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

(ii) Equation (1) has a solution T with eaT b ! if and only if it has an upper solution ] with
ea] b !.
14
Proof. The necessity in both cases is trivial. To show the sufficiency in both cases, use Proposition 8 to
conclude that equation (1) has an upper solution ] (lower solution ^ ) with ea] b ž! ( ea^ b !). So
equation (1) has a solution T with eaTb ž! ( !, respectively) by Theorem 11. ¨
Theorem 12 is not true under the weaker assumptions ea^ b ! or ea] b Ÿ !. Consider the
scalar case of equation (1) with Eœ "ß FœKœ R œ"ß GœHœVœWœ!ß C œ! ; that is,
w
T €" #T œ!ß T a> b œ" . Obviously, ! is a lower solution. It is feasible because ea! b œ fa! b œ! .
"
" "
However, the solution T ab > œ
#
€
#/ #> is not feasible because eaTb œ! but faTb
œT.
It is proved in
X
[8, Thm 4.1] that (1) has a solution T with eaT b ž ! if Wœ! , H Hž! and ! is a strict lower solution
with ea! b œV ! (e.g., V !, K !, R ! and either K ž ! or R ž! ). Next theorem generalizes
this result to equation (1) with an arbitrary lower solution. Our proof is based on Theorem 12.
Theorem 13. Recall eaR b œ V a> b€H a> bR Ha> b. Suppose HHž!
X X
" " "
.
(i) If eaRb ž! and ^ is a strict lower solution to (1) with ea^ b !, then (1) has a solution T with
eaT b ž !.
(ii) If eaR b ! and ] is a strict upper solution to (1) with ea] b Ÿ !, then (1) has a solution T with
eaT b !.
Proof. By Remark 3, we show (i) only. We first assume Wœ! and ^ œ! . In this case the assumption
implies that V ! and either K ! or R ! is strict. We will show that (1) has another lower solution
^+ ž ! in Ò> ! ß> " Ñ with ea^+
b ž ! in Ò> ! ß>
" Ó. Then by Theorem 12, (1) has a solution T with eaT b ž !.
!a>
>
If R ž! , then let ^+ œ &/ " bI8, where I8
is the 8‚8 unit matrix, & is the minimum
!a> >
eigenvalue of R and ! is an undetermined number. Using that e a^ b &/
" b X
HHž! , we have
+ ´ +
w !a> + +
> " b
X a^ b ^ € LQ a^ b€ Ca^ b &/ Q€K,
X X X X X X X X
where Qœ ! I 8 €E €E€G G aF €HGbaHHb aF €HG b€ CaI
8 bÞ Taking a
sufficiently large ! so that Qž! , we have Xa^+ b ž! as desired.
"ß_ 8
If K ž! , then let ^+ œ & ^ ! ab > , where & ž! and ^! ­P aMß’
b such that ^ ! a> " b œ R and
^ ab > ž! for >­Ò> ß> Ñ.
Using that e a^ b &ea^ b ž! in Ò> ß>
Ó, we have
! ! " +
! ! "
w
+ +
+ +
"
+
X a^ b ´ ^ € LQa^ b€ C a^ b œ & Q€K, (39)
X X X
where Qœ ^! €E ^! € ^! E€G ^! G fa^! b ea^! b fa^! b€
Ca^!
b Þ Since Kž! we can
make X a^+ b ž! by taking a sufficiently small & ž! .
"