Riccati Differential Equations (Mathematics in Science & Engineering, volume 86)

RlCCATl DIFFERENTIAL EQUATIONS

This is Volume 86 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

RICCATI DIFFERENTIAL EQUATIONS WILLIAM T. REID Department of Mathematics University of Oklahoma, Norman, Oklahoma

@

ACADEMIC PRESS

New York and London

1972

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CONTENTS

Preface

...............................

Chapter O n e

.Scalar Riccati Differential Equations

1 . Introduction . . . . . . . . . . . . . . 2 Related Linear Differential Systems . . . 3. Notes and Remarks . . . . . . . . . . .

.

Chapter Two

ix

............. . . . . . . . . . . . . . . .............

1

2 7

.Basic Properties of Solutions of Riccati Matrix Differential Equations

.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Related Riccati Equations and Linear Systems . . . . . . . . . . . . . Variation of Solutions . . . . . . . . . . . . . . . . . . . . . . . Transformations for (2.1) and (2.3M) . . . . . . . . . . . . . . . . . Associated Riccati Matrix Differential Equations . . . . . . . . . . . . Normality and Abnormality . . . . . . . . . . . . . . . . . . . . Distinguished Solutions of a Riccati Matrix Differential Equation . . . . Generalized Linear Differential Systems and Kiccati Matrix Integral Equa.............................. tions .9. A Class of Monotone Riccati Matrix Differential Operators . . . . . . 10. Results Related to the Perron-Frobenius Theorem . . . . . . . . . . 11 A Dissipative Property of Solutions of Riccati Matrix Differential Equations 12 Certain Results for Equation (2.1) for the Case m = n . . . . . . . . . 13 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . References ...........................

1 2. 3. 4. 5. 6. 7. 8.

. . .

9 10 19 28 34 36 45 49 61 74 77 84 87 88

Chapter Three . Involutory Riccati Matrix Differential Equations 1. 2. 3. 4. 5. 6.

Basic Definitions and Preliminary Concepts . Examples of Involutory Systems . . . . . . Properties of Solutions of Involutory Systems . Transformations for (1.2) and (1.3J . . . . . Obverse Involutory Differential Systems . . . Notes and Remarks . . . . . . . . . . . . . vii

. . . . . . . . . . ..

. . . . . .

. . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......

89 94 99 104 107 108

viii

CONTENTS

Chapter Four 1. 2. 3. 4 5. 6. 7. 8. 9. 10 11.

.

.

.Hermitian Riccati Matrix Differential Equations

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary Properties of Solutions . . . . . . . . . . . . . . . . . Distinguished Solutions of (1.2) . . . . . . . . . . . . . . . . . . . Hermitian Riccati Equations Which Are Definite . . . . . . . . . . . A Fundamental Property of Solutions of Riccati Matrix Differential Equations A Class of Monotone Matrix Differential Equations . . . . . . . . . . Properties of Hermitian Systems (1.3J Which Are Definite . . . . . . . Existence of a Principal Solution for an Hermitian System (1.31) . . . . Hermitian Systems with Constant Coefficients . . . . . . . . . . . . . A Matrix Method of Atkinson . . . . . . . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter

Five

.Applications

109 110 115 118 124 129 134 153 160 162 167 167

of Riccati Matrix Differential

Equations

.

1 2. 3. 4. 5. 6. 7 8. 9 10

. . .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Applications . . . . . . . . . . . . . . . . . . . . . . . . . Applications Arising from Partial Differential Equations . . . . . . . . An Introduction to Hamilton-Jacobi Theory . . . . . . . . . . . . . A Linear Regulator Problem . . . . . . . . . . . . . . . . . . . . A Problem in Linear Filtering and Prediction Theory . . . . . . . . . . The Mycielski-Paszkowski Diffusion Problem . . . . . . . . . . . . . The #-Product of Redheffer . . . . . . . . . . . . . . . . . . . . A Problem in Invariant Imbedding . . . . . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

207

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

213

References Index

169 169 172 176 184 195 197 201 203 206 206

PREFACE

The study of scalar Riccati differential equations dates from the early period of modern mathematical analysis, since such equations represent one of the simplest types of nonlinear ordinary differential equations. Because of their intimate connection with the theory of Bessel functions, these equations have occurred frequently in problems of mathematical physics. In pure mathematics they have proved to be of basic importance in such diverse disciplines as the projective differential geometry of curves and the calculus of variations. Within recent years there has been wide occurrence of matrix Riccati differential equations, notably in variational theory and the allied areas of optimal control, invariant imbedding, and dynamic programming. The purpose of this volume is to present in a unified form a compendium of results for such matrix differential equations, and to indicate briefly various problems in which they occur. Chapter I presents a brief survey of basic properties of scalar Riccati differential equations. Chapter I1 is devoted to the Riccati matrix differential equation in a general finite-dimensional m x n matrix function. Many of the results of this chapter have been presented previously for only the special case where m = n. Chapter I11 is concerned with a class of equations in n x n matrix functions which are either symmetric (i.e., involutory with respect to the transpose operation), or hermitian (i.e., involutory with respect to the conjugate transpose operation). Chapter IV treats hermitian Riccati matrix differential equations, with results on their significance in the Sturmian theory for linear Hamiltonian differential systems, the characterization of distinguished solutions, and various monotoneity properties possessed by the solutions of certain classes of such equations. Finally, in Chapter V there are described some occurrences of Riccati matrix differential equations in various fields, including transmission line phenomena, theory of noise and random processes, variational theory per se, optimal control theory, diffusion problems, and invariant imbedding. ix

X

PREFACE

There is no claim that the subject coverage is universal, or that the list of references at the end of the book is complete. In particular, the subject content has been conditioned greatly by certain aspects of the field in which the author has made personal contributions. It is felt that the bringing together of these results will be of use to the pure mathematician, and also to workers in areas of application such as those mentioned above. From the standpoint of generality, the subject material of Chapters 11-IV has been presented in the context where “solution” is taken in the CarathCodory sense as an absolutely continuous matrix function which satisfies the differential equation almost everywhere. For the reader not familiar with the Lebesgue integral, however, there should be no difficulty in following the exposition if he assumes that all occurring matrix functions are continuous, and integration is in the sense of the Riemann integral of elementary calculus. References to numbered theorems and formulas in a chapter other than the one in which the statement appears include an adjoined Roman numeral indicating the chapter of reference, while references to similar items in the current chapter do not contain the designating Roman numeral. For example, in Chapter 111 a reference to Theorem 3.1 or formula (2.15) of Chapter I1 would be made by citing Theorem II:3.1 or formula (11:2.15), whereas a reference to Theorem 1.1 or formula (2.1) would mean the designated theorem or formula in Chapter 111. Considerable portions of the material in Chapters I1 and IV are to be found in papers written by the author in past years when he was the recipient of research grants from the Air Force Office of Scientific Research, and to this supporting agency grateful acknowledgment is made. The author is also deeply grateful to Mrs. Carolyn Johnson for her typing of the manuscript.

SCALAR RlCCATl DIFFERENTIAL EQUATIONS

1. Introduction

An ordinary differential equation of the form

(1.1)

t [ w ] ( t )= w'(t) + f ( t ) w ( t )

+ b(t)W2(t)- c ( t ) = 0

is known as a Riccati equation, or a generalized Riccati equation, deriving its name from Jacopo Francesco, Count Riccati (1676-1754), who, in 1724 (see Riccati [l] of Bibliography), considered the particular equation

(1.2)

w ' ( t ) + t-"w2(t)- ntrn+"-l= 0,

where m and n are constants. At an early stage the occurrence of such equations in the study of Bessel functions led to its appearance in many related applications, and to the present time the literature on scalar equations of the form (1.1) has been extensive. Within recent years much attention has been directed to the study of the qualitative nature of solutions of this scalar equation and its matrix generalizations, and it is in this latter spirit that the present volume is written. For several reasons a differential equation of the form (1.1)) and generalizations thereof, comprise a highly significant class of nonlinear ordinary differential equations. First, equations of the form ( 1 . 1 ) are intimately related to ordinary linear homogeneous differential equations of the second order. In the case of t a real variable, Riccati equations are significant for the oscillation and comparison theorems for the related linear second-order homogeneous differential equations. An associated occurrence is that as the so-called "Legendre differential equation" for a simple integral problem of the calculus of variations (see, for example, 1

2

I. SCALAR RlCCATl DIFFERENTIAL EQUATIONS

Bolza [ l , Secs. 9, lo]). Second, the solutions of an equation (1.1) possess a very particular structure in that the general solution is a fractional linear function of the constant of integration. Third, when (1.1) is a differential equation in the complex plane with analytic coefficients, it has movable singularities which are simple poles (see Ince [l, Chap. 121). Fourth, the solution of an equation of the form (1.1) is involved in the reduction of an nth-order linear homogeneous ordinary differential equation to the “Forsyth-Laguerre canonical form” in which the coefficients of the derivatives of order n - 1 and n - 2 are zero (see Wilczynski [l, Chap. 11, Sec. 41). I n applications, Riccati differential equations (1.1) and its generalizations appear in the classical problems of the calculus of variations, and in the associated disciplines of optimal control and dynamic programming. 2.

Related Linear Differential Systems

I n this section we shall consider the relation of a scalar Riccati differential equation (2.1)

t [ w ] ( t )= w ’ ( t )

+ [ a ( t )+ d ( t ) ] w ( t )+ b ( t ) w * ( t )- c ( t ) = 0,

to a linear homogeneous first-order differential system (2.2)

L,[u, w](t) = - v ’ ( t ) LJU, v ] ( t )=

+ c ( t ) u ( t )- d ( t ) v ( t )= 0,

u ’ ( t ) - a ( t ) u ( t )- b ( t ) v ( t )= 0,

where it will be assumed that the coe8cientfunctions a ( t ) , b ( t ) , c ( t ) , d ( t ) are continuous on a given interval I on the real line. In terms of the twodimensional vector function y = (z)(a = 1, 2), with y1 = u, y2 = w, the system (2.2) may be written as (2.2‘) where

2Y’(t)

2 is the

+d ( t l Y ( t )

real skew matrix

and d ( t ) is the 2 x 2 matrix function

= 0,

2. RELATED LINEAR DIFFERENTIAL SYSTEMS

3

I t is to be remarked that if b ( t ) # 0 for t E I then the system may be written as

+ q1(t)u(t)l' - [qdt).'(t> + p ( t ) u ( t ) l = 0,

(2.2")

[r(t)u'(t)

where the coefficient functions a ( t ) , b ( t ) , c ( t ) , d ( t ) of (2.2) and r ( t ) , p ( t ) , q l ( t ) , q 2 ( t ) of (2.2") are related as follows:

(2.4')

r

=

l/b,

b

=

l/r,

q1 = -a/b,

a

=

q2 = -d/b,

-ql/r,

d = -q

2

/T ,

p

=c

c=

+ ad/b,

P - q1q2/y.

I n particular, if r ( t ) and q l ( t ) are continuously differentiable then (2.2") may be written as

(2.2'")

Pdt)u"(t)

+ Pl(t)u'(t)+ P"(t14t)

where

(2.5)

P d t ) = r(t),

Pl(t)

Po(t)

=

r'(t)

= 0,

+ q1(4 - Q d t ) ,

= 41'(t) - P ( t ) *

I n the above equation (2.1), one may obviously replace a ( t ) + d ( t ) by a single functionf(t). From the standpoint of transformations involving solutions of the associated linear differential system (2.2), however, there is an advantage in presenting the coefficient of w ( t ) in Eq. (2.1) as a(t) 4 t h For example, if f(t) is a given continuous function on I , and s E I , then under the transformation

+

(2.6)

w ( t ) = w o ( t )exp

we have

where

(2.8) with

+

+

a(t)

+ d ( t ) -f(t),

fo[wO](t) = ~ " ' ( t ) a0(t)wo(t) b0(t)wo2(t)- ~ " ( t ) ,

a"t)

=

b"(t) = [exp{-

4

1. SCALAR RlCCATl DIFFERENTIAL EQUATIONS

Correspondingly, for arbitrary a(t), d ( t ) continuous on I and such that a(t) d ( t ) = f(t), under the substitution

+

(2.10) o ( t ) = v o ( t )exp

we have

(2.11)

{J:

L,[u, v ] ( t ) = exp ~ [ uv l, ( t ) = exp{

J: a(.)

{J: -

1

d(t)d t Ll0[uo,v o ] ( t ) , d r ) ~ , o [ u ov, o ] ( t ) ,

where L,O[uO, vO](t) = - v " ( t )

(2.12)

Lz0[uo,v o ] ( t )=

I-

d(t) d t ,

+ cO(t)zrO(t),

~ " ( t) bO(t)vO(t),

and bo(t), co(t) are given by (2.9). T h e fundamental relation between the Riccati differential equation (2.1) and the linear differential system (2.2) is presented in the following theorem, which generalizes the result that if u ( t ) is a solution of the differential equation u " ( t ) - p ( t ) u ( t )= 0,

(2.13)

which is such that u ( t ) # 0 for t on a subinterval I, of I, then the logarithmic derivative w ( t ) = u ' ( t ) / u ( t ) is a solution on I, of the Riccati differential equation

(2.14)

w'(t)

+ w"t)

- p ( t ) = 0.

Theorem 2.1

The dtyerential equation (2.1) has a solution w ( t ) on a subinterval I , and only if there is a solution y ( t ) = ( u ( t ) ; v ( t ) ) of (2.2) such that of I u ( t ) # 0 and w ( t ) = v ( t ) / u ( t )for t E I , . In general, if u ( t ) and v ( i ) are functions which are continuously differentiable on a subinterval I , , and u ( t ) # 0 for t E I,, then w ( t ) = w ( t ) / u ( t )is a continuously differentiable function, and one may readily verify the identity

(2.15)

u"t) t [ w ] ( t ) 3

-u(t)L,[u, v ] ( t ) - w(t)L2[u,v ] ( t ) .

5

2. RELATED LINEAR DIFFERENTIAL SYSTEMS

Consequently, if such a pair of functions ( u ( t ) ; v ( t ) ) is a solution of (2.2), then w ( t ) = v ( t ) / u ( t )is a solution of (2.1). Conversely, if w ( t )is a solution of (2.1) on a subinterval I,, let u ( t ) be a solution of the first-order differential equation (2.16)

+ b(t)w(t)Iu(t),

[a(t)

u'(t) =

with u(s) # 0 for same s

E

I,. Then

for t E I,, so that u ( t ) # 0 for all t E I,. Moreover, from the identity (2.15), it follows that the functions u ( t ) , v ( t ) = w ( t ) u ( t )satisfy the equation L,[u, v ] ( t )= 0 on I , , so that ( u ( t ) ; v ( t ) ) is a solution of (2.2) on this subinterval. Theorem 2.2

If w o ( t ) is a solution of the Riccati diflerential equation (2.1) on a subinterval I , of I , and for s E I , the functions g = g ( t , s I w,), .h = h(t, s I w0) and f(t, s I w,) are dejned as

(2.19)

f ( t , s I w,) =

s:

g(., s I w,)b(t)h(t,s I wo) d t ,

then w ( t ) is a solution of (2.1) on I , if and only if the constant y - wo(s) is such that 1 y f ( t , s 1 w,) # 0 for t E I o , and

+

= w(s)

(2.20) The functions g and h of (2.17) and (2.18) may be characterized as the solutions of the respective differential systems (2.17')

g'(t)

(2.18')

h'(t)

+ [ d ( t ) + w,(t)b(t)lg(t)

= 0,

+ [ ~ ( +t ) b(t)w,(t)]h(t)= 0,

g(s) = 1,

h(s) = 1.

Consequently, if wo(t) is a solution of (2.1) on I,, and w ( t ) is a con-

I. SCALAR RlCCATl DIFFERENTIAL EQUATIONS

6

tinuously differentiable function, then w ( t ) is a solution of (2.1) on Z, if and only if on this subinterval the function

is a solution of the special Riccati differential equation

where, as in Theorem 2.2, y 7 w ( s ) - w,(s). Moreover, since the function f = f ( t , s 1 w,) of (2.19) is the solution of the differential system

it may be verified really that if r ( t ) is a solution of (2.21) then r l ( t ) = r(t)[l yf(t, s I w , ) ] - y satisfies the differential system

+

and hence on I , the function r l ( t ) is identically zero and r(t)= r/[l

+ rf(t,s I

WJl.

I n particular, if a ( t ) E d ( t ) on Z then g(t, s I w,) Eq. (2.20) may be written as

E

h ( t , s I w,), and

(2.20‘)

If m , , m , , m 3 , m, are four numbers, and we introduce the notations (1111

, nlf , m 3 } = (m3 - m l ) / ( m 3- m2),

(m19m2,m3,m,}=

{m1,m,,m3}{m,,m1,m,},

then { m , , m , , m 3 , m,} is called the anharmonic ratio, or cross ratio, of m, , m , , m 3 , m,. One of the most important properties of solutions of the Riccati differential equation (2.1) is the result of the following theorem. Theorem 2.3

Zf w , ( t ) (a = 1, 2, 3, 4) are four solutions of (2.1) on a subinterval I, of I, with w 3 ( t ) f: w 2 ( t ) and w,(t) $ w l ( t ) on this subinterval, then { w l ( t ) ,w , ( t ) , w 3 ( t ) ,w 4 ( t ) }is constant on I , .

7

3. NOTES AND REMARKS

From the uniqueness theorem for (2.1) it follows that two solutions of this equation are identically equal on I,, if and only if there exists some value on this subinterval at which they are equal. Consequently, if w n ( t ) and w , ( t ) (u = 1 , 2, 3, 4 ) are solutions of (2.1) on I,, and for a given s E I , we set y r = wz(s) - w n ( t ) ,then y3 - y 2 # 0 and y4 - y1 # 0. Moreover, from Theorem 2.2 it follows that

Consequently, for a , B

3.

=

1 , 2, 3, 4 we have

Notes and Remarks

As mentioned in Section 1 the literature on the scalar Riccati differential equation has been of long and continuing duration. Historically, the cited paper of Glaisher [ l ] provided a comprehensive survey of results on such equations as of 1884. At present the most readily accessible general account is that in Chapters I and IV of the classical work of Watson [ l ] on Bessel functions. Also a fairly comprehensive bibliography is to be found in Davis [l]. As pointed out in Chapter I of Watson [l], the work of Riccati was preceded some twenty years by results of James Bernoulli (1654-1705) on the equation

(3.1)

w’(t) = t z

+ w*(t).

I n particular, in a letter to Leibniz in 1703, James Bernoulli expressed the solution of (3.1) as the quotient of two infinite series. According to Watson, the procedure of James Bernoulli was, effectively, to consider an equation in a new variable u related to w by the formula -urlu = w . In particular, this result of James Bernoulli is a true precursor of the much later work of C. G. J. Jacobi (1804-1851), in relating the solution of the “Legendre differential equation” of variational theory to that of

8

I. SCALAR RlCCATl DIFFERENTIAL EQUATIONS

the “accessory,” or “Jacobi,” linear homogeneous differential equation of the second order; in this connection the reader is referred to Bolza [ l , Sec. 101. In conformity with the general study of differential equations in the eighteenth century, much of the early work was concerned with the study of particular classes of scalar Riccati equations with the aim of determining the solutions in finite form, or with the expression of solutions in terms of specified types of functional transforms. Many mathematicians of that period contributed to the study of such differential equations, including James Bernoulli, John Bernoulli (1667- 1748), Daniel Bernoulli (1700-1782), Leonhard Euler (1707-1783), Jean-le-Rond d’Alembert (1717-1783), and Adrien Marie Legendre (1752-1833). According to Watson [l, p. 31, the designation of (1.1) as “Riccati’s equation” was made by d’Alembert in 1763.

BASIC PROPERTIES OF SOLUTIONS OF RlCCATl MATRIX DIFFERENTIAL EQUATIONS

I. Introduction

The present chapter will be devoted to the study of a nonlinear matrix differential equation in an m x n matrix function, which for m = n = 1 reduces to the scalar Riccati differential equation considered in Chapter I. As in the case of the scalar equation, the theory of such matrix differential equations is intimately related to that of an associated linear matrix differential system. The results presented in this chapter are of basic significance, both as individual results and also as properties which will be used in later chapters for the study of particular types of such equations and in applications. Matrix notation is used throughout; in particular, matrices of one column are termed vectors, and for a vector y = ( y a )(a = 1,. . . , n), the norm I y I is given by (I y 1 l2 . . I y , I 2)1/2 ; the linear vector space of ordered n-tuples of complex numbers, with complex scalars, is denoted by Q, . The n x n identity matrix is signified by E n , or by merely E when there is no ambiguity. If M = [Map](a = 1, . . . , m ; = 1, . . . , n) is an m x n matrix, then i? denotes the conjugate of M whose elements are the complex conjugates of the elements of M , &? denotes the n x m transpose [MBa],and M" the conjugate transpose fi. The m x n zero matrix is denoted by Om,, or by merely 0 when there is no ambiguity as to its dimensions. If M is an m x n matrix the symbol v [ M ] is used for the maximum of I M y I on the unit ball { y I I y 1 5 l } in Q,, which is also the maximum of 1 f i z 1 or 1 M*z 1 on the unit ball {z 1 1 z 1 5 1) in Q,. The notation M 2 N { M > N } is used to signify that M and N are hermitian matrices of the same dimensions and M - N is a non-

+ - +

9

10

II. BASIC PROPERTIES OF SOLUTIONS

negative {positive} definite hermitian matrix. If M 2 0, then signifies the unique nonnegative definite hermitian square root of M ; if M > 0, then M-l12 denotes the inverse of M112.If an hermitian matrix function M ( t ) , t E I , is such that M ( s ) - M ( t ) 2 0, { S O } for (s, t ) E I x I , s < t, then M ( t ) is said to be nonincreasing {nondecreasing} hermitian on I . If the elements of a matrix M ( t ) are a.c. (absolutely continuous) on an interval [a, b ] , then M ' ( t ) signifies the matrix of derivatives at values for which these derivatives exist and the zero matrix elsewhere ; correspondingly, if the elements of M ( t )are (Lebesgue) integrable on [a, b ] then J: M ( t ) dt denotes the matrix of integrals of respective elements of M ( t ) . If M ( t ) and N ( t ) are equal a.e. (almost everywhere) on a common domain of definition we write simply M ( t ) = N ( t ) . A matrix function M ( t ) is called continuous, integrable, etc., when each element of the matrix possesses the specified property. Also, M ( t ) is said to be locally a.c. or locally of b.v. (bounded variation) on an interval I if it is a.c. or of b.v. on arbitrary compact subintervals [a, b ] of I . If M = [ M u j ] , N = [N,,], (a = 1, . . . , n ; j = 1, . . . , I ) , are n x r matrices, for typographical simplicity the symbol ( M ;N ) is used to denote the 2n x r matrix whosejth column has elements MIj, . . . , Mnj, N l j , . . . ,N n j . For a given compact interval [a, b ] on the real line the symbols @,,[a, b ] , t n r [ a , b ] , p g r [ a , b ] , 1 < p < 03, PE[a, b ] , %,,[a, b ] , and BBnr[a,b ] are used to denote the classes of n x r matrix functions M ( t ) = [Mus(t)] (a = 1, . . . , n ; /? = 1, . . . , r ) , which on [a, b ] are respectively continuous, (Lebesgue) integrable, (Lebesgue measurable with I Mus(t)I p integrable, measurable and essentially bounded, a.c., and of b.v. on [a, b ] . For brevity the symbols @,[a, b ] , f?,[a,b ] , t n p [a ,b ] , t n m [ a ,b ] , %,[a, b ] , and 9Bn[a, b] are written for the respective classes specified by indices n, r with r = 1.

2.

Related Riccati Equations and Linear Systems

I n the following discussion the matrix functions A , B , C , D will be supposed to be defined on a given interval I on the real line, and satisfy the following hypothesis :

. - I

D ( t ) E P.mm[a,b ] for arbitrary compact subintervals [a, b ] o f I .

11

2. RELATED EQUATIONS A N D LINEAR SYSTEMS

T h e Riccati matrix differential equation to be considered is of the form

(2.1)

+

+

R[wl(t) = W'(t) -

+ W(t)B(t)W(t)

W(t)A(t) D(t)W(t) C ( t )= 0.

By definition, a solution of (2.1) is an m x n matrix function W ( t )which is locally a.c. and such that R[w(t)= 0 for t on a subinterval I , of I . If W ( t ) ,t E I , , is a solution of (2.1), for s E I , let U ( t ) denote the n x n matrix function that is the solution of the linear matrix differential equation

(2.2)

U ' ( t )= [ A ( t )+ B ( t ) W ( t ) ] U ( t ) ,

U ( S )= M ;

where M is nonsingular; i.e., U ( t ) is a locally a.c. n x n matrix function which satisfies the initial condition U(s)= M , and U'(t) is equal to [ A ( t )+ B ( t ) W ( t ) ] U ( t )a.e. on I . Then the m x n matrix function V ( t )= W ( t ) U ( t )is such that ( U ( t ) ;V ( t ) )is a solution on I,, of the corresponding linear (Hamiltonian) matrix differential system

(2.3,)

+

L,[U, V ] ( t )3 - V ( t ) C ( t ) U ( t )- D ( t ) V ( t )= 0, L,[U, V ] ( t )= V ( t )- A ( t ) U ( t )- B(t)V(t)= 0.

Conversely, if ( U ( t ) ;V ( t ) )is a solution of (2.3nl) with U ( t ) nonsingular on a subinterval I , of I , then W ( t )= V( t ) U - I(t)is a solution of (2.1) on I,,. Associated with the matrix differential system (2.3,) is the vector differential system

(2.31)

as

L,[u,v ] ( t )= -v'(t)

+ C(t)u(t)- D(t)v(t)= 0,

L J U , v ] ( t )= u'(t) - A ( t ) u ( t )- B(t)v(t)= 0.

If the (m + n) x (m

+ n) matrix functions 3 and a(t) are defined +

and we set y = (y o )= ( u ; v ) (a = 1, . , . , m n), with ya = u, (a = 1, . . . , n),y,z+s= vs (/3 = 1, . . . , m),then (2.3,) may be written as

(2.3,')

L 7 [rl(t) = Y y ' ( t )

+ 'Ir(t)y(t)

= 0.

Correspondingly, if Y = ( Y o j ) (a = 1, . . . , 2n; j = 1, . . . , Y), and Yaj= Uaj(a = 1. . . . , n), Yn+s,j = Vsj (/I = 1, . . . , m),then the gen-

11

II. BASIC PROPERTIES OF SOLUTIONS

era1 matrix system (2.3,)

may be written as

+

5? [ Y ] ( t )= ZY'(t) % ( t ) Y ( t )= 0.

(2.3')

I t is to be noted that if W ( t )and Wo(t)are m x n matrix functions which are locally a.c. on a subinterval I , of I , then the matrix function Y(t)= W ( t )- W,(t) satisfies the identity

(2.4) R [ W l -

R[Wo]

=

Y'

+ Y ( A + BWo) + ( D + W$)Y + YBY.

Lemma 2.1

If Wo(t)is a solution of (2.1) on a subinterval I , of I , and for s E I , the matrix functions G ( t ) = G ( t , s I W,), H ( t ) = H ( t , s I W,) are dejned as the solutions of the linear matrix differential equations

+ ( D + W,J)G = 0, H' + H ( A + BW,) = 0,

(2.5 1

G'

(2.6)

G(s)= E m ,

H(s)= En,

and

then an m x n matrix function W ( t ) is a solution of (2.1) on I , q and only qthe constant matrix r = W(s)- W,(s)is such that E,+F(t, s 1 W,)r is nonsingular on I,, and (2.8)

w(t)= Wo(t)+ G ( t , s I Wo)r[&+ F(t, s I Wo)r]-'H(t,s I Wo)*

In particular, it is to be noted that if W o ( t )is a solution of (2.1) on I,, and ( U o ( t ) ;V,(t)) is a solution of (2.3) with U o ( t )nonsingular and such that Wo(t)= V,(t)U;'(t) on this interval, then for s E I , the solution of (2.6) is given by

H ( t , s I W,) = U,(s)U;'(t).

(2.91

If R[W,](t) = 0 for t E I,, and for an arbitrary m x n matrix function W ( t ) we set Y ( t )= W ( t )- Wo(t),then in view of the relations (2.4), (2.5), and (2.6) it follows that W ( t ) is a solution of (2.1) on I , if and only if the m x n matrix function R ( t ) defined by Y ( t ) = G ( t , s I W,)R(t)H(t, s I W,) is a solution on I , of the special Riccati matrix differential equation (2.10)

R'

+ R [ H B G ] R= 0,

R(s) = r = W(S) W,(S).

13

2. RELATED EQUATIONS A N D LINEAR SYSTEMS

Now if R ( t ) is a solution of (2.10) on I,, and F ( t , s I W,) is defined by (2.7), then the matrix function R , ( t ) = R ( t ) [ E n F ( t , s I W,)I'] - I' satisfies the linear homogeneous matrix differential equation

+

(2. 10')

R1' = - ( R H B G ) R , ,

R,(s) = 0,

and consequently R , ( t ) = 0 on I,. Moreover, if r E I, and q is a vector of (5, such that [En F(r, s I W,)I']q = 0, then 0 = R l ( r ) q= -Tq, and hence q = [En F(r, s I W,)I']q - F(r, s I W,)(I'q)= 0. Consequently, the matrix function En F ( t , s I Wo)I'is nonsingular for t E I , and

+ +

(2.11)

R(t)= r [ E ,

+

+ F ( t , s I W,)I']-l

for t E I,.

+

Conversely, if I'is a constant m x n matrix such that En F ( t , s I W,)I' is nonsingular throughout I,, then R ( t ) defined by (2.11) is the solution of the differential system (2.10) on this interval, and W ( t )given by (2.8) is the solution of (2.1) satisfying W(s)= W,(s) I'. Now for F and I' arbitrary matrices of respective dimensions n x m and m x n, it follows readily from the identity

+

(2.12)

+

I'(En

+ FI') = (Em+ I'F)I' + +

that En, I'F is nonsingular if and only if En FI' is nonsingular, and in this case we have [Em I'FI-lI'= r [ E , , FI'1-l. Consequently, in the discussion of the preceding paragraph we have that the nonsingularity of En F ( t , s I W,)F for t E I , is equivalent to the nonsingularity of Em I'F(t, s I W,) for t E I,, and we have the following result.

+

+

+

Corollary 1

Under the hypotheses of Lemma 2.1 an m x n matrix function W ( t ) is a solution of (2.1) on I, if and only if the constant matrix F = W(s) - W,(s) is such that Em I'F(t, s I W,) is nonsingular for t E I,, and

+

(2.8') W ( t )= Wo(t)

+ G(t,

s

I W,)[Em + W

t , s I Wo)l-lTqt, s I Wo).

From (2.8) or (2.8') it is immediate that for arbitrary t E I , the rank of W ( t )- W o ( t )is equal to the rank of I',and therefore we have the following result. Corollary 2

If W ( t )and W o ( t )are solutions of (2.1) on a subinterval I , of I , then W ( t ) - W o ( t ) is of constant rank on this subinterwal.

II. BASIC PROPERTIES OF SOLUTIONS

14

Also, in case m = n and r = W(s)- W,(s) is nonsingular, then r [ E , + F ( t , s 1 W,)r]-l = [r-l F ( t , s I W,)]-l, and we have the following representation formula.

+

Corollary 3

If m = n, while W ( t )and W o ( t )are solutions of (2.1) on a subinterval I , of I with r = W(s)- W,(s) being nonsingular for some s E I,, then r-1+ F ( t , s I W,) is nonsingular for t E I , and

The result of Corollary 3 was established by Sandor [l], and in this instance he termed W ( t ) representable with the aid of Wo(t)by (2.8"). The above derivation of relations (2.8) and (2.8') shows that this concept of representability may be given a form independent of the restriction that m = n and W(s)- W(s,) nonsingular. Lemma 2.2

If Wo(t),W,(t) (a = 1, . . . , k), are solutions of (2.1) on a subinterval I , of I , while s E I. and I', = W,(s) - Wo(s),then for a, B = 1, . . . , k and t E I. we have the relation

In view of Lemma 2.1, for t E I , the matrix functions G = G(t,s I W,), H = H(t, s I W,), F = F ( t , s I Wo),and W, - W, = W,(t) - W,(t) are such that

W, - Wo= Gr,[E,

+ Fr,]-'H

=

G[E,

+ r,F]-lr,H,

and for given a,B, Eq. (2.13) is an immediate consequence of the relation

The result of the following lemma may be verified directly.

2. RELATED EQUATIONS A N D LINEAR SYSTEMS

15

Lemma 2.3

If ( U ( t ): V ( t ) )is a solution of (2.3,) with U ( t ) nonsingular on a subinterval I , of I , and W ( t )= V (t )U -l (t ),then for s E I , and

(2.15)

(a) K(t, s I W ) = H - l ( t , s I W ) F ( t ,s I W ) , (b)

L(t, s I W ) = W ( t ) K ( ts, W )

I

+ G(t,s I W ) ,

the fundamental matrix solution of (2.3,) which is the identity for t

(2*16)

U(t)U-l(s) - K ( t , s I W ) W ( s ) - L ( t , s I W)W(s)

[ V(t>U-l(s)

' w'l

=s

is

K ( t 9s I W ) . L(t,

Lemma 2.4

If (U,(t); V,(t)) is a solution of (2.3,) with U,(t) nonsingular on a subinterval I , of I , and W,(t) = V,(t)U;'(t), then for ( U ( t ) ;V ( t ) )any solution of (2.3,) and ( t , s) E I , x I , , we have

(2.17) V ( t )- W,(t)U(t)= G(t, s I Wo)[V(s)- W ~ ( S ) ~ ( S ) ] , (2.18)

u(t)= uo(t)Uil(s){U(s) + F ( t , s I Wo)[V(s)- WO(S)U(S)II.

Conclusion (2.17) is a direct consequence of the fact that M ( t ) = V ( t ) Wo(t)U ( t ) satisfies the matrix differential equation

-

+

so that M ( t ) = G(t,s I W,)M(s). Similarly, N ( t ) = U ( t ) F(s, t I W,) x [ V ( t )- W,(t)U(t)] may be verified to be a solution of the matrix differential equation

"(t) = [ 4 t )

+ B(t)WO(t)lN(t),

t E I,,

so that N ( t ) = U,(t)Uil(s)N(s)= U , ( t ) U ; l ( ~ ) U ( ~ which ) , is the relation (2.18) with t and s interchanged. In the consideration of Riccati matrix differential equations (2.1) it is to be noted that if the coefficient matrix functions satisfy hypothesis ( 6 )and W ( t )is an m x n matrix function satisfying (2.1), then for k the greater of the values m and n the matrix W ( t )may be considered as a component of a k x k matrix function which satisfies an equation corresponding to (2.1), and whose coefficient matrix functions are all of dimension k x k.

II. BASIC PROPERTIES OF SOLUTIONS

16

Indeed, for n < m, consider the m x m matrix functions Ao(t),Bo(t), C o ( t ) ,Do(t) defined as follows:

Ao(t)= diag{A(t), 0}, Do(t)= D ( t ) , and

B!,(t) = B,,(t) (a = 1,

(2.19)

c;,(t)= C,&)

. . . , n ; /? = 1,

. . . , m),

E,(t)= 0 and CjJl(t)= 0 ( a = n + 1,

. . . , m ; /?=1, . . . , m).

I t then follows readily that an m x m matrix function Q ( t ) is a solution of the Riccati matrix differential equation

(2.20)

Q'(t)

+ Q(t)AO(t)+ D"t)Q(t) + Q(t)BO(t)Q(t)- P ( t ) = 0

on a subinterval I, of I if and only if there exists an m x n matrix function Wo(t),and also an m x (m - n) matrix function R ( t ) such that Q ( t ) = [Wo(t) R ( t ) ] ,where W = W o ( t )is a solution of (2.1) on I o , and R ( t ) satisfies the linear differential equation R'(t) Consequently, if s

+ [ W t ) + W,(t)B(t)lR(t)

EI,

= 0.

then

where G(t,s I W,) is the solution of the differential system (2.5). In particular, if there exists an s E I , such that R(s) = 0, then R ( t ) 0 on I , and Q ( t ) is of the form [Wo(t) 01. Correspondingly, if m < n define n x n matrix functions A1(t),B 1 ( t ) , C 1 ( t ) ,D 1 ( t )as follows: A ' ( t ) = A ( t ) , D ' ( t ) = diag{D(t), O } ,

(2.21)

B&(t) = B,,(t), and cjz(t)= cbz(t) ( a = 1, . . . , n ; / ? =1, . . . , m), Ba,(t) = 0, and Cj3(t)= 0 ( a = 1, . . . , n ; / ? = m + l ,

. . . , n).

It then follows that an n x n matrix function Q ( t ) is a solution of the Riccati matrix differential equation,

(2.22)

+

sz'(t) + Q ( t ) A l ( t ) D'(t)Q(t)

+ Q ( t ) B l ( t ) Q ( t) cyt)= 0,

2. RELATED EQUATIONS A N D LINEAR SYSTEMS

17

on a subinterval I, of I if and only if there exists an m x n matrix function Wo(t)and an (n - m) x n matrix function S ( t ) such that

) a solution of (2.1) on I,, and S ( t ) satisfies the linear where W = W o ( t is differential equation S'(t)

+ S ( t ) [ A ( t )+ B(t)W,(t)l

= 0.

Consequently, if s E I, then

S ( t ) = S ( s ) H ( t ,s I WO), where H ( t , s I W,) is the solution of the differential system (2.6). In particular, if there exists an s E I, such that S(s) = 0 then S ( t ) Z E 0 on I, and Q ( t ) is of the form

I t is to be noted that the matrix differential equations of (2.5) and (2.6) are intimately related to the equation of variation of (2.1). Suppose that conditions (6)holds on an open interval I, and for given t E I and a given m x n matrix M = [,u,J (a = 1, . . . , m ; = 1, . . . , n), let W = W(t ; t,M) be the solution of (2.1) satisfying the initial condition W ( t )= M . The maximal subinterval of I on which W ( t ;t,M) exists is then an open subinterval

I { t , M } = { t I a ( t ,M ) < t < b ( t , M ) } . For fixed to,M o = b;,] with T O E I and AM = M - MO, it follows from (2.8) that for t E I { t o , M ) n I{tO, MO} we have

(2.23)

W ( t ;to,M ) = W ( t ;TO, MO)

+ G(t, toI W(

*

; to,M o ) ) N ( t ,to,MO, M )

x H ( t , t o I W(* ; to, MO)),

where

(2.24)

N ( t , to,MO, M ) = A M [ E , + F(t, t o I W(* ; to, MO))AM]-'.

Moreover, if I, is a compact subinterval of I { P , MO}, then in view of the

II. BASIC PROPERTIES OF SOLUTIONS

18

result of Corollary 1 to Lemma 2.1 we have that there exists a constant > 0 such that if v[ilM] 5 x then I , c I { t o ,M } and also En +F(t, zo I W( ; to,M o ) ) d M is nonsingular for t & I , . Therefore, the elements of N ( t , to,M a , M ) given by (2.24) have continuous partial derivatives with respect to the individual components pzoof M in a neighborhood of the p2s of M O , and the partial derivatives ?V91,01,(t; t,M ) of W ( t ;t,M ) with respect to pzo exist for x

= Ma,

M

t = to,

t E I { t o ,MO},

and are given by (2.25)

W p z P (to, t; M a ) = G(t, to[ W(* ; to,M O ) ) W H ( t ,to I W(* ; to,Ma)),

where DxS is the m x n matrix [D;!](i = 1, . . . , m ; j = 1, . . . , n), with = 1 if ( i , j ) = (a, B), 0 if ( i , j ) # (a, /?). Stated in a different form, T ( t )= WPap(t; to, MO) is the solution of the linear matrix differential equation

Pf.=

qf

(2.26)

T'

+ T [ A ( t )+ B(t)Wo(t)]+ [o(t)+ Wo(t)B(t)IT

=0

satisfying the initial condition, (2.27)

T(t0) = DOIS,

where in (2.26) we have written W o ( t )for W ( t ;to,M a ) .T h e differential equation (2.26) is the equation of variation of (2.1) along the solution W = W ( t ;to,MO). I n this connection, the reader is referred to Reid [15 ; Problem 11.4.21. If t = to,M = MO are such that the matrix function S(t, W ) = -WA(t) - D(t)W - WB(t)W C ( t )

+

satisfies the condition

1

yo

lim

h+O

ro

fh

S(t, MO) dt

=

S(t0, Ma),

then one may also establish that at t = t o ,M = M a the matrix function W ( t ;t,M ) has a partial derivative with respect to t given by

W,(t; to, MO) = -G(t,

I W(- ; to, MO))S(to,Mo)H(t,t o I W(

to

*

; to,M a ) ) .

3. VARIATION OF SOLUTIONS

3.

19

Variation of Solutions

For given matrix functions A(t), B ( t ) , C ( t ) , and D ( t ) satisfying hypothesis ($), let N = m + n and define the N x N matrix functions d ( t ) , 9 ( t ) , @‘(t), 9 ( t ) on I as follows:

Clearly &’(t), 9 ( t ) , @ ( t ) , 9 ( t ) are all of class .CNN[a, b] for arbitrary compact subintervals [a, b] of I , so that the corresponding hypothesis (6) is satisfied by these matrix functions. Moreover, it may be verified readily that an N x N matrix function F ( t ) is a solution of the Riccati matrix differential equation

(34

+

+

X [ % q ( tEZ)“ P ( t ) Y ( t ) d ( t ) 9(t)PP(t) Y ( t ) . q t ) W ( t )- @ ( t ) = 0

+

on a subinterval I, of I if and only if

(3.3) where W ( t ) ,G(t),H ( t ) , F ( t ) are matrix functions of proper dimensions which provide on I , a solution of the differential system (a) (b) (3.4) (.) (d)

+

+

+

W ’ ( t ) W ( t ) A ( t ) D (t )W (t ) W ( t ) B ( t ) W ( t) C ( t ) = 0, G‘(t) [ q t ) W(t)B(t)lG(t) = 0, H ‘ ( t ) H ( t ) [ A ( t ) B(t)W(t)l= 0, F ’ ( t ) - H ( t ) B ( t ) G ( t )= 0.

+ +

+

+

I n particular, if W = W o ( t is ) a solution of (2.1) onI,, and G(t,s I Wo), H ( t , s I W,) and F ( t , s I W,) are defined by (2.5), (2.6), and (2.7) for s a given value on I , , then the solution W = K(t)of (3.2) satisfying the initial condition

(3.5) is given by

II. BASIC PROPERTIES OF SOLUTIONS

20

Moreover, for this solution K(t)of (3.2) the matrix functions V(t,s I 7 0 be such that 6 < min{t, - so, so - tl} and the closure of the oriented, rectangular region 9o with opposite vertices (so - 6, tl), (so 6, t,) lies in 9. Since on e0the matrix function Qo(s: t) = sZ(so: t)Q-l(so: s) is a solution of the matrix differential system

+

we have the basic relation

(3.20)

Q(s:

t)

= Q(so:

tl E

t)Q-l(s,: s), it19

for

I s - so I I 6,

t2l.

As Q(so, t) is a.c. in t on [tl , t,], and Q-l(s0: s) is a.c. i n s on I s - so I I 6, it follows that on B0the matrix function Q(s: t) is continuous in (s, t), and is a.c. in each of its variables, separately. Indeed, since [Q-'(s,:

s)];=

-Q-l(so:

s)[Q(so:

S ) ] ~ Q - ~ ( s) S~:

for

I s - so I 5 6,

24

II. BASIC PROPERTIES OF SOLUTIONS

it follows that

(3.21)

Q,(s:

t ) = Q(s: t ) Y "

a($),

(s, t ) E so.

for

I n view of the arbitrariness of the point (so, t o ) in 9 it follows that Q(s: t ) is continuous in (s, t ) on 9, and on this region is locally a.c. in each of its arguments. I n terms of the component matrices PJs: t ) , QJs: t ) (a = 1, 2), the vector differential equation (3.21) becomes

PI&: t ) = -P,(s: t ) A ( s )- P,(s: t)C(s), P,,(s: t ) = -P,(s: t ) B ( s )

(3.22)

Qids: t ) =

-Q

+ P,(s: t)D(s),

(S: t ) A ( s )- Q Z ( s :t)C(s),

+ Qz(s:

Qzds: t ) = -Q (s: t ) B ( s )

t)D(s).

Now the system (3.15) is equivalent to the system

H ( s : t ) = P;'(s : t ) ,

W(s:t ) = Q1(s: t)P;'(s: t ) ,

(3.23)

F ( s : t ) = P;'(s: t)P,(s: t ) ,

and in view of (3.22) we have the following result. Theorem 3.3

If 9 is a region satisfying the above described conditions, and W(s:t ) , G(s: t ) , H ( s : t ) , F(s: t ) are solutions of the dzfferential system (3.12) in

9, then throughout this region these matrix functions are continuous in (s, t ) , locally a.c. in each argument, and satisfy the following difjerential system :

(a)

(b) (3.24)

(c) (d)

+ G(s:t)C(s)H(s:t ) = 0, G,(s: t ) - G(s: t ) [ D ( s )+ C(s)F(s:t ) ] = 0, H,(s: t ) - [A(s)+ F ( s : t)C(s)]H(s:t ) ] = 0, W , ( S t: )

F,(s: t ) - F(s: t)D(s)- A(s)F(s: t ) - F(s: t)C(s)F(s:t ) B(s) = 0.

+

3. VARIATION

OF SOLUTIONS

25

I t is to be commented that the functional equations (3.13’a-d,) or (3.13’a-c, d,) may be used to characterize solutions of matrix differential systems of the form (3.12). Specifically, we shall prove the following result. Theorem 3.4

Suppose that 9 2 is a region of the type specijied above, and that on 9 the matrix functions W(s:t ) , G(s: t ) , H ( s : t ) , F(s: t ) are of respective dimensions m x n, m x m, n x n, n x m, with G(s: t ) and H ( s : t ) nonsingular for (s, t ) E 92.Moreover, suppose that the functional equations (3.13’a-d,) or (3.13’a-c, d,) hold for all points (s, t ) , ( I , t ) , ( Y , s) of this region, and (3.25)

W ( S S) : = Om,,

G(s:S) = E m , F(s: s) = o,,

H ( s : S)

= En,

while the partial derivatives W,(s:t ) , G,(s: t ) , H,(s: t ) , F,(s: t ) exist and arejnite at points of the line t = s in 92.Then throughout 9 the functions W(s:t ) , G(s:t ) , H(s: t ) , F(s: t ) have j n i t e partial derivatives with respect to t and s, are continuous in (s, t ) , and there exist matrix functions A ( t ) , B ( t ) , C ( t ) , D ( t ) such that the differential systems (3.12) and (3.24) hold in 92. The above theorem is devoid of any specific continuity hypothesis. However, existence of the partial derivatives W , , G,, H,, F , at points of the line t = s i n 9 implies the continuity of W(s:t ) , G(s: t ) , H(s: t ) , F(s: t ) as functions of t at such points, and relations (3.13’) require that throughout 9 these matrix functions are continuous in t for fixed s. Indeed, since F(s: t ) F(s: s) = On, and H(s: t ) H(s: s) = E, as t s, it follows from (3.13’~)that H ( r : t ) + H ( r : s) as t s, for arbitrary (r, s) E 92. By similar arguments it may be established that each of the matrix functions W,G, H, F is continuous in t for fixed s throughout the region 9. I t will be shown that the result of this theorem holds with -+

-+

(3.26)

-+

-+

A(s) = -H,(s: s), B(s) = F,(s: s), D(s) = - G ~ ( s s:),

C(S)= W,(s:s),

for arbitrary s such that (s, s) E 9. For brevity, if M(s, t ) is a matrix function defined for (s, t ) E 92,and ( Y , t ) , ( I , s) are points of 9 2 with

26

t

II. BASIC PROPERTIES OF SOLUTIONS

# s, we shall denote by M ( Y :s, t ) the difference quotient ( t - s ) - l

x [M(Y, t ) - M(Y, s)]. If (s, s) E 9, then the relations (3.26) are equiv-

alent to the following limit relations as t + s,

H ( s : s, t ) --t

(3.26’)

--A(s),

F ( s : s, t ) + B(s),

W(s:s, t ) + C(s),

G ( s :S, t ) -+ --D(s).

Now if a matrix function N ( t ) is nonsingular throughout a neighborhood of t = s, and has a finite derivative Nf(s)= lim { ( t - s ) - l [ N ( t )- N ( s ) ] } l+a

the relation N ( t ) - N(s) = N ( t ) hence has a derivative “(s) equal to -fl(s)N’(s)fl(s). Consequently, if A(s:t ) = H-’(s: t ) and e(s: t ) = G-’(s: t). the rdations of (3.26) involving H and G may be written as

at t

x

=

“(s)

(3.26”)

s, then

N ( t ) = N - l ( t ) satisfies

- N ( t ) ] A ( s ) ,and

A ( $ )= A&: s),

D ( s ) = C,(s:

s).

Corresponding to the above notation, we write A(Y:s, t ) for the difference quotient ( t - s > - l [ A ( r ,t ) - A(Y: s)], with a similar meaning for G ( r :s, t ) . Since F(s: s) = 0, from (3.13’d1) it follows that if ( Y , s) and ( I , t ) are points of 9 then F(Y:S, t ) = H ( Y s)F(s: : S, t)G-’(s: ~ ) G ( Y t ):. Consequently, upon taking the limit as t + s, it follows that the partial derivative of F ( Y :t ) with respect to t exists at t = s, and the value F,(Y:s) of this derivative is given by (3.27)

F t ( y : S) =

H ( Y~ : ) B ( s ) G (sY ),:

which is equivalent to the differential equation of (3.12d). T h e equation (3.27) might have been deduced from relation (3.13’d2) by a similar argument. Now the equation (3.13’~)is equivalent to the equation

A(r:t ) = A(s:t ) [ E * + F(s: t ) W ( r :s ) ] A ( r :s), which in turn implies the relation A ( r : s, t ) =

[A(s:s,

t)

+ A(s:t)F(s:s, t ) W ( y : s ) ] A ( r :s).

3. VARIATION OF SOLUTIONS

27

Upon taking the limit as t + s, in view of the first relation of (3.26”) and (3.27), it follows that A(Y:t ) has a partial derivative with respect to t at t = s, and the value At(y:s) of this partial derivative satisfies the equation A t ( y : S)

=

+

[A($) B(s)W(r:s)]A(r:s).

Since A(Y:t ) = H - ~ ( Y t ) ,: and A t ( y : t ) = -Z?(r: t ) H t ( y :t)Z?(r: t ) at values of t where I?[(Y: t ) exists, it follows that H ( Y :t ) has at t = s a partial derivative, and the value H t ( y : s) of this partial derivative satisfies the equation Ht(Y:

s)

+ H ( r : s ) [ A ( s )+ B(s)W(r:s)]

=

0,

which is equivalent to the differential equation (3.12~). Correspondingly, Eq. (3.13’b) implies that G(Y:t ) = G-](Y:t ) satisfies the equation

G ( Y : t ) = e(Y:s ) [ E ,

+ W ( r :s)F(s: t)]G(s:t ) ,

and a similar argument yields the result that G(Y:t ) has a partial derivative with respect to t at t = s, and the value G,(Y:s) satisfies the equation G t ( y : S) =

G(Y:s ) [ D ( s )+ W ( Y :s ) B ( s ) ] ,

which is equivalent to the differential equation (3.12b). Finally, with the aid of the equations that have been established for the partial derivatives of G(Y:t ) and H ( Y t: ) with respect to t at t = s, direct differentiation of (3.13’a) yields a differential equation that is equivalent to (3.12a). That is, under the hypotheses of the theorem we have established that throughout 9the matrix functions W(s:t ) , G(s: t ) , H ( s : t ) , F(s: t ) have finite partial derivatives with respect to t , and for the matrix functions A , B, C, D defined by (3.26) the differential equations and boundary conditions of (3.12) hold for (s, t ) E 9. I t is to be emphasized that there is no assurance that the coefficient matrix functions satisfy hypothesis ($), or that as functions o f t the matrix functions W(s:t ) , G(s: t ) , H ( s : t ) , F ( s : t ) are locally a.c. However, if the matrix functions P , ( s : t ) , Q,(s: t ) , P,(s: t), Q p ( s :t ) are defined by (3.15), and Q(s: t ) is as in (3.16), then throughout 9 the matrix function Q(s: t ) has a finite partial derivative with respect to t which satisfies (3.17). Moreover, Q(s: t ) is nonsingular, with inverse given by (3.18). Consequently, on 9 the matrix function

II. BASIC PROPERTIES OF SOLUTIONS

28

SZ-'(s:

t ) has a finite partial derivative with respect to t , and [Sz-'(s:

t)It =

-a-ys:

t)Q,(s: t ) Q - ' ( s :

t ) = m-ys: t)y"%(t).

Suppose that 9, is a subregion of the sort specified in the discussion following Lemma 3.1, and that D = O(s: t ) is a matrix function which on 9, has a finite partial derivative with respect to t , and which satisfies (3.17) on 9,. Then on this subregion we have [SZ-'(s: t ) f i ( s :t ) l t = 0, and hence on 9, the matrix function a(s: t ) is the 9 ( s : t ) with elements given by (3.15) in terms of the W(s:t ) , G(s: t ) , H ( s : t ) , F(s: t ) satisfying the corresponding system (3.12). Now Q(s,: t>Q-l(s,: s) is such a matrix function Q ( s : t ) , and hence Q(s: t ) = sZ(s,: t)Q-'(s,: s) for (s, t ) E 9,. Consequently, in view of the arbitrariness of the subregion ST,, we have that on 9the matrix function Q(s: t ) has finite-valued partial derivatives sZ,(s: t ) and Q,(s: t ) ; also, Q(s: t ) is continuous in (s, t ) on 9. Proceeding as in the argument preceding Theorem 3.3, we have that the matrix functions P,(s:t ) , P,(s: t ) , Q1(s: t ) , Qz(s: t ) satisfy the differential system (3.22), and using the relations (3.23) it follows that the matrix functions W(s:t ) , G(s: t ) , H ( s : t ) , F ( s : t ) also satisfy the system (3.24) on 9. 4.

Transformations for (2.1) and (2.3,)

Suppose that T (t ) and S(t ) are nonsingular matrix functions of respective dimensions n x n and m x m which are locally a.c. on I . If W ( t ) is an m x n matrix function which is locally a.c. on a subinterval I , of I , and

(4.1)

Wo(t)= S-'(t)W(t)T(t),

t E I,,

then W o ( t )is locally a.c. on I , . Moreover, it follows readily that

(4.2)

R[Wl(t)

=

S(t)3?o[Wo](t)T-'(t) for t E I , ,

where

(4.31

3?0[WO]=

+

+

WO'(t) WO(t)AO(t) DO(t)WO(t) Wo(t)Bo(t)Wo(t) - CO(t),

+

and the matrix functions Ao(t),B O ( t ) Co(t), , D o ( t )are defined as

(4.4)

A' = T-'[AT

T'], Bo = T-IBS, Co = S-'CT, DO = S-l[S' DS].

-

+

29

4. TRANSFORMATIONS FOR (2.1) and (2.3,)

Corresponding to the transformation (4.1), if we set

(4.51

U y t ) = T-'(t)U(t),

VO(t)= S - ' ( t ) V ( t ) ,

then we have the identities

and the coefficient matrix functions in (4.7) are defined by (4.4). In particular, W ( t ) is a solution of (2.1) on a subinterval I, of I if and only if the associated function W o ( t )defined by (4.1) is a solution of

fio[Wo](t) =0

(2.10)

for

t

E I,.

If Wo= Woo(t)is a solution of (2.1,) on I,, and Go (t )= GO(t,s I Woo), H o ( t ) = H o ( t ,s I Woo)and Fo(t,s I Woo)are defined by the equations

(2.5,)

GO' + (Do + WooBo)Go = 0,

G0(s)= E m ,

(2.6O)

HO'

= 0,

H'(S) = E n ,

+ H o ( A o+ BOW,')

and W o ( t )is the solution of (2.1) onI, such that Woo(t)=S-'(t)Wo(t)T(t), it may be verified readily that

In particular, if T ( t ) and S ( t ) are fundamental matrix solutions of the respective linear homogeneous matrix differential equations

(4-9)

(a)

T' - A(t)T = 0,

(b) S'

+ D ( t ) S = 0,

II. BASIC PROPERTIES OF SOLUTIONS

30

then the matrix functions of (4.4) are given by (4.10)

A' = 0,

CO = S-'CT,

Bo = T-'BS,

DO

=

0.

I n order to prove additional results on special transformations for (2.3.,,), we shall establish the following auxiliary result. Lemma 4.1

If M ( t ) is an n x n matrix function which is continuous and v [ M ( t ) ]5 1 for t on a compact subinterval [a, b ] of I , then the matrix function (4.11)

F(t)= E -

C c,M'(t), 03

P=l

with

c1 =

4, ck = [l . 3

. (2k - 3 ) ] / ( k ! 2 ' )(k = 2 , 3 , . . .), is such that

(i) F ( t ) is continuous on [a, b ] ; (ii) F2(t)= E - M ( t ) ; (iii) i f M ( t ) is hermitian and M ( t ) 2 0 for t E [a, b ] , then F ( t ) is such that F ( t ) 2 0 for t E [a, b ] ; (iv) i f v [ M ( t ) ]< 1 for t E [a, b ] , and M ( t ) is continuously dzfferentiable {u.c.} on [a, b ] , then F ( t ) is also continuously differentiable {u.c.} on this interval. Conclusion (i) is a direct consequence of the fact that cP > 0 (k = 1 , 2, . . .), and the series CEl ck is convergent, so that the matrix series of (4.11) converges uniformly on [a, b ] . Indeed, it follows readily that the convergence of this series is also uniform on the class of matrix functions U ( t ) which satisfy v [ U ( t ) ]5 1 for t E [a, b]. Conclusion (ii) follows from the fact that if f(z)= ZE, c k z k , then 1 - f ( z ) is the Maclaurin series expansion for the branch of (1 - z)lI2 that is equal to 1 at x = 0. Indeed, if K j ( t ) = c k M k ( t ) , then the fact that ck > 0 (k = 1, 2, . . .), and the identity f(z)= [z + f ' ( z ) ] / 2 , imply that [ M ( t ) K j 2 ( t ) ] / 2= ZEl d P j M k ( t ) with d , = ck for k = 1, . . . ,j 1, while 0 < dPj < cP for j + 1 < k 5 2j, and d P j= 0 for k > Zj. Consequently, we have that

xf=,

+

+

+

y[4{[E - Kj(t)I2- [ E - M ( t ) ] } ]= v [ $ { M ( t ) K,'(t)} - K j ( t ) ] m

for arbitrary matrix functions M ( t ) satisfying v [ M ( t ) ]5 1.

31

4. TRANSFORMATIONS FOR (2.1) and (2.3~)

Now if M ( t ) is hermitian and M ( t ) >_ 0 for t E [a, b ] , then it follows readily that K J ( t )2 0 f o r j = 1, 2, . . . . Moreover, since c, = f ( l ) = 1, we have that 0 5 K , ( t ) 5 E, and F ( t ) 2 0, thus establishing conclusion (iii). If M ( t ) is continuously differentiable on [a, b ] , then for k = 2, 3, . . . , the matrix function M L( t ) is also continuously differentiable on this inter' the sum of k terms M " ( t ) M ' ( t ) M P ( t )with @ = k val, and { M A ( t ) } is - 1 - a, ct = 0, 1, . . . , k - 1, and hence

, : z

k

v [ { M k ( t ) } '5 ] k(v[M(t)])"-'v[M'(t)],

=

1, 2, . . . .

Conscquently, if v [ M ( t ) ]< 1 for t E [a, b ] , the uniform continuity of M ( t ) on the compact interval [a, b] implies the existence of a value Y < 1 such that v [ M ( t ) ]5 Y i 1 for t E [a, b ] , and the uniform convergence of the infinite series zT=,kc,z"-' on { z I 0 51 u" 15 r } implies that the infinite series c , { M A ( t ) } 'converges uniformly on [a, b ] , so that F ( t ) is continuously differentiable on [a, b] and F ' ( t ) = c,{M,(t)}'. If M ( t ) is merely a.c. on [a, b ] , and v [ M ( t ) ]5 Y < 1 for t E [a, b ] , then the above argument shows that v [ K J ' ( t ) ]5 v [ M ' ( t ) ] ( x y =kc,r"-') , ( j = 1 , 2, . . .). a.e. on [a, b ] , and consequently that F ( t ) is a.c. on this interval with F ' ( t ) = c , { M L ( t ) } 'for t a.e. on this interval.

xr=,

,:x

-z&

Lemma 4.2

Suppose that N ( t ) is an n x n matrix function which is continuous on a compact subinterval [a, b] of I , and there exists a continuous scalar function x ( t ) which is nonzero on [a, b ] , and such that (4.12)

v[E - x ( t ) N ( t ) ]5 1

for

t

E

[a, b].

Then there exists a continuous matrix function G ( t ) on [a, b] such that G 2 ( t )= N ( t ) ; moreover, if (4.12')

v [ E - x ( t ) N ( t ) ]< 1

for

t E [a, b ] ,

while N ( t ) and x ( t ) are individually continuously dzgerentiable {a.c.} on [a, b] then G ( t ) may be chosen to be continuously differentiable { a x . } on this interval. Under the hypotheses of Lemma 4.2 the matrix function M ( t ) = E - x ( t ) N ( t ) satisfies the hypotheses of Lemma 4.1, and consequently there is a continuous matrix function F ( t ) such that F 2 ( t )= E - M ( t )

II. BASIC PROPERTIES OF SOLUTIONS

32

= x ( t ) N ( t ) .Since x ( t ) is a continuous scalar function which is nonzero, there exists a scalar function A(t) which is continuous and nonzero on [a, b], and such that A2(t)= x ( t ) . It then follows that G(t)= [l/A(t)]J’(t) satisfies the conclusions of Lemma 4.2.

Lemma 4.3

If H ( t ) is a continuous hermitian matrix function on a compact interval [a, b], and H ( t ) > 0 on this interval, then the positive definite hermitian square root matrix function P ( t ) of H ( t ) is continuous on [a, b] ; moreover, if H ( t ) is continuously differentiable {a.c.} on [a, b] then P ( t ) is also continuously diperentiable {a.c. } on this interval.

If H ( t ) is a continuous hermitian matrix function satisfying H ( t ) > 0 on the compact interval [a, b], then there exists a constant K > 0 such that 0 < K2H(t) < E for t E [a, b]. Consequently, the hermitian matrix M ( t ) = E - k 2 H ( t ) satisfies the conditions of Lemma 4.1, and F ( t ) defined by (4.11) is such that P ( t ) = ( l / K ) F ( t )is a positive definite hermitian square root matrix of H ( t )which is continuously differentiable, or a.c., whenever H ( t ) possesses the corresponding property. Now it is well known (see, e.g., Riesz and Nagy [ l , pp. 263-2651) that there is a unique nonnegative definite hermitian square root matrix of a given nonnegative definite hermitian matrix, so that P ( t ) = ( l / k ) F ( t )is the unique positive definite hermitian square root matrix of H ( t ) . I n particular, if H ( t ) > 0 for t on an interval I , and H ( t ) is locally a.c. on I , then the positive definite hermitian square root matrix of H ( t ) is also locally a.c. on I. Consequently, we have the following result. Theorem 4.1

Suppose that n = m, and B ( t ) is a positive definite hermitian matrix function which is locally a.c. on I . If T ( t ) = B1I2(t),the positive definite hermitian square root matrix of B ( t ) , then under the transformation

(4.13)

U o ( t )= T-’(t)U(t) = B-l”(t)U(t), V o ( t )= T * ( t ) V ( t )= B’”(t)V(t),

the system (2.3,w) is transformed into the system (4.14)

L,O[UO, VO](t)= 0,

L,O[UO, VO](t)= 0,

defined by (4.7), and the matrix functions A o ( t ) ,Bo(t), Co(t),D o ( t )are

33

4. TRANSFORMATIONS FOR (2.1) and ( 2 . 3 ~ )

Now let

(4.15')

dl= &(A0+ AO"),

d

2

=

&(A0- AO"

19

+

so that Ao = dl -o", and Ao" = -d, - d2. If A,(t) is locally a.c., then under the substitution

vqt)= vyt) + d l ( t ) U 0 ( t )

uqt)= UO(t),

the differential system (4.14) becomes

(4.14')

+ @'l(t)U'(t)- 91(t)vA(t) = 0,

-V.'(t)

U.'(t) - d 2 ( t ) U A ( t )- P ( t ) = 0.

where

Let A ( t ) be the solution of the differential system

(4.17)

A ' ( t ) - &'z(t)k(t)

= 0,

A(t0)=

E,

where to is some fixed point of I. Since -d2(t) = - d 2 * ( t ) the matrix function A ( t ) is unitary on I , so that A ( t ) A " ( t ) E E. Under the substitution

(4.18) the system (4.12')becomes

(4.19)

-Y

+ @(t)% - 9 ( t ) Y ( t )= 0,

'(t)

% ' ( t )- Y ( t ) = 0,

where

In particular, ( & ( t ) ; Y ( t ) )is a solution of (4.19) if and only if P ( t ) is

II. BASIC PROPERTIES OF SOLUTIONS

34

a solution of the second-order linear matrix differential equation

W”t)

(4.19‘)

+ 9 ( t ) P ’ ( t )- s ( t ) & ( t ) = 0,

and T ( t ) = P’(t). If we have also that D = A*, then in (4.15) we have Do = /lox. Proceeding to the system (4.16) we have g1 = Do -. J311 -- Aox- dl = d.*, and consequently from (4.20) it follows that in (4.19) we have 9 ( t ) = 0 and this system becomes

-F-‘(t)

(4.21)

+ @(t)!P(t)= 0,

P ’ ( t ) - Y ( t ) = 0,

which is equivalent to the second-order differential equation

9’‘ - &T(t)%(t)= 0.

(4.19”)

For simplicity of reference, the above results are assembled in the following theorem. Theorem 4.2

Suppose that n = m, B ( t ) is a positive de$nite hermitian matrix function on I , while B ( t ) and d l ( t ) are locally a.c. on this interval. If d ( t ) is the solution of the dzyerential system (4.17), then ( U ( t ); V ( t ) )is a solution of (2.3,) if and only if &(t)= &(t)B-1/2(t)U(t)

is a solution of the second-order linear matrix dzyerential equation (4.19‘), with

+

P ( t )= .,4x(t)[B1/2(t)V(t) d l ( t ).A(t)&(t)]; moreover, if D ( t ) = A # ( t )for t E I , then B ( t ) = 0 on this interaal and (4.19’) reduces to (4.19”). 5. Associated Riccati Matrix Differential Equations

T h e vector differential system adjoint to (2.3,‘) is (5.1)

9 * [ y ] ( t )= - Z * y ’ ( t )

+ %*(t)y(t)= 0,

5. ASSOCIATED DIFFERENTIAL EQUATIONS

35

which may be written also as (5.2)

-v'(t)

+ C"(t)u(t)- A"(t)v(t)= 0,

u ' ( t )- D"(t)u(t) - B'(t)v(t) = 0,

where u ( t ) and v(t) are vector functions of respective dimensions m and n. For (5.1) the corresponding Riccati matrix differential equation in the n x m matrix function W ( t ) is (5.3)

+

+

Jt*[W](t)= W ' ( t ) W(t)D*(t) A"(t)W(t) W(t)B"(t)W(t) - cyt)= 0.

+

I t is to be remarked that y ( t ) is a solution of (5.1) if and only if [ ( t ) = y ( t ) is a solution of the related system (5.1')

9A[c](t)

=

-$['(t)

+ g(t)c(t)

= O,

or, correspondingly, [ ( t ) = ti(t), q ( t ) = V ( t ) is a solution of (5.2')

-q'(t) 5'(t)

+ Qt)E(t) - &)ll(t)

= 0,

- D(t)E(t)- B(t)q(t)= 0.

Similarly, W ( t ) is a solution of (5.3) if and only if Q ( t ) = W ( t ) is a solution of the Riccati matrix differential equation (5.3')

fiA[Q](t)

.nr(t)

+ Q ( t ) f ) ( t )+ a(t)Q(t)

+ Q ( t ) B ( t ) Q ( t )- C(t)= 0.

Because of these simple relations existing between solutions of (5.1 '), (5.2'), (5.3') and (5.1), (5.2) and (5.3), specific formulation of results are limited to the systems (5.1), (5.2) and (5.3). The following interrelations may be established readily. Lemma 5.1

If y ( t ) = ( u ( t ) ; w ( t ) ) and y ( t ) = ( u ( t ) ;v(t)) are solutions of (2.3,) and (5.2), respectierely, then

(5.4)

y " ( t ) Z y ( t )= v " ( t ) u ( t ) - u*(t)er(t)

is constant. Similarly, if y ( t ) = ( u ( t ) ; o ( t ) ) and ( ( t ) = ( E ( t ) ;q ( t ) ) are solutions of (2.3,) and (5.2'), respectierely, then

(5.4') is constant.

E(t)2yct)= r'(t)u(t) - t ( t ) o ( t )

II. BASIC PROPERTIES OF SOLUTIONS

36

Lemma 5.2

A matrix function W ( t )is a solution of 8[Wl = 0 on a subinterval I , of I if and onZy if W ( t )= W*(t) { Q ( t ) = @ ( t ) } is a solution of 8,[w] = 0 {$?t,[Q] = o} on I , . Moreover, the corresponding matrix functions G*(t,s I W),H*(t, s I W), F*(C s I W) and G,(t, s I Q), HA(t, s I Q), FA(t,s I 9)de$ned by the analogs of (2.5)-(2.7) are

I JQ= A(t,s I W ) , HA(t, s I @) = C(t,s I W ) , (5.6) H*(t, s I W") = G"(t,s I W), FA(t,s I @) = P(t, s I W ) , (5.7) F*(t,s I W") = F*(t, s I W), where H*(t, s I W) and A(t,s I W) denote [ H ( t ,s [ W)]"and [ H ( t ,s I W)]-, with similar meanings for G"(t,s I W ) , e(t,s 1 W), F*(t, s I W ) and P(t, s I W). (5.5)

6.

G*(C s I W") = H"(t, s I W),

GA(t,s

Normality and Abnormality

Two distinct values 1, and t , on I are said to be (mutualzy) conjugate, with respect to (2.3,), if there exists a solution ( u ( t ) ;~ ( t )of) this differential system with u ( t ) f 0 on the subinterval with endpoints t , and t,, while u ( t l ) = 0 = u(t,). The system is called disconjugate on a subinterval I , of I provided no two distinct values on this subinterval are conjugate. Also, (2.3,) is said to be disconjugatefor large t if there exists a subinterval ( a , m) of I on which this system is disconjugate. Corresponding to the terminology of the calculus of variations for the accessory differential system of a Bolza type problem, (see, e.g., Bliss [l, Section Sl]), a system is termed identically normal 071 I , or normal 071 every subinterval, if whenever (u(t) 3 0 ; w ( t ) ) is a solution of this system on a nondegenerate subinterval I, of I then also v ( t ) = 0 on I,. For a nondegenerate subinterval I, of I, let A(I,,) denote the linear space of m-dimensional vector functions o ( t ) which are solutions of the vector differential equation v ' ( t ) D ( t ) v ( t )= 0, and satisfy B(t )v(t )= 0 for t E I,. Clearly v ( t ) E A(Io) if and only if ( u ( t ) = 0 ; v ( t ) ) is a solution of (2.3,) on I , . If A(I,) is zero-dimensional, then (2.3,) i s said to be normal on I,, or to have order of abnormality zero on I , , whereas if A(I,) has dimension d = d(I,) > 0 the system (2.3,) is said to be abnormal, with order of abnormality d on I,. The symbol &(I,) will denote for (5.1) the set corresponding to A(1,); that is, A*(Io)is the linear space

+

37

6. NORMALITY A N D ABNORMALITY

+

of vector functions v(t) which are solutions of v'(t) A"(t)v(t)= 0 and satisfy B * ( t ) v ( t )= 0 on I,. T h e dimension of .4,(Io) is denoted by d,(I,). If I , = [s, t ] , for brevity we write d[s, t ] and d,[s, t ] instead of the more precise d( [s, t ] ) , d,( [s, t ] ) , respectively, with similar contractions in case I , is of the form [s, t ) , (s, t ] , or (s, t ) . In the following, whenever intervals [s, t ] , (s, t ) , etc. occur we shall always understand that the intervals are nondegenerate. For I , a subinterval of I , clearly 0 5 d(I,) 5 m. Moreover, if s E I then d[s, t ] is an integral-valued monotone nonincreasing function on { t I t E I , t > s}, with at most m points of discontinuity, at each of which d[s, t ] is left-hand continuous. I n particular, if [s, b,) c I then d[s, b,) is the minimum of d[s, t ] for s < t < b,, and do = d,{I}, the maximum of d[s, b,) for s E I , is the limit of d[a, b,) as a b,. Moreover, if a is such that d[a, b,) = d o , then d[b, b,) = do for arbitrary b E [a, b,), and there exists a b E (a, b,) such that d[a, t ] = do for t E [b, b,). If d[s, r ] = d > 0, a E [s, r ] , and A = A ( a ) is an m x d matrix such that the solution ( U ( t ); V ( t ) )of ( 2 . 3 J f ) ,determined by the initial conditions U ( a )= 0, V ( a ) = A , has U ( t ) = 0 for t E [s, r ] , and the column vectors of V ( t ) form a basis for d[s, r ] , then for brevity we write A ( a ) d[s, r ] ; if the column vectors of A are required to be mutually orthogonal (i.e., d"d = E d ) as may be done without loss of generality, we write A ( a ) = A[$,T I . For (5.1) we write, correspondingly, A , ( a ) -d,[s, r ] and A,(a) = d,[s, r ] . If [a, b] c I , we shall denote by Q,[a, b] the linear space of (m n)dimensional vector functions ( u ( t ) ; v ( t ) )which are solutions of the differential system (2.3,) and satisfy the end conditions .(a) = 0 = u ( b ) ; the corresponding space for (5.1) will be denoted by Q,,[a, b ] . If K[a, b] is the dimension of Q,[a, b ] , then clearly k[a, b] 2 d[a, b ] , and K[a, b] > d[a, b] if and only if a and b are mutually conjugate, in which case the integer k [ a , b ] - d[a, b] is called the order of b { a } as a point conjugate to a { b } . If k,[a, b ] is the dimension of Q,,[a, b ] , then classical results from the theory of two-point boundary problems (see, e.g., Reid [15, Chapter 1111) imply that m k,[a, b] = n k [ a , b ] , and for given n-dimensional vectors ua, ub there exists a solution ( u ( t ) ; ~ ( t )of) (2.3,) satisfying .(a) = ua, u(b) = ub if and only if --f

-

+

+

(6.1) v*(a)ua - v*(b)ub = 0

+

for arbitrary (u(t);v(t)) E D,,[a, b ] .

If n < m, then clearly for an arbitrary subinterval [a, b ] of I there exists a nonidentically vanishing solution ( u ( t ) ; v ( t ) )of (2.3,) satisfying

II. BASIC PROPERTIES OF SOLUTIONS

38

the end conditions .(a) = 0 = u(b). That is, the dimension of Q[a, b] is positive, and if (2.3,) is normal on [a, b] then a and b are conjugate points relative to (2.3,). Correspondingly, if m < n then the dimension of Qn,,[a,b] is positive, and if (5.1) is normal on [a, b] then a and b are conjugate points relative to (5.1). If n = m and (2.3,) is disconjugate and identically normal on a subinterval I,, then the relation k,[a, b] = k[a, b] for arbitrary [a, b] c I implies that (5.1) is also disconjugate and identically normal on I,. In particular, in case n = m the hypothesis that (2.3,) and (5.1) are both disconjugate on a subinterval I , is weaker than the hypothesis that (2.3,) is disconjugate and identically normal on I , . Before presenting certain general properties of differential systems involving the concepts of normality and abnormality, we shall consider briefly a special case. Suppose that there is an r x m constant matrix 8 and an n x r matrix function P ( t ) such that B ( t ) = P ( t ) B , and for t a.e. on I we have that the rank of B ( t ) is equal to the rank of B. Stated in an alternative fashion, for t a.e. on I we have

{v I v

€a,,

B ( t ) v = O}

=

{v I v

€am, Bv

= O}.

I

When this condition holds we shall write B ( t ) 8, and say that “ B ( t )is null-space equivalent to 8 a.e. on I”. Two important specific instances

w

in which B ( t ) = l? are: (i) B ( t ) = P ( t ) R , where P ( t ) is an n x n matrix function that is nonsingular for t a.e. on I ; (ii) m = n, and B ( t ) = R*Q(t)R, where Q ( t ) is an n x n hermitian matrix function that is definite for t a.e. on I . It is to be emphasized that in this latter case it is not required that Q ( t ) be positive definite for t a.e. on I , or be negative definite for t a.e. on I , but merely that for t a.e. on I this matrix function is definite. Now if there are constant matrices D and 8 of respective dimensions m x m and Fn Y x m such that D ( t ) = D and B ( t ) = 8 on a nondegenerate interval I , of I,then clearlyy(t) = ( u ( t ) ;v ( t ) )is a solution of (2.3,) with u ( t ) = 0 on a nondegenerate subinterval [a, b] of I , if and only if v ( t ) = [exp{-tD}]e, where e is a constant m-dimensional vector and e*[exp{--t~*}]R* = 0 on [a, b]. Since by the Cayley-Hamilton theorem we have d , ( D ) = 0, where dD(l)is the characteristic polynomial det(tE, - D)of D,it then follows that e satisfies this latter condition if

39

6. NORMALITY A N D ABNORMALITY

and only if ~ *=80, Q*D*B* = 0, . , . , e*(D*)m-l8* = 0, and consequently o(t) can be nonidentically vanishing on [a, b] if and only if the m x mr matrix

(6.2')

[B* D*B* (D*)28

(D*),-18*]

* . *

has rank less than m. Similarly, if there are constant matrices A and 8, of respective dimensions n x n and n x s such that A(t) = A and w B*(t) = B1* on a nondegenerate subinterval I, of I, then y(t) = (u(t); v(t)) is a solution of (5.1) with u(t) = 0 on a nondegenerate subinterval [a, b] of I, if and only if v(t) = [exp{-tA*}]a, where a is a constant n-dimensional vector and a*[exp{-tA}]8, = 0 on [a, b]. Corresponding to the preceding result, v(t) can be nonidentically vanishing on [a, b] if and only if the n x ns matrix

(6.2")

[8,

AB1 A28,

.. .

An-'8,]

is of rank less than n. Moreover, since y(t) is a solution of (5.1) if and only if ((t) = f ( t ) is a solution of (5.1'), then clearly (5.1) is normal on a subinterval [a, b] if and only if (5.1') is normal on this subinterval. I n particular, the above comments imply the following result. Lemma 6.1

If there are constant matrices D and B of respective dimensions m x m w and r x m such that D(t) = D and B(t) = 8 on I, then the system (2.3,) is identically normal on I if and only if the m x mr matrix (6.2') is of rank m. Correspondingly, if there are constant matrices A and 8, of respective

w

dimensions n x n and n x s such that A(t) = A and B*(t) = 8,* on I, then each of the systems (5.1) and (5.1') is identically normal on I if and only if the n x ns matrix .(6.2") is of rank n. I n particular, if there are constant matrices A, B, D such that A(t) = A, B(t) = B and D(t) = D on I, then 8 and 8, may be chosen us B, in which case (6.2') and (6.2") become, respectively,

(6.200

[Be D*B* (D*)zB*

(6.2;)

[B A B A2B

...

(D*)m-lB*

I,

A"-'B].

For [a, b] c I, let B'[a, b] denote the class of n-dimensional vector functions q(t) for which there is a corresponding m-dimensional vector function ((2) which is (Lebesgue) measurable, with B(t)((t) E P,[a, b]

II. BASIC PROPERTIES OF SOLUTIONS

40

and L,[q, [ ] ( t )= q'(t) - A ( t ) q ( t )- B ( t ) [ ( t )= 0 on [a, b ] ;for brevity, this association of q ( t ) and [ ( t ) is denoted by q E 9 [ a , b] : [. The corresponding class of m-dimensional vector functions q(t ) associated with (5.1), and involving the differential equation q ' ( t ) - D"(t)q(t) - B*(t)c(t)= 0, will be denoted by 9 , ' [ a , b], and we write q E &*'[a, b] : The following results are ready consequences of the general results mentioned above, and are listed here for direct use in the following sections. Here, 4nd elsewhere throughout this discussion there is obvious modifcation by deletion of some matrices in case certain spaces are of dimension 0 or of maximum dimension, n or m. The results of the following three lemmas are ready consequences of the definitions of the concepts involved, and the above comments.

c.

Lemma 6.2

I f [a, b] c I , then there exists a solution ( u ( t ) ; v ( t ) )of (2.3,) satisfying .(a) = ua, u(b) = ub i f a n d only ifcondition (6.1) holds. I f 1;1 E 9 ' [ a , b] : [, and v(t) E A*[a, b], then v * ( t ) q ( t )is constant on [a, b] ; moreover, i f a and b are not mutually conjugate for (5.1), then there exists a solution of (2.3,) satisfying .(a) = ua, u ( b ) = 0 i f and only if v"(a)ua = 0 for arbitrary v(t) E A*[& b]. Lemma 6.3

+ +

+

I f (2.3,) is disconjugate on I , then m d,[a, b] 5 n d[a, b] for arbitrary [a, b] c I ; i f m d,[a, b] < n d[a, b] then a and b are mutually conjugate points for (5.1). In particular, i f (2.3,) is disconjugate on I then m d,[a, b] = n d[a, b] for all [a, b] c I i f and only if (5.1) is also disconjugate on I.

+

+

+

Lemma 6.4

-

Suppose that [a, b] c I , and c is a value on [a, b ) such that d[a, t ] t E (c, b],while d ( a ) A [ a , b ] ,and Q is an nzx ( m - d ) matrix such that [ A ( a ) Q ] is nonsingular. I f ( U , ( t ); V , ( t ) )is the solution of (2.3,) satisfying the initial conditions U,(a) = 0, V,(a)= Q, then a value t , E (c, b] is conjugate to a, with respect to (2.3,), i f and only i f U ( t , ) has rank less than m - d. I f , in addition, m d,[a, b] = n d[a, b ] , d , ( a ) A,[a, b ] , and ( U , ( t ) ; V 2 ( t ) )is the solution of (2.3,,f) satisfying the initial conditions U,(a) = d , ( a ) , V,(a) = 0, then a value t , E (c, b] is conjugate to a, with respect to (2.3,), if and only i f the n x n matrix [ U , ( t , ) U,(t,)] is singular. = d[n, b] = d for

-

+

+

41

6. NORMALITY A N D ABNORMALITY

-

If [a, b] is a subinterval of I with d = d[a, b] > 0, while s E [a, b] with d(s) A [ a , b ] , and W ( t )is a solution of (2.1) on [a, b ] , then from (2.5) and the definition of A [ a , b] it follows that the column vectors of G(t,s I W) d(s) from a basis for A [ a , b]. Correspondingly, if d,(s) &[a, b],then the column vectors of G,(t, s I W x )d,(s) = H*(t, s I W) x d,(s) form a basis for A*[a, b].

-

Lemma 6.5

If [s, r ] c I , d(s)-A[s, subinterval, then

and W ( t ) is a solution of (2.1) on this

F(t, s I W )d(s) = 0

(6.3') and the (n

I],

+d) x

for

t E [s, r ] ,

m matrix

(6.3") has rank less than m if only if and r is a conjugate point to s for (2.3,). Relation (6.3') follows readily from (2.7), and the fact that the column vectors of G(t,s I W) d(s) form a basis for A[s, r]. The further conclusion of Lemma 6.5 results from the fact that Lemma 2.3 implies that a constant m-dimensional vector t is such that F(r, s I W)E = 0 if and only if ( u ( t ) ; v ( t ) ) = ( K ( t ,s I W ) t ;L(t, s I W)E)is a solutionof (2.3,) satisfying the end conditions u(s) = 0 = u ( r ); moreover, u ( t ) = 0 on [s, I ] if and only if F ( t , s I W)E z 0 for t E [s, Y ] , in which case

and v ( t ) = G(t,s I W ) t is an element of A [ s ,r ] . Lemma 6.6

-

Suppose that [s, r] c I , c E (s, r ) with d[s,t ] = d[s, I ] = d for t > c, d(s) A [ s , I ] , and W(t)is a solution of (2.1) on [s, r]. If for (2.3,) there is no point on (c, I ] which is conjugate to s, then there exists an n x ( n - m d ) matrix V(t,s) independent of W ( t )which is continuous in t on (c, r ] , with (6.4) P*(t, s)F(t, s I W ) = 0,

+

V*(t,s)V(t,S) = E,l--m+d,

t E (c, Y ]

II. BASIC PROPERTIES OF SOLUTIONS

42

and the (n

+ d ) x (n + d ) matrix

is nonsinguzur for t E (c, I ] . If for (2.3,) there is no point on (c, r ] that is conjugate to s, then Lemma 6.5 implies that F(t, s I W) is of constant rank m - d on (c, r ] . Consequently, there exists an n x ( n - m d ) matrix function Y = Y(t,s) which is continuous in t on (c, Y] and satisfies (6.4), and the nonsingularity of the matrix function (6.5) for t E (c, Y] then follows from (6.4) and the fact that (6.3") is of rank m on this interval. Moreover, if W ( t ) nd Wl(t)are solutions of (2.1) on [s, I ] , and rl = Wl(s) - W(s),then by relation (3.10~)we have that

+

(6.6)

F ( t , s I Wl)

=

1

+

[En F ( t , s I W ) r I ] - l F ( t ,s I W )

= F(t,s

I W)[E,+

I

s W)I-',

and consequently a Y(t,s) which satisfies (6.4) for any one W ( t ) also satisfies this relation for arbitrary solutions of (2.1) on [s, r ] . For t E (c, r ] the reciprocal of (6.5) is of the form

[ W;(;,IsV

(6.7)

1

A(s) 0 '

and the m~ n matrix F # ( t , s I W )is the generalized inverse of F(t, s I W ) in the sense of E. H. Moore. For a brief discussion of the E. H. Moore generalized inverse of an m x n matrix the reader is referred to Reid [15, Appendix B, Sec. 31. If W ( t )and W l ( t )are solutions of (2.1) on [s, I ] , and F = F ( t , s I W ) , F # = F # ( t , s I W ) , Fl = F ( t , s I W l ) , F l # = F # ( t , s I Wl), A = A ( s ) , Y = Y(t,s), then the fact that (6.7) and (6.5) are reciprocals for t E (c, r ] implies for t on this interval that

(6.8)

F1#

- F'

+ Fl'(F1-

F ) F # = 0.

Also, (6.6) implies that

Fl - F = -FJ1F = - F r l F , ,

with

rl = W,(s)- W(s),

so that we have the relation

(6.9)

F l # - F # = F l # F l r l F F # = (Em- Ad")Fl(En- YY").

6. NORMALITY A N D ABNORMALITY

43

If ( U ( t ); V ( t ) )is a solution of (2.3,) with U ( t )nonsingular on [s, r ] , while W ( t )= V( t ) U - l ( t ) , and V (t,s) = G(t,s 1 W) d(s) is an m x d matrix function whose column vectors form a basis for A[s, Y ] , then for N an arbitrary d x n constant matrix we have that the matrix function (Ul(t) ; V,(t))= ( U ( t ); U t )

+ v t , s)N)

is a solution of (2.3,,f) on [s, r ] with corresponding Wl(t) = V,(t)U;l(t) such that

r, = W,(s) - W(s)= d(s)NU-l(s).

Moreover, since F(t, s I W)r,= 0 for t E [s, r ] , from (3.1Oa-c) it follows that for t E [s, r ] we have

I pv1) = G(t,s I W)[Em + riF(t,s I VI-', H ( t , s I W,) = H(t, s I W ) , F(t, s I W,) = F(t, s I W ) . G(t,s

(6.10)

In particular, the choice of N such that d"(s)W,(s)= 0 specifies W,(t) as the solution of (2.1) determined by the initial condition W,(s) = [Em - d ( s ) d " ( s ) ]W(s). If both (2.3,) and (5.1) are disconjugate on [s, r ] , and A,($) A,[$, Y ] , then one may choose Y(t,s) = d,(s), and combining the above results for both (2.3,) and (5.1) yields the following theorem.

-

-

Theorem 6.1

If both (2.3,) and (5.1) are disconjugate on [s, r ] , while d(s) A[s, r ] , d,(s) = A,[s, Y ] , and W ( t )is a solution of (2.1) on [s, I ] , then the solution W J t ) of (2.1) determined by the initial condition (6.11)

W,(s) = [Em - d(s) d * ( s ) l ~ ( s ) [ E ?-l d*(s) d**(s)l

exists on [s, Y ] , and

W,(s)d*(s) = 0,

(6.12)

d"(s)W,(s) = 0,

(6.13)

F(t, s I W,) = F(t, s I W )

f o r t E [s, Y].

A solution W J t ) of (2.1) which satisfies the conditions (6.12) will be termed a solution of (2.1) on [s, I ] which is normalized at s. With the aid of (6.9), one may establish the following result.

44

I I . BASIC PROPERTIES OF SOLUTIONS

Corollary

If both (2.3,) and (5.1) are disconjugate on a subinterval

[s, b,) of

I,

while c E. [s, b,) is such that d[s, t ] = d[s, b,) and d,[s, t ] = d,[s, b,) for t E (c, b,), and W,(t), Wo(t)are solutions of (2.1) on [s, bo), each of which is normalized at s. then

I n particular, if F # ( t , s I W,) tends to a limit as t b,, then F # ( t , s I W , ) also tends to a limit as t + b, , and these two limits are distinct unless W,(t) = Wo(t). -+

Theorem 6.2

Suppose that (2.3,) is disconjugate on [s, b,), and W ( t ) is a solution of (2.1) on this interval such that F # ( t , s I W )+ 0 U S t + b,. I f c E (s, b,) is such that d[s, c] = d[s, b,), and d[c, b,) = d o , then F # ( t , s, I W )+ 0 as t -+ b, for all s, E [c, b,).

-

+

If d(s) A [ s , 00), then by Lemma 6.6 there is a n x ( n - m d) matrix Y = Y'(t, s ) which is continuous in t and satisfies (6.4) for t E [c, b,), while the matrix (6.5) is nonsingular on this interval. If s, E [c, b,) then d[s, , b,) = d[c, b,) = d o , and there exists an m x do matrix d ( s , ) A [ C ,bo). If p ( t , s) denotes the minimum of I F ( t , s I W)E on the set

I

C(s) = ( 5 I E

E

am, I 5 I = 1,

and p ( t , sl) denotes the minimum of C(S,) =

(5 I 5

E

a,

dW(s)5= O } ,

I F ( t , s, I W ) [ ]

I5[

on the set

= 1, d'(s,)E = O},

then the conclusion of Theorem 6.2 is equivalent to the following statement: If p ( t , s) + 03 as t + b,, then also p ( t , s,) -+oo as t + b,, for all s1 E [c, b,). For basic relation for this result, see Reid [8, Sec. 61. From the definition of F ( t , s I W ) it follows readily that (6.15) q t , s,

I W ) = F(s, I W ) + H(s, I W ) F ( t ,s I W ) G ( s , 1 W ) . $1

$1

$1

7. SOLUTIONS OF A RlCCATl EQUATION

45

and consequently if E E C(sl)then F(t, s I W)G(s,s, I W)F=F(t, s I W)Fo, with Fo = [Em- d(s) d*(s)]G(s,s1 I W)F. Now d*(s)Lfo= 0, and if to were equal to 0, then Lf = G(sl, s 1 W )d(s)e with e = d*(s)G(s, s1 I W)F, and as G(t, s I W ) = G(t,s1 I W ) G ( s l ,s I W ) we would have that

G(t,s1 I W)G(s,,s I W ) d ( s ) e= G(t,s I W ) d ( s ) e E A[$,bo). As this relation implies that G ( t ,s1 I W ) t E A [sl, b,), it would then follow that there exists a do-dimensional vector 0 such that [ = d(sl)a. On the other hand, the condition 5 E C(sl) implies 0 = d*(sl)t = u, and hence the contradictory result = 0. Consequently, we have that to# 0, and therefore there exists a k(s, sl) > 0 such that

I [Em- d(s) d"(s)lG(s,s1 I W)F I L 4,sl>,

for

t E W,).

If k,(s, sl) and k,(s, sl) are positive constants such that

Hence we have the inequality and if p ( t , s) -,00 as t + b,, and s, E (c, b,), then also p(t, sl) -+ 00 as t -+ b, ,which in view of an above comment is equivalent to the conclusion of the theorem. 7.

Distinguished Solutions of a Riccati Matrix Differential Equation

If (2.3,) is disconjugate on a subinterval ( a , , b,) of I , then a solution = ( u b o ( t ) ; v b o ( t ) ) is termed a principal solution at b, of (2.3,) if there is an interval ( a , b,) on which U b o ( t ) is nonsingular and for W b , ( t ) = v,,(t)U;:(t) there is at least one value s on (a, b,) such that F # ( t , I Who) + 0 as t + b,; the corresponding Wb,(t)is called a distinguished solution at b, of (2.1). Theorem 6.2 implies that if Y b o ( t ) is a Yb,(t)

46

11. BASIC PROPERTIES OF SOLUTIONS

principal solution at b, of (2.3,) then there exists a subinterval ( a l , b,) such that F # ( t , s I W,) --+ 0 as t + b,, for every s E ( a , , b,). Moreover, from the Corollary to Theorem 6.1 it follows that if both (2.3,) and (5.1) are disconjugate on ( a , , b,) then (2.1) admits at most one distinguished solution at b, which is normalized at a value s. Suppose that (2.3,) is disconjugate on (a, b,), and for s a point on this interval let d(s) A[s, b,), and choose c E [s, b,) with d[s, t ] = d[s, b,) = d , d,[s, t ] = d,[s, b,) = d , for t E (c, b,). For r E (c, b,) let Y,,(t) = (U8,(t); V,,(t)) be an (n m ) x (d n - m ) matrix whose column vectors are K , = d n - nz linearly independent solutions (u(t); ~ ( t ) ) of (5.1) satisfying the boundary conditions u(s) = 0 = u(Y). According to the convention previously followed, if K , = 0 then statements regarding the existence of Y,,(t) and its properties are to be deleted. I t will be assumed that this basis for Q,,[s, Y ] has been so chosen that V :(s)V,,(s) = Ek., and by Q(s, I ) we shall denote an n x (n - K , ) = n x ( m - d ) matrix function such that N

+

+

+

In view of Lemma 6.2, for r E (c, b,) there is a unique solution Ysr(t) = (Ua,(t); Var(t))of (2.3M) such that (7.1)

US,(y) =

0,

usr(s) = Q(s, r ) ,

d"(s)T/pr(s)= 0.

Theorem 7.1

Suppose that (2.3,) is disconjugate on (a, b , ) ; moreover, for s E (a, b,) and d(s), Y,,(t)= (U,,(t);V,,(t)), Q(s, I ) determined as described above, let Y,,(t)= ( U , , ( t ) ; Vs,(t))be the solution of (2.3,v) satisfring the conditions (7.1). If W ( t )is a solution of (2.1) on (a, b,) such that d'(s)W(s) = 0 and F # ( t , s I W )+ 0 as t + b,, then for any sequence { r j } c (a, b,) such that yj b, and { Q ( s , Y ~ ) } converges to a limit matrix Q(s) as j -+ 00 the sequence {V6r,(s)}also converges to a limit matrix P(s), and if Y&,(t) = (Uabo(t); Vab0(t))is the so2ution of (2.3M) determined by the initial conditions --f

then

47

7. SOLUTIONS OF A RlCCATl EQUATION

Q 2 ( s ) is an n x k, matrix such that the n x n matrix [Q(s) Q2(s)]is nonsingular, while P2(s)= W(s)Q2(s), and Ys2(t) = (Usz(t); V82(t))is the solution of (2.3,) determined by the initial conditions

If

then ~ , ~ (=t W(t)U,,(t) ) on (a, b,). Moreover, Y ( t )= ( U ( t ) ;V ( t ) )with

is a solution of (2.3,) with U ( t ) nonsingular and

(7.6) For r

W ( t )= V(t)U-l(t), E

for t E (a, b,).

(c, b,), the condition (2.18) of Lemma 2.4 implies that

and since d*(s)[V8,,(s)- W(s)Q(s,r ) ] = 0 we have that

-

-

As Q(s, I) is bounded for r E (c, b,), if F # ( r , s I W ) 0 as I -+ b, then Var(s)- W(s)Q(s,I ) 0 as r + b,. Moreover, there exist sequences { r j } such that r j b, and {Q(s, r j ) } converges to a limit matrix Q ( s ) as j .+00, and for any such sequence we have that {Vsrj(s)} P(s) = W(s)Q(s)as j - w. If Yab0(t) = (Usbo(t); Vab0(t)) is the solution of (2.3,) determined by the initial conditions (7.2) then (7.3) holds for t = s, and hence for all t E (a, b,) by (2.17) of Lemma 2.4. Similarly, for Ysz(t)= ( U s P ( t )V ; a 2 (t )) the solution of (2.3,) determined by (7.4) we have that VSz(t) = W(t)Us2(t). and consequently for Y ( t )= ( U ( t ) ; V ( t ) )defined as in the statement of the theorem we have V ( t )= W ( t ) U ( t ) for t E (a, b,). If U(b)[ = 0 for a value b E (a, b,), it would then follow that V(b)[ = 0 and ( U ( t ) [ ; V ( t ) [ )is the zero solution of (2.3,), whereas U ( s ) is nonsingular, and hence [ = 0. Consequently, U ( t )is nonsingular on (a, b,) and relation (7.6) holds throughout this interval. If both (2.3,) and (5.1) are disconjugate on (a, b,), while s E (a, b,), d[s, 6,) = d o , where do is the maximum of d [ t , b,) for t E (a, b,), d,[s, b,) = d,,, where d,, is the maximum of d,[t, b,) for t E (a, b,), d(s) A[s, b,), A,(s) A,[s, b,), then V,,(s) may be chosen as d,(s) and Q(s, r ) as an n x (n - d,,) = n x (m - do) matrix Q(s) independent

-

-

-

II. BASIC PROPERTIES OF SOLUTIONS

48

of

Y,

and such that

(7.8)

d,*(s)Q(s) = 0,

Q*(s)Q(s) = Em-do*

If c E ( a , b,) is such that d[s, t ] = d[s, b,) for t E (c, bo), then the initial condition (7.1) determining for Y E (c, b,) the solution Ys,(t)= ( U s , ( t ) ; V8,(t))of (2.3,) becomes (7.9)

U87(y)

= 0,

usr(s) = Q(s),

d*(s)Var(s)= 0.

Now if there exists a solution W ( t )of (2.1) on [s, b,) which is such that F # ( t , s 1 W )+ 0 as t + b, , then by Theorem 6.1 the corresponding solution WY(t)determined by (6.11) satisfies the conditions (6.12), (6.13), and consequently also F # ( t , s I W,) + 0 as t -+ b,. Theorem 7.1 then implies that Va,(s)converges to a limit matrix P ( s ) as Y + b,, and the solution YsbO(t) = (Usao(t) ; Vsb,(t)) determined by (7.2) is such that v 8 b , ( t ) = W,(t)Usb,(t)on [s, b,). The matrix Q 2 ( s ) of Theorem 7.1 may be chosen as d,(s), and if Ys2(t) = ( U s 2 ( t )Vs ; 2 (t ))is the solution of (2.3,) determined by the initial conditions (7.10)

Us2(s) = d,(s),

Vs2(s) = 0,

and Y ( t )= ( U ( t ) ;V ( t ) )is defined by (7.5), then the final conclusion of Theorem 7.1 implies that U ( t ) is nonsingular on [s, b,), and V ( t ) U - l ( t ) = W y ( t )throughout this interval. On the other hand, suppose that the solution Y,,(t)= ( U a r ( t ) Var(t>) ; of (2.3,) determined by (7.9) is such that Vs,(s)converges to a limit matrix P(s) as Y b,. Let Yeb,(t) = (U8a,(t);Vsb,(t))and ys2(t) = (u82(t); Vsz(t))be the solutions of (2.3,) determined by the respective initial conditions (7.2) and (7.10), and define Y ( t )= ( U ( t ) ;V ( t ) ) by (7.5). As the condition that d*(s)Vm(s)= 0 for r E (c, b,) implies that d*(s)P(s)= 0, it follows that d*(s)V(s) = 0. Moreover, since V(s)U-l(s) = P(s)Q*(s),the solution W ( t )= V(t)U-l(t)of (2.1) satisfies A*(s)W(s) = 0, W(s),4,(s) = 0. Consequently, if U ( t ) is nonsingular throughout [s, b,), then W ( t )is a solution of (2.1) on this interval which is normalized at s. Also, since UaT(s) = Q ( s ) and W(s)Us7(s) = P(s)Q*(s)Q(s)= P(s), from (2.18) of Lemma 2.4 if follows that -+

(7.11)

F(Y,s

I v[vs,(s) - P(s)] == -Q(s)

and V8,(s)- P(s)

-+

0 as

Y -+ b,.

Now if

for

Y E

(c,

bo),

49

8. GENERALIZED LINEAR DIFFERENTIAL SYSTEMS

where ~ ( r+ ) 0 as r -+ b,. Consequently, if ,u*(r,s) denotes the minimum of I F"(r, s I W ) t I on Z,(s), then ,u"(r, s) -+ 03 as r + b,, and this latter condition is equivalent to the condition that F # ( r , s I W )+ 0 as r -+ b,. Thus we have established the following result. Theorem 7.2

Suppose that both (2.3,) and (5.1) are disconjugate on ( a , b,), and for s E ( a , b,) let d (s), d *(s) and Q(s) be determined as above. Then a necessary and suflcient condition for there to exist a solution W ( t )of (2.1) on [s, b,) which is a distinguished solution at b, is that the solution Yar(t) = (Uar(t); Va,(t))of (2.3,) determined by conditions (7.9) is such that VSr(s) converges to a limit matrix P(s) as r b,, and for the solutions Yabo(t) = (Uabo(t); K b , ( t ) ) , Ysz(t) = (Uaz(t) ; VszW), Y ( t )= ( W ); of (2.3,) determined respectively by (7.2), (7.10), and (7.5) the matrix U ( t )is nonsingular on [s, b,,); moreover, in this case W,,(t)= V ( t ) U - l ( t )satisfies on [s, b,) the relation F ( t , s I W,,) = F(t, s I W ) ,and W J t )is the unique distinguished solution at b, of (2.1) satisfying the conditions d*(s)W,,(s) = 0, W,(s)d*(s) = 0. -+

w>)

It is to be remarked that (2.3,) may be disconjugate on an interval ( a ,b,) without this system possessing a principal solution at b,. Indeed, this phenomenon may exist even when m = n and (2.3,) is hermitian, as will be discussed more fully in Chapter IV.

8. Generalized Linear Differential Systems and Riccati Matrix Integral Equations

Throughout most of the present section it will be supposed that on a given interval I on the real line the matrix functions A, B, C, D, M satisfy the following hypothesis:

SO

II. BASIC PROPERTIES OF SOLUTIONS

I n particular, the requirements of (6’)on A(t), C ( t ) , and D ( t ) are the same as those of hypothesis (G),but the hypothesis on B ( t )is strengthened so the functional appearing in (8.5) below is considered in a “Hilbert space” setting. For c E I, U ( t ) E Gnk[a, b ] and V ( t )E .r?Kk[[a,b] for arbitrary compact subintervals [a, b ] of I , the integral operator Q[c, t I U , V ] is defined for t E I as

(8.1)

Q[c,

t I U, VI =

[C(s)u(s)- D(s)V(s)l ds

+

s

t

[dM(s)lU(s),

where the last integral in (8.1) is a Riemann-Stieltjes matrix integral. We shall consider now the matrix generalized linear differential system

+

[CU - D V ] dt dL,[U, q ( t ) -dV (8*2M) L,[U, V ] ( t )3 U’ - A U - BV = 0,

+ [dM]U

=

0,

t E I.

By a solution of (8.2,) is meant a pair of matrix functions ( U ( t ) ;V ( t ) ) with U ( t )e a n k [ ab,] , V(tj E ’132Il,,[a,b ] on arbitrary compact subintervals [a, b] of I such that

(8.3)

d[c, t I U, V ] ( t )

-V(t)

+ Q(c, =

t I U, V ) -V(c) for (c, t ) E I x I,

and L,[U, V ] ( t )= 0 on I. It is to be remarked that the validity of (8.3) is readily seen to be equivalent to the existence of a co E I and a constant m x k matrix Vo such that

For the special case of K = 1, the system (8.2,,) reduces to the vector generalized differential system

(8.2)

~L,[u, v ] ( t ) -dv

+ [CU- Dv]dt + [dM]u= 0,

L,[u, v ] ( t ) 3 u‘ - Au - Bv = 0,

t

E

I.

If u,, and v o are given vectors of respective dimensions n and m,and c E I, there is a unique solution of (8.2) such that .(a) = u o , .(a) = v o , and this result for (8.2) clearly implies the corresponding existence and uniqueness theorem for (8.2.,,). I n view of Theorem 8.3 below, this result for (8.2) is a corollary to the corresponding result for an ordinary linear differential system. It may also be derived from the existence of a unique

8. GENERALIZED LINEAR DIFFERENTIAL SYSTEMS

51

solution of a Riemann-Stieltjes integral equation similar to that considered by the author in [ll, Section 21. Specifically, let U o ( t )be the solution of the differential system Uo'(t)- A(t)U,(t) = 0, Uo(c)= E , , and let V o ( t ) be the solution of the differential system Vo'(t)- D(t )V o (t )= 0, Vo(cj = E , . Then ( u ( t ) ; v ( t ) ) is the solution of (8.2) which satisfies the initial conditions u(c) = u o , v ( c ) = vo if and only if u ( t ) = Vo(t)uo(t), v ( t ) = Vo(t)vo(t),where uo(t) is the unique solution of the integral equation

and

where the matrix functions Bo(t)and M o ( t ) are defined as

Now for a compact subinterval [a, b] of I the symbol 9 [ u , b] will be used to denote the class of vector functions q ( t ) E '%,,[a,b] for which there is a corresponding [ ( t ) E i?,!,r12[a, b] such that L,[q, [ ] ( t )= 0 on [a, b], and this association of [ ( t ) with q ( t ) is indicated by 7 E 9 [ a , b ] : [. Also, the subclass of 9 [ a , b] on which q ( a ) = 0 = ~ ( b is) denoted by g o [ a ,61, with a similar meaning for q E g o [ a ,b] : 5. Correspondingly, 9 * [ a , b] denotes the class of vector functions p ( t ) E % ~ [ U ,b] for which there is an associated a(t) E L?,2[u,b] such that

L,*[@,o ] ( t )= @'(t)- D " ( t ) @ ( t ) B"(t)a(t)= 0 on [a, b ] , and

52

II. BASIC PROPERTIES OF SOLUTIONS

rb

in terms of which we have the following result. Theorem 8.1

If [a, b] c I , and ( u ( t ) , v ( t ) ) E Q,[a, b] x Cm2[a,b ] , then the following two conditions are equivalent :

(8.6)

(a) J [ u , v ; e, (71 = 0 for e E 9 * o [ a , bl : u ; (b) there exists a constant vector y and a v o ( t )E b%l,[a,b] such that B ( t ) [ v ( t )- v O ( t ) ]= 0 on [a, b ] , and for t E [a, b ] . d [ a , t u, vo] = y

I

If ( u ( t ) , v ( t ) ) E @,[a,b] x Em2[a,b ] , and (8.6b) holds, then for e 9 * o [ a ,b ] : u we have Ja

= e"(t)vo(t)

I

t-b t-a

E

= O?

so that (8.6a) holds also. Conversely, if ( u ( t ) , ~ ( t )E) Q,[a, b ] x ECm2[a, b] and (8.6a) is satisfied, for w(t)=SZ(a, t I u, v ) we have that w ( ~ )BBm[a, E b] and if e E 9 * o [ a ,b ] : u then

Now if Y ( t )is the solution of the differential system Y ( t ) = D"(t)Y(t), Y ( a )= E m , then ( e ( t ) ,u ( t ) ) E%,[u, b] x Cn2[a,b] with L2*[e,a](t)=O on [a, b ] , and e ( b ) = 0 if and only if

53

8. GENERALIZED LINEAR DIFFERENTIAL SYSTEMS

Consequently, e E 9*,,[a, b ] : u if and only if u ( t ) E tn2[a,b ] , e ( t ) satisfies (8.8), and

0=

(8.91

J

b a

Y-l(s)B*(s)a(s)ds.

Now the vector function f = z, - w is such that f(t) E tm2[a,b ] , and from relations (8.7)-(8.9) it follows that if e E 9*,,[a, b ] : u then the vector function g ( t ) = - Y*-'(t) J c Y*(s)D(s)f(s) ds a

is such that g

E

5Lnr2[a, b ] , and

I*

(8.10) if

+f ( t )

a

u*(t)B(t)g(t)dt = 0,

Jl

u E Cn2[u,b] and

u*BY*-~ dt = 0.

Condition (8.10) implies the existence of a constant n-dimensional vector A such that B(t)g(t)= B ( t )Y*-l(t)Aon [a, b ] (see e.g., Reid [15, Problem F : 2 . 7 ] ) . Hence h ( t ) = g ( t ) - Y*-l(t)A is such that h (t ) E Cm2[a,b ] , B(t)h(t)= 0 on [a, b ] , and (8.11)

f(t)-J1

~ * ( s ) ~ ( s ) f d~ (s)

~ * - 1 ( t )

a

= h(t)

+ Y*-l(t)I.

for t E [a, b ] .

That is, f(t) is determined as the solution of the Volterra vector integral equation (8.1 l ) , with kernel matrix function

Y*-'(t)Y*(s)D(s) = Y*-'(t)Y*'(s)

for s E [a, t ] , t E [a, b ] .

It may be verified directly that the resolvent matrix kernel for this integral equation is - D ( s ) for s E [a, t ] , t E [a, b ] , and hence f(t) = h ( t )

+ Y*-l(t)A+ J' D(s)[h(s)+ Y+-l(s)A]ds, a

t

E [a, b ] .

As D(t)Y*-'(t) = -[Y*-l(t)]' on [a, b ] , it follows that v ( t ) - w ( t ) = f(t) satisfies the equation

54

II. BASIC PROPERTIES OF SOLUTIONS

Consequently, if vo(t)is the vector function defined as vo(t) = w ( t )

s:

+

t

D(s)h(s)ds - y ,

E [a, bl,

then zyo(t) E d?Bl,[a,b ] , B ( t ) [ v ( t )- v O ( t )= ] 0 on [a, b ] , and vo(4 = Q [ a , t I u, vl

+ J D(s)h(s)ds t

- y,

I t I u, vO] - y.

= Q [ a , t u, v - h ] - y , = Q[U,

This latter relation states that d [ a , t I u, vO] = y , thus completing the proof of the theorem. As a ready consequence of the result of Theorem 8.2 we have the following theorem on the existence of solutions of (8.2). Theorem 8.2

If u ( t ) E %,[a, b ] there exists a v ( t )such that ( u ( t ) ; v ( t ) ) is a solution of (8.2) i f and only ij there exists a v , ( t ) such that u E 9 [ a , b ] : v l , and J [ u , v,; @, u] = 0 for @ E 9 * o [ a , b ] : u.

(8.12) J [ q , 5'; e,

U] =

s:

{u*B[- e*'Mq I

and consequently, if q relation

(8.13) J [ %5 ; e, = J6 a

e*Mq'

+ p*Cq} dt

I=(I

and e

E 9 * o [ a , b ] : u,

we have the

01

((U-M*Q)*B(~-M~)+~*[C-DM-MA-MBM]~} dt.

For brevity, let

(8.14)

E 9 [ a ,b]:[

--

A(t), s(t),&'(t),b(t)be defined on

A = A + BM, &' = C - D M - M A

- MBM,

[a, b] as

B = B, fi = D f M B ,

and denote by @a, b ] , &,,[a, b], &*[a, b ] , &,,[a,

b] the above-defined

55

8. GENERALIZED LINEAR DIFFERENTIAL SYSTEMS

classes 9, gn,p',, g 3 * 0 for the interval [a, b ] when the matrix functions A , B, C,D are replaced respectively by a, A, D.It follows readily that q E 9[0, b ] : 5', or 7 E g O [ a ,b ] : 5 if and only if f ( t ) = 5 ( t ) - M ( t ) q ( t ) is such that 7 E B [ a ,61: f, or 11 E G n [ a , 61: f. Similarly, e E -@',[a, b ] : u or p E 2 * , , [ a ,61: cr, if and only if ci(t) = o ( t ) - M"(t)p(t) is such that p E B*[a,61: ci, or Q E .G',n[a,b ] : 6. Moreover, if u E - 2 [ a , b ] : v, then ( d , 6 ) = ( u , ZJ - Mu) is such that d E B [ a , b ] : 8 and (8.6a) holds if and only if

e,

(8.15)

j [ d , 6 ; 6, a] b

=

n

{(;+A6+ (*ed}dt

Also, an integration by parts of ing precise derivative result.

=0

for ij

E

B,,[a,

b ] : 3.

Jl [ d M ( s ) ] U ( sin) (8.1) yields the follow-

Theorem 8.3

The vector function ( u ( t ) ;v ( t ) ) is a solution of (8.2) i j and only i j the associated vector function ( d ( t ) ;6 ( t ) ) = ( u ( t ) ;v ( t ) - M ( t ) u ( t ) )is locally a.c. on I , and is a solution of the vector ordinary dzflerential system -@(t)

(8.16)

+ C(t)C(t)- B(t)d(t)= 0,

t

d ' ( t ) - A(t)d(t)- B ( t ) f i ( t )= 0,

E

I.

I t is to be emphasized that in order for the system (8.16) to be well defined and have coefficients given by (8.14) which satisfy hypothesis (6)or (6')it is not necessary that M ( t ) be locally of b.v. on I. In particular, the coefficients of (8.16) satisfy (6)or (6')if the matrix functions A(t), B ( t ) , C ( t ) , D ( t ) satisfy the conditions of the respective hypothesis and M ( t ) E i!gn[[a,b ] for arbitrary compact subintervals [a, b ] of I . For matrix functions A, B, C, D, M satisfying hypothesis (6') the vector generalized differential system

(8.17)

+

[C'U - A+] dt dL,*[U, v ] ( t ) -dv L,*[u, v ] ( t )= ur- D+u - B"v = 0,

+ [dM*]U = 0,

t E I,

is said to be "adjoint to (8.2)." Corresponding to the above notations, Q,(c, t I u, u ) and d,(c, t I ZL, v ) are defined as Q*(c, t

I U, V )=

[C"(S)U(S) - A+(s)v(s)] ds

+

II. BASIC PROPERTIES OF SOLUTIONS

56

I n particular, (8.17) is obtained from (8.2) upon replacing A , B , C, D, M by D*,B*, C*, A*, M * , respectively, so that (8.2) is also adjoint to (8.17) in the sense thus defined. Under this substitution the classes 9 [ a , b], 9 , [ a , b ] for a compact subinterval [a, b] are interchanged with the respective classes 9 , [ a , b], 9,,[a, b ] . Moreover, for (rl(t),5 ( t ) ) E w a , bl x

bl

and (e (t), 4 t ) ) E QJa,

bl x

%z2[a7

bl

the functional

is such that

J[r,5 ; e, I a, bl

is the complex conjugate of

Consequently, results for (8.17) corresponding to those of Theorems 8.1, 8.2, 8.3 for (8.2) may be stated as follows in terms of the functional J [ %5 ; e, 01 = J[%5 ; e, 0 I a, bl defined by (8.5). Theorem 8.4 c I, and (u(t),v(t)) E Qm[a,b] X Cn2[a,b], then the following conditions are equivalent:

If [a, b] two

(a)

J[r,5 ; u , v] = 0,

for

Ego[%

bl: 5 ;

(8.19) (b) there exists a constant vector y, and a vo(t)E 2)Bl,[a,b] such that B*(t)[v(t) - v,(t)] = 0 on [a, b ] , and d,[a, t I u, v,] = y*, for t E [u, b ] . Theorem 8.5

If u ( t ) E 21m[a,b ] there exists a v(t) such that ( u ( t ) ;v(t)) is a solution of(8.17) on [a, b ] ifand only ifthere exists a vl(t)such t h a t u E 9 , [ a , b]: v, and J [ q , 5 ; u,vl]= 0 for q E 9 , [ a , b]: 5.

57

8. GENERALIZED LINEAR DIFFERENTIAL SYSTEMS

Theorem 8.6

The vector function y ( t ) = ( u ( t ) ;v ( t ) ) is a solution of (8.17) if and only the associated vector function ? ( t ) = ( a ( t ) ;+(t))= ( u ( t ) ;v ( t ) - M * ( t ) u ( t ) ) is locally U.C. on I , and is a solution of the vector ordinary diperential system C'"(t)ii(t)- A"(t)+(t) = 0, -+t(t) (8.20) t E I. 3 ( t ) - D * ( t ) a ( t )- B(t)+(t)= 0,

+

As (5.20) is adjoint to (8.16), the following result for solutions of (8.2) and (8.17) is a direct consequence of the corresponding result for ordinary differential systems. Corollary

If y ( t ) = ( u ( t ); v ( t ) ) and y ( t ) = ( u ( t ); v ( t ) ) are solutions of (8.2) and (8.17) respectively, then y " 3 y = v " ( t ) u ( t )- u " ( t ) v ( t ) is constant on [a, b].

Fgr the study of properties of solutions of (8.2) and (8.17), the result of the following theorem is of basic importance. Theorem 8.7

Suppose that [a, b] c I , (IlP),

w)

E %z[a,

bl x L2[a, bl, ( e ( t ) ,+))

E '%[a,

bl x G 2 [ a ,bl,

and there exist matrix functions ( U ( t ) , V ( t ) )E 21np[a,b] x 2f,[a, b], ( U ( t ) ,v ( t ) ) E almq[a, b] x %,[a, bl and h ( t ) E %,[a, b], k ( t ) E %Ja, b] such that ~ ( t=) U ( t ) h ( t )and e ( t ) = U ( t ) k ( t )on [a, b]. Then fur a 5 c < d 5 b the value of the functional

(8.21) J [ V C, ;

e, c 1 c, d ] = J

d

{U*BC

+ e*cr)dt +

e*[d~]r

is equal to each of the following expressions: (8.22)

J:

{(u - V k ) # B ( t- V h ) - (L,*[e, u])"Vh

+ k"V"(L,[U, V I h + k"(V"U -

-

L [ r ,C I )

- U " V ) h ' } dt

J: k*U*{dd[a, t 1 U,V ] } h+ k"U*Vh 1:

;

II. BASIC PROPERTIES OF SOLUTIONS

58

Id

The fact that J [ q , [; e, a I c, d ] is given by (8.22) may be verified by comparison of the involved integrand functions. I n turn, the conclusion that (8.21) is also equal to (8.23) is a consequence of the preceding result applied to the adjoint functional J*[e, a ; q, iI c, d ] , and the fact that J [ r l , 5 ; e, a I c, d ] is equal to the complex conjugate ofJ*[e, a ; q, 5 I c, d ] . The above general results for the special instance of h and K arbitrary constant vectors yield the following matrix results.

(8.25)

U"V - V"U =

U"(s){dd[n,s I

u, V ]

In view of Theorems 8.3 and 8.6, results concerning solutions of (8.2) and (8.17) are equivalent to corresponding results for the respective systems (8.16) and (8.20). If the matrix functions A, 8, b are defined by (8.14), and

e,

(8.26)

P ( t ; W ) = e(t)- @A(t) - b(t)E'- E'B(t)cG.,

then in view of the introductory comments of Section 2 we have that for [a, b ] c I there exists a @(t) E amn[a,b ] which is a solution of the Riccati matrix differential equation (8.27)

l[r/G7(t)3 W ( t )- q t ; E'(t))= 0

59

8. GENERALIZED LINEAR DIFFERENTIAL SYSTEMS

(o(t);

on [a, b] if and only if there is a solution P ( t ) ) of the differential system -P'(t) C(t)O(t)- D ( t ) P ( t )= 0, (8.28) O'(t) - A(t)O(t)- B ( t ) P ( t )= 0,

+

with o(t)nonsingular on [a, b ] , and such that m(t)= B ( t ) o - l ( t )on this interval. Indeed, in accord with the remark following the statement of Theorem 8.3, the validity of the above comments does not depend upon M ( t ) being of b.v. on [a, 61, and these comments are true under the following weaker assumption :

(Go') A ( t ) , B ( t ) , C ( t ) and D ( t ) satisfy hypothesis (G') and M ( t ) E 2,zn[a, b] for arbitrary compact subintervals [a, b] of I . For (8.29)

A^, B, e, D

related to A, B , C, D as in (8.14), and

F ( t ; W ) = C ( t ) - W A ( t )- D ( t ) W - WB(t)W,

it may be verified readily that if W ( t ) and W ( t ) are matrix functions satisfying

(8.30)

W ( t )=

W ( t )- M ( t )

on a subinterval [a, b ] of I then P ( t ; m(t))= F ( t ; W ( t ) )for t E [a, b ] . Consequently, m(t)is a solution of (8.27) satisfying the initial condition W(s)= Y for a value s E [a, b ] if and only if W ( t )= m(t) M ( t ) is the solution of the Riccati matrix integral equation

+

I n particular, we have the following result. Theorem 8.8

If hypothesis (Go') holds, and s E [a, b ] , then an m x n matrix Y is such that (8.31) has a solution on [a, b ] ifand only ifthe solution (O(t); P ( t ) ) of (8.28) determined by the initial conditions o(s)= E n , P(s) = Y is such that o(t)is nonsingular on [a,b ] , and in this case W(t)=M(t)+P(t)o-'(t). r f M ( t ) E 9!Bm,[a, b ] , then an equivalent condition is that the solution ( U ( t ); V ( t ) )of (8.2,) determined by the initial conditions U ( s ) = E n , V ( s )= M ( s ) Y is such that U ( t ) is nonsingular on [a, b ] , and in this case W ( t )= V ( t ) U - l ( t ) .

+

60

II. BASIC PROPERTIES OF SOLUTIONS

Now if @,(t) is a solution of (8.27) on a subinterval [a, b] of I , for I @,), Z?(t)=Z?(t, s I @,) of linear differential systems of the form (2.5) and (2.6) with A , B , D,W, replaced by A^, 8,D, @,, respectively, together with the associated matrix function p ( t , s I @,) given by (2.7) with B , H , G replaced by 8,A, G,respectively. Moreover, for (8.27) one has the result of Lemma 2.1 providing necessary and sufficient conditions for an m x n matrix function w ( t ) to be a solution of (8.27) on [a,b ] , together with representation formulas corresponding to (2.8) and (2.8'). Now if equations (8.14) hold, and W ( t ) , W o ( t )are related to respective matrix functions w ( t ) , @,(t) by (8.30), it may be verified directly that

s E [a, b] one has matrix solutions G ( t ) = G ( t , s

(8.32)

A + B@, = A

+ BW,,

W(S) -

D

+ @,B

=D

W,(s) = W(s)- W,(s).

+ W&?,

In particular, these relations imply that if G ( t ) = G ( t , s I Wo), H ( t ) = H ( t , s I W,), and F ( t , s I W,) are determined by the differential systems (2.5), (2.6) and integral relation (2.7), then H ( t , s I W,) = A ( t , s I G(t,s I W,) = e ( t , s I Po),and F ( t , s I W,) = p(t, s I In view of the above discussion, we then have the following result for the integral equation (8.31).

w0).

m,),

Theorem 8.9

Suppose that hypothesis (6,') holds, s E I , and W = W o ( t )is a solution of (8.31) for !P= !Po on a subinterval I , of I containing s. If G ( t ) = G ( t , s I W,), H ( t ) = H ( t , s I W,) are defined by the dzfferential systems (2.5) and (2.6), and F ( t ) = F ( t , s I W,) is defined by the corresponding integral relation (2.7), then an m x n matrix function W ( t ) is a solution of (8.31) on I , if and only if the constant matrix T = !P - !Pois such that En F ( t , s I W,)T is nonsingular on I,, and

+

moreover, in this latter case the matrix function Em nonsingular on I , , and

+ l'F(t, s 1 W,) is also

61

9. A CLASS OF MONOTONE OPERATORS

9. A Class of Monotone Riccati Matrix Differential Operators

For a general m x n matrix M = [Map] (a = 1, . . . , m ;b = 1,. . . , n), the symbol M 2 0 { M > . 0} will signify that the elements of M are real and Mas 2 0 {Mas> 0} for a = 1, . . . , m, = 1, . . . , n. Correspondingly, the symbol M x 2 c 0 will signify that the elements of M are real and Mms2 0 for a # b. T h e symbols M 5 0, M < 0, and M c 5 x 0 will denote the respective conditions ( - M ) 2 0, (-$2) > 0, and ( - M ) x 2 c 0. Also, M 2 N will denote the condition M - N 2 0, with similar meanings for M > N , M 5 N , N < . M , etc. T h e positive orthant { y I y = ( y u )E %, y , 2 0, (a 1, . . . , n)} will be denoted by %,+, M,, will signify the set of m x n real matrices { M I M = [M,,], M a p real, a = 1, . . . , m, B = 1 , . . . , n}, and Mf,,= { M I M E M , , , M . > . O } . For y E %, we set I y I l = CE=l I y , I ; in particular if y E %+, then I y = CtSly a . Also, for M E Mf,, we write

-

-

-

-

-

-

-

-

-

-

-

-

-

Il

Pm[M1= SUP1 MY

11,

yAM1 = infl MY I1 on { y I y E %? I yI+ I1 , = 11.

If M(l),. . . , M(,) denote the column vectors of a matrix M belonging to Mf,, , then clearly pm[M]is the maximum, and v,[M] is the minimum, of the values I M(l)1 1 , . . . , I M ( n )I l . If the elements of M ( t ) are continuous, measurable, or locally integrable on I , then the scalar functions pbo[M(t)] and Y, [M(t)] are correspondingly continuous, measurable, or locally integrable on this interval. If M E Mf,, and c 2 0, then clearly cM E ML, and pm[cM]= cp,[M], v,[cM] = CY,[M].Also if M , E Mf,, (a = 1, Z), then M , M2 E MLn and Pco[Ml+ M21 L Pm"M1l Pm[M21,

+

+ YAM1 + M21 2 YAM11 + vm[M*l.

I n the following, we shall assume that I is an open interval (a, b), and that the m x n matrix function F(t, W )= [Fas(t,W)] (a = 1, . . , m ; = 1, . . , n) is defined on I x M,, and satisfies the following

.

.

II. BASIC PROPERTIES OF SOLUTIONS

62

CarathCodory type” condition :

“

(a) for $xed W E M,, , F is (Lebesgue) measurable on I ; (b) f o r $xed t

(C)

E

I , F is continuous in W on M,, ;

(c) if [a, b] is a compact subinterval of I , and k > 0, there exists a nonnegative real-valued (Lebesgue) integrable function x ( t ) = x k ( t )on [a, b ] such that p m [ F ( t , W ) ]5 x ( t ) f o r t E [a, b ] and W E M,, with p m [ y 5 k.

Unquestionably, the most important class of matrix functions F ( t , W ) satisfying (C)is the class of m x n matrix functions which are continuous on I x M,, ,and the reader unfamiliar with Lebesgue integration may restrict attention to this class. For this restricted class the essential steps of the following discussion remain materially the same, although there are rather obvious simplifications of some details ; in particular, references to Chapter I1 of Reid [15] may be replaced by references to corresponding results in Chapter I of that work. If [a, b] is a compact subinterval of I, and Wo(t)is an m x n matrix function continuous on [a, b ] , then (C)implies that F ( t , Wo(t))E2,,[a, b]. Moreover, under this condition there is a local existence theorem for the differential equation (9.1)

W’(t)= F ( t , W(2)).

That is, if (s, K ) E I x M,, there exists an E,, > 0 such that [s - E ~ E ~ c ] I , and on this interval there is a solution W = W ( t ) of (9.1) satisfying the initial condition

s

+

(9.2)

W(s)= K.

In particular, the condition that W ( t )is a solution of (9.1) and (9.2) on [s - E ~ s , E ~ is ] equivalent to the condition that W ( t ) is continuous and

+

(9.3 1

W ( t )= K

+

+

s‘

F ( r , W ( r ) )dr

8

for t E [s - E ~ s, 4. We shall also suppose that the following condition is satisfied.

(C,)

On I x M,,

the solutions of (9.1) are locally unique.

I n particular, (C,)holds if in addition to condition (C)the matrix function F ( t , W ) is locally Lipschitzian in W in the sense that if (s, K ) E I

,

9. A CLASS OF MONOTONE OPERATORS

63

x M,, there is a neighborhood 0 = {(t, W ) 1 I t - s I I E , pm[W-K] 5 S}, with 0 c I x M,,, and a nonnegative real-valued integrable function Ao(t) on [s - E , s + E] such that Ilm[F(t,W,) - F(4 Wdl

5 Ao(t)rum[Wz - Wll for ( t , W , , W,) E I x M,, x M,,.

For given (s, K ) E I x M,,, the solution W of (9.1) satisfying the initial condition (9.2), and of maximal extent on I, will be denoted by

(9.4)

W = W(s:t ; K),

a ( s ;K ) < t

a or P ( s ; K ) < b, then pum[ W(s:t ;K)] -,00 as t approaches such an end-point. For a proof of these results, and related properties of solutions of vector or matrix differential equations, see, e.g., Reid [15 ; Chapter 11, Sections 1-31. We shall be concerned primarily with conditions under which the solutions of (9.1) possess the following property.

P. If (s, K ) E I x ML,, then W(s:t ; K ) E M+,, for t

E [s, P ( s ;

K)).

Of special interest for the following discussion are equations (9.1) for which one of the following conditions holds.

(C,) If W E Mk,, then F ( t , W ) . 2 0 for t a.e. on I . (C,) If Wj E M;,, ( j = 1 , 2 ) and W, - 2 - W,, then F ( t , W,) - 2 - F ( t , W,) for t a.e. on I .

\

If F ( t , W) satisfies condition (C),then by a well-known type of argument, (see, e.g., Reid [15 ; Chapter 11, Lemma 2.11, it may be established that if W ( t ) is an m x n continuous matrix function on a subinterval I, of I then F(t, W ( t ) )is locally integrable on I , ; moreover, if condition (C,)also holds, and W ( t ) 2 . 0 on I, then F(t, W ( t ) ). 2 0 for t a.e. on I,. Correspondingly, if conditions (C)and (C,)hold, and W j ( t ) ( j = 1, 2) are continuous m x n matrix functions on a subinterval I , with W,(t) 2 . W l ( t ) 2 0 for t E I , , then

-

-

-

~ ( t~,, ( t ) ) 2 F(t, ~ , ( t ) ) for t a.e. on I,.

-

II. BASIC PROPERTIES OF SOLUTIONS

64

For s E I , and 6 a positive value such that [s - 6, s [ ( t , 6) be defined as follows: C(t,

8) =s =t-6 =t+6

+ 61 c I , let

+

for t E [s - 6, s 61, for t > s + 6 , ~ E I , for t < s - 6 , tEI.

For K E M,,, and F(t, W ) an m x n matrix function satisfying condition (C),it follows from the above remarks that there exists a unique continuous matrix function W ( t ,6) satisfying the integral equation with retarded limit

Moreover (see, e.g., Reid [15; Chapter 11, Theorem 3.7]), if conditions (C)and (C,)hold we have that

(9.6) W(t,6) + W(s:t ; K )

-

as 6

-

---f

0 for t E (a(s; K ) , B ( s ; K)).

I n particular, if K 4 0 and F(t, W) satisfies conditions (C),(C,)and (C2), then (9.5) implies that W ( t ,6) 2 . 0 for t 2 s, t E I,and in view of (9.6) we have that W(s:t ; K ) . 2 0 for t E [s, B ( s ; K ) ) . If conditions (C),(Cl),(C2),and (C,) hold, then the right-hand member of (9.3) is an example of an operator which is monotone in the sense of Collatz [l, 21. I n this case, for (s, K ) E I x M+,, one may prove by mathematical induction. that the Picard iterates W j ( t )= Wj(s:t ; K ) defined by

(9.7)

Wo(t)= K ,

are such that (a) 0

(9.8)

(b)

-

Wj(t)= K

-

+

J:

- 5 . W j ( t ). 5 - W j + l ( t )

Wj(t) * 5

-

W ( S :t ; K )

F(Y,W j - l ( ~ dr )) ( j = 1, 2,

. . .),

for j = 0, 1, . . . and t L s , t €1; for j = 0, 1, .. . and t E [s, B ( s ; K ) ) .

These conditions imply the existence of a matrix function W J t ) on [s, B ( s ; K ) ) such that the sequence { W j ( t ) }converges monotonically ( 2 ) in the sense of (9.8a) to Wm(t).Moreover, for s 5 t , < t ,

-

9. A CLASS

65

OF MONOTONE OPERATORS

< P ( s ; K) we have 0 - 5 * W j ( t 2 )- W j ( t l ) =

J::

F ( Y , Wj-l(~))dr

*

5

-

F ( Y , W ( S rt; : K))dr,

and consequently the matrix functions of the sequence { W j ( t ) }are uniformly equicontinuous and bounded on arbitrary compact subintervals of [s, B ( s ; K)). Hence, the convergence of { W j ( t ) }to W m ( t )it uniform on such subintervals, and therefore Wm(t)is continuous and a solution of (9.1) satisfying WJs)= K , so that Wco(t)= W(s:t ; K ) for t E [s, B ( s ; K ) ) . For later reference, the above results are presented in the following theorem. Theorem 9.1

If conditions (C),(C,),(C,)hold, then the solutions of the dz@rential equation (9.1) possess property P.If in addition condition (C,)holds, then for (s, K ) E I x Mkn the sequence (9.7) of Picard iterates converges monotonically ( - > .) to W(s:t ; K ) on [s, B ( s ; K ) ) , and the conv+e is uniform on arbitrary compact subintervals of [s, B(s ; K ) ) . The following result is then a ready consequence of the above theorem, and well-known extension theorems. Corollary

Suppose that F,(t, W) and F,(t, W) are two m x n matrix functions on I x M,, which satisfy conditions (C),(C,),(C,),(C,),and furthermore

(9.9) if W E

wa then

-

-

F,(t, W ) 2 . Fl(t, W ) for t a.e. on I.

K , 2 . K , > . 0, then the solutions W j ( t )= Wj(s: t ; Kj), t E [s, /lj(s; K,)), of the differential systems

If

(9.10)

Wj’(t) = Fj(t, W j ( t ) ) ,

W~(= S )Kj

( j = 1, 2),

are such that B 2 ( s ; K , ) 5 /ll(s; K l ) , and

(9.11)

-

W,(S:t ; K2) 2 * W,(S:t ; K , )

for

t E [s, / ? Z ( S ; K2)).

Attention will be directed now to a Riccati matrix differential equation (2.1), for which the basic result concerning property P is presented in the following theorem.

66

II. BASIC PROPERTIES OF SOLUTIONS

Theorem 9.2

Under hypothesis (G),a necessary and su$cient condition for the solutions of (2.1) to possess property P is that (9.12)

C ( t ) * 2 * 0,

B ( t ) * 5 - 0,

A(t)t 5

t

0, D ( t ) t 5 t 0, for t a.e. on I .

For brevity, let (9.13)

F(t, w)= C ( t ) - W A ( t ) - D ( t ) W - WB(t)W,

so that equation (2.1) is of the form (9.1) with F(t, W) defined by (9.13). As a consequence of hypothesis (G),conditions (C)and (C,)hold for this particular F(t, W). Let Is denote the set of points on I at which the indefinite integrals of the matrix functions A ( t ) , B ( t ) , C ( t ) ,D ( t ) possess finite derivatives equal to the functional values of these respective matrix functions. It follows readily that if W = W(s:t ; K ) , t E ( a ( s ;K ) , P ( s ; K ) ) ,is the solution of (2.1) satisfying the initial condition W(s)= K , and to E Is n (a(s;K ) , P ( s ; K ) ) , then at to this solution possesses a finite derivative equal to the value of the function (9.13) at ( t o , W(s:t o ;K ) ) . Consequently, if the solutions of (2.1) possess property P and s E I s , K E M+,,, then for indices a,, Po such that Kuoso= 0 we must have Fuoso(s,K ) 2 0. This simple principle applied for suitable matrices K establishes the result that if the solutions of (2.1) possess property P then the conditions of (9.12) hold for t E Is,thus providing the conclusion (9.12) since I - Is is of measure zero. Indeed, for K = 0 this principle yields immediately C ( t ) 2 0 for t a.e. on I . For proof of the necessity of the other conditions of (9.12) it will be assumed that m 2 n. It is to be remarked that the alternative case of n 2 m is reducible to this case by the consideration of the corresponding Riccati matrix differential equation which is satified by the transpose of a solution of (2.1), and noting that the conditions (9.12) are invariant under such a substitution. If m = n, then for K = 2En with 2 > 0, we have

- -

F(t, 223,) = C ( t ) - 2 [ A ( t )

+ D(t)]- PB(t),

and in view of the arbitrariness of 2, it follows that B ( t ) rw 5 t 0 for t E Is. If m > n, for y = 0, 1, . . . , m - n let KY= [ K z ~(a ] = 1, . . . , m ; = 1, . . . ,n ) be defined as

K&,=O, K;+j,B= SjB

for a < y + 1 or a > y + n ; for j , /?= 1, . . . ,n.

9. A CLASS

67

OF MONOTONE OPERATORS

We then have

Therefore, for y = 0, 1, . . . , m - n, and j , B = 1, . . . , n with j # 19 we have Bj,Y+s(t)5 0 for t E Is, and consequently for t E Is we have (9.14)

for a = 1,

Bas(t)5 0

with

(a, B )

. . . , m, /?= 1, . . . , n, # (1, 1) and (a,B ) # (n,m).

N o w f o r i = 1, . . . , m a n d j = 1, ..., n l e t M i j = [ M $ ] ( a = 1, . . . ,m; B = 1, . . . , n) be the m x n matrix with Mi$ = 1 for a = i, B = j , and M$, = 0 otherwise ; i.e., M$ = ai, ai,. If m = n, then for i = 1, . . . , n - 1 we have

and from these relations it follows that B,,(t) 5 0 (a = 1, . . . , m = n) for t E Is. That is, for the case m = n, we have established the necessity of the condition B ( t ) 5 . 0 for t a.e. on I. For the case m > n, then the above relation (9.15b) for i = 1 implies that Bll(t) 5 0 for t E Is. Also, corresponding to (9.15a) we have the relation

-

Fm-1 ,n-1

(t, l[Mm-lon

- Cm-l,n-l(t)

+ Mm.n-1 1)

- J[An,n-l(t>

+

om-1,n(t)l

- J2Bnm(t),

from which it follows that B,,(t) 5 0 for t E Is. These results, together with (9.14), then imply that B ( t ) 5 0 a.e. on I for the case m > n. Now for i = 1, . . . , m, and j = 1, . . . , n we have

-

-

Fkj(t,lMij) = C k j ( t )- lDk.(t)

for k # i,

Fik(t,lMij) = Cik(t)- J A j k ( t )

for k

#j ,

and in view of the arbitrariness of the positive parameter 1 we have A ( t ) e 5 x 0 and D ( t ) x 5 x 0 for t E Is.Thus we have established that the conditions (9.12) are necessary for the solutions of (2.1) to possess property P.

II. BASIC PROPERTIES OF SOLUTIONS

68

If the matrix coefficient functions of (2.1) satisfy the condition (9.12’) C ( t ) * 2 * 0,

-

A ( t ) 5 * 0, D(t) * 5 0 for t a.e. on I ,

B ( t ) * 5 * 0,

which is stronger than (9.12), then the matrix function F ( t , W )defined by (9.13) satisfies the above conditions (C),(Cl), (C,), (C,), and consequently by Theorem 9.1 we have that the solutions of (2.1) possess property P.Now if e ( t ) is a real-valued function locally integrable on I , and s E I , it follows by direct substitution that an m x n matrix function W ( t ) is a solution of (2.1) on a subinterval I,, of I if and only if W,(t)= exp{Ji e ( r ) d r } W ( t ) is a solution of the Riccati matrix differential equation (2.11) W,’(t)+ ~ , ( t ) A l ( t ) + o l ( ~ ) ~ l ( tWl(wl(t)Wl(t))+ C,(t)=O, where

In particular, let e ( t ) be such that for t e ( t ) 2 2A,(t),

E

I we have

e ( t ) P 2Daa(t), (a = 1,. . . , m ;

B = 1,. . . , n ) ;

+ xr!l

for example, one might choose e ( t ) = 2[xi=, I A,(t) I I DJt) I]. Then whenever the matrix functions A(t),B ( t ) ,C ( t ) ,D ( t ) satisfy conditions (9.12) the matrix functions A,(t), B,(t), C l ( t ) ,D l ( t ) satisfy conditions (9.12‘), so that by the above discussion the solutions of (2.1,) possess property P. Since the solutions of (2.1) possess property P if and only if the solutions of (2.11) possess property P,however, it then follows that conditions (9.12) are sufficient for the solutions of (2.1) to possess property P. Corollary

Suppose that A,(t) E P,,[a, b ] and C l ( t )E P,,[a, b] for arbitrary compact subintervals [a, b ] of I ; moreover, that s E I , and

(9.16)

A,(t)+IxO,

6C,(t).>.O,a.e.

where 8 , = (OPm) is them-dimensionalvector with 8,

on { t l t ~ I ,t > s } , =

1 (B = 1, . . . , m).

69

9. A CLASS OF MONOTONE OPERATORS

If W,(t) is an m x n matrix function satisfring the dzferential system (9.17)

+ U.‘,(t)A,(t)- cl(t)

W~(= S )0,

= 0,

Wl’(t)

then emWl(t) 2. - 0 for t

E

I, t 2 s.

/

The result of this corollary follows from the above theorem, upon noting that the 1 x n matrix function W ( t ) = $,W,(t) is the solution of the corresponding differential equation (2.1) with A(t)= Al(t),B(t)= 0, C ( t ) = 8,Cl(t), D ( t ) = 0, and which satisfies the initial condition W(s)= 0. For the special case of (2.1) in which A ( t ) = 0, C ( t )= 0, and D ( t ) = 0 on I , we have that W = Wo(t)= 0 is a particular solution of the Riccati matrix differential equation

+

(9.18)

W ( t ) W(t)B(t)W(t) = 0,

with corresponding solutions of the equations (2.5) and (2.6) given by G(t,s I W,) EE E m , H ( t , s 1 W,) = En. Consequently, in view of Lemma 2.1 we have that the solution W ( t )= W,(s:t ; K ) of (9.18) satisfying the initial condition W(s)= K has maximal interval of extent a,(s; K ) < t < Bl(s; K ) , where a,(s; K ) and Bl(s; K ) are the respective infimum and supremum of values t on I such that Em K B ( r ) dr is nonsingular on the interval with endpoints s and t, and moreover,

+

(9.19) W,(s:t ; K )

+K

= [Em

Jl

B ( r ) dr]-’K

s:

for t E (a,(s;K ) , Bl(s; K ) ) .

In particular, when condition (9.12’) holds, and F ( t , W ) is defined by (9.13), then F ( t , W ) 2 -WB(t)W for ( t , W ) E I x Mhn, and if K E Mkn then in view of the Corollary to Theorem 9.1 we have that B(s; K ) I B l k K ) , and

-

Now if B,(t) = KB(t), then the m x m matrix function Wo(t) W,(s: t ) defined by Wo(t)= [Em J: B,(r) dr1-l is the solution of the differential system =

+

with maximal interval of extent a,(s; K ) < t < Bl(s: K ) , and if K E MZn

II. BASIC PROPERTIES OF SOLUTIONS

70

then by Theorem 9.1 the sequence of Picard iterates defined by

Wodt) = E m

9

converges monotonically (

-

2 . ) to Wo(t)on

-

[s, Bl(s; K)). Moreover,

zm(t) on

by induction it follows that Wom(t) 2 for t E [s, pl(s, K)) the series

if

[s, Bl(s; K ) ) . Hence,

converges and has sum S(s: t) which satisfies the relation

(Em

+ J: Bo(r)d r ) S ( s : t ) = Em.

Consequently, we have established the results of the following theorem. Theorem 9.3

If the matrix coeficient functions of (2.1) satisfy condition (9.12'), K E MLn, and W ( t )= W(s:t ; K ) , a ( s ; K ) < t < B(S; K ) , is the solution of (2.1) satisfying the initial condition W(s)= K , then on [s, p ( s ; K ) ) the m x m matrix Em + K B ( r ) dr is nonsingular, and (9.20) holds. Moreover, the matrix series

S:

(9.21)

Em

+

(-lY(K

3=1

Jt

B ( r )d r y

1)

+ K Jf B ( r ) dr1-l for t E [s, p ( s ; K ) ) .

is convergent and has sum [Em

In particular, suppose that 4 ( t ) = 4(s: t ; k), ao(s;k) < t < p 0 ( s ; k), is the solution of maximal extent of the scalar Riccati system @(t)

+ 2a(t)4(t) +

b ( t ) P ( t ) - c ( t ) = 0,

+(s) = k,

where a ( t ) , b(t), c ( t ) are real-valued, locally integrable functions on I satisfying the inequalities

(9.22) c ( t ) 2 0,

b ( t ) 5 0,

a ( t ) 5 0,

for t a.e.

on I ,

9. A CLASS OF MONOTONE OPERATORS

71

and k 2 0. The above Theorem 9.3 then implies that 1 and 4(t)

2

1

+k

k

J:

+ k Jt b(r) dr > 0

for t E [s, Po($; k)).

b(r)dr

Moreover, if y ( t ) is a nonnegative continuous function such that on a subinterval [s, sl) of I we have

(9.23) y ( t >2 K

+ J' [ c ( r ) - 2a(r)y(r)- b(r)yz(r)]dr

for t E [s, sl),

8

then by classical comparison and continuity methods for real-valued scalar differential and integral equations (see, e.g., Kamke [l ; pp. 81-86], Hartman [ 2 ; Chapter 111, Section 41, Reid [ 1 5 ; Chapter I, Section 4 ; Chapter 11, Section 31) it follows that Bo ( s; k) 2 s,, and Y(t)2 4(t)

for t E [s, s1).

Correspondingly, if ~ ( tis) a nonnegative continuous function such that

(9.24) X(t) 5 k

+J

I

for

[C(r)- 2a(r)X(r)- b(r)Xz(r)ldr

8

then ~ ( t5) 4 ( t )

for s 5 t

< min{s,,

Bo(s;

t

E

[s, $11,

k)}.

Whenever the coefficient matrix functions of (2.1) satisfy (9.12') and K E Mkn, the solution W ( t )= W(s:t ; K ) , a(s; K ) < t < B ( s ; K), of (2.1) is such that ~ ( t=) p m [ W ( t ) ]satisfies (9.24) with = ~ m [ K l , c ( t ) = ~m[C(t)], 2a(t) = - ~ m [ - A ( t ) ] - prn[-D(t)l,

b ( t ) = -~m[--B(t)],

$1

= B(s;

K).

and therefore one has the following result. Theorem 9.4

If the coefiient matrix functions of (2.1) satisfy condition (9.12') on (-00, m) and also K E Mkn, then the solution W ( t ) = W(s:t ; K ) ,

72

II. BASIC PROPERTIES OF SOLUTIONS

a(s; K )

< t < P ( s ; K ) , of (2.1) is such that:

(i) i f P ( s ; K) = 00 then the solution a ( t ) = a(s: t ; vD3[K]) of the scalar Riccati system

+

(9.25)

d ( t ) - (vm[-A(t)J vm[-D(t)])a(t) - vm[-B(t)]0’(t) 4 s ) = vm[KI, - vm[C(t)3= 0,

exists on [s, co); moreover, if either v,[K] > 0 or vm[C(t)]is not the null function on [s, co), then vm[-B(r)] dr is Fnite and

J-r

(ii)

if

(9.26)

exists on For n a positive integer, let 8, = (Oan) be the n-dimensional vector with €la, = 1 (a = 1 , . . . , n), and denote by r, the convex set in 3, consisting of vectors v = (v,) with v, 2 0 (a = 1, . . . ,n ) and 6,v 5 1. An m x n real matrix M will be said to have property A,, if Mv E r,,, whenever v E rn; if m = n, property A,, is the condition called ‘‘ dissipative” by Redheffer [ S ] . The result of the following lemma is immediate. Lemma 9.1

An m x n matrix M has property A , if and only if M 0, - OmM * 2 - 0.

-

2 . 0 and

The result of the following theorem will be of use in Section 7 of Chapter V, in the characterization of the “scattering matrix’’ considered therein. Theorem 9.5

Suppose that the matrix coeficient functions of (2.1) satisfy hypothesis s E I let W ( t ) = Wo(s:t I M o ) be the solution of (2.1) satisfying the initial condition Wo(s:s I MO) = MO, where M o is a given m x n matrix such that MO - 2 - 0 and 0, - OmMO = 0. If for each s E I , the solution Wo(s:t I M o ) exists throughout I + ( s ) = ( t I t E I ,

(6)on an interval I , and for

73

9. A CLASS OF MONOTONE OPERATORS

t 2 s} and possesses property A,, for each t E I + ( s ) , then (9.27)

T ( t I M o ) = C ( t ) - M o A ( t ) - D ( t ) M o- M o B ( t ) M o

is such that: (9.28)

(a) if a and fi are indices such that Tms(tI MO) 2 0 for t a.e. on I ; (b) e,T(t

1 Mo). 5

= 0,

then

0 for t a.e. on I.

In particular, i f m = n then for Mo = En condition (9.28a) becomes (9.28a)O

C ( t ) - A ( t )- D ( t ) - B ( t ) IC 2 IC 0

for

t a.e. on I ,

Moreover, i f conditions (9.12) hold on I , and (9.29)

BnA(t) - B,C(t)

2 - 0,

B,B(t)

+ B,D(t)

*

2* 0

for t a.e. on I ,

then for M o such that Mo . 2 - 0, 0, - BmMo= 0, and each s E I , the solution Wo(s:t I M o ) of (2.1) exists on I+(s) and possesses property A,, for each t E I+(s). Since the condition that Wo(s:t I MO) has property A,, is equivalent to the condition (9.30) Wo(s:t I MO) - 2 . 0

and

8,

- B,Wo(s: t

for t E I+(s)

I MO) . 2

0 for t E I+(s), *

the necessity of condition (9.28a) at each point s of the set I g is a ready consequence of the fact that Wo(s:s I MO) = M o and Wo,(s:s I MO) = T(s I MO). Correspondingly, the necessity of condition (9.28b) at each s E Ig follows from the fact that u ( s : t ) = - 6,Wo(s: t I MO) satisfies the initial condition ~ ( s s) : = 0 and u t ( s : s) = -6,T(s I MO) at such values s. Now suppose that conditions (9.12) and (9.29) hold, and for a given s E I let Wo(s:t I MO), a(s I MO) .< t < f i ( s I MO), be the solution of (2.1) satisfying the initial condition Wo(s:s I MO) = Mo, and of maximal extent (a(s I Mo), /?(s I MO)). In view of the conditions (9.12) from Theorem 9.2 it follows that Wo(s:t I MO) . 2 0 for t E [s, /?(s I MO)). Moreover, W l ( t )= W,(s:t I MO) = 1 3 ~ BmWo(s:t I MO) is a 1 x n matrix func-

on

II. BASIC PROPERTIES OF SOLUTIONS

r4

tion satisfying the differential system (9.17) with coefficient matrix functions given by

(9.31)

A,(t) = A ( t )

+ B ( t ) W o ( s :t I MO),

C,(t) = B,A(t)

- O,C(t)

-

+ [O,B(t) + B,D(t)]Wo(s: t I MO). -

Since A ( t ) x 5 wr 0 and B ( t ) 5 0 for t a.e. on I , and Wo(s:t I MO) . 2 . 0 for t E [s, B ( s I MO)), we have that A , ( t ) x 5 x. 0 and C , ( t ) 2 0 a.e. on [s, B ( s I Mo)), and from the Corollary to Theorem 9.2 it follows that W,(s:t I MO) = 6, - O,Wo(s: t I MO) . 2 0 for t E [s, B ( s I MO)). That is, it has been established that whenever conditions (9.12) and (9.29) hold, and M o 2 0, On - 6,Mo = 0, then for s E I the matrix function Wo(s:t I MO) possesses property A,, for each t E [s, B ( s I MO)). Finally, in view of the uniform boundedness of Wo(s:t I MO) on [s, B ( s I Mo)), it follows by a well-known extension argument, (see, e.g., Reid [ 1 5 ; Chapter I, Theorem 5.8, and Chapter 11, Theorem 3.9]), that [s, B ( s I MO)) = I+(s), thus completing the proof that if s E I, and MO satisfies the prescribed conditions, then Wo(s:t I MO) exists on I + ( s ) and has property A,, for each t E I+(s).

-

-

-

10.

Results Related t o the Perron-Frobenius Theorem

Certain results of the preceding section will be applied now to the particular Riccati equation (9.18) to obtain a result which includes a proof of the classical results of Perron and Frobenius on the dominant proper value and corresponding proper vector of an irreducible nonnegative constant matrix. For simplicity of notation, in the following we shall write 9 ( s ; t ) for the integral J: B ( r ) dr. Theorem 10.1

Suppose that the n x n matrix function B ( t ) is of class &,,,[a, b] for arbitrary compact subintervals [a, b] of [s, w), and B ( t ) 2 - 0 for t a.e. on [s, w). If b denotes the supremum of values s, such that E - 3 ' ( s ; t ) is nonsingular for s 5 t 5 s, , then:

(i) [ E -.59(s; t ) ] - l . 2 - 0 for t E [s, b); v[B(r)] dr = 00, then b < w; (ii) if (iii) if b < w, then E - 2 ( s ; b) is singular,

Jr

-+ 00

(10.1)

,urn[{ E - g ( s ; t)}-'] as t -+ b-, and there is a corresponding proper vector y such that

[E-S'(s;b)]y=O,

y -2.0;

75

10. PERRON-FROBENIUS THEOREM

(iv) if 9 ( s ; b) is irreducible in sense of Frobenius, then 1 = 1 is the Perron-Frobenius proper value of this matrix; if y is such that either [ E - 9 ( s ;b)]y 2 0 or [E - 9 ( s ;b)]y 5 0, then [ E - 9 ( s ; b)]y = 0 ; in particular, I = 1 is a simple proper value of 9 ( s ; b) ; ( v ) if G ( t ) = [G,,(t)] is a continuous complex-valued matrix such that I Gms(t)I 5 B,,(t) (a, B = 1, . . . , n ) and F(S; t ) = G ( r )dr, then E - F ( s ; t ) is nonsingular for s 5 t < b ; moreover, i f 9 ( s ; b ) is irreducible and E - F ( s ; b) is singular, then there exists a constant diagonal matrix K with diagonal elements of unit moduli such that G ( t )= KB(t)K-' on [s, b).

-

-

-

-

Ji

It is to be noted that the hypotheses of Theorem 10.1 require B ( t ) to satisfy the condition B(t) 2 0, in contrast to the condition B ( t ) . 5 . 0 in (9.12). This change seems appropriate, in order that the notation be in form more consistent with that frequently used in considering the Perron-Frobenius theorem (see, e.g., Beckenbach and Bellman [l ; Chap. 2, Secs. 38,391, Bellman [2; Chap. 161, or Gantmacher [l ; Chap. 1111). In effect, this change in notation merely replaces B ( t ) by - B ( t ) for the applications of the results of the preceding section. Now the matrix W = Wl(s: t ; E) is the solution of the matrix differential system

-

(10.2)

-

W ' ( t )- W ( t ) B ( t ) W ( t= ) 0,

W ( S )= E,

and conclusion (i) is an immediate consequence of Theorem 9.3, while conclusion (ii) follows from (i) of Theorem 9.4. If b < 00, then pm[W,(s:t ; E)] + w as t -+ b-, and hence if w ( t ) is a column vector of Wl(s: t ; E) satisfying I w ( t ) I = ,ucDIWl(s:t ; E)], then w"t) = (11 I w ( t ) I)w(t)

-

is such that I wo(t)I = 1, wo(t) 2 - 0, [ E - 9 ( s ; t)]wO(t) -+ 0 as t + b-. Consequently, if {t,} is a sequence on [s, b) such that t , + b and {wo(t,)} converges to a vector y, then y is a proper vector satisfying (10.1). Now if 0 5 k < 1 there exists an s1 E (s, b) such that < 8 ( s ; sl) 2 k B ( s ; b), and since by Theorem 9.1 the Picard iterates for (10.2) converge to W,(s:t ; E) for t E [s, p ( s : E)), it follows that for 0 5 k < 1 the matrix E - k g ( s : b) is nonsingular with inverse given by its Maclaurin expansion. Consequently, E - 1 2 ( s ; b) is nonsingular for all complex A with I 1 I < 1, or alternately, all proper values I of 3 ( s ; b) are such that I 1 I 5 1.

-

II. BASIC PROPERTIES OF SOLUTIONS

76

If - 6 ( s ; b) is irreducible in the sense of Frobenius (see, e.g., Gantmacher [ l ; Chap. 111, Sec. l]), then any nonnull vector y satisfying (10.1) must be such that y . > . 0, and by the argument of Wielandt [l ; (g), p. 6451, it follows that the dimension of { y I [E - 9 ( s ; b)]y = 0} is one. Moreover, since [ B(s;b)]* = B * ( r ) dr is irreducible whenever %(s; b) is irreducible, there exists a vector z > 0 which satisfies z"[E - .%'(s; b)] = 0. Consequently, if y is such that [E - B (s ; b)]y = u, with either u 2 0 or u . 5 . 0, it follows that 5% = 0 and hence u = 0, so that [E - 9 ( s ; b)]y = 0. I n particular, in case [E - 9 ( s ; b)]y = u, and [E - 9 ( s ; b)]u = 0, from the above discussion it follows that either u 2 . 0 or u . 5 0, and hence u = 0, so that 1= 1 is a simple root of the characteristic equation det[1E - B ( s ; b)] = 0. I n particular, the conclusions (i)-(iv) for a nonnegative constant matrix B provide a proof of basic results initially due to Perron and Frobenius, and the variational nature of 1= 1 as a proper value of 9 ( s ;b) is a ready consequence of (iv). I n order to establish conclusion (v), it is to be noted that in view of Theorem 9.3 the inequalities I Ga,(t) I 5 Ba,(t) (a, p = 1,. . . , n) imply that for t E [s, b) the matrix E - . F ( s ; t ) is nonsingular and its inverse is given by E zgl [ V ( s; t)]j. Now if E - Y ( s ; b) is singular, and o is a nonzero vector satisfying [E - F ( s ; b)]v = 0, then for o+ = (I o, I), (a = 1, . . . , n) and G+(t) = [I G,,(t) I] (a, /I= 1, . . . , n), we have the relations

s,"

-

-

-

-

-

1

+

(10.3) From conclusion (iv) it then follows that [E - 9 ( s ; b)]v+ = 0, so that o = ( y aexp{i8,}) (a = 1, . . . , n), with the 8, real and o+ = y , where y is a proper vector satisfying (10.1) and hence y 2 0, Moreover, the second relation of (10.3) implies that (J,"[ B ( r )- G+(Y)]dr)y = 0, and since B ( t ) . 2 G+(t) it follows that B ( t ) = G+(t) on [s, b). If K = [S,, exp{i8,}] (a, /?= 1, . . . , n), then the relations [E - F ( s ; b)]o = 0, o = Ky and [E - 9 ( s ; b)]y = 0 imply

-

-

[

0 = E - Jb K-'G(r)K d r ] y = a

[

Jb

[ B ( r ) - K--IG(r)K]d r ] y ,

8

and as [K-'G(t)K]+= G+(t). 5 . B ( t ) on [s, b) it follows that B ( t ) = K-'G(t)K and G ( t )= KB(t)K-' on this interval.

11. A DISSIPATIVE PROPERTY OF SOLUTIONS

n

11. A Dissipative Property of Solutions of Riccati Matrix Differential Equations

In this section we shall consider some properties of solutions of Riccati matrix differential systems (3.4) that are of significance in the.study of “scattering” processes (see Redheffer [3] and Reid [6, Secs. 1-71), in which the treated systems are of the form considered herein, with m = n. It will be assumed that the interval I of consideration is an open interval ( a o ,bo), and that the coefficient matrix functions A ( t ) , B ( i ) , C ( t ) , D ( t ) satisfy hypothesis ($) on I . Also, as in Section 9, the set of points on I at which the indefinite integrals of A ( t ) , B ( t ) , C ( t ) , D ( t ) possess finite derivatives equal to the functional values of these respective matrix functions will be denoted by Is. As in past discussions, we shall set I+(s) = { t I t E I , t 2 s}, I-(s) = { t I t E I , t p s}; also, for brevity in the formulation of various conditions and statement of results, we introduce the notations J A = I x I , J+ = {(s, t ) I (s, t ) E J A , t E I+(s)} and J- = {(s, t ) I (s, t ) E JAY t E I-($)}. For a given m x n matrix K , and s E I , the solution of (2.1) satisfying the initial condition W(s)= K will be denoted by W ( t )= W(s:t ; K ) , and I { s ; K } will designate the maximal interval of existence of W(s:t ; K ) containing t = s. One of the properties of solutions to be considered is the following.

(9+) I’v[K] < 1, then W(s:t ; K ) exists and satisfies v[W(s:t ; K ) ] < 1 on J+. If i n 9 + we replace J + by J- or J A , the resulting condition is denoted

by 9or 9 ’ A . If an m x n matrix M is such that M*M = E n , for brevity we shall say that M is “column-unitary.” If M is column-unitary then m >_ n ; moreover, if m = n then M is column-unitary if and only if it is unitary. By 2+we shall denote the following condition.

(2+)If K is column-unitary, then W(s:t ; K ) exists and is columnunitary on J+. If in %+ we replace J+ by J - or J A , the resulting condition is denoted by

8-or PA.

As in Section 3, the symbols W(s:t ) , G(s: t ) , H ( s : t ) , F ( s : 2) will denote the solution of the differential system (3.12), which for ready

II. BASIC PROPERTIES OF SOLUTIONS

78

reference is here repeated. (a)

(11.1)

+

+

W,(s:t ) W(s:t)A(t) D(t)W(s:t ) + W(s:t ) B ( t ) W ( s t: ) - C ( t )= 0,

(b) G,(s: t )

W(s:s) = om,,

+ [ D ( t )+ W ( S :t)B(t)]G(s:t ) = 0,

G(s: S)

(c)

H,(s: t ) + H ( s : t ) [ A ( t )+ B(t)W(s:t ) ] = 0, H ( s : S)

(d)

F,(s: t ) - H ( s : t)B(t)G(s:t ) = 0,

= Em, = En,

F ( s : S) = Onm.

Also, as in (3.3), for a solution of (11.1) we denote by W ( s : t ) the corresponding (n m ) x (n m ) matrix function

+

(11.2)

+

V ( s :t ) =

[

W(s:t ) H(s:t ) -F(s: t )

In particular, we shall be concerned with the following properties of the matrix function V ( s : t).

(%+) W ( s : t ) exists and satisfies v [ W ( s : t ) ]5 1 on J+. (%+) W ( s : t ) exists and is unitary on J+.

If in %+and %+ we replace J+ by J-, the resulting conditions are denoted by 3- and g-;correspondingly, if J + is replaced by J A the and %A. resulting conditions are designated by I n the consideration of the above stated properties a central role is played by the hermitian form (11.3)

+

+

Q ( t ; p ,q ) = -q"[D(t) D*(t)lq q"[C(t) - B"(t)lp P"[C"(t)- B(t)lq - p"[A(t) A"(t)lp

+

+

+

in the (n m)-dimensional vector ( p ; q), wherep = ( p m (a ) = 1, . . . , n), and q = (qp) (b = 1, . . . , m ) . I n particular, the presented results involve the following properties.

For t a.e. on I the hermitian form Q(t ;p , q ) is nonpositive (go+) definite. For t a.e. on I the hermitian form (9+)

Q(t;p , q)

is nonpositive definite on the set of ( n m)-dimensional vectors ( p , q ) with p E Q,, q E Q,, and I q l2 - I p l2 = 0.

+

79

11. A DISSIPATIVE PROPERTY OF SOLUTIONS

If in go+ and 9+ we replace “nonpositive definite” by “nonnegative definite” the resulting conditions are denoted by go-and 9; if “nonpositive definite” is replaced by “zero,” the resulting conditions are designated go’ and 9 A . If K is a given m x n matrix, and for s E I we denote by U ( t ) = U(s:t ; K ) , V ( t )= V(s:t ; K) the solution of the linear matrix differential system (11.4)

-V’(t) + C ( t ) U ( t )- D ( t ) V ( t )= 0) U ’ ( t )- A ( t ) U ( t )- B ( t ) V ( t )= 0,

V(S)= K ,

U ( S )= E n s

then I { s ; K} is the maximal subinterval of I containing s on which U(s:t ; K) is nonsingular, and (11.5)

W(s:t ; K ) = V(s:t ; K)U-’(s: t ; K)

for t E I { s ; K } .

Also, if we set

(11.6) Y ( s t: ; K ) = V ” ( S :t ; K ) V ( s :t ; K ) - U ” ( S :t ; K)U(s:t ; K ) then for t E I ( s ; K} we have

for (s, 2)

E JA,

(11.7)

P(s:t ; K ) = U ” ( S :t ; K)[W*(s:t ; K)W(s:t ; K ) - En] x U(s: t ; K ) .

Consequently, for t E I { s ; K} we have that v [ W ( s :t ; K ) ] 5 1 if and only if P(s: t ; K ) 5 0. Now if 5 E &, and s E I , then for t E I2 the scalar function 5”P(s: t ;K)5 has a derivative with respect to t given by

Prefatory to the proofs of Theorems 11.1 and 11.2, we shall establish the following lemmas. Lemma 11.1

If m 2 n and K is a given m x n column-unitary matrix such that W(s:t ; K ) exists and satisjies v[ W(s:t ; K ) ] 5 1 {is column-unitary } on J+, then for arbitrary 5 E an the hermitian form Q ( t ; 5, K 5 ) is nonpositive {zero} for t a x . on I .

80

II. BASIC PROPERTIES OF SOLUTIONS

If W(s:t ; K ) exists and satisfies v [ W ( s :t ; K)]5 1 on J+, then for arbitrary 5 E Qn we have E V ( s : t ; K)[ 5 0 on J+. As Y(s:s; K )= 0 whenever K is an m x n column-unitary matrix, it follows immediately from (11.8) that Q(s; 5, KE)5 0 for s E 4 , and hence Q ( t ; 6,K[)is nonpositive for t a.e. on I . Alternately, if W(s:t ; K ) is column-unitary on J+ then Y ( s : t ; K ) = 0 on J+, so that for s E Is and 5 E Qn we have Q(s; 5, KE)= 0 by (11.8). Lemma 11.2

If m 2 n then S Y A is a necessary condition for %+, and condition for PA.

g 0 A

is a suficient

If Condition %+ holds, then for an arbitrary column-unitary m x n matrix K the solution W(s:t ; K ) exists and is column-unitary on J+, so that Lemma 11.1 implies that Q ( t ; 5, KE)= 0 for arbitrary columnunitary K, [ E En, and t E I s . Now for m > n and (p,q) EQ, x Q, with I q l2 - I p l2 = 0 there exists a column-unitary matrix K such that q = Kp. First, if p = 0 then q = 0 and the result holds for K an arbitrary m x n column-unitary matrix. On the other hand, if p # 0, then q # O and there exists an m x m unitary matrix Q whose first column vector is (1/1 q 1)q, and an n x n unitary matrix P whose first column vector is (1/1 p l)p. If Qo is the m x n matrix whose column vectors are the first n column vectors of Q in given order, then K = Q$* is an m x n column-unitary matrix such that Kp = (I p 111 q 1)q = q. Hence for m 2 n the condition that Q(t ; E, KE)= 0 for arbitrary columnunitary K,5 E Q, ,and t E I s , is equivalent to the condition that Q(t ;p, q ) = 0 for arbitrary (p,q) E Q, x Q, with I q 12 - I p l2 = 0 and t E I s . On the other hand, if g o A holds and K is a column-unitary matrix, then the initial condition Y(s:s; K ) = 0 and (1 1.8) for arbitrary [ E Qn implies that W(s:t ; K )is column-unitary for t E I { s ; K}.By a well-known type of extension argument (see for example Graves [l, Chap. IX, Theorem 41 or Reid [15, Theorem I : 5.71) it then follows that I { s ; K } = I for each s; that is 9 , 'implies PA. Lemma 11.3

If m 2 n then 9+ is a necessary condition for .P+, and go+ is a suficient condition for 9+. If 9 ' + holds and K is an arbitrary column-unitary matrix, then for 0 < r < 1 the solution W(s:t ; rK)exists and v [ W ( s : t ; rK)]< 1 on J+.

11. A DISSIPATIVE PROPERTY OF SOLUTIONS

81

By the continuity of W(s:t ; r K ) as a function of r it then follows that W(s:t ; K ) exists and satisfies v [ W ( s :t ; K ) ] 5 1 on J+. Consequently, by Lemma 11.1 we have that for t E Is the form Q ( t ; 5, K 5 ) is nonpositive for K an arbitrary m x n column-unitary matrix, and 6 E En. Since m 2 n, by the same argument as used in the proof of the preceding lemma it follows that this latter condition is equivalent to 9+. Conversely, if 9 , ' holds and v[K]< 1, let I , = [s, p ( s ; K ) ) . It follows from (11.8) and (11.7) that if is an arbitrary nonnull vector of En then [*!P(s:

s;

K ) t = I K5

l2

-I5

l2

< 0, and

[t*Y(s:t ; K ) E ] ,5 0 for t E I, n Is.

Consequently, E"Y(s: t I K)6 < 0 and v [ W ( s :t ; K ) ] < 1 for all t E I , . By an extension argument similar to that cited in the above proof of Lemma 11.2 it then follows that I, contains all t E I with t 2 s ; i.e., 0,' implies 9+. Theorem 11.1

(i) go+, 9,-, or 9 , 'is a necessary and sujicient condition for the respective property %+, 3- or 9,';(ii) 0,' is a necessary condition and a sujicient condition for U ,'. for either g+or g-,

As in Section 3, let

and

Y ( s : t ) defined by From the discussion of Section 3 it follows that Y = s) : (11.2) is the solution of (11.10) satisfying the initial condition Y ( S = A? where (11.11) Corresponding to (11.3), for the Riccati differential equation (1 1.10)

II. BASIC PROPERTIES OF SOLUTIONS

a2

we have the hermitian form

(11.12)

+

+

R ( t ; n, a) = - a " [ 9 ( t ) 9 " ( t ) ] a a+[@(t)- 3 " ( t ) ] n +n*[FF+(t) -3 ( t ) ] a- n " [ d ( t ) d " ( t ) ] n ,

+

+

where now n 3 (nJ and a 3 (a,) are vectors of dimension m n. Moreover, if a and n are vectors of a+ ,, and we write a = ( q l ; p l ) , n = ( p z ;q2), where p a E Q, and qa E Q, for Q = 1 , 2 , then one may verify directly that

(11.13)

R ( t ; n, 0) = W t ;P z ,

41).

If condition %+holds, then application of Lemma 11.1 to the solution Y ( s : t) of (11.10) yields the result that for arbitrary n = ( p ; q ) with p E Q,, q E Q, we have R(t ; n,A n ) 5 0 for t a.e. on I. I n particular, for n = ( p ;q ) with p E Q, , q E Em,we have A n = (q ;p ) and from (11.13) it follows that for p E Q,, q E Em we have Q ( t ; p , q ) 5 0 for t a.e. on I. That is, go+ is a necessary condition for 9$+.Conversely, if 9 , 'holds then (11.13) implies that if n and a are arbitrary vectors of then R ( t ; n, a) 5 0. That is, for the Riccati matrix differential equation (1 l.lO), the corresponding condition go+ holds. The sufficiency conclusion of Lemma 11.3 applied to the differential equation (11.10) shows that this differential equation has the corresponding property .9'+ ; i.e., if W ( s : t ;X ) is the solution of this equation with W ( s : s; 3) =3 and ,[XI < 1, t h e n Y ( s : t ; Z )existsandsatisfies~[Y(s:t ; X ) ] 21, 3 z ( t ) = 8[3(t)- 223P)31

Moreover, if 3(t) is real-valued, and U,(t), U 2 ( t )are n x n matrix functions such that 3(t)= [ U , ( t ) U , ( t ) ] ,then & ( t ) = Y ( t ) , where W ( t )= g[U,(t) - i U 2 ( t ) ] ;also, if 3(t)is symmetric, then 3,(t) and &(t) are also symmetric, and W ( t ) is hermitian. In particular, if 3(t) is a real-valued symmetric locally a.c. matrix function such that 3 [ 3 ] ( t5) 0 a.e. on a subinterval I , of I, and B ( t ) 2 0 a.e. on I,, then also 3 [ 3 , ] ( t ) 5 0 a.e. on I, and the corresponding W ( t )is an hermitian locally a.c. n x n matrix function such that $t[Wj(t)5 0 a.e. on I,. A class of differential systems which satisfies the involution condition with 0= -, and for which the coefficient matrix functions are nonreal, is given by certain differential equations in the complex plane. For example, consider the second-order linear homogeneous differential equation

(2.16)

L( r ( z )$)- p ( z ) h = 0, dz

where ~ ( zand ) p ( z ) are holomorphic functions of the complex variable z in a simply connected region %, and ~ ( z#) 0 for z E 91. Under the substitution dh h d z ) = .(z) -&-I hdz) = h(z), this equation is equivalent to the first-order system

dhl --

(2.17)

dz

-

1

~ ( z h)2 ’

If ( h , ( z ) ) (a = 1, 2) is a solution of (2.17) in %, and I‘ is a rectifiable arc in 91 of absolutely continuous representation z = [ ( t ) , t , 5 t 5 t , ,

99

3. PROPERTIES OF SOLUTIONS

then u ( t ) = hl( C(t)),v ( t ) = hz( [ ( t ) )is a solution of the differential system (2.18)

-v’(t)

+ P(C(t))C’(t)u(t)

= 0,

t , 5 t 5 2,.

u ’ ( t ) - [C’(t)/r(C(t))]v(t) = 0,

This system is then of the form (1.3L)with% = 1, and C ( t ) = p(C(t))C’(t), B ( t ) = [‘(t)/r(C(t)).Clearly (2.18) is in general symmetric, but is hermitian only in the very special instance of functions ~ ( z p) (, z ) and curve T such that C ( t ) and B ( t ) are real-valued for t , 5 t 5 t,.

3.

Properties of Solutions of lnvolutory Systems

If the involutory condition (1.1) is satisfied, and W = W ( t ) is a solution of the Riccati matrix differential equation (1.2) on a subinterval I,, then in view of the fact that W = W @ ( tis ) also a solution of this equation, one may verify readily that the matrix functions defined in general as in Lemmas 1I:Z.l and 115.2 are related as follows: G@(t,s I W ) = H ( t , s I W @ ) ,

H@(t,s I W ) = G ( t , s I W @ ) ,

(3.1)

F@(t,s I W ) = F ( t , s I W@).

A fundamental property of solutions of (1.2) and associated systems ( 1.3M) is presented in the following theorem. Theorem 3.1

Suppose that W = Wo(t)is a solution of an involutory Riccati matrix dz@ential equation (1.2) on a subintervalI, of I , and Yo(t)=( U o ( t ) ;V o ( t ) ) is a corresponding solution of (1.3M) with U,(t) nonsingular and W o ( t )= V,(t)U;’(t)

for

t

E

I,.

If K is the constant matrix such that { Y oI 0I Y o } ( t )= --K on I,, and for s E I , the matrix function T = T ( t , s I Y o )is the solution of the diTerential system (34

T ’ ( t ) = - UG’(t)B(t)U$-’(t)KT(t),

T ( s )= E,

then G ( t , s I W,), H ( t , s I W,,),F ( t , s I W,) dejined by the respective systems

111. INVOLUTORY RlCCATl EQUATIONS

100

(I1 :2.5), (I1 :2.6), (11:2.7) are given by (a)

s I Y,)U,@(S) G(t,s W,) = H@(t,s 1 W,)U@-'(s)T@-'(t, = Ly-'(t)T@-l(t,s I Yo)U0@(s),

(3'3) (b) H(t, s W,) = Uo(S)U;l(t), (c)

F(t, s I W,) = U,(s)S@(t, s I Yo)Uo@(s),

where

(3.4)

S(t, s I Y o )=

t

T-l(r, s I Y o ) U , l ( r ) B ( r ) U : - l ( rdr. )

8

Moreower, a matrix function Y ( t )= ( U ( t ) ;V ( t ) )is a solution of (1.3M) on I , if and only on this subinterwal

+ U?-'(t)[K1- K N ( t ) ] ,

(3.5) U ( t )= U,(t)N(t), V ( t )= V,(t)N(t) where K , is a constant matrix and

(3.6)

+ S(t,s I Y0)KlI ;

N ( t ) = T ( t ,s I YO)[KO

in particular, if Y ( t )= ( U ( t ); V ( t ) ) and Yo(t)= (Uo(t); Vo(t)) are ) -Kl. related by (3.5), then { Y I 0I Y o } ( t =

The value of H ( t , s I W,) is given by (3.3b), in view of equation (I1 :2.9). Now W, = W,@ U8-lKUi1, (U8-l)' = -(A@ + W0@B)U8-',and (To-,)' = -KUi1BU8-'T@-l, and it may be verified directly that the matrix function

+

G,(t, S) = H@(t,s I W,)U$'-'(s)T@-'(t, s I Y,)U,@(s), = U$-'(t)T@-'(t,s I Yo)Uo@(s),

+

+

is the solution of the matrix differential equation G,' (A@ W&G, = 0 satisfying G, = E for t = s, so that G,(t, s) = G(t,s I W,) and (3.3a) holds. In turn, Eq. (11:2.7) now reduces to ( 3 . 3 ~ )with S(t, s 1 Y o ) given by (3.4). If we set

U ( t )= Uo(t)N(t),

+

V ( t )= Vo(t)N(t) W t ) ,

then substitution in the system (1.3M) yields the result that ( U ( t ) ;V ( t ) ) is a solution of that system if and only if the matrix functions N ( t ) and R(t) satisfy the related system

R',c)

+

Uo(t)"(t) = B ( t ) R ( t ) , V , ( t ) W ( t )= --A@(t)R(t).

3. PROPERTIES OF SOLUTIONS

101

+

Since { Y oI 0 I Y o } ( t = ) -K, and Uo'(t)= A(t)Uo(t) B(t)Vo(t), these equations imply that [ K N ( t ) Uo@(t)R(t)]' = 0. Consequently, there exists a constant matrix K , such that K N ( t ) Uo@(t)R(t) = K,, so that R ( t ) = U?-l(t)[K, - K N ( t ) ]

+

and Eqs. (3.5) hold for t we have that

E

+

I o . Moreover, since N' (t ) = U;'(t)B(t)R(t),

+

(3.7) " ( t ) = - U~'(t)B(t)U?-l(t)KN(t) U;'(t)B(t)U?-l(t)K,. The substitution N ( t ) = T(t,s I Y o ) P ( t )then yields the relation P'(t) = T - l ( t , s I YO)U;;'(t)B(t)U~-'(t)K,, so that P ( t ) is of the form P ( t ) = K O S(t,s 1 Y o ) K , where K O= N(s).Consequently, we have established that if ( U ( t ) ; V(t)) is a solution of (1.3M) on I . then Eqs. (3.5) must hold, where K , is a constant matrix and N ( t ) is of the form (3.6). Conversely, if K , and K Oare constant matrices and U ( t ) , V ( t )are defined by (3.5), it follows by direct substitution that Y ( t )= ( U ( t ) ;V ( t ) )is a solution of (1.3M) on 1,. Moreover, a direct computation verifies that when Y ( t )= ( U ( t ) ;V ( t ) )and Y o ( t )= ( U o ( t ) ;V o ( t ) )are related as in (3.5) then { Y 1 0 I Y o } ( t = ) -Kl.

+

Corollary 1

Suppose that Wo(t),Y o ( t )= ( U o ( t ) ;Vo(t))and K = - { Y o I 0I Yo} are as in Theorem 3.1 on a subinterval I , of I . If Y ( t )= ( U ( t ); V ( t ) )is any solution of (1.3M), then for (s, t ) E I , x I. one has the relation

(3.8)

{Y 1

0I yo} + K U ; l ( t ) U ( t ) = T@-'(t,s I YO)[{YI 0 I

Yo}

+ KUi'(s)U(s)l,

where T ( t ,s I Y o )is the solution of the diflerential system (3.2). Since K = -KO, it follows readily that (3.2) implies that the matrix function M ( t ) = T@-l(t,s 1 Yo)is the solution of the differential system

M ' ( t ) = - K U ~ ' ( t ) B ( t ) U ~ - ' ( t ) M ( t ) , M ( s ) = E. Now for N ( t ) = U;l(t)U(t)and K , E - { Y I 0 1 Y o } ,it may be verified that the differential equation (3.6) implies that N , ( t ) = K , - K N ( t ) is the solution of the differential system

N,'(t) = -KU~'(t)B(t)U?-'(t)N,(t),

N,(s) = K , - K N ( s ) .

102

111. INVOLUTORY RICCATI EQUATIONS

Therefore, we have that N l ( t ) = M ( t ) N , ( s ) , or

K , - K N ( t ) = T@-'(t,s I Yo)[Kl- K N ( s ) ] , and this relation is equivalent to (3.8). For the results of the following corollary, it is to be recalled that in accordance with the definitions of Section 11.6 the system (1.31)is normal on a subinterval I , of I whenever this system possesses on I , no nonidentically vanishing solution of the form ( u ( t ) = 0 ; v ( t ) ) , and that (1.31) is identically normal on I if this system is normal on every nondegenerate subinterval of I. Corollary 2

(a) Suppose that Y o ( t )= ( U o ( t ) ;V o ( t ) ) ,K = - { Yd I 0I Y o } ( t ) ,and S ( t , s I Y o )are as in Theorem 3.1 on a subinterval I , of I. If s E I,, and (l.31) is normal on every subintercal of I , that has s as an endpoint, then for t E I , and t # s the matrix function S ( t , s I Y o )is singular i f and only t is conjugate to s, relative to (l.31). (b) If I is an open interval (a,, b,) (-co 5 a, < b, 5 co) on which (1.31) is identically normal, while this system is disconjugate on a subinterval I , = ( c , , b,) of I and Y o ( t )= ( U o ( t ) ;V o ( t ) )is a solution of (1.3M) with U o ( t )nonsingular on I , , then for s E I , the matrix S ( t , s I Y o )is nonsingular for t E I,, t # s. Moreover, if there exists a value s E I , such that S-l(t, s I Y o )+ 0 as t + b,, then S-l(t, b I Y o ) 0 as t b, for arbitrary b E I,. --f

--f

I n order to establish conclusion (a), for s E I , let Y ( t ) = ( U ( t ) ; V ( t ) ) be the solution of (1.3M) with U(s)= 0, V(s)= E . I n view of the condition that (1.31) is normal on every subinterval of I , that has s as an endpoint, it follows that if t E I,, and t # s, then t is conjugate to s if and only if U ( t ) is singular. For this Y ( t ) it then follows that the N ( t ) in (3.5) satisfies the initial condition N(s) = 0, so that in the corresponding equation (3.6) we have K O= 0. Moreover, K , = - { Y I 0 I Y o } is the nonsingular matrix - V@(s)U,(s),and conclusion (a) is an immediate consequence of the representation

U ( t ) = U,(t)T(t, s I Y,)S(t, s I Yo)K*. T h e first part of conclusion (b) is a direct consequence of conclusion (a). For the conclusion of the last sentence of (b), it is to be noted that

3. PROPERTIES OF SOLUTIONS

103

if s and b are values on I, then the fundamental matrix solution of (3.2) satisfies the well-known relation T ( t , s I Y o )= T ( t , b I Y o ) T ( b s, I Yo), and by direct computation it follows that

(3.9)

S ( t , s I Yo) = T(s,b I Y o ) [ S ( t ,b I Yo) - S(s, b I Yo)].

Now for a general nonsingular n x n matrix M , denote the supremum and infimum of I M q I on the unit ball { q I q E a,, I q I 5 l } of (5, by,u(M) and A(M), respectively. From the relation

A M - ' ) I M q I 2 I M-l(Mq) I = I 7

I = I WM-lq) I 2 i ( M ) I M-'q I

- ---

?

it then follows that 1 = A(M),u(M-l). Since the condition that S-l(t, s I y o ) o as t -+ 03 is equivalent to ,u(S-'(t, s I Y o ) ) O as t -+03, this condition holds if and only if A(S(t, s I Yo)) 03 as t 03. From the relation (3.9) we have readily that A(S(t, s I Yo)) 03 as t 03 if and only if A(S(t, b I Yo)) -03 as t +03, thus completing the proof of conclusion (b). I t is to be remarked that if W o ( t )is a a-symmetric solution of (1.2) then the associated solution Y o ( t )= ( U o ( t ) ;V,(t)) of (1.3Al) occurring in Theorem 3.1 is a a-conjugate base for (1.31) and the matrix K = - { Y o I 0 I Y o }is zero. In this case T ( t , s I Y o )= E , the matrix function S ( t , s I Y o )becomes --f

--f

and S ( t , s I Y o )is 0-symmetric for (1, s) E I , x I,. I n view of the general result of Lemma I1 :2.1, we have the following theorem. Theorem 3.2

Suppose that W = W o ( t ) is a a-symmetric solution of an involutory Riccati matrix diflerential equation (1.2) on a subinterval I , of I , and that Y o ( t )= (U,(t); V o ( t ) ) is the corresponding a-conjugate basis for (1.3l) determined by the initial conditions Y,(s) = ( E ; Wo(s)),where s E I,. Then U o ( t )is nonsingular on I , and a matrix function W = W ( t ) ,t E I , ,is a solution of (1.2) on this interval if and only if the corresponding matrix Tl = U,@(s)[W(s) - W,(s)]V,(s)is such that E S ( t , s I Yo)Tlis nonsingular for t E I,, and

+

(3.11)

+ U$'-l(t)T1[E + S ( t , s I Y o ) T l ] - l U z l ( t )

W ( t )= W o ( t )

for

t E I,.

104

111. INVOLUTORY RlCCATl EQUATIONS

Corresponding to the result of Corollary 1 to Lemma 11:2.1, we have that E S( t , s I Yo)rlis nonsingular if and only if E+T,S(t, s I Y o ) is nonsingular, and that an alternate expression for W ( t )is

+

(3.11')

+

W ( t )= Wo(t) U$-'(t)[E

+ r , S ( t , s I YO)]-'TlU~'(t) for

t E Io.

4. Transformations for (1.2) and (1.3,)

We shall continue to suppose that in addition to the conditions of hypothesis (6) the algebraic properties (1.1) hold on a given interval I on the real line, and for the a-involutory system herein considered proceed to develop some results on transformations in the context of Section 11.4. If T ( t ) is an n x n matrix function which is nonsingular and locally a.c. on I , then under the transformation

(4.1)

u0(t)= T-'(t)u(t),

v o ( t )=

T@(t)v(t)

and (1.S2) is equivalent to the system

(1.3;)

L,O[uO, oO](t)= -vO'(t)

+ CO(t)uO(t)- AO@(t)vO(t)= 0,

L,O[uO, W O ] ( t ) = u"'(t) - AO(t)uO(t)- BO(t)vO(t)= 0,

where the matrix functions Ao(t),Bo(t), C o ( t )are defined as

(4.3 )

A0

=

T-'[AT

-

T'],

BO

2

T-'BT@-l,

CO = T@CT.

If y J t ) = ( u a ( t ) ;vJt)) (a = 1, 2) are solutions of (1.3,), and y:(t) = (u,"(t); vaO(t)) are the associated solutions of (1.3;) given by the

a

corresponding equations (4.1), then it follows readily that { y , I I yz}(t) E {y10I @ I yzO}(t);in particular, y l ( t ) and y z ( t )are a-conjugate solutions of (1.32)if and only if the correspondingy,O(t) andyzo(t)are @-conjugate solutions of (1.3/-'). Corresponding to (1.3;) we have the matrix differential system

L,O[UO; VO](t)= -VO'(t)

+ CO(t)UO(t)- AO@(t)VO(t)= 0,

(1-3M0) L20[UO;VO](t)I UO'(t) - AO(t)UO(t)- BO(t)VO(t)= 0.

105

4. TRANSFORMATIONS FOR (1.2) AND (1.3i)

Now Y ( t )= ( U ( t ) ;V ( t ) )is a a-conjugate basis for (1.3l) if and only if is the corresponding Yo(t)= ( U o ( t ) ;V o( t ) )= ( T - ' ( t ) U ( t ) ;T@(t)V(t)) a a-conjugate basis for (1.3;). Moreover, if W ( t )is a solution of (1.2) and Y ( t )= ( U ( t ) ;V ( t ) ) is an associated solution of (1.3M) such that W ( t )= V ( t ) U - l ( t ) ,then for

Y"t)

=

(UO(t); VO(t)) = ( T - ' ( t ) U ( t ) ;T @ ( t ) V ( t ) )

the associated solution of (1.3,O) we have that Wo(t)= Vo(t)Uo-l(t) = T @ ( t ) W ( t ) T (ist )a solution of the involutory Riccati matrix differential equation

(1.20)

+

+

@O[WO](t)= WO'(t) WO(t)AO(t) AO@(t)W(t) WO(t)BO(t)WO(t)- CO(t) = 0.

+

Also, W ( t )is a a-symmetric solution of (1.2) if and only if the corresponding Wo(t)= T @ ( t ) W ( t ) T (ist ) a a-symmetric solution of (1.2O). Corresponding to the matrix functions G(t,s I Wo), H ( t , s I Wo), F ( t , s I W,) occurring in (1.5), for a solution Wo= Woo(t)of (1.2O) we now have matrix functions Go = Go(t,s I Woo),H o = Ho(t,s I Woo), FO = FO(t, s I WOO)defined by the linear differential systems

(4.4)

GO' HO'

+ (Ao@+ WooBo)Go= 0, + H"(AO + BOWoO)= 0,

G'(S)= E,

H"(s)= E,

to which the systems (I1 :2.6O), (I1 :2.7O)reduce, together with the equation

(4.5)

FO(t, s I WOO)=

s:

H"(Y,s I W o o ) B o o ( ~ ) Gs oI Woo) ( ~ , dr.

With Wo(t)the solution of (1.3M) such that Woo(t) = T@(t)Wo(t)T(t), Eqs. (I1 :4.8) become (a) Ho(t,s I WOO)= T-'(s)H(t, s I Wo)T(t),

(4.6)

(b) (c)

Go(t,s I Woo)= T@(t)G(t, s I W0)T@-'(s), FO(t, s I WOO)= T-'(s)F(t, s I Wo)T@-'(s).

+

If Z ( t ) is a fundamental matrix solution of Z'(t) A@(t)Z(t) = 0, then T ( t )= Z@-l(t)is a fundamental matrix solution of T'(t) - A ( t ) T ( t ) = 0, and with this choice of T ( t ) the matrices of (4.3) are given by (4.7) AO(t) = 0, BO(t) = Z @ ( t ) B ( t ) Z ( t ) ,Cyt) = Z-'(t)C(t)Z@-'(t). For brevity, such a T ( t ) will be referred to as a reducing transformation for (1.31), and the resulting system (1.3,O) as a reduced system.

106

111. INVOLUTORY RlCCATl EQUATIONS

If (1.31) has order of abnormality equal to d on a subinterval I, of I, let the fundamental matrix solution Z ( t ) of 2' A @ ( t ) Z= 0 be chosen such that the last d column vectors of Z ( t ) form a basis for A(I,), the linear vector space of n-dimensional vector functions v ( t ) which are solutions of the vector differential equation v ' ( t ) A @ ( t ) v ( t= ) 0, and satisfy the condition B ( t ) v ( t )= 0 on I,. Then T ( t )= Z@-l(t)is such that the resulting matrix function Bo(t)of (4.7) is of the form B o ( t )= diag{B(t); 0}, where B ( t ) is an (n - d ) x ( n - d ) a-symmetric matrix function. For brevity, such a choice of T ( t ) will be referred to as a preferred reducing transformation for (1 .31). In particular, if Co(t)= T@(t)C(t)T(t) is written as

+

+

= eE(t)is ( n - d ) x ( n - d ) , c,,(t) = cg(t)is (n - d ) where e,,(t) x d and czz(t) = &'$(t) is d x d, then in terms of the vector functions q ( t ) = (u,(t)), C(t) = (v,(t)) (a= 1 , . * 12 - d ) and e ( t ) = (un-d+&)), a ( t ) = ( ~ ~ - ~ + (B ~ (=t 1, ) ). . . , d ) , the vector differential system (1.31) becomes q'(t) = B(t)C(t),

-

9

(4.9)

Moreover, t , and t , are conjugate points with respect to (1.31) if and only if these points are conjugate with respect to the truncated preferred reduced system

(4.10) Corresponding to (4.10) one has the truncated preferred reduced matrix system Z'(t) Qll(t)H(t)= 0, t E I, H ' ( t ) - B ( t ) Z ( t )= 0,

+

and the truncated preferred reduced Riccati matrix differential equation (4.11)

9'+ 9 B ( t ) 9 - C,,(t) = 0,

t E I.

Indeed, for Ao(t)= 0, Bo(t)= diag{B(t); 0}, and Co(t)given by (4.8),

101

5. OBVERSE SYSTEMS

upon expressing Wo(t) as the corresponding partitioned matrix

the Riccati matrix differential equation (1.2,) may be written as the system

(4.12)

W:j

+ W:$(t)W& - eola(t) = 0,

t

E

I , a, /l = 1, 2.

Clearly the interval of existence of W o ( t )is that determined by the equation in Wfl(t)given by a = l , /l = l in (4.12), which is the truncated preferred reduced Riccati matrix differential equation (4.11 ). With the value of W:l(t)thus determined, the matrix functions W:z(t)and W&(t) are solutions of related linear matrix differential equations, and W&(t) is obtained by integration. Now since d is the order of abnormality of (1 .31) on I, , for 0 5 d < n, it follows that if is a nonnull (n - d)-dimensional vector then s(t)lis not the null vector throughout I,. In the special important case wherein ( 1.31)has the same order of abnormality on all nondegenerate subintervals of I, the corresponding truncated preferred reduced system (4.10) is identically normal on I. 5.

0 bverse lnvolutory Differential Systems

The discussion of the preceding sections clearly gives preferential treatment to one of the component vector functions u ( t ) , v ( t ) , and this is to be expected in view of the individual roles assumed by these vector functions in such important applications as the canonical accessory differential equations for a variational problem, (see, e.g., Bliss [l ; Chaps. 111, IV, VIII]). From the formal point of view, however, in a a - i n volutory system (1.31) one may interchange the roles of u ( t ) and ~ ( t ) , leading to the differential system

(5.1) in

25'(t)+ %(t)Jyt) = 0,

y ( t ) = (u'(t); 5 ( t ) ) ,

t E I,

where

with (5.2)

4 2 ) = --A@(t),

B ( t )= C ( t ) ,

q t ) = B(t).

108

111. INVOLUTORY RlCCATl EQUATIONS

For each of the individual interpretations of the involution 0 in (l.l), it is evident that (5.1) is @-involutory whenever (1.3J is @-involutory. For brevity, (5.1) will be referred to as the system obverse to (1.31). Clearly (u'(t); 6 ( t ) )is a solution of (5.1) if and only if ( u ( t ) ; v ( t ) ) = ( 6 ( t ) ;u'(t)) is a solution of (1 .31). The Riccati matrix differential equation related to (5.1) in the manner that (1.2)is related to (1.31)is given by

(5.3 1

&[w-j(t)=

W ( t )+ W(t)A(t)+ A@(t)W(t)

+ W ( t ) B ( t ) W ( t-) q t ) = 0.

If W ( t ) is a nonsingular solution of (1.2)on a subinterval [a, b] of I, then W(t) = W - l ( t )is a nonsingular solution of (5.3) on this subinterval. Whenevery,(t) = ( u , ( t ) ; v,(t)) (a = 1, 2), are solutions of (1.31), and $ ( t ) = (u',(t); 6 , J t ) )= ( v a ( t )u,(t)) ; are the corresponding solutions of

In particular, Y ( t )= ( U ( t ) ;V ( t ) )is a @-conjugate basis for (1.31) if and only if the corresponding matrix function p ( t )= ( o ( t )r(t)) ; = ( V ( t ) ;U ( t ) ) is a a-conjugate basis for (5.1). 6.

Notes and Remarks

The results of this chapter for involutory systems as defined in Section

1 follow a well-known pattern for systems which are real symmetric or

complex hermitian. For the relation of such real symmetric systems to variational theory, the reader is referred to Bliss [l], notably Secs. 11, 23,36,81.I n particular, for the real symmetric case the result of Theorem 1.1 and the basic solution formulas (3.5) of Theorem 3.1 in the special instance of Yo(t)a conjugate basis are present, either specifically or in essence, in the papers of von Escherich [l], Radon [l, 21, Sternberg [l], Hartman [l ; 2, Chap. XI, Sec. 101. For complex hermitian systems the material of this chapter is to be found in the listed papers [l-3, 141 of the author; also, the beginning sections of Chap. VII of Reid [15], together with some of the problems listed for these sections, cover a major portion of the content of this chapter for the complex hermitian case. Results for generalized differential systems are given in Reid [17].

HERMITIAN RlCCATl MATRIX DIFFERENTIAL EQUATIONS

1. Introduction

The present chapter will be devoted to the discussion of hermitian Riccati matrix differential equations and the associated linear system ; i.e., equations and systems which are involutory in the sense of Chapter I11 with @ = X. Specifically, the coefficient matrix functions of Eq. (112.1) satisfy the conditions of hypothesis (6)on a given interval I on the real line, and

(1.1) B ( t ) = B*(t),

C ( t )= C*(t),

D ( t ) = A"(t)

for t E I.

Consequently (112.1) is of the form

+

(1.2) 8[w]( t )= W'(t ) W(t)A( t ) + A "( t )W(t ) + W ( t ) B ( tW ) ( t ) - C(t)=O where A(t),B ( t ) , and C ( t ) are n x n matrix functions, and if W = W ( t ) is a solution of (1.2) on a subinterval I, of I then W = W"(t)is also a solution of (1.2) on this subinterval. Moreover, W ( t ) is an hermitian solution of (1.2) on I, if and only if W ( t )exists on this subinterval and there is a value s E I, such that W(s)= W*(s). The corresponding linear vector differential system (I1 :2.31) or (111:1.3J is then

(1.31)

LJu, v](t) = -v'(t)

+ C(t)u(t)- A"(t)v(t)= 0,

L,[u, o ] ( t ) = u ' ( t ) - A ( t ) u ( t )- B ( t ) v ( t )= 0,

and corresponding linear matrix differential system

(la3hf)

L J U , V ] ( t )= -v,(t) LJU, V ] ( t )3

+ C ( t ) U ( t )- A*(t)V(t)= 0,

U ( t )- A ( t ) U ( t )- B(t)V(t)= 0. 109

IV. HERMITIAN EQUATIONS

110

2.

Preliminary Properties of Solutions

Since all the general results of Chapter I11 are valid for the differential equations and systems (1.2), (1.31) and (1.3M)with the interpretation 0= #, these results will be so used without further comment. For simplicity of statement in dealing with #-conjugate, or conjoined, solutions y J t ) = ( u a ( t ) ; v a ( t ) ) (a = 1, 2) of (1.31) we shall write {yl I y z } ( t ) instead of the more precise {yl 1 X 1 y z } ( t ) In . particular, a solution Y ( t ) = ( U ( t ) ;V ( t ) )of (1.3M) is a conjoined basis for (1.31) if { Y I Y } ( t ) = V * ( t ) U ( t )- U*(t)V(t)= 0 on I. Corresponding to Theorem III:1.2, we have that if W = W ( t )is a solution of the hermitian Riccati matrix differential equation (1.2) on a nondegenerate subinterval I, of I , and Y ( t )= ( U ( t ) ;V ( t ) ) is a corresponding solution of (1.3M) with U ( t ) nonsingular and W ( t )= V(t)U-'(t)on I,, then Y ( t )is a conjoined basis for (1.31) if and only if W ( t ) is an hermitian solution of (1.2) on I,. Now (1.2) may be written also as

(2.1)

W ( t )= (E;W(t)).%(t)(E;W ( t ) ) CW"(4 - W(t)l[A(t) B(t)W(t)l,

+

+

where % ( t ) is the hermitian matrix

In particular, if W ( t )is an hermitian solution of this equation on a subinterval I, of I, then W ' ( t )= (E; W(t))*%(t)(E;W ( t ) )and

(2.31

+

W ( t )= W(s) for

It R

(s, t ) E I ,

(E;W ( r ) ) * % ( r ) ( E W ; ( Y ) dr ) x I,.

Also, if W ( t )is an hermitian solution of (1.2) onI, and Y ( t )= ( U ( t ); V ( t ) ) is a conjoined basis for (1.3) with U ( t )nonsingular and W ( t ) = V ( t ) U - * ( t ) , then U * ( t ) W ' ( t ) U ( t= ) Y*(t)%(t)Y(t) for t E I,. Moreover, since (1.3M) may be written

$Y'+

(2.4)

%Y = 0,

and % ( t )= % * ( t ) , it follows that

Y*%Y

+ Y*'%Y = Y * % W % Y ]+ [-Y"%Z]%Y= 0.

2. PRELIMINARY PROPERTIES OF SOLUTIONS

111

Therefore, if %(t)is locally a.c. on I,, then U*(t)W’(t)U(t)is also locally a.c. on this interval and

[U”(t )w’(t )U(t ) ]’ = Y”(t ) % ( t )Y (t ) for t a.e. on I , , and hence

+

(2.5) W ( t )= U*-l(t)[ U*(s)W’(s)U(s)

s:

1

Y”(r)%’(r)Y(r)dr U-l(t)

for (s, t ) E I , x I,. I n particular, (2.3) and (2.5) imply the following results. Theorem 2.1

If W ( t )is an hermitian solution of an hermitian Riccati matrix dzTerentia1 equation (1.2) on a subinterval I , of I , and Y ( t )= ( U ( t ) ;V ( t ) )is a conjoined basis for (1.3J with U ( t ) nonsingular and W ( t )= V ( t ) U - l ( t )on I,, then: (a) i f % ( t ) 2 0 { % ( t ) 5 O}for t a.e. on I , , then W ( t )is a nondecreasing {nonincreasing} hermitian matrix function on I , ; (b) q q I ( t ) is locally a.c. on I , with ‘tr’(t) 2 0 {%’(t) 5 0 ) for t a.e. on I,, then U * ( t )w’(t )U ( t ) is a nondecreasing {nonincreasing}, hermitian matrix function on I,; in particular, if s E I , and

+

W‘(s)= -W(s)A(s) - A”(s)W(s)- W(s)B(s)W(s) C(s) satisjies W’(s)2 0 { w ‘ ( s ) 5 0 } , then W ( t ) is a nondecreasing {nonincreasing} hermitian matrix function on I,+(s) = {t I t E I , , t 2 s}. For the special important case in which A ( t ) 3 0 and B ( t ) is positive definite on I the system (1.31) is equivalent to the second-order linear matrix differential equation [R(t)u’(t)]’- C ( t ) u ( t )= 0, where R ( t ) = B-’(t). I n this instance, % ( t )= diag{C(t); - R - l ( t ) } and % ( t ) 5 0 if and only if C ( t ) 5 0 ; moreover, if B ( t ) and C ( t )are locally a s . then %’(t)>_ 0 {%’(t)5 0} if and only if C ’ ( t )2 0 and R’(t) 2 0 { C ’ ( t )5 0 and R ’ ( t ) 5 O}. For an hermitian system (1.31) the transformation (111:4.1) is

(2.6)

u y t ) = T-’(t)u(t),

and the system (111:1.3:)

(2.7)

v“t) =

T * (t)v(t),

now becomes

+ C O ( t ) U O ( t ) - AO*(t)wO(t) = 0,

L,O[UO,

vO](t)= - v ” ( t )

LzO[UO,

vO](t)3 u”(t) - AO(t)uO(t) - BO(t)WO(t)= 0,

112

IV. HERMITIAN EQUATIONS

where

( 2 . 8 ) A' = T-'[AT

-

T'],

Bo = T-'BT*-',

Co = T"CT.

+

If Z ( t ) is a fundamental matrix solution of Z ' ( t ) A * ( t ) Z ( t ) = 0, then T ( t ) = Z * - l ( t ) is a fundamental matrix solution of T ' ( t )- A ( t ) T ( t )= 0, the matrix coefficients (2.8) become

(2.8')

A0

= 0,

B" = Z*BZ,

CO = z-'CZ"-'

and, in particular, we have the following result. Lemma 2.1

If the hermitian system (1.31) is such that B ( t ) 2 0 for t a.e. on I and [a, b] is a compact subinterval of I , then (1.31) is normal on [a, b] if and only if (2.9 1

Jb a

B o ( t )dt

=

j bZ * ( t ) B ( t ) Z ( t )dt > 0. a

If (1.31) has order of abnormality d on a subinterval I , of I , and T ( t ) is chosen as a preferred reducing transformation for (1.31) on I , as defined in Section 111:4, then T ( t ) = Z*-l(t>, where Z ( t ) is a fundamental matrix solution of Z ' ( t ) A # ( t ) Z ( t )= 0 whose last d column vectors form a basis for A(I,). Moreover, if as in (111:4.8) we write Co(t)= T " ( t ) C ( t ) T ( t )in the form

+

eI1(t)

= e$(t) is ( n - d ) x ( n - d ) , e,,(t)= eZ(t)is ( n - d) where x d , and eZ2(t) = &'g(t) is d x d , then the truncated preferred reduced system (111:4.10) is

(2.10)

-C'(t)

+ ell(t)V(t)

= 0,

$ ( t ) - B(t>C(t)= 0.

T h e associated truncated preferred reduced matrix system (111 :4.10,w) is

(2.1OM)

-Z'(t)

H ' ( t ) - B ( t ) Z ( t ) = 0,

and

(2.11)

+ C,,(t)H(t) = 0,

Q'(t)

+ Q ( t ) B ( t ) Q ( t) C,,(t) = 0

113

Z.-PRELIMINARY PROPERTIES OF SOLUTIONS

is the truncated preferred reduced Riccati matrix differential equation. Consequently, if B ( t ) 2 0 for t a.e. on I , , then B ( t ) 2 0 for t a.e. on I , ; moreover, if I , is a nondegenerate compact interval [a, b] and 0 5 d < n, then J: B(t)dt > 0. The results of Section I11 :5 imply corresponding results for the system -G'(t)

(2.12)

G'(t)

+ B(t)u'(t)+ A(t)G(t)

= 0,

+ A*(t)u'(t)- C(t)G(t)= 0

obverse to the hermitian system (1.S2). In particular, if C ( t ) 2 0 for t a.e. on I , and [a, b] is a compact subinterval of I , then (2.12) is normal on [a, b] if and only if

It is to be noted that relations (2.1) and (2.3) are intimately related to the "trigonometric transform'' of a linear Hamiltonian system (1.32) as developed by Barrett [ l ] and Reid [4]. Basic to this trigonometric transform (see Reid [13; 15, Problems II.3:6 and VII.2:6]), is a related nonlinear differential system (a) Ale[@, Y ] ( t )= - Y ' ( t )- Q(t, @, Y)@(t) = 0, (2.14)

(b) A,'[@, Y](t) (c)

(10

@ ' ( t )- Q(t, @, Y)Y(t)= 0,

[@, Y, R ] ( t )= R'(t) - M ( t , @, Y ) R ( t ) = 0,

where (2.15) (2.16)

Q(t, @, Y)

=

+

Y(t)B(t)Y*(t) Y(t)A(t)@*(t)

+ @(t)A*(t)Y*(t) - @(t)C(t)@*(t), M ( t , @, Y) = @(t)A(t)@*(t) + Y(t)C(t)@*(t) + @(t)B(t)Y*(t)- Y(t)A*(t)Y*(t).

If t E I , and Y ( t )= ( U ( t ) ;V ( t ) )is a conjoined basis for (1.31) satisfying the initial condition Y(t)= ( U o ;V,), then (2.17)

u,*u, + VO*VO> 0,

Moreover, if

Q0,

Vo*U0- U,*V, = 0.

Yo, R, are matrices satisfying

+ Vo*Vo,

(2.18) R,*RO = Uo*Uo

Uo = @,*I?,,

V , = !J',*R,-,,

114

IV. HERMITIAN EQUATIONS

then

and the solution (@; !P; R) of system (2.14) satisfying the initial conditions (2.20)

@(t)

= @o,

Y ( t )= !Po,

R ( t ) = R,,

is such that (2.21)

U ( t )= @*(t)R(t), V ( t )= !P*(t)R(t) for t E I .

Conversely, if (@; !P; R) is a solution of (2.14) with initial values (@,; !Po;R o ) at t = t, where R, is nonsingular and (@,, !Po)satisfies conditions (2.19), then Y ( t )= ( U ( t ) ;V ( t ) )defined by (2.21) is a conjoined basis for (1.31) with (2.22)

+ V*(t)V(t)

R*(t)R(t)= U*(t)U(t)

for t E I .

Indeed, the solution of (2.14a, b) satisfying the initial condition (@(t) ; !P(t))= ( G o ;!Po) has maximal interval of existence equal to I , and throughout this interval one has the identities (2.23)

+

@(t)@*(t) !P(t)Y*(t)= E,

!P(t)@*(t)- @(t)!P*(t)= 0,

@*(t)@(t) !P*(t)Y(t) E ,

@*(t)!P(t)

+

-

!P*(t)@(t) 0.

Clearly Y ( t )= ( U ( t ) ;V ( t ) )is a conjoined basis for (1.31) with U ( t ) nonsingular on a subinterval I , of I if and only if for the corresponding solution ( @ ( t ) !P(t); ; R ( t ) ) of (2.14) satisfying (2.21) the matrix function @(t)is nonsingular on I,. Moreover, if W ( t )= V(t)U-l(t) is the associated solution of the Riccati matrix differential equation (1.2), then also W ( t )= !P*(t)@*-l(t)for t E I,. Utilizing the differential equations (2.14a, b), and certain of the relations (2.23), it may be verified that (2.24)

W'(t)= - @-1(t)Q(t,@, !P)@*-l(t);

consequently, (2.25)

W ( t )= W ( S )-

s:

@-'(Y)Q(Y,

for (s, t ) E I , x I , .

@, !P)@*-'(Y)dr

3. DISTINGUISHED SOLUTIONS OF (1.2)

115

I t is to be noted that relation (2.25) is equivalent to Eq. (2.3). Indeed, from (2.14) and (2.21) it follows immediately that

Q ( t , @, Y ) = - R*-1Y*(t)%(t)Y(t)R-l(t), where % ( t ) is defined by (2.2), and as W ( t )= V(t)U-l(t)is hermitian, the relations (2.24) and (2.25) are equivalent to (2.1) and (2.3), respectively. 3.

Distinguished Solutions of (1.2)

If I = ( a , , b,) and an hermitian system (1.31) is disconjugate on a subinterval [s, b), then whenever W ( t )is a distinguished solution of (1.2) at b, the corresponding normalized solution W Y ( t )defined by condition [IT :6.11) is also a distinguished solution of (1.2) at b, which satisfies the Wnditions t11: 6.12), (11:6.13). Moreover, in the case of an hermitian Riccati matrix hifferential equation (1.2), W = Wy"(t)is also a distinguished solution of (1.2) at b, satisfying the conditions d*(s)W,,"(s) = 0, W,,*(s)d(s) = 0, and the Corollary to Theorem II:6.1 implies that W , ( t ) I W,,*(t).That is, if for an hermitian Riccati matrix differential equation (1.2) we have that W = W Y ( tis ) a distinguished solution at b, which is normalized at s in the sense of Theorem 11:6.1, then W,,(t) is hermitian for t E [s, b,), and a principal solution Y ( t )= ( U ( t ) ;V ( t ) ) of (1.31) at b, which satisfies the relation W J t ) = V ( t ) U - l ( t )is a conjoined basis for (1.3J. If an hermitian system (1.31)is disconjugate on ( a , b,), while s E (a, b,), d[s, b,) = d, d(s) A [ s , b,), and Q(s) is an n x (n - d ) matrix such that

-

(3.1)

d*(s)Q(s) = 0,

Q*(s)Q(s) = E n - d ,

then by Theorem II:7.2, a necessary and sufficient condition for there to exist a solution W ( t )of (1.2) on [s, b,) which is a distinguished solution at b, is that the solution Ysr(t)= (U,,(t); V,,(t))determined by the conditions (11:7.9) is such that Vsr(s)converges to a limit matrix P(s) as Y -+03, and for the solutions Ysb,(t)= (u,b,(t);Vsb,(t)), Ysz(t) = ( U s z ( t ;) V s z ( t ) )Y, ( t )= ( U ( t ) ;V ( t ) )of (1.3M)determined respectively by (11:7.2), (11:7.10) with A,($) = d(s), and (11:7.5), the matrix U ( t ) is nonsingular on [s, b,). Moreover, in case U ( t ) is nonsingular on [s, b,) we have that W,,(t)= V(t)U-l(t)is the unique distinguished solution at b, of (1.2) satisfying (11:6.12). The fact that W,,(t)is hermitian on

116

IV. HERMITIAN EQUATIONS

[s, b,) follows from the above stated result. Also, it may be derived directly from the fact that the condition U,,(s) = 0 implies that { Y,, I Y,,}(t) = 0, so that U$(s)V,,(s) = Q"(s)V,,(s) is hermitian, and

u~,~,ly =bQ"(s)P(s) ,(s) = hm u W,(a) { W ( a ) 2 W o ( a ) }then , the interwal of existence of W ( t ) includes [a, b] and W ( t )> Wo(t){ W ( t )2 W o ( t ) } for t E [a, b]. Similarly, if W = W ( t ) is a solution of (1.2) with W(b) < W,(b) {W(b)5 W,(b)},then the interval of existence of W ( t )includes [a, b] and W ( t ) < W,(b) { W ( t )5 W o ( t ) for } t E [a, b ] .

If W ( a ) 3 W o ( a ) ,then T, = U,+(a)[W(a)- W,(a)]U,(a) > 0, and in view of the above remarks the result is a ready consequence of equation (111: 3.11) for s = a and I , = [a, b] since r;l S(t, a [ Y o )2 G1> 0 for t E [a, b ] , and

+

[T;l

+ S (t, a I Y0)]-l= rl[E + S(t, a I YO)Tl]-l.

Now if we have merely W ( a )- Wo(a)2 0, for E 2 0 let W ( t ;E ) be the solution of (1.2) satisfying the initial condition W(a; E ) = W( a ) EE. By the result just established, the interval of existence of W(t ; E ) includes [a, b ] , and W ( t ;E ) > Wo(t)for t E [a, b] and E > 0. Indeed, if 0 < c1 < E ~ then , application of this result with Wo(t)replaced by W(t ; el), and W ( t ;E ) replaced by W ( t ;E ~ ) yields , the conclusion that Wo(t)< W ( t ;E ~ < W ( t ;E ~ )for t E [a, b] and 0 < el < E ~ .From this boundedness condition it then follows that for E , > 0 the family of solutions W ( t ;E ) , 0 < E 5 E ~ t , E [a, b ] , of (1.2) is uniformly bounded and equicontinuous. Consequently, by the Ascoli theorem there is a monotone decreasing sequence { E , } converging to zero, and such that { W ( t ;E , ) } converges uniformly on [a, b] to a limit matrix function W l ( t ) ,which is therefore such that W l ( t )2 Wo(t)for t E [a, b]. Moreover, by a classical argument W = W l ( t )is a solution of (1.2) on [a, b], and since W l ( a )= W(a) it

+

)

120

IV. HERMITIAN EQUATIONS

follows that W l ( t )E W ( t )on [a, b ] , so that the interval of existence of W ( t )includes [a, b] and W ( t )2 Wo(t)on this interval. The final conclusion may be proved by a similar argument. Theorem 4.2

Suppose that I is an open interval ( a , , b,) (-m 5 a, < b, I m), on which (1.3J is identically normal, and hypothesis $ , { I } holds, while W = Wo(t)is an hermitian solution of (1.2) on a subinterval I , = (c,, b,) of I ; moreover, if Yo(t)= ( U o ( t ) ;V , ( t ) ) is a conjoined basis for (1.31) with Uo(t)nonsingular and W o ( t )= V,(t)U;l(t) on I , , it is supposed that S-l(t, s I Yo)-+ 0 as t + b,. If W = W ( t ) is an hermitian solution of (1.2) on a neighborhood of t = s, then the interval of existence of W ( t ) includes [s, b,) if and only if r = W(s)- W,(s) is nonnegative deJnite. I t is a direct consequence of the result of Theorem 4.1 that if W= W ( t ) in an hermitian solution of (1.2) with F = W(s)- W,(s) >_ 0, then the interval of existence of W ( t ) includes every compact subinterval [s, I ] of [s, b,), and hence also includes [s, b,). This part of the conclusion of the theorem uses neither the identical normality of (1.3J on I nor the condition that S-l(t, s I Yo)-+ 0 as t -+ b,. Now the existence of an hermitian solution W = Wo(t)of (1.2) on I , , and the identical normality of (1 .31) on I , imply that this system is disconjugate on I , , and hence, in view of Corollary 2 to Theorem III:3.1, for s E I , the hermitian matrix matrix function S(t, s I Yo) is nonsingular for t E I , , t # s. Since B ( t ) >_ 0 for t a.e. on I we also have that S(t, s I Y o )> 0 for t E (s, b,). Moreover, in view of the initial condition S(s, s I Y o )= 0, we have that A(S-l(t, s I Y o ) )-+m as t + s+, where, in general, for an hermitian matrix M the symbol A(M) denotes the smallest proper value of M . If W = W ( t )is an hermitian solution of (1.2) on a neighborhood of t = s and r= W(s)- W,(s), then the hermitian matrix rl = U,*(s)rU,(s) is such that the hermitian matrix function S - l ( t , s I Y o ) rl is positive definite for t > s and sufficiently close to s. Moreover, if rl fails to be nonnegative definite, and S-l(t, s I Yo)+ 0 as t + b,, it then follows that there exists a value Y E (s, b,) such that S-l(r, s I Y o ) rl fails to be nonnegative definite and hence there is a value t , E (s, r ) such that S-l(tl,s I Yo) rl is singular. I t then follows that E S ( t l ,s I Y O ) r l is singular, and from the result of Theorem III:3.2 we have that the interval of existence of W ( t )does not include the compact interval [s, t l ] , and therefore does not include [s, b,).

+

+

+

+

121

4. DEFINITE HERMITIAN EQUATIONS

The result of the following lemma is prefactory to the derivation of some comparison results for Riccati matrix differential equations. Lemma 4.1

Suppose that B ( t ) 2 0 and C ( t ) 2 0 for t a x . on a subinterval [a, b] of I . If QQ> 0 {Pa 2 0}, then the hermitian solution W ( t )of (1.2) determined by the initial condition w ( ~=)QQexists on [a, b] and W ( t )> 0 {w(t)2 o} for t E [a, b ] . Correspondingly, f Qb > 0, {Qb 2 o}, then the hermitian solution W ( t ) of (1.2) determined by the initial condition W ( b ) = -Qb exists on [a, b] and w(t)< 0 {w(t)5 o} for t E [a, b ] . Let Y ( t )= ( U ( t ) ;V ( t ) )be the solution of (1.3M) satisfying the initial condition Y ( a ) = (E; QQ),and consider first the case of QQ> 0. If c is a value on (a, b ] , and n is an n-dimensional vector such that either U(c)n= 0 or V(c)n= 0, let ( ~ ( t v) (;t ) ) = (U(t)n;V(t)n). Then ( ~ ( t v) (;t ) ) is a solution of (1.31), with either U ( C ) = 0 or v ( c ) = 0, while v ( a ) - Qu(a) = 0. For t E (a, b] andJ[q; a, t 1 Qa] the functional

(4.1) J [ q ;a, t I QuI

we then have the relation

Since QQ> 0, and B ( t ) 2 0, C ( t ) 2 0 for t a.e. on [a, c ] , it follows that .(a) = 0, B(t)w(t)= 0 and C ( t ) u ( t )= 0 on [a, c]. In particular 0 = ~ ( a ) = n,thus showing that U ( t )and V ( t )are both nonsingular for t E (a, b ] . Consequently, W ( t )= V(t)U-'(t) is a nonsingular hermitian matrix on [a,b] ; in particular, for t E [a, b] all proper values of W(t),which are necessarily real, are different from zero. As W ( a )= Qa > 0, all proper values of W ( a )are positive, and hence by continuity all proper values of W ( t ) are positive throughout the interval [a, b ] , and W ( t ) is positive definite for t on this interval. If we have merely QQ2 0, for E > 0 let Qm = Qa EE and denote by We(t)the solution of (1.2) satisfying W,(a)= Q m . By the result just established the solution W,(t)exists on [a, b ] and W.(t) > 0 for t E [a?b ] .

+

122

IV. HERMITIAN EQUATIONS

Combining this result with that of Theorem 4.1, we have that if 0 < E , < E , then 0 < Wc,(t)< WBa(t) for t E [a, b ] . Moreover, by an argument similar to that employed in the proof of Theorem 4.1 it follows that if W ( t )is the solution of (1.2) determined by the initial condition W(a)=Q, then W ( t ) exists on [a, b] and W E ( t-+ ) W ( t ) as E -+ 0, so that also W ( t )2 0 for t E [a, b ] . The second conclusion of the lemma is established by a similar argument, employing the functional J o [ q; t,b I Qb]defined for t E [a, b ) as

With the aid of the result of the above lemma, one has a ready proof of the following comparison theorem. Theorem 4.3

Suppose that B ( t ) 2 0 for t a.e. on a subinterval [a, b ] of I , that W = Wo(t)is a hermitian solution of (1.2) on this subinterval, and that C , ( t ) E P.,,[a, b] with C,(t) 2 C ( t )for t a.e. on [a, b ] . If W = W,(t) is a solution of the Riccati matrix diflerential equation

(4.3) n,[W,] = W,'

+ W,A(t)+ A"(t)W, + W,B(t)W,- C,(t) = 0

with W,(a) > Wo(a){W,(a)2 Wo(a)},then W,(t) exists on [a, b] and W l ( t )> Wo(t) {Wl(t)2 W o ( t ) }for t E [a, b ] . Correspondingly, if W = W l ( t )is a solution of (4.3) with W,(b) < Wo(b){ W,(b)I Wo(b)}, then W l ( t ) exists on t E [a, b ] .

If we set W l ( t )= I where 82[J472]

=

Wz't

with A,(t) = A ( t ) t - Co(t)2 0 for t E [a, b ] , and the conclusions of the theorem are ready consequences of the corresponding results of the above lemma. Now if W ( t )is a nonsingular solution of (1.2) on a subinterval [a, b ] of I , then w(t)= W-l(t) is a nonsingular solution on this subinterval of the related Riccati differential equation (4.4)

P(t)- W ( t ) A * ( t )- A ( t ) W ( t )+ W ( t ) C ( t ) W ( t) B ( t )= 0

123

4. DEFINITE HERMITIAN EQUATIONS

for the linear differential system -."(t)

(4.51

$'(t)

+ B(t)u'(t)+ A(t)v'(t)= 0,

+ A"(t)v'(t) - C ( t ) $ ( t )= 0.

Moreover, w ( t ) is positive definite if W ( t ) is positive definite. Consequently, if the conditions of (&) hold for (1.2) and C ( t ) 0 for t a.e. on I,, for (4.4)one has a comparison theorem corresponding to that of Theorem 4.3 for (1.2). I n particular, this result interpreted in terms of the original equation (1.2) yields the following result.

>

Corollary

Suppose that C ( t ) 2 0 for t a.e. on a subinterval [a, b] of I , and that B,(t) E .Cnn[a,b] with B 2 ( t )2 B ( t ) for t a.e. on [a, b]. I f W = W o ( t ) is a positive definite hermitian solution of (1.2) on this subinterval, and W = Wz(t)is a solution of the Riccati matrix dzfferential equation

(4.6) fiz[Wz]

Wz'

+ W z A ( t ) + A"(t)Wz + WzBz(t)Wz

-

c(t)= 0

with 0 < W z ( a )< W,(a) (0 < W z ( a )5 W o ( a ) } ,then W z ( t ) exists on [a, b] and 0 < W z ( t )< W o ( t )(0 < W z ( t )5 W o ( t ) )for } t E [a, b]. Correspondingly, if W = W o ( t )is a negative dejnite hermitian solution of (1.2) on [a, b ] , and W = W z ( t )is a solution of (4.6)with 0 > Wz(b)> Wo(b) (0 > Wz(b)> W,(b)},then W z ( t )exists on [a, b] and 0 > W z ( t )> W o ( t ) , (0 > W z ( t ) W o ( t ) }for t E [a, bl.

>

The following comparison theorem is a consequence of Lemma 4.1 and the combined results of Theorem 4.3 and its Corollary, together with a limit argument similar to that employed in the proof of Lemma 4.1 to treat the particular case in which we have W o ( a )2 0, but do not have W,(a) > 0. Theorem 4.4

Suppose that (1.2) is an hermitian Riccati matrix diyerential equation for which B ( t ) > 0 and C ( t ) 2 0 for t a.e. on I , and that on a compact subinterval [a, b] there exist B 3 ( t )E .C,,[a, b] and C 3 ( t )E .Cnn[a,b] with B 3 ( t )2 B ( t ) and C 3 ( t )p C ( t )for t a.e. on [a, b]. If W,(t) is a solution of (1.2) on [a, b] with W,(a) 2 0, and W 3 ( t )is a solution of

(4.71

83[W3]

+

+

W3'(t) W 3 ( t ) A ( t ) A"(t)W3(t) W3(t)B3(t)W3(t)(t) - Cdt)= 0

+

124

IV. HERMITIAN EQUATIONS

satisfying W3(a)> Wo(a){W3(a)2. W o ( a ) } ,then W3(t)exists on [a, b] and W3(t)> Wo(t){W3(t)2 W o ( t ) for } t E [a, b ] . Correspondingly, i f Wo(t)is a solution of (1.2) on [a, b ] with Wo(b)5 0, and W3(t)is a solution of (4.7) satisfying W3(b)< Wo(b){ W3(b)5 Wo(b)},then W3(t)exists on [a, b] and W3(t)< Wo(t) { W 3 ( t )5 Wo(t))for t E [a, bl. 5.

A Fundamental Property of Solutions of Riccati Matrix Differential Equations

We shall consider further an hermitian Riccati matrix differential equation (1.2) with matrix coefficients satisfying hypotheses (&) and g o { I } . For brevity, let (5.1)

F(t, W ) = C ( t ) - WA(t)- A"(t)W- WB(t)W,

so that W = W ( t ) is a solution of (1.2) on a subinterval I . of I if and only if

(5.2)

W ( t )= F(t, W ( t ) ) ,

t E I,.

Now if Z is an arbitrary n x n matrix, and W is an hermitian n x n matrix, the matrix function

(5.3 1

Go(t,2, W) = Z"B(t)Z - Z"B(t)W - WB(t)Z

is hermitian for t

E

I. Moreover,

Go(t,2, W) = [Z"

-

w ] B ( t ) [ Z - w]- WB(t)W,

and in view of hypothesis $,{I} we have

(5.4)

Go(t,2, W) 2 -WB(t)W

for t E I ,

with Go(t,2, W ) = -WB(t)W if and only if B ( t ) [ Zsequently, the matrix function

(5.5)

w]= 0.

+

G(t,2, W ) = C ( t )- WA(t)- A"(t)W Go(t, 2, W ) = C ( t ) Z"B(t)Z - W[A(t) B ( t ) Z ] - [A"(t) Z"B(t)]W

+

+

+

is also hermitian and such that if W is hermitian then

G(t,2, W) 2 F(t, W ) ,

for t E I ,

with G(t, 2, W ) = F(t, W) if and only if B ( t ) [ Z -

w]= 0.

Con-

5. A FUNDAMENTAL PROPERTY OF SOLUTIONS

125

Now for Z ( t ) a matrix function such that Z ( t ) E i!g[u, b ] for arbitrary compact subintervals [u, b ] of I , the matrix differential equation (5.6)

Q'(t) =

G(4 Z(t),Q ( t > )

is the linear equation (5.6')

Q'(t) = C,(t) - Q(t)A,(t)- A,"(t)Q(t),

where

If s E I and M = M ( t , s I 2 ) is the solution of the linear homogeneous matrix differential system

then N ( t ) = M-'(t) is the solution of the N(t)A,(t) = 0, N ( s ) = E, and Q ( t ) = Q(t if and only if Q ( t ) = N " ( t ) Y ( t ) N ( t ) and = M*(t)C,(t)M(t). Consequently, Q ( t ) is a only if

+

(5.9)

Q ( t ) = MI-'@)[Q(s)

differential system N ' ( t ) 12) is a solution of (5.6) Y' = N*-'(t)C,(t)N-'(t) solution of (4.6) if and

+ s' M*(s)C,(s)M(s)ds]M-'(t); a

in particular, Q ( t ) is an hermitian matrix function on I if Q(s) is hermitian. Now suppose that W = W ( t )is an hermitian solution of (1.2) on a subinterval I , of I , and s E I , . Since for any given Z ( t ) E i!rn[u,b] for arbitrary compact subintervals [a, b] o f 1 we have G(t, Z ( t ) , W )2 F ( t , W) for arbitrary hermitian n x n matrices W, it follows that the solution Q ( t ) = Q ( t 12) of (5.6) is a solution of the Riccati matrix equation f i , [ Q ] ( t= ) 0, where f i l [ Q ] is of the form (4.3) with

C,(t) = C ( t )

+ G(t, z(t),

Q(t))

-

F(t, Q ( t ) ) L c(t)-

In particular, if Q(s) is hermitian it follows from Theorem 4.3 that Q ( t ) > W(2) on I,+($) = {t I t E I,, t 2 s } if Q(s) > W(s), and Q ( t ) 2 W ( t ) on I,+(s) if Q(s) 2 W(s). Also, Q ( t ) < W ( t ) on Io-(s) = {t I t E I , , t 5 s} if Q(s) < W(s), and Q ( t ) I W ( t ) on Io-(s) if Q(s) 5 W(s).In particular, if Q(s) = W(s),then Q ( t ) 2 W ( t )on I,+(s) and Q ( t ) 5 W ( t ) on I,-(s).

126

IV. HERMITIAN EQUATIONS

In the following we shall suppose that the solution Q ( t ) = Q(t 1 2 ) of (5.6) satisfies the initial condition Q(s) = W(s) so that Q(t 1.7) is hermitian for t E I, and Q(t 2) 2 W ( t )on I,+(s) and Q(t 1 2 ) 5 W ( t ) on I,-($). We shall now consider a sequence of matrix functions Z j ( t ) , ( j = 0, 1,. .), defined recursively as Z , ( t ) = Z ( t ) , Zj+l(t)= Q ( t I Z j ) the solution of the system (5.6) with Z ( t ) = Z j ( t ) and satisfying the initial conditions Zj+,(s) = W(s).From the above discussion we have that for j = 1, 2, . . . the matrix function Z j ( t ) is hermitian for t E I , and Z j ( t ) 2 W ( t ) for t E I,+($), Z j ( t ) 5 W ( t ) for t E I,-($). From the equations (5.6) satisfied by SZ = Z j ( t ) and 52 = Z j + l ( t ) it then follows that R j ( t )= Z j ( t )- Z j + l ( t )satisfies the differential system

I

(5.10)

Rj'(t) = [Z.:-l(t) - Z;*(t)]B(t)[zj-l(t) - zj(t)] - R j(t)A j(t)- Aj"(t)Rj(t),

Rj(S) = 0,

( j = 1, 2,

. . .),

where (5.10')

A,(t) = A ( t )

+ B(t)Zj(t).

Now let M = M j ( t ) be the solution of the corresponding linear homogeneous differential system (5.11)

M ' ( t ) - A j ( t ) M ( t )= 0,

M ( s )= E.

As in the above discussion of solutions of (5.6), we have

1

1

(5.12) R j ( t )= Y-l(t)[ M,*(r)Cj(r)Mj(r) dr M;'(t), U

t

E

I,,

where for brevity we set (5.13)

C j ( t )= [ Z j q t ) - Zj*(t)]B(t)[zj-l(t) - Zj(t)].

Since C j ( t )2 0 for t a.e. onI, we have from equation (5.12) that Rj(t)LO for t E Io+(s),and R j ( t )9 0 for t E I,-(s). Combining these inequalities with the preceding ones we then have for j = 1,2,. . . , that Z j ( t ) is an hermitian matrix function on I , such that (5.14)

W ( t )5 Zj+l(t) 5 Z j ( t )

for t

W ( t )2 Z j + l ( t )2 Z j ( t )

for t E I,-(s).

E

Io+(s),

Thus the sequence { Z j ( t ) } of hermitian matrix functions is monotone

127

5. A FUNDAMENTAL PROPERTY OF SOLUTIONS

on Zo, and therefore this sequence converges on I , to a limit matrix function Z,(t). Moreover, since W ( t )5 Z j ( t ) 5 Z , ( t ) for t E I,+(s) and W ( t )2 Z j ( t ) 2 Z , ( t ) for t E Zo-(s), from the specific form of G(t,2, W) we have that there exists a scalar function x ( t ) on I, such that x ( t ) is integrable on arbitrary compact subintervals [a, b] of I , and

we then have that the matrix functions Z j ( t )are equiabsolutely continuous on arbitrary compact subintervals of I,, and hence the sequence { Z j ( t ) } converges uniformly to a limit matrix function Z , ( t ) on arbitrary compact subintervals. As

+

z j ( t ) = W(s)

we then have that

s‘

G ( t , Z1-,(y),zj(y))dr,

1E

I”,

+ J G(~,-&(r),Zm(r))dr

z,(t)= W(s)

1

8

so that Z,(t) is also a solution of the Riccati matrix differential equation (1.2) which has the same value as W ( t )at t = s, and hence Z,(t) = W ( t ) on I,. Returning to the differential equations satisfied by W ( t ) and Z j ( t ) we have that Si(t) = Z j ( t ) - W ( t ) is a solution on I , of the differential system

(5.15)

Sj’ = [Zi*-l(t)- W ( t ) ] B ( t ) [ Z j - l ( t) W(t)] --S,(t)Aj-,(t) - Aj+,(t)Sj(t), Sj(S) = 0,

+

where Aj-l(t)= A ( t ) B(t)Zi-,(t), as defined in (5.10’). In terms of the solution M j - l ( t ) of the associated system (5.11) we then have for t E I , the relation

Now in view of the bounds (5.14) it follows that if [u,b] is a compact

IV. HERMITIAN EQUATIONS

128

subinterval of I , containing t = s then there exists a constant x = %[a,b] such that

~[M;-!~(t)l 5x

~ [ M j - ~ ( t5) ]x ,

t

for

E

[a, b ] , j = 1, 2,

. .. .

From (5.16) it then follows that

v [ S J ( t ) ]5 x4

b a

v2[Sj-l(r)]v[B(r)] dr

for

t

E

[a, b].

In particular, if

v,[Sj;

a, b] = m a { v [ S j ( t ) ]1 t

and

k[a, b] = x 4 [ a , b] we have that

J:

E

[a, b ] } ,

v [ B ( r ) ]dr,

v,[Sj; a, b] 5 k[a, b]v,2[Sj-l; a, b]. The above results provide the following result. Theorem 5.1

Suppose that the matrix coeficients of an hermitian Riccati matrix differential equation (1.2) satisfy hypotheses ($) and $,{I}. Let Zo(t)be a matrix function on I which belongs to PZ[a, b] for arbitrary compact subintervals [a, b] of I , and for a given s E I define the sequence of matrix functions { Z j ( t ) }recursively as the solutions of the respective di@?rential systems Zj'(t) = C ( t )

(5.17)

+ Z&(t)B(t)Zj-,(t)

+

+

- Z j ( t ) [ A ( t ) B(t)Zj-,(t)l - [A"(t) Z:-l(t)B(t)lzj(t) Zj(s) = W,, where W, = W,#.

If W ( t )is the solution of the hermitian Riccati matrix diflerential equation satisfying W(s)= W,, and I , is the subinterval of I on which W ( t ) exists, then for j = 1, 2, . . . we have

I,+(s) = {t I t

W ( t )I zj+l(t) 5 Zj(t),

t

W ( t ) 2 z j + l ( t )2 q t ) ,

t E I,-(s)

E

The sequence { Z j ( t ) }converges to W ( t )for t

=

E

{t I t

E

I,, t 2 $1,

E

I,, t I s}.

I,, and this convergence is

6. A CLASS OF MONOTONE EQUATIONS

129

unljrorm on arbitrary compact subintervals [a, b] of I , containing t = s. Moreover, for such subintervals the convergence is quadratic in the sense that there exists a constant K = K[a, b] such that max v [ Z j ( t )- W ( t ) ]5 K{ max v[Zj-l(t)- W ( t ) ] } z .

astsb

astsb

6. A Class of Monotone' Matrix Differential Equations

For brevity, let !Illn denote the class of n x n complex-valued matrices, and !Ill,+the subclass of !Illn consisting of the nonnegative definite hermitian matrices. I n the following we shall be concerned with a matrix differential equation

(6.1) W ' + W A ( t ) + A"(t)W+ W B ( t ) W - C ( t ) - F ( t , W) = 0, where F ( t , W) is a function on I x !Illn to !Illn which possesses the following properties:

(61)

(a) F is continuous in W, for $xed t E I ; (b) F is Lebesgue integrable on compact subintervals [a, b] of I , for $xed W E !Ill,; (c)

if

W E !Ill,+then F ( t , W) E !Illm,+, for t E I .

In particular, if W ( t ) is a continuous hermitian matrix function with W ( t ) E !Illm,+ for each t on a compact subinterval [a, b] of I , then F(t, W ( t ) )E Cnn[a,b] and F ( t , W ( t ) )2 0 for t E [a, b]. Moreover, for I, a subinterval of I we shall denote by&{I,} the following condition :

The matrix functions A ( t ) , B ( t ) , C ( t ) satisfy hypothesis

&.{I,} ( 5 )and the conditions(l.1);moreover B ( t ) 2 0 and C ( t ) 2 0 for t a.e. on I,.

Lemma 6.1

Suppose that a E I , hypotheses (&a-c) and & { I } hold, and that W = W o ( t )is a solution of (1.2) with W,(a) 2 0. If W = W ( t )is a solution of (6.1) on an interval [a, c ) c I and W ( a )> Wo(a),then W ( t )> W o ( t )2 0 for t E [a, c).

130

IV. HERMITIAN EQUATIONS

In view of Lemma 4.1 we have that the solution W = Wo(t )of (1.2) exists and satisfies W o ( t )2 0 for t E I + ( a )= { t I t E I , t 2 a}. If W = W ( t )is a solution of (6.1) on an interval [a, c) with W ( a ) > Wo(a) and W ( t )2 W o ( t )for t on a subinterval [a, b,] of [a, c), then C , ( t ) = C ( t ) F(t, W ( t ) )>_ C ( t ) for t E [a, b , ] , and from Theorem 4.3 it follows that W ( t ) > Wo(t)for t E [a, b,]. In view of the arbitrariness of [a, b,] as a subinterval of [a, c) we then have that W ( t )> Wo(t)for all t E [a, c). For further considerations it will be supposed that the function F(t, W) satisfies some of the following additional conditions :

+

(d) the solution of (6.1) satisfying given initial data is unique; that is, if ( t o , WO)E I x !Illn there is a unique solution W = W ( t ;to, W o ) of (6.1) such that W(to)= W o ;

(6,)

(e)

if 0 I W , 5 W 2 ,then

0I F(t, W,) I F(t, Wz);

(f) there exist nonnegative real-valued functions p l ( t ) , p 2 ( t ) which are Lebesgue integrable on arbitrary compact subintervals of I , and

(6.2) y[F(t, W)] I Pl(t)

+

p2(t)y[V

for

(4 W) E I x

ml.

Condition (&e) is a rather restrictive condition, which is not satisfied by such a simple function as F(t, W) = W2.On the other hand, all the conditions (a)-(f) of are satisfied by such a function as F(t, W ) = Fo(t) G*(t)WG(t),where Fo(t)2 0 for t a.e. on I and Fo(t) E cnn[a,b ] , G ( t ) E P.;,[a, b] for arbitrary compact subintervals [a, b] of I . These conditions also hold for a nonlinear functional such as Fk(W) = k I WI ( E k I W()-l = E - ( E + k I Wl)-I, wherekisapositive integer and 1 W I denotes the nonnegative definite hermitian square root matrix of W* W. Indeed, 0 5 Fk(W) 4 E, and consequently conditions (a)-(f) of (6,) also hold for F(t, W) = #k(t)Fk(W), where the # k ( t ) are nonnegative Lebesgue measurable functions on I such that # ( t ) = CE1~ $ ~ (ist Lebesgue ) integrable on arbitrary compact subintervals [a, b] of I .

{a,}

+

+

xP=,

Theorem 6.1

Suppose that a E I , hypotheses ($,a-d) and $ , { I } hold, and that W Wo(t)is a solution of (1.2) with Wo(a)2 0. If W = W ( t )is a solution of (6.1) on an i n t m a l [a, c ) and W ( a )2 Wo(a),then W ( t )2 Wo(t)2 0 for t E [a, c). =

6. A CLASS

OF MONOTONE EQUATIONS

131

For E > 0, let W = W(t; E ) be the solution of (6.1) satisfying the initial condition W ( a ;E ) = W ( a ) EE.If [a, c ( E ) ) is the maximal righthand interval of existence of W ( t ;E ) , then from Lemma 6.1 we have that W ( t ;E ) > Wo(t)for t E [a, c ( E ) ) . Moreover, from well-known continuity properties of solutions of ordinary differential equations (see, e.g., Reid [15; Chaps. I, 11]), we have that if b, E [a, c) then [a,bl] c [a,c ( E ) ) for E sufficiently small, so that W ( t ;E ) + W ( t ) on [a,b,] as E 0, and hence W ( t )>_ Wo(t)2 0 for t E [a,c).

+

-

Theorem 6.2

Suppose that a E I , hypotheses ($,a-e) and & { I } hold, and that W = W o ( t )is a solution of (1.2) with Wo(a)2 0. If 0 5 Q15 Qzand W = Wa(t)(u = 1, 2), is the solution of (6.1) satisfying the initial condition Wa(a)= Wo(a)+ Qa and with right-hand maximal interval of existence [a, ca), then c2 5 c1 and

Moreover, if0 < Q, or Q1< Qz,then the respective relation W,(t)> Wo(t) or Wz(t)> W , ( t ) holds for t E [a,c2). From Lemma 4.1 it follows that the solution W = Wo(t)of (1.2) exists and satisfies the condition Wo(t)2 0 for t E I + ( a ) = ( t I t E I , t 2 a } . From the results of Lemma 6.1 and Theorem 6.1 it then follows that if Q 2 0 and W = W ( t )is the solution of (6.1) satisfying W ( a )= Wo(a) Q, then the relation W ( t )2 Wo(t)2 0 holds for t E [a, c ) , where [a, c ) is the right-hand maximal interval of existence of W ( t ) ;moreover, W ( t )=. Wo(t)for t E [a, c) whenever Q > 0. Now suppose that 0 5 Q1 5 Qz, and let W = Wa(t)(a = 1, 2) be the solution of (6.1) with Wa(a)= Wo(a) Qa; the right-hand maximal interval of existence of Wa(t)is denoted by [a, ca). Then the matrix function @ ( t ) = Wz(t) - W ,(t) is a solution of the matrix differential system

+

+

(6.4)

W' + WA(t)+ A*(t)W + WB(t)W - Fz(t, W ) = 0, W ( a )= Qz - Pi.

where

Since the matrix function F,(t,

W ) satisfies conditions (&a-c), applica-

IV. HERMITIAN EQUATIONS

132

tion of the result of Lemma 6.1 to W = W(t)and the solution W 0 ( t )= 0 of the corresponding Riccati system

+ WoA(t)- A*(t)Wo+ WoB(t)Wo= 0,

(6.5) Wo’

W ( U )= 0,

yields the conclusion that W ( t ) 2 0 for t E [a, c,) n [a, cz), and indeed W(t)> 0 for t on this interval if W ( a )= Q2 - Q , > 0. That is, the conclusions of the theorem have been established for t E [a, c,) n [a, cz). Now if [a, c,) were a proper subinterval of [a, c z ) , then c, would be an interior point of I , the relation 0 5 W,(t)5 W z ( t )would hold on [a, c,) with W 2 ( t )continuous and satisfying 0 5 W 2 ( t )on [a, c2), so that Wl(t) would remain bounded as t + cl-, a condition which contradicts the assumption that [a, c,) is the right-hand maximal interval of existence of W l ( t ) (see, e.g., Reid [15, Chaps. I and 113). Consequently, we have that c1 2 cz, thus completing the proof of the theorem. Theorem 6.3

Suppose that hypotheses (&,a hold, and that W = Wo(t) is a solution of (1.2) with Wo(a)2 0. If Q 2 0 and W = W ( t ) is the solution of (6.1) satisfying the initial condition W(a)= Wo(a) Q, then the right-hand maximal interoal of existence of W ( t ) is I+(a).

+

Since B ( t ) 2 0 for t a.e. on I, it follows from the differential equation (6.1) that if the right-hand maximal interval of existence of W ( t ) is [a, c) then 0 IW ( t )IW(a)

for t E (a, c). I n view of the matrix inequalities

134

IV. HERMITIAN EQUATIONS

it then follows that

+

+

where p,,(t) = p l ( t ) v[C(t)],p ( t ) = p z ( t ) 2 v [ A ( t ) ] . With the aid of the Gronwall inequality it then follows from (6.9) that

Consequently, if c were an interior point of I it would follow that W ( t ) remains bounded as t -+ c-, so that c would not be the endpoint of the right-hand maximal interval of existence of W ( t ) .Hence the right-hand maximal interval of existence of W ( t ) is I + ( a ) , and the result of the theorem is established. 7. Properties of Hermitian Systems (1.3J Which Are Definite

Corresponding to the discussion of Section II:S, we shall now restrict attention to systems (1.31) for which the coefficient matrix functions satisfy the strengthened condition

("I

A ( t ) E 2,,[a, b ] , B ( t ) E &:[a, b ] , C ( t ) E Pan[a,b] trary compact subintervals [a, b ] of I .

f0Y

arbi-

Also, as in Section II:S, the symbol 9 [ a , b] will be used to denote the class of n-dimensional q ( t ) E%,[u, b] for which there is an associated ( ( t ) E En2[u,b] such that L,[q, ( ] ( t ) = q'(t) - A(t)q(t)- B(t)C(t)= 0 for t a.e. on [a, b ] . The subclass of 9 [ a , b ] on which q(u) = 0 = q ( b ) will be designated by g o [ a ,b ] . As in Section I1 :8, this association will be denoted by the symbols q E: 9 [ a , b]:C and q E g o [ a ,b ] : ( . If ( q = ,C,) E @,[a,b] x 2,Z[a, b] for CL = 1, 2, we define

As B ( t )and C ( t )are hermitian, under the conditions of (6')the functional (7.1) is an hermitian form on @,[a,b] x Cn2[a,b ] . That is, if (qs,Cp)

7. PROPERTIES OF DEFINITE SYSTEMS (1.3z)

E &[a,

b ] x Efi2[a,b ] for t9

=

135

1,2, 3, then

Now if q, E 9 [ a , b ] :5, (a = 1, 2) the vector functions 5, are in general not determined uniquely. The value of the corresponding functional J[ql:c1, q2:c2; a, b ] is independent of the particular Ca's, however, and consequently the symbol for this integral is reduced to J [ q l , q2; a, b ] . That is, if qolE 9 [ a , b ] : 5 , (a = 1, 2), then we write rb

Also, if q

E

9 [ a , b ] : 5 we write J [ q ; a, b ] for J [ q , q ; a, b ] ; i.e.,

The following results are ready consequences of the definitions of the involved functions. Lemma 7.1

(a) If 7,

(7.4')

E

A171

(7.4") (b) If q,

9 [ a , b]:5, (a = 1, 2)

E

, 72; a, bl

=

r/Z"tl

a, bl

=

%*51

9 [ a , b ] :C, and 5,

a

Ja

E %,[a, b ]

for

a = 1, 2, then

If a 5 t , < t , 5 b, and t , , t , are mutually conjugate with respect to (1.31), then there exists a solution ( u ; v ) of this system with u ( t ) 0 on [tl, t,] and u ( t l ) = 0 = u(t,). If ( q ( t ) , C ( t ) ) = ( u ( t ) , o ( t ) ) for

+

IV. HERMITIAN EQUATIONS

136

t E [tl , 4 , and (q(t),C ( t ) ) = (0, 0) for t E [a, t , ) u ( t z , b ] , then q E g o [ a ,b] :c, and from relation (7.4”) with (ql,5,) = (u, v ) on [tl , t,] it follows that J [ q ;a, b] = J [ u ; t , , t 2 ] = v*u

I

ta

tl

= 0.

I n particular, we have the following result. Corollary

If [a, b] c I , and there is a pair of points on [a, b] which are conjugate with respect to (1.3J, then there exists an 17 E g o [ a ,b] such that q ( t ) f 0 on [a, b] and J [ q ; a, b] = 0. Lemma 7.2

If [a, b] c I , and the functional J [ q ; a , b] is nonnegative deJinite on B o [ a ,b], then condition $,[a, b] holds. This result may be established by indirect argument. Suppose that J [ q ;a , b] is nonnegative on g o [ a ,b ] , but that it is not true that B ( t ) is nonnegative definite for t a.e. on [a, b ] . Then there exists a constant vector 5, with I 5, I = 1, and positive constants x , , x , such that So = {t I t E [a, b ] , v [ B ( t ) ] < xl, co*B(t)S, < - x , } is of positive measure. Let $,(t) be the characteristic function of S o ; i.e., $,(t) = 1 for t E S o , and $,(t) = 0 for t 4 S o. Moreover, let to be a point of the open interval (a, b ) such that the function $,(s) ds, t E [a, b ] , has derivative equal to 1 at t = t o ; consequently, if to E (c, d ) n [a, b] then the set of points So n (c, d ) has positive measure. I n particular, if the matrix function B ( t ) is continuous on [a, b] then for to any point of (a, b ) at which B(t,) fails to be nonnegative definite there exists a vector toand constants x , , x , satisfying the conditions specified above. For @ ( t ) a fundamental matrix solution of @ ’ ( t )- A(t)rD(t)= 0, let x3 be a constant such that ~ [ r D ( t ) @ - ~ ( 5 s ) ]x 3 for (t, s ) E [a, b] x [a, b ] , and let (c, d ) be an open subinterval of [a, b] such that to E (c, d ) and

Jk

b

( d - c)-I > ( x 1 2 x 3 2 / x z ) v [ C ( t ) ]dt. a

Now let q ( t ) be a continuous scalar function which is not identically zero on [c, d], and such that the solution u ( t ) of the differential system L,[u, t0$04]= 0, u(c) = 0, satisfies u ( d ) = 0. In particular, q ( t ) may be

137

7. PROPERTIES OF DEFINITE SYSTEMS (1.3J

+ + - - + c,tn with

chosen as a polynomial q ( t ) = co clt I c, l 2 = 1. Then u ( t ) f 0 on [c, d ] , and

+

I co l2 + -

*

-

Also, in view of the definitive properties of x1 and x 3 , we have

If q ( t ) = u(t), [ ( t ) = 50+o(t)q(t)for t E [c, d], and q ( t ) = 0, t ( t ) = 0 on [a, c ) u (d, b ] , then q E g o [ a ,b ] : t and

J[v ; a, bl L - x2

yI

bo(tMt)

l 2 dffX12X32(

yI

+o(t)dt) I

q(

J-:.[C(t)l.).

Now the Schwarz inequality implies that

also, J,"I +O(t)q(t)l 2 dt > 0 in view of the above described choice of to and (c, d). Hence it follows that

J h ; a, bl x2

- X12X32(d - c )

y

a

Y[C(t)]d t }

yI

+"(t)q(t)12 dt

< 0,

C

contrary to the hypothesis that J [ q ; a, b] is nonnegative definite on g o [ % b].

Lemma 7.3

Suppose that [a, b] c I and U ( t ) , V ( t ) are n x k a.c. matrix functions on [a, b ] . If q, E %,[a, b ] , C, E Cf12[a,b] for a = 1,2, and there exist a.c. k-dimensional vector functions h,(t) such that q,(t) = U(t)h,(t)on [a, b ] , then on [a, b] we have the identity (7.6)

52"Btl

+ qz"Cq1 = {C, - Vh,}"B{C, Vhl} hZ"V"L2[171, Cll (Lz[rle, + h,"(V"LZ[U, V ] + U*L,[U, V ) h l -

-

- h,*{U"V

-

- V*U}h,'

t 2 1 ) * ~ ~ 1

+ [h,*U*Vh,]'.

As an immediate consequence of the identity (7.6) we have the following result.

138

IV. HERMITIAN EQUATIONS

Corollary

I f [a, b] c I , and the column vectors of Y ( t ) = ( U ( t ) ; V ( t ) )form a bm*s for a k-dimensional conjoined family of solutions of (1.31), while q E 0 [ a , b]:5 and there exists a k-dimensional a x . vector function h such that q ( t ) = U ( t ) h ( t )for t E [a, b], then

(7.7)

J [ q ; a, b] = q*Vh

1 + Sb b

a

a

[5 - Vh]"B[5- Vh]dt.

If [a, b] is a compact subinterval of I, we shall denote by $+[a, b] the condition that the functional J [ q ; a, b] of (7.4) is positive definite on 9 ! , [ a , b] ; i.e., if q E 9 , [ a , b] then J [ q ; a, b] 2 0, and the equality sign holds only if q ( t ) = 0 on [a, b]. The basic results concerning disconjugacy of (1.31) on [a, b], the condition $+[a, b], and the existence on [a, b] of an hermitian solution of the hermitian Riccati differential equation (1.2), are given in the following theorem. Theorem 7.1

For [a, b] c I the condition $+[a, b] holds if and only if $!,[a,b] is satisfied and one of the following conditions holds: (a) (1.31) is disconjugate on [a, b]; (b,) there exists no point s E (a, b] which is conjugate to t = a ; (b,) there exists no point s E [a, b ) which is conjugate to b ; (c) there exists a conjoined busis Y ( t )= ( U ( t ) ; V ( t ) )for (1.31) with U ( t ) nonsingular on [a, b]; (d) there exists on [a, b] an hermitian solution W ( t ) of (1.2). If (1.3J is identically normal on I then the proof of the results of Theorem 7.1 are particularly simple, and for simplicity details of proof will be restricted here to that important case. For the proof of these results without the assumptions of identical normality the reader is referred to Reid [8; Theorem 5.11, [ll, Theorem 5.11, or [15, Sections VII.4,5]. Assuming identical normality, this theorem will be established by proving the following sequence of statements (9 (ii) (iii) (iv) (v) (vi)

-

- --

(c) (d); (d), $!,[a, bl -+$+[a, bl ; $+[a, bl (a), $!,[a, bl ; (a) (bd ; (bd, @!,[a,bl (b,); (bz),Sio[a, bl (c). +

139

7. PROPERTIES OF DEFINITE SYSTEMS (1.3,)

Statement (i) is immediate, since if Y ( t )= ( U ( t ) ;V ( t ) )is a conjoined basis for (1.31) with U ( t ) nonsingular on [a, b ] , then W ( t )= V ( t ) U - l ( t ) satisfies (d). T o prove (ii), if W ( t )satisfies (d) and U ( t ) is the solution of the differential system (7.8)

U ( t )= [ 4 t )

+ B(t)W(t)lU(t),

U ( a ) = E,

then Y ( t ) = ( U ( t ) ;W ( t ) U ( t ) )is a conjoined basis for (1.31) with U ( t ) nonsingular on [a, b ] . If q E 9 J a , b ] : [ and h ( t ) = U--l(t)q(t)then h ( t ) is a.c. on [a, b ] and satisfies the end conditions h ( a ) = 0 = h(b). In view of the Corollary to Lemma 7.3 we then have (7.9)

J [ q ; a, b]

=

J:

[C

-

Vh]"B[5- Vh] dt.

Since B ( t ) 2 0 for t a.e. on [a, b] it then follows that J [ q ; a, b] 2 0 with the equality sign holding if and only if B ( t ) [ [ ( t )- V ( t ) h ( t ) ]= 0 a.e. on [a, b ] . As L,[q, [ ] ( t )= 0, L,[U, q ( t ) = 0 and q = Uh, it follows that this latter condition holds if and only if U(t)h'(t)= 0 a.e. on [a, b] so that h ( t ) E h(b) = 0 on [a, b ] . Statement (iii) is a direct consequence of the Corollary to Lemma 7.1 and Lemma 7.2, while statement (iv) is immediate. For the proof of (v), let Y , ( t ) = ( U , ( t ) ; V,(t)) be a solution of (1.3M) with U,(a) = 0, V,(a) nonsingular. Then Y,(t) is a conjoined basis for (1.3J and condition (b,) implies that u,(t)is nonsingular on (a, b ] . Now let Yb(t) = ( U , ( t ) ; Vb(t)) be a corresponding conjoined basis for (1.31) determined by the initial conditions Ub(b)= 0, V,(b) nonsingular. Since the values t conjugate to b are those for which Ub(t) is singular, from conclusions (i) and (ii) applied to subintervals [al, b] of (a, b] it follows that ub(t) is nonsingular on (a, b ) . Moreover Ub(a) is also nonsingular, since a and b are not mutually conjugate in view of (bl). Finally, in order to establish statement (vi) it is to be noted that an argument similar to that employed for statement (v) yields the result that conditions (b,) and &,[a, b] imply (b,). Consequently, if Y , ( t ) = ( U , ( t ) ; V , ( t ) ) and Yb(E)= (ub(t); v b ( t ) ) are conjoined bases for (1.31) as defined above in the proof of (v), we have that { Y , I Y , } = 0, {Yb I Yb} = 0, and { Y , I Yb} is the nonsingular constant matrix M = Vb"(b)U,(b). Upon replacing y , ( t ) by -Y,(t)M-' it results that { Y , I Yb} = -E. It then follows that ( U ( t ) ;V ( t ) )= (U,(t) U , ( t ) ; V , ( t ) Vb(t))is a conjoined basis for (1.3), and we shall proceed to show that U ( t ) is nonsingular on [a, b ] . Since u(a)= Ub(a) and U ( b )

+

+

140

IV. HERMITIAN EQUATIONS

U,(b), the matrices U ( a ) and U ( b ) are nonsingular. If c 5 is an n-dimensional vector such that U ( c ) t = 0, define =

( q ( t ) ,c ( t ) )= (Ua(t)t, V,(t)t) =

(- Ub(t)t,- W ) E

E

(a, 6 ) and

for t E [a, C I , ) for t E (c, b ] .

Then q E g o [ a ,b ] : c ,and upon applying (7.4") to the individual intervals [a, c ] and [c, b] it follows that

+ t*U,*(C)Vb(C)t,

J [ q ; a, b] = E"V,"(C)U,(C)t - t " U b * ( C ) V b ( C ) E ,

-

--E"Vu"(c)Ub(c)E

=

-t"E 5 0.

On the other hand, if h ( t ) = t on [a, c ] , and h ( t ) = -Ul1(t)Ub(t)t on [c, b ] , then h ( t ) is a.c. and y ( t ) = U,(t)h(t) on [a, b]. From the Corollary to Lemma 7.3 we then have

J [ v ;a, b] =

sb

h#'(t)U,#(t)B(t)U,(t)h'(t) dt,

C

and hence J [ q ; a, b] 2 0 in view of condition $jo[a,b] of (vi). Consequently, we have t = 0, and therefore Y ( t )= ( U ( t ) ;V ( t ) ) is a conjoined basis for (1.31)with U ( t ) nonsingular on [a, b]. In view of the equivalence of the conditions of the above theorem, one has as immediate consequences the following additional results involving differential inequalities.

Corollary If [a, b] c I and$,[a, b] is satisfied, then (1.3J is disconjugate on [a, b] if and only if one of the following conditions holds: (i) there exists on [a, b] a nonsingular n x n matrix function U E 9 [ a , b ] : V with a V ( t )E % ~ , [ U , b ] , while { U ; V I U ; V } ( t )= 0 and U"(t)L,[U,V ] ( t )2 0 for t a.e. on [a, b];

(ii) there exists an n x n hermitian matrix function W ( t )E a n n [ u b] , which satisfies the matrix dtflerential inequality ,@[T(t) 5 0 for t a.e. on [a, b]. If U ( t ) and V ( t ) are a.c. matrix functions with U ( t ) nonsingular on an interval [a, b ] , then W ( t )= V ( t ) U - l ( t )is also a.c. on this interval and

,@[T(t) = - W(t)JL[U, Vl(t)U-'(t) -

Vl(t)u-l(t);

141

7. PROPERTIES O F DEFINITE SYSTEMS (1.3J

also,

V * ( t ) U ( t )- U*(t)V(t)= U # ( t ) [ W # ( t> W(t>]U(t), and hence { U ; V I U ; V } = 0 on [a, b] if and only if W ( t )is hermitian on [a, b ] . Moreover, if L , [ U , V ] ( t )= 0 on [a, b ] , then

(7.10)

M(t)

U"(t)L,[U,V ] ( t )= - U*(t){R[W](t)}U(t),

so that U # ( t ) L , [ U ,V ] ( t )2 0 if and only if R [ q ( t ) 5 0 a.e. on [a, b ] , in which case the matrix function M ( t ) of (7.10) satisfies M ( t ) 2 0 a.e. on [a, b ] . Consequently Y ( t )= ( U ( t ) ; V ( t ) )is a conjoined basis for the matrix differential system

(7.11)

L,O[U, U ( t )= -vl(t)

+ CO(t)U(t)- AO"(t)V(t)= 0,

L,O[U, V ] ( t ) U '(t ) - AO(t)U(t)- BO(t)V(t)= 0,

with coefficient matrix functions

(7.12)

BO(t) = B ( t ) , U*-'(t)M(t)U-'(t) = C ( t )

AO(t)= A ( t ) ,

cyt)= C ( t )-

+ a[w](t).

Since Ao(t)= A ( t ) , Bo(t)= B ( t ) , the linear vector spaces 9 [ a , b] and g o [ a ,b ] are the same for the system (7.11) and the system (1.31), and from the preceding result applied to (7.11) we have that

is positive definite on g o [ a ,b ] . Since the condition M ( t ) 2 0 implies that C o ( t )5 C ( t ) , it then follows that

is nonnegative for q E g 0 [ a ,b ] , and hence J [ q ; a, b] is also positive definite on g o [ a ,b ] . In particular, in terms of the now involved matrix functions U ( t ) , V ( t )we have corresponding to (7.7) the relation

(7.14') J [ q ; a, b]

=

J: {[C

-

Vh]*B[5- Vh]

+ q*[C- C o ] q }&.

The following auxiliary result will be of use in certain future considerations.

I42

IV. HERMITIAN EQUATIONS

f emma 7.4 Suppose that [a, b] c I , and the functional J [ q ; a, b] is positive dejinite on B 0 [ a ,b]. If q E 9 [ a , b ] : 5 then there exists a unique solution y ( t ) = ( u ( t ) ;v ( t ) )of (1.31) such that .(a) = q(a), u(b) = q(b), a n d J [ q ;a, b] > J [ u ; a, b ] , with the equality sign holding if and only if q ( t ) = u ( t ) on [a, bl. As the positive definiteness of J [ v ;a, b] on g 0 [ a ,b] implies that a and b are not mutually conjugate relative to (1.31), the existence of a unique solutiony(t) = ( ~ ( t;)v ( t ) ) o f (1.31) satisfying the end conditions

~ ( a= ) q(a), u ( b ) = q(b) is a consequence of Lemma II:6.2. The final

conclusion of the lemma then follows from the identity

J [ v; a, bl For a given subinterval [a, b] of I , of I , let 9 # o [ a ,b] denote the subspace of 9 [ a , b] on which q(b) = 0. The fundamental relation between the existence of an hermitian solution of (1.2) on [a, b] and the extremum of an associated hermitian functional is presented in the following theorem (see, e.g., Reid [ l l , Theorem 5.5; 15, ChapterVII, Section 61). Theorem 7.2

If [a, b] c I , and Qa is a given n x n hermitian matrix, then the functional

is positive dejinite on 53+,,[a, b] if and only if$,[a, b] is satisjied, and one of the following conditions holds: (a) if Y ( t ) = ( U ( t ) ; V ( t ) ) is the solution of (1.3M) satisfying the initial condition Y ( a ) = ( E ; Qa), then U ( t ) is nonsingularr on [a, b ] ; (b) the hermitian solution W ( t )of (1.2) determined by the initial condition W ( a )= Qa exists on [a, b] ; (c) there exists an n x n hermitian matrix function W ( t ) which is of class qnn[a,b ] , and satisjies the conditions

(7.16)

Qa 2 W(a),

$t[WJ(t)5 0

for

t a x . on [a, b ] .

143

7. PROPERTIES OF DEFINITE SYSTEMS (1.3J

Corresponding to the above proofs of Theorem 7.1 and its corollary, if W ( t ) is a solution of (1.2) on [a, b] with W ( a )= Qa, and Y ( t ) = ( U ( t ) ;V ( t ) ) is the corresponding solution of (1.3&,) satisfying U ( a ) = E, V ( a ) = Q u , then Y ( t )is a conjoined basis for (1.3i) and we now have the formula

Since q ( b ) = 0 and B ( t ) 2 0 a.e. on [a, b ] , by the same argument as in the proof of Theorem 7.1 one establishes the positive definiteness of J [ q ; a, b I Qu] on 9 , 0 [ a , b ] . I n turn, if W ( t ) is an hermitian matrix function which is of class 91nn[a,b] and satisfies (7.16), and the matrix function is M ( t ) defined as in the above proof of the Corollary to Theorem 7.1, then in view of the previous result for the related system (7.11) with coefficient matrix functions (7.12) one obtains for q E B w o [ a b]:C , the relation (7.18)

+ y {C*B05'+ q*COq) dt

q*(a)W(a)q(a)

[C - Vh]*B[5- Vh] dt.

= Jb U

Also, corresponding to (7.14') we have the relation (7.19)

J [ v ;a, b I QJ

= q*(a)[QU

+

-

W(a)lq(~)

{[C - Vh]*B[5- Vh]

Jb U

+ q*[C- CO]q}dt.

In particular, conclusion (a) is equivalent to the condition that the conjoined basis Y ( t ) determined by Y ( u )= ( E ; Q u ) has no focal point on [a, b ] . Now if q E 9 [ a , b]:5, t E ( a , b ] , and y ( t ) = ( u ( t ) ; w ( t ) ) is a solution of (1.31) such that ~ ( t=)q ( t ) and Quu(a) - .(a) = 0, then 7 - u E B w o [ at,] : ( C - w ) and J [ q - u ; a, t I Qu] > 0,

J[u,7 - u ; a, I QuI

=

unless

q ( t ) = u ( t ) on

[a, t ] ;

[u*(a>Qa- w*(a)l[q(a) - 4a)I

+ w * ( t ) [ r ] ( t ) - u(.)] = 0.

Arguments similar to those employed in the proof of the above theorem

1u

IV. HERMITIAN EQUATIONS

then yield the relation

and the following result. Lemma 7.5

Suppose that [a, b] c I and the functional (7.15) is positive dejinite on g b 0 [ a ,b], while Y ( t )= ( U ( t ) ; V ( t ) ) and W ( t )are determined as in (a) and (b) of Theorem 7.2. If 17 E 9 [ a , b]:5, and z E (a, b], then there is a unique solution

A t ) = ( 4 t ); 7 4 ) ) = (w)u-1(M49 W)Wz)17(4) of (1.3J such that

Q d a ) - 4.1

and J [ q ; a, T

= 0,

4.)

=~

( t ) ,

I Qal LJ[u; a, T I Pal = v*(z)u(z),

with the equality sign holding if and onZy if q ( t ) = u ( t ) on [a, z]. Consequently, if (T,e ) E (a, b] x Q, and g,[a, then J [ q ;a, z

TI

I

= {'I rl E

q . 9

I Qal 2 e*W(T)e

21,17(2)

for

=

el,

E g,[a,

71,

with the equality sign holding if and onZy if q ( t ) = U(t)U-l(r)e on [a, TI. Since some of the applications of matrix Riccati differential equations of the form (1.2) involve initial conditions at the second endpoint of the interval of consideration, we will state without proof the following duals of the above Theorem 7.2 and Lemma 7.5, involving the subspace g W [ a ,b] of 9 [ a , b] on which q ( a ) = 0. Theorem 7.3

If [a, b] c I and Qb is a given n x n hermitian matrix, then the functional (7.20) JOLT; a, b I QbI = q*(b)Qbdb)

14s

7. PROPERTIES OF DEFINITE SYSTEMS (1.38)

is positive definite on go# [a, b ] if and only if$,[a, b ] is satisfied, and one of the following conditions holds: if Y ( t ) = ( U ( t ) ; V ( t ) )is the solution of (1.3M)satisfying the initial condition Y(b)= ( E ; - Q b ) , then u ( t ) is nonsingular on [a, b ] ; (b) the hermitian solution W ( t ) of (1.2) determined by the initial condition W(b)= -Qb exists on [a, b ] ; (c) there exists an n x n hermitian matrix function W ( t ) which is of class ann[a,b ] , and satisfies the conditions (a)

(7.21)

W(b)2

-

Q b , f i [ q ( t )5 0

fOY

t a&. On [a, b ] .

Lemma 7.6

Suppose that [a, b] c I and the functional (7.20) is positive dejinite on go# [a, b ] , while Y ( t ) = ( U ( t ) ; V ( t ) )and W ( t ) are determined as in (a) and ( b )of Theorem 7.3. l f q E 9 [ a , b ] : 5 , and t E [a, b), then there is a unique solution

Y ( t ) = ( 4 t ) ;4 t ) ) = ( ~ ( t ) W + ? (~ 4( t~) ~ - W 1 1 ( 9 ) of (1.3i) such that Qbu(b) -k v ( b ) = 0,

and Jo[rl;'G, b

I

Qb]

>_Jo[u;t, b

= T(t),

I Qb]

= -v*(t)U(t),

with the equality s&n holding i f and only if q ( t ) = u ( t ) on sequently, if (t,e ) E [a, b ) x Q, and

gqt,bl

= {1;1

I 11 E g[t,bl,

[t,b].

Con-

q(t) = el,

then Jo[17;

t, b

I QbI

b], L - @"W(t)@ for 7 E gp[t,

with the equality sign holding if and only

if q ( t ) 3 U(t)U-l(t)e on

[z,b].

In various applications of the result of this lemma one finds the final inequality stated in terms of the matrix function W(t)= -W(t), which is characterized as the solution of the differential system

P(t)

+ W(t)A(t)+ A * ( t ) W ( t ) - W ( t ) B ( t ) W ( t )+ C ( t ) = 0,

W(b)= Q b -

f46

IV. HERMITIAN EQUATIONS

Now let 8, and 8 b be n X f a and n X i b matrices with 0 5 ra 5 n, 0 5 rg 5 n, it being understood that if either I , = 0 or i b = 0 the respective matrix 8, or 8 b does not occur, and that if 0 < ra 5 n or 0 < rb 5 n then the corresponding 8, or 8 b has rank ra or r b . For Q, and Q b given n X n hermitian matrices we now consider the conditions under which the functional

(7.22)

f [ q ; Q, bI = q+(a)Qaq(a)

'V"(b)Qb'?(b)

is positive definite on the class 9 e [ a , b ] consisting of those q such that q E 9 [ a ,b ] : 5 and

(7.23)

8,"q(U)

= 0,

8b"q(b) = 0.

In particular, g o [ a ,b ] = g o [ a ,b ] if 8, = 8 b = E n , g e [ a , b ] = g [ a , b ] if 8 , and €Jb are nonexistent, g @ [ ab, ] = g x o [ a ,b ] if 8 b = En with 8, nonexistent, and g e [ a , b ] = g o + [ a ,b ] if 8 , = En with 8 b nonexistent. Theorem 7.4

If [a, b ] c I , while Q , , Qb are n x n hermitian matrices and 8,, Ob are matrices as described above, then the functional (7.22) is positive dejinite on [a, b ] q a n d only if&,[a, b ] is satisfied, and one of the following conditions holds. (a) There exists a real constant x such that there is an hermitian solution W ( t )of (1.2) satisfying the conditions (7.24)

Qa

+ X8,8,"

- W ( U ) > 0,

Q b

+

x8b8b"

+ W ( b ) > 0.

(b) There exists an n x n hermitian matrix function W ( t ) which is of class'U,,[a, b ] and satisjies with a real constant x the conditions (7.25)

+

" W ( a )> 0, Qb (a) Qa -I- x ~ a ~ a( b ) R [ W j ( t )5 0 for t a.e. on [a, b ] .

xebeb"

+ W ( b ) > 0,

Since g o [ a ,b ] c 9 e [ a , b ] , in view of Theorem 7.1 the positive definiteness of (7.22) on g e [ a , b ] implies $,[a, b ] , and also that t = b is not conjugate to t = a. If the order of abnormality of (1.3z) on [a, b ] is equal to d then there exists a set of 2n linearly independent solutions y ( j ) ( t )= ( u ( j ) ( t )d; j ) ( t ) ) , ( j = 1 , . . . , 2n) of (1.3z) such that d U ) ( t = ) 0

147

7. PROPERTIES OF DEFINITE SYSTEMS (1.3i)

on [a, b] for a = n - d

[

;3

+ 1, . . . ,2n,

( i = 1,

and the 2n

x

(2n - d) matrix

. . . , n ; k = 1, . . . ,2n - d )

is of rank 2n - d. I n view of Lemma 7.4, if lution

E 9 [ a , b]:5 there

is a so-

and

with equality in (7.28) if and only if T,I - up = 0 and B[5 - a,] = 0 on [a, b ] . As q ( t ) and ,u,(t) have the same end values at t = a and t = b, one has equally well that

f [ r ;a, 61 2 fb,; a, bl,

(7.28')

with equality if and only if q - up = 0 and B[5 - a,] = 0 on [a, b ] . It follows, therefore, that the functional (7.22) is positive definite on S$[a, b] if and only if the hermitian form

is positive definite on the linear vector space of (2n - d)-dimensional vectors ,u = (pk)satisfying the (la t b ) conditions

+

&*u,((2)

2n-d k-1

(7.30) eb"&!,,(b)

zn-d k-1

[ea"u'k'(a)]pk= 0, [6b*U'k'(b)]pU,= 0.

Application of a theorem on pairs of hermitian forms then yields the result that this latter condition is equivalent to the existence of a real constant x such that the (2n - d) x (2n - d) hermitian form

91.p;

b1

+

ea*up(a)12

+1

eb*Up(b)

1'1

148

IV. HERMITIAN EQUATIONS

is positive definite, and in view of Lemma 7.4, this result is equivalent to the condition that

is positive definite on a [ a , b]. That is, we have established that the positive definiteness of (7.22) on S o [ a ,b] is equivalent to the condition that (7.31) be positive definite on the class 9 [ a , b] which involves no ) ~(b). restriction on the end values ~ ( aand Since 9 * , [ a , b] and 9,+[a, b] are linear subspaces of B [ a , b], we have that (7.31) is positive definite on each of these classes. Let Y,(t) = (U,(t); V a ( t ) )and Yb(t) = ( U b ( t ) ;V b ( t ) )be the solutions of (1.3M) satisfying the respective initial conditions

(7.32)

(a)

U,(a) = E, (b) Ub(b) = E,

V,(a) = Qa vt,(b)

+ x4$,*, + xebeb*].

= -[Qb

Then Y,(t) and Yb(t) are conjoined bases for (1.3J with u,(t)and Ub(t) nonsingular on [a,b]. Now V,*U, - U,*V, = 0 and Vb*ub - ub*vb = 0, and the constant matrix M such that V,*ub - u,*vb = M is nonsingular. Indeed, if 5 is an n-dimensional vector such that M t = 0 then ( u ( t ) ; v ( t ) ) = (ub(t)t; Vb(t)t) is a solution of (1.3M) such that V,*(b)u(b)- U,#(b)o(b)= 0, and as the matrix [U,*(b) V,*(b)] is of rank n, and V,*(b)U,(b) - U,*(b)V,(b) = 0, it then follows that there exists an n-dimensional vector to such that ( u o ( t ) ;v o ( t ) ) = (U,(t)[O; V a ( t ) t o )is a solution of (1.32) satisfying the conditions u,(b) = u(b), w,(b) = w(b), and hence uo(t)= u ( t ) , oo(t)= w ( t ) . In par- .(a) = 0 and [Qb ticular, [Q, x€I~B~*]u(u) xO&,*]u(b) w(b) = 0, and as a result of a simple integration by parts, we have that s , [ u ; a, b] = 0. In view of the positive definiteness of Yo[q;a, b] on S [ a , b], it then follows that 0 E u ( t ) = Ub(t)t,so that 5 is the zero vector and M is nonsingular. Now upon replacing (Ub(t); V b ( t ) )by (Ub(t)M-l; Vb(t)M-') we obtain a conjoined basis for (1.32),which will still be denoted by (Ub(t); Vb(t)), that satisfies in place of (7.32b) the related condition

+

+

+

and for which the associated constant matrix function V,*ub - u,*Vb

I49

7. PROPERTIES OF DEFINITE SYSTEMS (1.3J

is the identity matrix. I t then follows readily that ( U ( t ) ;V ( t ) ) = (U,(t) U b ( t ) ;V a ( t ) V b ( t ) )is a conjoined basis for (1.3J; i.e., V # U - U x V = 0. We shall proceed to show that U ( t ) is nonsingular for t E [a, b ] . If c E [a, b] and 5 is an n-dimensional vector such that U(c)t = 0, define

+

+

(q(t),W ) )= (uuwt, V a ( t ) t ) = (- Ub(t)t, - Vb(t)E)

for t for t

E

[a, c ] ,

E

(c, b ] .

Then q E 9 [ a , b ] : [ and if c = b then q E 9+,,[a, b ] : 5 ; similarly, if c = a then q E 9 & [ a , b ] : 5 . We shall consider specifically only the case c E (a, b), since whenever c = a or c = b the same type of argument applies with obvious simplifications. Upon using appropriate integration by parts for the integrals J [ q ;a, c ] and J [ q ;c, b] it follows that .,Te[q ; a, b] = t * V a * ( ~ ) U a ( ~ ) t*Ub*(c)Vb(c)t, t-

+ t*ua"(c)Vb(c)ti

= -t"Va"(c)ub(~)t =

-t"t.

Consequently, the positive definiteness of .,Te[q; a, b] on 9 [ a , b] implies that t = 0, and hence the nonsingularity of U ( c ) .I n view of the condition Va"Ub- U,*Vb = E, one may verify directly that for an arbitrary n-dimensional vector we have

+ U(a) U*(a)V(a)}t t"t.+ t"{Ub"(a)[Qa + t"t + I [Ubt ; a, b] 2 5"t.

t"{U*(a)[Qa =

=

~eaea"]

-

xeaea"]ub(a) -

Ub*(a)Vb(a)}t

Consequently, W ( t )= V(t)U-l(t) is an hermitian solution of (1.2) satisfying

+ xBaBa*

n*{Qa

-

W(u)}n2 n + U + - 1 ( ~ ) U - l ( ~ ) n

1SO

IV. HERMITIAN EQUATIONS

Therefore, W ( t )= V(t)U-l(t)is an hermitian solution of (1.2) satisfying

+

+

for arbitrary n-dimensional vectors n, and hence Qb dbob* W(b) > 0. If W ( t ) satisfies condition (a) of Theorem 7.4, then clearly W ( t ) also satisfies condition ( b ) . Conversely, if W ( t ) satisfies condition ( b ) of Theorem 7.4 then for U ( t )the solution of the differential system (7.8) and M ( t ) = -U*(t)R[W](t)U(t), we have that Y ( t )= ( U ( t ) ; V ( t ) = W ( t ) U ( t ) )is a conjoined basis for the system (7.11) with coefficient matrix functions (7.12). By argument similar to that used in the proofs of the corollary to Theorem 7.1, and of Theorem 7.2, it follows that for 7 E 9 [ a , b ] : 5 and h ( t ) = U-'(t)q(t) we have the i-elation

+ jl {[C

-

Vh]*B[C- V h ]

+ vy[C

-

C o ] q } dt,

Ie[v;

which implies the positive definiteness of a, b] on 9 [ a , b ] . Finally, we shall present a proof of the following theorem on the existence of a conjoined basis Y ( t )= ( U ( t ) ;V ( t ) )of (1.31) with U ( t ) nonsingular on a compact interval [a, b ] . In particular, this result will be used in the following section in the proof of the existence of a distinguished solution of an hermitian equation (1.2) which satisfies the condition of definiteness. Theorem 7.5

Suppose that [a, b] c I , and J [ v ; a, b] is positive dejnite on g o [ a ,b ] . If d [ a , b] = d , d ( a ) s A[a, b ] , and Q is an n x (n - d ) matrix such that Q* d ( a ) = 0 and [ d ( a ) Q ] is nonsingular, then there exists a unique solution Yb(t) = (ub(t);Vb(t)) of (1.3M) such that

(7.34)

Ub(a)= Q,

Ub(b)= 0,

Vb'(a) d ( U ) = 0.

The column vectors of Yb(t)form a basis for a conjoined family of solutions of (1.31) of dimension n - d , and if Y4(t)= ( U 4 ( t ) ;V 4 ( t ) )is a second solution of (l.3M) whose column vectors form a basis for a conjoined family

151

7. PROPERTIES O F DEFINITE SYSTEMS (1.3J

of solutions of (1.3J of dimension n

-d

and satisfy

then U 4 ( t )is of rank n - d o n [a, b ] . Moreover, i f Y z ( t )= ( U , ( t ) ; V , ( t ) ) is the solution of (1.3Af) satisfying the initial conditions U,(a) = d ( a ) , V 2 ( a )= 0, then

Y ( t ) = ( W ) ;W ) )= ([U4(t) U2(2)1;[V4(t)

K(t)l)

is a conjoined basis for (1.31) with U ( t ) nonsingular on [a, b ] . The existence of a unique solution Yb(t)= (Ub(t); v b ( t ) ) of (1.3M) satisfying (7.34) is a consequence of Lemma II:6.2, and the condition Ub(b)= 0 implies that the column vectors of Yb(t) are mutually conjoined solutions of (1.31), Moreover, since Ub(a) = Q is of rank n - d, the column vectors of Yb(t) are linearly independent solutions of (1.3l), and hence they form a basis for a conjoined family of solutions of dimension n - d . Now if Y 4 ( t )= ( U 4 ( t ) ;V 4 ( t ) )is a solution of (1.3M) satisfying (7.35) then the matrix U4(b)is of rank n - d . Indeed, if U4(b)t = o then ( ~ ( 2 ) ; v ( t ) ) = ([u4(t) - Ub(t)]E;[ V 4 ( t )- vb(t)]t) is a sohtion of (1.31) with u ( a ) = 0 = u(b). As b and a are not mutually conjugate it then follows that u ( t ) = 0 on [a, b] and v ( t ) E A [ a , b ] . In turn, (7.34) and (7.35) imply that d * ( a ) v ( a ) = 0, and since d ( a ) = A [ a , b ] it follows that v ( t ) = 0 on [a, b ] . Then

and 5 = 0 in view of the last condition of (7.35). Now if c E (a, b ) , and there exists a nonzero 5 such that U4(c)t = 0, let (q(t),[ ( t ) )= ([U4(t) - U b ( t ) ] t ,[ v 4 ( t ) - vb(t)lt) for La, cl, and (q(t),c ( t ) )= (- u b ( t ) t ,-vb(t)t) for t E (C, b ] . Then 11 E 9 o [ a , b ] : 5 , and appropriate integrations by parts on the individual subintervals [a, c ] and [c, b] yield the relation

(7.36)

J [ q ; a, b]

=

t*[u4*(c) -

- ub*(c)l[v4(c)- vb(C)Il t*ub*(c)v b ( c ) l .

Since U,(c)t = 0, the right-hand member of (7.36) is equal to

152

IV. HERMITIAN EQUATIONS

Moreover, since { Y4[ Y b } ( t )is constant we have that ub"(c)V4(c)

-

Vb"(c)u4(c)

=

ubx(u)v4(a)

-

Vb+(a)U4(a)

= U4"(a)V4(a) - Ub"(a)V*(a),

where the last relation follows from the condition Ub(a)= U4(a)= Q, and the fact that the matrix VbX(a)Ub(a) is hermitian. Thus under the assumption that U4(c)E= 0 we have, in view of (7.35), that

J [ q ;a, bl = -E"*[:U,"(a)V4(a)- ub"(a)Vb(a)lE < 0, whereas the positive definiteness of J [ q ; a, b] on g o [ a ,b] implies that J [ q ;a, b] 2 0. Consequently the assumption that there is a value on [a, b] at which U4(t)has rank less than n - d has led to a contradiction. Now if Y z ( t )= ( U , ( t ) ; V,(t)) and Y ( t )= ( U ( t ) ;V ( t ) )are defined as in the statement of the theorem, the matrix function U ( t )is nonsingular on [a, b]. Indeed, if c E [a, b] and U ( c ) were singular, then there would exist vectors and (T of respective dimensions d and n - d, with e and (T not both zero and ( u o ( t ) ;o o ( t ) )= (U,(t)e U 4 ( t ) u ;Vz(t)e V 4 ( t ) 0 ) a solution of (1.3J for which uo(c) = 0. As e)"(t)uo(t)= 0 on [a, b] for arbitrary ~ ( tE)A[a, b] by Lemma II:6.2, it would then follow that 0 = d"(a)u,(a) = d # ( a )d ( a ) e = e, so that u ( t ) = U4(t)aand U4(c) has rank less than n - d, contrary to the preceding result. Therefore, we have that U ( t ) is nonsingular on [a, b]. Finally, since U4*(u)V4(u) = Q"V4(a) is hermitian by hypothesis, one may verify directly that V # ( u ) U ( a )- U+(u)V(a)= 0, and hence Y ( t )= ( U ( t ) ;V ( t ) )is a conjoined basis for (1.31) with U ( t ) nonsingular on [a, b ] . The preceding resulG of this section provide a basis for the study of comparison and oscillation phenomena for solutions of hermitian differential system (1.3z). For extension of the Sturmian theory for real linear homogeneous differential equations of the second order to such differential systems, the reader is referred to Morse [ l ; 21, Hartman [2, Chap. XI], and Reid [15, Chap. VII]. I n this connection, transformations of the sort considered in Section I1 :4 and Section 2 of this Chapter are of importance, since they may be employed to reduce a given selfadjoint system to other equivalent systems. In particular, if in the hermitian system (1.3z) the matrix function B ( t ) is a positive definite hermitian matrix function, while B ( t ) and the matrix dlof (11:4.15') are locally a.c., then under the transformation of Theorem II:4.2 the differential system (1.3M) is reduced to a secondorder matrix differential equation (I1 :4.19"), or the equivalent system

+

+

153

8. EXISTENCE OF A PRINCIPAL SOLUTION

(11 :4.21), where now the matrix coefficient @(t) is hermitian. Moreover, if ( U ( t ) ; V ( t ) ) is a conjoined basis for (1.3Aw) then the corresponding solution ( % ( t ) ;T ( t ) ) of (11:4.21) is also a conjoined basis for that system, and consequently results on the disconjugacy of one of these systems are immediately translatable into results for the corresponding system. It is to be remarked that this particular reduction of (1.3M) to (11:4.21) is a generalization of the well-known reduction of the selfadjoint scalar differential equation

+

[r(t)u'(t)lr p ( t ) u ( t )= 0 to a canonical form

Y"(t)

+ g ( t l y ( t )= 0,

where

g ( t ) = [r'"t)

- 2r(t)r"(t)

+ 4r(t)p(t)]/[4r2(t)].

In connection with this particular result, the reader is referred to Birkhoff and Rota [ l , Exs. 4 and 6, p. 381.

8.

Existence of a Principal Solution for an Hermitian System (1.3J

The basic result of the present section is presented in the following theorem. Theorem 8.1

Suppose that an hermitian system (1.31) satisJes hypotheses (6') and $ , { I } , where I = ( a o ,b,), -m 5 a, < b, 5 00. If this system is disconjugate on a subinterval (a, b,), then there exists a principal solution at b, .

As in the earlier general discussion of Section 11:6, let do denote the maximum value of the index of abnormality d[t, b,) for t E (a,, b,), and for a value to E (a, b,) such that d[t,, b,) = do choose s E (to,b,) such that d[to, s] = do. If d (s) = A [ t o ,b,), denote b y Q an n x (n - do) matrix satisfying d*(s)Q = 0 and Q*Q = En+, and let Yo8(t) = (UOs(t);V o s ( t ) )be the solution of ( 1 . 3 M ) determined by the initial conditions

154

IV. HERMITIAN EQUATIONS

Moreover, if yo is such that ro E (s, b,) and d[s, y o ] = d o , for Y E ( y o , b,) let YsT(t) = ( U s r ( t ) ;V s r ( t ) be ) the solution of (1.3M) determined by tb.e two-point conditions

(8.2)

usr(s) = Q,

usr(r) = 0,

d"(s)J'sr(s) = 0.

In view of the conditions Uos(to) = 0 and U S r ( y = ) 0, it follows that each of the matrix functions U$(t)Vos(t)and U$(t)Vsr(t)is hermitian on (a, bo). We shall proceed to show that

(8.3)

U:(s)Vos(~)>

uz(s)vs,(s) 2 uE(s)vsr(s)

for

YO

I Y I e < b,,

with inequality in the last relation provided e is sufficiently large. arbitrary nonnull (n - do)-dimensional vectors 5, let ( q ( t ) ;5 ( t ) = (uOs(t)t; vos(t)t) for t E [to, s), and ( ~ ( t )C ;( t ) ) = (Usr(t)E;v s r ( t ) t for t k [s, Y ] . Then q ( t ) 0 on [ t o ,Y ] and q E 9 , [ t o ,r ] :5, so that from Theorem 7.1 it follows that

+

and hence the hermitian matrices U$(s)V,,(s) and U$(s)Vsr(s)satisfy the condition

so that

Moreover, the equality sign in (8.4) holds if and only if q ( t ) = u ( t ) and B ( t ) [ c ( t )- w ( t ) ] = 0 for t E [s, el. In particular, if e is so large that d[r, Q ] = d[r, b,] = d o , then equality in (8.4) implies Vs,(t)5E A [ Y ,el, and consequently Us,(t)E 3 0 on [ t o ,bo), Vse(t)EE A [ t o ,b,), whereas the condition 0 = Use(s)E= .QE implies that 5 = 0. Hence if E # 0 the inequality sign holds in (8.4) whenever ro 5 r < e and e is so large that d[r, Q ] = d o , thus establishing (8.3).

8. EXISTENCE OF A PRINCIPAL SOLUTION

155

From this result we have that U$(s)V,,(s) = Q*V,,(s) is a monotone nondecreasing bounded family of hermitian matrices for r E [yo, b,), and hence there exists an hermitian matrix H such that Q*V,,(s) -+ H as r -+ 00. As d*(s)V,,(s) = 0 and [ A ( $ ) Q ] is nonsingular, it follows that Vabo= limr+boVm(s)exists; moreover, d*(s)Vsbo = 0, Q*VEbois hermitian, and we have Q*Vos(s)> Q*V8bo> Q*V,,(s) for r E [r,, b,). If Y:b,(t)= (U'J,,(t);V'&,(t))is the solution of (1.3fi1)determined by the initial condition Y&,(s)= ( Q ; VEbo), then for t E [s, b,) we have that Y,,(t)-+ Y$,,(t) as t -+ 0 0 ; moreover, d*(s)V$,,(s) = 0 and also U:$,(S)V;~,(S) > U$(s)V,,(s)for r E [ y o , bo).Hence for y 2 s ( t ) = (U2,(t); V z E ( t )the ) solution of (1.3,w)determined by the initial conditions Y2,(s) = (d(s); 0) it follows from Theorem 7.5 that Yabo(t) = (Usbo(t);Vsbo(t)) with U,b,(t) = [U%,(t) UZ&)I, V8bo(t)= [V%,(t) V2dt)l is such that Ysb,(t)is a conjoined basis for (I.&) with Usbo(t)nonsingular on [s, I ] for r > r , , and hence Usb,(t)is nonsingular throughout [s, b,). In view of the remarks in the initial paragraphs of Section 3 it then follows that Yabo(t) is a principal solution of (1.3J at b,, and correspondingly W,b,(t) = VEb,(t)U&(t)is the unique distinguished solution at b, of (1.2) satisfying the normalizing condition (I1:6.12). The behavior of (1.31) and (1.2) on an interval (a,, a) is obviously equivalent under the transformation (8.5)

UA(t) = U ( - t ) ,

V'(t) = - V ( - t ) ,

WA(t)= - W ( - t )

to the behavior of the respective systems

-V.'(t) (1.3MA)

(1.2A)

+ C(-t)UA(t) + A"(-t)VA(t) = 0,

UA'(t)

W.'(t)

+ A(-t)UA(t) - B(-t)VA(t) = 0,

- WA(t)A(-t) - A*(-t)WA(t)

+ WA(t)B(--t)WA(t)- C ( - t ) = 0

on (-a, -a,). A principal solution of (1.32)at a,, and the associated distinguished solution of (1.2) at a,, may be defined as the images under the above transformation (8.5) of a principal solution for (1.3MA)at -ao and the distinguished solution W s , ( t ) of (1.2,) at -ao. For intervals of the form (a,, a) the analogues of earlier results for differential systems (1.3J, (1.3M) and equations (1.2) on intervals (a, b,) are immediate, and will not be repeated in detail. Whenever (1.32) is identically normal on a subinterval (a, b,) there are decided simplifications of detail, and form of final statements, for

IV. HERMITIAN EQUATIONS

1%

the preceding results on disconjugacy and existence of principal solutions. These aspects of the theory also admit considerable simplifications under certain assumptions of normality of intermediate strength. For brevity, the notations %+(I)and %-(I) are introduced for the following conditions.

'+(I)

I is an open interval, and for s E I there exists a b(s) E I such that s < b(s) and (1.31) is normal on [s, b ( s ) ] .

'-(I)

I is an open interval, and for s E I there exists an a(s) E I such that a(s) < s and (1.31) is normal on [a(s),s].

An equivalent formulation of %+(I)is that d(I+(s))= 0 for arbitrary s E I. In terms of the preferred reducing transformation introduced in Section 111:4, and discussed further at the end of Section 2 of the present chapter, if B ( t ) >_ 0 for t a.e. on I the condition %+(I)might also be phrased as the condition that for s E I there exists a b(s) E I such that s < b ( s ) and the matrix B o ( t )of (2.8) determined by a preferred reducing transformation is such that J: B o ( t )dt > 0 if b E I and b > b(s). The corresponding equivalent formulations of %-(I)will not be stated specifically as they should be obvious to the reader. The argument used in the proof of Theorem 8.1, with simplifications now arising from the fact that do = 0, yields the following result. Theorem 8.2

Suppose that an hermitian system ( 1.31)satisjies hypotheses (6')and&,{I} For r E I , let Y r ( t ) = (U,(t); V,(t)) be the conjoined basis for (1.31) satisfying the initial conditions U,(r) = 0, V r ( r )= E. If condition %+(I)holds, then for s E I and r > b(s) the matrix U r ( t ) is nonsingular for t E I-(s) = {t I t E I , t 5 s}, the matrix function W r ( t )= V?(t)U;'(t) tends to a limit W b o ( t ) as r -,b, for t E I , and Wb,(t)is the distinguished solution of (1.2) at b,. Correspondingly, if %-(I) holds, then for s E I and r < a(s) the matrix U,(t) is nonsingular for t E I+(s)= {t I t E I , t 2 s}, the matrix function W,(t) = Vr(t)U;'(t) tends to a limit W a o ( t )as r -,a, for t E I , and Wa,(t) is the distinguished solution of (1.2) at a,. on I = (a,, b,), and this system is disconjugate on I.

For the continued discussion of solutions of hermitian Riccati matrix differential equations (1.2) we shall now suppose that the following hypothesis of intermediate strength is satisfied.

8. EXISTENCE OF A PRINCIPAL SOLUTION

157

The n x n matrix functions A ( t ) , B(t), C ( t ) satisfy hypotheses (8’)and $ , { I } on I = (a,, b,), and there exists ‘duo’bo) a nonnegative integer d such that d[a, b] = d for arbitrary [a, bl = ( U o , bo). In all cases, the argument will be presented for the case d > 0, since in the alternate case the system is identically normal, and arguments simplify through the nonexistence of certain matrices. For such systems there is an n x d matrix Vdo(t) which satisfies VL;,(t) A*(t)Vdo(t)= 0, B(t)Vdo(t) = 0 for t E ( a o ,b,), and such that the column vectors of F/ao(t)form a basis for A ( I l ) , where Il is any nondegenerate subinterval of I. For S E I k t d(S) = V ~ O ( S ) [ ~ ~ ( S ) V ~ Owith ( S ) ][-V” $~O, ( S ) ~ ~ O ( S ) ] - ~ ’ ~ the inverse of the positive definite square root matrix [V % ( s ) V d ~ ( s ) ] ~ ’ ~ of the positive definite d x d hermitian matrix V$,(s)Vd,(s).Then Vd8(t) = Vao(t)[V~(s)VdO(S)]-l/z is an n x d matrix function whose column vectors form a basis for A ( I l ) on arbitrary nondegenerate subintervals Il of I, and Vds(s)= d(s) where d*(s)d(s) = Ed. Moreover, let Q(s) be an n x ( n - d ) matrix such that d*(s)Q(s) = 0 and Q*(s)Q(s) = En4, so that for s E I the n x n matrix [Q(s) d ( s ) ] is unitary. In particular, the matrix Q(s) may be chosen to be locally a.c. on I, although this property will not be used in the following discussion. The argument used in the proof of Theorem 8.1 now provides the following theorem.

+

Theorem 8.3

Suppose that an hermitian system (1.31) satisjies hypothesis 8 ~ ( a , b,), , and is disconjugate on a subinterval I , = (a, b,). For s E I , let the n X d matrix functions d(s), Vd,(t), and the n x (n - d ) matrix Q(s), be de= (Utb0(t); v;a0(t)) and Y28(t) termined as indicated above. If Y:bo(t) = (Uz8(t); V28(t)) are as in the aboveproof, then Y8bo(t) = (U8b(t);v 8 b 0 c t ) ) = [Vtbo(t) Vz8(t)]is such that: with Uho(t)= [Qb0(t) U2s(t)],V8bo(t) (i) Y8bo(t) is a conjoined basis for (1.31)with U8bo(t)nonsingular on I , , and Vz8(t)Ufbo(t) = 0 for t E I,; correspondingly, W8bo(t) = V8bo(t)UGi(t) is an hermitian solution of (1.2) satisfying d*(s)W8bo(s)= 0. (ii) Ysbo(t) is a principal solution of (1.31) at b,, and Wsbo(t)is a distinguished solution of (1.2) at b, , so that for

158

IV. HERMITIAN EQUATIONS

and

As Vz8(t)U:bo(t) = 0 and B ( t ) V d 8 ( t= ) 0 for t n x (n - d ) matrix D 8 ( t ) such that

E I,

there exists an

Consequently, U&(r)B(r)U$;l(r) = diag {@8*(r)B(r)@8(r) ; 0 }, and (8.9) where

s(t,s [ S(t, s I Ysb0)is

Ysb,) = diag{S(t,

I

yabo);

O},

the (n - d ) x ( n - d ) matrix function

(8.10) Moreover, if Y&(t)= (Uzs(t);V:,(t)) is the solution of (1.3,~)determined by the initial condition Y&(s)= (0 ; Q(s)), and Y38(t)= (U3s(t);V 3 8 ( t )is) the solution of (1.3h1) with U3a(t)= [UL(t) Uz8(t)], , from the representation formula (111:3.5) V38(t)= [V:8(t) V 4 t ) ] then of Theorem III:3.1 it follows that

As (1.31) is disconjugate on I, it then follows that U&(t)is of rank n - d for t # s, and consequently the ( n - d ) x (n - d ) matrix function S(t, s I ysb,)is nonsingular for t s. Now, in general, if K is an n x n hermitian matrix of rank n - d, and Y is an n x (n - d ) matrix whose column vectors form an orthonormal basis for the linear subspace of (5, spanned by the column vectors of K, then there exists a nonsingular ( n - d ) x (n - d ) matrix x such that K = Y x Y * . I n terms of these component matrices Y , x , it follows readily that the generalized inverse K # is given by K" = !Px-'Y*. Also,

+

8. EXISTENCE OF A PRINCIPAL SOLUTION

159

if N is a nonsingular n x n matrix and K , = NKN", then K , = YlxlYl*, where Yl= NY1-1/2and x1 = 11/2x11/2with 1 = Y*N*NY, satisfy the conditions specified above for Y, x , and hence Kl# = Y1 x-?P 1 1 = N!P1-lx-ll-lYfN*. In particular, if x = x ( t ) for t E I and Y is independent of t, then K # ( t )= Y x - l ( t ) Y * -+ 0 as t + b, if and only if x-l(t) + O as t + b o ; moreover, if N is also independent of t then K l # ( t ) 0 as t + b, if and only if K # ( t )-+ 0 as t + b,. From these remarks and the results of Section 2 following Lemma 2.1 we have the additional results. -+

Corollary 1

s(t,s I Ysbo)is nonsingular for t # s, and s - l ( t , s 1 Ysbo)

+

0 as t

b,.

-+

The following results are ready consequences of Thecxems 4.1 and 4.2, and will be employed to obtain more precise results on the behavior of hermitian solutions of (1.2). Theorem 8.4

If (1.3[) is an hermitian differential system satisfying hypotheses (6')and &,{I} on I = (a,, b,), and which is disconjugate on a subintmal I, = (a, b,), then: (a) if W,(t) is an hermitian solution of (1.2) on a subinteroal [s, b,) of I,, and W ( t ) is the solution of (1.2) satisfying W(s)= W,(s) r, where r 2 0, then throughout [s, b,) the solution W ( t )exists and satisfies W ( t ) - Wo(t)2 0 ; (b) if in addition to the abooe hypotheses the system (1.3J is identically normal on (a,, b,), and Wbo(t)is the distinguished solution of (1.2) at b, ,

+

then for r an hermitian matrix that fails to be nonnegatioe definite the solution W ( t )of (1.2) which satisfies the initial condition W(s)= W,(s) r does not exist throughout [s, b,).

+

IV. HERMITIAN EQUATIONS

160

Now suppose that (1.31) satisfies hypothesis $ j N ( a 0b,), , and (2.7) is related to this system by the transformation (2.6) with T ( t ) = Z*-l(t), with Z ( t ) a fundamental matrix solution of Z’ A*(t)Z = 0 such that the last d column vectors of Z ( t ) form a basis for A(a,, b,). Then the resulting truncated preferred reduced system (2.10) is disconjugate and identically normal on ( a , , b,), so that the above theorem is applicable to this system. These results, together with the corresponding results for distinguished solutions of (1.2) at a , , yield the following conclusions.

+

Theorem 8.5

Suppose that an hermitian system (1.31) satisjies hypothesis $ j N ( a ,b,) , and is disconjugate on ( a , , b,). For r E (a,, b,) let the solution of a corresponding truncated preferred reduced matrix system (2. l o M )satisfying the initial conditions H ( r ) = 0, Z ( r ) = En-d be denoted by H,(t), Zr(t). Then H,(t) is nonsingular for t # r, the matrix function Q,(t) = Z,(t)H;l(t) is such that Qbo(t)= limr+bof2r(t)and QUo(t) = lim,,uo~r(t)exist, are the distinguished solutions of (2.11) at b, and a,, respectively, andpossess the following properties : (i) I f Q ( t ) is an hermitian solution of (2.11) which exists on (u,, b,), then Q ( t ) - Qbo(t)2 0 and Quo(t)- Q ( t ) 2 0 throughout (a,, b,) while if Q ( t ) is an hermitian solution of (2.11) for which at some value s the matrix Q(s) - QbO(s),{Qa0(s) - Q(s)} fails to be nonnegative dejinite then Q ( t ) does not exist throughout the interad [s, b,) {(a,, s]}. @ ) I f also > 0for t a.e. on (a,, b,), then Qbo(t)5 0 and Q u o ( t ) 2 0 for t E ( a , , bo).

e,,(t)

9.

Hermitian Systems with Constant Coefficients

If the coefficient matrix functions A , B, C of an hermitian differential system (1.j1) are constant, and Y ( t )= ( U ( t ) ;V(t))is a solution of (1.3M), then Y ( t - c) = ( U ( t - c ) ; V ( t - c)) is also a solution for arbitrary real values c. Consequently, (1.31) is disconjugate on an interval ( a , co) or (-03, a ) if and only if it is disconjugate on the whole infinite line (-m, a). Moreover, if Y,(t) = (U,(t); V , ( t ) ) is the solution of (1.3M) satisfying U,(r) = 0, V,(r) = E, then

and the corresponding solution W r ( t )= V,(t)U;l(t) of (1.2) exists on

9. SYSTEMS WITH CONSTANT COEFFICIENTS

161

a given interval [c, d ] if and only if W,(t) = Vs(t)U;l(t) exists on [c s - I , d s - I ] . Also, from the general result of Lemma I1:6.1 we have the following algebraic criterion for identical normality.

+

+

Lemma 9.1

I f for an hermitian dzflerential system (1.31) there are constant matrices A and 8, of respective dimensions n x n and n x s such that A ( t ) = A Yl and B ( t ) = 8," on (-m, m), then (I.&) is identically normal on (--00, -00) if and only if the n x ns matrix

is of rank n. I n particular, if A ( t )and B ( t ) are both constant on (-CO, m), then 13, may be chosen as B, in which case (9.1) becomes the n x n2 matrix (9.1')

[B AB A2B

A"-'B].

For systems with constant coefficient matrix functions the following result is a consequence of Theorem 8.5. Theorem 9.1

A n hermitian system (1.31) with constant matrix coeficients, and which is identically normal and disconjugate on (-03, m), has a principal solution at co, (at -co} if and only if the solution Yo(t)= ( U o ( t ) ;Vo(t)) of (1.3M)satiifying the initiaZ conditions Uo(0)= 0, Vo(0)= E is such that Wo(t)= Vo(t)U(;'(t)converges to a limit W , {W-,} as t + - m { t - o o } ; the corresponding distinguished solution of (1.2) at co {at -GO} is W,(t) = (W-,(t) = W-,).

w,

Furthermore, Theorems 8.4 and 8.5 imply the following results for systems with constant coefficients. Theorem 9.2

A n hermitian system (1.S1) with constant coeficient matrices satisfying = B 2 0, and which is identically normal, is disconjugate on (-m, -00) if and only if there exists an hermitian constant matrix W satisfring the algebraic equation

C" = C, B"

(9.2)

WA + A" W

+ WBW - C = 0.

162

IV. HERMITIAN EQUATIONS

Moreover, if such a system is disconjugate on (-m, m) then there exist hermitian matrices W , and W-, which are individually solutions of (9.2), and are extreme solutions for (1.2) in the sense that if W ( t )is any hermitian solution of (1.2) on (-00,oo) then W,< W ( t ) < W,, for t E (-00,m); in particular, i f W is any hermitian solution of the algebraic matrix equation (9.2), then W , 5 W 5 W-,.

If B and C are constant matrices the differential system -v'(t)

(9.3)

+ C u ( t ) = 0,

d ( t )- Bv(t)= 0

is identically normal on (-co, m) if and only if B is nonsingular, and consequently the following result follows immediately from the above theorem. Corollary

If B and C are constant hermitian matrices with B > 0, then (9.3) is disconjugate on (-00, co) ;f and only if C 2 0, and whenever this latter condition holds the extreme solutions of

WBW - c = 0

(9.2') are given by (9.4)

W,

=

W-, = - W ,

-B--1/2[B1'2CB1/2]1/2B-l/2,

.

It is to be remarked that this corollary provides a differential equation algorithm for the nonnegative definite square root matrix of a nonnegative definite matrix C; namely,

(9.5 1

C1/2 = lim Y,(t)U,-'(t)

=

t-kO

- lim Y,(t)U;'(t), t+-w

where Y&t)= (U,(t); Y o ( t ) )is the solution of the matrix differential system U ' ( t ) = V(t), V'(t)= C U ( t ) satisfying the initial conditions U(0) = 0, Y ( 0 )= E. 10. A Matrix Method of Atkinson

For the discussion of oscillatory properties of solutions of an hermitian system (1.31)in reduced form with A ( t ) = 0, and the study of associated two-point boundary problems, Atkinson [l, Chap. 101 has employed a

10. A MATRIX METHOD OF ATKINSON

163

procedure, which he refers to as a “matrix method.” This method is materially different from that presented in earlier sections of this chapter, although there are various interrelations between the methods. The results of the matrix method which are basic for the study of oscillation phenomena for an hermitian system (1 .31)are summarized in the following theorem, using the notation and terminology of this chapter.

Theorem 10.1

If (1.3,) is an hermitian direrentid system satisfying hypothesis (8) and Y ( t )= ( U ( t ) ;V ( t ) )is a conjoined basis for this dzfferential system, then : (i) the matrix functions

O(t) = ~

(10.1)

( t) iU(t),

P(t)= ~

+

( t ) iU(t)

are nonsingular for t E I ; (ii) the matrix function

(10.2)

e(t

I

Y )=

P(t)O-yt)

is unitary for t E I ; (iii) t = t is a focal point of the conjoined basis Y ( t ) if and only if 1 = 1 is a proper value of O(t I Y); (iv) O ( t ) = O(t Y ) is a solution of the di@mntiaZ equation

I

(10.3)

O ’ ( t ) = iO(t)N(t

I

Y)

I

where N ( t Y ) is the hermitian matrix function

(10.4)

N(t I Y ) = -20-’(t)Y*(t)%(t)Y(t)O-’(t),

and % ( t )is deJined by (2.2). Since Y ( t )is a conjoined basis for (1.31)we have

U * ( t ) V ( t )- V * ( t ) U ( t )= 0 arid

U * ( t ) U ( t )+ V*(t)V(t)> 0

for t E I,

and conclusion (i) is a ready consequence of the identities

(10.5)

O*(t)O(t) = [ ~ “ ( + t ) i ~ * ( t ) ] [ V (t )iU(t)] =

+

V*(t)V(t) U * ( t ) U ( t ) ;

IV. HERMITIAN EQUATIONS

164

(10.6)

+

P * ( t ) P ( t )= [V*(t)- i U * ( t ) ] [ V ( t ) iU(t)] = V*(t)V(t) U * ( t ) U ( t ) .

+

Also, from relations (10.5) and (10.6) we have ??*(t)O(t)= V*(t)P(t), so that E = u * - l ( t ) P * ( t ) P ( t ) u - l ( t= ) O*(t I Y)O(t I Y)and O(t I Y)is unitary for t E I. I n particular, the unitary nature of O ( t I Y) implies that for t E Z all proper values of O ( t I Y) are of absolute value one. Moreover, since t = t is a focal point of Y ( t )if and only if U ( t )is singular, conclusion (iii) is a ready consequence of the identity [O(t

I

Y ) - AE]t = [(l

-

+ i(1 + A ) U ( t ) ] i ,

A)V(t)

i = ??-l(t)t. Finally, by direct computation it follows that O(t) I Y) satisfies the differential equation (10.3) with iN(t I Y) = { [ O * ( t I Y ) - E ] V ' ( t ) + i[O*(t I Y ) + E ] U ' ( t ) } O - ' ( t ) ,

for

= O(t

=

O*-'(t){[P*(t)- U * ( t ) ] ~ ( + t ) i[P*(t) x U-'(t),

+ O*(t)]U'(t)}

I

from which the expression (10.4) for N(t Y) is immediate. It is to be noted that in terms of the matrix functions Q(t, @, Y ) and R ( t ) appearing in the trigonometric transform (2.21) of the conjoined basis Y ( t ) as defined by equation (2.15) we have also (10.7)

N ( t I Y)= 2U*-'(t)R*(t)Q(t;

@, Y ) R ( t ) U - ' ( t ) .

Moreover, since the last member of (10.5) is equal to R * ( t ) R ( t )we have E = O*-'(t)R#(t)R(t)O-'(t), so that the matrix function R(t)O-'(t) is unitary for t E I. Since (10.7) states that N(t I Y) and 2Q(t; @, Y ) are unitarily similar we have that these matrix functions have equal traces for t E I. In particular, the basic determinantal formula (10.8)

det O(t I Y) = det O(s I Y)exp for

( t ,s) E I x I ,

may be written equally well as (10.9)

det O ( t I Y)= det O(s

I Y)exp

T r Q(Y;@, P) dr for

( t ,s)

E

I x I.

As the formula (10.8) is of central importance for Atkinson's matrix

10. A MATRIX METHOD OF ATKINSON

165

method, the import of the above remark is that for many of the results derived by his method there is a ready connection with equivalent results obtained through the use of the trigonometric transform. In this connection, the reader is referred to Probs. 13, 14, 15 of Sec. VII.5, and Prob. 4 of Sec. VII.7, of Reid [ 1 5 ] . I n regard to the above function O(t [ Y ) it is to be remarked that if U ( t )is nonsingular for t on a subinterval I,, of I , then on this subinterval O(t I Y )is the Cayley transform [- V(t)U-l(t)-iiEj[- V(t)U-l(t)+iE]-' of the hermitian matrix function - V ( t )U-l(t). Correspondingly, if V ( t ) is nonsingular on a subinterval I,, , then on this subinterval e(t I Y ) is the negative of the Cayley transform [ U ( t ) V - l ( t )- iE][U(t)V-'(t) iE1-l of the hermitian matrix function U ( t ) V - l ( t ) . Moreover, although the differential equation (10.3) highlights the unitary character of O(t I Y ) , it is to be noted that this matrix function is also a solution of a Riccati matrix differential equation. Indeed, in view of ( 1 . 3 1 ) it may be verified readily that P ( t ) = (f)(t); P ( t ) ) is a solution of the differential system

+

(10.10)

-P'(t)

+ C ( t ) U ( t )- d(t)P(t)= 0,

O(t)- A(t)O(t)- B ( t ) P ( t )= 0,

where

A(t) = H{i[C(t)- B ( t ) ] (10.11)

+ A ( t )- A*(t)},

B(t) = ${- i[C(t)+ B ( t ) ] - A ( t )- A*(t)},

+

C(t) = B(i[C(t) B ( t ) ] - A ( t )- A*(t)},

B ( t ) = i { i [ C ( t )- B ( t ) ] - A ( t ) + A*(t)}.

Consequently, we have that O(t) = e(t I Y )is a solution of the corresponding Riccati matrix differential equation

I n the study of two-point boundary problems Atkinson [ l , Sec. 10.71 also utilizes a Riccati matrix differential equation. Again, discussion is limited to formulation of the problem with indication of the fashion of occurrence of the Riccati equation, and the notation used is at variance with that of Atkinson so that certain resulting differential systems are in somewhat agreement with earlier notations of the present work.

166

IV. HERMITIAN EQUATIONS

The boundary problem involves a first-order vector differential equation (10.13)

Sf ' ( t )- R ( t ) f ( t )= 0,

t E I,

where : (10.14)

(a) S is a complex-valued k x k nonsingular skew-hermitian matrix, i.e., S* = - S ; (b) R ( t ) is a k x k hermitian matrix function, which is locally Lebesgue integrable on I.

Associated with (10.13) are two-point boundary conditions at t = a and t = t, presented parametrically as (10.15)

f ( a ) = PE,

f(t) =

Qt,

where P and Q are k x k matrices satisfying the following conditions:

( 10.16)

(a) P and Q have no common null vectors; i.e., the k X 2k matrix [P* Q*] is of rank k ; (b) the matrices S, P, Q satisfy the condition P"SP - Q"SQ = 0.

Conditions (10.14) imply that the vector differential operator S'-R(t)f is formally self-adjoint, and conditions (10.14) and (10.16) specify that the boundary problem (10.13), (10.15) is self-adjoint. In this connection the reader is referred to Reid [15, Chapter 1111. Now let F ( t ) be a solution of the differential system (10.17)

S F ' ( t ) - R ( t ) F ( t )= 0,

F(u) = E k ,

and define the k x k matrix functions U ( t ) , V ( t ) as (10.18)

U ( t )= F ( t ) P

+ Q,

V ( t )= S [ F ( t ) P- Q].

It is to be commented specifically that in (10.18) the symbols U and V do not agree with their usage in the corresponding equations (10.7.4) and (10.7.5) of Atkinson [l], but are interchanged. In view of condition (10.14), we have that F*(t)SF(t) = F*(a)SF(a)= S, and by direct computation it follows that Y ( t ) = ( U ( t ) ;V ( t ) ) is a solution of the

11. NOTES AND REMARKS

167

hermitian matrix differential system (1.3M) with (10.19) A ( t )= - iS*-'R(t), B ( t ) = - 4S*-'R(t)S-',

C ( t ) = +R(t).

Moreover, in view of the identity F*(t)SF(t) = S it follows by direct computation that (10.20)

U*(t)V(t)- V*(t)U(t)= 2[P*SP - p S Q 3 = 0.

That is, Y ( t )= ( U ( t ) ;V ( t ) )is a conjoined basis for the Hamiltonian system (1.31) with coefficient matrix functions given by (10.19). Consequently, on any subinterval where U ( t ) is nonsingular the hermitian matrix function m(t)= V(t)U-l(t)is a solution of an hermitian Riccati matrix differential equation, which may be written as

11. Notes and Remarks

The major portion of the subject matter of this chapter is devoted to properties of Riccati matrix differential equations that are related to the Sturmian theory for linear Hamiltonian systems which are hermitian in the sense defined in Chapter 111. Consequently, these results have many of their historical roots in the basic work of Morse [l, 21, which showed that variational principles provided the proper environment for the extension to self-adjoint differential systems of the classical Sturmian theory for linear homogeneous ordinary differential equations of the second order. The treatments of Atkinson [l, Chap. 101, Hartman [2, Chap. XI], and Reid [15, Chaps. V, VI, and VII] provide general references of relevant material. For individual sections, specific references are as follows. REFERENCES

Sec. 2. Atkinson [l, Chap. lo], Barrett [l, 21, Etgen [l, 21, Hartman [2, Chap. XI, Appendix], Radon [l, 21, Reid [4, 13; 15, Sec. 2 of Chap. VII].

Sec. 3. Hartman [l ; 2, Chap. XI, Appendix], Reid [3, 7, 8; 15, Secs. 3, 5 of Chap.

VII], Sandor [l]. For the case of real, second-order linear differential equations, see Hartman and Wintner [l], Leighton [l], and Leighton and Morse [l]. Sec. 4. Ahlbrandt [l],Bucy [l, 21, Bucy and Joseph [l], Reid [2, 7, 131, Sandor [l]. Sec. 5. Aoki [l], Bellman [l, 41, Kalaba [l].

168

IV. HERMITIAN EQUATIONS

Sec. 6. Jackson [l], Reid [14], Wonham [l]. See. 7. Ahlbrandt [l], Hartman [2, Chap. XI, Appendix], Reid [7, 8, 11; 15, Secs.

4, 6 of Chap. VII], Sternberg [l, 21. For the important instance of accessory differential systems for simple integral variational problems, see Bliss [l , Secs. 11, 23, and 811. Sec. 8. Hartman [ l ; 2, Chap. XI, Appendix], Reid [a, 7, 8, 14; 15, Secs. 3, 5 of Chap. VII], Sandor [l]. For the case of real, second-order linear differential equations, see Hartman and Wintner [l], Leighton [l], and Leighton and Morse [l]. Sec. 9. Reid [7, Sec. 9; 91. Sec. 10. Atkinson [l, Ch. 101, Coppel [l], Etgen [3-51, Reid [13; 15, Chap. VII, Probs. 13-15 of Sec. 5 and Prob. 4 of Sec. 71.

APPLICATIONS OF RlCCATl MATRIX DIFFERENTIAL EQUATIONS

1.

Introduction

Many of the results of Chapters 11, 111, and IV may properly be classified as applications of Riccati matrix differential equations to the study of the qualitative nature of solutions of various types of differential systems. In particular, many such applications to related linear differential systems center around the oscillatory or nonoscillatory (disconjugate) nature of solutions. The present chapter is devoted to the consideration of various instances of the occurrence of Riccati matrix differential equations which in large measure are outside the realm of mathematics per se. In most cases, explicit detail of discussion is limited to a level that will afford an introductory knowledge of the problem setting, and a brief account of obtained results as related to the theory of Riccati equations as presented in the preceding chapters. There is deviation from this procedure in the case of the linear regulator problem treated in Sec. 5 , however, since this type of problem is prevalent in many areas of application. It is to be emphasized that no claim is made as to the comprehensive nature of the coverage afforded by these examples, nor the completeness of the associated list of references in the Bibliography. 2.

Direct Applications

Some instances are concerned with phenomena which in their mathematical formulation lead to direct applications of some of the mathematical theory as presented in the earlier chapters. Such occurs, for 169

170

V. APPLICATIONS

example, in the paper by Kaufman and Sternberg [l] on multiple nonuniform transmission lines in steady state which, in particular, extends earlier results of Rice [l] on multiple uniform transmission lines in steady state. Specifically, the authors treat a multiple nonuniform transmission line consisting of n wires with ground, or additional single wire return, extending from x = a to x = b, 0 5 a < b 5 00. The differential equations of the system in steady state are written in the form (2.1)

- v ’ ( x ) - Z ( x ) u ( x ) = 0, u’(x)

+

Y(.).(X)

= 0,

) (uu(x)), w ( x ) = ( v u ( x ) ) (a = 1, . . . , n) are the complex where ~ ( x 3 current and voltage vector functions with the actual instantaneous currents and voltages the real vector functions

(2.2) i(x, t) = %e[u(x) exp{iot}],

e(x, t) = % e [ v ( x ) exp{iwt}],

and where Y(x) and Z(x) are, respectively, the “specific shunt admittance” and “specific series impedance” n x n matrix functions (2.3 1

Y(x) = G(x)

+ iwC(x),

Z(X) = R ( x )

+ iwL(x).

In (2.3) it is supposed that G(x), C(x), R ( x ) , L ( x ) are real-valued, continuous, symmetric matrix functions on [a, b] ; when all these matrix functions are constant, the system reduces to that considered by Rice. In the above problem description the notation of the authors has been retained, except for denoting the pure imaginary unit by “i” instead of by “j,” and substituting i(x, t), e(x, t), Y(x), Z(x) for i ( x , t), e(x, t ) , Y(X),+). I n notation similar to that employed in the earlier chapters, we shall write the system (2.1) in terms of the 2n-dimensional vector function A t ) = ( y m ) ( 0 = 1, * * , 2n), with Y&) = u&), m+&) = (a = 1, . . . ,n), as (2.1’) where

Z Y ’ ( 4 + %(xlr(x)= 0-

3 and %(x) are the 2n x

2n matrix functions

As 2 is skew, and %(x) is complex symmetric, in the sense of Section

2. DIRECT APPLICATIONS

171

1II:l the system (2.1) is a-symmetric, with 0= -. In particular, if yl(x) = ( u I ( x ) ;v z ( x ) ) and y 2 ( x ) = ( u , ( x ) ; v z ( x ) ) are solutions of (2.1’) then (2.5 )

Y z ( x ) 2 Y l ( x ) = W).,(.) - W ) v , ( x )

is such that [ j j z ( x ) ~ y l ( x ) = ] ’ 0, and hence Y p ( x ) ~ y l ( is x )constant on [a, b ] . Following the terminology used for similar real differential systems that are the Jacobi equations, or accessory differential systems, for a variational problem, the authors define two solutions of (2.1’)as conjugate if the value of the constant function (2.5) is zero. As emphasized in Chapter 111, the concept of two solutions of (2.1) being conjugate according to this definition is distinct from that of being conjoined as used for hermitian systems in Chapters I11 and IV, although the two concepts are the same for real-valued solutions of real self-adjoint systems of the form (2.1). Corresponding to (2.1‘) we have the matrix differential system

(2.1M’)

P’( + W.Y) ( x )= 0,

and if Y l ( x )and Yz(x)are solutions of this system, then the matrix function P , ( x ) ~ Y , ( xis) constant on [a, b ] . Also, if Y ( x )= ( U ( x ) ;V ( x ) )is a solution of (2.lnl’)then the condition P ( x ) z Y ( x )= 0 is the requirement that the matrix function P ( x ) U ( x ) be symmetric. If for a solution Y ( x )= ( U ( x ) ; V ( x ) )of (2.1M‘)we have that V ( x ) is nonsingular on a subinterval I , of [ a , b ] , then K ( x ) = V(x)U-’(x)is a matrix function which satisfies on I , the Riccati matrix differential equation

(2.6)

+

K ’ ( x )- K(x)Y(x)K(x) Z(x) = 0 ;

in particular, if P(x)$?’Y(x) = 0 then K ( x )= V(x)U-’(x)is a symmetric solution of (2.6) on I,. Similarly, if Y ( x )= ( U ( x ) ;V ( x ) )is a solution of (2.1M’) with V ( x ) nonsingular on a subinterval I , of [a, b] then H ( x ) = U(x)V-’(x)is a solution of the Riccati matrix differential equation

(2.7)

+

H ’ ( x ) - H ( x ) Z ( x ) H ( x ) Y(x) = 0

on I , and H ( x ) is symmetric if P ( x ) z Y ( x )= 0. If in (2.3) the matrices G(x) and R ( x ) are identically zero, then the multiple transmission line has been termed “lossless” by the authors. In this case, if the matrix functions C(x) and L ( x ) are nonsingular then

In

V. APPLICATIONS

(2.1) is equivalent to either of the following linear homogeneous secondorder vector differential equations (2.8) (2.9)

+ 0 2 L ( x ) u ( x )= 0 [L-’(x)v’(x)]’+ w2C(x)er(x) = 0

(current equation),

[C-l(x)u‘(x)]’

(voltage equation).

If (2.9) is disconjugate on [a, b] then the lossless multiple line is said to be “nonnodal with respect to voltages at the angular frequency w.” Similarly, whenever (2.8) is disconjugate on [a, b] then the lossless multiple line is said to be “nonnodal with respect to currents at the angular frequency 0.”I n particular, if C(x) and L(x) are symmetric matrix functions with L ( x ) positive definite then in view of Theorem IV:7.1 we have that (2.9) is disconjugate on [a, b] if and only if on this interval there exists a real symmetric solution W ( x )of the Riccati matrix differential equation (2.10)

+ W(x)L(x)W(x)+ wZC(x) = 0.

W‘(x)

Correspondingly, if C(x) and L ( x ) are symmetric matrix functions with C(x) positive definite then (2.8) is disconjugate on [a, b] if and only if on this interval there exists a real symmetric solution Wl(x) of the Riccati matrix differential equation (2.11)

+

+

W1’(x) W1(x)C(x)W1(x) o2L(x) = 0.

Expressed in terms of H ( x ) = io-lW and K = iw-’W1, in the language of Kaufman and Sternberg we have the following result. AJinite lossless multiple line is nonnodal with respect to voltages {currents}, at the angular frequency o if and only i f there exists a pure imaginary symmetric matrix function H = H ( x ) { K = K ( x ) } , satisfying the Riccati matrix dif/erential equation

+

H ’ ( x ) - i o H ( x ) L ( x ) H ( x ) i o C ( x ) = 0,

+ i o L ( x ) = O}.

{ K ’ ( x )- i o K ( x ) C ( x ) K ( x ) 3.

Applications Arising from Partial Differential Equations

Consider the linear partial differential equation n

3. PARTIAL DIFFERENTIAL EQUATIONS

173

in (t, 6) = ( t , E l , . . . , En), where Tus(t), Sus(t),&(t) are real-valued functions on an interval I on the real line and

(3.2)

(a, = 1, . . . , n).

Pus(t)3 Psu(t)

If ye denotes the vector function (yau)(a = 1, . . . , n), and yet the n x n matrix function [ytuas],then in matrix notation, the partial differential equation (3.1) may be written

and (3.2) requires the real-valued matrix function P ( t ) to be symmetric for t E I. Now one may seek a solution of (3.1) of the form

with 1;1 an n-dimensional vector (qu) (a = 1,

+(t,E , 7) = -il[fj@(t)q

(3.4)

. . . , n), and

+ 2fjH(t)E + fW(t)EI,

where @(t),H ( t ) , and W ( t )are n x n matrix functions with @(t)3 &(t) and W ( t )= m(t)on I. For a solution of the form (3.3) and (3.4), we have

~ ( 06,) = eo e x ~ { + d Eq)), ,

(3.5)

where q50(E, q) is of the form

and

+ 2fjHOE +

+o(E, q) = -il[fj@oq

(3.6)

W O E 1 9

Woare real symmetric matrices. Now

and consequently by direct computation it may be verified that a y of the form (3.3), (3.4) is a solution of (3.1) if and only if

(3.8)

e' ~

+

4 - !~[fj@'(t)q

X P

+

+

r?H'(t)~ W ( t ) q lw'(t)~] [ f j H ( t ) ew(t)]s(t)E - Ty { P ( t )W(t)>+{ fjH(t)+FW(t)}P(t) {&t)7+ W(t)E1).

= y ( f T ( t ) E-

+

In particular, there exists a solution of (3.1) of the form (3.3), (3.4)

174

V. APPLICATIONS

which satisfies the initial conditions (3.5) and (3.6) if W ( t )is a symmetric solution of the Riccati matrix differential equation (3.9a)

+

+

--3wf(t)= + [ ~ ( t )p(t)I - i[w(t)S(t) s(t)W(t)I

+ W(t)P(t)W(t),

with W(0)= W o , while H ( t ) and @(t)satisfy the matrix differential equations

+ H(t)P(t )W(t ),

(3.9b)

-i H ' ( t ) = -+ H ( t ) S ( t )

(3.9c)

-&@'(t) = H ( t ) P ( t ) A ( t ) ,

H(O) = H , ,

@(O) = @o,

and e ( t ) is the solution of the scalar differential equation

e'

(3.9d)

-

=

-e

Tr{P(t)W(t)}.

I t may be verified readily that W(t), H ( t ) , G ( t )= A(t), and F ( t ) -@(t) form a solution of the differential system

where

(3.11)

A ( t )= -S(t),

B ( t ) = 2P(t),

C ( t )= - [ T ( t )

+ T(t)].

That is, the thus defined matrix functions W(t),H ( t ) , G(t),F ( t ) form a solution of a differential system of the form (I1 :3.4), with real coefficients and D ( t ) = act). From (3.10~)we have that d(x) = det H ( x ) satisfies the differential equation d'(x) = - [Tr A ( x )

+ T r { B ( x ) W ( x ) }d(x). ]

Moreover, Eq. (3.9d) implies that

3. PARTIAL DIFFERENTIAL EQUATIONS

175

and, therefore, e-2(x)d (x) = ei2det Ho exp{ -

s' 0

T r A(s)ds} .

I n particular, if Ho is nonsingular, so that d(x) = det H ( x ) is nonzero for all x, we have that e(x) = e o

[ detdetHH( x, ) 1,'

exp{ 0

T r A(s)ds} .

An instance of a Riccati matrix differential system appearing in the context described above is treated in the paper of Siegert [l, Part 1111 concerned with a class of problems in the theory of noise and other random phenomena. Specifically, an n-dimensional Gaussian process X ( t ) = { x l ( t ) , . . . ,x n ( t ) }with mean zero is described by the correlation matrix R(t) with components ~

~

= ~( x k (( t )tx i ) (t

+

for

t)>Av

t

2 0;

R(-t),

~ ( t =)

and for a stationary Markoffian Gaussian process the matrix R ( t ) has the form

R ( t ) = exp{Qt}

for

t

2 0 if R(0) = E.

T o obtain the characteristic function for the distribution of the functional rt

(3.12)

n

where K ( t) = [K,,(t ) ] is a symmetric matrix, we define the characteristic function for the joint distribution of X(O), X ( t ) , and u,

(3.13)

+(%,

- - ,rln; tl, *

* * *

, ttl,t, A)

where il is taken to be real if 5 > 0, and negative imaginary otherwise. The function P then is shown to satisfy the partial differential equation

and the initial condition at t = 0 for i is

(3.15)

V. APPLICATIONS

176

The differential equation (3.14) is then of the form (3.1) with (3.16)

Y=+,

T=Q,

S=Q,

P=IK.

Moreover, in view of (3.13), for the corresponding system (3.1) we have that Q o = H o = w o =E, e o = 1. The system (17a-d) of Seigert’s paper should then be the above system (3.9a-d) with T, S, P defined by (3.16), and W = c, H = b, Q = a, and =f; his equation (17c) should have been h = -2ilbK8, the error entering through an interchange of subscripts in the preceding substitution equation (16). 4.

An Introduction t o Hamilton-jacobi Theory

Attention will be limited to a concise formulation of a nonparametric variational problem of Lagrange, or Bolza, type in the most classical variational setting, with statements on the pertinent results of the Hamilton-Jacobi theory, and an indication of how the study of real hermitian Riccati matrix differential equations is related to this theory. For a more elaborate discussion of Hamilton-Jacobi theory, the reader is referred to books on the calculus of variations. I n particular, for various simple integral variational problems this topic is treated by Bliss [l ; Secs. 25-31, 38, 39, 75, S1, 911. For a treatment of Hamilton-Jacobi theory in “conirol theory formulation,” the reader is referred to Hestenes

PI.

The real-valued scalar functionf(t, x, p) = f ( t , x, , . . . ,x,, p , , . . . , p , ) and the m-dimensional real-valued vector function

4(t,x,p)=(4a(t,%

. . * , % l , p 1*, * . , p n ) )

(o.= 1, . * . , m < n )

are supposed to be of class C3 in an open region 9of real (2n dimensional (t, x, p)-space. If (4-11

c:x = x ( t ) ,

t,

+ 1)-

5 t 5 t,,

where the n-dimensional vector function x ( t ) = ( x a ( t ) ) (a = 1, . . . , n) is continuous, and has a piecewise continuous derivative x ’ ( t ) = (xa’(t)) on [tl , t 2 ] ,with (t, x ( t ) , x ’ ( t ) ) E 9 for t E [ t , , 4, then C is said to be a di#uentially admissible arc. I n this connection, it is to be noted that

in

4. HAMILTON-JACOB1 THEORY

if to E ( t l , t,) is a point at which x ’ ( t o ) does not exist, then the imposed condition requires the left- and right-hand derivatives, x’(to-) and %’(to+), to be such that both ( t o , x ( t o ) , x’(t,-)) and ( t o , x ( t o ) , x’(t,+)) are points of the open region 9. The variational problem to be considered is that of determining conditions that must be satisfied by a specific admissible arc C,: x = xo(t), t E [ t l , t,], that extremizes

in the class of differentiably admissible arcs (4.1) which satisfy the mdimensional vector differential equation

(4.31

#(t, 4 t h x’ct))

= 0,

t

E [tl

Y

4,

and join two given points ( t l , xl), (t,, x,). For A = (1,; A), where 1, is a real scalar and 1 = (Au) (a= 1, is a real-valued m-dimensional vector, we set

(4.4)

F(t, x , P, (1) = 1Of(C x, P )

. . . , m)

+ M t , x , PI.

By definition, an extremal for the above formulated variational problem is a pair E = {C; A ( t ) } ,where C is a differential admissible arc, 1, = con-

stant, with x ’ ( t ) and 1 ( t ) continuous, such that the vector function

Fp(t, x ( t ) , x ’ ( t ) , A(t)) = (Fp,(t, x ( t ) , x ’ ( t ) , .l(t))) (a = 1,

-

* 9

n),

is continuously differentiable, and the Eider-Lagrange dzflerential equations

(4.51

d F,(t, x ( t ) , x ‘ ( t ) , 4) -) -&- Fp(t,

x ’ ( t ) , 4 t ) ) = 0,

# ( t , 4 t h .’W) = 0,

hold along this arc. By the use of the implicit function theorem, it may be established that whenever the extremal E = {C; A ( t ) } is nonsingular in the sense that the (n m ) x (n m ) matrix

+

+

is nonsingular for all sets of arguments belonging to E, then the vector functions x ‘ ( t ) , 1 ( t ) have continuous derivatives of at least the first order.

V. APPLICATIONS

178

A (Mayer) Jield 5 for the above defined variational problem is a region d of (t, x)-space, together with slope functions y ( t , x) = (y,(t, x ) ) (a = 1, . . . ,n), and multiplier sets A ( t , x) = (Z,, Z(t, x ) ) with I , = constant and l(t, x ) = (Z@(t,x ) ) (/?= 1, . . . , m),such that (t, x, y ( t , x)) EST whenever (t, x) E A , the vector functions y ( t , x ) , Z(t, x) are of class C’ in d, the auxiliary relations (4.7)

4(t, x, y o ,

4)= 0,

for ( t , x ) E 4

hold, and the (Hilbert) integral 3* = 3*(t,6; t, x ) defined as (4.8)

3* =

(l.2)

[F - Rpy] dt

(r.8

+ f l p dx

is independent of the path in A, where in (4.8) the arguments of F and Fp are (t, x, y ( t , x), A(t, x ) ) . Moreover, if x ( t ) is a solution of the associated first-order vector differential equation (4.9 )

x’(t) = y(t, x ( t ) ) ,

1 E [tl,

t21,

with (t, x ( t ) ) E d for t E [ t l , t 2 ] ,then E = (C, Act)), with C defined by x ( t ) and A ( t ) = (A, = 1, A ( t ) = Z(t, x ( t ) ) ) is called a trajectory of the field; alternate terminologies are extremal arc or characteristic arc of the field. In view of the independence of path of integration, the function 3* = 3*(t,6; t, x) defined by (4.8) is seen to have continuous partial derivatives with respect to its arguments on d x A . Moreover, (4.10)

(a) 3,* =

[F- Ppy](t,z),

(b) 3,* = -[F - f ; a y ] ( 7 J ) ,

3,* = [ F p ] ( l . Z ) , at* = - [Fp](7*0,

where the superscripts denote the involved arguments ; for example,

In particular, we have the following derivative identities

179

4. HAMILTON-JACOB1 THEORY

Also, one may establish readily the following properties of trajectories of a Mayer field.

(C, A ( t ) ) , where C: x ( t ) , t E [ t l , t 2 ] , and A ( t ) 1, A ( t ) = l ( t , x ( t ) ) ) is u trajectory of a Mayer f e l d 5,then E is a solution of the Euler-Lagrange equations (4.5) ; moreuwer, 3(C) = 3*(C).

If E

('I

=

= ('lo=

+

Now let 9 be an open region in (2n 1)-dimensional ( t , x, y)-space such that if ( t , x, y ) E 9 then there is a unique solution

of the equations (4.13) with ( t , x, p ) €9, and such that the (n

+ m) x ( n + m) matrix

(4.14) is nonsingular. I n view of the initial assumptions on f and 4, it follows that the solutions (4.12) of (4.13) are of class C2in 9. For ( t , x , y ) E 9, the Hamiltoniun H ( t , x, y ) is defined as (4.15)

H ( t , x, y ) = {Pp(t,x, p , A)P - F ( t , X, p , jl)}p-P(',z,y).'-L(t,z,y) = p ( t , x, y ) - F(t, x, P ( t , x, Y ) , Ut,x, Y ) ) .

From the above stated properties we have the following results. 1. If

5 is a given M a y e r f e l d with slope functions y ( t , x) and A ( t , x) = (1, 3 1, 3, = l(t, x) = (I&,

multipliers

x))},

then the function W = W ( z ,5 ; t, x) = 3*(z, 5 ; t , x) dejined by (4.8), us a function of ( t , x) satisfies the equations (4.16)

Fp(4x, y ( t , x), A(t, x)) 4(t, x, y ( t ,

=

Wz(t,x),

4) = 0.

180

V. APPLICATIONS

is a solution of the Hamilton-Jacobi partial differential equation

(4.17)

W,

+ H ( t , x, W,) = 0.

(t,t

) we have

so that y ( t , 5 ) = P(r, 5, -Wt(t, 5)), l ( t , t )= L ( t , 5, -W&,

t)),and

In a similar fashion, it follows that as a function of

(4.18)

F&,

5 9

y ( r , t),4 t , 5 ) ) +(T,

(4.19)

- W t ( t , t),

=

6,y ( t , t ) )= 0,

Wr - H ( t , t, -W.) = 0.

I n passing, it is to be commented that frequently only one of the forms (4.17) or (4.19) is given for the Hamilton-Jacobi equation. In t,t),(t,E ) E A , is a soluone sense only one is required, since if W = tion of (4.19), then W = -@(t, x), ( t , x) E A is a solution of (4.17). However, the value of considering both equations has been appreciated for some time, especially in such areas as geometric optics. Jn this connection, see, for example, Synge [l ;Chap. 11,Sec. 5 and Chap. V, Sec. 211.

m(

2. For ( t , x, y ) E 9, the following partial derivatives of H have the given representations:

H,,(t, x, Y)= -F&

x, P(t, x, Y ) , Jqt, x, Y))9

Hy,(t, x, Y ) = Pa(t, x, Y ) , H,(t, x, Y ) = - q t , x, P ( t , x, y ) , u t , x, A)

9

Hgazp(t,x, Y)= Pazp 9 Hvau&t, x, Y)= PWP. 3. If E = (C, A(t)) is an extremal arc with C : x = x ( t ) , t,‘ I t 5 t2’, and A(t)= (1, = 1, A(t)=(&(t)) (u=l, . . . , m ) } , while y(t)=Fp(t, x ( t ) , x’(t), A ( t ) ) is such that (t, x ( t ) ,y ( t ) ) E 9for t E [tl’,t 2 ’ ] ,then on [tl’,tz‘] we have the relations

(4.20‘)

4 4 = L (t, x ( t ) ,A t ) ) r’(t)= F z ( t , x(t),p(t, x ( t ) ,r(‘)) L ( t , x ( t ) , rct>)) ’ x‘(t) = P ( t , x ( t ) , A t ) ),

9

J

4. HAMILTON-JACOB1 THEORY

181

I n particular, we have the following CANONICAL EQUATIONS OF THE ExTREMALS,

4. I f x = x ( t , d ) , y = y ( t , d ) is a parameter family of solutions of (4.21) of class C2 in ( t , d ) on an open neighborhood of values t , 5 t 5 t , , d = d o , then the vector functions

(4.22)

u!t) = %(t, do),

t

v ( t ) = Y d t , do),

E

[tl t21, 9

satisfy the linear differential system

where the arguments of the second order partial derivatives of H occurring in (4.23) are (t, x ( t , do),y ( t , do)). 5. For a Mayer jield 5 with slope functions y ( t , x) and multipliers A ( t , x) = {A, = 1, A = Z(t, x) = (Zp(t,x)) >,let x = @(t,c) = @,(t, c l , . . . ,c,J),

be an n-parameter family of trajectories of thejield of class C2in an open region of ( t , c) space, and where the parameters c l , . . . , c, have been so chosen that the n x n matrix @,(t, c) = [@,,(t, c)] (a,/3 = 1, . . . , n), is nonsingular for some initial value o f t . Then the vector functions

(4.24)

@(t,c), Y ( t , c) = q t , @(t,c), y ( t , @(t,c)), A ( t , @(t,4))

satisfy the following conditions. (a) The matrix functions 0, = 0,(t, cj, Y, = Y,(t, c ) are solutions of the linear dt&ential system

(4.25)

-.!P,' - Hz.O, - HwY,

= 0,

0,'- HYz@,- HmYC= 0,

where the arguments of the partial derivatives of H in (4.25) are ( t , @(t,c),

Y(t,c)).

( b ) 0,(t, c) is nomingular for t E [ t l ,t,]. ( c )Since for,fixed (zo, l o )the function W ( t ,x) the relation

w(c@ ( t ,4) = Fp(t, @(t,c), y ( t , @(t,c)),

=3

* ( ~l o~;t,, x) satisfies

A ( t , @(t,4)= Y(t,c),

182

V. APPLICATIONS

it follows that (4.26)

w,(t,

@(t,c)) = YJc(t,C ) % l ( t ,

c).

(d) A s a function of t the matrix function S(t,c) = W,(t, @(t,c)) is a real symmetric solution of the Riccati matrix differential equation

(4.27)

8'

+ Ha$ + SH,,

4-SH,,S

+ Hzz

= 0,

where the arguments of the partial derivativesof H in (4.27) are (t, @(t,c), q t , 4)' If E = {C,A ( t ) } , with C : x = x ( t ) and A ( t ) = { I , = 1, I,(t)}, t, 5 t 5 t,, is an extremal, let

(4.28)

4 t , 'I,n) = 4{71"[Fzz'I+ FGl

+

+

ji[FPZ'I

+ Fpp.7zI1,

@(t, 'I,n) = &'I 4p9

where the arguments of the partial derivatives of F and 4 in (4.28) are the t , x ( t ) , x'(t), A ( t ) of E. The second variation, or accessory, problem for the functional (4.2) at the extremal E involves the quadratic functional

(4.29) for arcs I-': 7 = q ( t ) , t E [t,, t,], with the vector function q ( t ) = ( V J t ) ) , (a = 1, . . . , n ) , continuous and piecewise continuously differentiable, and subject to the vector differential equation

@(t, ' I ( t ) ,'I'(0) = 0.

(4.30) as

If the matrix (4.6) is nonsingular along E, and Q ( t , 7 , n,p ) is defined Q ( t , 'I,n,p ) = o ( t , 'I,n)

+ ii@(t,'I,n),

then the system

(4.31) is linear in n = (nJ,p = (p,,), and has unique solutions

4. HAMILTON-JACOB1 THEORY

183

and the corresponding Hamiltonian function %(t, q, 5 ) is given by

(4.33) %(t, q, 5 ) = {jiQ*(t, q, n,p ) - Q ( t , q, n,~ ) ) n = n ( t , t l , C ) , I r = M ( t . t l , c ) =

+

S{7?[Hzzl;l Hm51

+ t[H,zr + H,,51>.

For the above defined accessory problem the Hamilton-Jacobi partial differential equation corresponding to (4.17) for the original problem is then

Moreover, for the accessory problem the canonical equations q1 = zc,

(4.350)

5'

=

-2q

for the extremals are

(4.35) The following properties are ready consequences of the above formulas and definitions. 6. The canonical equations for the extremals of the accessory problem are the equations of variation of the canonical equations for the extremals of the original problem.

7. I f S ( t ) is a real symmetric n x n matrix function, then Y ( t , q ) &fjS'(t)qis a solution of the Hamilton-Jacobi partial diflerential equation (4.34) for the accessory problem if and only if S ( t ) is a solution of the Riccati matrix diflerential equation (4.27). For the variational integral (4.2) subject to the differential equation restraint (4.3), the Weierstrass excess function involving the vector multiplier A = { A o , A = (A8) (p = 1, . . . , m ) } , is =

(4.36)

x, p , r, A ) = F(t, x, r, A ) - F(t, x, P, A ) - i q t , x , P , A ) ( r - PI.

For a Mayer field 5 with slope function y ( t , x ) and multiplier A ( t , x) = {lo = 1 , 1 = (lp(t,x ) ) } , and ( t o ,xo) a fixed point of A , let J$' (t,t)be defined as equal to 3*(t , t; t o ,x,) ; that is,

184

V. APPLICATIONS

(t,E ) E d, where arguments of F and Fp in (4.37) are (t, x, ~ ( tx), , A ( t , x)). If the Weierstrass excess function satisfies the condition

for

(4.38)

x, r(t,x), I, 4)> 0 for all (t, x, Y) E 9 satisfying

( t , x, I) # (t, x, y ( t , .I),

$(t, x, y ) = 0 and then for an admissible arc C: x ( t ) , (4.3) we have

tl 5

t 5 t2lying in d, and satisfying

Moreover, equality in (4.39) holds if and only if C is an arc of a trajectory of the field joining (tl , x(tl)) to (t2,x ( t 2 ) ) . In particular, the d , y ( t , x), A ( t , x), m(t,x) thus defined are the elements of a particular optimal j e l d , as introduced by Hestenes [l ; Chap. 3, Sec. 11 ; Chap. 6, Sec. 101. Moreover, as emphasized by Hestenes [ l ; Chap. 61 the optimality principle of the general theory of optimal fields is the basis for the theory of dynamic programming as developed by Bellman [l]. 5.

A Linear Regulator Problem

As an example of the occurrence of Riccati matrix differential equations in the variational context of optimal control, we shall consider the linear regular problem as formulated by Kalman [I]; see also Kalman et al. [l]. It is to be commented that in order for certain differential systems of the treatment to agree in notation with that used in preceding chapters, the notation employed is at variance with that used by Kalman. The model studied is represented by a matrix system of the form

where x ( t ) , y ( t ) , and h(t) are vector functions of respective dimensions n, k, and Y , and all matrix coeficient functions and vector functions are assumed to be real-valued. I n the terminology of Kalman, system (5.1) is a “plant” or “model” involving the components x of “state,” y of control” or “input,” and h of “output”; also, for brevity in notation, a pair ( t o ,xo) is called a “phase.” 6


W@il;l(t)

t0

where @

(5.10)

2

diag{Q, O k k ) ,

q t ) = diag{Onn, H ( t ) ) ,

9 ( t ) = diag{S(t), O k k } . For this variational problem the Lagrangian involving the k-dimensional multiplier vector ,u may be written as

j’H(t)p‘ + x”S(t)x+ 2 P [ d - G(t)p‘- F ( ~ ) x ] ,

and if (y, 4) denotes the canonical variables corresponding to ( x , p ) we have P =Y* (5.11) H(t)p‘ = 4,

W),

X‘

- G(t)pr- F ( ~ )= x 0.

187

5. A LINEAR REGULATOR PROBLEM

The matrix corresponding to (4.6) is then Onn

(5.12)

0,

En

Onk

En

W t ) -W)

--G(t)

Onn

which is nonsingular in view of the hypothesis that H ( t ) is positive definite. It may be verified readily that the reciprocal of (5.12) is

G(t)H-'( t)G(t ) H-1( t )G(t )

(5.13)

G (t)H-'( t ) H-'(t) On,

En Okn].

Onn

Moreover, in view of the conditions on H ( t ) and G ( t ) appearing in hypothesis (&), it follows that the elements of (5.12) and its reciprocal (5.13) are essentially bounded on arbitrary compact subintervals [a, b] of I. In terms of the (x, p) and canonical variables ( y , q ) the Euler-Lagrange equations are

+ W ) H - ' ( t ) W ) y ( t )+ G(t)H-'(t)q(t), p'(Q = H-'(t)G(t)y(t)+ H-'(t)q(t),

(a) x ' ( t ) = F ( t ) x ( t ) (5.14)

(b)

(c) r'(4= m w w x ( t ) - J w Y ( t > , (d) q'(t) = 0.

+

These equations may be written in terms of the (a K)-dimensional vector functions u ( t ) = ( x ( t ) ; p ( t ) ) and v ( t ) = ( y ( t ) ; q ( t ) ) as (5.14')

-s'(t)

+ @ ( t ) u ( t ) - d ( t ) o ( t ) = 0,

u ' ( t ) - d ( t ) u ( t ) - 9(t).(t) = 0,

where

Moreover, the given boundary conditions at t = t o , together with the transversality (or natural boundary), conditions for the problem of

188

V. APPLICATIONS

minimizing (5.8) subject to (5.6), (5.7) may be written as u(t0) -

(5.16)

v(ti)

+ [diag{Q,

(“0;

okk)lU(ti)

0) = 0, = 0.

(G),

In view of hypothesis the real quadratic functional %(x, y 1 t o , tl) defined by (5.8) is nonnegative definite on the class of vector functions ( ~ ( t y) (, t ) ) E % [ t o , t1l x pkqto, tll. Therefore, for - d ( t ) , B(t),s(t> defined by (5.15), we have that the functional (5.8’) is nonnegative definite on the corresponding class Bo[t0, tl], consisting of those (n k)dimensional vector functions ~ ( t satisfying ) with some corresponding [ ( t ) E 2i+k[to, tl] the differential equation

+

$ ( t ) - d ( t ) q ( t )- . 9 ( t ) [ ( t )= 0. In particular, (5.8‘) is positive definite on g o * “ t o , t11 = {V

IV

E q t o , tll, V ( t 0 ) = 01,

and if y(t)= ( P ( t ) ;Y . ( t ) ) is the solution of the matrix differential system corresponding to (5.14’) which satisfies the initial conditions P(tl)= Enfk, T ( t l )= -@, then % ( t ) is nonsingular on [ t o ,tl]. Equivalently, if Y ( t ) is the solution of the associated Riccati matrix differential equation

+ W ” ( t ) - d ( t )+J(t)W”(t)

Y’(t) (5.17)

+ Y ( t ) @ ( t ) W ” ( t-) 8 ( t )= 0, Y ( t i ) = diag{-Q,

Okk).

+

then the interval of existence of Y ( t )includes [ t o ,tl]. Let the (n k) x (n k) matrix functions P(t),T ( t ) , Y ( t )be written in partitioned form as

+

where the matrix elements denoted by subscripts 11, 12, 21 and 22 are of respective dimensions n x n, n x k, K x n and k x k. In view of the boundary conditions

5. A LINEAR REGULATOR PROBLEM

189

we have that

while U,,(t), U,,(t), Vll(t) are solutions of the matrix differential system

with the boundary conditions

Either from (5.21), or from the system (5.22), (5.23), together with the representation F ( t ) = F ( t )%-l(t), one finds that

and that

190

V. APPLICATIONS

is the solution of the Riccati matrix differential system wil(t)

(5.26)

+ Wll(t)F(t) + m w l l ( t )

+ ~11(t)G(t)H-'(t)G(t)w,,(t-)E(t)T(t)L(t)

= 0,

wii(ti) = -Q*

In view of Lemma IV:7.6 we have that if (Ull(t); Vll(t)) has been chosen as a solution of the differential system (5.21a, b) then the minimizing element for B(x, y 1 t o , t l ) in the class of all solutions of (5.1) is x = xm(t), y = ym(t), where (a) (5.27)

Xm(t)

(b) y,(t)

=

~ l I ( ~ ) ~ G Y ~ O ) ~ O ,

= Gl(t) = H - ' ( t ) ~ ( t ) ~ l l ( t ) U ~ l ' ~ o, ) x o = H-1( t

)G(t )w,, (t)xm(t ).

Moreover, the corresponding minimum value B(xm,y m cost functional (5.3) is given by (5.28)

w m , Ym

I to,t,)

of the

I t o , tl) = -~owll(to)xo.

In particular, (5.27b) presents the answer to the quest for a control law (5.2); that is x ( t , x) = H-'(t)G(t)W,,(t)x.

In the above discussion, the determination of a solution of the differential system (5.14) satisfying the prescribed boundary conditions has been reduced to the consideration of solutions of the differential system (5.21). Now the vector differential system (5.29)

-w'(t)

+ E(t)T(t)L(t)u(t)- E(t)a(t)= 0,

u ' ( t ) - F(t)u(t)- G(t)H-'(t)G(t)w(t)= 0,

corresponding to the matrix system (5.21a, b), is of the form (IV:1.31) with A ( t ) = F ( t ) , B ( t ) = G(t)H-l(t)G(t) and C ( t ) = E(t)T(t)L(t). Moreover, these real-valued matrix functions are such that B ( t ) and C ( t ) are symmetric and nonnegative definite for t a.e. on I. Now let @= @ ( t ; T) be the fundamental matrix solution of @'(t) - F ( t ) @ ( t )= 0 satisfying the initial condition @(T) = E. The semigroup property of @ states that @ ( t ;T) = @ ( t ;u ) @ ( u ;T)

for

(t, z, u)

EI

x I x I,

5. A LINEAR REGULATOR PROBLEM

191

and, in particular, @ - l ( t ; t) = @(t; t). I n view of Lemma IV:2.1, the differential system (5.29) is normal on a compact subinterval [a, b] c I if and only if (5.30) for some one value t,and hence for all values t on I. Moreover, by the remarks following Lemma IV:2.1 we have that the differential system obverse to (5.29) is normal on a subinterval [a, b] c I if and only if (5.31) for some one value t,and hence for all values t,on I. We shall proceed to discuss interrelations between these and certain other concepts. I n the terminology of Kalman [l], a state xo is said to be controllable at to if there exists a t , > to and some control y ( t ) which transfers the phase ( t o ,xo) to ( t , , 0) ; in general, when such t , and y ( t ) exist they are dependent upon to and xo. If every state .xo is controllable at t o , then (5.1) is said to be completely controllable at t o ;if every state is controllable at all values to E I, then for brevity the plant is said to be completely controllable. Now the solution of (5.la) is given by (5.32) and hence y ( t ) transfers the phase ( t o ,x,) to (1, , 0) if and only if (5.33)

If the nonnegative definite hermitian matrix (5.34)

is singular, and 5 is a nonzero n-dimensional vector such that Q(t,, t,)E = 0, then I G(s)&(t,;s ) t l2 ds = 0, and in view of (5.33) we have that tx, = 0 if ( t o ,x,) is transferrable to ( t l , 0). I n particular, there is no y ( t ) which transfers ( t o ,t )to ( t l ,0). On the other hand, if Q(t,, tl) is nonsingular, then for a given xo the phase ( t o ,x,) is transferrable to (tl ,0) by the control

Ji:

(5.35)

y ( t ) = -G(t)&(to; t)Q-l(t,, tl)x,.

192

V. APPLICATIONS

Now in view of the positive definite character of H - l ( t ) we have that the matrix (5.34) is nonsingular if and only if the condition (5.30) holds for a = t o , b = t , , z = t o . That is, the plant defined by system of (5.1) is completely controllable at a fixed value to if and only if there exists a t , > to such that the system (5.29) is normal on [ t o , t , ] . Also the plant defined by system (5.1) is completely controllable if and only if for each to E I there is a t , E I such that t , 2 to and (5.29) is normal on [ t o ,t , ] , which is the condition Yl+(I)as introduced in Section IV:8. If the matrix functions F ( t ) and G ( t ) occurring in (5.1) are constant on I, then in view of the positive definite character of H ( t ) we have that 8l

e.

GH-'(t)G = Consequently, in view of the above remarks and Lemma IV:9.1 we have the following result.

(5.36)

If the matrix functions F and G of (5.la) are constant on I , then this plant is completely controllable i f and only i f the n x nk matrix

[G FG F2G

* * *

F"-'G]

is of rank n.

Now since E ( t ) T ( t ) L ( t )2 0 and G ( t ) H - ' ( t ) e ( t ) 2 0 for t a,e. on ( a , , b,), the system (5.29) is disconjugate on I, and there exists a principal solution Wbo(t)of the Riccati matrix differential equation (5.26). Moreover, for the case of a completely controllable plant we have satisfied the condition !N+(I) of Section IV.8, and from the result of Theorem IV:8.2 on the determination of Wbo(t)it follows that W b o ( t ) 0 for t E I. If for t E I we denote by W = W ( t ; t) the hermitian solution of (5.26) satisfying the initial condition W ( t )= 0, in view of Lemma IV:4.1 we have that W ( t ;t) exists throughout I, with W ( t ; t) 2 0 for t E I , + = { t I t E I , t L z } , and W ( t ; t ) ( O for t E I , - = { t I t E I , t 5 z}. Also, if a, < t, < t2 < b,, it follows as a consequence of the result of Theorem IV:4.3 that

I

=

Hence there exists an hermitian matrix function W l b o ( t )on I such that W ( t ;t)-+ W l b o ( t as ) t b,, and in view of the boundedness condition W a 0 ( t )5 Wlbo(t)5 0, and the monotone character of the convergence of W ( t ;t) as t b,, it follows that the convergence of W ( t ;t) to Wlb0(t)is uniform on arbitrary compact subintervals of I, and therefore ---f

---f

5. A LINEAR REGULATOR PROBLEM

193

W = Wlb,(t)is also an hermitian solution of (5.26) on I. In terms of the matrix function Wlb,(t)one obtains the performance index of a problem appearing as a limit of the one considered above, upon setting t , = bo and Q = 0, so that the cost functional is now (5.38)

21°(x, y

I to) = =

j {?(t)H(t)y(t)+ www)dt, bo

10

to be considered for the class of all solutions of (5.1) on [ t o ,b,) such that the integral of the right-hand member of (5.38) exists. In view of the nonnegativeness of the integrand, clearly this integral exists only if the function of t , defined as Jtl 10

{WWN

+ www dt

is bounded on [ t o ,bo). For a given t E ( t o ,bo), the above treatment implies that the corresponding problem with cost functional

has its minimum attained for y(t)=H-’(t)G(t)W(t;r ) x ( t ; t), and x ( t ; t) = U(t)U-l(to)x0,where Y ( t )= ( U ( t ) ; V ( t ) ) is the solution of the differential system (5.21a,b) satisfying U(t)= En, V(t) = 0 and W(t,t) = V ( t ,t > U - l ( t ,t). Moreover, the minimum value of ?Bo(x,y I t o , t) is -ZoW(t,,; t ) x 0 , and in view of the above discussion we have that if to < tl < t2 < b,, then

-zoW(to; t i ) x o 5 -zoW(to;

.z)xo

5 -$Wib,(to)Xo-

That is, for the cost functional (5.38) the minimum value is given by B0(xm,ym I to) = -X”oW~b,(to)Xo, and the control law is x ( t , x) = H-’(t)G(t)W,b,(t)x.

In comparing the above results with those of Kalman [l], it is to be noted that his Riccati matrix equation (6.3) is not the equation of the above system (5.26), but rather the equation that is satisfied by W = -W,,(t). To characterize a plant according to its “output” properties, Kalman [l] utilized the concept of the “dual plant,” specified by the equations (5.39)

x‘(t) = F(t)x(t)

+ G(t)Y(t),

h(t) = L(t)x(t),

194

V. APPLICATIONS

related to (5.1) by the substitution (5.40)

t

=

F(t)

-t,

= P(t),

G(t) = E ( t ) ,

L(t) = G ( t ) ,

where x, y, h are vector functions of respective dimensions n, r, and k. For the original system (5.1) concepts of “observability at to,” “complete observability at t,,” and “complete observability” are defined as the concepts for (5.39) of “controllability at to = -to ,” “complete controllability at to = -to ,” and “complete controllability.” Moreover, for this dual system one may introduce a “dual cost functional” involving

H(t) = T-’(t),

T(t) = H-’(t).

Now if @(t) = o(t,7) is the solution of the differential system

a(.)

o’(t)= F(t)@(t), and

T = -t,

= En,

then *(t, T) = @ ( r ,t ) .

Also, the condition that (5.1) is complete observable at to may be expressed as the condition that there exists a t , < to such that

jll

6 ( s , t ) ~ ( s ) L ( s ) @ (rs ), ds

> 0.

Moreover, in view of the positive definiteness of T ( t ) , this condition is equivalent to the existence of a t , < to such that

s“

6 ( s , z)L(s)T(s)L(s)O(s,z) ds > 0

1,

for some z, and hence for all t, belonging to I . As pointed out above, this latter condition states that the differential system obverse to (5.29) is normal on [ t , , t o ] . Correspondingly, complete observability of (5.1) is equivalent to the condition that for each to E I there is a t , < to such that the system obverse to (5.29) satisfies condition 31-(I) as introduced in Section IV:8. Finally, corresponding to (5.36) we have the following result.

If the matrix functions F and L of (5.1) are constant on I , then this plant is completely observable i f and only i f the n x nr

(5.41) matrix

is of rank n.

[ E PE PL

*

*.

En-IE]

6. LINEAR FILTERING

6.

AND PREDICTION THEORY

195

A Problem in Linear Filtering and Prediction Theory

T h e problem described in this section is the one that has been presented by Kalman and Bucy [l], and whose mathematical treatment is reduced to the consideration of a system of the same form as that treated in the preceding section. T h e present discussion is limited to a brief statement on hypotheses and the attainment of the variance equation as a Riccati matrix differential equation. For more detailed discussions on the formulation of the problem, and derivation of results, the reader is referred to Kalman [l, 21, Kalman and Bucy [l], and Wonham [2]. In particular, the following definitions and assumptions are pertinent to the discussion.

(1)T h e “message” is a random process (6.1)

~‘(= t )F(t)x(t)

x(t)

generated by the “model”

+G(t)~(t)

and the ‘(observed signal” is of the form

(6.2)

z ( t ) = y(t)

+ v(t)

= H(t)x(t)

+ v(t>.

I n (6.1)and (6.2) the matrix functions F ( t ) , G ( t ) ,H ( t ) are of respective dimensions n x n, n x m, p x n, and the vector functions u(t), v(t) are independent random processes (white noise) with identically zero means and covariance matrices

~ ( t )= ] Q ( t ) * d(t - t), COV[U(~), (6.3)

COV[V(~), ~ ( t )= ] R ( t ) * d ( t - t), cov[u(t), v(.)] = 0

for all t, t, where d is the Dirac delta function and Q ( t ) ,R ( t ) are real symmetric nonnegative definite matrices of respective dimensions m x m and p x p , which are supposed to be continuously differentiable functions of t. (2) T h e matrix R(t) is positive definite for all t. (3) T h e measurement of z ( t ) starts at some fixed time t o , which may be to = -co, at which time cov[x(to), x(t,)] is known. T h e topic of principal concern is the optimal estimation problem ; i.e., given known values of z(t) in the time interval to 6 t 5 t, find an estimate 2(t, I t) of x(tl) of the form

(6.4)

2(t, I t ) =

t t0

A(t,,

t)z(t)dt

V. APPLICATIONS

196

such that the expected squared error a[x#, x(tl) - %(t,I t)12 encountered in estimating any real-valued linear function x* of the message is minimized. I n (6.4), the kernel matrix function is of dimension n x p , and is assumed to be continuously differentiable in both arguments. Firstly, it is shown that if %(tl 1 t) is defined by (6.4) then a necessary and sufficient condition for [x*,%(tl I t)] to be a minimum variance estimator of [x*, x ( t , ) ] for all x# is that the kernel matrix function satisfy for all 0 E [ t o ,t) the Wiener-Hopf equation

(6.5)

cov[x(tl), z(a)] -

j'A(t,,

t)cov[z(t),

z(a)] d t = 0.

t0

Upon considering t, = t, computing the derivatives of the left-hand members of (6.5) with respect to t, and interchanging the operations of differentiation with respect to t and computation of expected values, it is shown that the optimal estimate %(t I t) is generated by a linear dynamical system of the form

(6.6)

(d/dt)%(t1 t ) = F(t)%(t I t )

I

+ K ( t ) i ( tI t ) , I

Z ( t t ) = ~ ( t-) H(t)%(t t ) ,

where K(t) = A(t, t) and the initial state %(toI t o ) is zero. Moreover, for optimal extrapolation the relation

(6.7)

%(t,I t ) = Q , ( t l ,t)%(tI t)

is adjoined, where

Q, = Q,(t, t o ) is

for

t , 2 t,

the solution of the differential system

Q,'(t) = F(t)Q,(t),

@ ( t o )= En.

If %(tI t) = x(t) - %(t I t) then

+

) K(t)[v(t) H(t)%(t t)]. (6.8) (d/dt)%(tI t ) = F ( t ) % ( t [ t ) + G ( t ) u ( t -

I

Moreover, if P ( t ) denotes the matrix function cov[%(t I t),%(t[ t)], then

(6.91

K ( t ) = P(t)fi(t)R-'(t),

and the variance equation satisfied by P ( t ) is the Riccati matrix differential equation

(6.10)

+

P'(t) = F ( t ) P ( t ) P ( t ) F ( t ) - ~ ( t ) ~ ( t ) ~ - l ( t ) ~ ( t ~) (~t () t~)( t ) ? ; ( t ) .

+

7. MYCIELSKI-PASZKOWSKI DIFFUSION PROBLEM

1w

Also, the solution P(t) of (6.10) is subject to the initial condition (6.11)

P(t0) = cov[x(to), x(t0)l.

Under assumptions (l), (2), and (3), the solution of the optimal estimation problem with to > -co is giveii by relations (6.6)-(6.11). For a detailed discussion of this problem and related matters, dealing with the case of to = -co and treating the concepts of observability and controllability in a fashion equivalent to that appearing in the problem of the preceding section, the reader is referred to the papers cited in the initial paragraph of this section.

7. The Mycielski-Paszkowski Diffusion Problem

In the simplest instance of n = 1, the probability problem to be considered was formulated by C. Ryll-Nardzewski in a paper presented to the SocietC Polonaise de MathCmatique, and was discussed in more general form by Mycielski and Paszkowski [l] and Paszkowski [l]. The formulation presented in this section is that of Redheffer [ S ] , and the treatment follows that of Reid [6, Sec. 81. Let I, , I, , . . . ,1, be parallel lines each endowed with mass ; for mathematical description, we shall suppose that the set of points on each individual line is defined by t belonging to an open interval I = (a,8) on the real line. If [s, t] c I,then for brevity we setI+(t) = {t I t E I , t 2 t } and I-(s) = {t I t E I , t 5 s}. Moreover, for i = 1, . . . , n, the points on Zi defined by t E [s, t ] , t E I + ( t ) ,and t E I-(s), will be denoted respectively by [s, t l i , I i + ( t ) ,and Ii-(s). Now suppose that a molecule moves on this system of lines in such a way that i a n d j are any of the values 1, 2, . . . , n,and when the molecule moving on line Zi encounters the segment [s, tIi from the right then it may be reflected on the segment I j + ( t ) , transmitted to the segment Ij-(s), or absorbed on [s, tIi. Similarly, if the molecule in moving on line Zi encounters the segment [s, tIi from the left then it may be reflected to the segment Ij-(s), transmitted to the segment I j + ( t ) ,or absorbed on [s, tIi. I n case of absorption, it is understood that the molecule does not again start to move. Moreover, it is assumed that the encounter of the molecule with line segments is a random phenomenon, which is independent with respect to segments whose projections are nonoverlapping, and that the probabilities of transmission, reflection, and absorption associated with each segment depend only upon

V. APPLICATIONS

198

the direction of motion of the molecule, and on the material mass of the segmect. Now when the molecule is incident to segment [s, tIi from the left, let

p i j ( s : t ) = probability of transmission to Ii+(t), qij(s:t ) = probability of reflection to Ii-(s),

(7.1)

eij(s, t ) = probability of absorption on [s, tIi.

Correspondingly, when the molecule is incident to the segment [s, tIi from the right, let

Pi,($: t ) = probability of transmission to Ii-(s), Qij(s: t ) = probability of reflection to Ii+(t), *

(7.2)

Rii(s, t ) = probability of absorption on [s, t I i . As shown in Redheffer [ 5 ] , these probability functions satisfy the functional equations : (a)

(7.31

(7.4)

(b)

p ( r ; t ) = p ( s : t)[l - Q ( r :s ) q ( s : t ) ] - l p ( r : s), q ( r : t ) = q ( r : S) P ( Y :s)[l - q(s: t ) Q ( r :s)]-'~(s: t ) p ( ~s):,

+

(c)

e ( r : t ) = R(r: s)[1 - q(s: t)Q(r:s)]-'q(s: t ) p ( r : s) e(s: t ) [ l - Q ( Y :s ) ~ ( s : t ) ] - ' p ( ~S) : e ( r : s),

(a)

P ( r : t ) = P ( r : s)[l - q(s: t ) Q ( r :s)]-lP(s: t ) ,

+

(b) Q ( Y :t ) = Q ( Y :s) (c)

+

+ p ( s : t ) [ l- Q(r :s ) q ( s : t)]-IQ(r:s)P(s:t ) ,

R ( r : t ) = e(s: t ) [ l - Q ( Y :s ) q ( s : t)]-'Q(r: s)P(s:t ) f R ( r : s)[l - q ( s : t ) Q ( r :s)]-'P(s: t ) R(s: t).

+

The physical interpretation of the above described phenomenon suggests the initial conditions (7.5 1

$(S:

S) =

E n , P ( s :S) = E , , e(s: s) = 0,

p(s:

S) = 0,

Q ( s :S)

= 0,

R(s: s) = 0.

It will be supposed that the initial conditions (7.5) hold, and also that the matrix functionsp(s: t),P(s: t ) are nonsingular for arbitrary [s, t ] c I ; moreover, it is supposed that the matrix functions p ( s : t ) , q ( s : t), e(s: t ) , P(s: t ) , Q(s: t ) , R(s: t ) have finite partial derivatives with respect to t

199

7. MYCIELSKI-PASZKOWSKI DIFFUSION PROBLEM

at the value t = s. If we set

(7.6)

A ( $ )= Q t ( s : s),

M ( s ) = R t ( s : s),

K(s) = P,(s: s),

A(s) = q&: s),

k ( s ) = p , ( s : s), p(s) = Q t ( s : s), then as in the proof of Theorem II:3.4 it may be shown that the matrix functions p , q, e, P, Q, R satisfy the following differential system, which corresponds to system (I1:3.12).

+ k ( t ) Q ( s : t ) + Q(s: + Qb:t)A(t)Q(s: t ) ,

(a) Qt(s: t ) = A ( t ) (b)

(7.7)

+

W ( t )

P,(s: t ) = P(s: t ) [ K ( t ) A(t)Q(s: t ) ] ,

+

+

( c ) R,(s: t ) = M ( t ) p(t)Q(s: t ) R(s: t ) K ( t ) R(s: t)A(t)Q(s:t ) ,

+

+ Q ( s : t)A(t)l~(s:t ) , P ( s : t)A(t)p(s:t), [ W s : t)A(t) + p(t)lp(s: 0.

(d)

pt(s: t ) = [k(t)

(e)

q t ( s : t) =

(f) et(s: t ) = Also, corresponding to (II:3.24) we have the system (a) Q8(s:t ) = -p(s: t)A(s)P(s:t ) ,

(7.8)

+ q(s: t)A(s)]P(s:t ) ,

(b)

P8(s: t ) = -[ K ( s )

(c)

R,(s: t) = -[e(s:

(d)

P&:

(e)

q&: t ) = -[A(s)

(f)

eds: t ) =

t ) = --P(s:

t)A(s)

+ M(s)]P(s:t),

+ A(s)q(s: t)l, + K(s)q(s:t ) + q ( s : W s )

t)[&)

+ d s : t ) 4 s ) d s : 01, -b ( s ) + M(s)q(s: t ) + + e(s: M s ) q ( s : 01.

e(s : t ) W )

The conditions for physical realizability require that for (s, t) E I x I the elements of the matrix functions p, q, e, P, Q,R satisfy the following conditions:

200

V. APPLICATIONS

In particular, conditions (7.9) and (7.10) may be stated as the condition that the 3n x 3n matrix function

I:

i

p(s: t ) Q(s:t ) Y ( s : t ) = q ( s : t ) P(s: t ) e(s: t ) R(s: t ) E

(7.11)

has for arbitrary s E I the property A,,,, for each t E I+(s), where the dissipative property A,,,,, is defined as in Section II:9. That is, if O,, = = 1 (a = 1, . . . , 3n), and r,, is the convex set in with % ,, consisting of all real 3n-dimensional vectors w = ( w a ) with wa 2 0, (a = 1, . . . , 3n), and 6,%w 5 1, then for s E I and t E I+(s) we have 9 ( s : t)w E T,, whenever w E F3n. Theorem 7.1

Suppose that the functions A(s), K(s), M(s), A(s), k(s), p(s) dejined by (7.6) are of class L?,,[a, b] for arbitrary compact subintervals [a, b] c I . Then a necessary and suflcient condition for the matrix function 9 ( s : t ) defined by (7.1 1) to exist and have property A,, ,,, for t E I+($)and arbitrary s E I , is that the 3n x 3n matrix A ( t ) dejined by (7.12) be such that (7.13)

A ( t ) wr 2

wr

0,

Q3,A(t)- 5 - 0 for

t a.e. on

I.

Indeed, to deduce the result of Theorem 7.1 from Theorem II:9.4 it suffices to show that Y ( t )= Y ( s : t ) is the solution of the Riccati matrix differential equation (7.14) Y’(t)+Y(t)d(t)+=0(t)Y(t)+Y(t)S’(t)Y(t)--(t)=O, satisfying Y ( s ) = E,,, where

(7.15)

0 s(t) = [o 0

4t) 0 M(t)

q, 0

9 ( t )=

[-44 -k(t) 0

0 0 0 01. 0 0

201

8. THE X-PRODUCT OF REDHEFFER

Moreover, the corresponding matrix E ( t ) = @ ( t ) - d ( t ) - 9 ( t ) - 9 ( t ) is equal to the matrix & ( t ) defined by (7.12), the condition d ( t )x 2 x 0 is equivalent to the set of conditions

-

@(t) 2

. 0,

9 ( t )* 5

*

d ( t )* 5

0,

x

0,

9 ( t )x 5

x

0,

8. The X-Product of Redheffer

As indicated in the initial paragraph of the preceding section, the formulation of the problem considered therein was due to Redheffer, but the presented treatment followed that given in a paper by the author. T h e consideration of this problem by P-edheffer was materially different, utilizing in particular the concept of a *-product of two matrices. Consequently, the present section is devoted to the definition of this product and brief comments on its significance for the theory of Riccati matrix differential equations. For a comprehensive discussion of this concept the reader is referred to the papers of Redheffer [2, 3, 5 , 6, 7, 81. In particular, an extension of the concept to closed linear operators in Hilbert space is presented in Redheffer [6]. In the following we shall be concerned with matrices A which belong to the class !PIm+,of (m n) x (m n) matrices, which for our purpose will be written in partitioned form

+

+

[

G W -F HI’

where

F, G, H , W a r e of respective dimensions n x m, m x m, n x n, (8*2) m x n. For typographical simplicity, we shall write = [G ; W ;- F ; HI instead of the usual matrix symbol as displayed in (8.1). If d,= [G, ; W, ; -Fl ; H I ] E mm+,, then we shall denote by 1 the collection of matrices d, = [G, ; W, ; -F, ; H,] E !lJlm+n such that the m x m matrix Em W,F, is nonsingular. As in the discussion of Section 11.2, it follows that Em W,F, is nonsingular if and only if the

r{dl

+

+

202

V. APPLICATIONS

n x n matrix En

+ FzWl is nonsingular, and in this case

That the *-product is of basic significance for the theory of Riccati matrix differential equation is evidenced by the following fact, which follows directly from equations (I1:3.13). Theorem 8.1

If I ,

c

I , and for (r, s )

E I,

x I,, the solutions W ( t ) = W(s:t ) and W ( t ) I , , then for

= W ( r : t ) of (II:3.12a) exist on

IID(s: t ) =

G(s:t ) W(s:t ) ] t ) H(s:t )

[ -F(s:

and t E I , we have that IID(t: t ) E l'{m(s:t ) } and (8.5 1

m ( r : 1 ) = m ( r :s )

*

m(s: t ) .

That is, the semigroup property of the solutions of system (I1:3.12) is expressible in t e r m of the *-product. I n general, it is to be noted that if A E Wm+, then 8 = diag{E, = Em+,,belongs to l'{4 and

A? = A?*8

=8

,E n }

* A.

Moreover, if as usual the symbol A? * denotes the conjugate transpose of the matrix A?,we have that whenever dl6 I'{A} then A?* E l'{Al*} and [.A* All*= .Al*X A?*.

203

9. INVARIANT IMBEDDING PROBLEM

Further properties of the *-product lemmas.

are presented in the following

Lemma 8.1

If 44,= [G,; W,; - F a ; Ha] (a = 1, 2, 3) belong to !lR,,,+n, while AzE J'{Al}, and A3E r{Alx Az}, then A2x A3E J'{Jl}, and 4

1

x

( A 2

x43)

=

x J2) x 4 4 3 .

(dl

f emma 8.2 Suppose that 44,= [G,; W,; -F,; Ha] (a= 1 , 2 ) and A2E r{dl}. If v [ L l ] 5 1 and .[A2] 5 1, then v{Al x A2}5 1 ; correspondingly, if each 44, is unitary then dlx J2is unitary. 9. A Problem in Invariant Imbedding

I n this section we shall consider a model of a transport process by the method of invariant imbedding that leads to a Riccati matrix differential system which is intimately related to the system considered in Section 7. For details of the specific problem, the reader is referred to the paper [l] of Bellman, Cooke, Kalaba, and Wing. The process considered is that of an idealized neutron transport occurring in a one-dimensional, homogeneous isotropic rod extending along the real line from z = 0 to z = x. It is supposed initially that there are only a finite number, N, of different types of particles moving along the rod; these possible states will be labelled i = 1, . . . , N, and may be thought of as energy levels. It is assumed that as a particle in state i transverses a segment of the rod it is subject to interactions with the material forming the rod, and that these interactions may produce forward or backward scattering into any of the N possible states, or absorption. Also, it is assumed that there is no spontaneous generation of new particles ; that is, it is supposed that fission does not occur. In view of these hypotheses the total number of particles in the process, including those absorbed as well as those scattered, is altered only by addition from an external source. Finally, the possibility of collisions or interactions between neutrons is excluded in the model. It is assumed that when a particle in state j ( j = 1, . . . , N), enters an infinitesimal segment [x, x S] from either direction, then:

+

204

V. APPLICATIONS

(a) the expected number leaving the segment in state j , and moving in the same direction, is 1 djj(x)8 O(P);

+

+

(b) the expected number leaving the segment in state i, i # j , and moving in the same direction, is dij(x)6 O ( P ) (forward scattering) ;

+

(9.1)

(c) the expected number leaving the segment in state i, and moving in the opposite direction, is bij(x)6 O ( P ) (back scattering) ;

+

(d) the expected number absorbed in the segment is fjj(S)

+O ( V .

The N x N matrix functions D ( x ) = [dij(x)], B ( x ) = [ b i j ( x ) ] ,F ( x ) are called the forwurd scattering, backward Scattering, and absorption matrices, respectively. On physical grounds, it is assumed that for x 2 0 we have the inequalities = [fii(x)Sij]

(a) dij(x) 2 0,

i#j;

(b) bij(x) L 0 ;

(9.2)

(c)

fii(.)

2 0.

The basic conservation assumption then requires the equations

In.particular, (9.3) implies that d j j ( x )5 0 for j = 1, . . . ,N. Moreover, if O N denotes the N-dimensional vector ( O a N ) with OoN = 1 (CT = 1, . . . , N), then (9.3) may be written as

(9.4)

ON[B(x)

Now for i,j = 1, values of z, let:

+ D ( x ) + F ( x ) ]= O

for x >_ 0.

. . . , A', and a rod extending from x = 0 to positive

rij(x) = expected flux of neutrons in state i, reflected from a rod of length x, resulting from an incident flux at x of unit intensity in state j ; t i j ( x ) = expected flux of neutrons in state i , transmitted through a rod of length x resulting from an incident flux at x of unit intensity in state j;

9. INVARIANT IMBEDDING PROBLEM

205

Zij(x) = expected flux of neutrons in state i, absorbed within a rod of length x, resulting from an incident flux at x of unit in-

tensity in state j .

The N x N matrix functions R ( x ) = [ r i j ( x ) ] , T ( x ) = [ t i j ( x ) ] , L(x) = [Zij(x)]are termed the reflection, transmission, and absorption matrices. Using invariant imbedding techniques, the authors show that R(x), T ( x ) , L ( x ) satisfy the differential system

+ D ( x ) R ( x )+ R(x)D(x)+ R(x)B(x)R(x), T'(x) = T ( x ) [ D ( x )+ B ( x ) R ( x ) ] , L'(x) = L(x)[D(x) + B(x)R(x)l + F(x" + W)l,

(a) R'(x) = B(x) (9.5) (b) (c)

and the initial conditions (9.6)

R(0) = 0,

T ( 0 )= E N ,

L(0) = 0.

Upon identifying x with t, B ( x ) with A ( t ) and A ( t ) , D ( x ) with K ( t ) and k ( t ) , and F ( x ) with M ( t ) and ,u(t),the solution matrix functions R(x), T(x), L(x), of the differential equations (9.5a-c) correspond to the respective solution matrix functions Q(s: t), P(s: t ) , R(s: t) of the differential equations (7.7a-c). The matrix corresponding to (7.12) is

oj

D(x) B(x) 0 .A(.) = B ( x ) D ( x ) 0 [F(x) F(x)

and clearly d ( x ) * 2 x 0. Moreover, 03,+?(x) = 0 in view of (9.4). From Theorem 7.1 we have that the associated matrix (7.11) of solutions of the augmented differential system involving (9.5) and associated equations of the form (7.7d-f) possesses property d g N , 3for N x E I+(O). Consequently, in particular, we have that the matrix functions R(x), T(x), L(x) satisfying (9.5) and (9.6) have nonzero elements for x L: 0 and all elements of the 1 x N matrix function %(x) = & [ E N

- R(x) - T ( x ) - L(x)]

are nonnegative. Indeed, in view of (9.4) we have that g(x) satisfies the differential system %'(XI

= Y(x)[D(x)

+ B(x)R(x)],

%(O) = 0,

206

V. APPLICATIONS

and hence 9(x)

3

0. That is, we have the conservation law

+

BN[R(x) T ( x )

+ L(x)] = ON

for x 2 0.

As the above argument shows that the elements of the matrix functions R(x), T ( x ) , L ( x ) satisfying (9.5) and (9.6) are bounded on any interval [0, xo) of existence, the usual continuation theorem for solutions of differential equations provides the result that the solutions of (9.5) specified by the initial conditions (9.6) exist on [0, 00). 10.

Notes and Remarks

Following is a limited list of pertinent references for the individual sections of this chapter. REFERENCES

Sec. 2. Kaufman and Sternberg [l], Rice [l]. Sec. 3. Bellman [3, 51, Bliss [l, Secs. 28, 29, 30, 381, Dreyfus [l], Hestenes [l,

Sec. 5. Sec. 6. Sec. 7. Sec. 8.

Sec. 9.

Chap. 3, Sec. 11 ; Chap. 6, Sec. 101. In this connection, mention is made of the papers of Dresden [l, 21 on the second derivatives of the extremal integral, as these papers are antecedents of procedures appearing in present day dynamic programming and invariant imbedding. Athans and Falb [l, Chap. 91, Citron [l, Chap. 81, Kalman [l], Kalman, Ho, and Narendra [l], Lee and Markus [l, Chaps. 2, 31, Pierre [l, Chap. 71. Kalman [l, 21, Kalman and Bucy [l], Wonham [2]. Mycielski and Paszkowski [l], Paszkowski [l], Redheffer [l-5, 7, 81, Reid [6]. Kaplan and Stock [l], Redheffer [2, 3, 5-81. In particular, the paper of Kaplan and Stock generalizes the *-product to the case of matrices whose component entries are not required to be of like dimensions ; consequently, when each factor is a solution matrix of the form (8.4) for a related differential system (11:3.12), the differential systems need not be the same. Bellman, Cooke, Kalaba, and Wing [l], Bellman, Kabala, and Wing [l], Bellman and Kalaba [l], Wing [l].

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Riccati, Count J. F. 1. Animadversationes in aequationes differentiales secundi gradus, A c t m m Eruditorum quae Lipsiue publicantur. Supplementa 8 (1724), 66-73. Rice, S. 0. 1. Steady state solutions of transmission line equations, Bell. System Tech. J . 20 (1941), 131-178. Riesz, F., and Sz.-Nagy, B. 1. Functional Analysis, Ungar, New York, 1955.

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Sternberg, R. L. 1. Variational methods and non-oscillation theorems for systems of differential equations Duke Math. J. 19 (1952), 311-322. 2. A theorem on hermitian solutions for related matrix differential and integral equations, Portugaliae Math. 12 (1953), 135-139. Synge, J. L. 1. Geometrical Optics, an Introduction to Hamilton's Method, Cambridge Tracts in Mathematics and Mathematical Physics, No. 37, Cambridge Univ. Press, 1937. von Escherich, G. 1. Die zweite Variation der einfachen Integrale, Wiener Sitzungsberichte, (8) 107 (1898), 1191-1250. Watson, G. N. 1. Theory of Bessel Functions, Cambridge Univ. Press, 1922; 2nd ed., 1944. Whyburn, W. M. 1. Matrix differential equations, Amer. J. Math. 56 (1934), 587-592. Wielandt, H. 1. Unzerlegbare, nicht negative Matrizen, Math. 2. 52 (1949-SO), 642-648. Wilczynski, E. J. 1. Projective Dzfferential Geometry of Curves and Ruled Surfaces, Teubner, Leipzig, 1906. Wing, G. M. 1. An Introduction to Transport Theory, Wiley, New York, 1962. Wonham, W. M. 1. On a matrix Riccati equation of stochastic control, S I A M J. Control 6 (1968), 681-697. 2. Random differential equations in control theory, Probabilistic Methods in Applied Mathematics, Vol. 11, 131-212, Academic Press, New York, 1970.

INDEX

A Abnormality, 36 order of, 36 Absorption matrix function, 198, 205 Accessory problem, 94-98, 182, 183 canonical equation for extremals of, 183 Adjoint system, 55 Ahlbrandt, C. D.,167, 168, 207 d’Alembert, J., 8 Anharmonic (cross) ratio, 6, 85-87 Aoki, M.,167, 207 Associated Riccati matrix differential equation, 34-36 Athans, M., 206, 207 Atkinson, V., 162, 165-168, 207 Atkinson matrix method, 162-167

v.

B Barrett,J. H., 113, 167, 207 Basis conjoined, 110 @-conjugate, 93 Beckenbach, E. F., 75, 207 Bellman, R., 75, 167, 184, 203, 206, 207 Bernoulli, Daniel, 8 Bernoulli, James, 7, 8 Bernoulli, John, 8 Birkhoff, G., 153, 208 Bliss, G.A., 36,88,94, 107, 118, 168,176,

206, 208

Bolza, O., 2, 8,208 Bolza, problem of, 176-184 Boundary problem, two-point, 37, 166,

167

Bucy, R. S., 167, 195, 206, 208, 209 213

Canonical form of Euler-Lagrange equation, 181 of extremals of accessory problem, 183 Canonical variables, 94, 96, 186 Carathtodory type condition, 62 Cayley transform, 165 Chiellini, A., 88, 208 Citron, S. J., 206,208 Clebsch condition, 118 Coles, W. J., 88,208 Collatz, L., 64, 208 Comparison theorems, 122, 123, 142-150 Conjoined (conjugate) basis, 93, 110 Conjugate point, 38 criteria for, 4142 order of, 37 Control, 184 Controllable, 191 completely, 191 Cooke, K. L., 203,206,207 Coppel, W.A., 168,208 Cost functional, 185 dual, 194 Cross (anharmonic) ratio, 6, 84-87

D Davis, H. T., 7, 208 Definite hermitian system, 134-153 Differential inequalities, 140,142,145,146 Differential system, controllable, 191 completely, 191 generalized, 49-60, 108 hermitian, 90

214

INDEX

observable, 194 completely, 194 obverse, 107, 108, 191, 194 reduced, 105 symmetric, 90 @-symmetric, 90 truncated preferred reduced, 106 Differentially admissible arc, 176 Diffusion problem, 197-201 Dimension of @-conjugate family, 92 Disconjugacy, 36 criteria for, 41,42, 138-143 Dissipative property of solutions, 72-74,

77-84

Distinguished solution of Riccati matrix differential equation, 4549, 115-1 18 Dresden, A., 206,208 Dynamic programming, 184

E 8-function, 183, 184 Etgen, G. J., 167,168,208 Euler, L., 8 Euler and Euler-Lagrange equation, 94,

G Gantmacher, F. R., 75, 76,209 Generalized inverse, 42,44,158 Glaisher, J. W.L., 7, 209 Graves, L. M., 80, 209

H Hamiltonian, 179 Hamilton-Jacobi partial differential equation, 180 for accessory problem, 183 Hartman, P., 71, 88, 108, 152, 167, 168,

209

Hermitian Riccati matrix differential equation, 90, Ch. IV definite, 118-129, 134-160 with constant coefficients, 160-162 Hestenes, M. R., 176, 184, 206, 209 Hilbert integral, 178

209

177, 187

Extensibility of solutions, 65, 70-74,

131-1 34

Extremal, 177 arc of a field, 178

F Falb, P., 206, 207 Field (Mayer), 178 multipliers of, 178 optimal, 184 slope functions of, 178 trajectory (or extremal arc) of, 178 Focal point, 93 criteria for nonexistence of, 142-153 Forsyth-Laguerre canonical form, 2 Functional hermitian, 134-138 quadratic cost, 185 Fundamental matrix solution, 15

I Identical normality, 36 Ince, E. L., 2, 209 Index of performance, 185 Invariant imbedding, 203-206 Involutory Riccati matrix differential equation, Ch. I11

J Jackson, D. H., 168, 209 Jacobi, C. G. J., 7 Jacobi differential equation, 8 Joseph, P. D., 167, 208

INDEX

215

K Kalaba, R., 167, 203, 206, 207, 209 Kalman, R. E.,184,191,193,195,206,209 Kamke, E., 71, 209 Kaplan, E. J., 206, 209 Kaufman, H., 170,172,206,210

Nonoscillation, see Disconjugacy Normality, 36 criteria for, 38-39, 161 identical, 36 %(I), 156 %+(I), 157 0

L Lee, E. B., 206, 210 Legendre, A.-M., 8 Legendre (or Clebsch) condition, 118 Legendre differential equation, 1, 7 Leibniz, G. W., 7 Leighton, W., 167, 168, 210 Levin, J. J., 86, 88, 210 Linear differential system, generalized, 49-60, 108 Linear filtering problem, 195-197 Linear optimal regulator problem, 184194 iplant for, 184' dual, 193 state for , 184

M Markus, L., 206, 210 Matrix functions, notation for, 9, 10 absorption, 198, 205 reflection, 198, 205 square root, 30-32 transmission, 198, 205 Matrix method of Atkinson, 162-167 Mayer field, 178 McCarty, G. S., Jr., 88, 210 Monotone matrix differential equation, 129-1 34 Monotone Riccati matrix differential equation, 61-74 Moore, E. H., 42, 210 Morse, M., 152, 167, 168, 210 Mycielski, J., 197, 206, 210

N Narendra, K. S., 206, 209 Natural boundary conditions, 187

Observable, 194 completely, 194 Obverse differential system, 107, 108, 191, 194 Optimal estimation problem, 195-197 Optimal field, 184

P Partial differential equation, . applications arising from, 172-176 Paszkowski, S., 197, 206, 210 Performance index, 185 Perron-Frobenius theorem, 74-76 Pierre, D.A., 206, 210 Principal solution, 45, 49 of definite herrnitian Riccati matrix differential equation, 153-160 Property .,A 72 Properties .P+,9-,V+,V-, 77 Properties go+, go-, goA, e0+,e0-, goA; Bo+, B+, 78

+,

P,

Q Quadratic cost functional, 185

R Radon, J., 108, 167, 210 Redheffer, R. M., 72, 77, 84, 88, 197, 198, 201,206,210 #-product of, 201 Reduced system, 105, 156 truncated preferred, 106, 160 Reflection matrix function, 198, 205 Reid, W. T., 18, 37,42,44, 53,62, 63,64, 71, 74, 77, 80, 83, 84, 88, 94, 108, 113, 131, 132, 142, 152, 165, 166, 167, 168, 197, 206, 211

216

INDEX

Riccati, Count J. F., 1, 211 Riccati matrix integral equation, 49-60 Rice, S. O., 170, 206, 211 Riesz, F., 32, 211 Rota, G.-C., 153, 208 Ryll-Nardzewski, C., 197 S

Sandor, S., 14, 86, 88, 167, 168, 211 Second variation problem, 182 canonical equations for extremals of, 183 Scalar Riccati differential equation, Ch. I, 70-72 Siegert, A. J. F., 175, 211 Sol-(@-) symmetric, 90 Solutions of Riccati matrix differential equation, conjoined, 92 normalized, 43 preliminary properties of, 110-115 representation formulas for, 12-1 8, 99-104 @-symmetric, 93 Square root matrix function, 30-32 Sternberg, R. L., 108, 168, 170, 172, 206, 210, 212 Stock, D. J. R., 206, 209 Symmetric Riccati matrix differential equation, 90

Synge, J, L., 180, 212 Sz.-Nagy, B., 32, 210

T Trajectory of field, 178 Transformations, for linear differential equation, 28-34, 104-107 preferred reducing, 106, 156 reducing, 105 Trarismissionline problem, 170-172 Transmissionmatrix function, 198, 205 Transversality conditions, 187

V Variation equation of, 17, 18 of solutions, 19-28 Variational problem of Bolza, 176-184 von Escherich, G., 91, 108, 212

W Watson, G. N., 7, 8, 212 Weierstrass g-function, 183, 184 Whyburn, W. M., 88, 212 Wielandt, H., 76, 212 Wilczynski, E. J., 2, 212 Wing, G. M., 202, 206, 207, 212 Wintner, A., 167, 168, 209 Wonham, W. M., 168, 195, 206, 212