1 Upper-Lower Solution Method for Differential ... - Bradley Bradley
1 Upper-Lower Solution Method for Differential ... - Bradley Bradley
1 Upper-Lower Solution Method for Differential ... - Bradley Bradley
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1<br />
<strong>Upper</strong>-<strong>Lower</strong> <strong>Solution</strong> <strong>Method</strong> <strong>for</strong> <strong>Differential</strong><br />
Riccati Equations from Stochastic LQR Problems<br />
Libin Mou<br />
Department of Mathematics<br />
<strong>Bradley</strong> University<br />
Peoria, IL 61625<br />
Abstract<br />
We use upper and lower solutions to study the existence and properties of solutions to differential<br />
Riccati equations arising from stochastic linear quadratic regulator (LQR) problems. The main results<br />
include an interpretation of upper and lower solutions, comparison theorems, an upper-lower solution<br />
theorem, necessary and sufficient conditions <strong>for</strong> existence of solutions, an estimation of maximal<br />
existence intervals of solutions and an approximation of solutions. Many of the results are new, while<br />
others are generalizations of some known results.<br />
1. Introduction, Notations and Definition<br />
In this paper we study the following matrix differential equation<br />
Ú w X<br />
X<br />
Ý T €E T€TE€G TG€ K € CaTb<br />
Û<br />
X X X<br />
X " X X<br />
aF T€HTG€WbaV€HTHb aF T€HTG€Wb<br />
œ! ß<br />
Ý<br />
Ü Ta> b œ R<br />
"<br />
(1)<br />
X<br />
w<br />
of Riccati type on a fixed interval Mœ c>ß> d or Ð _ß>Ó, where E is the transpose of E, T œ , R<br />
! "<br />
"<br />
is a symmetric matrix, C is a linear map of symmetric matrices, and EßFßGßHß K ßVßW are matrix<br />
functions in<br />
M satisfying the conditions stated in (4).<br />
.T<br />
.><br />
As a motivation <strong>for</strong> equation (1), we consider a stochastic<br />
linear quadratic regulator ( LQR)<br />
problem with noise depending on both of the state and control. For =Mß let [ be a standard Brownian<br />
motion with<br />
[ ab = œ! almost surely and hc=ß> " d be the set of all square integrable control processes<br />
defined on c=ß > d and adapted to the 5-field generated by [. For D ‘ 8 and ? hc=ß<br />
> d , consider the<br />
" "<br />
following state equation and cost function N a?<br />
b:<br />
.B œ aEB€F? b.>€ aGB€H? b.[ß>Mà B= abœD, (2.1)<br />
ž<br />
X ><br />
N a? b œIš B a b B a b € ' X X X<br />
> R ><br />
"<br />
aB KB€#? WB€?V? b.><br />
›, (2.2)<br />
" "<br />
where Ief represents the expectation of the enclosed variable. The problem is to<br />
=<br />
(2)
maximize/ minimize N a? b <strong>for</strong> ? h c=ß> " dÞ<br />
(3)<br />
See [27, Ch. 6] <strong>for</strong> a detailed description of this problem. This problem leads to equation (1) with C œ! .<br />
For a derivation, see [3], [5], [7], [27] or Theorem 1 below. The inclusion of the term C is important <strong>for</strong><br />
application of (1) to stochastic control problems with Markovian jumping noises and differential game<br />
problems with state-dependent noises; see [23] and [16], <strong>for</strong> example. Although the term<br />
X<br />
G TG<br />
2<br />
may be<br />
considered as a part of CaT<br />
b, we will keep them separated <strong>for</strong> generality. Readers who are interested in<br />
the equation associated with problem (3) may assume that C œ! .<br />
Riccati equations (differential, difference and algebraic) appear in various control and min-max<br />
problems. The classical differential Riccati equation is (1) with<br />
GœHœWœ C œ! . For this case, the<br />
existence, comparison, approximation and other properties of solutions have been extensively studied; see<br />
[2], [4], [6], [14], [15], [21], [24], [28] and many references therein. Many results <strong>for</strong> classical Riccati<br />
equations have been extended to equation (1) with HœWœ! in [26], [11], [12] and [10]. Existence and<br />
approximation of solutions to (1) with Gœ C œ! have been obtained in [7], [ 8] and [27].<br />
We will prove comparison theorems, an upper-lower solution theorem, necessary and sufficient<br />
conditions <strong>for</strong> existence of solutions, an estimation of maximal existence intervals of solutions and an<br />
approximation of solutions <strong>for</strong> equation (1) under general settings. In particular, K and R are allowed to<br />
X<br />
be indefinite and V€HTH may be either positive or negative semidefinite.<br />
The method of upper and lower solutions has been introduced <strong>for</strong> differential equations as early as<br />
1940's. Although Riccati inequalities have been studied or used in some literature of Riccati equations<br />
(e.g., [24], [9] and [22]), this paper, together with [18], [17] and [16], appear to be the first systematic<br />
application of the method of upper and lower solutions to Riccati equations. It turns out this method has<br />
many desired merits. For example, it naturally links equation (1) with C œ! with the LQR problem (3).<br />
It derives the main results under general assumptions. It gives verifiable necessary and sufficient<br />
conditions <strong>for</strong> the existence of solutions. It also gives algorithms <strong>for</strong> approximating solutions. It can be<br />
used to estimate the maximal existence intervals of solutions to differential Riccati equations without<br />
solving the equations. It applies to Riccati equations of different types.<br />
This paper is organized as follows. Notations and assumptions are introduced in this section<br />
followed by the definitions of upper and lower solutions. In Section 2, we prove a relationship between<br />
upper and lower solutions and the well-posedness of the LQR problem (3). In addition, some intrinsic<br />
structural properties of equation (1) are proved in Section 2. In Section 3 we prove some general<br />
comparison theorems and an upper-lower solution theorem <strong>for</strong> equation (1). As an application, we obtain<br />
some necessary and sufficient conditions <strong>for</strong> the existence of solutions to (1). Furthermore, we show that
3<br />
the solution can be approximated by a sequence of solutions to linear equations. In Section 4, we apply<br />
the upper-lower solution theorem to estimate the maximal existence intervals of solutions to (1). Finally,<br />
in Section 5 we mention a generalization of equation (1).<br />
The author would like to thank Stan Liberty and Mike McAsey <strong>for</strong> stimulating discussions.<br />
Notations. Denote by ’ 8 the set of all real symmetric 8‚8 matrices. We write Q R ( QžR) if Q,<br />
R ’ 8 and Q R is a positive semidefinite (definite). For a map C À ’ 8 Ä ’ 8 we write C ! if<br />
_<br />
CaQ b ! <strong>for</strong> each Q !. For a Hilbert space — and an interval Mœ c>ß><br />
d or Ð _ß>Ó, P aMß—<br />
b is<br />
! "<br />
"<br />
the space of all bounded and measurable functions from M to —. Furthermore, we define P "ß_ aMß—<br />
b œ<br />
ÖT P _ aMß— bßT w P _ aMß — b×Þ<br />
Additional notations will be introduced later. For reader's convenience, we list below the<br />
frequently used notations in this paper.<br />
w<br />
XaT b is a short hand <strong>for</strong> T € LQ aT b€ C aTb, which is the left-hand side of (1).<br />
LQaTb œK€ LaTb Q aTb, where L aTb, Q aTb<br />
are defined in (5). Here K is "constant" term,<br />
L aT b is the linear term, while QaT b is the "quadratic" term.<br />
and f.<br />
eaTb and faT b are defined in (7). The facts ea! b œV and fa! b œWexplain the choices of e<br />
O is a generic notation <strong>for</strong> a feedback matrix. ŠaTb<br />
is the set of all feedback matrices associated<br />
with T; O s ŠaTb; ^aTb ŠaTb<br />
is the unique feedback matrix defined in (7) and (8) in terms of the<br />
" €<br />
inverse eaTb or psuedoinverse eaTb of eaTb.<br />
ZaO<br />
b is defined in (19). The fact Za! b œKexplain the choice.<br />
_ aOàT b is defined in (22). Note that _ a!àTb œ LaTbÞ<br />
R ’ 8 is the boundary value of (1); Tß]ß^P "ß_ aMß<br />
’ 8<br />
b often represent a solution, an upper<br />
solution and a lower solution to (1)Þ EßFßGßHßK ßV and W are coefficient matrix functions of (1).<br />
Assumption. The basic assumption <strong>for</strong> EßFßGßHßKßVßWßR and C in (1) is<br />
Ú C P<br />
_ 8 8<br />
a’ ß’<br />
b is linear and C !,<br />
_ 8‚8 X _ 8‚5<br />
Û EßGP aMß‘ b, FßHßW P ÐMß‘<br />
Ñ,<br />
Ü _ 5 _ 8 8<br />
VP ÐMß ’ Ñ, K P aMß’ bß R ’ .<br />
To write equation (1) concisely, we denote, <strong>for</strong> T P<br />
_ a<br />
Mß’ 8 b,<br />
Ú<br />
Û<br />
X<br />
X<br />
LaTb<br />
œE T€TE€G TG,<br />
X X X<br />
Q aTb<br />
œ aF T€HTG€Wba V€HTHb " X X<br />
aF T€HTG€Wb,<br />
Ü LQaTb œK€ LaTb<br />
QaT<br />
b<br />
(4)<br />
(5)
4<br />
with parameters EßFßGßHßK ßV and W satisfying (4). Thus (1) becomes<br />
To indicate the parameters, we may say<br />
equation (1) with parameters EßFßGßHßKßVßWßC and RÞ<br />
w<br />
T € L Q aT b€ C aTb œ!ßTa> b œ R.<br />
(1)<br />
"<br />
LQ a† b with parameters EßFßGßHßK ßV and W, and<br />
say<br />
X<br />
We remark that Q aTb and LQ aTb<br />
may be well-defined even if V€HTH is singular. If<br />
V€HTH<br />
X is nonsingular, then Q aTb<br />
can be written as<br />
where eaTb, faT b and ^aTb<br />
are<br />
" X<br />
Q aTb œ faTbeaTb faTb<br />
œ ^aTb eaTb^aTb, (6)<br />
X<br />
eaT<br />
b œV€HTH, (7)<br />
X<br />
X<br />
faTb<br />
œ F T€HTG€W,<br />
^aTb œ eaTb faTb.<br />
Here eaT b and faTb<br />
can be considered as perturbations of V and W with respect to T. The term<br />
"<br />
^aTb œ eaTb faTb<br />
appears frequently as the optimal feedback matrix <strong>for</strong> problem (3) when T is a<br />
solution to (1) with<br />
C œ! ; see Theorem 1 below.<br />
If eaT<br />
b is singular, then we define<br />
"<br />
€<br />
^aTb œ eaTb faTb, (8)<br />
€ €<br />
where eaTb is the pseudoinverse of eaTb. Recall that any matrix Q has a unique pseudoinverse Q<br />
with the following properties (see [19] and [1]).<br />
€ € € €<br />
QQ QœQßQ QQ œQ . (9)<br />
8 € 8 € €<br />
If Q ’ , then Q ’ , and QQ œQ Q.<br />
€<br />
Q ! if and only if Q !.<br />
€<br />
Note that ^aTb œ eaTb faTb always exists, but it may not satisfy faTb œ eaTb^aTb. This<br />
leads to the following definition.<br />
Definition 1. If T P<br />
_ aMß’ 8 b satisfies f a T b œ e a T b ^ a T b , then T is said to be feasible .<br />
_ 5‚8<br />
Denote ŠaTb œ eO P aMß‘ bß faTb œ eaTbOf.<br />
Obviously, if T is feasible, then<br />
^aT b ŠaTb and so ŠaTb<br />
Ág. The converse is also true. We have<br />
Proposition 1. If ŠaTb Ág, then T is feasible and <strong>for</strong> each O ŠaTb,<br />
X<br />
X<br />
Q aTb<br />
œ ^aTb eaTb^aTb œO eaTbO,<br />
(10)
5<br />
where ^aTb œ eaTb faTb.<br />
€<br />
Proof. If OŠaT b Ág, then faTb œ eaTbO. Since eaTb œ eaTbeaTb eaTb, we have<br />
€ €<br />
eaTb^aTb œ eaTbeaTb faTb œ eaTbeaTb eaTbO œ eaTbO œ faTb. This shows that T is<br />
€ €<br />
feasible. Furthermore, from that ^aTb<br />
œ eaTb faTb œ eaTb eaTbO<br />
and property (9), we have<br />
€ €<br />
^aTb X eaTb^aTb œO X eaTbeaTb eaTbeaTb eaTbOœO X eaTbO.<br />
This shows (10). ¨<br />
Proposition 1 shows that if ŠaTb Ág, then Q aTb is well-defined by (10). Each O s ŠaTb<br />
will<br />
"ß_ 8<br />
be called a feedback matrix associated with TP aMß’ b. This term is motivated by the fact (see<br />
"ß_ 8<br />
Theorem 1 below) if TP aMß’ b is a feasible solution to (1) with C œ! , then each O s ŠaTb<br />
is an<br />
optimal feedback matrix.<br />
€<br />
Definition 2. Suppose T P "ß_ aM, ’ 8 b is feasible.<br />
T is an upper solution to (1) if<br />
T is a lower solution to (1) if<br />
w<br />
T € LQ aTb€ CaTb<br />
Ÿ!à T a> b R.<br />
w<br />
T € LQaTb€ CaT b !à Ta> b Ÿ RÞ<br />
T is a solution if it is both an upper solution and a lower solution. An upper or lower solution is called<br />
strict if at least one of the inequalities in the definition is strict.<br />
"<br />
"<br />
All differential equations and differential inequalities in this paper are considered pointwise <strong>for</strong><br />
almost every > in an interval indicated by context. For brevity, we will write, <strong>for</strong> example, "K !" or<br />
" K ! in M " to mean that " Kab<br />
> ! <strong>for</strong> almost every >M."<br />
basic results are still of interest if it is assumed that<br />
The variable > is often suppressed. The<br />
EßFßGßHßK ßV and W are all continuous (or<br />
piecewise continuous) and bounded in M, and it is assumed that T is (piecewise) continuously<br />
differentiable in M. The equations and inequalities will be then pointwise (except at finite points).<br />
Note that ! ’ 8 X<br />
is a lower solution to (1) if and only if K W V " W !, R ! and that ! is<br />
X "<br />
an upper solution to (1) if and only if K W V WŸ! , R Ÿ! . These are the common cases studied in<br />
the classical literature of Riccati equations; see [7] and [8] <strong>for</strong> remarks. Also, ! is a strict upper or lower<br />
solution if one of the inequalities is strict. See Proposition 6 <strong>for</strong> an equivalent description.<br />
§2. Interpretation of <strong>Solution</strong>s and Structure of LQaT<br />
b
6<br />
We start with an interpretation of upper and lower solutions to (1) with<br />
Using the notation of the LQR problem (2), we have<br />
w<br />
C œ! , that is,<br />
T € L Q aTb œ!à Ta> b œ R.<br />
(11)<br />
"<br />
Theorem 2. Suppose T P<br />
"ß_ "<br />
8<br />
ac=ß> d ß’ b is feasible.<br />
(i) If T is a lower solution to (11) with eaTb !, then N a? b D X T ab =D<strong>for</strong> all ? h c=ß><br />
" d.<br />
(ii) If T is an upper solution to (11) with eaT b Ÿ !, then N a? b Ÿ D X T ab =D <strong>for</strong> all ? h c=ß><br />
" d.<br />
X<br />
(iii) If T is a solution to (11) with eaTb !(<br />
eaT b Ÿ !), then D T ab =Dis the minimum (maximum,<br />
respectively) value of J a? b over hc=> ß " d, which occurs at ?œ OB s , where O s ŠaTb<br />
and B satisfies<br />
.B œ E FOs‘ B.>€ G HOs‘ B.[ßB= ab œDÞ<br />
(12)<br />
"ß_ 8<br />
Proof. Suppose TP ac=ß> d ß ’ b and B is the solution to equation (2.1) with ? hc=><br />
ß d.<br />
By the<br />
" "<br />
Fundamental Theorem of calculus and Ito's <strong>for</strong>mula, applied to B X<br />
ab >T ab >B> ab, we obtain<br />
"<br />
X X<br />
.<br />
X<br />
IeB a> " bT a> " bB> a " bf D T ab =DœIœ(<br />
B ab >T ab >B>.> ab<br />
.><br />
><br />
X X X X X X<br />
œI( eB aT € L aTbbB€#? aF T€HTGB€?HTH? b<br />
f.><br />
,<br />
=<br />
"<br />
w<br />
=<br />
><br />
(13)<br />
X<br />
X<br />
where L aTb œE T€TE€G TG as defined in (5). Adding (13) to N a?<br />
b and using the notations<br />
X X X<br />
eaTb œV€HTH and faTb<br />
œ F T€HTG€Win (7), we obtain<br />
X<br />
X<br />
N a? b D T ab =D€IeB a> baT a> b RB> b a bf<br />
><br />
" " "<br />
X X X<br />
œI( eB aT € LaT b€KB€#? b faTbB€? eaT b? f.>Þ<br />
=<br />
"<br />
w<br />
Since T is feasible, faTb œ eaTbOs<br />
<strong>for</strong> each O s ŠaTb. By completing the squares and using that<br />
Os<br />
X ea TOœ b s Q a T b as in (10) and LQaTb<br />
œK€ L aTb<br />
Q a T b in (5), we have<br />
(14)<br />
X<br />
X<br />
N a? b D T ab =D€IeB a> baT a> b RB> b a bf<br />
><br />
" " "<br />
X w<br />
X<br />
œI( š B aT € LQaT bbB€ ˆ ?€OB s ‰ e aT bˆ ?€OB s ‰›.>.<br />
=<br />
"<br />
(15)<br />
w<br />
In case (i), we have T a> b RŸ! , T € LQ aT b ! and e aT b !. So (15) implies that<br />
"<br />
X<br />
X<br />
N a?<br />
b D T ab =D <strong>for</strong> every ? h c=ß> " d. Similarly, in case (ii), (15) implies that N a? b Ÿ D T ab =D<strong>for</strong><br />
every ? hc=ß> d. In case (iii), (15) implies that <strong>for</strong> every ? hc=ß><br />
d,<br />
" "<br />
><br />
X<br />
X<br />
N a? b D T ab =DœI ( š ˆ ?€OB s ‰ eaT<br />
bˆ ?€OB s ‰›.>Þ<br />
=<br />
"
X<br />
It follows that N a? b has a minimum (maximum) D T ab =D at ?œ OB s if eaT b !ÐeaTb<br />
Ÿ!Ñ.<br />
Equation (12) is precisely the state equation with ?œ OB s . ¨<br />
7<br />
Setting ? œ! in (2), we obtain<br />
.B œEB.>€GB.[ß> c=ß><br />
" dàB= abœ<br />
D, (16.1)<br />
ž<br />
X<br />
> "<br />
N ´ Iš B a> bR B a> b € ' X<br />
B KB.><br />
› (16.2)<br />
!<br />
" "<br />
=<br />
.<br />
(16)<br />
By (14) with ?œ!, we obtain a representation <strong>for</strong> N ! by each TP "ß_ ac=ß> d ß’ 8 b with T a> b œ :<br />
"<br />
" R<br />
X<br />
X<br />
N!<br />
œD T ab =D€I( B aT €K€ L aTbbB.>Þ<br />
(17)<br />
=<br />
><br />
"<br />
In particular, if TP "ß_ =ß> ß 8<br />
w<br />
ac d ’ b is the solution to the linear equation T € LaT b€Kœ!<br />
with<br />
"<br />
X<br />
Ta> " b œ R, which exists by Proposition 7 below, then N! œ D T ab. =D There<strong>for</strong>e, if N ! œ ! <strong>for</strong> every<br />
a=ßDb<br />
M‚V 8 , then T´! in M, which implies that R œ! and K´! in MÞ This leads to the following<br />
proposition, which will be used in the proof of Proposition 4.<br />
_<br />
Proposition 3. Suppose K ßK P aMß’ 8 > "<br />
b and Iš ' X<br />
> "<br />
B K B.> › œIš ' X<br />
B K B.> › <strong>for</strong> the<br />
" #<br />
w<br />
= " = #<br />
solution B to equation (16.1) with each a=ßDb M‚V<br />
8 . Then K ´ K in MÞ<br />
" #<br />
Proposition 3 follows from the case of (16) with R œ! and KœK" K#<br />
. The assumption of<br />
Proposition 3 implies that N! œ! <strong>for</strong> all a=ßDb<br />
M‚V 8 . There<strong>for</strong>e K´! , or equivalently, K " ´K# in<br />
MÞ<br />
Note that equation (16.1) is independent of K" and K# . Proposition 3 shows that K " ´K#<br />
in M if<br />
> "<br />
Iš ' X<br />
> "<br />
B B.> › œIš ' X<br />
K B K B.> › <strong>for</strong> the solution B to certain linear stochastic differential equation<br />
= " = #<br />
like (16.1) with each a=ßDb<br />
M‚V 8 ; see the proof of Proposition 4.<br />
Suppose O P _ aMß ‘ 5‚8 b and let ?œ OB. Then (2) reduces to<br />
.B œ aE FObB.>€ aG HObB.[ßB= abœ<br />
Dß<br />
ž<br />
X<br />
><br />
N ´ Iš B a> bR B a> b € ' " X<br />
B aObB.> › ß<br />
O<br />
where ZaOb<br />
are defined as<br />
" "<br />
=<br />
Z<br />
(18.1)<br />
(18.2)<br />
(18)<br />
X X X<br />
ZaOb<br />
œO VO O W W O€K. (19)<br />
Note that (18) is (16) under the following replacement:
8<br />
aEßGßKb Ä aE FOßG HO, ZaObb. (20)<br />
There<strong>for</strong>e, a representation <strong>for</strong> N O follows from (17) under replacement (20); that is,<br />
NO<br />
œ D X T ab =D€I( B X aT € Z aO b€ _ aOàTbbB.><br />
, (21)<br />
where _aOàT b is L aT<br />
b defined (5) under replacement (20), namely,<br />
=<br />
><br />
"<br />
w<br />
X<br />
X<br />
_aOàTbœ aE FOb T€TaE FO b€ aG HOb TaG HObÞ<br />
(22)<br />
Propositions 4 and 5 below reveal some important structural properties of LQaT b.<br />
Proposition 4. Suppose TP "ß_ a 8<br />
Mß b _ 5‚8<br />
’ is feasible and O P aMß‘<br />
b. Let LQ a T b , Z a O b and<br />
_aOàT b be defined in (5), (19) and (22), respectively. Then<br />
(i) LQ aT b€ a^aTb Ob X e aTba^aTb<br />
Ob<br />
œ Z aO b€ _ aOàTb,<br />
(23)<br />
(ii)<br />
Ú LQaTb Ÿ Z aO b€ _ aOàTb if eaT b !à (24.1)<br />
Û LQaT b Z aO b€ _ aOàTb if eaTb<br />
Ÿ!à (24.2)<br />
Ü LQaTb œ Z a^aTbb€ _ a^aT bàTb.<br />
(24.3)<br />
(24)<br />
Remark 1. From the proof below, it follows that identities (23) and (24.3) hold with ^aT b replaced by<br />
any O s ŠaTb. Identity (24.3) has been verified directly in [27, Proof of Thm 7.2, Ch. 6]. Here it<br />
follows from (23) with Oœ^aT<br />
b.<br />
Inequalities (24.1) and (24.2) follow directly from (23) and the<br />
assumptions eaT b ! and eaTb<br />
Ÿ! . So we only need to verify (23), which could be done directly.<br />
Our derivation of (23) is based the fact that both sides of (23) represent the cost N a?<br />
b in (2) with<br />
RœT a> " b and ?œ OB.<br />
Proof of (23). For a=ßDb<br />
M‚V 8 "ß_ 8<br />
and a given TP aM ß’ b, consider the cost N a?<br />
b in (2) with<br />
RœT a> " b and ?œ OB. Then N a? b is precisely NO<br />
in (18), which is represented in (21). On the other<br />
hand, by (15) with RœT a> b, Oœ s aT b and ?œ OB, we have<br />
" ^<br />
X X w<br />
X<br />
N œD T ab =D€I( ˜ B ˆ T € LQaT b€ a^aTb Ob e aTba^aTb<br />
Ob<br />
‰ B .>Þ (25)<br />
O<br />
=<br />
><br />
"<br />
So the two integrals in (25) and (21) are identical <strong>for</strong> the solution<br />
a=ßDb<br />
M‚‘ 8 . By Proposition 3, (23) must hold. ¨<br />
B<br />
to equation (18.1) <strong>for</strong> each
9<br />
Proposition 5. Suppose ] ß^ P aMß’ b are feasible, O P aMß‘<br />
5‚8 b. Denote Tœ] ^,<br />
EœE s F^a^bßGœG s H^a^b and Vœ s ea^b. Then<br />
(i) LQ a] b LQa^b<br />
(26)<br />
‡ ‡<br />
œET€TE€GTG<br />
s s s s ˆ<br />
X X<br />
X<br />
F T€HTGs‰ˆ V€HTH s X €<br />
‰ ˆ<br />
X X<br />
F T€HTGs‰<br />
(ii) If ^ is given, then ] satisfies equation (1) if and only if T œ] ^ satisfies<br />
Ú<br />
Ý w<br />
‡ ‡<br />
T € K€ s ET€TE€GTG€ s s s s CaTb<br />
X €<br />
Û ˆ X X<br />
Ý<br />
F T€HTGs‰ˆ V€HTH s X ‰ ˆ X X<br />
F T€HTGs‰<br />
œ!ß<br />
Ü T a> b œ R ^> a b,<br />
" "<br />
(27)<br />
where Ks œ^ w € LQa^b€ Ca^bÞ<br />
Proof. Let Tœ] and ^ in Proposition 4 (i), respectively. Then we get<br />
LQa]<br />
b LQa^b<br />
œ<br />
_ aOàTb<br />
a^a] b<br />
X<br />
O b ea] ba^a] b O b€ a^a^b X<br />
Ob ea^ba^a^b<br />
OÞ b<br />
(28)<br />
Setting Oœ^a^b<br />
in (28), we obtain<br />
X<br />
LQa] b LQ a^ b œ _ a^a^ bàTb<br />
a^a] b ^a^ bb ea] ba^a] b ^a^bb.<br />
(29)<br />
To continue the proof, we first simplify ea] bc^a] b ^a^ bd. Note that<br />
X X X<br />
ea] b ea^ b œHTH and fa] b fa^b<br />
œF T€HTG. The feasibility of ] and ^ implies that<br />
ea] b^a] b œ fa] b and ea^b^a^b œ fa^b. There<strong>for</strong>e<br />
ea] bc^a] b ^a^ bd œ ea]<br />
b^a] b ea] b^a^b<br />
X<br />
œ ea] b^a] b ea^b^a^b HTH^a^b<br />
X X X X X<br />
œF T€HTG HTH^a^b<br />
œF T€HTGs .<br />
€<br />
Now using the fact ea] bea] b ea] b œ ea]<br />
b and (30), we obtain that<br />
a^a] b ^a^ bb X ea] ba^a] b ^a^bb œ ˆ X X X<br />
F T€HTGs‰ €<br />
ea]<br />
b ˆ F X T€HTG<br />
X s‰<br />
. (31)<br />
Substituting (31) and _a^a^bàTb<br />
‡ ‡<br />
œ ET€TE€GTG<br />
s s s s<br />
into (29), we obtain (i).<br />
To show (ii), denote Xa] b œ ] w € LQ a] b€<br />
Ca]<br />
b.<br />
Then ] is a solution to equation (1) if and<br />
only if Xa] b œ! and ] a> b œ . Note that<br />
" R<br />
Xa] b Xa^b<br />
œ T w € LQa] b LQ a^ b€<br />
CaTb,<br />
where LQa] b LQ a^b is expressed in terms of T as in (26). If ^ is given, then Xa] b œ! and<br />
] a> " b œ R if and only if Tœ] ^ satisfies (27). ¨<br />
(30)
Remark 2. Note that equation (27) is (1) with parameters<br />
10<br />
EßFßGßHßKß s s s Vß!ß s C and R ^> a b. It<br />
follows that ] is a solution (upper solution, lower solution) to (1) if and only if T œ] ^ is a solution<br />
(upper solution, lower solution, respectively) to equation (27). For example, if ^ œ! , then (1) is<br />
equivalent to (27) with Es œE FV " Wß GœG s HV " Wß KœK s X<br />
W V " WßVœVÞ s In (27) the<br />
"crossing term" Wœ! .<br />
If ^ is a lower solution to (1), then ! is a lower solution to (27) because K s !,<br />
R ^> a b !<br />
"<br />
" .<br />
Similarly, if ^ is an upper solution to (1), then ! is an upper solution to (27). For convenience, we will<br />
say equation (1) is a standard case if Wœ! and ! is a lower solution or an upper solution. We have<br />
Proposition 6. Equation (1) can be reduced to a standard case if and only if it has a lower or upper<br />
solution.<br />
Remark 3. Let ]œ ^, then it is easily seen that ^ is a solution (upper solution, lower solution) to (1)<br />
if and only if ] is a solution (lower solution, upper solution) to the following equation<br />
] w € ] € ] K X X<br />
La b Ca b aF ] €H ] G Wb<br />
X X X X<br />
a V€H ] Hb aF ] €H ] G Wb<br />
œ!<br />
with ] a> " b œ R. This is same as (1) with Kß Vß W and R replaced by Kß Vß W and R,<br />
respectively. When C œ!, this equation is precisely the differential Riccati equation associated with the<br />
X<br />
X<br />
problem maximizing N a?<br />
b. Also note that V€H^H ! if and only if V€H ] HŸ! . As a<br />
result, if a lower solution ^ (if it exists) to (1) with ea^b ž! has certain property, then an upper<br />
solution ] (if it exists) to (1) with ea]<br />
b ! also has the corresponding property. Because of this, we<br />
will state properties of upper solutions without proof.<br />
"<br />
§ 3. Comparison Theorems, <strong>Upper</strong>-<strong>Lower</strong> <strong>Solution</strong> Theorem, Existence and Approximation of<br />
<strong>Solution</strong>s<br />
We first consider the linear matrix differential equation<br />
w "<br />
X<br />
X<br />
T € E T€TE€G TG€ CaT b€ K œ !ß T a> b œR<br />
(32)<br />
where EßGßK and C are as in (5) and R’ 8 .<br />
Proposition 7. (i) E quation (32) has a unique solution TP<br />
"ß_ aMß’ 8 b.<br />
(ii) If ! is a lower solution to (32) then T !. If ! is strict, then and Tž! .<br />
(iii) If a]ß^ b is a pair of upper-lower solutions to (32), then ] ^ . If one of ] and ^ are strict, then<br />
]ž^.
11<br />
Proof. Following the idea in [26], we let Ga>ß=<br />
b be the fundamental matrix of E, that is,<br />
`<br />
Ga>ß= b œE> abGa>ß=ß b Ga>ß> b œE> ab, > ! Ÿ=ß>Ÿ > " Þ<br />
`><br />
" `<br />
`><br />
Note that Ga>ß= b œ Ga=ß> b and Ga=ß> b œ E> abGa=ß><br />
b. It follows that (32) is equivalent to<br />
><br />
"<br />
X X X<br />
T ab > œ Ga> " ß> b R Ga> " ß> b€ ( Ga=ß> beCaT ab = b€ G ab = Tab = G ab = € Kab<br />
= fGa=ß>.=<br />
b (33b5)<br />
><br />
The Volterra equation (33) has a unique solution T which can be found by successive approximations;<br />
say, eT À œ!ß"ß#ßâf<br />
with T œ! . This shows (i).<br />
/ / !<br />
8<br />
In case ! ’ is a lower solution (i.e., K ! and R !), we have T / ! <strong>for</strong> all / !, which<br />
implies that T ab > œ lim / Ä_ T / ab > !. If ! is a strict lower solution (i.e., K ! or T a><br />
" b ! is strict),<br />
then<br />
#<br />
X<br />
X<br />
T ab > T ab > ´ Ga> ß> b RGa> ß> b€ ( Ga=ß> b Kab = Ga=ß>.= b ž! . (34)<br />
" "<br />
This proves (ii). If a]ß^b<br />
is a pair of upper-lower solutions to (32), then Tœ] ^ satisfies (32) with<br />
some K ! and T a> b œ] a> b ^> a b !. Applying the conclusions in (ii) to T, we obtain (iii).<br />
><br />
><br />
"<br />
" " " ¨<br />
As an application of Proposition 7, we consider the linear matrix equation<br />
w<br />
T € _^ a aT<br />
b; Tb€ Z^ a aT<br />
bb€ C aTb œ! , T a> b œ R<br />
(35)<br />
! !<br />
where T P "ß_ aMß’ 8 b is a given matrix function and _ is defined in (22). Denote by [ aT<br />
b the unique<br />
! !<br />
T [aT ! b<br />
solution to (35). The following property of will be used in the proofs of Theorem 12 and<br />
Theorem 14.<br />
"<br />
Proposition8. Let T œ [aT<br />
b be the solution to (35).<br />
!<br />
(i) If (1) has a lower solution ^ with ea^b ž! , then T is an upper solution to (1) and T ^.<br />
(ii) If (1) has an upper solution ] with ea] b !, then T is a lower solution to (1) and TŸ] .<br />
Proof. (i) We first show that eaTb<br />
ž! . By Proposition 4 (i) with T œ ^ and Oœ^aT! b, we have<br />
w<br />
^ € _ a^aT bà ^b€ Za^aT bb€<br />
Ca^b<br />
w<br />
! !<br />
X<br />
œ ^ € LQa^b€<br />
c^a^b ^aT bd ea^bc^a^ b ^aT bd€ Ca^b<br />
!,<br />
! !<br />
where the inequality follows from that ea^ b ! and that ^ is a lower solution to (1). So ^ is also a<br />
lower solution to (35). Proposition 7 applied to (35) implies that T ^. In particular,<br />
eaTb ea^b<br />
ž! . By (24.1) with Oœ^aT! b and (35), we have<br />
w<br />
w<br />
Ÿ ! !<br />
T € LQaTb€ C aTb T € _^ a aT b; Tb€ Z^ a aT bb€ CaTb<br />
œ! .
12<br />
So T is an upper solution to (1). The proof of (ii) is similar; it also follows from Remark 3. ¨<br />
Now we prove a general comparison theorem <strong>for</strong> equation (1).<br />
Theorem 9. Denote XaT b ´ T w € LQ aTb€ CaTb. Suppose ], ^ P "ß_ aMß’ 8 b are feasible and satisfy<br />
Xa] b Ÿ Xa^ bß<br />
] a> " b ^> a " b, (36)<br />
and either ea^ b ! or ea] b Ÿ!. Then ] ^ . If one of (36) is strict, then ] ž ^.<br />
Proof. Let Tœ ] ^ . Then T satisfies T a> " b !. Suppose ea^ b !. Setting Oœ^a]<br />
b in (29), we<br />
have<br />
w<br />
! Xa] b Xa^ b œ T € LQa] b LQa^b€ CaTb<br />
w<br />
œT € _^ a a] bàT b€<br />
CaT b€<br />
a^a^b X<br />
^a] bb ea^ba^a^b<br />
^a]<br />
bb<br />
w<br />
T € _^ a a]<br />
bàT b€<br />
CaTbÞ<br />
(37)<br />
So aTß! b is a pair of upper-lower solutions to the equation T w € _^ a a] bàT b€CaTb<br />
œ! with<br />
T a> " b œ! . By Proposition 7 (iii), we have ] ^. If one of (36) is strict, then T will be a strict upper<br />
solution, which implies Tž! , or equivalently, ]ž^.<br />
If ea] b Ÿ! , then set Oœ ^a^ bin<br />
(29) to get<br />
w<br />
! Xa] b Xa^ b œ T € LQa] b LQa^b€ CaTb<br />
w<br />
œT € _^ a a^ bàT b€<br />
CaTb<br />
a^a^ b<br />
X<br />
^a] bb ea] ba^a^b<br />
^a]<br />
bb<br />
w<br />
T € _^ a a^bàT b€<br />
CaTbÞ<br />
The rest of the proof is the same as the case ea^ b !¨ .<br />
The conclusion that ] ^ implies that either ea] b ea^ b ! or ! ea] b ea^b.<br />
In<br />
other words, ] and ^ turn out to have the same definiteness. There<strong>for</strong>e, ea^ b ! and ea]<br />
b Ÿ!<br />
usually do not occur at the same time.<br />
Theorem 9 immediately extends to two equations like (1):<br />
w<br />
X 3 aT b ´T € LQ 3 aTb€ aTb œ! , T a> b œT , (38)<br />
C 3 " "3<br />
where the parameters EßFßGßHßK<br />
ßVßW , C and T satisfy (4) <strong>for</strong> 3œ"ß# , and T T .<br />
X<br />
Denote e 3 aT b ´ V 3 €HTH 3 , <strong>for</strong> 3œ"ß#Þ<br />
3 3 3 3 3 3 3 3 "3 "" "#<br />
3<br />
Theorem 10. Suppose ], ^ P "ß_ aMß ’ 8 b are feasible <strong>for</strong> both equations, ] is an upper solution to<br />
(38)" while ^ is a lower solution to (38)#. If one of the following conditions holds, then ] ^ in M.<br />
(i) X a] b X a] b and either e a^ b ! or e a]<br />
b Ÿ! .<br />
" # # #<br />
(ii) X a^b X a^ b and either e a^ b ! or e a]<br />
b Ÿ! .<br />
" # " "<br />
3
13<br />
Proof. The proof is almost trivial. The assumptions imply that X<br />
have that X<br />
a] b Ÿ!Ÿ X a^b; that is,<br />
# #<br />
a] b Ÿ!Ÿ X a^b. Then in case (i), we<br />
" #<br />
a]ß^b is a pair of upper-lower solutions to (38)#. The conclusion<br />
follows from Theorem 9 applied to X# . Similarly, in case (ii), we have that X" a] b Ÿ ! Ÿ X"<br />
a^b. So<br />
a]ß^ b is a pair of upper-lower solutions to (38)" and the conclusion follows from Theorem 9 applied to<br />
X " .¨<br />
Theorems 9 and 10 are very general comparison results. Comparison theorems have been proved<br />
(e.g., in [11], [12], [20] and [25]) <strong>for</strong> solutions to (1) with GœHœWœ! under the assumptions that<br />
" X<br />
E3 FV 3 3 F3<br />
L " L # and C" C# , where L3<br />
œ Œ<br />
X<br />
. These assumptions imply that<br />
K E<br />
X<br />
3 3<br />
aTb X aT b <strong>for</strong> all T’ 8 , which is stronger than the conditions (i) and (ii). In this special case, the<br />
" #<br />
conditions on e a] b and e a^b<br />
are not necessary; see [18].<br />
3 3<br />
Theorem.<br />
From Theorem 9 and the local theory of ODE we prove the following upper-lower solution<br />
Theorem 11. Suppose that a] , ^b<br />
is a pair of upper-lower solutions to (1).<br />
(i) If either ea^ b ! or ea] b Ÿ! , then ] ^ . In addition, if one of ] and ^ is strict, then ] ž ^.<br />
(ii) If either ea^ b ž! or ea] b !, then equation (1) has a unique solution T with ] T ^.<br />
Proof. Part (i) follows immediately from Theorem 9. For part (ii), we first consider the case Mœc>ß<br />
! > " d.<br />
The local existence theory of ODE implies that equation (1) has a solution that exists in a maximal<br />
interval Ð7ß> Ó§M . Part (i) implies that ] T ^ in Ð7ß><br />
Ó. By using the equation, we see that<br />
" "<br />
T ´ lim T ab > exists and ] a7b T ^a7b. In particular, eaT b ea^ a7bb<br />
ž! . If > ž 7 ž> ,<br />
7 7 7<br />
>Ä 7€<br />
then the local existence theory of ODE again shows that T can be extended further left beyond 7. This<br />
would contradict the definition of 7. There<strong>for</strong>e, 7 œ > ! and so T ab > exists in c>ß<br />
! > " d. For the case<br />
MœÐ _ß> Ó, the same argument shows that the solution T exists on c>ß > d <strong>for</strong> every > > . So the<br />
" ! " ! "<br />
solution exists on Ð _ß> " Ó.<br />
¨<br />
"<br />
!<br />
Theorem 11 (ii) implies that a necessary and sufficient condition <strong>for</strong> the existence of a solution T<br />
to (1) with either eaTb ž! or eaT b ! is the existence of a pair a] , ^b<br />
of upper-lower solutions to<br />
(1) satisfying either ea^ b ž! or ea]<br />
b !, respectively. Proposition 8 shows that, in fact, the existence<br />
of one of ] and ^ is sufficient <strong>for</strong> the existence of a solution. We have<br />
Theorem 12.<br />
(i) Equation (1) has a solution T with eaT b ž ! if and only if it has a lower solution ^ with ea^b<br />
ž !.
(ii) Equation (1) has a solution T with eaT b ! if and only if it has an upper solution ] with<br />
ea] b !.<br />
14<br />
Proof. The necessity in both cases is trivial. To show the sufficiency in both cases, use Proposition 8 to<br />
conclude that equation (1) has an upper solution ] (lower solution ^ ) with ea] b ž! ( ea^ b !). So<br />
equation (1) has a solution T with eaTb ž! ( !, respectively) by Theorem 11. ¨<br />
Theorem 12 is not true under the weaker assumptions ea^ b ! or ea] b Ÿ !. Consider the<br />
scalar case of equation (1) with Eœ "ß FœKœ R œ"ß GœHœVœWœ!ß C œ! ; that is,<br />
w<br />
T €" #T œ!ß T a> b œ" . Obviously, ! is a lower solution. It is feasible because ea! b œ fa! b œ! .<br />
"<br />
" "<br />
However, the solution T ab > œ<br />
#<br />
€<br />
#/ #> is not feasible because eaTb œ! but faTb<br />
œT.<br />
It is proved in<br />
X<br />
[8, Thm 4.1] that (1) has a solution T with eaT b ž ! if Wœ! , H Hž! and ! is a strict lower solution<br />
with ea! b œV ! (e.g., V !, K !, R ! and either K ž ! or R ž! ). Next theorem generalizes<br />
this result to equation (1) with an arbitrary lower solution. Our proof is based on Theorem 12.<br />
Theorem 13. Recall eaR b œ V a> b€H a> bR Ha> b. Suppose HHž!<br />
X X<br />
" " "<br />
.<br />
(i) If eaRb ž! and ^ is a strict lower solution to (1) with ea^ b !, then (1) has a solution T with<br />
eaT b ž !.<br />
(ii) If eaR b ! and ] is a strict upper solution to (1) with ea] b Ÿ !, then (1) has a solution T with<br />
eaT b !.<br />
Proof. By Remark 3, we show (i) only. We first assume Wœ! and ^ œ! . In this case the assumption<br />
implies that V ! and either K ! or R ! is strict. We will show that (1) has another lower solution<br />
^+ ž ! in Ò> ! ß> " Ñ with ea^+<br />
b ž ! in Ò> ! ß><br />
" Ó. Then by Theorem 12, (1) has a solution T with eaT b ž !.<br />
!a><br />
><br />
If R ž! , then let ^+ œ &/ " bI8, where I8<br />
is the 8‚8 unit matrix, & is the minimum<br />
!a> ><br />
eigenvalue of R and ! is an undetermined number. Using that e a^ b &/<br />
" b X<br />
HHž! , we have<br />
+ ´ +<br />
w !a> + +<br />
> " b<br />
X a^ b ^ € LQ a^ b€ Ca^ b &/ Q€K,<br />
X X X X X X X X<br />
where Qœ ! I 8 €E €E€G G aF €HGbaHHb aF €HG b€ CaI<br />
8 bÞ Taking a<br />
sufficiently large ! so that Qž! , we have Xa^+ b ž! as desired.<br />
"ß_ 8<br />
If K ž! , then let ^+ œ & ^ ! ab > , where & ž! and ^! P aMß’<br />
b such that ^ ! a> " b œ R and<br />
^ ab > ž! <strong>for</strong> >Ò> ß> Ñ.<br />
Using that e a^ b &ea^ b ž! in Ò> ß><br />
Ó, we have<br />
! ! " +<br />
! ! "<br />
w<br />
+ +<br />
+ +<br />
"<br />
+<br />
X a^ b ´ ^ € LQa^ b€ C a^ b œ & Q€K, (39)<br />
X X X<br />
where Qœ ^! €E ^! € ^! E€G ^! G fa^! b ea^! b fa^! b€<br />
Ca^!<br />
b Þ Since Kž! we can<br />
make X a^+ b ž! by taking a sufficiently small & ž! .<br />
"
If ^Á! or WÁ! , then consider \œT ^. Equation (1) <strong>for</strong> T is equivalent to (27) <strong>for</strong> \.<br />
Now (27) has a strict lower solution ! because ^ is a strict lower solution to (1). From what we just<br />
X<br />
proved, (27) has a solution \ such that ! ea^ b€H \ Hœ ea^ € \ b. There<strong>for</strong>e, Tœ ^ € \ is a<br />
solution to (1) with eaT<br />
b ž!.¨<br />
15<br />
Next we show that the solution to (1) can be approximated by solutions ] œ [a]<br />
b to (35).<br />
3 3 "<br />
Theorem 14. Suppose Mœ c>ß ! > " d and ^ is a lower solution to (1) such that ea^b<br />
ž! . Let<br />
"ß_ 8<br />
] P aMß’ b and ] œ [ a]<br />
b <strong>for</strong> 3 ". Then<br />
! 3 3 "<br />
(i) ]" ]# ] $ â ^.<br />
(ii) ] œ lim ] is a solution to (1) and there exist constants 5 and - such that<br />
3Ä_ 3 _ 4 #<br />
-<br />
4 #<br />
l ] 3ab > ] ab >lŸ5 " a><br />
" > b , >MÞ<br />
(40)<br />
a 4 #x b<br />
4œ3<br />
Proof. Proposition 8 implies that each ] 3 is an upper solution and ] 3 ^ <strong>for</strong> 3 ". To show other<br />
conclusions, we first derive an equation satisfied by ? 3 œ ] 3 ] 3€" . By the definitions of ] 3 and ] 3€" ,<br />
they satisfy<br />
] € _^ a a] bà] b€ Z^ a a] bb€ C a]<br />
b œ!Þ<br />
(41)<br />
3 w 3 " 3 3 " 3<br />
w<br />
3€" 3 3€" 3 3€"<br />
] € _^ a a] bà] b€ Z^ a a] bb€ C a]<br />
b œ!Þ<br />
(42)<br />
By Proposition 4 (i) with Tœ ] and Oœ^a]<br />
b, we have<br />
3 3 "<br />
X<br />
_^ a a] bà] b€ Z^ a a] bb œ a] b€ a^a] b ^a] bb ea] ba^a] b ^a]<br />
bbÞ<br />
3 " 3 3 " LQ 3 3 3 " 3 3 3 "<br />
However, LQa] b œ _^ a a] bà] b€ Z^ a a]<br />
bb<br />
by (24.3) Þ Thus (41) becomes<br />
3 3 3 3<br />
w<br />
3 3 3 3 3 3 3 " 3 3 3 "<br />
X<br />
] € _^ a a] bà] b€ Z^ a a] bb€ Ca] b€ a^a] b ^a] bb ea] ba^a] b ^a]<br />
bb<br />
œ! .(43)<br />
Subtracting (42) from (43) we obtain<br />
w<br />
3 3 3 3 3 3 " 3 3 3 "<br />
X<br />
? € _^ a a] bà ? b€ Ca?<br />
b€<br />
a^a] b ^a] bb ea] ba^a] b ^a]<br />
bb<br />
œ!Þ<br />
(44)<br />
X<br />
Note that a> b œ! and a^a] b ^a] bb ea] ba^a] b ^a] bb<br />
! because ea] b !Þ By<br />
? 3 " 3 3 " 3 3 3 " 3<br />
Theorem 9, ? 3 !. This finishes the proof of (i).<br />
Now (i) implies that e] 3fß eea] 3bf and e^a]<br />
3bf<br />
are all uni<strong>for</strong>mly bounded in M. It follows that<br />
X<br />
la^a] b ^a] bb ea] ba^a] b ^a] bblŸ5l? lß l_^ a a]<br />
b à ? b€ C a?<br />
blŸ5l?<br />
l,<br />
3 3 " 3 3 3 " 3 " 3 3 3<br />
3
16<br />
<strong>for</strong> some constant<br />
5ž! . Thus from (44) we obtain that<br />
><br />
l ? 3ab >lŸ5( al ? 3ab =l€l ? 3 " ab =l.= b . (45)<br />
><br />
"<br />
The rest of the proof follows exactly the argument in [27, p.324]. We repeat it here <strong>for</strong> reader's<br />
><br />
convenience. Denote @ ab > œ ' " l ? ab =l.= . Then (45) reduces t<br />
which implies that<br />
3 > 3<br />
w<br />
3<br />
@ ab > €5@ ab > €5@ ab > !ß<br />
5><br />
3 3 "<br />
> ><br />
" "<br />
"<br />
@ 3ab > Ÿ5/ ( @ 3 " ab =.= œ- ( @ 3 " ab =.=ß<br />
> ><br />
where -œ5/<br />
Þ By induction, we deduce that<br />
5> "<br />
3<br />
-<br />
3<br />
@ 3€" ab > Ÿ a> " > b @" a> ! bÞ<br />
3x<br />
It follows then from (45) that<br />
- 3 " -<br />
l ? 3 ab >lŸ5 œ a> " > b € a> " a 3 "x b a 3 #x b<br />
3<br />
> b<br />
#<br />
@ " a> ! bÞ<br />
This yields (40). ¨<br />
By Remark 3, we also have an approximation theorem <strong>for</strong> the solution ^ with ea^ b !.<br />
"ß_ 8<br />
Theorem 15. Suppose ] is an upper solution to (1) with ea] b !. Let ^!<br />
P aMß’<br />
b and<br />
^<br />
œ [a^<br />
b <strong>for</strong> 3 ". Then<br />
3 3 "<br />
(i) ^ Ÿ ^ Ÿ ^ ŸâŸ]<br />
.<br />
" # $<br />
(ii) ^ œ lim ^ is a solution to (1) and there exist constants 5 and - such that<br />
3Ä_ 3 _ 4 #<br />
-<br />
4 #<br />
l ^3 ab > ^ ab >lŸ5 " a> " > b , >Mœ c>ß ! > " dÞ<br />
a 4 #x b<br />
4œ3<br />
§ 4. Estimates of Maximal Intervals of Existence<br />
Suppose ] ( ^Ñ is an upper (lower) solution to the equation<br />
w<br />
XaT b ´ T € LQ aT b€ CaTb œ!ßTa> b œ R<br />
(46)<br />
on an interval M . If ea] b ! or ea^b<br />
ž! , then Theorem 11 implies the existence of a solution T in M<br />
with eaT b ! or eaTb ž! , respectively. If ea] b ž! or ea^b<br />
!, then it is well-know that the<br />
"
solution may blow up in M. This happens, <strong>for</strong> example, to Riccati equations from differential games [2]. In<br />
this case, it would be of interest to estimate the maximal existence interval Ð7 ß> " Ó of the solution to (46).<br />
17<br />
By Theorem 11, the solution to (1) exists in the intersection of the existence intervals of a pair of<br />
upper-lower solutions. So the maximal existence interval of the solution to (1) can be estimated by<br />
constructing their upper and lower solutions. Theoretically, such estimates can be made as accurate as<br />
possible if we can find upper and lower solutions that are close enough to the solution. <strong>Upper</strong> and lower<br />
solutions can be constructed by using the solutions of comparison equations like those in Theorem 10. In<br />
Theorems 16, 17 and Proposition 18 below, we demonstrate how to construct scalar autonomous<br />
differential equations in terms of ] and ^ to obtain explicit estimate <strong>for</strong> Ð7 ß><br />
" Ó.<br />
By (27) and Proposition<br />
6 we need to consider only the standard cases with Wœ! , and ] œ! or ^ œ! .<br />
Denote by -a† b, Aa† b and 5a†<br />
b the minimum eigenvalue, maximum eigenvalue and maximum<br />
singular value of a matrix, respectively. Suppose Wœ! , Vž! and ] œ! is an upper solution. Let<br />
+ß,ß-ß.ߟ > " ß<br />
œ<br />
:> a b œ : " ,<br />
"<br />
(48)<br />
#<br />
+ :<br />
.:<br />
€< " "<br />
where 2: a b œ €, : €- . Note that + !, . !ß -Ÿ !, : Ÿ! . Suppose .: € " Ó. Since 2 a! b œ-Ÿ!, ! is an upper solution to<br />
(48), which implies that :> abŸ! in Ð5ß> Ó by Theorem 11. Furthermore, .: ab > €Ð5ß><br />
Ó.<br />
" "<br />
Theorem 16. Suppose ! is an upper solution to (46) with V ž! such that .:<br />
€ " Ó with .:<br />
€ " Ó with eaTb<br />
ž! .<br />
"<br />
Proof. Let ^ œ :I . We only need to show that ^ is a lower solution in Ð5ß> Ó to (46) with ea^b<br />
ž! .<br />
Note that<br />
8 "<br />
^> a " b œ :I " 8 Ÿ Rß<br />
X<br />
ea^ b œ V € : HH a
18<br />
have<br />
w<br />
Xa^b œ ^ € LQa^b€<br />
Ca^b<br />
w<br />
X<br />
X<br />
œ :I € : aE<br />
€E€ G G € CaI bb€<br />
K<br />
8 8<br />
# X<br />
"<br />
X X X X X<br />
: aF €H GbaV € : HHb aF €H Gb<br />
#<br />
w<br />
+ :<br />
w<br />
”: €, : €- • I 8 œ c: € 2: a bdI8<br />
œ!Þ<br />
<br />
" Ó.<br />
¨<br />
such that<br />
Similarly, suppose Wœ! , V ! and ! is a lower solution. Let +ß,ß-ß.ß<br />
" Ó.<br />
Since<br />
" "<br />
2 a! b œ- !, ! is a lower solution to (48), which implies that : ab > ! in Ð5 ß><br />
" Ó by Theorem 11 and<br />
.: ab > €< ! <strong>for</strong> >Ð5 ß><br />
" Ó.<br />
Theorem 17. Suppose ! is a lower solution to (46) with V ž!. If (48) with coefficients in (50) has a<br />
solution : in an interval Ð5ß> Ó with .: €< !, then (46) has a solution T in Ð5ß><br />
Ó with eaT b !.<br />
" "<br />
Proof. The proof is similar to that of Theorem 16. One only needs to show that :I 8 is an upper solution.<br />
¨<br />
We now give an integral representation <strong>for</strong> the maximal existence interval Ð5 ß>Ó " of the solution<br />
:> ab to (48) <strong>for</strong> a general rational function 2: a b. Assume <strong>for</strong> some 5 !2: , a b has 5€# distinct zeros<br />
and poles _œD! D" â D5 D5€"<br />
œ_ , including „_.<br />
Proposition 18. Suppose : aDßD b <strong>for</strong> some 3œ!ß#ßâß5 . Then the solution :> abof (48) exists on<br />
" 3 3€"<br />
a maximal interval Ð5 ß>ӧР" _ß>Ó " with<br />
> 5 œ (<br />
"<br />
:<br />
:<br />
"<br />
"‡<br />
.:<br />
,<br />
2: a b<br />
where : œ :> ab; that is, : œD if 2: a b ž! and : œD if 2: a b !.<br />
"‡ lim "‡ 3€" " "‡ 3 "<br />
>Ä5<br />
Proof. If 2: a b ž! , then the solution :> abis strictly increasing at > decreases in Ð5ß>Ówith range<br />
" "<br />
" 3€" lim ab .:<br />
3€" " 3€"<br />
>Ä 5 2: a b<br />
Ò: ßD Ñ and :> œD . Write (48) as €.> œ! . Integrating this equation from : to D <strong>for</strong>
19<br />
: and from > " to 5 <strong>for</strong> > , we get<br />
(<br />
:<br />
D<br />
"<br />
3€"<br />
.:<br />
"<br />
2: a b<br />
€ 5 > œ!Þ<br />
If 2: a b !, then the solution :> abis strictly decreasing at > decreases in Ð ß>Ówith range ÐD ß:Ówith<br />
" 5 "<br />
3 "<br />
.:<br />
3 2: a b " 3 " 5<br />
lim :> ab œD . Integrating €.> œ! from : to D <strong>for</strong> : and from > to <strong>for</strong> > , we obtain<br />
>Ä5 The Proposition is proved.¨<br />
(<br />
:<br />
D<br />
"<br />
3<br />
.:<br />
"<br />
2: a b<br />
€ 5 > œ!Þ<br />
Example. We take a simple example from [27, Ch. 6, Example 7.8] to illustrate an application of<br />
Proposition 17. Consider a scalar case of problem (3) with EœGœKœWœ!ß<br />
FœHœ R œ> " œ" and Vœ < with decreases to < as<br />
"<br />
>Ä 5 . By Proposition 18, 5 œ " '<br />
: <<br />
< :<br />
.: œ#
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