’ b and B is the solution to equation (2.1) with ?­ hc=>
ß d.
By the
" "
Fundamental Theorem of calculus and Ito's formula, applied to B X
ab >T ab >B> ab, we obtain
"
X X
.
X
IeB a> " bT a> " bB> a " bf D T ab =DœIœ(
B ab >T ab >B>.> ab
.>
>
X X X X X X
œI( eB aT € L aTbbB€#? aF T€HTGB€?HTH? b
f.>
,
=
"
w
=
>
(13)
X
X
where L aTb œE T€TE€G TG as defined in (5). Adding (13) to N a?
b and using the notations
X X X
eaTb œV€HTH and faTb
œ F T€HTG€Win (7), we obtain
X
X
N a? b D T ab =D€IeB a> baT a> b RB> b a bf
>
" " "
X X X
œI( eB aT € LaT b€KB€#? b faTbB€? eaT b? f.>Þ
=
"
w
Since T is feasible, faTb œ eaTbOs
for each O­ s ŠaTb. By completing the squares and using that
Os
X ea TOœ b s Q a T b as in (10) and LQaTb
œK€ L aTb
Q a T b in (5), we have
(14)
X
X
N a? b D T ab =D€IeB a> baT a> b RB> b a bf
>
" " "
X w
X
œI( š B aT € LQaT bbB€ ˆ ?€OB s ‰ e aT bˆ ?€OB s ‰›.>.
=
"
(15)
w
In case (i), we have T a> b RŸ! , T € LQ aT b ! and e aT b !. So (15) implies that
"
X
X
N a?
b D T ab =D for every ?­ h c=ß> " d. Similarly, in case (ii), (15) implies that N a? b Ÿ D T ab =Dfor
every ?­ hc=ß> d. In case (iii), (15) implies that for every ?­ hc=ß>
d,
" "
>
X
X
N a? b D T ab =DœI ( š ˆ ?€OB s ‰ eaT
bˆ ?€OB s ‰›.>Þ
=
"