1 General Algebraic and Differential Riccati ... - Bradley Bradley
1 General Algebraic and Differential Riccati ... - Bradley Bradley
1 General Algebraic and Differential Riccati ... - Bradley Bradley
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1<br />
<strong>General</strong> <strong>Algebraic</strong> <strong>and</strong> <strong>Differential</strong><br />
<strong>Riccati</strong> Equations from Stochastic LQR Problem<br />
Libin Mou<br />
Department of Mathematics<br />
<strong>Bradley</strong> University<br />
Peoria, IL 61625<br />
Abstract. In this paper we consider a class of matrix <strong>Riccati</strong> equations arising from stochastic<br />
LQR problems. We prove a monotonicity of solutions to the differential <strong>Riccati</strong> equations, which<br />
leads to a necessary <strong>and</strong> sufficient condition for the existence of solutions to the algebraic <strong>Riccati</strong><br />
equations. In addition, we obtain results on comparison, uniqueness, stabilizability <strong>and</strong> approximation<br />
for solutions of the algebraic <strong>Riccati</strong> equations.<br />
§ 1. Introduction<br />
The following differential <strong>Riccati</strong> equation has been studied in [11] by using the method of<br />
upper <strong>and</strong> lower solutions.<br />
Ú w X<br />
X<br />
Ý T €E T€TE€G TG€ K € CaTb<br />
X X X<br />
X " X X<br />
Û aF T€HTG€WbaV€HTHb aF T€HTG€Wb<br />
œ!ß<br />
Ý<br />
Ü T a><br />
b œR,<br />
"<br />
(1)<br />
where E is the transpose of E, T œ<br />
.><br />
, R is a symmetric matrix, C is a linear map of symmetric<br />
matrices, <strong>and</strong> EßFßGßHß K ßV <strong>and</strong> W are bounded <strong>and</strong> measurable matrix functions with appropriate<br />
dimensions. The main results in [11] include an interpretation of upper <strong>and</strong> lower solutions,<br />
comparison theorems, an upper-lower solution theorem, necessary <strong>and</strong> sufficient conditions for<br />
existence of solutions, an estimation of maximal existence intervals of solutions <strong>and</strong> an approximation<br />
of solutions.<br />
X w .T<br />
(1):<br />
This paper is a continuation of<br />
X<br />
X<br />
E T€TE€G TG€ K € CaTb<br />
[11]. The focus here is the algebraic equation associated with<br />
X<br />
X X X X X<br />
aF T€HTG€WbaV€HTHb aF T€HTG€Wb<br />
œ! ,<br />
"<br />
(2)<br />
where EßFßGßHßK ßV <strong>and</strong> W are constant matrices with appropriate dimensions; see (5) below.<br />
The inclusion of the term C is important to stochastic control problems with Markovian<br />
jumping noises <strong>and</strong> differential game problems with state-dependent noises; see [17] <strong>and</strong> [10], for<br />
example.<br />
Equation (2) with C œ! becomes<br />
X<br />
X<br />
E T€TE€G TG€ K<br />
X<br />
X X X X X<br />
aF T€HTG€WbaV€HTHb aF T€HTG€Wb<br />
œ! .<br />
"<br />
(3)<br />
Equation (3) arises from the following stochastic<br />
linear quadratic regulator ( LQR) problem of infinite-
horizon with control-dependent noises:<br />
Ú minimize/maximize Na b for hÒ!ß_Ñ, where<br />
_<br />
Û a b e' X X X<br />
N œI<br />
!<br />
aB KB€#B W€V b.><br />
f subject to (4.1)<br />
Ü .B œ aEB€F b.>€ aGB€H b.[ß> 0 àBa!<br />
b œD,<br />
(4.2)<br />
2<br />
(4)<br />
where [ ab > is a st<strong>and</strong>ard Brownian motion on a complete probability space with [ a!<br />
b œ! almost<br />
surely, Ief is the expectation of the enclosed variable, <strong>and</strong> hÒ!ß_Ñ is the set of all admissible control<br />
processes ; see Section 2 for a further description. For a detailed account of this problem, see [1]<br />
<strong>and</strong> [22]. Through approaches of convex optimization <strong>and</strong> semidefinite programming, interesting<br />
relationships between equation (3) <strong>and</strong> the LQR problem (4) have established in [1] <strong>and</strong> [22]. These<br />
approaches are especially attractive for purpose of approximating solutions.<br />
A monotonicity property for solutions to equation (1) will be proved, which leads to necessary<br />
<strong>and</strong> sufficient conditions for the existence of solutions to (2). For solutions to equation (2), we will<br />
prove results on comparison, uniqueness, stabilizability <strong>and</strong> approximation. We use a method of upper<br />
<strong>and</strong> lower solutions, which satisfy associated <strong>Riccati</strong> inequalities. This method is closely related to the<br />
methods of linear matrix inequalities in [1] [18] [22] <strong>and</strong> semidefinite programming in [22]. As<br />
pointed out in [11], our method has several desirable merits. For example, it directly links equation<br />
(3) with the LQR problem (4); see Theorem 2. It derives the main results under general assumptions.<br />
It gives verifiable necessary <strong>and</strong> sufficient conditions for the existence of solutions [Theorems 8 <strong>and</strong><br />
9]. It also gives algorithms for approximating solutions. It can be used to estimate the maximal<br />
existence intervals of solutions to differential <strong>Riccati</strong> equations without solving the equations [11]. It<br />
applies to differential as well as algebraic <strong>Riccati</strong> equations.<br />
The paper is organized as follows. In Section 2, we set up the notations used in this paper <strong>and</strong><br />
define upper <strong>and</strong> lower solutions. In addition, we describe the LQR problem (4) <strong>and</strong> interpret upper<br />
<strong>and</strong> lower solutions to (2) in Theorem 2.<br />
Section 2 is ended with some results from [11] that are needed in this paper. Specifically,<br />
Theorem 5 is an upper-lower solution theorem <strong>and</strong> Propositions 3 <strong>and</strong> 4 are some structural<br />
properties of equations (1), (2) <strong>and</strong> (3).<br />
In Section 3, we prove a monotonicity [Theorem 7] for solutions to (1), which leads to a<br />
general necessary <strong>and</strong> sufficient condition [Theorem 8] for the existence of solutions to (2). A refined<br />
existence result [Theorem 9] for (2) is proved under ms-stabilizability. Theorem 9 generalizes a main<br />
result in [1, Theorem 10] for (2) with C œWœ! . The definitions of ms-stability, ms-stabilizability<br />
<strong>and</strong> ms-detectability for equation (2) are given in this section.<br />
In Section 4, we study the comparison, uniqueness, stabilizability <strong>and</strong> approximation of<br />
solutions to (2). Theorem 11 is a comparison theorem for ms-stabilizing solutions. Theorem 12 gives<br />
the uniqueness <strong>and</strong> extreme properties of stabilizing solutions. <strong>General</strong>izing a classical relationship<br />
between detectability <strong>and</strong> stability, Theorem 14 shows that ms-detectability implies the msstabilizability<br />
of solutions to (2). Theorem 16 gives an algorithm for approximating solutions to (2).<br />
The results here generalize known results for equation (2) with GœHœWœ! obtained in [5], [6]<br />
<strong>and</strong> [21]. For results for equation (2) with GœHœWœ C œ! , see [2], [3], [4], [8], [9], [14],<br />
[15], [16], [18], [19], [20] <strong>and</strong> [24]. The proofs are reduced to proving the same results for linear<br />
equations in Theorems 10 <strong>and</strong> 13, which may be of interest in their own right.<br />
§ 2. Preliminary Results
3<br />
Denote ’ 8 ‘<br />
8‚8 X<br />
X<br />
œ eT ÀT œTf, where T is the transpose of T. We write Q " Q#<br />
8 8 8<br />
( Q" žQ# ) if Q" Q#<br />
’ is a positive semidefinite (definite). For a map C À ’ Ä ’ we write<br />
C ! if CaQ b ! for each Q .<br />
" " !<br />
Assumption. We assume that EßFßGßHßKßVßW <strong>and</strong> R in equations (1), (2) <strong>and</strong> (3) are constant<br />
matrices <strong>and</strong> C À ’ 8 Ä ’<br />
8 is a linear map, which satisfy<br />
8‚8 8‚5 8<br />
X<br />
EßG ‘ àFßHßW ‘ àV ’ 5 ; K, R ’ ; C 0 Þ<br />
(5)<br />
For a Hilbert space — <strong>and</strong> an interval M, P<br />
_ aMß—<br />
b is the space of all bounded <strong>and</strong> measurable<br />
"ß_ _ w _<br />
functions from M to —. Furthermore, we define P aMß— b œÖT P aMß— bßT P aMß — b×Þ<br />
The<br />
solution T to (1) is assumed to be in P "ß_ aMß’ 8 b. Since all matrices in (1) are constant, a solution T<br />
on M is actually smooth. The solution T to (2) <strong>and</strong> (3) is assumed to be in ’ 8 , which may be<br />
considered as a constant solution to the associated differential equation.<br />
As in [11], we abbreviate (1), (2) <strong>and</strong> (3) as<br />
Ú T w € LQaT b€ CaTb œ!ß T a> " b œ Rß (1)<br />
Û LQaT b€ CaTb<br />
œ!ß<br />
(2)<br />
Ü LQaTb<br />
œ!ß<br />
(3)<br />
where LQaTb œK€ L aTb<br />
QaT<br />
b <strong>and</strong><br />
X<br />
X<br />
L aTb<br />
œE T€TE€G TG,<br />
œ<br />
Q a b<br />
X X<br />
aF T€HTG€Wba X<br />
X " X X<br />
T œ<br />
V€HTHb aF T€HTG€Wb<br />
(6)<br />
X<br />
We remark that Q aTb <strong>and</strong> LQ aTb<br />
make sense even if V€HTH is singular. To see this, we<br />
introduce the following notations<br />
X X X<br />
eaTb œV€HTH, faTb<br />
œ F T€HTG€W. (7)<br />
"<br />
e f e<br />
^a b œ a T b a T b if a T b is nonsingular,<br />
T œ<br />
€<br />
eaTb faTb if eaTb<br />
is singular,<br />
(8)<br />
€<br />
where eaTb is the pseudoinverse of eaTb. Recall that any matrix Q has a unique Moore-Penrose<br />
pseudoinverse Q € with the following properties (see [13] <strong>and</strong> [1]).<br />
€ € € €<br />
QQ QœQßQ QQ œQ . (9)<br />
8 € 8 € €<br />
If Q ’ , then Q ’ , <strong>and</strong> QQ œQ Q.<br />
Q<br />
€<br />
! if <strong>and</strong> only if Q !.<br />
€<br />
Although ^aTb œ eaTb faTb<br />
always exists, but it may not be bounded nor satisfy<br />
faTb œ eaTb^aT b<br />
(10)<br />
on the interval M. We recall some definitions in [11]. A function TP "ß_ aMß’ 8 b is said to be feasible<br />
if ^ aTb P _ aMß‘<br />
5‚8 b <strong>and</strong> (10) holds . Afunction O P _ aMß‘<br />
5‚8 b is called a feedback matrix<br />
associated with TP<br />
"ß_ aMß’ 8 w<br />
b if it satisfies faTb œ eaTbO.<br />
The set of all such O= is denoted by
4<br />
_ 5‚8<br />
ŠaTb. Suppose ^aTb<br />
P aMß‘ b, then T is feasible if <strong>and</strong> only if ŠaTb Ág, <strong>and</strong> in this case,<br />
X<br />
X<br />
Q aTb<br />
œ ^aTb eaTb^aTb œO eaTbO,<br />
(11)<br />
for each OŠaTb; see [11] for proof. In other words, Q aTb<br />
is well-defined by (11) whenever T is<br />
feasible.<br />
If T ’ 8 , then T is feasible as long as (10) holds. The term "feasible" is consist with that<br />
defined in [22] for the associated semidefinite programming problem.<br />
Definition 1. T P<br />
"ß_ a<br />
Mß’ 8 bis a solution (upper solution, lower solution) to (1) if<br />
LQaTb€ L Q aT b€ CaTb<br />
œ! ( Ÿ! , !ßT ) a> b œR ( R,<br />
ŸRÑÞ<br />
An upper or lower solution is strict if one of the inequalities is strict. Similarly, T’ 8 is a solution<br />
(upper, lower solution) to (2) if L QaT b€ CaTb<br />
œ! ( Ÿ! , !, respectively).<br />
Upper <strong>and</strong> lower solutions can be interpreted in terms of the well-definedness of the LQR<br />
Problem (4). Let P<br />
# 5 (same as P<br />
# 8 <br />
a‘ b<br />
Ya‘<br />
b in [1]) be space of ‘ 5 -valued processes on Ò!ß_Ñ that<br />
are adapted to the 5 -field generated by [ ab > such that I' _ #<br />
!<br />
ll .> _(square-integrable). Each<br />
P<br />
# a‘ 5 b is an open-loop control. We say that P<br />
# a‘<br />
5 b is ms-stabilizing if the solution B to<br />
equation (4.2) satisfies IelBab<br />
> l # f Ä! as >Ä_ . The state equation (4.2) is stabilizable if there<br />
exists a feedback matrix O ‘ 5‚8 such that œ OB is stabilizing, where B is the solution to<br />
.B œ aE FObB.>€ aG HO bB.[ßBa! b œ D, (12)<br />
which is induced from (4.2) by œ OB. In this case, O is called a stabilizing feedback matrix. A<br />
8<br />
solution T ’ to (3) is said to be stabilizing if œ ^aTbB<br />
is stabilizing.<br />
These concepts will be generalized for equation (2) without referring to the state equation<br />
(4.2).<br />
Let h Ò!ß_Ñ be the set of all P # a‘<br />
5 b such that the solution B to equation (4.2) is P<br />
# -<br />
_<br />
integrable; that is, Ie'<br />
#<br />
!<br />
lB ab >l.> f _. Clearly N a b is well-defined for each hÒ!ß_Ñ.<br />
Furthermore, the following relationship holds.<br />
"<br />
Proposition 1. Each hÒ!ß_Ñ is stabilizing. Conversely, if O‘ 5‚8 such that œ OB<br />
stabilizing, then hÒ!ß_Ñ.<br />
is<br />
_<br />
Proof. If hÒ!ß_Ñ, then the solution B to (4.2) satisfies Ie'<br />
#<br />
!<br />
lB ab >l.> f _, which implies<br />
#<br />
that IelBab<br />
> l f Ä! as > Ä_ . In other words, is stabilizing. Conversely, if œ OB is<br />
stabilizing, then by [1, Thm 1] , there exists T ’ 8 , Tž! , such that _aOàTb<br />
!, where<br />
_aOàTb<br />
œ aE FOb T€TaE FO b€ aG X<br />
HOb TaG HOb, (13)<br />
which is L aTb<br />
with E <strong>and</strong> G replaced by E FO <strong>and</strong> G HO, respectively. By Ito's lemma <strong>and</strong><br />
the fundamental theorem of calculus, we have<br />
"<br />
X X<br />
.<br />
X<br />
IeB a> " bTB> a " b f œD TD€I(<br />
B ab >TB>.> ab<br />
.><br />
><br />
!<br />
><br />
X<br />
X<br />
œD TD€I( B _ aOàTbB.><br />
.<br />
!<br />
"<br />
(14)
5<br />
#<br />
From (14) <strong>and</strong> the fact that IelBa> " bl f Ä! as > " Ä_ , one has that<br />
I' _ X<br />
X<br />
_<br />
!<br />
B _ aOàTbB.> œ D TD. Since _ aOàTb<br />
!, it follows that Ie'<br />
#<br />
!<br />
lB ab >l.> f _. So<br />
h Ò!ß_Ñ. ¨<br />
Theorem 2. Suppose T’ 8 is feasible <strong>and</strong> stabilizing.<br />
(i) If LQ aT b ! <strong>and</strong> eaTb<br />
!, then Na b D X TD for hÒ!ß_Ñ.<br />
(ii) If LQ aTb<br />
Ÿ! <strong>and</strong> eaT b Ÿ !, then Na b Ÿ D X TD for hÒ!ß_Ñ.<br />
X<br />
(iii) If LQ aTb<br />
œ! <strong>and</strong> eaTb !(or<br />
eaT b Ÿ !), then D TD is the minimum (maximum,<br />
respectively) value of Na b over hÒ!ß_Ñ , which occurs when œ OB, where O ŠaTb<strong>and</strong><br />
B<br />
satisfies (12).<br />
Proof. Suppose h Ò!ß_Ñ <strong>and</strong> B is the solution to equation (4.2).<br />
By the Fundamental Theorem of<br />
calculus <strong>and</strong> Ito's formula, we have<br />
"<br />
X X<br />
.<br />
X<br />
IeB a> " bTB> a " b f œD TD€I(<br />
B ab >TB>.> ab<br />
.><br />
><br />
!<br />
><br />
X X X X X X X<br />
œD TD€I( eB L aTbB€# aF T€HTGB€HTH b<br />
f.>ß<br />
!<br />
"<br />
X X<br />
where L aTb<br />
œE T€TE€G TG as defined in (6). Taking limits in (15), we obtain<br />
_<br />
X X X X X X X<br />
!œD TD€I( eB L aTbB€# aF T€HTGB€HTH b<br />
f.><br />
. (16)<br />
!<br />
Adding (16) to Na b<strong>and</strong> using the notations eaTb <strong>and</strong> faTb<br />
in (7), we obtain<br />
_<br />
X X X X<br />
Na<br />
b œD TD€I( eB aL aT b€KB€# b faTbB€ eaT b f.><br />
. (17)<br />
!<br />
Since faTb œ eaTbO for each O ŠaTb, we can write<br />
# X faTbB€ X eaT bœ a€OBb X<br />
eaT ba€OBb O X eaTb O.<br />
Recall that LQaTb œK€ L aTb O X<br />
eaTbO. It follows that<br />
_<br />
X X X<br />
Na<br />
b œD TD€I( ˜B LQaTbB€ a€OBb e aT ba€OB b.>Þ<br />
(18)<br />
!<br />
X<br />
In case (i), we have LQaTb 0 <strong>and</strong> e aT b !, so (18) implies that Na<br />
b D TD for every<br />
X<br />
hÒ!ß_Ñ. Similarly, in case (ii), (18) implies that N a b Ÿ D TD for every hÒ!ß_Ñ. In case<br />
(iii), (18) implies that for every hÒ!ß_Ñ,<br />
_<br />
X<br />
X<br />
N a b œD TD€I ( ˜a€OBb eaT ba€OB b.>Þ<br />
!<br />
X<br />
It follows that N a b has a minimum (maximum) D TD at œ OB if eaT b !Ðif eaTb<br />
Ÿ!Ñ.<br />
Equation (12) is precisely the state equation (4.2) with œ OB.<br />
¨<br />
Note that when œ OBß the cost N a b becomes N œ ' B X<br />
ZaOB.><br />
b , where<br />
O<br />
_<br />
!<br />
(15)<br />
X X X<br />
ZaOb<br />
œ O VO O W W O€K. (19)
6<br />
The following two propositions <strong>and</strong> Theorem 5 are proved in [11].<br />
Proposition 3. Suppose TP "ß_ 8 _ 5‚8<br />
aMß’ b is feasible <strong>and</strong> O P aMß‘<br />
b. Let LQ aTb,<br />
_ aOàTb<br />
<strong>and</strong> ZaOb<br />
be defined in (6), (13) <strong>and</strong> (19), respectively. Then<br />
(i) LQ aT b€ a^aTb Ob X e aTba^aTb<br />
Ob<br />
œ Z aO b€ _ aOàTb,<br />
(20)<br />
(ii)<br />
Ú LQaTb Ÿ Z aO b€ _ aOàTb, if eaT b ! (21.1)<br />
Û LQaT b Z aO b€ _ aOàTb, if eaTb<br />
Ÿ! (21.2)<br />
Ü LQaTb œ Z a^aTbb€ _ a^aTbàTb. (21.3)<br />
(21)<br />
Proposition 4. Suppose ] ß^ P "ß_ aMß’ 8 b are feasible, O P _ aMß‘<br />
5‚8 b. Denote Tœ] ^,<br />
EœE s F^a^bßGœG s H^a^b <strong>and</strong> Vœ s ea^b. Then<br />
(i) LQ a] b LQa^b<br />
(22)<br />
X X X<br />
œEs T€TE€G s s TGs<br />
X X X €<br />
ˆ ‰ˆ ‰ ˆ<br />
X X<br />
F T€HTGs V€HTH s F T€HTGs‰<br />
(ii) If ^ is given, then ] satisfies equation (1) if <strong>and</strong> only if T œ] ^ satisfies<br />
Ú<br />
X<br />
X<br />
Ý w<br />
T € K€ s Es T€TE€G s s TG€ s CaTb<br />
X<br />
€<br />
Û ˆ X X X X X<br />
Ý<br />
F T€HTGs‰ˆ V€HTH s ‰ ˆ F T€HTGs‰<br />
œ!ß<br />
Ü T a> b œ R ^> a b,<br />
" "<br />
(23)<br />
where Ks œ^ w € LQa^b€ Ca^bÞ<br />
Note that ^ is a lower solution to (1) if <strong>and</strong> only if K s ! <strong>and</strong> R ^> a " b !, which are<br />
equivalent to that ! is a lower solution to (23). Similarly, ^ is an upper solution to (1) if <strong>and</strong> only if !<br />
an upper solution to (23). In other words, equation (1) has an upper or lower solutions is equivalent<br />
to that (1) can be translated to a st<strong>and</strong>ard problem that has ! as an upper or lower solution.<br />
Also note that Propositions 3 <strong>and</strong> 4 hold, in particular, for Tß]ß^ ’ 8 <strong>and</strong> O‘ 5‚8 Þ<br />
Theorem 5 (upper-lower solution theorem). Suppose that a] , ^b<br />
is a pair of upper-lower solutions<br />
to (1).<br />
(i) If either ea^ b ! or ea] b Ÿ! , then ] ^ . In addition, if one of ] <strong>and</strong> ^ is strict, then<br />
] ž ^.<br />
(ii) If either ea^ b ž! or ea] b !, then equation (1) has a unique solution T with ] T ^.<br />
Remark 6. As noted in [11], if a lower solution ^ (if it exists) to (1) with ea^b ž! has certain<br />
property, then an upper solution ] (if it exists) to (1) with ea]<br />
b ! also has the corresponding<br />
property. Same is true for equations (2) <strong>and</strong> (3). Because of this, we will state properties of upper<br />
solutions without proof.<br />
§3. Monotonicity of Solutions to (1) <strong>and</strong> Existence of Solutions to (2)<br />
Now we consider equations (1) <strong>and</strong> (2), which are
w<br />
XaT b ´ T € LQaT b€ CaTb<br />
œ!ß Ta> " b œ R, (1)<br />
XaT b ´ LQaT b€ C aTb<br />
œ! . (2)<br />
By the local theory of differential equations, equation (1) has a unique solution in a maximum interval<br />
Ð> ! ß> " Ó. We show that this solution is monotone if <strong>and</strong> only if R is a lower or upper solution to (2);<br />
that is, XaR b ! or XaRb<br />
Ÿ! .<br />
Theorem 7 (Monotonicity of Solutions) . Suppose T is the feasible solution of (1) in Ð> ! ß><br />
" Ó. Then we<br />
have<br />
(i) XaR b ! if <strong>and</strong> only if T is increasing in Ð> ! ß><br />
" Ó as > decreases.<br />
(ii) XaR b Ÿ ! if <strong>and</strong> only if T is decreasing in Ð> ! ß><br />
" Ó as > decreases.<br />
Proof. (i) If XaRb !, then R is a lower solution to (1). By Theorem 5, T ab > R for all<br />
8<br />
>Ð> ! ß> " ÓÞ For any number 7 a!ß> " > ! b, define T‡ ÀÐ> ! € 7ß> " € 7ÓÄ ’ by T ‡ ab > œT a><br />
7b.<br />
Since (1) is time-invariant, T ‡a> b is a solution to ( 1 ) with T* a> " b œT a> " 7b R œTa><br />
" b. By<br />
Theorem 5 again, T ‡a> b T ab > for >Ð> ! € 7ß><br />
" Ó, or equivalently, T a> 7b T ab > for every<br />
7 a!ß> " > ! b. In other words, T ab > is increasing in Ð> ! ß><br />
" Ó as > decreases. Part (ii) is proved<br />
similarly by using the fact that R is an upper solution.<br />
¨<br />
Theorem 7 implies that if T is a bounded solution to (1) on Ð _ß>Ó " with R being an upper<br />
or lower solution to (2), then T ´ lim T ab > exists <strong>and</strong> T is a solution to (2). As a result, we have<br />
_<br />
>Ä _<br />
the following necessary <strong>and</strong> sufficient existence condition for solutions to (2).<br />
Theorem 8. Equation (2) has a solution T ’ 8 with e a T b ž! a eaT b ! b if <strong>and</strong> only if it has an<br />
upper solution ] <strong>and</strong> a lower solution ^ such that ] ^ with ea^b<br />
ž!<br />
ae a] b !, respectively b.<br />
Proof. The necessity is obvious by choosing ] œ ^ œT. For the sufficiency, consider equation (1)<br />
with Rœ] <strong>and</strong> ^ respectively. Since ] is an upper solution <strong>and</strong> ^ is a lower solution (1) in<br />
Ð _ß>" Ó, by Theorem 5, there exist solutions T] <strong>and</strong> T^ on Ð _ß>"<br />
Ó such that T ] a> " b œ] ,<br />
T ^ a> " b œ^ <strong>and</strong> ] T ] T ^ ^. By Theorem 7, both T] <strong>and</strong> T^<br />
are monotone. So<br />
]_ œ lim>Ä _ T ] ab > <strong>and</strong> ^_ œ lim>Ä _ T ^ ab > exist. Clearly ]_ <strong>and</strong> ^_<br />
are constant solutions to<br />
(2) satisfying ] ]_ ^ _ ^. If ea^b ž! then ea] _ b ea^_ b ž! . If ea] b ! then<br />
!žea]_<br />
b ea^_ b. ¨<br />
In fact, ]_<br />
<strong>and</strong> ^_<br />
are the maximal <strong>and</strong> minimal solutions of (2) in the "interval"<br />
c^ ß] d œ eT ’ 8 ß^ŸTŸ] f, respectively. Indeed, if Q c^ ß]<br />
d is a solution to Ð2, Ñ then<br />
] T ab > Q T ab > ^, which imply that ] Q ^ .<br />
] ^ _ _<br />
We now define stabilizability <strong>and</strong> detectability for equation (2). Since equation (2) may arise<br />
from problems with different state equations, we will avoid using the state equations in our definitions<br />
of stabilizability <strong>and</strong> detectability.<br />
Definition 2. Suppose EßFßGßHßKß C are as in (5) <strong>and</strong> T ’ 8 . We define the following<br />
terms.<br />
_<br />
7
aEßGßCb<br />
is ms-stable if there exists Y ’ 8 ß Y ž! such that<br />
X<br />
X<br />
L aYb€ CaYb œE Y € YE€G YG€ CaYb<br />
!Þ<br />
(24)<br />
In other words, aEßGß Cb is ms-stable if L aTb€ CaTb<br />
œ! has a strict upper solution Yž! .<br />
aEßFßGßHßCb<br />
is ms-stabilizable if there exists O‘ 5‚8 such that<br />
aE FOßG HOßC b is ms-stable, or equivalently, _ aOàT b€ CaTb<br />
has a strict upper solution<br />
Yž! . Such a matrix O is also called an ms-stabilizing feedback matrix.<br />
Š È KßEßGß C ‹ is ms-detectable if KœJ<br />
X J for some J‘<br />
;‚8<br />
<strong>and</strong> there exist<br />
;<br />
Q" ßQ# V 8‚ such that aE Q" JßG Q#<br />
JßCb<br />
is ms-stable.<br />
8<br />
T ’ is ms-stabilizing if aE F^aTbßG H^aTbß Cb<br />
is ms-stableÞ<br />
As proved in [1, Theorem 1], the ms-stabilizability of aEßFßGßHß!<br />
b is equivalent to the<br />
existence of a matrix O ‘ 5‚8 such that the solution B to (12) satisfies IelBab<br />
> l # f Ä! as >Ä_ ;<br />
see the paragraph before Proposition 1. When GœHœ! , the ms-stabilility, ms-stabilizability <strong>and</strong><br />
ms-detectability defined here are equivalent to those defined in [5, Definitions 3.1 <strong>and</strong> 3.2].<br />
Assuming that aEßFßGßHß C b ms-stabilizble, we show that (2) has an upper or lower<br />
solution <strong>and</strong> we obtain a refined necessary <strong>and</strong> sufficient condition for solutions to (2).<br />
Theorem 9. Suppose aEßFßGßHß Cb<br />
is stabilizable.<br />
(i) Equation (2) has a solution T with eaTb ž! if <strong>and</strong> only if it has a lower solution ^ with<br />
eaZ b ž! .<br />
(ii) Equation (2) has a solution T with eaT b ! if <strong>and</strong> only if it has an upper solution ] with<br />
ea] b !.<br />
Proof. (i) The necessity is trivial. For sufficiency, suppose that aEßFßGßHß Cb<br />
is ms-stabilizable <strong>and</strong><br />
that ^ is a lower solution with ea^b ž! . Then _ aOà Yb€ CaY b ! for some O‘ 5‚8 <strong>and</strong><br />
Y ž! . Let ]œ ! Y. Choose an ! ž! such that ] ^<strong>and</strong><br />
_ aOà] b€ Ca] b€ ZaO b !ß<br />
where ZaO b is defined in (19). It follows that ea]<br />
b ea^b<br />
ž! . By Proposition 3 (ii) with Tœ] ,<br />
we have that<br />
LQa] b€ C a] b Ÿ _ aOà] b€ Ca] b€ Z aOb<br />
!Þ<br />
That is, ] is a strict upper solution to (2). By Theorem 8, (2) has a solution in c^ß]<br />
d. The proof of<br />
(ii) is similar or it follows from Remark 6. ¨<br />
8<br />
§ 4. Comparison, Uniqueness, Stabilizability <strong>and</strong> Approximation of Solutions to (2)<br />
Now we consider the uniqueness, ms-stabilizability <strong>and</strong> approximation of solutions to (2). We<br />
first prove some equivalent descriptions of the ms-stability.<br />
Theorem 10.<br />
(i) The following are equivalent.<br />
(a) aEßGßCb<br />
is ms-stable.
8 8<br />
(b) Re -aE b ! <strong>and</strong> < 5a7b ", where 7 : ’ Ä ’ is defined as 7aY<br />
b œ<br />
_<br />
' E> E><br />
!<br />
/ X<br />
ÐG X<br />
YG€ C aYbÑ/ .>.<br />
(c) LaYb€ C aYb€ K œ! has a unique solution for every K ’ 8 . If K ! then Y !<br />
<strong>and</strong> if K ž! then Yž! .<br />
(ii) Suppose aEßGßCb is ms-stable <strong>and</strong> aYßZb<br />
is a pair of upper <strong>and</strong> lower solutions to<br />
L aT b€ CaT<br />
b€Kœ! , then Y ZÞ<br />
Proof. (a) Ê (b). Suppose Y ’ 8 ß Y ž! satisfies L aY b€ CaY b !. Let<br />
L œ cL<br />
aYb€ CaYbd<br />
ž! , then we have<br />
X<br />
X<br />
E Y € YE€G YG€ CaYb € L œ! . (25)<br />
In particular, E X<br />
Y € Y E !, which implies that E must be stable in the sense that all eigenvalues of<br />
E have negative real parts; that is, Re-aE b !. Now (25) can be rewritten as<br />
_<br />
X<br />
Y œ ( / Ð G X YG€<br />
CaYb€ LÑ/ .> œ 7 aYb€ Qß<br />
!<br />
E> E><br />
_ X<br />
where Q œ ' E> E> 5€"<br />
/ L/ .> ž!Þ It follows that for all integers 5 !, Y œ 7 aYb€0 aQb,<br />
!<br />
5<br />
3<br />
where 0 aQ b ´ !7 aQb. Since Q ž! , 0 aQb is an increasing sequence <strong>and</strong> 0 aQb<br />
ŸY.<br />
5 5 5<br />
3œ!<br />
_<br />
0 _ a Q b ´ ! 7 3a Q b 0_<br />
a Q b œ YÞ T’<br />
8<br />
3œ!<br />
€ „ _ _<br />
_<br />
_ a b _ a 3<br />
€ b _ a b - 7 !-<br />
3œ!<br />
Therefore, the series converges <strong>and</strong> Since each can be<br />
written as TœT T where T ! <strong>and</strong> 0 is linear, 0 is well-defined for all T’ 8 by<br />
0 T œ0 T 0 T . Now if is an eigenvalue of , then must converge to an<br />
eigenvalue of 0. Therefore, l- l " <strong>and</strong> so < 5a7 b ".<br />
_<br />
(b) Ê (c) Suppose V/ -aE b ! <strong>and</strong> < a7b<br />
". Then 0 a T b œ ! 7 3a<br />
T b is defined for every<br />
_<br />
5 _<br />
X<br />
8 E> E><br />
T ’ . Take T œ '<br />
!<br />
/ K/ .>. It is easily checked that Y œ0aTb satisfies that Y œ T € 7aYb,<br />
which is equivalent to L aYb€ CaYb€ K œ! . Since any solution to L aYb€ CaYb€ K œ! is<br />
_ E> E><br />
represented as 0_<br />
aTb , where T œ ' / X<br />
!<br />
K/ .>, the solution has to be unique. If K ! or ž! ,<br />
then T ! or ž! , which implies that Y ! or ž! , respectively.<br />
(c) Ê (a).Let K œI , then L aYb€ CaYb€Iœ! has a solution Y ž!Þ So aEßGßCb<br />
is msstable.<br />
This finishes the proof of (i).<br />
To prove (ii), consider TœY Z . Then T satisfies L aT b€ CaT<br />
b€ L œ! for some<br />
L !. Part (i.c) implies that Y Zި<br />
Now we can prove the following comparison result for (2).<br />
Theorem 11. Suppose that ] is an upper solution <strong>and</strong> ^ a lower solution to equation (2). Then<br />
] ^ if either (i) ] is ms-stabilizing <strong>and</strong> ea^b ž! , or (ii) ^ is ms-stabilizing <strong>and</strong> ea]<br />
b !.<br />
Proof. Suppose ] is ms-stabilizing <strong>and</strong> ea^ b ž! . Let Tœ ] ^ <strong>and</strong> L œ Xa^ b Xa] b !.<br />
Then L œ CaT b€<br />
LQa]<br />
b LQ a^Þ<br />
b By Proposition 3 (i),<br />
3œ!<br />
5<br />
9
10<br />
Lœ CaT b€<br />
LQa]<br />
b LQa^b<br />
œ<br />
X<br />
X<br />
CaT b€<br />
_ aOàTb<br />
a^a] b O b ea] ba^a] b O b€ a^a^b Ob ea^ba^a^b<br />
Ob,<br />
(26)<br />
where _aOàTb<br />
œ aE FOb X<br />
X<br />
T€TaE FO b€ aG HOb TaG HOb<br />
Setting Oœ^a]<br />
b in (26), we get<br />
as defined in (13).<br />
X<br />
_ a^a] bàT b€ CaT b€a^a^b ^a] bb ea^ba^a^ b ^a] bb€ L œ! .<br />
Since ea^ b ! <strong>and</strong> L !, T is an upper solution to _ a^a] bàT b€ CaTb<br />
œ!. By assumption,<br />
aE F^a] bßG H^a] bß<br />
Cb<br />
is ms-stable. Therefore, we can apply Theorem 10 (ii) to the<br />
equation _ a^a] bàT b€ CaTb<br />
œ! to conclude that T !; that is, ] ^Þ<br />
The proof (ii) is similar<br />
or it follows from Remark 6.¨<br />
8<br />
Denote by m€<br />
( m ) be set of all solutions T ’ to (2) with eaTb<br />
ž! ( !, respectively).<br />
By Theorem 11, if T m€ is ms-stabilizing, then T X for every Xm€<br />
. It follows that the msstabilizing<br />
solutions in m€<br />
must be maximal <strong>and</strong> so unique. Similarly, ms-stabilizing solutions in m<br />
are minimal <strong>and</strong> unique. Therefore, we have the following uniqueness of ms-stabilizing solutions.<br />
Proposition 12.<br />
(i) If T m€ is ms-stabilizing, then T is maximal in m€<br />
<strong>and</strong> T is the unique ms-stabilizing solution<br />
in m € .<br />
(ii) If T m is ms-stabilizing, then T is minimal in m <strong>and</strong> T is the unique ms-stabilizing solution<br />
in m .<br />
Now we consider the question on the<br />
linear equation associated with (2):<br />
<strong>and</strong> its generalization with O <br />
ms-stabilizability of solutions to (2). First consider the<br />
X<br />
X<br />
L aT b€ CaT b ´E T€ TE€G T G € CaT<br />
b€Kœ!Þ<br />
(27)<br />
‘ 5‚8<br />
where _aOàTb<br />
is defined as in (13). We have<br />
Theorem 13. Suppose that<br />
À<br />
X<br />
_ aOàT b€ CaT b€K€ O VOœ!<br />
, (28)<br />
Š È KßEßGßC ‹<br />
is ms-detectable.<br />
(i) If (27) has an upper solution T !, then aEßGßCb<br />
is ms-stable.<br />
5‚8 8<br />
(ii) More generally, if for some O V <strong>and</strong> V ’ with Vž! , (28) has an upper solution<br />
T !, then aE FOßG HOßC b is ms-stable.<br />
Proof. (i) The ms-detectability of Š È KßEßGßC ‹ implies that KœJ X J for some J ‘;‚8<br />
<strong>and</strong><br />
8‚ ;<br />
there exist Q ßQ V such that aE Q JßG Q JßCb<br />
is ms-stable; that is,<br />
for some<br />
" # " #<br />
X<br />
X<br />
aE Q J b Z € Z aE Q J b€ aG Q Jb Z aG Q J b€ C aZ<br />
b ! (29)<br />
" " # #<br />
! Z ’ 8 . Exp<strong>and</strong>ing (29) we obtain<br />
X X X X X X X<br />
L aZ b€ CaZ b aJ Q Z€Z Q J€J Q ZG€G Z Q J b€J Q Z Q J !Þ (30au)<br />
" " # # # #
X X X X<br />
where L aZ b œE Z € Z E€G Z G. With J Q# Z Q#<br />
J 0 dropped, (30) still holds. It follows<br />
that for some & ž! ,<br />
L aZ b€ C aZ b aJ X Q X Z€Z Q J€J X Q X ZG€G X Z Q J b€ % Z€ % G X ZG !.(31)<br />
" " #<br />
#<br />
Let + be the largest eigenvalue of aQ Z Q € Q Z Q bÎ<br />
X X<br />
" " # # & #<br />
. Then one has<br />
# #<br />
X X X X<br />
J aQ Z Q" € Q Z Q# bJÎ& # Ÿ+J JŸ + cL aT b€ CaTbd, (32)<br />
" #<br />
where the last inequality is just (27). Define [ œ+T €Z . Then [ž!. Combining (31) <strong>and</strong> (32),<br />
we obtain that<br />
La[ b€ Ca[ b œ+ aLaT b€ CaT bb€<br />
LaZ b€ CaZ<br />
b<br />
# X X # X X X X X<br />
& Z J Q Z Z J€ G Z J J ZG€+J J‘<br />
" Q" & G ZG Q#<br />
Q#<br />
X<br />
X<br />
Ÿ a& I Q JÎ& b Z a& I Q JÎ & b€<br />
a& G Q JÎ& b Za& G Q JÎ&<br />
b‘<br />
Ÿ !Þ<br />
" " # #<br />
This shows that aEßGßCb<br />
is ms-stable.<br />
J<br />
(ii) Let Y œ ” . Then Y Y œ<br />
By assumption,<br />
V<br />
O<br />
•<br />
J X J€ O VO. aE Q" JßG Q# JßCb<br />
is ms-stable for some Q" <strong>and</strong> Q# Þ Let `" œ Q" ß FV<br />
"Î#‘ <strong>and</strong> `# œ Q#<br />
ß HV<br />
"Î#‘. Then<br />
we have that<br />
E FO ` Y œE Q JßG HO ` Y œG Q J.<br />
" " # #<br />
So aE FO `" YßG HO `# YßCb œ aE Q" JßG Q#<br />
JßCb, which is ms-stable. In<br />
other words, ˆ ÈYXYßE FO ßG HOßC‰<br />
is ms-detectable. By (i) applied to (28),<br />
aE FOßG HOßC b is ms-stable. The proof (ii) is similar or it follows from Remark 6.¨<br />
Remark 3. In Theorem 13, a sufficient condition for Š È KßEßGßC ‹ to be ms-detectable is that<br />
#<br />
Kž! . Indeed, let Jž! such that KœJ <strong>and</strong> ! a!ß_ b such that ! I CaIÎ#<br />
b . Take<br />
" "<br />
Q" œ aE ! IJ b <strong>and</strong> Q#<br />
œ GJ . Then the left-h<strong>and</strong> side of (29) with Z œI becomes<br />
#! I€ CaI b !. So Š ÈKßEßGßC‹<br />
is ms-detectable.<br />
Now we prove the ms-stabilizability of solutions to (2).<br />
Theorem 14. Suppose a] ß ^ b is a pair of upper-lower solutions to (2) <strong>and</strong> ] ^.<br />
(i) If ea^b ž! <strong>and</strong> ˆ ÈXa^ bßE F^a^bßG H^a^bßC‰<br />
is ms-detectable, then ] is ms-<br />
stabilizing.<br />
(ii) If ea] b ! <strong>and</strong> ˆ È Xa] bßE F ^a] bßG H ^a]<br />
bßC‰<br />
is ms-detectable, then ^ is ms-<br />
stabilizing.<br />
In both cases, (2) has a unique solution T in c^ ß]<br />
d <strong>and</strong> T is ms-stabilizing.<br />
Proof. (i) We first assume Wœ! <strong>and</strong> ^ œ! . The assumptions imply that K !ß Vž! <strong>and</strong><br />
Š È KßEßGß C ‹ is ms-detectable. By (20) with Oœ ^ a T b , equation (2) is equivalent to<br />
_^ a aTbàT b€ CaT b€ Z^ a aTbb<br />
œ!. (33)<br />
11
X<br />
Since Wœ! , Z^ a aTbb œK€ ^aTb V^aTb. So (33) is precisely (28) with Oœ ^aTb.<br />
Because ]<br />
is an upper solution to (2) <strong>and</strong> so it also an upper solution to (33), Theorem 13 (ii) implies that<br />
aE F ^a] bßG H ^a]<br />
bßC b is ms-stable; that is, ] is ms-stabilizing. For the general case,<br />
consider T œ] ^. Then by (23), T satisfies equation (2) is equivalent to<br />
Ú<br />
X<br />
X<br />
Ý K€ s Es T€TE€G s s TG€ s CaTb<br />
X<br />
€<br />
Û ˆ X X X X X<br />
Ý<br />
F T€HTGs‰ˆ V€HTH s ‰ ˆ F T€HTGs‰<br />
Ÿ!ß<br />
Ü T a> b œ R ^> a b !,<br />
" "<br />
where EœE s F^a^bß GœG s H^a^b, Kœ s LQ a^ b€ Ca^ b ! <strong>and</strong> Vœ s ea^b<br />
ž! . The<br />
assumptions imply that (34) has a lower solution ! with ms-detectable Š È Kß s EG s ß s ß C ‹. This is<br />
precisely the case we just proved with EßGßKßV replaced by EßGßKßV s s s s , respectively. Therefore, T<br />
is ms-stabilizing in the sense that Š Es F^saT bßGs H^saT bßC‹<br />
is ms-stable, where<br />
"<br />
^s aTb œ ˆ V€HTH s X ‰ ˆ X X<br />
F X€HTG s‰ . Since ea] b ea^b<br />
ž! , it is directly checked that<br />
^s aTb œ^a] b ^a^ b; see [11, proof of Proposition 5]. It follows that<br />
aE F ^a] bßG H ^a] bßCb œ Š Es F^saT bßGs H^saT bßC‹<br />
is ms-stable. So ] is ms-stabilizing.<br />
In particular, all solutions of (2) in c^ ß ] d are ms-stabilizing <strong>and</strong><br />
so they must be unique by Proposition 12. The proof (ii) is similar or it follows from Remark 6.<br />
¨<br />
By Remark 3, the ms-detectability condition in Theorem 14 hold if ^ is a strict lower solution (i.e.,<br />
Xa^ b ž! ) or ] is a strict upper solution ( Xa]<br />
b !). Thus we have the following corollary.<br />
Corollary 15. Suppose a] ß ^ b is a pair of upper-lower solutions <strong>and</strong> ] ^.<br />
(i) If ea^b ž! <strong>and</strong> Xa^ b ž! , then ] is ms-stabilizing.<br />
(ii) If ea] b ! <strong>and</strong> Xa] b !, then ^ is ms-stabilizing.<br />
In both cases, (2) has a unique solution in c^ ß ] d <strong>and</strong> it is ms-stabilizing.<br />
Finally we show that the stabilizing solution to (2) can be approximated by solutions to linear<br />
equations.<br />
Theorem 16. Suppose a] ß ^ b is a pair of upper <strong>and</strong> lower solutions such that ] ^.<br />
(i) Suppose ea^b ž! <strong>and</strong> ˆ ÈXa^bßE F^a^bßG H^a^bßC‰<br />
is ms-detectable. Define<br />
L œ ] <strong>and</strong> L be the unique solution to<br />
" 3€"<br />
12<br />
(34)<br />
_^ a aL3bàT b€ Z^ a aL 3 bb € CaTb<br />
œ!<br />
(35)<br />
for 3 "Þ Then L L â ^ <strong>and</strong> Lœ lim L is the unique ms-stabilizing solution to (2)in<br />
" # 3<br />
3Ä_<br />
c^ ß ] d.<br />
(ii) Suppose ea] b ! <strong>and</strong> ˆ È Xa]<br />
bßE F ^a] bßG H ^a]<br />
bßC‰<br />
is ms-detectable. Define<br />
L œ ^ <strong>and</strong> H be the solution to (35) for 3 "Þ Then L ŸL ŸâŸ ] <strong>and</strong> Lœ lim L is<br />
" 3€" " # 3<br />
3Ä_<br />
the unique ms stabilizing solution to (2) in c^ ß ] d.
Proof. (i) We first show that L 3€" ^. Since ^ is a lower solution with ea^ b !. By (21.1) with<br />
Tœ ^ <strong>and</strong> Oœ ^aL<br />
b, we have<br />
3<br />
!Ÿ LQa^b€ C a^b<br />
Ÿ _^ a aL bà ^b€ Z^ a aL bb€<br />
Ca^b<br />
3 3 .<br />
It follows that aL3€" ß^b<br />
is a pair of upper-lower solution to (35). By Theorem 10 (ii), L 3€" ^.<br />
Consequently, L3€" is a solution to (35) with eaL 3€" b ea^b<br />
ž! . By (21.1) with TœL3€"<br />
<strong>and</strong><br />
Oœ ^aL<br />
3 b, we have<br />
LQaL b€ C aL b Ÿ _^ a aL bàL b€ Z^ a aL bb € CaL b œ!Þ<br />
3€" 3€"<br />
3 3€" 3 3€"<br />
So L3€" is an upper solution to (2). By Theorem 14, L3€"<br />
must be ms-stabilizing.<br />
Next we show that L L . By (20) with TœL, we have<br />
3 3€" 3<br />
! LQaL b€ C aL b œ _^ a aL bàL b€ Z^ a aL bb € CaL bÞ<br />
3 3<br />
3 3 3 3<br />
In other words, aLßL<br />
3 b is a pair of upper-lower solutions to (35). By Theorem 10 (ii) again,<br />
3€"<br />
L 3 L3€".<br />
Now it is clear that the limit Lœ lim L exists <strong>and</strong> satisfies (2). By Theorem 14, L is<br />
3Ä_ 3<br />
stabilizing. The proof of (ii) is similar or it follows from Remark 6.<br />
¨<br />
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