Global existence and stability of solutions of matrix riccati equations

doi:10.1006rjmaa.2000.7058, available online at http:rrwww.idealibrary.com on

Global Existence and Stability of Solutions of Matrix

Riccati Equations

Jonq Juang1

Department of Applied Mathematics, National Chiao-Tung Uni¨ersity, Hsinchu, Taiwan, R.O.C.

Submitted by Horst R. Thieme

Received February 16, 1999

We consider a matrix Riccati equation containing two parameters c and␣. The quantity c denotes the average total number of particles emerging from a collision,

Ž . Ž .

which is assumed to be conservative i.e., 0- c F 1 , and ␣ 0 F ␣ - 1 is an

Ž . 4

angular shift. Let Ss c,␣ : 0 - c F 1 and 0 F ␣ - 1 . Stability analysis for two steady-state solutions Xmin and Xmax are provided. In particular, we prove that

Ž .4

Xmin is locally asymptotically stable for Sy 1, 0 , while Xmax is unstable for Ž .4

Sy 1, 0 . For c s 1 and ␣ s 0, Xmins Xmax is neutral stable. We also show Ž .

that such equations have a global positive solution for c,␣ g S, provided that the initial value is small and positive. 䊚 2001 Academic Press

I. INTRODUCTION

This paper is concerned with the global existence and stability problem of the matrix Riccati equation of the form

XXs B y AX y XD q XCX [ FF X ,

Ž

.

Ž

1a

.

X 0

Ž .

s X .0

Ž

1b

.

Here, A, B, C, and D are matrices with the structure

1 1 1 As diag , , . . . , c␻ 1 q ␣1

Ž

.

c␻ 1 q ␣2

Ž

.

c␻ 1 q ␣n

Ž

.

1 _{c c c} . 1 2 n . y _. , , . . . , 2␻1 2␻2 2␻n 1

w

x

T T [ diag ␦ , ␦ , . . . , ␦ y eq [ D y eq ,1 2 n 1

Ž .

2 1_{Supported in part by NSC of R.O.C., Taiwan. E-mail: [email protected].}

1

0022-247Xr01 $35.00 Copyright䊚 2001 by Academic Press All rights of reproduction in any form reserved.

1 1 1 T Ds diag , , . . . , y qe

Ž .

3 c␻ 1 y ␣1

Ž

.

c␻ 1 y ␣2

Ž

.

c␻ 1 y ␣n

Ž

.

w

x

T T [ diag d , d , . . . , d y qe [ D y qe ,1 2 n 2

Ž .

4 Bs eeT_{, 5}

Ž .

Cs qqT ,

Ž .

6 Ž .

and the initial matrix X₀ is nonnegative, i.e., X_{0 i j}G 0 for all i, j. Ž .

Equation 1a contains two parameters, c an ␣. The quantity c denotes the average total number of particles emerging from a collision, which is

Ž . Ž .

assumed to be conservative i.e., 0F c F 1 , and ␣ 0 F ␣ - 1 is an

 4n  4n

angular shift. The data ␻i is1 and ci is1 are sets of the Gauss᎐Legendre

w x

nodes and weights, respectively, on 0, 1 with 1)␻ ) ␻ ) ⭈⭈⭈ ) ␻ ) 0,_{1 2 n} and n c s 1, c ) 0, is 1, 2, . . . , n.

Ý

i i is1 w x.

Such an equation is induced via invariant imbedding 2᎐5, 10, 11 , and the

w x

integration formula from an ‘‘angularly shifted’’ transport model 6, 7 in the slab geometry.

Ž .

The solutions of 1 exhibit interesting behavior with increasing slab thickness. The equation, with slab thickness z as a parameter, can be analyzed in the context of a dynamical equation.

Ž . The purpose of this paper is twofold. First, stability analysis of Eq. 1 for two steady-state solutions Xmin and Xmax is provided. In particular, we show that the steady state Xmin is locally asymptotically stable for all 0- c F 1 and 0 F␣ - 1, except that c s 1 and ␣ s 0, while Xma x is

Ž .

unstable for such c and ␣. For c s 1 and ␣ s 0, Xmins Xmax is neutral Ž .

stable. Second, we show that Eq. 1 has a global positive solution for all 0- c F 1 and 0 F␣ - 1. In Section 2, we recorded some of the needed

Ž .

results concerning the steady-state solutions of Eq. 1 . The main results are given in Sections 3 and 4.

II. STEADY-STATE SOLUTIONS

In the terminology of dynamical equations, the steady-state solutions to Ž .1 satisfy

Let the matrix H be defined in block form by

D yC

H[ ;

Ž .

8

B yA

Ž .

we shall call this matrix a Hamiltonian-like matrix of Eq. 1 . The complete

Ž . w x

solution bifurcation diagram of Eq. 2 has recently been obtained in 8 by considering the invariant subspace of H. Some of the results needed to

Ž .

study the stability of 1 are recorded in the following:

Ž w x. Ž .

THEOREM 2.1 Lemma 2.1 of 8 . The matrix H, as defined in 8 , has

 4

only real eigen_¨alues y␮ , . . . , y␮ , ␭ , . . . , ␭ , which are arranged in an_{n 1 1 n} ascending order. Those eigen_¨alues of H satisfy the following secular equation

Ž . f ␭ of H y ␭I: n _q n _q i i f

Ž .

␭ s 1 y

_Ý

y

_Ý

. diy␭ ␦ q ␭i is1 is1

Moreover, the following assertions and estimates hold:

Ž .i Let ␦ and d , i s 1, 2, . . . , n, be given as 2 and 4 , respec-i i Ž . Ž . tively. Then

y␦ - y␮ - y␦n n ny1- ⭈⭈⭈ - y␦ - y␮ - y␦ - y␮ F 0,2 2 1 1

0F␭ - d - ␭ - d - ⭈⭈⭈ - ␭ - d ._{1 1 2 2 n n} Ž .ii ␮ s 0 only if c s 1.₁

Žiii. ␭ s 0 only if c s 1 and ␣ s 0.1

Živ. For ␣ s 0, ␮ s ␭ , i s 1, 2, . . . , n._{i i}

Ž w x.

THEOREM2.2 Theorems 3.3 and 3.4 of 8 . Let 0- c F 1 and 0 F␣

Ž .

- 1. Equation 2 has a unique nonnegati¨e solution for cs 1 and ␣ s 0. Otherwise, it has two nonnegati¨e solutions, say X_min and X_max with X_maxG

Ž . 

Xmin) 0. Moreo¨er, the spectrum ␴ D y CXmin of Dy CXmin is ␭ , ␭ ,1 2

4  4

. . . ,␭ , and that of D y CXn max is y␮ , ␭ , . . . , ␭ .1 2 n

Ž w x.

THEOREM 2.3 see Theorem 5.4 of 8 . The minimum solutions Xmin

Ž . Ž . Ž

of Eq. 2 are strictly increasing in c for fixed ␣ and decreasing in ␣ for .

fixed c .

III. LINEARIZED STABILITY

Considering the linearized operator of FF at Xs X#, where X# is a Ž .

stationary solution of 1a , we have that F

X_{Ž .}

The eigenvalue problems of FF X# can then be formulated as F

FX

Ž

X# R s

.

␭R.

Ž

10

.

or, equivalently,

y A y X#C R y R D y CX# s

Ž

.

Ž

.

␭R.

Ž

11

.

To see the eigenvalues of Ay X#C, we need the following lemmas. Set

n n n n 1 c_k 1 c_k a# s

_Ý

cj

_Ý

Ž

X#

.

k j, b# s

_Ý

ci

_Ý

Ž

X# ,

.

i k 2 js1 ks1 ␻k 2is1 ks1 ␻k and n n n n 1 c_k 1 c_k ␣# s

_Ý

cj␻j

_Ý

Ž

X#

.

k j, ␤# s

_Ý

ci␻i

_Ý

Ž

X# .

.

i k 2 js1 ks1 ␻k 2is1 ks1 ␻k

Here) s min or max. Ž .

LEMMA3.1. i If cs 1 and␣ s 0, then a_mins b_mins a_maxs b_maxs 1.

1q␣ 1y␣ 1q␣

Ž .ii If cs 1, and ␣ / 0, then a_min-1y␣, b_mins1q␣, a_maxs1y␣, and

1y␣ _{Ž .} 1y␣

b_{ma x})1q␣. iii For all c and ␣ / 0, b_max)1q␣. Ž .

Proof. Consider the component form of 7 . We get that

1 1 q X_{i j}

ž

␻ 1 q ␣i

Ž

.

␻ 1 y ␣j

Ž

.

/

n n 1 c_k 1 c_k s c 1 q

ž

_Ý

Xi k

/ ž

1q

_Ý

Xk j

/

.

Ž

12

.

2ks1 ␻k 2 ks1 ␻k Ž .

Multiplying Eq. 12 by c c and summing the resulting equation, we have_{i j}

a# b# c

q s

Ž

1q a# 1 q b# .

. Ž

.

Ž

13

.

1q␣ 1y␣ 2

Ž .

The first assertion of the lemma now follows from 13 and the fact that Ž .

for cs 1 and ␣ s 0, X_mins X_max. After some algebra, 13 reduces to

1y␣ a# y 1 q ␣ 1q␣ b# y 1 y ␣

Ž

.

Ž

.

Ž

.

Ž

.

s 1 y c 1 y␣2 _{a# q 1 b# q 1 .}

14

Ž .

Noting that the right-hand side of 14 is nonnegative we thus conclude, via the fact that X_{ma x}G X , that_min

1q␣ 1y␣

aminF and bminF ,

Ž

15a

.

1y␣ 1q␣ and 1q␣ 1y␣ ama xG and bmaxG .

Ž

15b

.

1y␣ 1q␣ Ž .

Note also, via 14 , that for cs 1, we have

1q␣ 1y␣

U

␣ s or b# s .

1y␣ 1q␣

We next show that for cs 1 and ␣ / 0, it is impossible to have both

1q␣ 1y␣ _{Ž .}

a# s1y␣ and b# s1q␣. To see this, multiplying 12 by c c_{i j}␻ , and_i c c_{i j}␻ , respectively, and summing the resulting equations, we get, respec-_j tively, n n 2␤# 1 1 a# q

_{Ý Ý}

c c Xi j i js q␤# q q␤#a# , 16a

Ž

.

1y␣ 1q␣_{is1 js1} 2 2 and n n 2␣# 1 1 b# q

_{Ý Ý}

c c xi j i js q␣# q q␣#b# . 16b

Ž

.

1q␣ 1y␣ _{is1 js1} 2 2

We have used the property of Gauss᎐Legendre nodes and weights, i.e.,

1

n 1

Ž . Ž . Ž .

Ýis1 ic␻ s H ␻ d␻ s , to justify 16 . Now, multiplying 16a and 16b ,i 0 2

Ž . Ž .

respectively, by 1q␣ and 1 y ␣ , and taking the difference of the resulting equations, we get

1q␣ 1y␣ 2

_ž

␤# y ␣#

_/

1y␣ 1q␣ 1q␣ 1q␣ s␣ q 1 q ␣ ␤# y 1 y ␣ ␣# q

Ž

.

Ž

.

ž

/

a# y

ž

/

b0 2 2 q 1 q

Ž

␣ a# ␤# y 1 y ␣ ␣#b#.

.

Ž

.

Ž

17

.

1q␣ 1y␣ _{Ž .}

If a# s1y␣ and b# s1q␣, Eq. 17 would then yield that 4␣

␥ [ _1y␣2 s 0,

Ž

.

1q␣

and, hence, ␣ s 0, which is a contradiction. Hence, either a# /1y␣ or

1y␣

b# /1q␣. Moreover, for cs 1 and ␣ / 0, it is impossible to have

1q␣ 1y␣

amins , bmin- .

Ž

18

.

1y␣ 1q␣

Ž . Ž .

If these were the case, substituting 18 into 17 , we would have

1q␣ 1y␣

␬ [ ␣ q

_ž

_/

a_miny

_ž

_/

b_min- 0.

2 2

However, this is not possible since

2 2 1q␣ 1y␣

Ž

.

Ž

.

␬ ) ␣ q y s␥ ) 0. 2 1

Ž

y␣

.

2 1

Ž

q␣

.

Ž . We thus complete the proof of the first assertion of Lemma 3.1 ii . The

1y␣

Ž .

second part of Lemma 3.1 ii can be similarly obtained. If bma xs1q␣, it

1y␣

Ž .

follows from 14 that cs 1. However, for c s 1 and ␣ / 0, bma x)1q␣. Ž .

Hence, the assertion in Lemma 3.1 iii holds as claimed. We are now ready to study the eigenvalues of Ay X#C.

Ž . Ž . 

LEMMA3.2. i The spectrum, ␴ A y X C , of A y X C is ␮ , ␮ ,min min

˜ ˜

1 2

4

. . . ,␮ , where ␮ ) 0 for i s 2, 3, . . . , n. Moreo

_˜

_n

_˜

_{i ¨}er, ␮ s 0 at c s 1;

_˜

₁

Ž . Ž . 

otherwise, ␮ ) 0. ii The spectrum ␴ A y X

_˜

_{1 max}C of Ay X_maxC is ␮ ,₁

4 Ž

␮ , . . . , ␮ , where ␮ ) 0 for i s 2, 3, . . . , n. Moreo_{2 n i} ¨er, ␮ ) 0 resp.,₁

2

. Ž .

s 0, - 0 if pc,␣[c 1Ž q␣. y 1 ) bmax resp., s b , - bmax max . In particu-lar,␮ s 0 at c s 1 and ␣ s 0, for c s 1 and ␣ / 0, ␮ - 0, and ␮ ) 0_{1 1 1}

Ž .

if c 1q␣ - 1.

Ž . T

Proof. We rewrite Ay X#C as A y X#C s D y e q X#q q [₁

T w x _{Ž .}

D₁y q#q , where D s diag₁ ␦ , ␦ , . . . , ␦ is defined as in 2 . Using the_{1 2 n} Gaussian elimination technique, we see readily that ␦ , i s 1, 2, . . . , n, arei not eigenvalues of Ay X#C. Thus, for ␭ / ␦ , i s 1, 2, . . . , n, we havei that

det Ay X#C y␭I s det D y ␭I y q#qT

Ž

.

_Ž

1

_.

y1 T

s det D y

Ž

1 ␭I det I y D y ␭I

.

_Ž

Ž

1

.

q#q

_.

[ det D y

Ž

1 ␭I f ␭ ,

. Ž .

where n _{q q#}

Ž

.

i i f

Ž .

␭ s 1 y

_Ý

.

Ž

19

.

␦ y ␭i is1

Hence, finding eigenvalues of Ay X#C is equivalent to locating the roots Ž .

of f ␭ . Clearly, ␮ and ␮ , i s 2, 3, . . . , m, lie between ␦i

˜

i iy1 and ␦ , and,i hence, are all greater than zero. To see the sign of ␮ and ␮ , we note that1

˜

i

Ž .

c 1q␣

Ž . w x

f 0 s 1 y 2 1q b# . Clearly, for c s 1 and b# s b , it followsmin

Ž . Ž .

from Lemma 3.1 that f 0 s 0. The last assertion of Lemma 3.2 i follows Ž .

directly from Theorem 2.3 and Lemma 3.1 ii . For b# s b ,ma x

c 1

Ž

q␣

.

c 1

Ž

q␣ b

.

ma x

f 0

Ž .

s 1 y y .

2 2

Ž . Ž . Ž .

Hence f 0 - 0 if pc,␣- b ; f 0 s 0 if pmax c,␣s b ; f 0 ) 0 if pmax c,␣)

Ž .

b_{ma x}. The last assertions of Lemma 3.2 ii follow directly from Lemma 3.1. We are now ready to state our stability results. Let

Ss c,



Ž

␣ : 0 - c F 1, 0 F ␣ - 1 .

.

4

Ž .

THEOREM3.3. i The steady state Xmin is locally asymptotically stable for Žc,␣ g S y 1, 0 and is neutral stable for c s 1 and ␣ s 0. ii The. Ž .4 Ž .

Ž . Ž .4

steady state X_{ma x} is unstable for c,␣ g S y 1, 0 .

Ž w x. Ž XŽ ..

Proof. It is well known see, e.g., 1 that the spectrum ␴ FF X# of

X Ž . F F X# is equal to y␮ y ␭ : ␮ g ␴ A y X#C and ␭ g ␴ D y CX# . 20



Ž

.

Ž

.

4

Ž

.

Ž . Now, the first assertion of Theorem 3.3 follows from Lemma 3.2 i and Theorems 2.1 and 2.2. To complete the proof, it then suffices to show that

Ž . Ž .4

for c,␣ g S y 1, 0 ,

␮ ) ␮1 1 if ␮ G 0.1

Ž

21

.

Ž . Ž . Ž . Ž .

To this end, let g ␭ s f y␭ , where f ␭ is given as in Theorem 2.1 , and define n _q n _{q q y 1}

Ž

.

i i max i h

Ž .

␭ [ f ␭ y g ␭ s

Ž .

Ž .

_Ý

y

_Ý

. d_iq␭ ␦ y ␭_i is1 is1

For 0F␭ - ␦ ,₁ n _q n _q

_Ž

_q

_.

_{y 1} i i max i X h

Ž .

␭ s y

_Ý

₂ y

_Ý

₂ - 0, d q␭ ␦ y ␭

Ž

.

Ž

.

is1 i is1 i c c Ž . Ž . Ž . Ž .

and h 0 s 1 y2 ␣ y 1 q ␣ b2 _{ma x}- 0. We have used Lemma 3.1 iii

Ž . Ž .

to justify the last inequality. Therefore, f ␭ - g ␭ for all 0 F ␭ - ␦ .1

Ž .

Hence, 21 holds as claimed. We thus complete the proof of the theorem.

IV. GLOBAL EXISTENCE

Our objective in this section is to investigate the global solution of Eq. Ž .1 . We note that the local version of the main result, Theorem 4.2, in this

w x section is a direct consequence of Theorems 9.1 and 9.2 of Reid 9 . To

Ž . Ž .

study the global solution of Eq. 1 , we first rewrite 1 as an equivalent Ž .

integral formulation. To this end, we begin with writing Eq. 1 as

XXq D X q XD s B q eqT_{Xq Xqe}T_{q XCX. 22}

Ž

.

1 2

Ž .

Premultiplying and postmultiplying Eq. 22 by the integration factors eyŽ zys. D1 _{and e}yŽ zys. D2_{, respectively, and integrating the resulting}

equa-tion with respect to s from 0 to z, we obtain X z

_{Ž .}

s eyz D1_{X e}yz D2 0 z yŽ zys. D1 T T q

_H

e Bq eq X s q X s qe

Ž .

Ž .

0 yŽ zys. D2 qX s CX s e

Ž .

Ž .

ds 4 yz D1 yz D2 [ WX z [ e

Ž

. Ž .

X e0 q

_Ý

Ž

W Xi

. Ž .

z ,

Ž

23

.

is1 where the operators W , ii s 1, 2, 3, 4, are defined as

z

yŽ zys. D1 yŽ zys. D2

W X z s e Be ds,

Ž

1

. Ž .

_H

0

z

yŽ zys. D1 T yŽ zys. D2

W X z s e eq X s e ds,

Ž

2

. Ž .

_H

Ž .

0

z

yŽ zys. D1 T yŽ zys. D2

W X z s e X s qe e ds,

Ž

3

. Ž .

_H

Ž .

0

and

z

yŽ zys. D1 yŽ zys. D2

W X z s e X s CX s e ds.

Ž

4

. Ž .

_H

Ž .

Ž .

 Ž m ._{Ž .4}

Let us define the standard Picard iteration X z by

XŽ0. _z s 0 _24a

Ž .

Ž

.

XŽ mq1. z s WXŽ m . _{z . 24b}

Ž .

Ž .

Ž

.

Ž . Ž .

Notation: Let As a and B s b be two square matrices of the samei j i j size; we shall write AG B if a G b for all i, j.i j i, j

LEMMA 4.1. If X_minG X G 0, then W is a monotone operator and₀

Ž m ._{Ž .} Ž mq1._{Ž .} w _.

0F X z F X z F X_min for all zg 0, ⬁ and all m g N. More-o_¨er, XŽ m . _{is nondecreasing in z pro}¨_{ided that B}y D X y X D G 0.

1 0 0 2

Proof. It is clear that W is a monotone operator provided X₀G 0. The first two inequalities are a direct consequence of an induction. To see the

Ž my1._{Ž .}

last inequality, assuming that X z F X_min for all z, we have that XŽ m .

_{Ž .}

_z F eyz D1X eyz D2 0 z yŽ zys. D1 T T q

_H

e Bq eq X_minq X_minqe 0 yŽ zys. D2 qX CX_{min min} e ds z

yz D1 yz D2 yŽ zys. D1

w

x

yŽ zys. D2

s e X e₀ q

_H

e D X_{1 min}q X_minD₂ e ds 0 s X y eyz D1

_Ž

_X y X e

_.

yz D2 min min 0 F X .min Ž my1._{Ž .}

To complete the proof, we assume that X z is nondecreasing in z. Set

KŽ my1.

Ž .

z s B q eqTXŽ my1.

Ž .

z q XŽ my1.

Ž .

z qeT q XŽ my1. _{z CX}Ž my1. _{z ,}

Ž .

Ž .

Ž my1._{Ž .} Ž my1._{Ž .}

and, hence, K z is increasing. Differentiating WX z with

respect to z, one obtains that d Ž my1. WX

Ž .

z dz s yeyz D1 _{D X} q X D eyz D2q KŽ my1. _z

Ž

1 0 0 2

.

Ž .

z

yŽ zys. D1 Ž my1. Ž my1. yŽ zys. D2

y

_H

e D K1

Ž .

s q K

Ž .

s D2 e ds 0 G yeyz D1

_Ž

_{D X} q X D e

_.

yz D2q eyz D1_KŽ my1.

_{Ž .}

_{z e}yz D2 1 0 0 2 s eyz D1

_Ž

By D X y X D e

_.

yz D2G 0. 1 0 0 2

Ž my1._{Ž .}

The fact that K s are increasing in s has been used to justify the first inequality above.

Ž .

THEOREM 4.2. i Let 0- c F 1, and let 0 F␣ - 1. Moreo¨er, the initial ¨alue X₀ is so small that X_minG X G 0, B y D X y X D G 0._{0 1 0 0 2}

Ž m ._{Ž .}

Then the sequence X z con¨erges pointwise to a continuous function

Ž⬁._{Ž . w . Ž .} Ž⬁._{Ž . Ž . w .}

X z on 0,⬁ . ii X z is an nondecreasing function in z on 0,⬁ ,

Ž . Ž . Ž⬁._{Ž .}

which is a global solution of 1 . iii The limit of X z as zª ⬁ exists, say

Ž⬁. _{Ž .} Ž⬁. _{Ž .}

X . iv Moreo¨er, the limit X is a solution of steady-state Eq. 7 . Furthermore, XŽ⬁.s X .min

Ž . Ž . Ž .

Proof. The assertions of Theorem 4.2 i , ii , and iii follow from the Monotone Convergence Theorem and Lemma 4.1. To complete the proof of the last assertion of the theorem, we need to show that X is a solution_⬁

Ž .

of 7 , or, equivalently, X satisfies_⬁

⬁ ys D1 T T ysD2 X_⬁s

_H

e Bq eq X q X qe q X CX e_{⬁ ⬁ ⬁ ⬁} ds 0 n [

_Ý

lim W z X ,i

Ž .

⬁ zª⬁ is1 where z ys D1 ysD2 W z X1

Ž .

⬁s

_H

e Be ds, 0 z ys D1 T ysD2 W2

Ž .

z X⬁s

_H

e eq X e⬁ ds, 0 z ys D1 T ysD2 W3

Ž .

z X⬁s

H

e X qe e⬁ ds, 0 and z ys D1 ysD2 W z X4

Ž .

⬁s

_H

e X CX e⬁ ⬁ ds. 0 wŽ .Ž . Ž .Ž .x

To this end, we need to show that lim_zª⬁ W X_i z X_⬁y W X z s 0,₂

Ž .

is 1, 2, 3, 4. Here W are defined in 24a . We illustrate only i s 2, 4; the_i other limits can be similarly obtained. Now,

z

ys D1 T ysD2

W2

Ž .

z X⬁y W X z s

Ž

2

. Ž .

_H

e eq

Ž

X⬁y X z y s e

Ž

.

.

ds.

w x

Dividing the integration interval 0, z into two parts, we have the following estimates: zr2_{eys D}1_eqT

_Ž

_X y X z y s e

_Ž

_.

_.

ysD2_ds

H

⬁ 0 zr2 _{ys D T z ysD} 1 2 F

_H

e eq

_Ž

X_⬁y X

_{Ž .}

₂

_.

e ds 0 zr2 _{ys D y1 T z} 1 F

H

e D D1 1 eq

Ž

X⬁y X

Ž .

2

.

ds 0 z y1 T F D eq X y X1

Ž

⬁

Ž .

2

.

. Furthermore, z ys D1 T ysD2 e eq

Ž

X y X z y s e

Ž

.

.

ds

H

⬁ zr2 z ys D1 y1 T F

_H

e D D1 1 eq

Ž

X⬁y X ds0

.

zr2 F eyŽ zr2. D1_Dy1_qT

_Ž

_X y X .

_.

1 ⬁ 0 Ž .

The fact that X z is increasing in z has been used to justify the above inequalities. We now turn to the next estimate:

W z X4

Ž .

⬁y W X z

_Ž

4

_.

Ž .

z

ys D1 ysD2

s

_H

e

Ž

X CX⬁ ⬁y X z y s CX z y s e

Ž

.

Ž

.

.

ds.

0

We have, via similar estimates, that

zr2 _{ys D ysD} 1 2 e

Ž

X CX y X z y s CX z y s e

Ž

.

Ž

.

.

ds

H

⬁ ⬁ 0 z z y1 F D1

Ž

X CX⬁ ⬁y X

Ž .

2 CX

Ž .

2

.

, and that z ys D1 ysD2 e

Ž

X CX y X z y s CX z y s e

Ž

.

Ž

.

.

ds

H

⬁ ⬁ zr2 F eyŽ zr2. D1Dy1

_Ž

X CX y X CX .

_.

1 ⬁ ⬁ 0 0 Ž .Ž . Ž .

Therefore, W X_i z y W z X can be made arbitrarily small by choosing_{i ⬁} Ž .

Ž n._{Ž .}

be equal to X_min. Since 0F X z F X_minfor all n and z, and X_⬁F X ,_min then X_⬁s X ._min

REFERENCES

1. R. Bellman, ‘‘Introduction to Matrix Analysis,’’ 2nd ed., McGraw-Hill, New York, 1970. 2. R. Bellman, R. E. Kalaba, and C. Prestrud, ‘‘Invariant Imbedding and Radiative Transfer

in Slabs of Finite Thickness,’’ Elsevier, New York, 1963.

3. R. Bellman and G. M. Wing, ‘‘An Introduction to Invariant Imbedding,’’ Wiley, New York, 1975.

4. I. W. Busbridge, ‘‘The Mathematics of Radiative Transfer,’’ Cambridge Univ. Press, LondonrNew York, 1960.

5. S. Chandrasekhar, ‘‘Radiative Transfer,’’ Dover, New York, 1960.

6. F. Coron, Computation of the asymptotic states for linear half space kinetic problem, Ž .

Transport. Theory Statist. Phys. 19, No. 2 1990 , 89᎐114.

7. B. D. Ganapol, An investigation of a simple transport model, Transport Theory Statist. Ž .

Phys. 21, Nos. 1 and 2 1992 , 1᎐37.

8. J. Juang and W. W. Lin, Nonsymmetric algebraic Riccati equations and Hamiltonian-like Ž .

matrices, SIAM J. Matrix Anal. Appl. 20, No. 1 1998 , 228᎐243.

9. W. T. Reid, ‘‘Riccati Differential Equations,’’ Academic Press, New York, 1972. 10. A. Shimizu and K. Aoki, ‘‘Application of Invariant Embedding to Reactor Physics,’’

Academic Press, New York, 1972.