Dichotomies in stability theory

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Series: Australian National University, Canberra Adviser: M. F. Newman

629 W. A. Coppel

Dichotomies in Stability Theory

Springer-Verlag Berlin Heidelberg New York 1978

Author W. A. C o p p e l Department of Mathematics Institute of A d v a n c e d Studies Australian National University Canberra, ACT, 2 6 0 0 / A u s t r a l i a

AMS Subject Classifications (1970): 34 D05, 58 F15 ISBN 3-540-08536-X ISBN 0-387-08536-X

Springer-Verlag Berlin Heidelberg NewYork Springer-Verlag NewYork Heidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2140/3140-543210

PREFACE

Several years ago I formed the view that dichotomies, characteristic autonomous

exponents,

differential

are the key to questions

equations.

rather than Lyapunov's

of asymptotic behaviour

for non-

I still hold that view, in spite of the fact that

since then there have appeared many more papers and a book on characteristic exponents.

On the other hand, there has recently been an important new development

in the theory of dichotomies. accessible

Thus it seemed to me an appropriate

account of this attractive

The present lecture notes are the basis Florence

in May, 1977.

for a course given at the University

I am grateful to Professor R. Conti for the invitation

visit there and for providing the incentive to put my thoughts grateful to Mrs Helen Daish and Mrs Linda Southwell typing the manuscript.

time to give an

theory.

in order.

for cheerfully

of

to

I am also

and carefully

CONTENTS

Lecture I .

STABILITY

Lecture 2.

EXPONENTIAL

AND

ORDINARY

Lecture 3.

DICHOTOMIES

AND

FUNCTIONAL

Lecture 4.

ROUGHNESS

Lecture 5.

DICHOTOMIES

Lecture 6.

CRITERIA

Lecture 7.

DICHOTOMIES

Lecture 8.

EQUATIONS

Lecture 9.

DICHOTOMIES

Appendix

THE METHOD

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

DICHOTOMIES

ANALYSIS

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

i0

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

20 28

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

AND

REDUCIBILITY

FOR AN EXPONENTIAL

AND

ON

LYAPUNOV

R

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

47

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

59

DICHOTOMY

FUNCTIONS

AND ALMOST

PERIODIC

...........

67

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

74

EQUATIONS

AND THE HULL OF AN EQUATION

OF PERRON

38

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

87

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

Notes

92

References

95

Subject Index

.

.

.

.

.

.

.

.

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.

.

.

.

.

.

.

.

.

.

.

98

I.

STABILITY

A dichotomy, exponential or ordinary,

is a type o f c o n d i t i o n a l stability.

Let us

begin, then, b y r e c a l l i n g some facts about u n c o n d i t i o n a l stability. The c l a s s i c a l definitions o f s t a b i l i t y and a s y m p t o t i c stability, due to Lyapunov, are w e l l s u i t e d for the study of autonomous d i f f e r e n t i a l equations.

For non-

autonomous equations, however, the concepts o f u n i f o r m s t a b i l i t y and u n i f o r m a s y m p t o t i c s t a b i l i t y are more appropriate. Let

x(t)

be a solution o f the vector d i f f e r e n t i a l equation x ~ = f(t, x)

w h i c h is d e f i n e d on the h a l f - l i n e

uniformly stable any s o l u t i o n s ~ 0

if for each

x(t)

0 ~ t < ~

e > 0

The s o l u t i o n

and for each for some

s ~ 0

then

a corresponding

6 = 6(e) > 0

for all

for all

such that if

such that for some t ~ s .

if in addition there is a

T = T(e) > 0

Ix(t) - x(t) I < ~

is said to be

Ix(s) - x(s) I < 6

Ix(t) - k(t) I < ~

uniformly asymptotically stable

e > 0

x(t)

there is a c o r r e s p o n d i n g

o f (i) w h i c h satisfies the i n e q u a l i t y

is d e f i n e d and satisfies the i n e q u a l i t y

It is s a i d to be

(i)

60 > 0

Ix(s) - x(s) I < 6 0

t ~ s + T .

We w i l l be i n t e r e s t e d in the a p p l i c a t i o n of these notions to the linear differential equation x ~

where

A(t)

is a continuous

f u n d a m e n t a l m a t r i x for (2). x = 0

n x n

=

A(t)x

,

m a t r i x function for

(2)

0 ~ t < ~

Let

X(t)

be a

It may be shown w i t h o u t d i f f i c u l t y that the solution

o f (2) is u n i f o r m l y stable if and only if there exists a constant

K > 0

such

that Ix(t)x-l(s)i

~ K

for

0 ~ s ~ t
0

such that

K > 0 ,

''IX(t)x-l(s)I ~ Ke -~(t-s)

for

0 ~ s ~ t


0

then, for all

t ~ 0

~

for all

C

.

,

/~e-~(t-S)y(s)ds

< (I - e-~)-ic

,

<e

Z

.

-~(s-t)y(s)ds

Proof.

t ~ 0

(1

-

e-a)-lc

From

ft-m -a(t-s)y(s)ds t_m_l e

~

e-atea(t-m) C

=

e-amc

we obtain

:t e -~(t-s) yts)ds . JO ~ ~

e-~mc = (i - e-~)-ic

.

m=O

The second inequality

is proved similarly.

Suppose next that (i) has an ordinary dichotomy on = 0 .

Then for any locally integrable

corresponding

inhomogeneous

equation

function

f(t)

[+ .

Thus (3) holds with

with

~oIf(t)Idt

(2) has a bounded solution,

< ~

the

defined by the same

formula (4). We intend to establish

converses to these results.

space of all bounded continuous

vector functions IIfllc = sup t~o

Let which

M

C

denote the Banach

If(t) I .

denote the Banach space of all locally integrable ft+!If(s)Ids ~t

Let

f , with the norm

vector functions

f

for

is bounded, with the norm

IlfllM : supft+llf(s ) I d s . tz0 ~ (The same space intervals Finally,

M , with an equivalent norm, would have been obtained if instead of

of length let

L

1

we had used intervals

of any fixed length

denote the Banach space of all vector functions

Lebesgue integrable

on

~+

, with the norm

IIfllL = ~oIf(t)Idt Then we will prove

•

h > 0 ) . f

which are

22

PROP05ITION 1.

The inhomogeneous equation (2) has at least one bounded solution for

every function

f E L

if and only if the homogeneous equation (i) has an ordinary

dichotomy. PROPOSITION 2.

The inhomogeneous equation (2) has at least one bounded solution for

every function

f E M

if and only if the homogeneous equation (1) has an exponential

dichotomy.

PROPOSITION 3.

Suppose (i) has bounded growth.

Then the inhomogeneous equation (2)

has at least one bounded solution for every function

f ~ C

if and only if the

homogeneous equation (i) has an exponential dichotomy. Let of

V

V

be the underlying

consisting

vector space

of the initial

be any fixed subspace

of

V

values

or

~n)

of all bounded

supplementary

to

V1

solutions

Also let

of (i), and let

P

denote

Suppose the equation (2) has a bounded solution for every function

solution

y(t)

r B = rB(P)

of (2)

with

E V2

y(O)

Then there exists

f E B , the unique bounded

such that, for every

> 0

is

L , M , C .

denotes any one of the Banach spaces

V2

the project-

PROPOSITION 4.

B

The basic result

be the subspaee

VI

f E B , where

V2 .

VI .

, let

ion with range

a least constant

and nullspace

(~n

satisfies

Hyllc -< rBllfllB • Proof. equation the map fn + f

It follows

from the super~osition

(2) does have a unique bounded : : f ÷ y in

8

and

is linear. Yn = Yf

n

principle solution

We will show that

+ y

in

y(O)

C .

that,

y(t) :

f E 8 , the

for every

with

y(O)

E V2 .

has a closed graph.

Then

= lim Yn(O)

6 V2

n~¢o and, for any fixed

t ,

So.(S>ds: lim Sh<s>ds. n->oo Hence y(t) - y(O) = lim SoYn(S)ds n-~m = lira /~{A(s)Yn(S) n-~O

+ fn(s)}ds

Moreover Suppose

23

.tJo{A(s)y(s) + f(s)}ds . Thus

y(t)

y = If

is a solution of the equation

(2)°

Therefore,

since it is bounded,

.

It now follows continuous.

from the Closed Graph Theorem that the linear map

This proves the existence

of some constant

T

is

r B > 0 , and the existence o f

a least one can be deduced immediately. Put "IX(t)Px-l(s)

for

0 ~ s < t ,

[-X(t)(l

for

0 ~ t < s .

G(t, s)

Then If

G f

is continuous is a function

- P)X-I(s)

except on the line

in

B

s = t , where

which vanishes

for

t > tI

it has a jump discontinuity. then - - cf.

(4) - -

tI y(t) = /0 G(t, u)f(u)du is a solution o f (2).

Moreover it is bounded,

since

t y(t) = X ( t ) P f o I X - l ( u ) f ( u ) d u

for

t e tI ,

and t y(O) -- -(I - P ) f 0 Z x - l ( u ) f ( u ) d u Therefore,

b y Proposition

~ V2 .

4, llyllC ~ rBIlfl]B •

We can now complete and let

f

the p r o o f of Proposition

be the function

L0 s ~ 0

and

h > 0 .

Then

ly(t)l Dividing by

h

and letting

Let

be any constant vector

defined by

f(t) = I ~

where

i.

for

s _< t < s + h ,

otherwise,

f ~ [

= I~+hG(t,

and

IIfllL = hiE[.

u)~dul ~ rLhl~l •

h + 0 , we obtain

for any

IG(t, s)~l ~ rLl~l

t ~ s

Therefore

24

Hence,

since

~

is arbitrary,

IG(t, s)r ~ r L Thus

(S) holds with

the excepted

case

r = rC .

and

~ = 0 .

By continuity,

(3) remains valid also in

s = t .

The deduction that the equation

K = rL

of Propositions

2 and 3 is not quite so immediate.

(2) has a b o u n d e d solution

for every function

We suppose now

f ( C

and put

Take f(t) = ~(t)×(t)/Ix(t) I

where ~(t)

x(t) = X(t)~

t ~ 0 , of

is any nontrivial

is any continuous

¢

~(t) = 0

for

real-valued t ~ tI .

solution

function

of the homogeneous

such that

IIfllC < i

Then

equation

0 ~ ~(t) ~ i

(i) and

for all

and hence, by the arbitrary nature

_

,

tI Ift0G(t , u)x(u)Ix(u)I-idu;'''

Putting

t I = t , resp.

J r

for

0 ~ tO ~ tI

and

t ~ 0 .

t O = t , we obtain

IX(t)P~II~

IX(u)~l-ldu ! r

for

t ~ tO ~ 0 ,

0

(5) t Ix(t)(l - P ) ~ I f t l l x ( u ) ~ l - l d u Replacing

~

by

P~

, resp.

(I - P)~

~ r

for

t ~ tI ! =

, it follows by integration

I s IX(u)P

s

,

_< Le - ~ ( s - t )

for

s

>_ t

,

46

where

K , L

are positive

constants

independent

~

of

I~o~ ~1.

~

and

6.

CRITERIA FOR AN EXPONENTIAL DICHOTOMY

Proposition 1.5, with asymptotic stability.

We intend to derive an analogous sufficient condition for an

exponential dichotomy.

LEMMA I. n

x

n

a < 0 , gives a sufficient condition for uniform

We first establish some preliminary results.

There exists a numerical c o n s t a n t

matrix

k

n

> 0

A ,

IA-II ~ knIAIn-i/Idet

Proof.

such that, for every n o n - s i n g u l a r

AI .

Let

det(II - A) = I n - alln-i + ... + (-l)nan

: (I - I l)

...

(I - I n )

.

Since a I : Zj lj , a 2 : j 0 , then any m a t r i x

IB - AI ! h n E ( g / I A [ ) where

o < e < ~ , has

k

eigenvalues

eigenvalues with real part

Proof.

If

IR~f

< ~ -

I~I

>

21AI

and

B

n-1

if

n - k

A

is an

n x n

eigenvalues

with

satisfying

,

with real p a r t

-< -a + g

and

n

- k

>- ~ - e.

and

E

Ixl ~ 21AI

I(A - kI)-lI if

such that,

then,

b y Lemma 1,

< kn(31AI)n-~e

n

then

I(A -

XI)-ll

= I%l-il(l

- %-iA)-I 1

< (21AI)-I(I

+ 2-I

+ 2 -2 + . . , )

= IA1-1 < kn(31AI)n-/s since

g < ~ ~

IAI

and

k

~ i .

n

It f o l l o w s

n ,

that

if

IB - A 1 ~ g Y k n ( 3 1 A I ) then

B - hi

is i n v e r t i b l e

I(B Thus

aii eigenvalues

B(8)

= 0B + (i - 8 ) A

-

l I )

of

for

(A

-

B

Since

the eigenvalues B

LEMMA 3.

have

the same

have

of

I :

real

- n I !

B(0)

number

part

are

IB - A I

for

continuous

-

kz)-l[-I

or

~ ~ - E .

If

n > i

0 ~ @ ~ i .

functions

in t h e

constant

e

n

>

~ -a

, since

the result

n-I

of

8

it f o l l o w s

that

left half-plane.

such that,

0

or

for the left half-plane

IPI ! c ( a - l l A I ) n We a s s u m e

< I(A

~ -~ + g

all have real p a r t

is its spectral p r o j e c t i o n

Proof.

, since

IB - AI

of eigenvaiues

~here exists a'numerical

m a t r i x w h o s e eigenvalues P

lI)

-

< ~ - g

then IB(e)

and

IR%I

n-I

if

A

~ ~ , for some

is an

for

n = i .

x n

~ > 0 , and if

then

.

is t r i v i a l

n

We h a v e

49

p = 2~]y(z!where

y

A)-Idz ,

is any rectifiable simple closed curve in the left half-plane which contains

in its interior all eigenvalues of Y : YI U Y2 ' where

YI

A

with negative real part.

is the left half of the circle

segment of the imaginary axis.

On

YI

J(zl - A)-IL

We will take

Izl : 21A I and

Y2

is a

we have : IzI-~l(I

- z-iA)-ll

_< 2-iIA{-l(i + 2 -i + 2 -2 + ...) = IAl-i

.

There fore I/yl(z~ - A)-idzt

~ ~ . 21Ai.

IAt -1 : 2~ .

On the other hand, by Lemma i, l(zl - A)-i{ ! knlZl - Aln-i/Idet(zl - A) I

= knlZl

where

kl, ..., kn

- AI n-

are the eigenvalues of

•//!11~jl z -

A .

,

Therefore n

Ify2(Zl - A)-idzl ! kn(31Al)n-lff~

~

{iy - ljl-idy .

j=l

But

fly - k j l t

~

and, by Schwarz's inequality, 2

ff~ [ fly _ ljl-ldy _< ff ( 2 + y2)-idy : - i j=l

.

Hence i/y2(Z I _ A)-idzl < ~kn(3-iiAi)n-i Thus for the projection

P

.

we obtain finally Jml ! i + ½

3n-ik (~-llAl)n-i . n

Since

~ < IAI , this inequality can be written in the required form. Suppose now that

A(t)

is a continuously differentiabie matrix function

satisfying the conditions of Lemma 3 for every the formula

t

in some interval

J .

Then from

50

P'(t) = 2 ~ / y [ z I

- A(t)]-IA'(t)[zI

- A(t)]-idz

we obtain in the same w a y

IP'(t)l where

d

n

> 0

~ dn(a-llA(t)I)2n-llA'(t)l/IA(t)l

is a n u m e r i c a l constant.

We are now ready for our main result.

PROPOSITION I . interval

(i)

J

Let

has

A(t)

IA(t)l - 6 > o

for all

for all

< -a < 0

and

n - k

eigenvalues

t ( J ,

t E J . e < min(a

there exists a positive constant

, 8)

such that, if

IA(t 2) - A ( t l ) I _< 6 h > 0

matrix function defined on an

eigenvalues with real part

For any positive constant 6 = ~(M , ~ + 8 , e)

where

n x n

such that

with real part (ii)

be a continuous

A(t)

for

It 2 - tzl _< h

,

is a fixed number not greater than the length of

J

,

then the equation

x' = A ( t ) x

has a fundamental matrix

X(t)

satisfying

Ix(t)#x-Z(s)l

(i)

the inequalities

_< Ke - ( C ~ - e ) ( t - s )

for

t~s,

T] ~ 120K4e -(Y-g/6)h l(I - ~)T-II If

h > 0

A(u)

is so large that

has

k

eigenvalues

P .

- a + e

and

n - k

eigenvalues

8 - e . A(t)

(i) has an exponential

arbitrary projection

then it follows from Lemma 4 that

with real part less than

It is evident that if the matrix then the equation

,

A 120K4e -(Y-e/6)h

120K 4 < 2-½e 5Eh/6

with real part greater than

, we have

is continuous

dichotomy on

J

on a compact interval corresponding

Thus the concept of exponential

interval might appear to be without if the interval is sufficiently dichotomy and the coefficient

interest.

J

to an

dichotomy on a compact

Nevertheless,

Proposition

long relative to the constants

matrix is slowly varying then

2 shows that

of the exponential

k , and hence

P , is

uniquely determined. Finally we give another simple sufficient

PROPOSITION

3.

Let

on the half-line

~+

condition

A(t) = (a..(t)) be a bounded, 13 and suppose

for an exponential

continuous

there exists a constant

n x n

6 > 0

dichotomy.

matrix function

such that

n IRaii(t) I > ~ +j~llaij(t)l

(7)

j#1 for all

t E ~+

then the equation

where

K

and

i = i .....

n .

If

Raii(t) < 0

(i) has a funda~nental matrix

X(t)

for exactly satisfying

IX(t)PX-I(s)I

-< Ke -s(t-s)

for

t > s ,

IX(t)(l - P)X-I(s)I

< Ke -6(s-t)

for

s > t ,

is a positive

constant

and

k

subscripts

the inequalities

i

56

[iiI Proof.

We use the norm

l~r : su%1 0

ii

for

without

i > k .

ii

For any nontrivial solution

x(t)

of (i) we have

½ d/dtlxi 12 : Rx.x.'1 l

= Raiilxi 12 + Rj#i I a..x.x. l] I ] and hence Ixilj~ilaijIlxjl ~ ½ d/dtlxi 12 - Raiilxi 12

If

Ix(s)l : Ixi(s) I

for some

s ( ~+

it follows that

- ~ laij(s)l ~ ½1xi(s)I-2d/dtlxil2(s)

- Rail(S)

j#i

~ fa..(s)J j#i Therefore, by (7), It follows that

d/dtlxil2(s) Ix(t)I

±] is negative if

and positive if

i > k .

does not have a local maximum in the interior of

For suppose a local maximum occurred at ;xi(t)I 2

i ~ k

t = s , say, with

would also have a local maximum at

Ix(s)l = Ixi(s) I .

t = s

and hence

for some

i > k

+ Then

d/dtlxiI2(s) = 0 ,

contrary to what has just been proved. We show next that if

Ix(s); = Ixi(s) I

and some

s ( ~+

then

57

Ix(t)[

is strictly

sufficiently [X(tl) I ~

increasing for

small

h > 0 , and hence

Ix(t2) I ~ where

on the interval

t h s .

In f a c t

Ix(s)l

< Ix(s+h)l

s ~ t I < t 2 , then

[s~ t 2]

at an interior

Ixi(s) I < lxi(s+h) I

Ix(t)l

point,

.

for all

If we had

would assume

its maximum

value

and we have seen that this is

impossible. If

Ix(s) I : Ixi(s) I

ly small

h > 0 .

Then for small

for

For let

i > k

no

[

denote

then

Ix(s+h)I

the set of all

i

< Ix(s)I such that

for all sufficientIx(s)I

= Ixi(s) I •

h > 0

lxi(s+h)

lxi(s)t

L
k

increasing

U m = x-l(sm)V1

of the unit sphere

and vectors

there exists

k-dlmensional

decreasing

i > k , and let

vector space

the compactness

a

i : i(h) ( I .

for no

and strictly

be the k-dimensional

for every

(i) such that

m~ + ~

VI

: Ixi(s) I

paragraph,

t ~ c

We now show that there exists that

for some

.

independent.

By

58

then

x m (t) + x(t) ~)

Xm(O ) ( Um

as

~ + ~

Therefore

definition of

V1 .

increasing on

IXm(t) I

Thus

IXmv(tl) j > [Xmv(t2) I ~{+ .

for every

if

tl,

and henee

t ( ~{+

We have

Xm(S m) ( VI , since

is strictly decreasing for t 2 E IR+

Ix(tl)

and

t 1 < t 2

I >-Ix(t2)

I .

It is actually decreasing on

t < Sm " by the

then,

Thus

for

all

Ix(t)l

~{+ , since if

large

v

is nonIx(t)l

were

constant over a subinterval it would have a local maximum in the interior of There exists also an Ix(t)l vectors x(t)

(n - k)-dimensional

is strictly increasing on ~ = (~i)

with If

s ( R+ ,

such that

x(0) 6 V 2

x(t)

~+

~i = 0

for every

V2

x(t)

[R+ such that

is the subspace of all

i ~ k , the nontrivial solutions

have this property.

is a solution such that

Ix(s)l

subspace of Solutions

In fact, if

,

= Ixi(s) 1 for some

Ix(t)I

i > k .

increases on

Since

~+

d/dtlxil2(s)

then, for every

> 261xi(s)l 2 , by

(7), it follows that lim h--~0 I Ix(s+h)12 _

Ix(s)l

I/h

lim ~I xi(s+h)l >-h--~7+O

2

-I×i(s)l

h

I/

261x(s)J 2 Integrating this differential inequality for the continuous

function

''Ix(t)l2 , we

get Ix(t)l ~ {x(s)le 6(t-s) A similar argument shows that if ~+

x(t)

for

t ~ s .

is a solution such that

Ix(t)l

decreases on

then

Ix(t)l ~ Ix(s)le -6(t-s)

for

t ~ S .

Since the equation (i) has bounded growth, the result now follows immediately. Proposition line

~+

3 remains valid, with only minor changes in the proof, if the half-

is replaced by the whole line

~ .

7.

L y a p u n o v functions

DICHOTOMIES AND LYAPUNOV FUNCTIONS

are a standard tool in stability

will consider their relationship with linear differential

with dichotomies.

equations

theory.

In this lecture we

Since we will be concerned only

it is natural to restrict attention

to quadratic

Lyapunov functions. Let xmH(t)x

A(t)

be a continuous

, where

function,

H(t)

matrix function

for

t ~ 0 .

is a b o u n d e d and continuously

will be a L y a p u n o v function

The Hermitian

differentiable

for the linear differential

form

Hermitian

matrix

equation

x' = A(t)x if its time-derivative exists a constant

along solutions

q > 0

norm.

x(t)

By replacing

solution

x(t)

H(t)

of (i) is negative

+ H(t)A(t)

of (i). by

is arbitrary,

+ A*(t)H(t)]x(t)

n-iH(t)

we can assume condition

+ A*(t)H(t)

~ -I

We are going to show that this is almost equivalent

I.

i.e.,

if there

~ -nlx(t)l 2

Here, and in what follows, we use the Euclidean

the p r e c e d i n g

Hr(t) + H(t)A(t)

PROPOSITION

definite,

such that

x*(t)[HF(t) for all solutions

(i)

q = i .

Then, since the

is equivalent for

to

t ~ 0 .

(2)

to an exponential

If the equation (l) has an exponential dichotomy on

dichotomy.

~+ , then there

exists a bounded, continuously differentiable Hermitian matrix function

H(t)

which

satisfies (2). Proof.

Let

hypothesis

X(t)

be the fundamental

there exists a projection

matrix for (l) such that P

and positive

constants

IX(t)PX-I(s)I

_< Ke -~(t-s)

for

t >- s -> 0 ,

IX(t)(l - P)X-I(s)I

_< Ke -a(s-t)

for

s _> t >- 0 .

X(0) = I . K , c~

By

such that

60

For convenience of writing put

Xl(t , s) = X(t)PX-I(s) X2(t , s) = X(t)(I and let

H(t)

,

- P)X-I(s)

be the matrix function defined by

½H(t) = J~tXl*(S,

t)Xl(S , t)ds

- X2*(O, t ) X 2 ( O , - f~X2*(s,

Then

H(t)

,

t)

t ) X 2 ( s , t)ds .

is Hermitian and .2rt -2e(t-s ½1H(t)I S K2~te-2e(s-t)ds + K2e -2~t + ~ ]0 e )ds

K2(I + - I ) Moreover

H(t)

.

is continuously differentiable and H'(t) = -H(t)A(t) - A*(t)H(t) - 2Xl*(t, t)Xl(t, t) - 2X2*(t, t)X2(t, t) .

But

Xl(t, t) = X(t)PX-I(t)

is a projection and

only remains to show that if

Q

X2(t , t) = I - Xl(t, t) .

Thus it

is a projection then

R = Q*Q + (I - Q*)(I - Q) h ½1 . Any vector

~

is in the range and

can be uniquely represented in the form ~2

in the nullspace of

Q .

{ = [i + $2 ' where

Then

~2*R~I = 0 . Hence

~*R( = ( 1 " ( 1

+ ~2'(2

~ ½(*(

.

This completes the proof. It may be noted that if

A(t)

is bounded then

Proposition 1 admits the following converse:

H'(t)

is also bounded.

~i

61

PROPOSITION 2.

Suppose (1) has bounded growth.

If there exists a bounded,

continuously differentiable Hermitian matrix function the equation (i) has an exponential dichotomy on ~+

Proof.

By hypothesis

there exists a constant

IH(t)l and a constant

C ~ 1

(The interval

length Assume

p > 0

for

ep

~ Clx(s)J

for

such that

t ~ 0

such that every solution

Ix(t)j

dichotomy).

~ p

x(t)

of (i) satisfies

0 ~ s ~ t ~ s + ep

is chosen to give a good exponent

Ix(O)I

= 1

which satisfies (2), then

H(t)

.

in the exponential

and put V(t) : x*(t)H(t)x(t)

.

Then V'(t) = x*(t)[H'(t) _~

+ H(t)A(t)

+ A*(t)H(t)]x(t)

-Ix(t)l 2

< 0 . Thus

V(t)

is a strictly

Suppose

first that

decreasing V(t) ~_ 0

function.

for all glx(u)

since for any

t _~ 0 .

Then

12du
0

-p : -~Ix(0)l 2 _< -v(0) _~ v(t) - v(0)

_< -/~lx(u) 12du

.

It follows that iimlx(t) I = 0

Let

t

m

be the least value such that

and, by the same argument

Ix(tm) I = e -m/2

.

Then

0 = tO < tI
T .

exists

V(t, ~v)

< ~

Then

and

and

value

b e the s u b s p a c e

o f (I), a n d let ~ ~ V 2 , let

V2

Ix(t • ~)I N ~ 1

be a n y

b e the

t~+ ~ v

.

T

that

t'

and

number

~

~ .

Then

v >_ v'

V(t',

, which

s u c h that

Since ~)

V(T, ~) ~ -p ~ ~ V2

Ix ( T , ~)I ~ N l x ( t ~

tm

tO S T •

such that Suppose

= x(t, ~)

.

IX(tm) I = e m/2

t ~ T .

If

Since .

~)I

Ix(t)I + ~

Then

t m _< t < tm+ I


1

there exists

0 ~ t ~ T . solution

of

fixed subspace

= x(t, $)

In fact, o t h e r w i s e

t _> t'

a positive

x(t)

we m a y s u p p o s e

~) < ~ for

V2

consisting

We w i l l s h o w t h a t

t + ~

in

V(t',

again a particular a greatest

VI

and a sequence

w e can c h o o s e

~ 6 V2

let

t = 0 . as

we have

~ 6 V2 .

Moreover,

0 ~ t o < t I < ...

$

Ix(t)l + ~

, ~ ) Z ~ for a l l

Thus t h e r e e x i s t s

unit vector Consider

there

t + ~

unit vector

at

o f the u n i t s p h e r e

, and hence

contradiction. every

solutions

For any unit vector

t a k e s the v a l u e

< 0 .

implies

vector space,

values of all bounded

= V(t, $) + - ~

sequence

t + ~

b e the u n d e r l y i n g

supplementary solution

as

~ p-lv(t*)

then

for and for

63

-PJX(tm+l)J 2 _< V(tm+ I)

V(tm+ I) - V(t) /tm+lj -~t

'x(u)J2du

-(tm+ I - t)JX(tm)I 2

JX(tm+l)J 2 = eJX(tm) J2

Since

tm+ 1 - T S ep

if

it follows

that

tm+ I

t m -
T

and

In either case

tm < T .

Jx(t)J S CNIX(tm) j . Suppose

T ~ t ~ s < ~

If

t m -< t < tin+I

and

s < tn+l(O ~ m Z n)

tn

Jx(t)J ~ CNJx(tm) j = e½CNe -(n+l-m)/2

, then

X(tn)]

e½CNe-(n+l-m)/21x(s)J e½CNe-~(s-t)Jx(s)J where

e = (2ep) -I

an exponential

as before.

dichotomy

Since

(i) has bounded

on the subinterval

IT, ~)

,

growth,

it follows

that it has

, and hence also on the half-line

+ We show next, by an example, omitted

in Proposition

function for

such that

t ÷ ~ , and

explicitly.

2.

Let

~(t) ~ 1 ~(n)/~(n

that the hypothesis

#(t) for all

t ~ 0 ,

- 2 -n) ~ ~

The differential

~0{~2(t)

x(t) = x(0)e-t#(t)

uniformly

On the other hand,

- l}dt < = ,

is continuously h'(t)

Moreover

h(t)

is bounded,

#(t)

= o(e t)

can easily be constructed

- l]x

and hence

is asymptotically

if we set

h(t) = { e 2 t / ~ 2 ( t ) } < e - 2 S ~ 2 ( s ) d s h(t)

cannot be

differentiable

equation

has the solutions

then

growth

continuously

Such a function

x' : [~'(t)/~(t)

stable.

of bounded

be a real-valued

differentiable

and

+ 2{~'(t)/~(t)

-

since

1}h(t)

= -1

.

stable,

but not

64

0 Z h(t) = {e2/~2(t)}]te-2Sds~/~

+ {e2t/~2(t)}<e-2S{%2(s)

- l}ds

9 + ~0{%2(s) - l}ds . The p r e c e d i n g results equations.

can he made more precise

in the special case of autonomous

Thus we now consider the equation x ~ : Ax ,

where

A

is a constant matrix.

can also be chosen constant.

We show first that the matrix

The argument

differs only slightly

H(t)

o f Proposition i

from the previous

one.

PROPO$1IION 3.

If the matrix

a Hermitian matrix

H

A

has no pure imaginary eigenvalues, then there exists

such that HA + A*H ~ -I .

PrOOf.

Let

half-planes

P+

and

P_

respectively.

he the spectral projections

of

A

for the right and left

Then

-J~oe-tA*p+Sp+e-tAdt

y = is defined and

YA + A*Y = ~ 0 d

, -tA )dt (e -tA* P+~:P+e

= -p *p + + Similarly Z = ~0etAep_*P

etAdt

is d e f i n e d and ZA + A*Z : -P *P The m a t r i x

H = 2Y + 2Z

is Hermitian and HA + A*H = -2P + +*P

Proposition ~nt

that the t i ~

matrix

2 can be s h a ~ e n e d ~rivative

C , the ordered pair

~r

autonomous

o f the ~ a p u n o v (A, C)

- 2P *P

~ -I . equations

by relaxing the require-

function be negative

is said to be controlZ~le

~*Akc = 0

for all

k ~ 0

if

de~nite.

For any

65

implies that the vector k

such that

~ : 0 .

By the Cayley-Hamilton

theorem we need only consider

0 ~ k < n .

PROPOSITION 4.

Suppose there exists a Hermitian matrix

H

such that

HA + A*H = -C ,

where

C = Cm ~ 0 . Then

singular if and only if

A

(3)

has no pure imaginary eigenvalues and is controllable.

(A*, C)

positive (negative) eigenvalues of

H

H

is non-

In this case the number of

is equal to the number of eigenvalues of

A

with negative (positive) real part. Proof.

Suppose

corresponding

first that

eigenvector

A

has a pure imaginary eigenvalue

< .

Then, by

-~*C~ : i ~ * H ~ Since

C ~ 0 , this implies

that

cAk IAI

for

P+

y

be the simple

Then the spectral projections

A , for the right and left half-planes,

Here the expression

Let

o f a segment of the imaginary axis and

P_ = 2~Tfy(Zl

- A)-idz

,

P+ = 2~T/x(zl

+ A)-Idz

.

are give D by

has been obtained b y the change o f variable

z + -z

from the more usual expression b y a contour integral over the curve symmetric to

Y

If we set G = H - P *H - HP then G = 2~fy{(zI

= 2~TI7(zl

On the circle

Izl = r

+ A*)-IH - H(zl - A)-l}dz

+A*)-IC(zl

the integrand

G : -T~JA

is

_ A)-Idz

O(r -2)

- i~If~-lc(A

.

.

Thus, letting

- i~I)-ld~

.

r -~ co we obtain

.

68

Consequently real

~

G ~ 0 .

, and hence

and only if

Moreover

if and only if

(A m, C)

For any vector

On the other hand,

In general,

if

G~ : 0

if and only if

cAk< = 0

for all

C(A - i~I)-l~ = 0 k h 0 .

Therefore

for all G
i

in

B(~)

in

B(~+)

.

.

and y(t)

is a solution of the c o r r e s p o n d i n g

= f~(u)du

equation

x(t)

(i).

Since

L,(u)du: fl*U)du: ½, we have y(t) : ½x(t) By hypothesis

are solutions respectively.

the equation

for

t ~ i , y(t) = -½x(t)

(I) has a solution

9(t)

z+(t)

: 9(t)

- y(t)

+ %x(t)

,

z (t)

= ~(t)

- y(t)

- ½x(t)

,

of the homogeneous Moreover,

for

since

equation y(0)

t ~ -I .

which is bounded on

(2) which are b o u n d e d on

= 0 ,

z+(o)

: ~(o)

+ ½x(O)

,

z (o)

: ~(o)

- ~(o)

.

.

[ .

is any fixed solution of (2) and

x(t)

¢(t) ~{1-oltl Then

, [

Then (i) and (ii) are clearly satisfied,

8(~ - ) , is the restriction Take

, y(t) : y_(t)

~ .

fl{+ , ~{

Then

,

69

Hence

x(O) = z+(0) - z (0)

and

By combining Proposition conditions.

x(t) = z+(t) - z_(t)

i with Propositions

PROPOSITION 2.

Suppose (2) has bounded growth.

~

the proof.

3.1-3.3 we can obtain more explicit

Let

~

Then the inhomogeneous equation ( i )

for every continuous function

f

which is

if and only if the homogeneous equation (2) has an exponential

dichotomy on

Proof.

T~is completes

We will consider only one important special case.

has a unique solution bounded on bounded on

.

~ . X(t)

be the fundamental matrix for (2) such that

there exist a projection

P

and positive constants

IX(t)Px-l(s)l

S Ke -~(t-s)

for

K , ~

X(0) = I

and suppose

such that

t ~ s , (3)

IX(t)(l - P)X-I(s)I Then for any bounded continuous

S Ke -e(s-t)

function

f

for

s Z t .

the corresponding

inhomogeneous

equation

(i) has the bounded solution y(t) = ft X(t)Fx-l(s)f(s)ds Moreover,

To prove the necessity bounded growth, ~

of the condition we use Proposition

3.3.

(Proposition

bounded growth is replaced by a hypothesis corresponding

projections.

Then

P+V

there exists a projection an exponential

dichotomy on

A matrix function of real numbers

P

A(t)

continuous

R on

as

of bounded decay). (I - P )V

P+ V

span

Let

on

P+ , P

{k~}

P .

such that the translates

This implies that

results can be strengthened.

A(t)

A(t)

{h }

ACt + k~)

is bounded and

of (2) is almost periodic the preceding

We first prove

Hence

(I - P )V , and (2) has

R .

When the coefficient matrix

of

be

is said to be almost periodic if every sequence

~ +

+

V , by (iii), and have

and nullspace

with projection

contains a subsequence

converge uniformly on

Since (2) has

dichotomies

(2) has no nontrivial bounded solution.

with range ~

i.

(2) has no

of bounded growth.

3.3 remains valid if the hypothesis

and

only the zero vector in common, because

(4)

.

equation

Here we have not used the hypothesis

(i) and (ii) imply the existence of exponential

, by Proposition

uniformly

- P)X-I(s)ds

this bounded solution is unique, since the homogeneous

nontrivial bounded solution.

and

- <X(t)(I

70

LEMMA 1.

A(t)

Let

be a continuous matrix function on

(2) has an exponential dichotomy (3) on A(t + h ) ÷ B(t)

~+

R

and suppose the equation

If, for some sequence

uniformly on compact subinter~oals of

R

then

X(h

h

,

)px-I(h~) -~ Q

~nd ~he equation

y' has an exponential dichotomy on

Proof.

: B(t)y

(s)

with projection

Q

and the same constants

K, a .

The translated equation x' : A(t + h )x

has the fundamental matrix

X (t) = X(t + hv)x-l(h

IX (t)P X -l(s)I

where

P

for

t Z s ~ -h v ,

I X ~ ( t ) ( I - P ) X - I ( s ) I ~ Ke -~(s-t)

for

s Z t ~ -h~

.

Since

we can assume that

X (t) + Y(t)

for every

Y(O) = I , it follows

P

IP I ~ K , by restricting + Q , where Y(t)

t , where

Q

attention

is a projection.

is the fundamental

for

-~ < s ~ t < ~

IY(t)(I - Q)Y-I(s) I ~ Ke -~(s-t)

for

-~ < t ~ s < ~

it follows

PROPOSITION 3.

to a

Since

matrix of (5) such that

~ Ke -~(t-s)

Since the projection

,

that

IY(t)Qy-I(s)I

determined

and

_< Ke -a(t-s)

= X(hv)px-I(hv)

subsequenee

)

corresponding that

Suppose

P

A(t)

to an exponential

* Q

dichotomy

without restriction

,

is uniquely

on

to a subsequence.

Then the

is an almost periodic matrix function.

following statements are equivalent: (i)

the homogeneous equation (2) has an exponential dichotomy on

(ii)

the homogeneous equation (2) has an exponential dichotomy on

(i£i)

(iv)

for every

the inhomogeneous equation (i) has a solution bounded on almost periodic function

f(t) ,

the inhomogeneous equation (i) has a solution bounded on almost periodic function

+ •

f(t)

.

+

for every

71

Proof.

(i) implies

sequence

h

÷ ~

(ii) implies Proposition

(ii):

This follows at once from Lemma i, since there exists a

such that

A(t + h ) ÷ A(t)

This was established

(iii):

(iv) implies

(iv): (i):

Since

A(t)

is bounded it is sufficient, by Proposition

if it has a solution bounded on The set

A

of all functions

of almost periodic

functions on

sup -~O

Let any

e

g(t)

such that

2~/m > T equation

be any function in 0 < ~ < 2~/T

C(~+)

which coincides with

T

g(t)

IIy~II ~ rIIgII.

f~(t)

Thus the sequence

subsequenee we can assume that intervals,

on

[0, T]

y(t)

with

a sequence of values

continuous periodic functions

it follows that

T

be any positive number.

we can find a continuous

(i) has a bounded solution

Now give

and let

T

function

and satisfies

÷ ~

y~(t) + y(t)

~fll ~ IIgll •

We obtain corresponding

is bounded.

y~(O) + N .

with period Then the

IIyll ~ rllgll •

and bounded solutions

{y~(O)}

f(t)

For

Since

for every

of the equation y' = A(t)y + g(t)

y~(t)

÷ g(t)

t , where

of

with

By restricting

%(t)

sequences

attention to a

uniformly on compact y(t)

is the solution

72

such that

y(0) = n •

Evidently

y(t)

is bounded and

IIyll ~ rIIgll .

This completes

the proof. The argument asserted. .

Let

in the last part of the proof establishes

S

Then the set

be any module, A(S)

i.e., an additive

of all almost periodic

rather more than has been

subgroup

functions

of

~ , which

f(t)

is dense in

with Fourier series

i~kt eke

, where

~

( S

for all

that in (iv) it is sufficient for every omitted.

f E A(S)

.

For example,

k

, is a closed subspace

to assume the existence

However,

the hypothesis

that

of

A .

The proof shows

of a solution b o u n d e d on S

is dense in

~

~+

cannot be

the scalar equation y' : ½iy + f(t)

has a solution function

of period

f(t)

2~ , and hence a b o u n d e d solution,

of period

for every continuous

27 .

It should be noted also that in (iii) the solution b o u n d e d on (ii).

Furthermore

it is almost periodic

joint frequency module of argument

A(t)

and

and its frequency

f(t)

•

is unique, by

module is contained

in the

This may be established by a similar

to that used to prove our next result.

PROPOSITION 4.

A(t)

Suppose

is an almost periodic matrix function and the equation

(2) has an exponential dichotomy (3), with fundamental matrix

P . Then

Let

uniformly

{h}

on

the normality U(t + h )

A(t)

and projection

.

be any sequence

[ , with limit properties

converges

x(t)

is almost periodic and its frequency module is contained in

X(t)PX-k(t)

the frequency module of Proof.

~

of real numbers

B(t)

say.

If we put

of almost periodic

uniformly on

[

By Lemma i and its proof, as

such that

A(t + h )

U(t) : X(t)PX-I(t)

functions,

it is sufficient

converges then, by to show that

.

v ÷

U(t + h v) : X ( t ) U ( h ) x v - l ( t )

÷ Y(t)Qy-l(t) where

Q

is a projection

such that

Y(O) : I .

and

{k }

of

is the fundamental

Moreover the convergence

it were not uniform on subsequence

Y(t)

= V(t)

matrix of the equation

such that~ IU(t

for all

+ k ) - V(t

(5)

is uniform on compact intervals.

there w o u l d exist a sequence {h }

, say,

V ,

)I ~ y > 0 .

{t }

of real numbers

If

and a

73

By restricting attention to a further subsequence we may suppose that converges

uniformly on

uniformly on

~ .

~ , with limit

C(t)

say.

By what we have already proved,

uniformly on compact intervals,

to

Z(t)pz-I(t)

Then also U(t + t

, where

Z(t)

A(t + t

B(t + t~) ÷ C(t) + k )

converges,

is the fundamental

matrix of the equation z' = C(t)z such that

Z(0) = I

and

V(t + t )

converges to

P

is a uniquely determined projection.

Z(t)pz-I(t)

.

Therefore

lu(t~ + k ) - v(tv) I + 0 , which is a contradiction.

+ k

Similarly,

9.

Let

A(t)

continuous

n × n

contains

locally

function limits

be an

A(t) A(t)

The set

uniformly,

A

{k }

hull

is translation

to an element o f

A .

~ x(s)

~)

of

as

~ If

theorem,

of

A(t)

h + 0 .

contains

x (t)

i.e.,

if

Moreover

of

A(t + k )

interval. The set

The set

A(t)

~ A

The limit A

o f all such

then also

A(t + h) ~ A(t) A

a subsequence

Aw(t) ÷ A(t)

, where

continuous.

{h }

.

invariant, h .

any sequence

such that the translates

i.e. uniformly on every compact

for any real n u m b e r

o f elements

x (s)

Then, by Ascoli's

is again b o u n d e d and ~niformly

(and even uniformly on

then

[ .

a subsequence

is called the

A(t + h) 6 A

sequence

m a t r i x function which is bounded and uniformly

on the whole line

real numbers converge

DICHOTOMIESAND THE HULL OF AN EQUATION

locally uniformly

is also compact, which converges

locally uniformly,

is the solution

~

i.e., any

locally uniformly

+ ~ , and

of the differential

s

~ s

equation

x' = A ( t ) x such that

x (0) = ~

and

x(t)

is the solution o f the differential

equation

x' = A(t)x such that

x(O) = ~

We propose

.

It is just these properties

to study the

family

of differential

x' = A(t)x We first prove

o f the hull that we w i l l actually

,

A 6

A

equations .

(l)

75

LEMMA I.

The equations

exists a constant

(i) have uniformly bounded growth and decay;

C > 0

such that every solution

that is, there

of an equation (I)

x(t)

satisfies

Ix(t)l ~ clx(s)l Proof.

Otherwise,

solutions

since

x(t)

A

for

is translation

|t

with

Ix (O)I = 1 , and a sequence

that

Ix~(s )I ~ ~

invariant,

s 1

.

there exists a sequence

of

By r e s t r i c t i n g

attention

locally uniformly,

x (0) + ~

-i ~ s ! i .

Then

, where

x (s)

x(O) = ~ .

+ x(s)

( A

o f real numbers

A (t) ~ A(t)

(i) v {s }

with

-i ~ s

to a subsequence

where x(t)

~ 1 , such

we may suppose that

|~I : i , and

+ s

s

where

is the solution of the equation

(i)

Thus we have a contradiction.

Until further notice we now make the following (H)

sl

of equations x' : A (t)x , A

such that

-

fundamental

NO equation (i) has a nontrivial solution bounded on

assumption:

~ .

With its aid we first prove

If

LEMMA 2.

holds, and if

(H)

that every solution

Otherwise,

T > 0

such

of an equation (1) satisfies

x(t)

Ix(t)l S 8

Proof.

o < @ < l , there exists a constant

since

A

sup lu-t |~T

Ix(u) I

is translation

s -1 ~

for

-~ < t
0

such that

Ix(u)l

lul~T for every solution sequence

t

÷ ~

x(t)

o f an equation

, and a sequence

(1) with

o f solutions

|x(0) I = 1 .

x (t)

Thus there exists a

of equations

(i) V

with

Ix (O)I = 1 , such that

sup

Ix (t) I < @-i •

Itl~t~ By restricting

attention

l~I = i , and

A (t) ÷ A(t)

for every real each fixed

t

Ix(t)l ~ @ -1 .

to a subsequenee

t , where we have

locally uniformly, x(t)

Ixg(t)l

Therefore,

we may suppose that

is the solution < 0 -1

where

A ( A .

Then

of (I) such that

for all sufficiently

by the hypothesis

xg(O) + ~ , where

large

(H) , x(t) £ 0 .

x (t) + x(t)

x(O) = ~ .

For

9 , and hence

Thus

~ = 0 , which

is

76

a contradiction. It follows positive which

from Lemma 2, by Proposition

constants

is bounded

K , ~

on

such that

for every solution

each equation

solution

x(t)

for

(i) has an exponential

of an equation

(i) which

(I) has an exponential

If, for a given equation [+

and a solution

hounded

dichotomy

independent

of the particular

have

projection,

on

(1)m

locally

uniformly,

then

{P }

equation

m .

to

P .

÷ P

has a convergent

equations

Thus

Consider uniformly

P

dichotomy

K

on

and

~

on

R

matrix which

on

hm ÷ ~

~

P

dichotomy

takes

P

I

at

If the equations and if

Am(t) ÷ A(t)

and the equation

In fact the bou~nded

is its limit then the limit

on

(i) and suppose

P , since the

~{ with constants

and so the whole

P+

are

and the

the value

equation.

P .

on

has an

that the norm of the

with projection

dichotomies

If

of the equation

If

bounded

, this equation

is again a projection,

subsequence.

equation

(H)

It follows

with projection

dichotomy

,

are independent

~{ with projections P

for every

,

is the sum of a solution

of the particular

determined,

now an arb&trary

[{

Similarly,

on

and decay.

have exponential

is uniquely

for some sequence

exponential

growth

where

dichotomy

on

[+

of the exponential

since

to the fundamental

(i) has an exponential

approximating of

(i)

•

-~ < t 0 .

and fix

~ > 0

so small that

% = £K[y -I + (~_y)-l] < i .

Let z(t)

~

be any vector in the range of be any continuous

function

from

P0

such that

R÷

into

~n

I~I < (i - 8)K-16 such that

, and let

85

fl~ll

= sup

eYtlz(t)l ~ 6

.

t~O

Consider the integral operator

Tz(t) : Yh(t)Ph[

+

/~Yh(t)PhYh-l(s)fh [s, z(s)]ds - ~tYh(t)(l - Ph)Yh-l(s)fh[s,

z(s)]ds

.

Using (5), it is easily verified that

Ilrzlt ~ Klel

* ell~ll < 6

and

tF h

- rz211 ~ ellz I - ~2tl.

It follows, by the contraction principle, T

has a unique fixed point

zh(t, $)

that for any given vector

in the ball

Hzh(t, ~)If-< ( i Moreover,

for

llzll~ ~ , and

e)-iK1~[

•

t = 0 Zh(O , ~) = Ph~ - g ( I

and hence, as

the operator

i~l + 0

- Ph)Yh-l(s)fh[s,

,

Zh(0, ~) : Ph ~ + o(I~I) The fixed point

Zh(S, ~)]ds

zh(t , ~)

uniformly in h .

is a solution of the differential equation

z' = Cx[Xh(t)]z + fh(t, z) and thus xh(t, ~) = zh(t, ~) + xh(t) is a solution of the original equation (3).

Thus for each

dimensional family of solutions of (3) which converge to

h Xh(t)

we have an

(n-m)-

exponentially as

t ~ We wish to show that the function origin onto a neighbourhood of the point

(h, ~) + ~n(O, ~) Xo(0) •

For

maps a neighbourhood of the

lhI + [~[ + 0

~(0, ~) : ~ + o(]~t) , Xh(O) = g(O) + gu(O)h + o(lhl)

,

we have

86

and hence Xh(0, 6) = Xo(0) + gu(0)h * ~ + o(lh I + I~l) • The linear map

(h, ~) -~ gu(0)h + ~

is invertible, since

the nullspaee and range of the projection

P

o

gu(0)h

and

~

run through

Therefore the assertion follows from

the topological inverse function theorem. Consequently, for each solution m-dimensional torus

x(t)

of (3) which passes sufficiently near the

g(u) , on which the given quasi-periodic solution

dense, there exists a vector

h = (h i , ..., h m)

x (t) o

is

such that the difference

x(t) - g(ml t + hl, ..., 0~mt + h m) tends to zero exponentially as

t + ~

orbital stability of periodic solutions.

This generalises a standard result on the

APPENDIX:

THE METHOD OF PERRON

We describe here an alternative in the Notes,

functional-analytic Let

x(t)

approach

to the problems

of Lecture

and give an application which does not seem amenable

A(t)

approach.

be a continuous

n x n

matrix function on the half-line

be a fundamental m a t r i x for the linear differential x' = A(t)x

By Gram-Schmidt

3, mentioned

to our previous

orthogonalisation

and let

equation

(l)

.

of the columns o f

column, we obtain a unitary matrix

~+

U(t)

X(t)

, starting with the last

and a lower triangular

matrix

Y(t)

such

that X(t) = U(t)Y(t) Moreover,

U(t)

main diagonal

of

differentiable,

and

Y(t)

Y(t)

are uniquely

determined

to be real and positive.

it is easily seen that then

The change of variables

x = U(t)y

B = U-IAu - U-~'

equation, since

U

.

Since

B(t) : Y'(t)y-l(t)

Y(t)

if we require

Since

U(t)

replaces

y' = B(t)y Where

.

and

X(t) Y(t)

the equation

the elements

is continuously are also. (i) by

,

(2)

is a fundamental m a t r i x of the transformed

is lower triangular with real main diagonal.

is unitary, (UU*)'

= U'U* + UU *t = 0

and hence U(B + B*)U* = A + A* . Therefore

lIB

+

B*II

in the

=

IIA

+

A*]I •

Moreover,

88

n x n

For any

matrix

C = (Cjk)

it is easily shown that

n-1Zlojkr 2~ llcll2 ~ Z j,k Since

B

is triangular and

lejkl2

j,k

bkk = b~k

, it follows

that

iiBi12~ j,k ibjk12 O

such that the inhomogeneous linear

differential equation (4) has a bounded solution for every bounded continuous function g(t)

if it has a bounded solution for

g(t) = gl(t) . . . . , gn(t)

gk(t): glk'(t) 1 gnk(t) 1

, where

89

is a continuous function such that, for all

t ~ 0 ,

Rgkk(t) h i , Igjk(t) I ~ 6 (j : 1 . . . . . k - i) . Proof.

Suppose first that

n = 1 .

Then

y(t) : exp I~B(u)du is a positive solution of the corresponding homogeneous equation (2). (4) with

g(t) = Rgl(t)

has a real bounded solution

The equation

Moreover

wl(t) .

wl(t)

has

the form wl(t) -- y(t){Wl(O) + foRgl(u)/y(u) du} . The expression between the braces has a finite or infinite limit as

t + ~

If this

limit is zero then wl(t) : -y(t) 0

du .

such that

for all large

t ,

and hence also y(t)f~Rgl(u)/y(u) du is a bounded solution of (4).

Since

Rgl(u) ~ i

it follows that either

y(t)/y(u) du or f~y(t)/y(u) du is bounded.

g(t)

Hence for any bounded continuous function

the equation (4) has the

bounded solution w(t) : -y(t)~g(u)/y(u)

du

or t w(t) = y(t)fog(U)/y(u) Suppose next that write

n > i

du

and that the result holds in lower dimensions.

We

go

[

~(t) 0 ] b(t) B(t)

B(t) =

where

B(t)

positive -~ 6 .

is an

constant

(n - i) x (n - i) corresponding

to

Then the inhomogeneous

B(t)

by Proposition

solution

w(t)

and

= ~(t)~

r

and

g(t) = gn(t)

N

~

be the

and suppose

continuous

function

g(t)

.

Moreover,

3.2,

are positive

the equation

hypothesis

Let

+ ~(t)

llw(t)II-< eNe-r-ltIlw(O)ll where

is a scalar.

by the induction

for every bounded

3.4 and Lemma

6(t)

equation

~' has a bounded

matrix

constants

+ r sup

depending

(4) has a bounded

llg(t)ll

only on

B(t)

, •

By hypothesis,

for

solution

[Wn (t)] Wn(t) Thus

~(t)

is a bounded

solution

: [c0(t) J "

of the scalar equation

00' = 8(t)co + b(t)Wn(t) We can choose

6 > 0

tO > 0

.

so small that r(n - i) ~ 6 sup t>0

and then

+ gnn(t)

llb(t)II -
t o .

n = i , the scalar equation co' : 8(t)~ + y(t)

has a bounded follows

solution

for every bounded

from the induction

every bounded

continuous

hypothesis

function

continuous

scalar

that the equation

g(t)

.

function

y(t)

(4) has a bounded

.

It now

solution

for

91

PROPOSITION 2. R+

and let

A(t)

Let

Xl(t) . . . . .

be a bounded continuous real

n

×

n

matrix function on

be a fundamental system of solutions of the linear

~(t)

differential equation (I). Then there exists a constant

0 > 0

such that (i) has an exponential dichotomy

if the corresponding inhomogeneous equation (3) has a bounded solution for f(t) : fl(t), ..., fn(t) , where

is a continuous function whose distance from

fk(t)

the subspace spanned by

Xk+l(t) . . . . ~ Xn(t)

the subspace spanned by

>~k(t). . . . . Xn(t)

Proof.

Let

X(t)

is at least is less than

i

and whose distance from

p , for all

be the fundamental matrix of (i) with columns

t ~ 0 .

xl(t), ..., Xn(t)

and form the corresponding system (2) with triangular coefficient matrix.

By

Proposition 3.3 and our previous remarks, it is sufficient to show that the inhomogeneous equation (4) has a bounded solution for every bounded continuous function

g(t) .

u-l(t)fk(t)

We know that (4) has a bounded solution for

(k = I . . . . . n) .

If we denote the columns of

Ul(t) .... , Un(t)

then

Uk(t)

xk(t) . . . . , Xn(t)

which is orthogonal to

g(t) : gk(t) = U(t)

by

is a unit vector in the subspace spanned by Xk+l(t) , ..., Xn(t) .

Since

fk(t) = glk(t)ul(t) + ... + gnk(t)Un(t) we have 2

2

glk(t)

+ ...

gl~(t)

+ gk_l,k(t)

+ ...

~ p ,

+ gk~(t)

~ I .

The result now follows from Proposition i. In the complex case we can prove in the same way

PROPOSITION 3.

A(t)

Let

be a bounded continuous

Then there exists a constant Ul(t), ..., Un(t)

with

n > 0

n

x n

matrix function on

+

and continuously differentiable functions such that the equation (k) has an exponential

llul (t)ll ~ i

dichotomy if the corresponding inhomogeneous equation (3) has a bounded solution for f(t) = f1(t), ...~ ~ ( t )

llfk(t)

- uk(t)ll

< n

, where

for

all

t ~

fk(t) 0

is a continuous function satisfying

.

In particular, when the coefficient matrix is bounded the inhomogeneous equation (3) has a bounded solution for every f

in some dense subset of

C .

f ( C

if it has a bounded solution for every

NOTES

Lecture I.

For the variation of constants formula, Gronwall's inequality~ and further

information about stability and nonlinear problems see, e.g.~ Coppel [i].

An

eigenvalue is said to be semisimple if the corresponding elementary divisors are linear, i.e., all corresponding blocks in the Jordan canonical form are

1 × 1 .

The

example of the failure of the eigenvalue characterisation in the non-autonomous case comes from Hoppensteadt [I].

The theory of functions of a matrix is discussed in

Dunford and Schwartz [i] and in Hille and Phillips [i]. Gel'land and Shilov [i], p.65. Coppel [i], p.l17.

For Proposition 6 see Coppel [3].

'logarithmic norm'

~(A)

Lecture 2.

For the inequality (4) see

A less precise version of Proposition 4 is given in The main properties of the

are described in Coppel [I], pp.41 and 58.

The terms 'exponential dichotomy' and 'ordinary dichotomy' are due to

Massera and Schaffer.

For definitions of the angle between two subspaces see Massera

and Sch~ffer [3], Chapter i, and Dalecki[ and Kre~n [i], Chapter 4.

Proposition 1 is

due to Massera and Seh~ffer [2], although it was suggested by earlier work of Krasovski[ [i]°

An indirect proof of Proposition 2 is given in Massera and Schaffer

[3], Chapters 4 and 6.

Lecture 3.

Questions of the type studied here were first considered by Perron [i],

although Bohl [i] had come very close.

Perron assumed the coefficient matrix

A(t)

bounded and showed that by a change of variables it could be supposed triangular, thus permitting induction on the dimension. using Perron's method.

Ma~zel' [I] first established Proposition 3,

An interesting application of this method, recently given by

Rahimberdiev [i], is discussed in the Appendix. For the case in which all solutions of (i) are bounded Kre~n [i] and Bellman [1] used instead the uniform boundedness theorem of functional analysis.

Massera and

Schaffer [i] used the closed graph theorem, as in Proposition 4, to establish both Propositions 1 and 3, and later also Proposition 2. generality in Massera and Schaffer [3].

Their theory is set out in full

For the closed graph theorem itself see,

93

e.g., Dunford and Schwartz [i] or Hille and Phillips [i].

Lecture 4.

The roughness of exponential dichotomies is treated in Massera and

Schaffer [3], Chapter 7, and in Dalecki~ and Kre~n [i], Chapter 4.

For the

successive approximations argument used in the proof of Lemma i el. Coppel [i], p.13.

lecture 5.

The term 'kinematic similarity' comes from Markus [1].

to Coppel [2].

Lemmas 1-3 are due

However, the use in Lemma 2 of the fundamental matrix of (4) follows

Dalecki~ and Kre[n [I].

The smoothness properties of the positive square root of a

positive Hermitian matrix follow at once from its representation by a contour integral. If, in Proposition i, the original coefficient matrix every

t

then the change of variables

x = T(t)z

sharper estimates for the various constants.

A(t)

commutes with

P

for

is superfluous and one obtains much

Propositions 2 and 3 are proved in

Coppel [3].

Lecture 6.

Schaffer [i] shows that in Lemma i one can take

conjectures that the best possible value is

kn = 2

for

k

n

: (en) ½

n > i .

extension, due to Chang and Coppel [i], of a result in Ceppel [2].

and

Proposition 1 is an The important

differential equation (4) was discovered independently by Dalecki[ and Kre[n [i] ~ and Kato [i].

Daleckii and Krein [I] give an extension of Proposition I to Hilbert space.

An approach which permits an extension to Banach space is contained in a recent article of ~ikov [i].

Proposition 2, which is new, generalises Theorem 6.4 of

Dalecki~ and Kre[n [i], Chapter 3. Proposition 3 is essentially due to Lazer [i].

It has been shown by Palmer [i]

that, for real systems, there is still an exponential dichotomy if, more generally, (7) holds with

Lecture 7.

6 = 0

and

infldet A(t) I > 0 .

The connection between Lyapunov functions and exponential dichotomies was

first considered by Ma~zel' [i].

The connection with ordinary dichotomies is also

studied in Massera and Schaffer [3], Chapter 9.

For Proposition 4 cf. Ostrowski and

Schneider [i], Dalecki~ and Kre~n [i], Chapter i, and Wimmer and Ziebur [i].

For the

concept of controllability see, e.g., Kalman et al. [i]. Propositions 2.1, 3.3, and 7.2 state properties equivalent to an exponential dichotomy under the assumption of bounded growth.

We pose the problem:

what are the

relations between these properties without this assumption?

Lecture 8.

Propositions i and 3 are contained in Massera and Schaffer [3], Theorems

81.E and I03.A.

For Proposition 4 see Coppel [2].

94

Lecture 9.

Propositions

i and 3 are due to Sacker and Sell [i], although their

original proofs used algebraic topology. found independently

by Kate and Nakajima

Sacker and Sell [2], [3]. formulation

Proposition

Proofs similar to those given here were [i].

Further results are contained in

4 is new.

analogous to that of Proposition

It would be desirable to have a

i, in which the hypothesis

is omitted and the conclusion applies to each equation Lemma 3 was suggested by Nakajima out in Coppel [4].

[i].

The application

[i].

of recurrence

m-limit set of (2).

to Hahn's problem was pointed

The result on orbital stability cannot be extended to almost

periodic solutions which are not quasi-periodic; Thomas

(i) in the

cf. Theorem 3.4 of Allaud and

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Remarks on inertia theorems for matrices,

Czechoslovak Math. J. 25 (1975), 556-561. V.V. ~ikov [i],

Some questions of admissibility and dichotomy, The method of

averaging (R), Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), 1380-1408.

INDEX

Almost periodic

.

.

.

Angle between subspaces Bounded growth

.

.

.

Exponential Hull

.

.

.

.

.

.

.

.

.

.

similar .

.

.

Locally uniformly

.

.

.

.

Logarithmic

.

.

.

.

.

.

.

.

Module

function .

.

.

.

.

.

.

.

.

.

.

Ordinary

.

.

.

.

projection

Projection

.

.

.

.

Quasi-periodic

.

.

..

.

.

Recurrent

.

.

.

.

.

.

Reducible

.

.

.

.

.

.

Semisimple

.

.

Uniform asymptotic U n i f o r m stability

.

.

64-65

.

.

.

.

.

.

.

.

i 0

.

.

.

.

74

. .

....

20

° ,

....

74

. . . .

9

....

59

° .

....

72

. °

.

.

.

.

83

° .

.

.

.

.

i 0

° °

. .

. °

.

.

.

.

39

, .

.

.

.

.

i 0

° .

.

.

.

.

83

° .

....

78

. °

....

38

, °

° .

. .

. °

° °

. . . .

° °

..

.

23, 92-93

° .

stability .

8

. °

° .

eigenvalue

-.

38

. °

. °

. . . .

....

. °

. .

Orbital stability

Orthogonal

° .

.

dichotomy

° .

69 ii, 92

....

. °

° .

..

.

norm

° .

..

° °

. . . . . . ....

° °

. .

.

.

Integrable

Lyapunov

° °

-.

.

. °

, °

.

dichotomy

Kinematically Locally

.

° .

.-

Closed graph theorem Controllable

. °

.

. °

° .

° .

. °

, °

° °

. . . .

. °

° .

.

.

.

3, -*

.

.

.

92 1

1