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

1 

Upper-Lower Solution Method for Differential 

Riccati Equations from Stochastic LQR Problems 

Libin Mou 

Department of Mathematics 

Bradley University 

Peoria, IL 61625 

Abstract 

We use upper and lower solutions to study the existence and properties of solutions to differential 

Riccati equations arising from stochastic linear quadratic regulator (LQR) problems. The main results 

include an interpretation of upper and lower solutions, comparison theorems, an upper-lower solution 

theorem, necessary and sufficient conditions for existence of solutions, an estimation of maximal 

existence intervals of solutions and an approximation of solutions. Many of the results are new, while 

others are generalizations of some known results. 

1. Introduction, Notations and Definition 

In this paper we study the following matrix differential equation 

Ú w X 

X 

Ý T €E T€TE€G TG€ K € CaTb 

Û 

X X X 

X " X X 

aF T€HTG€WbaV€HTHb aF T€HTG€Wb 

œ! ß 

Ý 

Ü Ta> b œ R 

" 

(1) 

X 

w 

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 

! " 

" 

is a symmetric matrix, C is a linear map of symmetric matrices, and EßFßGßHß K ßVßW are matrix 

functions in 

M satisfying the conditions stated in (4). 

.T 

.> 

As a motivation for equation (1), we consider a stochastic 

linear quadratic regulator ( LQR) 

problem with noise depending on both of the state and control. For =Mß let [ be a standard Brownian 

motion with 

[ ab = œ! almost surely and hc=ß> " d be the set of all square integrable control processes 

defined on c=ß > d and adapted to the 5-field generated by [. For D ‘ 8 and ? hc=ß 

> d , consider the 

" " 

following state equation and cost function N a? 

b: 

.B œ aEB€F? b.>€ aGB€H? b.[ß>Mà B= abœD, (2.1) 

ž 

X > 

N a? b œIš B a b B a b € ' X X X 

> R > 

" 

aB KB€#? WB€?V? b.> 

›, (2.2) 

" " 

where Ief represents the expectation of the enclosed variable. The problem is to 

= 

(2)

maximize/ minimize N a? b for ? h c=ß> " dÞ 

(3) 

See [26, Ch. 6] for a detailed description of this problem. This problem leads to equation (1) with C œ! . 

For a derivation, see [3], [5], [7], [26] or Theorem 1 below. The inclusion of the term C is important for 

application of (1) to stochastic control problems with Markovian jumping noises and differential game 

problems with state-dependent noises; see [22] and [15], for example. Although the term 

X 

G TG 

2 

may be 

considered as a part of CaT 

b, we will keep them separated for generality. Readers who are interested in 

the equation associated with problem (3) may assume that C œ! . 

Riccati equations (differential, difference and algebraic) appear in various control and min-max 

problems. The classical differential Riccati equation is (1) with 

GœHœWœ C œ! . For this case, the 

existence, comparison, approximation and other properties of solutions have been extensively studied; see 

[2], [4], [6], [13], [14], [20], [23], [27] and many references therein. Many results for classical Riccati 

equations have been extended to equation (1) with HœWœ! in [25], [10], [11] and [9]. Existence and 

approximation of solutions to (1) with Gœ C œ! have been obtained in [7], [ 8] and [26]. 

We will prove comparison theorems, an upper-lower solution theorem, necessary and sufficient 

conditions for existence of solutions, an estimation of maximal existence intervals of solutions and an 

approximation of solutions for equation (1) under general settings. In particular, K and R are allowed to 

X 

be indefinite and V€HTH may be either positive or negative semidefinite. 

The method of upper and lower solutions has been introduced for differential equations as early as 

1940's. Although Riccati inequalities have been studied or used in some literature of Riccati equations 

(e.g., [23] and [21]), this paper, together with [17], [16] and [15], appear to be the first systematic 

application of the method of upper and lower solutions to Riccati equations. It turns out this method has 

many desired merits. For example, it naturally links equation (1) with C œ! with the LQR problem (3). 

It derives the main results under general assumptions. It gives verifiable necessary and sufficient 

conditions for the existence of solutions. It also gives algorithms for approximating solutions. It can be 

used to estimate the maximal existence intervals of solutions to differential Riccati equations without 

solving the equations. It applies to Riccati equations of different types. 

This paper is organized as follows. Notations and assumptions are introduced in this section 

followed by the definitions of upper and lower solutions. In Section 2, we prove a relationship between 

upper and lower solutions and the well-posedness of the LQR problem (3). In addition, some intrinsic 

structural properties of equation (1) are proved in Section 2. In Section 3 we prove some general 

comparison theorems and an upper-lower solution theorem for equation (1). As an application, we obtain 

some necessary and sufficient conditions for the existence of solutions to (1). Furthermore, we show that

3 

the solution can be approximated by a sequence of solutions to linear equations. In Section 4, we apply 

the upper-lower solution theorem to estimate the maximal existence intervals of solutions to (1). Finally, 

in Section 5 we mention a generalization of equation (1). 

The author would like to thank Stan Liberty and Mike McAsey for stimulating discussions. 

Notations. Denote by ’ 8 the set of all real symmetric 8‚8 matrices. We write Q R ( QžR) if Q, 

R ’ 8 and Q R is a positive semidefinite (definite). For a map C À ’ 8 Ä ’ 8 we write C ! if 

_ 

CaQ b ! for each Q !. For a Hilbert space — and an interval Mœ c>ß> 

d or Ð _ß>Ó, P aMß— 

b is 

! " 

" 

the space of all bounded and measurable functions from M to —. Furthermore, we define P "ß_ aMß— 

b œ 

ÖT P _ aMß— bßT w P _ aMß — b×Þ 

Additional notations will be introduced later. For reader's convenience, we list below the 

frequently used notations in this paper. 

w 

XaT b is a short hand for T € LQ aT b€ C aTb, which is the left-hand side of (1). 

LQaTb œK€ LaTb Q aTb, where L aTb, Q aTb 

are defined in (5). Here K is "constant" term, 

L aT b is the linear term, while QaT b is the "quadratic" term. 

and f. 

eaTb and faT b are defined in (7). The facts ea! b œV and fa! b œWexplain the choices of e 

O is a generic notation for a feedback matrix. ŠaTb 

is the set of all feedback matrices associated 

with T; O s ŠaTb; âTb ŠaTb 

is the unique feedback matrix defined in (7) and (8) in terms of the 

" € 

inverse eaTb or psuedoinverse eaTb of eaTb. 

ZaO 

b is defined in (19). The fact Za! b œKexplain the choice. 

_ aOàT b is defined in (22). Note that _ a!àTb œ LaTbÞ 

R ’ 8 is the boundary value of (1); Tß]ß^P "ß_ aMß 

’ 8 

b often represent a solution, an upper 

solution and a lower solution to (1)Þ EßFßGßHßK ßV and W are coefficient matrix functions of (1). 

Assumption. The basic assumption for EßFßGßHßKßVßWßR and C in (1) is 

Ú C P 

_ 8 8 

a’ ß’ 

b is linear and C !, 

_ 8‚8 X _ 8‚5 

Û EßGP aMß‘ b, FßHßW P ÐMß‘ 

Ñ, 

Ü _ 5 _ 8 8 

VP ÐMß ’ Ñ, K P aMß’ bß R ’ . 

To write equation (1) concisely, we denote, for T P 

_ a 

Mß’ 8 b, 

Ú 

Û 

X 

X 

LaTb 

œE T€TE€G TG, 

X X X 

Q aTb 

œ aF T€HTG€Wba V€HTHb " X X 

aF T€HTG€Wb, 

Ü LQaTb œK€ LaTb 

QaT 

b 

(4) 

(5)

4 

with parameters EßFßGßHßK ßV and W satisfying (4). Thus (1) becomes 

To indicate the parameters, we may say 

equation (1) with parameters EßFßGßHßKßVßWßC and RÞ 

w 

T € L Q aT b€ C aTb œ!ßTa> b œ R. 

(1) 

" 

LQ a† b with parameters EßFßGßHßK ßV and W, and 

say 

X 

We remark that Q aTb and LQ aTb 

may be well-defined even if V€HTH is singular. If 

V€HTH 

X is nonsingular, then Q aTb 

can be written as 

where eaTb, faT b and âTb 

are 

" X 

Q aTb œ faTbeaTb faTb 

œ âTb eaTbâTb, (6) 

X 

eaT 

b œV€HTH, (7) 

X 

X 

faTb 

œ F T€HTG€W, 

âTb œ eaTb faTb. 

Here eaT b and faTb 

can be considered as perturbations of V and W with respect to T. The term 

" 

âTb œ eaTb faTb 

appears frequently as the optimal feedback matrix for problem (3) when T is a 

solution to (1) with 

C œ! ; see Theorem 1 below. 

If eaT 

b is singular, then we define 

" 

€ 

âTb œ eaTb faTb, (8) 

€ € 

where eaTb is the pseudoinverse of eaTb. Recall that any matrix Q has a unique pseudoinverse Q 

with the following properties (see [18] and [1]). 

€ € € € 

QQ QœQßQ QQ œQ . (9) 

8 € 8 € € 

If Q ’ , then Q ’ , and QQ œQ Q. 

€ 

Q ! if and only if Q !. 

€ 

Note that âTb œ eaTb faTb always exists, but it may not satisfy faTb œ eaTbâTb. This 

leads to the following definition. 

Definition 1. If T P 

_ aMß’ 8 b satisfies f a T b œ e a T b ^ a T b , then T is said to be feasible . 

_ 5‚8 

Denote ŠaTb œ eO P aMß‘ bß faTb œ eaTbOf. 

Obviously, if T is feasible, then 

âT b ŠaTb and so ŠaTb 

Ág. The converse is also true. We have 

Proposition 1. If ŠaTb Ág, then T is feasible and for each O ŠaTb, 

X 

X 

Q aTb 

œ âTb eaTbâTb œO eaTbO, 

(10)

5 

where âTb œ eaTb faTb. 

€ 

Proof. If OŠaT b Ág, then faTb œ eaTbO. Since eaTb œ eaTbeaTb eaTb, we have 

€ € 

eaTbâTb œ eaTbeaTb faTb œ eaTbeaTb eaTbO œ eaTbO œ faTb. This shows that T is 

€ € 

feasible. Furthermore, from that âTb 

œ eaTb faTb œ eaTb eaTbO 

and property (9), we have 

€ € 

âTb X eaTbâTb œO X eaTbeaTb eaTbeaTb eaTbOœO X eaTbO. 

This shows (10). ¨ 

Proposition 1 shows that if ŠaTb Ág, then Q aTb is well-defined by (10). Each O s ŠaTb 

will 

"ß_ 8 

be called a feedback matrix associated with TP aMß’ b. This term is motivated by the fact (see 

"ß_ 8 

Theorem 1 below) if TP aMß’ b is a feasible solution to (1) with C œ! , then each O s ŠaTb 

is an 

optimal feedback matrix. 

€ 

Definition 2. Suppose T P "ß_ aM, ’ 8 b is feasible. 

T is an upper solution to (1) if 

T is a lower solution to (1) if 

w 

T € LQ aTb€ CaTb 

Ÿ!à T a> b R. 

w 

T € LQaTb€ CaT b !à Ta> b Ÿ RÞ 

T is a solution if it is both an upper solution and a lower solution. An upper or lower solution is called 

strict if at least one of the inequalities in the definition is strict. 

" 

" 

All differential equations and differential inequalities in this paper are considered pointwise for 

almost every > in an interval indicated by context. For brevity, we will write, for example, "K !" or 

" K ! in M " to mean that " Kab 

> ! for almost every >M." 

basic results are still of interest if it is assumed that 

The variable > is often suppressed. The 

EßFßGßHßK ßV and W are all continuous (or 

piecewise continuous) and bounded in M, and it is assumed that T is (piecewise) continuously 

differentiable in M. The equations and inequalities will be then pointwise (except at finite points). 

Note that ! ’ 8 X 

is a lower solution to (1) if and only if K W V " W !, R ! and that ! is 

X " 

an upper solution to (1) if and only if K W V WŸ! , R Ÿ! . These are the common cases studied in 

the classical literature of Riccati equations; see [7] and [8] for remarks. Also, ! is a strict upper or lower 

solution if one of the inequalities is strict. See Proposition 6 for an equivalent description. 

§2. Interpretation of Solutions and Structure of LQaT 

b

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 

= 

"

X 

It follows that N a? b has a minimum (maximum) D T ab =D at ?œ OB s if eaT b !ÐeaTb 

Ÿ!Ñ. 

Equation (12) is precisely the state equation with ?œ OB s . ¨ 

7 

Setting ? œ! in (2), we obtain 

.B œEB.>€GB.[ß> c=ß> 

" dàB= abœ 

D, (16.1) 

ž 

X 

> " 

N ´ Iš B a> bR B a> b € ' X 

B KB.> 

› (16.2) 

! 

" " 

= 

. 

(16) 

By (14) with ?œ!, we obtain a representation for N ! by each TP "ß_ ac=ß> d ß’ 8 b with T a> b œ : 

" 

" R 

X 

X 

N! 

œD T ab =D€I( B aT €K€ L aTbbB.>Þ 

(17) 

= 

> 

" 

In particular, if TP "ß_ =ß> ß 8 

w 

ac d ’ b is the solution to the linear equation T € LaT b€Kœ! 

with 

" 

X 

Ta> " b œ R, which exists by Proposition 7 below, then N! œ D T ab. =D Therefore, if N ! œ ! for every 

a=ßDb 

M‚V 8 , then T´! in M, which implies that R œ! and K´! in MÞ This leads to the following 

proposition, which will be used in the proof of Proposition 4. 

_ 

Proposition 3. Suppose K ßK P aMß’ 8 > " 

b and Iš ' X 

> " 

B K B.> › œIš ' X 

B K B.> › for the 

" # 

w 

= " = # 

solution B to equation (16.1) with each a=ßDb M‚V 

8 . Then K ´ K in MÞ 

" # 

Proposition 3 follows from the case of (16) with R œ! and KœK" K# 

. The assumption of 

Proposition 3 implies that N! œ! for all a=ßDb 

M‚V 8 . Therefore K´! , or equivalently, K " ´K# in 

MÞ 

Note that equation (16.1) is independent of K" and K# . Proposition 3 shows that K " ´K# 

in M if 

> " 

Iš ' X 

> " 

B B.> › œIš ' X 

K B K B.> › for the solution B to certain linear stochastic differential equation 

= " = # 

like (16.1) with each a=ßDb 

M‚V 8 ; see the proof of Proposition 4. 

Suppose O P _ aMß ‘ 5‚8 b and let ?œ OB. Then (2) reduces to 

.B œ aE FObB.>€ aG HObB.[ßB= abœ 

Dß 

ž 

X 

> 

N ´ Iš B a> bR B a> b € ' " X 

B aObB.> › ß 

O 

where ZaOb 

are defined as 

" " 

= 

Z 

(18.1) 

(18.2) 

(18) 

X X X 

ZaOb 

œO VO O W W O€K. (19) 

Note that (18) is (16) under the following replacement:

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€ aâTb Ob X e aTbaâTb 

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 aâTbb€ _ aâT bàTb. 

(24.3) 

(24) 

Remark 1. From the proof below, it follows that identities (23) and (24.3) hold with âT 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 OœâT 

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€ aâTb Ob e aTbaâTb 

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

9 

Proposition 5. Suppose ] ß^ P aMß’ b are feasible, O P aMß‘ 

5‚8 b. Denote Tœ] ^, 

EœE s Fâ^bßGœG s Hâ^b and Vœ s ea^b. Then 

(i) LQ a] b LQa^b 

(26) 

‡ ‡ 

œET€TE€GTG 

s s s s ˆ 

X X 

X 

F T€HTGs‰ˆ V€HTH s X € 

‰ ˆ 

X X 

F T€HTGs‰ 

(ii) If ^ is given, then ] satisfies equation (1) if and only if T œ] ^ satisfies 

Ú 

Ý w 

‡ ‡ 

T € K€ s ET€TE€GTG€ s s s s CaTb 

X € 

Û ˆ X X 

Ý 

F T€HTGs‰ˆ V€HTH s X ‰ ˆ X X 

F T€HTGs‰ 

œ!ß 

Ü T a> b œ R ^> a b, 

" " 

(27) 

where Ks œ^ w € LQa^b€ Ca^bÞ 

Proof. Let Tœ] and ^ in Proposition 4 (i), respectively. Then we get 

LQa] 

b LQa^b 

œ 

_ aOàTb 

aâ] b 

X 

O b ea] baâ] b O b€ aâ^b X 

Ob ea^baâ^b 

OÞ b 

(28) 

Setting Oœâ^b 

in (28), we obtain 

X 

LQa] b LQ a^ b œ _ aâ^ bàTb 

aâ] b â^ bb ea] baâ] b â^bb. 

(29) 

To continue the proof, we first simplify ea] bcâ] b â^ bd. Note that 

X X X 

ea] b ea^ b œHTH and fa] b fa^b 

œF T€HTG. The feasibility of ] and ^ implies that 

ea] bâ] b œ fa] b and ea^bâ^b œ fa^b. Therefore 

ea] bcâ] b â^ bd œ ea] 

bâ] b ea] bâ^b 

X 

œ ea] bâ] b ea^bâ^b HTHâ^b 

X X X X X 

œF T€HTG HTHâ^b 

œF T€HTGs . 

€ 

Now using the fact ea] bea] b ea] b œ ea] 

b and (30), we obtain that 

aâ] b â^ bb X ea] baâ] b â^bb œ ˆ X X X 

F T€HTGs‰ € 

ea] 

b ˆ F X T€HTG 

X s‰ 

. (31) 

Substituting (31) and _aâ^bàTb 

‡ ‡ 

œ ET€TE€GTG 

s s s s 

into (29), we obtain (i). 

To show (ii), denote Xa] b œ ] w € LQ a] b€ 

Ca] 

b. 

Then ] is a solution to equation (1) if and 

only if Xa] b œ! and ] a> b œ . Note that 

" R 

Xa] b Xa^b 

œ T w € LQa] b LQ a^ b€ 

CaTb, 

where LQa] b LQ a^b is expressed in terms of T as in (26). If ^ is given, then Xa] b œ! and 

] a> " b œ R if and only if Tœ] ^ satisfies (27). ¨ 

(30)

Remark 2. Note that equation (27) is (1) with parameters 

10 

EßFßGßHßKß s s s Vß!ß s C and R ^> a b. It 

follows that ] is a solution (upper solution, lower solution) to (1) if and only if T œ] ^ is a solution 

(upper solution, lower solution, respectively) to equation (27). For example, if ^ œ! , then (1) is 

equivalent to (27) with Es œE FV " Wß GœG s HV " Wß KœK s X 

W V " WßVœVÞ s In (27) the 

"crossing term" Wœ! . 

If ^ is a lower solution to (1), then ! is a lower solution to (27) because K s !, 

R ^> a b ! 

" 

" . 

Similarly, if ^ is an upper solution to (1), then ! is an upper solution to (27). For convenience, we will 

say equation (1) is a standard case if Wœ! and ! is a lower solution or an upper solution. We have 

Proposition 6. Equation (1) can be reduced to a standard case if and only if it has a lower or upper 

solution. 

Remark 3. Let ]œ ^, then it is easily seen that ^ is a solution (upper solution, lower solution) to (1) 

if and only if ] is a solution (lower solution, upper solution) to the following equation 

] w € ] € ] K X X 

La b Ca b aF ] €H ] G Wb 

X X X X 

a V€H ] Hb aF ] €H ] G Wb 

œ! 

with ] a> " b œ R. This is same as (1) with Kß Vß W and R replaced by Kß Vß W and R, 

respectively. When C œ!, this equation is precisely the differential Riccati equation associated with the 

X 

X 

problem maximizing N a? 

b. Also note that V€H^H ! if and only if V€H ] HŸ! . As a 

result, if a lower solution ^ (if it exists) to (1) with ea^b ž! has certain property, then an upper 

solution ] (if it exists) to (1) with ea] 

b ! also has the corresponding property. Because of this, we 

will state properties of upper solutions without proof. 

" 

§ 3. Comparison Theorems, Upper-Lower Solution Theorem, Existence and Approximation of 

Solutions 

We first consider the linear matrix differential equation 

w " 

X 

X 

T € E T€TE€G TG€ CaT b€ K œ !ß T a> b œR 

(32) 

where EßGßK and C are as in (5) and R’ 8 . 

Proposition 7. (i) E quation (32) has a unique solution TP 

"ß_ aMß’ 8 b. 

(ii) If ! is a lower solution to (32) then T !. If ! is strict, then and Tž! . 

(iii) If a]ß^ b is a pair of upper-lower solutions to (32), then ] ^ . If one of ] and ^ are strict, then 

]ž^.

11 

Proof. Following the idea in [25], we let Ga>ß= 

b be the fundamental matrix of E, that is, 

` 

Ga>ß= b œE> abGa>ß=ß b Ga>ß> b œE> ab, > ! Ÿ=ß>Ÿ > " Þ 

`> 

" ` 

`> 

Note that Ga>ß= b œ Ga=ß> b and Ga=ß> b œ E> abGa=ß> 

b. It follows that (32) is equivalent to 

> 

" 

X X X 

T ab > œ Ga> " ß> b R Ga> " ß> b€ ( Ga=ß> beCaT ab = b€ G ab = Tab = G ab = € Kab 

= fGa=ß>.= 

b (33) 

> 

The Volterra equation (33) has a unique solution T which can be found by successive approximations; 

say, eT À œ!ß"ß#ßâf 

with T œ! . This shows (i). 

/ / ! 

8 

In case ! ’ is a lower solution (i.e., K ! and R !), we have T / ! for all / !, which 

implies that T ab > œ lim / Ä_ T / ab > !. If ! is a strict lower solution (i.e., K ! or T a> 

" b ! is strict), 

then 

# 

X 

X 

T ab > T ab > ´ Ga> ß> b RGa> ß> b€ ( Ga=ß> b Kab = Ga=ß>.= b ž! . (34) 

" " 

This proves (ii). If a]ß^b 

is a pair of upper-lower solutions to (32), then Tœ] ^ satisfies (32) with 

some K ! and T a> b œ] a> b ^> a b !. Applying the conclusions in (ii) to T, we obtain (iii). 

> 

> 

" 

" " " ¨ 

As an application of Proposition 7, we consider the linear matrix equation 

w 

T € _^ a aT 

b; Tb€ Z^ a aT 

bb€ C aTb œ! , T a> b œ R 

(35) 

! ! 

where T P "ß_ aMß’ 8 b is a given matrix function and _ is defined in (22). Denote by [ aT 

b the unique 

! ! 

T [aT ! b 

solution to (35). The following property of will be used in the proofs of Theorem 12 and 

Theorem 14. 

" 

Proposition8. Let T œ [aT 

b be the solution to (35). 

! 

(i) If (1) has a lower solution ^ with ea^b ž! , then T is an upper solution to (1) and T ^. 

(ii) If (1) has an upper solution ] with ea] b !, then T is a lower solution to (1) and TŸ] . 

Proof. (i) We first show that eaTb 

ž! . By Proposition 4 (i) with T œ ^ and OœâT! b, we have 

w 

^ € _ aâT bà ^b€ ZaâT bb€ 

Ca^b 

w 

! ! 

X 

œ ^ € LQa^b€ 

câ^b âT bd ea^bcâ^ b âT bd€ Ca^b 

!, 

! ! 

where the inequality follows from that ea^ b ! and that ^ is a lower solution to (1). So ^ is also a 

lower solution to (35). Proposition 7 applied to (35) implies that T ^. In particular, 

eaTb ea^b 

ž! . By (24.1) with OœâT! b and (35), we have 

w 

w 

Ÿ ! ! 

T € LQaTb€ C aTb T € _^ a aT b; Tb€ Z^ a aT bb€ CaTb 

œ! .

12 

So T is an upper solution to (1). The proof of (ii) is similar; it also follows from Remark 3. ¨ 

Now we prove a general comparison theorem for equation (1). 

Theorem 9. Denote XaT b ´ T w € LQ aTb€ CaTb. Suppose ], ^ P "ß_ aMß’ 8 b are feasible and satisfy 

Xa] b Ÿ Xa^ bß 

] a> " b ^> a " b, (36) 

and either ea^ b ! or ea] b Ÿ!. Then ] ^ . If one of (36) is strict, then ] ž ^. 

Proof. Let Tœ ] ^ . Then T satisfies T a> " b !. Suppose ea^ b !. Setting Oœâ] 

b in (29), we 

have 

w 

! Xa] b Xa^ b œ T € LQa] b LQa^b€ CaTb 

w 

œT € _^ a a] bàT b€ 

CaT b€ 

aâ^b X 

â] bb ea^baâ^b 

â] 

bb 

w 

T € _^ a a] 

bàT b€ 

CaTbÞ 

(37) 

So aTß! b is a pair of upper-lower solutions to the equation T w € _^ a a] bàT b€CaTb 

œ! with 

T a> " b œ! . By Proposition 7 (iii), we have ] ^. If one of (36) is strict, then T will be a strict upper 

solution, which implies Tž! , or equivalently, ]ž^. 

If ea] b Ÿ! , then set Oœ â^ bin 

(29) to get 

w 

! Xa] b Xa^ b œ T € LQa] b LQa^b€ CaTb 

w 

œT € _^ a a^ bàT b€ 

CaTb 

aâ^ b 

X 

â] bb ea] baâ^b 

â] 

bb 

w 

T € _^ a a^bàT b€ 

CaTbÞ 

The rest of the proof is the same as the case ea^ b !¨ . 

The conclusion that ] ^ implies that either ea] b ea^ b ! or ! ea] b ea^b. 

In 

other words, ] and ^ turn out to have the same definiteness. Therefore, ea^ b ! and ea] 

b Ÿ! 

usually do not occur at the same time. 

Theorem 9 immediately extends to two equations like (1): 

w 

X 3 aT b ´T € LQ 3 aTb€ aTb œ! , T a> b œT , (38) 

C 3 " "3 

where the parameters EßFßGßHßK 

ßVßW , C and T satisfy (4) for 3œ"ß# , and T T . 

X 

Denote e 3 aT b ´ V 3 €HTH 3 , for 3œ"ß#Þ 

3 3 3 3 3 3 3 3 "3 "" "# 

3 

Theorem 10. Suppose ], ^ P "ß_ aMß ’ 8 b are feasible for both equations, ] is an upper solution to 

(38)" while ^ is a lower solution to (38)#. If one of the following conditions holds, then ] ^ in M. 

(i) X a] b X a] b and either e a^ b ! or e a] 

b Ÿ! . 

" # # # 

(ii) X a^b X a^ b and either e a^ b ! or e a] 

b Ÿ! . 

" # " " 

3

13 

Proof. The proof is almost trivial. The assumptions imply that X 

have that X 

a] b Ÿ!Ÿ X a^b; that is, 

# # 

a] b Ÿ!Ÿ X a^b. Then in case (i), we 

" # 

a]ß^b is a pair of upper-lower solutions to (38)#. The conclusion 

follows from Theorem 9 applied to X# . Similarly, in case (ii), we have that X" a] b Ÿ ! Ÿ X" 

a^b. So 

a]ß^ b is a pair of upper-lower solutions to (38)" and the conclusion follows from Theorem 9 applied to 

X " .¨ 

Theorems 9 and 10 are very general comparison results. Comparison theorems have been proved 

(e.g., in [10], [11], [19] and [24]) for solutions to (1) with GœHœWœ! under the assumptions that 

" X 

E3 FV 3 3 F3 

L " L # and C" C# , where L3 

œ Œ 

X 

. These assumptions imply that 

K E 

X 

3 3 

aTb X aT b for all T’ 8 , which is stronger than the conditions (i) and (ii). In this special case, the 

" # 

conditions on e a] b and e a^b 

are not necessary; see [17]. 

3 3 

Theorem. 

From Theorem 9 and the local theory of ODE we prove the following upper-lower solution 

Theorem 11. Suppose that a] , ^b 

is a pair of upper-lower solutions to (1). 

(i) If either ea^ b ! or ea] b Ÿ! , then ] ^ . In addition, if one of ] and ^ is strict, then ] ž ^. 

(ii) If either ea^ b ž! or ea] b !, then equation (1) has a unique solution T with ] T ^. 

Proof. Part (i) follows immediately from Theorem 9. For part (ii), we first consider the case Mœc>ß 

! > " d. 

The local existence theory of ODE implies that equation (1) has a solution that exists in a maximal 

interval Ð7ß> Ó§M . Part (i) implies that ] T ^ in Ð7ß> 

Ó. By using the equation, we see that 

" " 

T ´ lim T ab > exists and ] a7b T â7b. In particular, eaT b ea^ a7bb 

ž! . If > ž 7 ž> , 

7 7 7 

>Ä 7€ 

then the local existence theory of ODE again shows that T can be extended further left beyond 7. This 

would contradict the definition of 7. Therefore, 7 œ > ! and so T ab > exists in c>ß 

! > " d. For the case 

MœÐ _ß> Ó, the same argument shows that the solution T exists on c>ß > d for every > > . So the 

" ! " ! " 

solution exists on Ð _ß> " Ó. 

¨ 

" 

! 

Theorem 11 (ii) implies that a necessary and sufficient condition for the existence of a solution T 

to (1) with either eaTb ž! or eaT b ! is the existence of a pair a] , ^b 

of upper-lower solutions to 

(1) satisfying either ea^ b ž! or ea] 

b !, respectively. Proposition 8 shows that, in fact, the existence 

of one of ] and ^ is sufficient for the existence of a solution. We have 

Theorem 12. 

(i) Equation (1) has a solution T with eaT b ž ! if and only if it has a lower solution ^ with ea^b 

ž !.

(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! . 

"

If ^Á! or WÁ! , then consider \œT ^. Equation (1) for T is equivalent to (27) for \. 

Now (27) has a strict lower solution ! because ^ is a strict lower solution to (1). From what we just 

X 

proved, (27) has a solution \ such that ! ea^ b€H \ Hœ ea^ € \ b. Therefore, Tœ ^ € \ is a 

solution to (1) with eaT 

b ž!.¨ 

15 

Next we show that the solution to (1) can be approximated by solutions ] œ [a] 

b to (35). 

3 3 " 

Theorem 14. Suppose Mœ c>ß ! > " d and ^ is a lower solution to (1) such that ea^b 

ž! . Let 

"ß_ 8 

] P aMß’ b and ] œ [ a] 

b for 3 ". Then 

! 3 3 " 

(i) ]" ]# ] $ â ^. 

(ii) ] œ lim ] is a solution to (1) and there exist constants 5 and - such that 

3Ä_ 3 _ 4 # 

- 

4 # 

l ] 3ab > ] ab >lŸ5 " a> 

" > b , >MÞ 

(40) 

a 4 #x b 

4œ3 

Proof. Proposition 8 implies that each ] 3 is an upper solution and ] 3 ^ for 3 ". To show other 

conclusions, we first derive an equation satisfied by ? 3 œ ] 3 ] 3€" . By the definitions of ] 3 and ] 3€" , 

they satisfy 

] € _^ a a] bà] b€ Z^ a a] bb€ C a] 

b œ!Þ 

(41) 

3 w 3 " 3 3 " 3 

w 

3€" 3 3€" 3 3€" 

] € _^ a a] bà] b€ Z^ a a] bb€ C a] 

b œ!Þ 

(42) 

By Proposition 4 (i) with Tœ ] and Oœâ] 

b, we have 

3 3 " 

X 

_^ a a] bà] b€ Z^ a a] bb œ a] b€ aâ] b â] bb ea] baâ] b â] 

bbÞ 

3 " 3 3 " LQ 3 3 3 " 3 3 3 " 

However, LQa] b œ _^ a a] bà] b€ Z^ a a] 

bb 

by (24.3) Þ Thus (41) becomes 

3 3 3 3 

w 

3 3 3 3 3 3 3 " 3 3 3 " 

X 

] € _^ a a] bà] b€ Z^ a a] bb€ Ca] b€ aâ] b â] bb ea] baâ] b â] 

bb 

œ! .(43) 

Subtracting (42) from (43) we obtain 

w 

3 3 3 3 3 3 " 3 3 3 " 

X 

? € _^ a a] bà ? b€ Ca? 

b€ 

aâ] b â] bb ea] baâ] b â] 

bb 

œ!Þ 

(44) 

X 

Note that a> b œ! and aâ] b â] bb ea] baâ] b â] bb 

! because ea] b !Þ By 

? 3 " 3 3 " 3 3 3 " 3 

Theorem 9, ? 3 !. This finishes the proof of (i). 

Now (i) implies that e] 3fß eea] 3bf and eâ] 

3bf 

are all uniformly bounded in M. It follows that 

X 

laâ] b â] bb ea] baâ] b â] bblŸ5l? lß l_^ a a] 

b à ? b€ C a? 

blŸ5l? 

l, 

3 3 " 3 3 3 " 3 " 3 3 3 

3

16 

for some constant 

5ž! . Thus from (44) we obtain that 

> 

l ? 3ab >lŸ5( al ? 3ab =l€l ? 3 " ab =l.= b . (45) 

> 

" 

The rest of the proof follows exactly the argument in [26, p.324]. We repeat it here for reader's 

> 

convenience. Denote @ ab > œ ' " l ? ab =l.= . Then (45) reduces t 

which implies that 

3 > 3 

w 

3 

@ ab > €5@ ab > €5@ ab > !ß 

5> 

3 3 " 

> > 

" " 

" 

@ 3ab > Ÿ5/ ( @ 3 " ab =.= œ- ( @ 3 " ab =.=ß 

> > 

where -œ5/ 

Þ By induction, we deduce that 

5> " 

3 

- 

3 

@ 3€" ab > Ÿ a> " > b @" a> ! bÞ 

3x 

It follows then from (45) that 

- 3 " - 

l ? 3 ab >lŸ5 œ a> " > b € a> " a 3 "x b a 3 #x b 

3 

> b 

# 

@ " a> ! bÞ 

This yields (40). ¨ 

By Remark 3, we also have an approximation theorem for the solution ^ with ea^ b !. 

"ß_ 8 

Theorem 15. Suppose ] is an upper solution to (1) with ea] b !. Let ^! 

P aMß’ 

b and 

^ 

œ [a^ 

b for 3 ". Then 

3 3 " 

(i) ^ Ÿ ^ Ÿ ^ ŸâŸ] 

. 

" # $ 

(ii) ^ œ lim ^ is a solution to (1) and there exist constants 5 and - such that 

3Ä_ 3 _ 4 # 

- 

4 # 

l ^3 ab > ^ ab >lŸ5 " a> " > b , >Mœ c>ß ! > " dÞ 

a 4 #x b 

4œ3 

§ 4. Estimates of Maximal Intervals of Existence 

Suppose ] ( ^Ñ is an upper (lower) solution to the equation 

w 

XaT b ´ T € LQ aT b€ CaTb œ!ßTa> b œ R 

(46) 

on an interval M . If ea] b ! or ea^b 

ž! , then Theorem 11 implies the existence of a solution T in M 

with eaT b ! or eaTb ž! , respectively. If ea] b ž! or ea^b 

!, then it is well-know that the 

"

solution may blow up in M. This happens, for example, to Riccati equations from differential games [2]. In 

this case, it would be of interest to estimate the maximal existence interval Ð7 ß> " Ó of the solution to (46). 

17 

By Theorem 11, the solution to (1) exists in the intersection of the existence intervals of a pair of 

upper-lower solutions. So the maximal existence interval of the solution to (1) can be estimated by 

constructing their upper and lower solutions. Theoretically, such estimates can be made as accurate as 

possible if we can find upper and lower solutions that are close enough to the solution. Upper and lower 

solutions can be constructed by using the solutions of comparison equations like those in Theorem 10. In 

Theorems 16, 17 and Proposition 18 below, we demonstrate how to construct scalar autonomous 

differential equations in terms of ] and ^ to obtain explicit estimate for Ð7 ß> 

" Ó. 

By (27) and Proposition 

6 we need to consider only the standard cases with Wœ! , and ] œ! or ^ œ! . 

Denote by -a† b, Aa† b and 5a† 

b the minimum eigenvalue, maximum eigenvalue and maximum 

singular value of a matrix, respectively. Suppose Wœ! , Vž! and ] œ! is an upper solution. Let 

+ß,ß-ß.ßŸ > " ß 

œ 

:> a b œ : " , 

" 

(48) 

# 

+ : 

.: 

€< " " 

where 2: a b œ €, : €- . Note that + !, . !ß -Ÿ !, : Ÿ! . Suppose .: € " Ó. Since 2 a! b œ-Ÿ!, ! is an upper solution to 

(48), which implies that :> abŸ! in Ð5ß> Ó by Theorem 11. Furthermore, .: ab > €Ð5ß> 

Ó. 

" " 

Theorem 16. Suppose ! is an upper solution to (46) with V ž! such that .: 

€ " Ó with .: 

€ " Ó with eaTb 

ž! . 

" 

Proof. Let ^ œ :I . We only need to show that ^ is a lower solution in Ð5ß> Ó to (46) with ea^b 

ž! . 

Note that 

8 " 

^> a " b œ :I " 8 Ÿ Rß 

X 

ea^ b œ V € : HH a

18 

have 

w 

Xa^b œ ^ € LQa^b€ 

Ca^b 

w 

X 

X 

œ :I € : aE 

€E€ G G € CaI bb€ 

K 

8 8 

# X 

" 

X X X X X 

: aF €H GbaV € : HHb aF €H Gb 

# 

w 

+ : 

w 

”: €, : €- • I 8 œ c: € 2: a bdI8 

œ!Þ 

 

" Ó. 

¨ 

such that 

Similarly, suppose Wœ! , V ! and ! is a lower solution. Let +ß,ß-ß.ß 

" Ó. 

Since 

" " 

2 a! b œ- !, ! is a lower solution to (48), which implies that : ab > ! in Ð5 ß> 

" Ó by Theorem 11 and 

.: ab > €< ! for >Ð5 ß> 

" Ó. 

Theorem 17. Suppose ! is a lower solution to (46) with V ž!. If (48) with coefficients in (50) has a 

solution : in an interval Ð5ß> Ó with .: €< !, then (46) has a solution T in Ð5ß> 

Ó with eaT b !. 

" " 

Proof. The proof is similar to that of Theorem 16. One only needs to show that :I 8 is an upper solution. 

¨ 

We now give an integral representation for the maximal existence interval Ð5 ß>Ó " of the solution 

:> ab to (48) for a general rational function 2: a b. Assume for some 5 !2: , a b has 5€# distinct zeros 

and poles _œD! D" â D5 D5€" 

œ_ , including „_. 

Proposition 18. Suppose : aDßD b for some 3œ!ß#ßâß5 . Then the solution :> abof (48) exists on 

" 3 3€" 

a maximal interval Ð5 ß>Ó§Ð " _ß>Ó " with 

> 5 œ ( 

" 

: 

: 

" 

"‡ 

.: 

, 

2: a b 

where : œ :> ab; that is, : œD if 2: a b ž! and : œD if 2: a b !. 

"‡ lim "‡ 3€" " "‡ 3 " 

>Ä5 

Proof. If 2: a b ž! , then the solution :> abis strictly increasing at > decreases in Ð5ß>Ówith range 

" " 

" 3€" lim ab .: 

3€" " 3€" 

>Ä 5 2: a b 

Ò: ßD Ñ and :> œD . Write (48) as €.> œ! . Integrating this equation from : to D for

19 

: and from > " to 5 for > , we get 

( 

: 

D 

" 

3€" 

.: 

" 

2: a b 

€ 5 > œ!Þ 

If 2: a b !, then the solution :> abis strictly decreasing at > decreases in Ð ß>Ówith range ÐD ß:Ówith 

" 5 " 

3 " 

.: 

3 2: a b " 3 " 5 

lim :> ab œD . Integrating €.> œ! from : to D for : and from > to for > , we obtain 

>Ä5 The Proposition is proved.¨ 

( 

: 

D 

" 

3 

.: 

" 

2: a b 

€ 5 > œ!Þ 

Example. We take a simple example from [26, Ch. 6, Example 7.8] to illustrate an application of 

Proposition 17. Consider a scalar case of problem (3) with EœGœKœWœ!ß 

FœHœ R œ> " œ" and Vœ < with decreases to < as 

" 

>Ä 5 . By Proposition 18, 5 œ " ' 

: < 

< : 

.: œ#

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