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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
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
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 <
It is u n i f o r m l y a s y m p t o t i c a l l y stable if and only if there exist constants > 0
such that
K > 0 ,
''IX(t)x-l(s)I ~ Ke -~(t-s)
for
0 ~ s ~ t <
For example, the scalar differential equation x" = -(t + l)-ix , which has the fundamental solution
x(t) = (t + i) -I , is uniformly stable and
asymptotically stable, but not uniformly asymptotically stable. The scalar differential equation x" = {sin log(t + i) * cos iog(t + i) - ~}x , which has the fundamental solution x(t) = exp{(t + i) sin log(t + i) - st} , is asymptotically stable and even 'exponentially stable' if stable if
a <
2½
~ > 1 , but not uniformly
.
Uniform asymptotic stability has the important property of being preserved under small perturbations of the coefficient matrix.
Suppose
PROPOSITION I .
A(t)
the fundamental matrix
X(t)
is a continuous matrix function on an interval
for
is a continuous matrix function such that
B(t)
the fundconental matrix
Y(t)
J
and
of the equation (2) satisfies the inequality
IX(t)x-l(s)l ~ Ke ~(t-s)
If
This follows from
t ~ s . IB(t)I ~ ~
for all
t E J
then
of the perturbed equation y" = [A(t) + B(t)]y
(3)
satisfies the inequality IY(t)y-l(s)I ~ Ke B(t-s)
where Proof.
for
t ~ s ,
B = ~ + ~K . If
y(t)
is any solution of (3) then, by the variation of constants formula, y(t) = X(t)x-l(s)y(s) + /~X(t)x-l(u)B(u)y(u)du
Hence, for
.
t ~ s , ly(t)l ~ Ke~(t-S)ly(s)l
Thus the scalar function
+ Kf~e~(t-u) IB(u)lly(u)Idu .
w(t) = e-at]y(t)l
satisfies the inequality
w(t) ~ Kw(s) + K/~IB(u)lw(u)du By Gronwall's inequality, this implies
for
t
S
w(t) S KW(S) exp Kf~IB(u)Idu
for
t ~ s
.
Consequently ly(t)l ~ KIy(s)le 8(t-s) Taking
y(t) = Y(t)y-I(s)~
, where
~
for
t { s .
is an arbitrary constant vector, we obtain the
result. Uniform stability, on the other hand, is preserved under absolutely integrable perturbations of the coefficient matrix.
The following proposition may be established
in a similar manner to the preceding one. PROPOSITION 2.
Suppose
the fundamental matrix
A(t) X(t)
is a continuous matrix function on an interval
for
fundamental matrix
fjIB(t)Idt ~ ~
then the
of the perturbed equation (3) satisfies the inequality
Y(t)
IY(t)y-I(s)I ~ L
where
t ~ s .
is a continuous matrix function such that
B(t)
and
of the equation (2) satisfies the inequality
IX(t)X-I(s)I ~ K
If
J
for
t ~ s ,
L : Ke 6K . For autonomous linear differential equations stability can be characterised in
terms of the eigenvalues of the coefficient matrix.
The equation
x" : A0x is (uniformly) asymptotically stable if and only if all eigenvalues of the constant matrix
A0
have negative real part.
eigenvalues of
A0
It is (uniformly) stable if and only if all
have non-positive real part and those with zero real part are
'semisimple'. We will now show by an example that this type of characterisation is no longer possible for non-autonomous equations.
Ao
=
,
Take
U(t)
A(t) = U-I(t)AoU(t)
=
-sin t Evidently
A(t)
has both eigenvalues
-i
, where
for every
cos t .
But the corresponding
equation (2) has the fundamental matrix et(cos t + ½sin t)
X(t) :
l
et(sin t - ½cos t)
and is consequently unstable.
e-St(cos t - ½sin t)] ] e-St(sin t + ½cos t)
Thus eigenvalues
fail as a general theoretical tool in the non-autonomous
case.
Nevertheless we will show that it is possible to salvage something.
PROPOSITION 3.
If
a
is an
n
(i)
euery eigenvalue of
(ii)
[AI ~ M ,
and if
0
<
e
<
2M
x
A
n
matrix such that
has real part
then
fetAl ~ (2M/e)n-le(a+e)t Proof.
~ a ,
for
t ~
0
.
We show first that [etA 1 S
Assume initially that polynomial of degree A I . . . . , An represent
A < n
e
at n - i ~ (2tM)k/k! k=0
for
has distinct eigenvalues
t ~ 0 .
AI,
(4)
..., hn
which takes the same values as
~5(I) = e
Then, by the theory of functions of a matrix, p(A)
by Newton's interpolation
Let t~
p(1)
be the
at the points
p(A) = e tA
If we
formula in the form
p(~) = e I + c2(h - ~i ) + e3(~. - ~I)(A - A 2) + ... + Cn(~ - Ii ) ... (I - In_ 1 ) then the coefficients
are given by
c I = ~(hl )
and, for
k > I ,
tl tk-2,(k-l)r~ + _ + _ Ck = foI /0 "'" fo ~ ahl (A2 il)tl "'" + (Ik hk-l)tk-l]dtl Here the argument of to the half-plane
~(k-l)
belongs to the convex hull of
R1 _< a .
Since
~(k-l)(1)
= tk-letl
ICk[ < e~ttk-I/(k_l) Also, for every eigenvalue
~k
Substituting these estimates eigenvalues A
of
A
we have
are not distinct then
with distinct eigenvalues
A
and by letting
it follows that, if
and hence
p(A)
where
0 < n < i .
Then
gn(t) + 0
-nt
as
~
IA - Aki I < 2[A I •
we obtain (4).
+ ~
If the
we see that the inequality
Now set
= e
t >- 0 ,
is the limit of a sequence of matrices
holds also in the general case.
gn(t)
and hence
,
I~kl -< [A I
in the formula for
hl' "'"' kn
"'" dtk-i
tk/k!
,
k=O t ÷ ~ , gn(O) = i , and
(4)
4 Thus
~'i(0) = I - q > 0
at an interior point. satisfies
and
gn(t)
.
Since
(n > i) .
assumes its maximum value in the interval
At this point
Pn 5 N-ipn_l
0 < e < 2M
= gn-i - Ngn
gn -- q-lgn-I
P0 = i
Therefore
it follows that
[0, ~]
~n = max gn(t)
Pn -< q-n
Hence if
then n-I (2tM)k/k ' <- (2M/g)n-le ~t
for
t >- 0 .
k=0 Combining this with the inequality It may be noted that, since
(4), we obtain the result. e _> -M , if
letA I -< e (~+£)t PROPOSITION
4.
Suppose
A(t)
e >_ 2M
for
then
t > 0 .
is a matrix function defined on an interval
J
such
that tl, t 2 ( J
for all
IA(t 2) - A(tl) I 5 ~It 2 - tll and
le{A(~l
~ Ke ~t
where
8 > 0
and
K > i .
for
T Z 0 , t ( J ,
Then the fundamental matrix
of the equation (2)
X(t)
satisfies the inequality IX(t)X-I(s)I
where Proof.
fl = ~
+
~ Ke g(t-s)
for
t ~ s
(SK log K) ½ .
For any fixed
u ( J
we can write (2) in the form
x' = A(u)x + [A(t) - A(u)]x . Hence, by the variation of constants x(t) = e(t-s)A(U)x(s) Therefore,
for Ix(t)l
By Gronwall's
+ /te(t-t')A(U)[A(t') -s
x(t)
satisfies
- A(u)]x(t')dt t
t ~ s , ~ Ke~(t-S)lx(s)l inequality,
u = (s + t)/2
+ Kf~e~(t-t')IA(t')
- A(u) llx(t')ldt'
this implies
Ix(t)l ~ K e a ( t - S ) I x ( s ) Taking
formula, any solution
t exp
K/~tA(t')
- A(u) Idt'
for
and using the Lipsehitz condition, we get
Ix(t) I <_ Ke&(t -s) e ~K(t-s)2/4 Ix(s)l
for
t ~ s •
t Z s
.
Put
, y -- %(6K l o g
h = 2
For
s -< t _< s + h
K) ½
we have 6K(t
-
s)/4 2 6Kh/4 = y
and hence
Ix(t)I ~ Ke(~+Y)(t-S)Ix(s)l In general, if
s + nh ~ t < s + (n + l)h
sit_<s+h
for
.
then
Ix(t)l ~ Ke(~+Y)(t-s-nh)Ix(s
+ nh)]
° . .
Kn+le(a+Y)(t-S)Ix(s)I Since
.
y = h -I log K , K n = e nYh < eY(t-s)
Hence
Ix(t)[ -< Ke(~+2Y)(t-S)Ix(s) I By combining Propositions PROPO$ITION
5.
Let
A(t)
for all
t ->
s
3 and 4 we obtain at once
be an
n x n
matrix function defined on an interval
such that (i)
every eigenvalue of
A(t)
IA(t)l <- M
t ~ J .
(ii)
Then for any
for all
~ > 0
has real part
there exists
6 = 6(M, ~) > 0
IA(t 2) - A(tl) I -< Slt 2 - ill the fundamentaZ matrix
x(t)
K
for all
for all
t E J ,
such that if tl, t 2 E J
of the equation (2) satisfies the inequality
IX(t)X-I(s)I where
<_ a
-< Kge(~+~)(t-s)
for
t _> s ,
= max{(4M/e) n-l, I} .
In particular, sufficiently
(2) is uniformly asymptotically
stable if
~ < 0
and
~
is
small.
Our next result shows that, if the coefficient matrices are bounded,
uniform
asymptotic stability is also preserved under 'integrally small' perturbations. PROPOSITION
6.
Let
A(t)
,
B(t)
be continuous matrix functions on an interval
J
such that IA(t)l
~ M , IB(t)I
and suppose the fundamental matrix
of the equation (2) satisfies the inequality
X(t)
IX(t)x-l(s)l S Ke~(t-S) where
~ M ,
for
t
~ s
,
K ~ i . If If ~ B ( t )dt, ~ ~
then the fundamental matrix
Y(t)
for
It 2 - tll ~ h
of the perturbed equation (3) satisfies the
inequality
IY(t)y-I(s)I ~ (1 + 6)Ke8(t-s) where
for
t
~ s
,
: a + 3MK6 + h -I log(l + 6)K .
Proof.
By the variation
integral
of constants
formula
any solution
of (3) satisfies
y(t)
the
equation y(t)
: X(t)x-l(s)y(s)
+ f~X(t)x-l(u)B(u)y(u)du
•
If we set C(s)
then,
on integrating y(t)
:
by parts and using
: X(t)x-l(s)[I
that for
(2) and (3), we obtain
+ C(s)]y(s)
- /:X(t)X-I(u)[A(u)C(u) It follows
I[B(u)du
- C(u)A(u)
- C(u)B(u)]y(u)du
.
s ~ t ~ s + h
ly(t)l S (i + @)Ke~(t-S)ly(s) I + 3MK@/tea(t-U) ly(u)Idu
.
--S
Therefore,
by Gronwall's
inequality,
w(t)
= e-~tly(t) I
w(t) ~ (i + ~)KeSMK@(t-S)w(s) For any
s ,
t ( J
with
t { s
s + nh 5 t < s + (n + l)h .
there
for
s S t S s + h .
is a unique non-negative
Then
w(t) S (i + ~)Ke3MK@(t-s-nh)w(s
+ nh)
,,°
[(i + 6)K]n+le3MK@(t-S)w(s) and
satisfies
integer
n
such that
ly(t)I 5 [(i + 6)K]n+Ie(~e3MK6)(t-S)Iy(s)I
If we put
y = h -I log(l + 6)K
-
then
[(i + @) K ]n : enyb
eY(t-s)
Hence ly(t)l 5 (i + @)KeS(t-S)ly(s)l
for
t >-- s
which is equivalent to the required inequality. It is clear that, given any and if
6 : 6(K, N, ~)
B < ~ + e
C > 0 , we will have
if
h > 2g -I l o g K
is sufficiently small.
The integrally small perturbations of Proposition 6 form a more extensive class than the small perturbations of Proposition I.
Suppose, for example, that
an almost periodic matrix function with mean value zero. h > 0 ,
6 > 0
there exists a corresponding
B(t)
is
We will show that for any
m0 = w0(h' 6) > 0
such that if
~ > ~0
then
I
In fact
B(t)
< ~
is bounded, say
for
IB(t)l _< M
It 2 - tll < h .
for all
t .
Hence if
It2 - tll _< 6/M
then fto lJtlB(~t)dt, _< 6 B(t)
On the other hand, since such that if
T > TO
m .
has mean value zero, there exists
T O = T0(~/h) > 0
then IT-1/S+TB(t)dtl < 6/h J
Therefore, if
for all
~S
6/M <- It2 - tll s h
for all
s
Q
W
and
~ >- ~0 = MT0/6 '
i f~t 2 ]f~21B(~t)dtl -< l~(t 2 - tl)- J~tlB(U)du]h < 6/h • h = ~ . Thus it follows from Proposition 6 that the uniform asymptotic stability of an autonomous system is not destroyed by sufficiently rapid oscillations.
This is a
counterpart to Proposition 5, which says that it is not destroyed by sufficiently slow variations. We will say that the differential equation (2) has J
if, for some fixed
h > 0 , there exists a constant
bounded growth on an interval C ~ i
such that every
solution
x(t)
of (2) satisfies Ix(t)I S Clx(s) I
The equation
for
s , t E J
and
s ~ t S s + h .
(2) has bounded g~owth if and only if there exist real constants
such that its fundamental matrix
X(t)
[X(t)x-l(s)I In fact we can take
K = C
and
~ Ke ~(t-s)
t e s .
of the choice of
(5) This shows that the
h .
(2) certainly has bounded growth if its coefficient matrix t+h /t IA(s)Ids
bounded, or even if
is bounded,
IX(t)x-l(s)I More generally,
for
e = h -I log C , resp. C = Ke ~h .
condition of bounded growth is independent The equation
K ,
satisfies
~ exp
it has bounded growth if
since by Gronwall's
A(t)
is
inequality
:~IA(u)IduI .
~[A(t)]
or
t+h :t ~[A(s)]ds
is bounded
above, where ~(A)
=
tim{If
+
~AI
- 1}/~
e->+0 is the logarithmic norm of the matrix Ix(t)x-l(s)I However,
A , since
~ expf~[A(u)]du
the preceding propositions
for
t
s
provide smaller values for the exponent
~
of (5)
in a number of cases. In these lectures the hypothesis of the more restrictive
hypothesis
of bounded growth will often be imposed instead
that the coefficient
matrix is bounded.
Actually,
the concept of bounded growth is just the restriction to linear differential equations of the concept of uniform continuous solution of an arbitrary differential Let let
h
x(t)
dependence
fop a
equation.
be a solution of the differential
be a fixed positive number.
on initial conditions
Then
x(t)
equation
(i) on an interval
J
may be said to be (right) - uniformly
dependent on its initial value if for each
e > 0
6 = ~(e) > 0
of (i) which satisfies the inequality
such that any solution
Ix(s) - x(s) I < ~
for some
Ix(t) - x(t) I < e
for all
s E J t E J
x(t)
there is a corresponding
is defined and satisfies the inequality with
and
s ~ t ~ s + h .
2.
Let
X(t)
EXPONENTIAL AND ORDINARY DICHOTOMIES
be a fundamental
m a t r i x for the linear differential x' = A ( t ) x
where the equation P
n x n
coefficient
-- that is, a matrix
e ,
8
matrix
(i) is said to possess P
an
A(t)
equation
,
(i)
is continuous
on an interval
J .
The
exponential dichotomy if there exists a projection
such that
p2 = p
_
and positive
constants
K ,
L ,
such that Ix(t)Px-l(s)I
s Ke -a(t-s)
for
t Z s , (2)
IX(t)(l - P)x-l(s)I It is said to possess 8
an
~ Le -6(s-t)
for
s e t .
ordinary dichotomy if the inequalities
(2) h o l d with
~
and
not positive but zero. The two cases o f most interest
the whole J=~
line
~ .
are w h e r e
J
is the positive half-line
~+
and
Until the final paragraph of this lecture we assume that
+ The autonomous
equation x'
has an e x p o n e n t i a l matrix
A0
dichotomy on
has zero real part.
~+
= AoX
if and only if no eigenvalue
It has an ordinary
dichotomy
of the constant
if and only if all tA 0
eigenvalues
of
and we can take
A0 P
with zero real part are semisimple. to be the spectral p r o j e c t i o n
y
is any rectifiable
simple
x(t)
= e
defined b y
p : 2~--~/y(zl - A0)-idz where
In each case
,
closed curve in the open left half-plane
which
11
contains in its interior all eigenvalues concepts of exponential
of
A0
with negative
and ordinary dichotomy generalize
real part.
The
to non-autonomous
equations
these two types of conditional stability. The equation
(i) has an exponential
uniformly asymptotically
to rewrite the conditions
(2) in the equivalent
for
jx(t)(I- p)~J ~ L'e-~(s-t)JX(s)(I IX(t)PX-I(t)I K' ,
vector.
Lp ,
M'
definiteness
assume
t + ~
! M'
are again positive
Suppose the projection
k-dlmensional
if and only if it is
~ ,
p = I
if and only
To see what a dichotomy means in the general case it is
Ix(t)P~l ~ K'e-~(t-S)Ix(s)P~I
where
p = I
stable, and an ordinary dichotomy with
if it is uniformly stable. convenient
dichotomy with
P
~ > 0 .
subspace of solutions
t ~ s ,
- P)~i
for all
for
s~
t ,
(2)'
t ,
constants and
has rank
form
~
is an arbitrary constant
k , i.e. trace
Then the first condition
P = k , and for (2)' says that there is a
tending to zero uniformly and exponentially
The second condition says that there is a supplementary
as
(n - k)-dimensional
subspace of solutions tending to infinity uniformly and exponentially
as
t +
The third condition says that the 'angle' between these two subspaces remains bounded away from zero. If
~
condition
or
8
is positive and the equation
(i) has ~ bounded growth then the third
(2)' is actually implied by the previous two conditions. IX(t)x-l(s)l
E Ce ~(t-s)
for
For suppose
t Z s , (3)
where
C ~ i
and
U > 0
are constants.
IX(t + h)Px-l(t)I IX(t + h)(l - P)X-I(t)I
For any
h > 0
we have
~ Kte-ehIx(t)Px-l(t)l ~ L'-leBhlx(t)(l
,
- P)X-I(t)I
Putting p : IX(t)(l - p)x-l(t)I
,
~ = IX(t)Px-l(t)I
,
we obtain Ip-Ix(t + h)(l - P)X-I(t) where
Y = L'-le ~h - K~e -~h
Choose
+ ~-ix(t + h)Px-l(t)l
h > 0
so large that
t y ,
y > 0
Then, by (3~ 10-1X(t)(I - P)x-l(t) The left side of this inequality
+ c-Ix(t)PX-I(t)I
a yc-le -Uh
can be written in the form
.
12
lo-il+
(p-i _ - i ) x ( t ) ( I _ p)x-l(t)i
! o-I + Ip-i _
= ~-i{i
2-i
Hence if
~ ~ 2y-iCe ~h
P # 0 , I
~id, by symmetry,
we can choose
h > 0 p ,
If
O~
and
8
I~ - P l }
.
p ~ 2y-iCe ~h .
It is readily shown that
so that
g ~ 2 e C u~ + a s ' (K'L ' ) ( ~ - ~ ) / ( ~ + G )
are positive but the equation
then the third condition example,
also
+
-ilp
(i) does not have b o u n d e d growth
(2)' is not implied by the previous
two conditions.
For
the second order system I
-x I + e2tx 2
xI
X 2r : X2 has the fundamental
matrix
e t
(e 3t _ e-t)/41
X(t) = 0 It may be verified w i t h o u t
then the third
If condition order
first
condition
e = B = 0 (2)'
difficulty that if
two c o n d i t i o n s is not
(2)'
are satisfied,
even if
implied
the coefficient
By t h e p r e v i o u s
~ = 1
matrix
A(t)
two c o n d i t i o n s .
system xI : x2 x 2' = 0
has the fundamental matrix
xti: :) If
with
and
8 = 3 , but the
satisfied.
then,
is not
et
is bounded, For example,
the third the second
13
then the first two conditions condition
interval ~+
with
d : 8 = 0 , but the third
is not satisfied.
If the equation
line
(2)' are satisfied,
[t 0, ~)
(i) has an exponential
then it also has an exponential
, with the same projection
can choose
or ordinary
P
dichotomy
(2) on a sub-
or ordinary dichotomy on the half-
and the same exponents
~ , 8 •
In fact we
N ~ i , for example
N
:
to exp fO IA(u) Idu '
so that
Ix(t)×-l(s)1 ~ N If
0 -< s _< t o _< t
for
0 s s , t s to .
then
IX(t)PX-I(s)I
~ NIx(t)PX-I(t0)I
-~(t-t 0 ) NKe
NKe If
0 -< s <- t -< t o
~t 0 -~(t-s) e
then
IX(t)PX-I(s)I
~ N21X(t0)PX-I(t0)I
N2K
N2Ke
~t 0 -e(t-s) e
Hence IX(t)PX-I(s)I
where
2 ~t0 K = N Ke
.
~ ~e -~(t-s)
OSsSt
<~
,
Similarly,
IX(t)(l - p)x-l(s)l
where
for
~ ~e -8(s-t)
for
0 < t < s < ~
L = N2Le Bt0 Suppose the equation
, 8 > 0 .
(I) satisfies
For any solution
x(t)
the first two conditions
of (i) write
(2) t
with
14
xl(t) = X(t)Px-l(t)x(t)
, x2(t) : X(t)(l - P)x-l(t)x(t)
,
so that
x(t) If
Ix2(s)l
>_
lXl(S)l
X(t)Px-l(s)xl(s)
+ X(t)(I
> {L,-leB(t-s) >_ ½ { L , - l e
Ix2(s)l _< Ixl(s)l
if
lx(t)l For any
8
such that
.
t > s ,
then, for
Ix(t) I >_ L'-le6(t-S)lx2(s)
Similarly,
- P)x-l(s)x2(s)
K'e-a(t-S)lxl(s) l
_ K,e-a(t-s)}lx(s)
then,
0 < 8 < i
_
_ K'e-e(t-s)}lx2(s)l
6(t-s)
> ½{K,-le
I
for
~(s-t)
I .
t < s , _ L'e-B(s-t)}l×(s)
we can choose
L,-le 6T _ K,e-~T > 2e -I
T > 0
I . so large that
K,-le ~T _ L,e-6T > 2e -I
Then in either case we will have Ix(s)I -< e sup
Ix(u) I
for every
s > T .
lu-sl_
for an exponential
PROPOSITION constants
I.
dichotomy
(i) has b o u n d e d growth then this necessary
is also sufficient.
The equation (i) has an exponentia~ dichotomy on
T > 0 , C > i
and
0 < e < i
such that every solution
~+
if there exist x(t)
of (i)
satisfies
Ix(t)l -< Clx(s)l
for
o <_ s <- t
<- s + T
and
Ix(t)l
-< e
sup
Ix(u) I
for
t >_ T .
(4)
lu-tl~
Suppose
first that
x(t)
is a nontrivial
b o u n d e d solution and for any
set p(s) = sup u~s Then
for
Ix(u)l
t ~- s + T
]x(t)l
= e
sup
lu-t/~T
Ix(u)l ~ e~(s)
s > 0
15
and hence
v(s) =
sup
rx(u)l •
s~U~s+T It follows that
lx(t)l If
~
s + n T ~ t < s + (n + I)T
ctx(s)l
for
0 < s -< t < ~
then
lx(t)t
~
sup
en
Ix(u)l
lu-ti~nT e%[x(s)l e-lce(t-s)/Tlx(s)l
.
Thus Ix(t)I ~ K e - ~ ( t - S ) I x ( s ) I where
K = @-Ic > 1
and
Suppose n e x t that define
t
> 0
n
0 ~ s ~ t <
~ = -T -I log @ > 0 . x(t)
is an u n b o u n d e d s o l u t i o n with
Ix(0)I : 1 .
We can
by
IX(tn) I = @-nc Then
for
T < t I < t 2 < .. .
and
,
Ix(t)I < @ - n c
tn ÷ ~
for
tn+l < - t n + T ' since
Moreover
IX(tn) I ~ e
0 ~ t < tn
sup
Ix(u) I
0~u~t +T n and
Ix(u)l < 8-11X(tn) I
tn < . s <. tn+ .1 (i .< m < n)
for
0 S u < tn+ 1 .
Suppose
t ~ s
and
t
m
-< t < t
m+l
'
Then
Ix(t) I < 8-m-ic = 8n-mlx(tn+l) I
cs-l@n-m+iIx(s)l ce-le(s-t)/TIx(s)I
.
Thus
Ix(t)l S K e - e ( s - t ) I x ( s ) I
Let
V
he the u n d e r l y i n g v e c t o r space
for
(~n
tI S t ~ s <
or
~n)
, let
VI
be the subspace
c o n s i s t i n g o f the initial values o f all b o u n d e d s o l u t i o n s o f (i), and let
V2
be any
16
fixed subspace x(t) = x(t, ~) is unbounded,
supplementary
VI .
and hence there is a least value •
We will show that the values
By the compactness
~V + ~ ' where
l~I = i .
< 6-Ic
+ x(t, ~)
for
which is a contmadiction Thus there exists x(t)
with
Consequently .
0 S t < tl(V)
because TI > 0
X(t) as
matrix.
X(O) = I .
projection
tl(<) ~ T I
for
dichotomy on
Suppose
,
for all
since
[TI, =)
on the subinterval
, and hence also on
(i) possesses
P , corresponding P'
or ordinary
,
and hence
Therefore,
for any vector
-
and all
P')
=
(P
-
s , t t 0
an exponential
for a
or matrix
is a projection with the same range
an exponential
p'p = p , PP' = p'
P(P
[+
in a dichotomy
to the fundamental
We have
:
and every
(i) has bounded growth.
P'
P'
~ ,
TI ~ t ~ s < =
the equation
(i) also possesses
-
=
we may suppose that
o f varying the projection
We will show that if
P
tI
t e 0 .
(2)' are satisfied
dichoton~y (2) with projection
then the equation
In fact,
(v) such that
V2
0 ~ t < ~
(2)' is redundant,
(i) has an exponential
with P
are bounded.
satisfies
the first two conditions
given fundamental
x(t, $)
it follows that
for
such that
We consider next the possibility
ordinary
Then
~ E V2 .
x(0) ( V 2
The t h i r d condition
Therefore
in
t = 0 . such that
~v ~ V2
for every
Ix(t) I ~ Ke-~(s-t)Ix(s) I
[TI, ~)
tl(~)
of the unit sphere
Ix(t , $)I ~ e-IC
solution
at
Then
x(t, ~ )
Ix(t • ~v)I
~ ~ V 2 , let
t I = tl(~)
there exists a sequence of unit vectors
tl(~ ~) + ~
Since
For any unit vector
be the solution which takes the value
Ix(t I, ~)I = e-IC otherwise
to
P')(I
-
P)
.
dichotomy with
]7
IX(t)(P
- P')~I
~ Ke-~tI(P
- P')~I
Ke-~tls'
- Pil
KLe-~te-~Slp' It follows
that for
- P)
_ PIIX(s)~l
0 ~ t ~ s
I X(t)(I
- P')~I
~ I X(t)(I
- P)~I
+ I X(t)(P
- P')~]
{i + KIP' - Pl}Le-6(s-t)IX(s)~l Similarly,
for
.
0 ~ s ~ t IX(t)P'~I
Therefore
.
LIP'
~ {1 +
- PI}Ke-~(t-S)Iz(s)~I
(i) has a dichotomy with projection
unaltered and the constants
K , L
P'
.
, the exponents
b e i n g m u l t i p l i e d by
~ , 8
i + LIP' - PI
being
, 1 + KIP' - PI
respectively. In the case of an exponential choice o f the projection determined
as the subspace
However, projection V
P
if
consisting
X(0) = I , the range of
consisting
Let
V
dichotomy
tend to zero as
PROPOSITION 2.
V1
be the underlying
consisting
t ÷ ~
vector space,
Q
V1
be the subspace of
o f (i), and let
of the initial values o f all solutions
be the fundamental matrix of ( 1 ) with
equation (Z) has a dichotomy (2) with projection projection
let
for the
V0
be
of (i) w h i c h
Then
X(t)
Let
in the
is uniquely
there may be other choices
o f the initial values o f all b o u n d e d solutions of
P
of the initial values of all b o u n d e d solutions.
in the case o f an ordinary
P .
the subspace
dichotomy this is the only indeterminacy
since,
P
×(0)
:
I .
If
the
then it also has a dichotomy with
if and only if V0 ! QV 2 Vi
Proof.
We may assume that
nontrivial t
~ 0 .
solution Since
(2) holds with
of (i) such that
x(t)
~ : ~ = 0
IX(tn) I + 0
is n o n t r i v i a l we must have
n
and
K : L .
for some sequence t
+ ~
Put
n
Xl(t) : X(t)Px(0) Then
(s)
x(t) = Xl(t ) + x2(t )
, x2(t) : X(t)(I - P)x(0)
and
,
Ixl(t) I ~ Klx(tn) I
for
t ~ tn
Ix2(t) I ~ K[x(tn) I
for
0 S t ~ tn
.
Let
x(t)
of numbers
be a
18
The first r e l a t i o n implies that x2(t) : 0 Thus
for all
t { 0 .
V0 ~ P V .
IXl(t) I + 0
Hence
x(O) ( P V
On the o t h e r h a n d
of
P .
P0
be a p r o j e c t i o n w i t h range
Then
V' 0 : (P - P o ) V
[X(t)¢[
and
V0
N > 0
Z N[¢ I
Ix(t)I ÷ 0
t
~ 0
a s u b s e q u e n c e we may assume that
it follows that
]X(t~)¢[ + 0 .
Thus for any v e c t o r
~ ( V
t ~ ~
IX(t)PI ~ K
PV
s u p p l e m e n t a r y to
~ 6 V0'
and
]X(t ) ~ I
$9 + ~ , w h e r e
Therefore
( V 0r
~
÷ 0 .
with
and
we have, for
Ni(P- P0)~I ~KIx(s)¢I
0 ~ s ~ t ,
+ [X(t)P0¢ 1
and all
forall
se 0 .
s , t ~ 0
[ X ( t ) ( P - PO)<[ ~ K](P -
P0)
N-IK21X(s)¢I Hence
IX(t)P0¢ I -< I X ( t ) P ¢ I
+ I X ( t ) ( P - PO)¢I
< (1 + N-1K)K[X(s)¢[ and
We w i l l
: i
Since
¢ ( V0 , w h i c h is a contradiction.
we o b t a i n
~ ( V
[~1
l~I = 1 •
KIx(s)~l + Ix(t)mo¢l
Therefore, for any v e c t o r
V0 .
and a
By r e s t r i c t i n g attention to
~ 6 V 0'
[X(t)P¢[
t ÷ ~
t ~ 0 .
t > 0 .
NI(P - P0)¢I ~ IX(t)(P - P0)
Letting
for
such that
for
such that
as
w h o s e nullspace contains the nullspace
In fact, o t h e r w i s e there exists a sequence of vectors s e q u e n c e o f numbers
and the s e c o n d implies that
(5).
is a s u b s p a c e o f
show that t h e r e exists a constant
t + ~
P V ~ V I , since
This proves the n e c e s s i t y o f the condition Let
as
for
0 -< s < t
19
lx(t)(I -
P0)~l
~
Ix(t)(l - P)~[ + IX(t)(P - P0)~I (i + N-IK)KIX(s)~]
Thus
(i) has a dichotomy with p r o j e c t i o n
P0 "
Suppose now that
satisfying
Q
is a projection
have just proved we may assume that N' > 0
constant
0 ~ t ~ s .
the condition
PV c QV , i.e.,
QP = P .
(5).
By what we
There exists a
such that IX(t)~l S N'I~ I
Therefore,
for
for any vector
~ ( V
for
~ E VI
and all
and
t ~ 0 .
s , t a 0
IX(t)(Q - P)~I ~ N'I(Q - P)$[
s N'IQ[I(~ - P)~l
It follows
that for
0 ~ s < t
~ ( l + N'IQI)KIXes)~I
Ix(t)Qgl and for
0 -< t -< s ]X(t)(l - Q ) ~ ]
Thus
~ (I + N ' I Q I ) K I X ( s ) ~
(i) has a dichotomy with projection
Q .
This completes
Finally we remark that in the case of an exponential
P ~+
the projection
P
is the subspace
o f initial values of solutions b o u n d e d
and the nullspace
the negative half-line ordinary, P
uniquely
is
dichotomy
of
P
is the subspace
~
P .
Also,
In fact, if
if the equation
or ordinary,
the proof.
dichotomy on the whole line X(0) = I , the range of on the positive half-line
of initial values of solutions b o u n d e d on
(2) on each of the half-lines
then it has an exponential,
projection
determined.
1 .
(i) has an exponential, ~+
dichotomy
, ~_
or
with the same projection
on the whole line
~
with
3.
DICHOTOMIES AND FUNCTIONAL ANALYSIS
Th~ougho,~t this lecture we will assume that function defined on the half-line
~+
matrix of the linear differential
X(O) = I .
if it is measurable f(t)
/jlf(t)Idt
is locally integrable
X(t)
matrix
the fundamental
equation
A vector function
and
is a continuous
and we will denote by
x' such that
A(t)
< ~
= A(t)x
(1)
f(t)
locally integrable
will be said to be
for every compact interval
then by a solution of the inhemogeneous
J c k+
If
equation
y' = A(t)y + f(t) we will mean an absolutely all
t
continuous
function
(2)
y(t)
which satisfies
(2) for almost
.
Suppose first that (i) has an exponential a projection
P
and positive constants IX(t)PX-I(s)I
dichotomy on
K , ~
Thus there exists
R+
such that
2 Ke - ~ ( t - s )
for
0 2 s 2 t
, (3)
IX(t)(I
- P)X-I(s)I
Then for any bounded continuous equation
function
(2) has a bounded solution. y(t)
2 Ke - ~ ( s - t )
: I~X(~)px-l(s)f(s)ds
f(t)
for
0 2 t 2 s .
the corresponding
inhomogeneous
In fact, it is easily verified that - <X(t)(I
- P)x-l(s)f(s)ds
(4)
is a solution of (2) and sup ly(t)l t~O
~ 2~-IK sup If(t)I t~O
.
Actually the formula (4) defines a bounded solution of (2) not only for any bounded continuous that
function
t+l It If(s)Ids
f(t)
but also for any locally integrable
is bounded.
This follows
from
function
f(t)
such
21 LEMMA I .
Let
be a non-negative locally integrable function such that
y(t)
f~+iy(s)ds if
~>
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
for
that
t h s ~ tO ,
(6) tI _r-l(s_t)ftll Is IX(u)(I - P)~l-ldu ~ e "t X ( u ) ( I We use these inequalities
LEMMA 2.
and let
Suppose r = rc .
the equation Let
x(t)
- P)~I-idu
for
t ~ s ~ tI .
to establish
(2) has a bounded
be a solution
solution
for every function
of the corresponding
homogeneous
(i) and put xl(t) : X(t)PX-I(t)x(t)
If, for some fixed
, x2(t) : X(t)(I
- P)X-I(t)x(t)
s ~ 0 , ]Xl(t) [ S
Nix(s)[
for
s ~ t
~ s
+ r
.
f ( C equation
25
then Ixl(t) I ~ eN)x(s)le -r-l(t-s)
If, for some fixed
for
s ~ t <
s ~ 0 ,
Ix2(t)1 ~ mlx(s)I
for
max (0 , s - r) ~ t ~ s
then
Ix2(t) I ~ eNIx(s)le -r-l(s-t) Proof. s
by
We can write s + r
x(t) = X(t)~
for
for some vector
in the first inequality
0 ~ t ~ s . ~ .
Replacing
(6) we obtain for
I2+rlXl(U)l
rIN lx(s)l
(5)~ this gives for
]Xl(t)l ~
by
s
and
t ~ s + r
ldu
e e-r-l(t-s)f~IXl(U)I-idu
Using the first inequality
tO
.
t ~ s + r
r E : ~ I X l ( U ) l - l d ~ -I
eN}x(s)le -r-l(t-s) Since the same inequality first assertion
evidently holds also for
The p r o o f of the second assertion by
s
in the second inequality We now conclude
equation
is similar,
the p r o o f of Proposition
(2) has a b o u n d e d
solution
for every
all the information
that we w i l l actually and
K = rL x(t)
~ = 0 .
of (i) and every
replacing
s
by
s - r
and
2.
Since
f ( C
use.
C c M
By Proposition
Hence in Lemm~ 2 we can take s ~ 0 , and obtain
and
and for every
(3) with
I and its proof,
C ~ i
and
~ > 0
are constants.
in the second inequality
for
Replacing
(5), this gives
for
K = er L
and
Thus
t ~ s , ~
by t ~ s
(3)
N = r L , for every
3 and assume that (i) has bounded growth.
ix(t)X -i(s )I ~ ce~(t-s) tI = ~
1
This is
-i
We turn next to Proposition
t
L c M , the f ( L .
~ = rC
where
the
(6).
holds with solution
s ~ t ~ s + r , this proves
of the lemma.
x-l(s)~
and putting
26
IX(t)(l
- P)X-I(s)~I
_< rI~slX(u)x-l(s)~l-ldul
-I
u]-1 _< r
Thus IX(t)( I _ p)x-l(s)l
2 prC
for
t -< s .
In the same way
IX(t)(I
- p)x-l(s)l
_< prCe ] J ( t - s )
for
t >_ s ,
~ (i + pr)Ce p(t-s)
for
t ~ s .
and hence ''IX(t)PX-I(s)I Similarly,
from the first inequality IX(t)Px-l(s)l
By using this inequality
~ prC~
~r
inequality
~r
IX(t)(l
t > s .
j
,
t - s ~ h , we get ~ (i + 2pr)C
- P)X-I(s)I
When the coefficient
for all
t ~ s .
~ eprCe -r-l(s-t)
for
0-<s-
,
for
O
.
which
implies
dichotomy. matrix
growth, we can obtain slightly
A(t)
sharper
We will show that if
IX(t)Px-l(s)I
for any unit vector
is bounded, values
IA(t)l S M
for the constants for
~ e(l + rM)e -r-l(t-s)
IX(t)(l - P)X-I(s)I In fact,
(l+2pr]
~ e(l + 2~r)Ce -r-l(t-s)
(i) has an e ~ o n e n t i a l
dichotomy.
for
from Lemma 2 that
IX(t)Px-l(s)I
Thus
p-I .
:
IX(t)Px-l(s)I It now ~ l l o w s
- e-~(t-s~-i
t - s ~ h , where h
and the previous
(5) we obtain
~
~ erMe -r-l(s-t)
put
t ~ 0
that
(i) has bounded
of the exponential
then
for
0 ~ s ~ t ,
for
0 ~ t S s .
27
g
]-X(t)(I - P)X-I(s)[ y(t)
y(O)
( V2
and
0 ~ t ~ s ,
for
t ~ s .
| [X(t)PX-l(s)~ - ~
Then
for
y(t
is a b o u n d e d s o l u t i o n o f the inhomogeneous e q u a t i o n (2)
with f(t) : I
0
LA ( t ) ~ Hence, e v e n though
f
for
0 ~ t ~ s ,
for
t > s .
is not continuous,
]y(t) I ~ rM
Ix(t)p×-l(s)f ~ 1 + rM IX(t)(I - P)X-I(s)I
~ rM
t { 0 . It follows that
for
for
0 _< s -< t ,
for
0 -< t -< s .
The r e s u l t n o w follows from L e m m a 2. We show finally, b y an example, that the h y p o t h e s i s of b o u n d e d growth cannot be o m i t t e d in P r o p o s i t i o n
3 (nor in P r o p o s i t i o n 2.1).
continuously differentiable
g {i/~(t)
- l}dt < ~
,
and
function such that ~ ( n ) / ~ ( n - 2 -n) + ~
e a s i l y be c o n s t r u c t e d explicitly.
Let
%(t)
0 < ~(t) ~ i as
be a r e a l - v a l u e d for all
n ÷ ~
t { 0 ,
Such a function can
The h o m o g e n e o u s e q u a t i o n (7)
x' : [~'(t)/~(t) - l]x has the solutions
x(t) = x(O)e-t~(t)
.
For all
t { 0
/0tlx(t)/x(s)Ids= foe-(t-s)~(t)/}(s)ds e-t~(t)/~eSds
+ /~{1/~(s) - 1}ds
1 + g{i/~(s)
- l}ds
Hence for any bounded, c o n t i n u o u s function y' = [~'(t)/~(t)
f(t)
.
the inhomogeneous e q u a t i o n
- l]y + f(t)
has the b o u n d e d s o l u t i o n y(t) = f ~ { x ( t ) / x ( s ) } f ( s ) d s The e q u a t i o n (7) is a s y m p t o t i c a l l y , bounded.
.
and even e x p o n e n t i a l l y ,
stable, since
On the o t h e r h a n d it is not uniformly stable, since x ( n ) / x ( n - 2 -n) ÷ ~
It seems desirable to exclude such behaviour.
as
n +
~(t)
is
4.
ROUGHNESS
One of the most important properties of exponential dichotomies is their
roughness. matrix.
That is, they are not destroyed by small perturbations of the coefficient
This was first proved by Massera and Sch~ffer (1958) under the assumption
that the original coefficient matrix is bounded. assumption.
Later Schaffer (1963) removed this
Both these proofs depend on the functional analytic characterisation of
exponential dichotomies and hence, ultimately, on the closed graph theorem. (1967) gave a completely elementary proof.
Coppel
The key idea was to show that the general
case can be reduced to the much simpler special case in which the coefficient matrix A(t)
commutes for every
t
with the projection
P
of the exponential dichotomy.
This is a useful general principle, but the resulting proof of roughness is somewhat indirect.
A direct and elementary proof was given by Dalecki~ and Kre~n (1970), but
under the assumption that
A(t)
is bounded.
We will show here that this assumption
can be quite easily removed. To avoid interrupting the argument we first prove
LEMMA I .
¢(t)
Let
be a bounded, continuous real-valued function such that
for all
¢(t) 2 Ke -~t + eo ~0e-al t-ul ¢(u)du
where
K
, ~
,
@
are positive constants.
¢(t)
S pKe - 6 t
If
e
then
< ½
for all
t
t ~ 0 ,
~
0
,
where B = ~(l
Proof.
- 2@) ~
p = e-1{1
-
(l
- 2e) ~}
Consider the corresponding integral equation
¢(t)
= Ke - a t + e ~ o e - a l t - u l g ( u ) d u
By separating the interval of integration
[0 , ~)
.
into two parts,
[0 , t]
29
and
[t , ~)
, we see that any bounded,
continuous
solution
is differentiable
$(t)
and t
$'(t)
= -aKe -at
- @a2foe-e(t-u)$(u)du
^ 2r ~ -~(u-t)
....
+ ~a it e Hence
$(t)
is differentiable
$''(t) It follows that
- 2ee2$(t)
is a solution
$(t)
c .
equation we obtain continuous
Substituting
c = oK .
solution
2e)$
-
= ce -Bt
this expression
for any constant
L >_ sup ~(t)
Let
$(t)
¢(t)
for
L >_ @-IK
for all
is uniquely
$(t)
such that
determined,
K ,
a ,
e
back in the integral
,
t > 0 .
be a continuous
@(s - t)
that
for all
follows.
we obtain at once
real-valued function such that
are positive constants.
~(t)
approximations
¢(t) -< $(t) < L
the result
¢(t) ~ Ke -a(s-t) + ea/~e-aEt-ul¢(u)du where
$(t)
it follows by the method of successive
By applying the p r e c e d i n g lemma to
LEMMA 2.
8 < ½ , that
$(t) = oKe -Bt
the integral equation has a solution Since
if
,
L >_ Ke -~t + e ~ 0 e - a l t - U I L d u
t >- 0 .
.
Thus the integral equation has the unique bounded,
It is easily verified that,
If we choose
.
equation
is b o u n d e d this implies, $(t)
for some constant
+ ea3~0e-elt-ul$(u)du
of the differential
= ~2(1
$'' Since we are assuming that
.
again and
= ~2Ke-at
$(t)
~ku)au
~ pKe - 6 ( s - t )
If
for
8 < ½
for
0 ~
o ~
t ~
s
,
then t
~
s
,
where B = ~(i
Now let fundamental
A(t)
-
2e) ½
be a continuous
p :
e-l{i
-
(i
matrix function
m a t r i x for the linear differential x F = A(t)x
-
for
2 e ) ½}
t ~ 0
and let
X(t)
be the
equation
(i)
30
such that
X(0)
= I .
Thus there exists
We assume
a projection
that the equation P
and positive
(i) has an exponential
constants
K
~ ,
IX(t)PX-I(s)I
~ Ke -~(t-s)
for
t ~ s ,
- P)X-I(s)I
~ Ke -~(s-t)
for
s { t .
dichotomy.
such that
(2) IX(t)(l Let
B(t)
be a bounded,
continuous
matrix
6 : sup th0 is sufficiently
small then the perturbed
We wish to show that if
function.
IB(t)I
equation
(3)
y, = [A(t) + B(t)]y also possesses
an exponential
For any bounded,
dichotomy.
continuous
matrix
function
Y(t)
set
IIYII : sup IY(t)l t>0 and
TY(t) = x(t)P + /~x(t)Px-l(u)B(u)Y(u)du
X(t)(I
- P)x-l(u)B(u)Y(u)du
•
Then
ITY(t)I
~ Ke -~t
+
KllYIl(/~e-~(t-u)lB(u)ldu + ~te-~(u-t)lB(~)lda}
It follows
that
TY(t)
is again bounded
and continuous,
.
and
IITYII ~ K + 2~-lK611YII . Similarly
we obtain
IITY Hence, by the contraction
- TYtl ~ 2e-IK611Y - Y I I -
principle,
if
8 : a-IK6 then the mapping
has a unique
< ½ ,
fixed point.
Denoting
this
fixed point by
Yl(t)
we have
Yl(t) : x(t)P + /~x(t)Px-l(u)B(U)Yl(U)du (4) - <X(t)(l
- P)x-l(u)B(U)Yl(U)du
.
3~
It follows that equation
(3).
Yl(t) Since
In particular,
if
is differentiable YI(t)P
and is a matrix solution of the differential
is also a fixed point of
T
we have
YI(t)P : Yi(t)
•
then
Q = YI(O)
Qp = Q • From (4), with
t
replaced by
X(t)Px-l(s)Yl(S)
s , we obtain also
= X(t)P + f~x(t)Px-l(u)B(U)Yl(U)du
.
(5)
Combining this with (4) we obtain Y!(t) = X(t)}×-l(s)Yl(S)
+ f~x(t)Px-l(u)B(U)Yl(U)du
(6) -
On the other hand, setting
t = s = 0
<X(t)(l
- F)x-l(u)B(U)Yl(U)du
.
in (5) we get PQ = p .
It follows that
Yl(t)Q
is also a fixed point of Yl(t)Q = Yl(t)
For
t = 0 If
this shows that
Y(t)
Q
Y
and hence
.
is a projection.
is the fundamental matrix of the equation
(3) such that
Y(0) = I
then Yl(t)
= Y(t)Q
.
Put
Y2(t) so that
Y(t) = Yl(t) + Y2(t)
Y2(t) Replacing
t
by
s
.
= X(t)(I
= Y(t)(z
By the variation
, of constants
formula,
- Q) + f ~ x ( t ) x - l ( u ) B ( u ) Y 2 ( u ) d u
and using the fact that
x(t)(I
- Q)
- p)x-l(s)y2(s)
+ f~x(t)(I
(I - P)(I - Q) = I - Q
= x(t)(z
- Q)
- P)×-l(u)B(u)Y2(u)au
Combining this with the previous equation we obtain
(7)
.
.
we get
32
Y2(t)
: X(t)(I
- P)X-I(s)Y2(s)
+ /SX(t)PX-I(u)B(u)Y2(u)du
(8) - /~X(t)(I - P)x-l(u)B(u)Y2(u)du It follows from (6) and (8) that, for any vector
IYl(t)~l
~ Ke-~(t-s)lyz(s)~l
.
~ ,
+ e~se-~lt-ullYl(U)
t ~ s ~ 0 ,
IY2(t)~ I ~ Ke-~(s-t)IY2(s)~ I + 8~/~e-~It-UIIY2(u)~Idu for
s h t ~ 0 .
Hence, by Lemmas i and 2, IYl(t)~ 1 ~ pKe-B(t-S)IYl(S)~l
for
t ~ s ~ 0 , (9)
IY2(t)~ I 2 pKe-B(s-t)lY2(s)~
I
for
s e t e 0 ,
where 6 = ~(1
- 26) % , p = e - l { i
To deduce from the inequalities
-
(i
- 2 e ) %} .
(9) that the differential equation (3) has an
exponential dichotomy it is only necessary to show that
Y(t)Qy-I(t)
is bounded.
From (4) we immediately obtain X(t)(l - P)x-l(t)Yl(t)
: -<X(t)(I
- P)x-l(u)B(U)Yl(U)du
.
Since, by (9), IYl(U)~ I ~ pKe-B(u-t)IYl(t)~l
for
u >- t
it follows that
IX(t)(l - P)X-I(t)YI(t)~I ~ @epKIYl(t)~l<e-(e+B)(u-t)du (io)
= nlYl(t)~ I , where
= (p - I)K .
From (7) we obtain similarly X(t)Px-l(t)Y2(t)
= /SX(t)Px-l(u)B(u)Y2(u)du
.
Since, by (9),
IY2(u)~ I
~ pKe-B(t-U) IY2(t)~I
for
0 ~ u ~ t
33
it follows that IX(t)PX-I(t)Y2(t)~I
_ it -(~+B)(t-U)du < e~pKIY2(t) ~ ~o e (ll)
NIY2(t)~ 1 • Now it is easily verified that
Y(t)Qy-l(t)
- x(t)Px-l(t)
= x(t)(l - P)x-l(t)Y(t)Qy-l(t) x(t)Px-l(t)Y(t)(I
Replacing
by
Y-l(t)~
- Q)y-l(t ) .
in (i0) and (ii), for an arbitrary vector
Iy(t)Qy-l(t)
- x(t)Px-l(t)I
~ , we obtain
~ n(Y I + Y2 ) ,
where Yl(t) = Iy(t)Qy-l(t)l
, Y2(t) = Iy(t)(l - Q)y-l(t)l
.
Hence YI ~ N(YI + Y2 ) + K . Since Y(t)Qy-I(t)
- x(t)Px-l(t)
= X(t)(l - P)x-l(t)
- Y(t)(l - Q)Y-I(t)
,
we have equally Y2 S n(Yl + Y2 ) + K . Adding, we obtain YI + Y2 ~ 2(1 - 2n)-iK if
n < ½ , and hence YI ' Y2 s (i - 2N)-IK .
If in (9) we replace
where
by
Y-l(s)~
we now obtain the exponential dichotomy:
IY(t)Qy-I(s)I
~ Le -B(t-s)
for
t > s >- 0 ,
IY(t)(I - Q)y-l(s)I
~ Le -s(s-t)
for
s > t > 0 ,
L = (I - 2N)-IQK 2
34
There is no special merit in the values of the constants. is certainly
satisfied
calculations
yield the following simpler
PROPOSITION ] .
if
8 < I/4K
.
Since necessarily
The condition
N < ½
K ~ i , some elementary
final statement:
Suppose the linear differential equation (i) has an ex~ponential
dichotomy (2) on
If
~+
= s~p
IB(t)l
a/4K 2
<
tE~+
then the perturbed equation ( 3 ) also has an exponential dichotomy:
[Y(t)(I
where
~ (5/2)K2e -(~-2Ks)(t-s)
for
t
- Q)Y-l(s)[
5 (5/2)K2e -(a-2K~)(s-t)
for
s ~ t ~ 0 ,
is the fundamental matrix for ( 3 ) such that
Y(t)
projection
]Y(t)Qy-t(s)[
Q
has the same nullspace as the projection
~
Y(O)
s
~
0
,
and the
= I
P .
Moreover 'IY(t)Qy-I(t) The roughness Suppose
- X(t)PX-I(t)I'
of exponential
(2) holds on
~
dichotomies
Then the p e r t u r b e d equation ~+
nullspace
, ~ as
Proposition
~
P , I - QP'
IB(t)I
projections
0 = ~ - l K 6 < 1/4K .
dichotomy on each of the half-
Q'
, Q"
has the same nullspace ,as
- 2e)-lK
, IQ"
- P[
- Q";
< i
.
is a projection
s
e(l
pQ' = p ,
Put , Q : SPS -I
S : I + Q' - Q " Then Q
Moreover
and SP : Q'
, S(I
-
P)
:
I
-
Q'
has the same
I - P , and (by the proof of
Thus
Q''p = p , PQ" = Q"
IQ'
can be deduced from this result.
i)
Q ' p = Q'
and
t ~ 0 .
< a/4K 2
(3) has an exponential
with c o r r e s p o n d i n g
IQ' - P[ ~ o ( 1
where
on
for all
and @ : sup t(R
lines
~ 4a-lK3~
Q,W
-
2e)-lK
,
35
It follows
that
Consequently ~+
, ~_
Q
has the same range as
the equation
with common projection
with projection
Q
3.1).
method,
dichotomy
(and exponent
Q~J
dic~otomy.~, on
R
~ - 2K6). of ordinary
dichotomies
and, to illustrate
characterisation
a
(Proposition
to prove
Suppose the linear differential equation (i) has an ordinary
dichotomy on the half-line
~+
If
~oIB(t)Idt
<
then the perturbed equation (3) also has an ordinary dichotomy on there exists an invertible matrix P
as
on each o 9 the half-lines
an exponential
we will use their functional-analytic
We propose
PROPOSITION 2.
and the same nullspace
Q , and therefore
We consider now the roughness different
Qf
(3) has an exponential
T
PROPOSITION 3 .
Moreover
such that (i) has a dichotomy with projection
if and only if (3) has a dichotonrj with projection In fact this will follow
~+
TPT -1 .
from
Suppose the linear differential equation (i) has an ordinary
dichotomy on the half-line
~+ .
If
CoIB(t)Idt
< ~,
<
~o]f(t)Idt
then there is a one-to-one affine mapping between the bounded solutions of the equation (i) and the bounded solutions of the inhomogeneous perturbed equation y' = [A(t) + B(t)]y
+ f(t)
(12)
,
such that the difference between corresponding solutions tends to zero as Proof.
Again let
X(t)
be the fundamental
there exists a projection
P
and a constant
IX(t)PX-I(s)I
Ix(t)(I Moreover, vector
by Proposition
~
Choose
matrix of (i) such that
- p)x-i(s)I
K > 0
for
t t s ~ 0 ,
~
for
s ~
2.2, we can suppose
that
t ~
so large that
e = ~ F t IB(t)Idt 0
0
Then
.
IX(t)P~I
.
tO ~ 0
= I .
such that
~ K K
X(O)
t ÷
< i o
~ 0
as
t * ~
for any
36
For any continuous
function
y(t)
with
IIyll : sup ly(t)l t~t 0
<
set
Ty(t) = /~oX(t)p×-l(s)[B(s)y(s) ~ + f(s)]ds
X(t)(I - P)X-I(s)[B(s)y(s) Then
Ty(t)
tinuous
is again bounded and continuous.
functions
Yl(t)
, Y2(t)
+ f(s)]ds
Moreover,
.
for any two bounded con-
we have
IITyI - Ty2!l S K~t01B(s)IlYl(S)
- y2(s)Ids
811yI - Y211 • Let
x(t)
be any bounded solution of the differential
contraction
principle,
equation
(1).
Then by the
the integral equation y(t) = x(t) + Ty(t)
has a unique bounded continuous
solution
is a solution of the differential solution of (12) then
x(t) = y(t) - Ty(t)
Thus (13) establishes (12) and (1).
equation
Given any
a
i-i
~ > 0
y(t)
.
(12).
(13)
It is easily verified that Conversely,
if
y(t)
y(t)
is a bounded
is a bounded solution of (i).
affine mapping between the bounded solutions we can choose
t I >_ t O
of
so that
K$1EJB(s)IIyCs)l + If(s)l~ds < ~ Then t
Iry(t)I ~ Jx(t)pil×-l(s)[B(s)y(s)
+ f(s)]dsf + E
tO < 2C Thus
y(t) - x(t) + 0 Proposition
equation
as
for all large t •
t +
3 shows incidentally,
for
(i) has an ordinary dichotomy on
B(t) s 0 , ~+
~olf(t)Idt
that if the homogeneous
then for any function
<
f(t)
such that
37
the inhomogeneous
equation y' = A(t)y + f(t)
has at least one b o u n d e d solution which tends to zero as
5.
DICHOTOMIES AND REDUCIBILITY
Consider the linear differential equation x' = A(t)x , where the coefficient matrix
A(t)
(i)
is continuous on an interval
J .
The equation
(I) is said to be kinematically similar to another equation y' = B(t)y
(2)
if there exists a continuously differentiable invertible matrix
S(t)
which satisfies
the differential equation S' = A(t)S - SB(t) and which is bounded, together with its inverse, on x = S(t)y
J .
The change of variables
then transforms (i) into (2).
We will say that the equation (i) is reducible if it is kinematically similar to an equation (2) whose coefficient matrix
B(t)
IBllt)
Bl(t)
and
B2(t)
B2(t0)
t 6 J .
,
being matrices of lower order than
terms this means that there is a projection for every
has the block form
P # 0 , I
B(t) .
In coordinate-free
which commutes with
B(t)
It may be noted that although this definition agrees with the
definition of reducibility in linear algebra, it differs from Lyapunov's use of the term.
(Lyapunov calls an equation (i) reducible if it is kinematically similar to an
autonomous equation.) We are going to show that if the equation (i) has an ordinary or exponential dichotomy with projection II ~II the
P # 0 , I
then it is reducible.
Euclidean norm of the vector
square root of the largest eigenvalue of
We will denote by
~ . Then the induced matrix norm A*A .
A projection
P
IIAII is the
is said to be
39 orthogonal if
P* = P .
Let
LEMMA ].
P
be an orthogonal projection and let
Then there exists an invertible matrix
S
X
be an invertible matrix.
such that
SPS -I : XPX -I and
Ilsll
s 2~ ,
1Is-Ill ~ {HxPxllV 2 + IIxx-lii2} ~ Moreover, if t
is a continuous, or continuously differentiable, function of
X = X(t)
on an interval
J
we can take
differentiable, function of PrOOf.
Since
exists
any p o s i t i v e
a unique
t
S = S(t)
on the same interval.
Hermitian
R = Re > 0
to be a continuous, or continuously
m a t r i x has a unique p o s i t i v e
Moreover, S
since
R2
has the
commutes
with
first r e q u i r e d
- P)
P , so also does
property.
for a n y v e c t o r
~
there
R
.
.
Thus
if we put
S = XR -I
Since
I : PS*SP + (I - P)S*S(I we have,
root,
such that
R 2 = PX*XP + (I - P)X*X(I
then
square
- P)
,
11s~I[2 ~ {li.sP~ll + iIscI - P)~il} 2 2{HsF~EI 2 + Ils(~
- P)~ll 2}
: 211~1I 2 ~us
I!sll ~ 2½
•
On the other hand, (S-I)*s -I = X ~ - I p x * x P X -I + X*-I(I
- P)X*X(I
- P)X -I
and hence
IIs-lll 2 < IIxPx-lll The last r e m a r k
follows
or c o n t i n u o u s l y
differentiable,
continuous,
2 + IIx(I
- P)x-lll
from the fact that the p o s i t i v e
or c o n t i n u o u s l y
positive
Hermitian
differentiable.
2
square
matrix
root of a continuous,
function
is again
40
LEMMA 2.
Let
X(t)
be a fundamental matrix for the linear differential equation
(i), and suppose there exists an orthogonal projection
bounded for
P
such that
X(t)Px-l(t)
is
t ( J .
Then the equation (i) is kinematically similar to an equation
z' whose coefficient matrix
C(t)
:
is Hermitian, commutes with
llC(t)II ~ IIA(t)ll Proof.
(3)
c(t)z
U n d e r the p r e s e n t circumstances
for
every
(i) into (2) , where
f u n d a m e n t a l matrix, Let
U(t)
B = R'R -I
commutes with
U(t 0) = I
for some
t ( J , since
change of variables
S(t)
of the
.
Since (2) has
R(t)
x = S(t)y as a
½[B(t) - B * ( t ) ] u
tO ( J .
Then
B*(t)
commutes w i t h
y = U(t)z
transforms
is H e r m i t i a n and commutes with For each fixed
t
P
commutes with
t ( J . I , U
such that
.
its least value then
IIA + A*II = m a x { I l l , R
U(t)
P
for
and the solutions o f (4) are
+ B*(t)]U(t)
11 S A + A* ~ ~I
From the definition o f
Moreover
t ( J ,
It is easily verified that the further
there exist real numbers
U
is unitary for e v e r y
(2) into (3), w h e r e
for every
has its greatest value and
(4)
U(t)
P* = P
C(t) = ½u-l(t)[B(t)
l
and
be the fundamental m a t r i x for the equation
u n i q u e l y d e t e r m i n e d by their initial values,
If
R(t)
P .
since the c o e f f i c i e n t m a t r i x is skew-Hermitian. every
.
The change of variables
B = S-I(As - S')
u' = such that
( J
the functions
previous lemma are c o n t i n u o u s l y differentiable. transforms
t
P , and satisfies
IUI} •
we have RR' + R'R = PX*(A + A*)XP + (I - P)X*(A + A*)X(I - P)
It follows that IR 2 ~ RR' + R'R ~ uR 2 and h e n c e ~I < R'R -I + R - I R ' < UI
•
.
41
Therefore
liB Since
U
is unitary
and
IIA*II
+
B*II ~ tl A
I1AII
:
+
A*II.
this implies
IIcll =@ltB + B*II ~ IIAIt • Finally, we have
x = T(t)z
, where
T(t) = S(t)U(t)
satisfies
[IT(t)II ~ 2 ½ , I]T-l(t)ll z {l]X(t)Px-l(t)ll 2 + l]X(t)(l - p)x-l(t)ll2} ½ . It follows at once from the definitions exponential
or ordinary dichotomy.
that kinematic
Lemma 2 provides
similarity preserves
an
the following more precise
result:
LEMMA 3.
If the linear differential equation
(i) has an exponential
or ordinary
dichotomy ""IIX(t)PX-I(s)II ~ Ke -~(t-s)
for
t ~ s , (S)
IIX(t)(I - P)x-l(s)U
where T(t)
s Ke -~(s-t)
for
P = P* , then there exists a continuously
s { t ,
differentiable
invertible matrix
with
llT-l(t)H < 2~K
ilT(t)ll ~ 2~ such that the change of variables whose coefficient matrix
IIC(t)ll ! llA(t)ll for every
The fundamental matrix for every
t 6 J
x = T(t)z
transforms
Z(t) = T-l(t)X(t)
(3)
and satisfies
of the equation (3) comnnltes with
P
and satisfies z 2K2e -a(t-s)
for
t ~ s ,
IIz(t)(l - P)z-l(s)ll ! 2K2e -a(s-t)
for
s ~ t .
The advantage
of this transformation
is that it eliminates
the 'big' solutions with the 'small' solutions. of exponential
dichotomies
similar to an orthogonal projection
P
the interaction of
It will now be used to establish the
on an arbitrary
consider the case where the projection
the form
P
t ( J •
NZ(t)pz-i(s)II
roughness
(i) into an equation
is Hermitian, commutes with
C(t)
interval.
is orthogonal,
It is sufficient
since any projection
to
is
and, in fact, to a uniquely determined matrix of
42
I: where
0 -< k -< n
PROPOSITION I . dichotomy
.
Suppose
(5) , where
the linear differential P = P*
and
6
=
K >- I ,
sup
IIB(t)H
equation
(i) has an exponential
o~ > 0 .
~ a/36K ~
t(J then the p e r t u r b e d equation y' : [A(t) + B ( t ) ] y
also has an exponential
Moreover,
dichotomy:
IIY(t)py-I(s)II
5 12K3e -(a-6K36)(t-s)
for
t ~ s ,
llY(t)(I - m)Y-l(s)ll
S 12K3e - ( ~ - 6 K 3 6 ) ( s - t )
for
s ~ t .
for all
t ( J , lly(t)py-l(t)
PrOOf.
We m a k e a l i n e a r c h a n g e
transforms
(i) i n t o
- X(t)Px-l(t)II
of variables
(3) a n d the p e r t u r b e d
~ 144a-IK6s
x = T(t)z
equation
D(t)
= T-l(t)B(t)T(t)
the e q u a t i o n commutes
with
.
Thus
(7) i n t o a k i n e m a t i c a l l y
This
(7) .
equation
E
O u r o b j e c t n o w is to t r a n s f o r m whose
coefficient
matrix
put E l = PEP + (I - P)E(I
variables
3.
P .
For a n y m a t r i x
so t h a t
, as in L e m m a
,
llD(t)II z 2K6 similar
.
(6) into
w' = [C(t) + D ( t ) ] w where
(6)
E = E1 + E2 . w = S(t)v
- P)
,
E 2 = P E ( I - P) + (I - P ) E P
,
Evidently
which
E1
transforms
commutes with
if
S(t)
We l o o k for a c h a n g e o f
(7) i n t o
v' = [C(t) + { D ( t ) S ( t ) } l ] V T h i s w i l l be a c h i e v e d
P .
satisfies
(8)
.
the d i f f e r e n t i a l
equation
43
(9)
S' = C(t)S - SC(t) + D(t)S - S{D(t)S} I or putting
S(t) = I * H(t)
, if
H(t)
satisfies
H' = C(t)H - HC(t) + {D(t)(l Consider
the integral
TH(t)
equation
= /:Z(t)pz-I(s)[I
and
J = (a , b)
,
+ H)} 2 - H{n(t)(l
H = TH
For any bounded
Using
Lemma
3 and
the
+ H(s)]Z(s)(I
- H(s)]D(s)[I continuous
IIHII = sup t(J
(10)
+ H)} 1
, where
- H(s)]D(s)[I
f~z(t)(I - e)z-l(s)[I
the equation
- P)z-l(t)ds
+ H(s) ]Z(s)pz-l(t)ds
matrix
IIH(t)II
function
H(t)
set
•
identity (I
-
G)D(I
+
G)
-
(I
-
H)D(I
+
H)
= (H - G)D - D(H - G) + (H - G)DH + GD(H - G)
HHJJ ~ ½ , bJJ ~ ~
we see that if
IIYH(t) - TG(t)ll
then ~ 12K4IIH - GII{fte-2~(t-s)llD(s)]Ids a
~b -2~(s-t) + Jt e
]lD(s)llds}
and hence JlTH - TGII ~ 12~-IK4;IH - GIIIIDII • Similarly
for
IIHII ~ ½
we obtain } + [be-2~(s-t)llD(s)ITds ~t "
HrH(t)ll -< 9K4{/te-2~(t-s)IID(s)llds -a and hence
frrHll ~ 9~-lK4bJl . Thus
if
itself.
6 S e/36K 5 Consequently,
has a unique
T
then
solution
is a contraction
by the contraction H(t)
in
and maps the ball
principle,
the integral
IIH]I s ½ equation
into H = TH
lIHII s ½ .
Moreover
IIHJJ ~ 18~-lKB~ The solution
H(t)
is differentiable
•
and satisfies
the differential
equation
44
H' : C(t)H - HC(t) Moreover
H(t)
= H
+ {(I - H)D(t)(I
satisfies PHP = 0 , (I - P)H(I
Hence
H = PH + HP , from which
differential b-ill
+ H)} 2
equation
(I0).
it follows
Then
- P) = 0 .
that
H(t)
S(t) = I + H(t)
is also a solution
satisfies
(9) and
of the
IIsU ~ 3/2
,
~ 2 . Since the coefficient
decompose
matrices
into two independent
Proposition
i.i with
t
of (3) and (8) commute with
systems
replaced
by
-t .
Hence
Y(t)
that the equation
i.i, or
(8) has a fundamental
~ 3K~
matrix
V(t)
satisfying
HV(t)PV-I(s)II
~ 2K2e -(~-6K3@)(t-s)
for
t ~ s ,
]]V(t)(I - P)v-l(s)]I
S 2K2e -(~-6K3@)(s-t)
for
s h t .
= T(t)S(t)V(t)
these equations
Since
ll{D(t)S(t)}l[ I ~ IID(t)S(t)ll it follows
P
to which we can apply Proposition
is a fundamental
matrix
for (6) satisfying
IIY(t)py-I(s)I[
~ 12K3e -(~-6K36)(t-s)
for
t ~ s ,
[IY(t)(I - P)Y-I(s)[I
~ 12K3e -(~-6K36)(s-t)
for
s ~ t .
Moreover y(t)py-l(t) and, since
IIHI] ~ ~
_ X(t)Px-l(t)
: T(t)[S(t)ps-l(t)
, IIS(t)PS-I(t)
It follows
The constants
Proposition
there
J
Pl[
~
41[H(t)ll.
in Proposition
- X(t)Px-l(t)ll
I applies
to exponential
and if, for some
fixed
small,
then
.
in Proposition
on the norm and the projection. dichotomies
Lemma
on
3.1 - - t h a t
a~bitrary
4.1 and, as However,
intervals.
if (i) has an exponential
dichotomy
h > 0 , : sup h t(J
is sufficiently
~ 144a-iK6~
i are cruder than those
is also a restriction
The proof shows too - - c f . on
-
that lly(t)py-l(t)
it stands,
- p]T-l(t)
-irt+h, ]t B(s)Ids
(6) has an exponential
dichotomy
on
J
with the same
45
projection.
Finally,
by appealing
to Proposition
1.6 instead of to Proposition
i.i,
we can prove
PROPOSITION
J
2.
Let
A(t)
and
B(t)
be continuous matrix functions on an interval
such that
IA(t)l and suppose there exist constants
X(t)
and a projection
< Ke - a ( t - s )
for
t
>- s
,
- P)x-l(s)t
-< Ke -a(s-t)
for
s >- t
,
is a fundamental matrix for the linear equation (1).
positive constant 6 : 6(P,
K > i , (~ > 0
-< M ,
IX(t)PX-I(s)I Ix(t)(I where
-< M , IB(t)l
e < a
there exist positive constants
T = T(P
P
such that
Then ~ r , K)
any
and
such that if
K, M, a, e)
t2
IftlB(t)dt
I <- S for
It 2
tll
-< e - I T
then
IY(t)py-I(s)I
IY(t)(z where
Y(t)
for
t ~ s ,
~ Ke - ( ~ - E ) ( s - t )
for
s Z t ,
is a fundamental matrix for the perturbed equation (6) and
As an immediate
PROPOSITION
- e)Y-Z(s)l
2 Ke - ( ~ - E ) ( t - s )
3.
Let
application
we obtain
be an almost periodic matrix function and suppose that all
A(t)
eigenvalues of its mean value A 0 : lim z±-~l/TTA(t)dt-
have real part different from zero. part
< -~ < 0
and
n - k
Then for all large
In fact, let
have
A0
eigenvalues with real part
~ > 0
k
> B > 0 .
the equation x' = A(cet)x
has a fundamental matrix
X(t)
such that
IX(t)PX-I(s)t
Ix(t)(z
- ~)x-Z(s)l
eigenvalues with real
-< Ke -O~(t-s)
for
t
>
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<~ ~ 1.1 ] k
and
Itjl ~ TAr
we
ave
<
,
...
, an
:
11
.-.
In
But, by the Cayley-Hamilton
a nA -I = (_A) n-I + al(-A)n-2 + ... + an_l I Since
a
n
= det A
it follows that
Idet AlIA-If < IA
Thus we can take
k
n
= 2n - 1 .
In-lnklIn ] = (2 n _ I)IAI n-I k:0[k]
theorem,
48
LEMMA
There exists a numerical
2.
m a t r i x w h i c h has
k
real p a r t
for some
~ ~ ,
eigenvalues
constant
h
> 0
n
with real part
~ - a
a > 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 <- M
k
>- 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,
<_ Le - ( 6 - e ) ( s - t )
for
s~t,
(2) IX(t)(I where
K , L
Proof.
- P)X-I(s)I
are positive constants depending only on
The change o f variables
A(t) + ½ ( ~ - 8)I
, ~
from the outset that
and
t ( J .
by
½ ( ~ + 8)
, and
replaces M
A(t)
2M .
by We may therefore assume
is c o n t i n u o u s l y d i f f e r e n t i a b l e with
By P r o p o s i t i o n 1.5 we may assume that
p r o j e c t i o n c o r r e s p o n d i n g to d i f f e r e n t i a b l e and
by
A(t)
8 = e •
Suppose first that all
8
x = e(~-8)t/2x
and
M,~+6,e
A(t)
as in L e m m a 3.
0 < k < n . Then
P(t)
IA'(t)l s 6 Let
P(t)
for
be the
is c o n t i n u o u s l y
51
IP(t)l ff c (~-iIA(t)l)n-i n
IP'(t)l ~ d ( ~ - I I A ( t ) I ) 2 n - l l A ' ( t ) I / I A ( t ) Since
P(t)
is a projection
W(t)
be the solution
, PPJP = 0 .
of the linear differential W' = [P'(t)P(t)
which takes
the value
every
t ~ J .
(4).
Therefore,
I
Using
Q
we have P' = P'P + PP'
Let
1
/
n
(3), one readily
equation
- P(t)P'(t)]W
at a fixed point
since the solutions
(3)
to ~ J .
verifies
(4)
Then
that
W(t)
P(t)W(t)
of (4) are uniquely
is invertible
for
is also a solution
determined
of
by their initial
values, P(t)W(t)
where
PO = P(to)
"
Thus
0 < k < n , it follows differentiable
= W(t)P 0 ,
P(t) = W(t)PoW-I(t)
from Lemmas
invertible
matrix
.
Since
IT(t)llT-l(t)l
the norm
function
~ alP(t) I , where
(here and in what
is similar to
and
5.1 and 5.2 that there exists a continuously T(t)
such that
P(t) = T(t)PT-I(t) with
PO
follows).
a
,
is a positive
Moreover,
if
constant
B(t)
depending
only on
is the coefficient
matrix of
(4), then C(t) = T-l(t)B(t)T(t) satisfies
IC(t)l
~ alB(t) I .
It follows
IT-l(t)Tf(t)l If in equation
- T-I(t)TP(t)
that
~ alP(t)IIP'(t) I .
(i) we make the change of variables
x = T(t)y
we obtain
the
equation y' : [D(t) - T-l(t)T'(t)]y where with
D(t) = T-l(t)A(t)T(t) P .
.
Since
A(t)
commutes
, with
(5) P(t)
,
D(t)
commutes
Thus the equation z' = D(t)z
decomposes
into a system of order
has the same eigenvalues
as
A(t)
k
Consequently
replaced
-t , to the two systems
by
and a system of order
and has
half-plane.
(6)
P
as spectral
we can apply Proposition into which
n - k . projection
1.5, or Proposition
(6) splits.
Since
Moreover
D(t)
for the left 1.5 with
t
52
D' : -T-IT'D + T-IA~T + DT-IT ' , we have
ID'(t)I ~
a(~-lIA(t)I)~n-31A'(t)I
It follows that there exists a positive constant
61 = 61(M , ~ , ~)
6 ~ 61 , the equation (6) has a fundamental matrix
where
Z(t)
for
t ~ s ,
IZ(t)(l - })z-l(s)I S Kle-(~-S/2)(s-t)
for
s { t ,
Therefore, by the roughness of exponential dichotomies
(Proposition 5.1), there exists a positive constant that, if
where
satisfying
IZ(t)pz-I(s)[ ~ Kle-(~-s/2)(t-s)
K 1 = KI(M , ~ , s) .
62 = @2(M , ~ , t) ~ 61
@ ~ 62 , the equation (5) has a fundamental matrix
Y(t)
IY(t)py-l(s)I ! K2e-(~-c)(t-s)
for
t ~ s ,
IY(t)(l - ~)Y-l(s) I ! K2e-(~-~)(s-t)
for
s ~ t ,
K 2 = K2(M ~ ~ , ~) .
such that, if
such
satisfying
Since the original equation (i) is kinematically similar
to the equation (5), this proves the result under the present hypotheses. We consider now the general case.
Put
Al(t) : h-lf~+hA(u)du , Then all t (J
is continuously differentiable,
Al(t) t
( J
A2(t)
.
On the other hand,
A2(t)
: A(t)
- Al(t)
.
IAl(t) I S M , and
is continuous and
IAl'(t) I ~ 6h -I
IA2(t) I ~ @
for all
. By Lemma 2 there exists a positive constant
~ 63 , then
Al(t)
has
eigenvalues with real part already proved, if
k
eigenvalues with real part { ~ - c/4
6 ~ 64(M , ~ , ~)
for all
Xl(t)
t ( J .
the equation
x I : Al(t)x has a fundamental matrix
a S : 63(M , E)
satisfying
such that, if
~ - ~ + ~/4
and
n - k
Hence, by what we have
for
53
I Xi ( t )PXI-I
(s )I
IXl(t)(I - P)X-I(s)I
where
K' : K ' ( M
see that if matrix
, ~ , E) .
@ = ~0(M
X(t)
~ K 'e-(a-c/2)(t-s)
for
t ~ s ,
~ K'e -(~-g/2)(s-t)
for
s ~ t ,
Using a g a i n the r o u g h n e s s of e x p o n e n t i a l d i c h o t o m i e s we
, ~ , E)
then the o r i g i n a l e q u a t i o n (i) has a f u n d a m e n t a l
s a t i s f y i n g the i n e q u a l i t i e s
P r o p o s i t i o n i admits a converse.
(2).
In p r o v i n g it we w i l l use the f o l l o w i n g lemma,
w h i c h depends on a result that w i l l be e s t a b l i s h e d (independently) later.
LEMMA 4.
Let
P
be an orthogonal projection of rank
k .
If
T
is an invertible
matrix such that
IIPTII < 2 -½ then
T
has
, ll(I
- p ) T- i Il
,
2 -%
eigenvalues of absolute value less than
k
eigenvalues of absolute value greater than Proof.
<
For a n y n o n - z e r o vector
1
and the remaining
n - k
i .
~ ,
¢*T*PT¢ < ½¢*¢ , ¢*(I
Put
H : 2P - I .
-
P)¢
< ½¢*T*T¢
.
Then $*T*HT~ = 2<*T*PT~ - ~*T*T~ < ~*~ - 2~*(I - P)~ =
Thus
T*HT < H .
Therefore, since
H
¢*H¢
.
has
k
eigenvalues
i
and
n - k
eigenvalues
-i , we can complete the p r o o f b y a p p e a l i n g to
LEMMA 5.
Let the matrix
T
be such that, for some Hermitian matrix
H ,
T*HT < H .
Then
T
has no eigenvalues on the unit circle,
positive (negative) eigenvalues of
H
of T .
Let
T .
A
is non-singular, and the number of
is equal to the number of eigenvalues of
with absolute value less than (greater than) Proof.
H
be a c o m p l e x n u m b e r o f absolute value
By r e p l a c i n g
T
by
l-IT
T
i .
we may s u p p o s e that
Then A = (T + I)(T - I) -I
I
w h i c h is not an eigenvalue I
is not an e i g e n v a l u e o f
54
is defined
and
HA + A*H < 0 .
z ÷ (z + l)(z - i) -I
maps the interior
(right) half-plane, The converse
PROPOSITION interval
2.
J
Since the linear
(exterior)
the result now follows to Proposition
Let
A(t)
and suppose
fractional
transformation
of the unit circle
from Proposition
onto the left
7.4.
1 states:
be a continuous the equation
n x n
matrix function
(i) has a fundamental
defined on an
matrix
X(t)
satisfying
the inequalities
where
IX(t)PX-I(s)I
<_ Ke -a(t-s)
for
t >- s ,
JX(t)(l - ~)x-l(s)l
_< Le -$(s-t)
for
s >- t ,
K , L , ~ , 6
are positive
constants
p
For any positive h = h(K, L, ~)
and
constant
=
,
e < min(a,
~ = 6(K, L, ~)
such
IA(t2> - A(tl) I < 6 then the matrix eigenvalues of
J
is at least
Proof. change and
A(t)
has
with real part
of variables
Suppose
by
constants
It 2 - tll -< h ,
with real part
for every
t
that the norm is Euclidean ~ = e (~-B)t/2 x
A(t)
6 s
there exist positive
for
eigenvalues
~- 8 - E
g)
that, i f
~- - ~ + E
whose distance
and
n - k
from both endpoints
h .
We may assume
A(t)
k
and
= A(t)
E/36K
5
+ ½(~
and
Then we can apply Proposition
the
and that
we can replace
- B)I
both
K = L > i . a
and
8
by
Also, by the y = (a + B)/2
.
interval
J
= [u
5.1 to the autonomous
- h,
u + hi
is
contained
in
equation
y' = A(u)y -- %(t)y + [%(u) - A(t)]y on the interval
J ,
Hence there exists
an invertible
matrix
particular,
and
ICPC-lehA(u) I ~ 12K3e -(Y-6K3d)h
,
IC(I - ~)c-le~hA(u) I ~ 12K3e -(¥-6K3~)h
,
C
such that,
in
d
.
55
ICPC -1 _ X ( u ) P X - l ( u ) ] The last inequality
~ 144y-1K66 .
implies
Ic~c-ll ~ sK By Lemma 5.1 we can write
c}c -I = s}s -1 , where
ISIIS-II ~ 10K .
Therefore,
putting
T : exp(hS-iA(u)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
~ = (~.) . 1
The induced norm of an
n x n
matrix
A = (a..) i]
is then n
IAI = s~pj~llaijl
•
The hypothesis (7) implies that the continuous function
Ra..(t)
has constant
ii
sign.
By a p p l y i n g
the
same permutation
loss of generality that
Ra..(t) < 0
to
for
rows
i ~ k
and
columns
and
we m a y a s s u m e
Ra..(t) > 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 <
for all
i ~ I
and Ix(s+h)I The set of all by the result is strictly
s
: Ixi(s+h) [
such that
Ix(s)I
of the previous
decreasing
for
Ix(t)I
m ÷ ~ ~i = 0
is strictly
Let
underlying
X(0) = I .
Then V .
El,
.Let.
.. .,~k
xl(t),
E 1 , .. ., (k
..., xk(t)
Consider
in
x (t) m
subspace
Choose
e
for
such that
of solutions so that
of all vectors
~ = (~i)
be an orthonormal
V , there exists
Hence Ix(t) I
x(t) sm + ~
such as
such that
matrix of the equation
is a k-dimensional
k ' ~m
R+
t h c .
s m 6 [+
a sequence
subspace basis
for
of the Qm
of integers
such that
xJ(0)
as
~ +®
orthogonal
(j
: 1 .....
k)
unit vectors.
: ~J(j = i, ..., k)
+...+ ~kxk(t)
of (i) such that
Xm(O)
1 k = ~l~m +...+ ~k~m
.
Hence the solutions
are linearly
linear combination
is the solution
is open in
a point
be the f~ndamental
x(t) = ~ixl(t) If
.
subspace
i .~m'
are mutually
such that
any nontrivial
~+
X(t)
~< +~J Evidently
on
i > 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 <
t > 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 <
as above,
-plx(tm) l2
-/~÷ll×(u)lhu m
< -(tm+ I - tm) IX(tm+l)l 2 •
62
Since
IX(tm) 12 = e l X ( t m + l ) l 2
0 < s < t < ~
If
it f o l l o w s
t m _< s < tin+ I
and
that
tm+ I
t m <_ ep
-
tn <. t . < t.n + l .( 0
.
Suppose
< m < n)
Ix(t)l ~ C l x ( t n ) I = e e e - ( n + l - m ) / 2 1 x ( t m + l )
, then
I
< eCe-(n+l-m)/21x(s)l ~ e C e - ~ ( t - s ) I x ( s )I where
~ = ~-(2ep)-I Suppose
.
next that
V(t*)
< 0
for s o m e
t* ~ 0 .
Then
for
t Z t*
-plx(t)l 2 ~ v(t) ~ v(t~) and hence V'(t) Therefore Let
V(t) + - ~ V
the i n i t i a l
V(t)
, which
to
which
VI .
~
uni~orenly in
of unit vectors
$~ 6 V 2 V(t
By the c o m p a c t n e s s V(t,
~) ÷ - ~
for
~ _> ~'
as
all ever 7
t > 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
<
is a
f o r e v e r y unit v e c t o r
so t h a t
x(t)
a
such that
for s o m e
> 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 -<
-
ep
if
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<= Thus
(A*, C)
Suppose next that
A
+
P
Therefore for all
k a 0 .
has no pure imaginary eigenvalues.
the left h a l f o f the circle and
= 0
: 0 .
is not controllable.
closed curve in the left half-plane
P
- i~*H~
C~ = 0 .
(i~)kc{
im , with
(3),
of
consisting
Izl = r , where
r > 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<
0
if
is controllable. ~ 6 V+ = P + V
we have
for any vector
~ = ~+ + <
(
, where
v
P ~ = 0
:
~+ ( V+
P
V
and hence
we have
and
~
P < : ~
and hence
( V_ , then
<*G~ : ~+*H~+ - < *H<_ Therefore only if If definite
H
is non-positive
H~+ = H~_ = 0 . G on
(negative)
Thus if
is non-singular, V
It follows
eigenvalues
n u m b e r of eigenvalues must be non-singular.
of is
on
H
then
V+
and non-negative
G
is singular,
H
is negative
on
then
H
definite
from the minimax principle is at least equal to
V
, and
G~ = 0
is also singular. on
V+
and positive
that the number of positive
dim V_ (dim V+) .
Since the total
dim V_ + dim V, , we must actually have equality
This completes
the proof.
if and
and
H
8.
In Lecture solutions
EQUATIONS ON IR AND ALMOST PERIODIC EQUATIONS
3 we considered,
of the inhomogeneous
for the half-line
[+
, the existence
of b o u n d e d
equation y' = A(t)y + f(t)
for, say, every b o u n d e d line
R
continuous
function
f .
(i) Corresponding
may be obtained by the change of variable
the same question
for the whole line
[
t + -t .
results
for the half-
We will now show that
can be answered in terms of the results
for
the two half-lines.
PROPOSITION I .
The inhomogeneous
solution bounded on the Banach spaces
~ C ,
linear differential equation ( l ) has at least one
for every function M ,
f (B(R)
,
where
B
denotes any one of
L , if and only if the following three conditions are
satisfied: (i)
the equation (l) has at least one solution bounded on function
(ii)
for every
~
for every
,
the equation (1) has at least one solution bounded on function
(iii)
f (B(~+)
R+
f ( B(~ ) ,
every solution of the corresponding homogeneous equation x' = A(t)x
is the sum of a solution which is bounded on
~+
(2)
and a solution which is bounded on
68
Proof.
Suppose
first that the three conditions
for any function b o u n d e d on
~+
f (B(R)
, [_
the homogeneous
the equation
respectively.
equation
y+(t)
By (i) and (ii), , y_(t)
By (iii) there exist solutions
(2) which are b o u n d e d x+(o)
are all satisfied.
(i) has solutions
- x_(O)
on
= y+(O)
[+ , ~
- y_(o)
which are
x+(t)
respectively
, x_(t)
of
and such that
.
Then y(t) : y+(t) - x+(t) are solutions Hence
o f (I) which are b o u n d e d on
y(t) = y(t) Conversely,
f (B(a) or
.
is a solution
suppose
~+
of (i) which
f(t) = ~(t)x(t)
to
~+
, where
, or
~
f (8(R)
- x_(t)
respectively
is bounded on
(i) has a solution b o u n d e d on
[
and
y(0) : y(O)
for every function
since every function
, of a function
for
Itl ~ i ,
for
Itl > 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 -~
~+
for every bounded continuous
which are restrictions
such that, for any function
+
C(~+)
.
f(t) ,
If(t) I = ~
If(t)I
t-~
~ , and its limit in
The proof of Proposition
f(t) .
to the half-line
is a closed subspace of the Banach space
function
3.3,
function
for every almost periodic function
sequence of almost periodic
sequence on
~+
[+
f(t) ~
In fact, for any almost periodic
bounded on
in the first part of the proof of
This is trivial.
to show that (i) has a solution bounded on
fundamental
~ .
2.
(iii) implies
f(t)
uniformly on
functions
C(~)
on
~+
is also a
is an almost periodic
3.4 shows that there exists a constant f 6 A , the equation
function.
r = rA > 0
(i) has a solution
y(t)
which is
and satisfies IIyll ~< rIIfll ,
where
IIyll = suply(t)I
•
t_>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 <
invariant,
sup
there is no
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 <- s < ~
dichotomy
equation,
independently
P
on
then, by the hypothesis
dichotomies
(i) has an exponential sequence
R_
bounded
corresponding
have exponential
for
~{ and the constants
uniformly
t = 0 , is also bounded
dichotomy
(i), each solution
on
exponential
equations
of an equation
-~ < s ~ t < ~
is bounded
Ix(t)I _< Ke-~(s-t)Ix(s) 1 and each equation
x(t)
+ •
Ix(t) I _< Ke-~(t-S)Ix(s) I Moreover
2.1 and its "proof, that there exist
sequence
{P }
A(t + h m) ÷ A0(t)
is a projection
(i) on the half-line
~+
independent converges
corresponding
locally to the
then, by Lemma 8.1,
the equation x' = A ( t ) x has an exponential Similarly, If
P
suppose
dichotomy
on
with projection
A(t + k m) ~ A (t)
is a projection
corresponding
locally
of the same rank as
uniformly
to the exponential
P+
for some sequence dichotomy
km÷
_oo
of the equation
(i)
77
on the half-line
~{
then the equation
x' : A~(t)x has an exponential
dichotomy
The preceding
PROPOSITION
results
on
[
with projection
establish,
X;
'
:
~(t)
has a nontrivial solution bounded on Then each equation (1) in the
x(t, A)
where
A
of the equation (2)
X
~ . a-
or
~-
limit set of (2) has an exponential
~ . Moreover there exist positive constants
K , a
such that, if
X(O, A) = I ,
is the yundamental matrix of (i) for which IX(t, A)P(A)X-I(s,
A) I ~ Ke -a(t-s)
for
-~ < s _< t < ~
IX(t, A)(I - ?(A))X-I(s,
A) I ~ Ke -~(s-t)
for
-~ < t _< s < ~
P(A)
P_
in particular,
Suppose that no equation (i) in the hull
I.
dichotomy on
of the same rank as
is a uniquely determined projection depending continuously on
A
(with
the topology of uniform convergence on compact intervals). We also have
PROPOSITION 2.
(i)
(ii)
The following statements are equivalent:
no equation (i) in the hull
A
of (2) has a nontrivial solution bounded on
the equation (2) has an exponential dichotomy on each of the half-lines R+ , ~
with corresponding projections
P+ , P
such that
P + P - = P- P+ = P+
Proof.
(i) implies
(ii):
Only the conditions
These say that the nullspace range of
P+
is contained
nullspaee
of
P
is satisfied
intersect
and then
(ii) implies hv ÷ h , where
(i):
P
of
trivially,
P_
Let
bounded
A(t + h v) ÷ A(t) If
h
in the nullspace
because
P+
condition locally
is finite then
solution
of
P+
But, since the range of
we can choose
so that the second
remain t o be proved.
on the projections
is contained
in the range of
-= E h ~ ~
(i) has no nontrivial
P_
and the P+
and the
so that the first condition is also satisfied.
uniformly.
We may suppose
A(t) = A(t + h)
the range of
P+
that
and the equation
and the nullspace
of
78
P
intersect trivially.
exponential
If
d i c h o t o m y on
h = ± ~
[
then, by Lemma 8.1, the equation
(i) has an
and the same c o n c l u s i o n holds.
The conditions o f P r o p o s i t i o n 2 do not guarantee the existence o f an e x p o n e n t i a l d i c h o t o m y on the w h o l e line
[ .
Consider,
for example, the scalar e q u a t i o n
x' = (tanh t)x , w h o s e solutions are given by and u n i f o r m l y continuous where
-~ < h < ~
x(t)
function
=
x(0) cosh t .
tanh
t
The h u l l
-i , i . R .
does not h a v e an e x p o n e n t i a l d i c h o t o m y on
[
This example can e a s i l y be generalised.
tanh(t + h)
,
Hence no equation in the h u l l has
On the other hand, the given e q u a t i o n
has no n o n t r i v i a l solution w h i c h is b o u n d e d on either
t ( ~<
of the b o u n d e d
consists of all functions
, and the two constants
a n o n t r i v i a l solution w h i c h is b o u n d e d on
function for
A
R+
or
[
, and t h e r e f o r e
. Let
A(t)
be a continuous m a t r i x
=
lim ~(t) t~-~
such that the limits
A+ :
lim A(t) t++oo
, A
e x i s t and have no pure imaginary eigenvalues.
If
A+
A
and
do not have the same
n u m b e r o f e i g e n v a l u e s w i t h n e g a t i v e real part, then the e q u a t i o n
(2) does not have an
e x p o n e n t i a l d i c h o t o m y on
(2) has no non-
trivial solution equation
~(t)
~
.
But if, in addition, the equation
such that
~(t) + 0
for both
t++ ~°
A b o u n d e d and u n i f o r m l y continuous m a t r i x function
recurrent if, for every A(t)
.
t+-~
A(t)
in the h u l l of
A(t)
A(t)
, also
, then n o [.
is said to be
A(t)
is in the h u l l of
For example, any almost p e r i o d i c function is recurrent.
The p r e c e d i n g p r o b l e m does not arise if b o t h an
and
(i) in the h u l l o f (2) has a n o n t r i v i a l solution b o u n d e d on
~-
and an
PROPOSITION 3.
dichotomy on
Suppose ~
A(t)
is recurrent.
For t h e n
A(t)
is
m - l i m i t o f itself, and from P r o p o s i t i o n i we obtain
A(t)
is recurrent.
Then the equation ( 2 ) has an exponential
if and only if no equation (i) in the hull
nontrivial solution bounded on
A
of
A(t)
has a
~ .
In the almost p e r i o d i c case this provides a n o t h e r e q u i v a l e n t to the statements o f Proposition
8.3.
In particular, statement
F a v a r d [i] the h y p o t h e s i s redundant. hull
A
(iii) shows that in a classical t h e o r e m o f
that the inhomogeneous e q u a t i o n has a b o u n d e d s o l u t i o n is
It may be n o t e d also that for almost p e r i o d i c
A(t)
the definition o f the
is not a l t e r e d if locally u n i f o r m convergence is r e p l a c e d b y u n i f o r m con-
v e r g e n c e on
~
.
Proposition
3 characterises r e c u r r e n t equations with e x p o n e n t i a l dichotomies.
We n o w turn o u r attention to r e c u r r e n t equations with o r d i n a r y dichotomies.
Thus we
79
drop the hypothesis
(H)
.
We first prove
LEMMA 3.
A(t)
Suppose
is continuous on
and the equation (2) has an
~
m-dimensional subspace of solutions such that, for each nontrivial solution
~(t)
in
this subspace, 0 < inf t(~+
Suppose also that Then the
Ix(t)l ~ sup t(R+
u-limit equation (i) has an
m-dimensional subspace of solutions such
s
sup
t(R Proof.
Let
xl(t),
By r e s t r i c t i n g
attention
(j : i, ..., m) solution
..., Xm(t)
.
Then
of the equation
solutions
xi(t),
Ix(t)t
<
t(~ be a basis
to a subsequence
for the given subspace of solutions we may suppose that
x.(t] + h ) ÷ xj(t) (i) such that
..., Xm(t)
h~ + ~
in this subspace,
x(t)
I×(t)!
0 < inf
<
locally uniformly for some sequence
A(t + h ) + A(t)
that, for each nontrivial solution
l~(t)l
for every real
x.(O) ]
are b o u n d e d on
= ~j [ .
o f (2).
xj(h ) ÷ ~j t , where
(j = i, ..., m) .
x.(t)]
is the
Evidently the
They are also linearly
independent.
For suppose Cl~ I + . . .
+ Cm~ m = 0
and put x(t) = Cl~l(t) + ... + CmXm(t) Since
x(h)
÷ 0
we must have
x(t) ~ 0 .
Hence
.
c I = .. . = a m = 0 .
Let x(t) : alxl(t) be a nontrivial
linear combination
of
xl(t) , ..., Xm(t)
x(t) = al~l(t) there is a
6 > 0
it follows that
PROPOSITION 4. dichotomy on
such that Ix(t)I ~ 6
Suppose ~+ .
A(t)
+ ... + amXm(t)
Ix(t)l ~ ~
-
Then if
+ ... + amXm(t) for every
for every real
t ~ 0 .
Since
x(t + h ) ÷ x(t)
t .
is recurrent and the equation (2) has an ordinary
Then there exist positive constants
K , ~
such that, if
X(t, A)
80
is the fundamental matrix of an equation (i) in the hull of (2) with
where
IX(t, A)Po(A)X-I(s , A) I ! K
for
_co <
8
~ t
< co
Ix(t, A)Pe(A)X-I(s, A) I ~ Ke-~(t-s)
for
_oo <
S <- t
< co
IX(t, A)P (A)X-I(s, A) I S Ke -a(s-t)
for
-~ < t ~ s < ~ ,
Po(A)
, P+(A)
Proof.
are uniquely determined supplementary projections
, P (A)
depending continuously on intervals)
(with the topology of uniform convergence on compact
A
.
Let
P
be a projection whose range is the subspace
solutions of (2) which are bounded on with projection is an
P , by Proposition
of
2.2.
of initial values of
(2) has a dichotomy on
Since every equation
(i) has a dichotomy on
P
is exactly the subspace ~+
satisfies
VI
~+
(i) in the hull of (2)
~
with projection
Hence either x(t)
Thus every solution
x(t) ÷ 0
inflx(t) 1 > 0
zero as
for
V+ c V I t ÷ ~
as
t ~ ~
either
x(t)
of an equation
(i) which is bounded on
for
-~ <
s
inflx(t) I > 0
~ ~
as
t ÷ -~
~
t
<
for or
t ( R+ x(t)
Moreover,
is bounded on
if R
and
t ( be the subspace of initial values of solutions of (i) which tend to
and let
~
be any subspace of
R .
If
V0
VI
supplementary
to
V+
It follows
(i) has a subspace of solutions with the same dimension
such that each nontrivial
from zero on
or
Ix(t)I
from Lemma 3 that the equation W
of the same rank as
an inequality
is nontrivial,
Let
P
It follows that the range
of initial values of solutions of (i) which are
Ix(t)I ! L1×(s)1
as
Toe equation
Thus we could have started with (i) instead of (2).
bounded on +
~+
e-limit of (2), the argument in the first part of the proof of Lemma 8.1 shows
that each equation .
X(0, A) = I ,
solution
in this subspace is bounded and bounded away
is the corresponding
subspace of initial values then
VI = V0 $ V+ .
Similarly, are bounded on
let ~_
V2 and
which tend to zero as
be the subspace of initial values of solutions V_ c V 2 t ÷ -~
of (i) which
the subspace of initial values of solutions of (i) Then in the same way we can write
V2 = Vo ' $ V_ ,
81
where
V 0,
is of the same nature as
dim V 0 S dim V 0' , and since Hence
dim V 0
P
= dim V 0
V2
V0 .
and
Since
V+
intersect
and we can take
space
V
~+
and
V_
intersect
trivially,
trivially,
dim V 0 ' ~ dim V 0
o
V0P = V 0 .
Finally, since (i) has a dichotomy on solution b o u n d e d on
VI
~
, every solution
and a solution b o u n d e d on
~
is the sum of a
Therefore
the entire vector
admits the direct decomposition
v : v° % v+ % v_ The preceding
argument
shows also that the dimensions
same for all equations value in both
(I).
V 0 $ V+
and
of the three components
Any solution which is bounded on V 0 $ V_ ,
and hence in
V0 .
space of initial values of solutions which are b o u n d e d on solution b o u n d e d on
~
is b o u n d e d away from zero.
trivial solution with initial value in Consider,
in particular,
There exist uniquely V0
' V+ ' V-
pPojection
PO + P+
K > 0
where
P+
and on
V0
is just the sub-
~ , and every nontrivial
Furthermore,
if
x(t)
is a non-
as
(2) and let
V. = Vj(A) ]
(j = O, +, -) .
projections
PO ' P+ ' >-
2.2, the equation
N_ , and h e n c e a l s o R .
Thus
Ix(t)l ÷ ~
supplementary
R+
has its initial
then
By Proposition
on b o t h
a dichotomy with projection
(V_)
the equation
determined
respectively.
Q+
~
are the
It follows
t ÷ -~ (+~) .
with ranges
(2) has a dichotomy with
on
~ .
Similarly
it has
that there exists a constant
such that
X(t) = X(t, A) Suppose
IX(t)P+X-I(s)I
~ K
for
-~ < s ~ t < ~
,
IX(t)P X-I(s)]
~ K
for
-~ < t S s < ~
,
IX(t)PoX-I(s) I ~ K
for
-~ < s , t < ~
,
.
A(t + h ) + A(t)
locally
we see that there exist supplementary same ranks as
P+ ' P
' P0
such that
uniformly.
projections
Then by passing to a subsequence P+(A)
, P_(A)
, P0(A)
with the
82
IX(t, A)P+(A)X-I(s, A) I ! K
for
-~ < s ! t < ~ ,
{X(t, A)P (A)x-l(s, A) I i K
for
-~ < t ~ s < ~ ,
IX(t, A)P0(A)X-I(s, A) I ~ K
for
-~ < s , t <
Any nontrivial solution
x0(t)
of the equation (i) with initial value in
is bounded, and bounded away from zero, on follows that is bounded on
Po(A)V
V0 = P0(A)V .
R .
Since the dimensions are equal, it
Any nontrivial solution
x+(t)
starting from
Therefore, by Lemma 3, there is a solution
+
Po(A)V
x0(t)
P+(A)V
starting from
such t h a t
Ix0(t) + x+(t)J : 0
inf t(R + Since
Ix+(t)f ~ Klx0(s) + x+(s)I this implies that that P (A)
<
x+(t) + 0
= P+(A)V .
as
Similarly,
t + ~ V
for t ~ s
Since the dimensions are equal, if follows
= P_(A)V .
Thus the projections
P0(A] , P+(A) ,
are uniquely determined and, without passing to a subsequence,
X(h )PjX-I(h~) ÷ Pj(A)
(j : 0, +, -)
We are now in a position to apply our previous arguments under the hypoihesis (H)
to the subspace
0 < 8 < i x(0) ( V
V_ $ V+
there exists $ V+
First of all we have the analogue of Lemma 2:
T > 0
such that every solution
x(t)
if
of (2] with
satisfies Ix(t)l ~ 0
sup
Ix(u)I
.
Eu-tE~T For otherwise there exists a sequence of solutions
%(t)
of (2) with
Moreover we may assume that where
I~l : i .
Then
~(0)
T
+ ~ , a real sequence 6 V
$ V+
A(t + t ) + A(t)
x~(t + t ) ÷ x(t)
and
{t } , and a sequence
I~v(t ) I = 1 , such that
locally uniformly and
for every
t ~ where
x(t)
$
is the solution
83
of the equation
(i) such that
the other hand, since
Hence
x(O) = ~ .
PoX (0) = 0
Ix(t)l S e -I
for
On
-~ < t < ~
also
Po(A)~ : lim X(t )PoX-I(t
) . X(t )x (0) : 0 .
Thus we have a contradiction. that , 'with respect
It follows an e x p o n e n t i a l , P+
dichotomy on
The remaining
Proposition
, the corresponding
assertions
4 provides
almost periodic, asymptotic
~
o f Proposition
fact w e then have P
on
projections
~+
linear differential
Thus if
It shows that for any
stability on
is any almost periodic
a(t)
difficulty.
system uniform stability
together imply u n i f o r m asymptotic
= PO = 0 .
(2) has
being n e c e s s a r i l y
4 now follow without
an answer to a p r o b l e m of Hahn.
or even recurrent,
stability
V _ $ V+ ' , the equation
to the subspace
~
and
.
In
function with
mean value zero such that tt
]Oa(S)ds + -~ then the scalar differential
as
t ÷ ~
equation x' = a(t)x
is asymptotically
stable but not uniformly
a(t) = -
Proposition periodic
and let
el,
suppose
that
of an autonomous
function from "''' ~ m
~m
nonlinear into
be real numbers
Xo(t) is a solution of the autonomous
~
is continuously
in
sin ~t/n
the
one can take
.
o~b£~Z 8tab{Z{ty of Let
g(u)
a quasi-
be a continuously
with period 1 in each coordinate
linearly
differential = ~(x)
differentiable
For example,
system.
= g(~l t,
x'
where
-3/2
4 may also be used to discuss
solution
dlfferentiable
~ n n=l
stable.
independent
over the rationals.
u. , ] We
..., ~ m t) equation
(3)
,
on the range of
g •
Then all functions xh(t) in the hull of
Xo(t)
= g(~it+hl,
are also solutions
..., ~mt+hm)
of (3), and the
(h 6 A m) partial derivatives
84
-..,
gu.(~it~ ]
~ m t)
are quasi-periodic
solutions
y' = Cx[Xo(t)]y We assume that the variational m
solutions
form a basis
equation
of the variational
equation
(4)
•
(4) is uniformly stable on
for the solutions
which are bounded on
+
and that these
~ .
The hull of (4) consists o f all equations
y, Let
Yh(t)
be the fundamental
Proposition
= Cx[Xh(t)]y
•
matrix of this equation such that
4 there exist positive IYh(t)PhYh-l(s)I
Yh(0)
constants
K , ~
~ Ke -~(t-s)
for
-~ < s ~ t < ~
Z K
for
-~ < s , t < ~ ,
= I .
By
such that ,
(5) IYh(t)(l - Ph)Yh-l(s)I where
Ph
is a uniquely determined projection
hypothesis,
the vectors
which depends
gu (0) (j = i, ..., m) ]
are a basis
continuously
on
h .
By
for the null space o f
P0 " If in (3) we make the change of variable
z' = ~x[Xh(t)]z
+
we obtain the equation
x = z + Xh(t)
fh(t,
z)
,
where
fb(t, Thus
fh
z)
= ¢[z
is a quasi-periodic
+ xh(t)]
- ¢[ym(t)]
function o f
t
- Cx[Xh(t)]z
for each fixed
Ifh(t , z I) - fh(t, z2) I ~ elz I - z21 if
IZll
~ ~
Choose
,
y
Iz21
~ 6
so that
, where
@ :
0 < y < ~
@(g)
for all
z , fh(t, 0) = 0 , and
t
and
h
> 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
= ~nlIA
+ A*II
2nHAH2 Thus
llB(t)ll The inhomogeneous
~ (2n)~llA(t)ll-
equation z' = A(t)z + f(t)
will have a b o u n d e d solution
z(t)
if and only if the transformed
w' = B(t)w + g(t) where
g(t) = u-l(t)f(t)
equation
function
is real then
g(t)
.
If
A(t)
is also bounded.
of a b o u n d e d coefficient
It follows
(4)
w(t) = u-l(t)z(t)
solution
B(t)
is also real, and if
that to prove Proposition
on the dimension
n .
Hence the function
for every b o u n d e d
f(t)
continuous
A(t)
is bounded
3.3, for the ease
matrix, we can assume that the coefficient
triangular and use induction
•
for every bounded continuous
(4) has a b o u n d e d
B(t)
equation
,
, has a b o u n d e d solution
(3) will have a bounded solution
if and only if the equation
then
(3)
m a t r i x is also
We will not carry out the details
here, but obtain instead a new result.
PROPOSITION I .
Let
B(t)
be a continuous
n
x
matrix function on
n
~+ , which is
lower tr4angular with real main diagonal and bounded off-diagonal elements. Then there exists a constant
6 = 6(B) > 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)
c > 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 -<
i
so large that
llb(t)~n(t) lI -< ~1 Then, by what we have proved
for
for
t -> 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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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