Asymptotic distribution of singularities of solutions of Matrix-Riccati differential equations

Abstract

In the present paper, we make use of the method of asymptotic integration to get estimates on those regions in the complex plane where singularities and critical points of solutions of the Matrix-Riccati differential equation with polynomial co-efficients may appear. The result is that most of these points lie around a finite number of permanent critical directions. These permanent directions are defined by the coefficients of the differential equation. The number of singularities outside certain domains around the permanent critical directions, in a circle of radius r, is of growth O(log r). Applications of the results to periodic solutions and to the determination of critical points are given.

Full text

J. Austral. Math. Soc. Ser.
B
34(1992), 112-131
ASYMPTOTIC DISTRIBUTION
OF SINGULARITIES OF SOLUTIONS
OF MATRIX-RICCATI DIFFERENTIAL EQUATIONS
GERHARD JANK'
(Received 28 June 1990; revised 15 January 1991)
Abstract
In the present paper, we make use
of
the method
of
asymptotic integration to get
estimates
on
those regions
in
the complex plane where singularities and critical
points
of
solutions
of
the Matrix-Riccati differential equation with polynomial co-
efficients may appear. The result
is
that most
of
these points lie around
a
finite
number
of
permanent critical directions. These permanent directions are defined
by the coefficients of the differential equation. The number
of
singularities outside
certain domains around the permanent critical directions,
in a
circle
of
radius
r ,
is
of
growth 0(log
r).
Applications of the results to periodic solutions and
to
the
determination
of
critical points are given.
1.
Introduction
Many problems
in
mechanics, control theory and system theory lead
to a
nonlinear system of differential equations of the form
W' = A(z)
+
B(z)W+WC(z) + WD(z)W. (1.1)
Here the coefficient matrices
A, B, C, D are
matrix polynomials
(in the
complex variable
z) of
type mxn,mxm,nxn,nxm, respectively.
According
to the
existence theorem (see e.g. [6]), every solution
W{z) of
(1.1)
is a
locally holomorphic
m x n
matrix. Following W.
T.
Reid [14] we
consider the linear system
'Lehrstuhl
II fur
Mathematik, RWTH Aachen, Templergraben 55, D-5100 Aachen, Federal
Republic
of
Germany.
© Copyright Australian Mathematical Society 1992, Serial-fee code 0334-2700/92
112

[2]
Matrix-Riccati differential equations
113
Y'
= £l(z)Y
(1.2)
associated with (1.1), where
-C(z)
~D(z)
A(z)
B(z)
As
Q is a
polynomial matrix
of
type (n + m)
x
(n
+ m),
we get from
[6]
that all solutions
of
(1.2) are entire, i.e. holomorphic
in
the complex plane.
Let
U, V
be matrices
of
type
n x n, m x n
respectively, where
•GO
(1.4)
is
a
solution
of
(1.2). Then
it is
easy
to
see that
W
—
VU is a
solution
of (1.1)
if U~
l
exists.
To
guarantee
the
existence,
we
may
for
example
take solutions
Y of
(1.2) with 7(0)
=
(
E
v
«),
where
E
n
is the n-dimensional
identity matrix and V
o
is arbitrary. On the other hand, given any solution
W
of (1.1) with
W(z
0
) = W
0
,
z
Q
eC, then we take the entire solution
Y - (y)
of (1.2) with
Y(z
0
) =
(^). According to the uniqueness of solutions of (1.1)
we get
W{z)
= VU~\ (1.5)
As
the
right hand side
is
holomorphic
in C,
except
for a
discrete
set of
singularities where detC/
= 0,
(1.5) gives
a
meromorphic extension
of
the
given solution
W.
For what follows we have the important
REMARK 1.1. All solutions of (1.1) are meromorphic in C, i.e. given
z
0
e C,
a solution
W
has locally the Laurent representation
w
k(
z
-
z
o)
k
>
W
k
€C
mx
",
peN
0
. (1.6)
k=-
P
For
p
>
0 in
(1.6) and
W /0
we say that
W
has
a
pole
of
order
p in
z
o-
In the scalar case
of
(1.1)
it
is easy to see that the order of almost all poles
is one (see [8,
p.
219]). This
is
not true for the higher dimensional case,
as
the following example shows.
The equation
has the solution

114 Gerhard Jank
[3]
/tanz\
-{tm
2
z)
which
has
infinitely many poles
of
second order.
It
is a
task
of
this paper
to get
information
on the
asymptotic distribution
of
the
poles
of the
solutions. Clearly,
it
depends
on the
coefficients
in
(1.1).
This
can be
done
by
characterising
the
regions where
no
singularities
can ap-
pear.
In
these regions,
we can
derive uniformly asymptotic estimates
for the
solutions.
The
method
we use is the
asymptotic integration which goes back
to
L. W.
Thome,
H.
Poincare
and G. P.
Birkhoff,
and was
further developed
by
W. B.
Jurkat [10], whose presentation
we
follow.
2.
Preliminaries
In this section,
we
want
to
describe
the
tool
of
asymptotic integration
for
linear systems
in the
irregular singular case,
and how it can be
applied
to
the asymptotic representation
of
solutions
of (1.1) in the
neighborhood
of
infinity. From
[10, p. 32] and [2, p. 11] we get for the
system
(1.2) the
existence
of a
normalised formal fundamental system
of the
form
H(z) =
T(z
i/p
)(z
l/p
)
L
e
W
(Q(z
l/p
)),
(2.1)
with a suitable peN; here T, L, Q, denote n + m x n + m matrices. T
is a formal power series of the form
T(t)
=
T
k
t
k
+ T
lc
_/~
i
+
--- + T
Q
+
T_
l
t~
i
+•••
, Tj
:
e
c"
+mxn+m
,
L
is a
constant matrix
in
Jordan canonical form
and Q is a
diagonal matrix
Q
=
diag(?,,
... ,
q
n+m
),
where
q
i
, j =
1,...,«
+ m , are
polynomials with
^(0)
= 0.
Furthermore
we
have from
[2, p.
11] that
QL = LQ.
Notice that
this condition implies that
the
polynomials
in Q
which belong
to one
specific
Jordan block
of L are
identical.
There
is an
algorithm
for
the determination
of
the polynomials
# and the
eigenvalues
of L up to an
integer, which makes
use of
transformations
of
the
system (1.2)
to an
ordinary differential equation
of
order
n+m ,
with rational
coefficients. This algorithm has been implemented using the computer algebra
system MAPLE
(cf.
[3], [4]).
If
all
polynomials
q
x
, ... ,
q
n+m
are
equal, then
(1.2)
can
easily
be
transformed
to the
regular singular case,
i.e. to the
case
where
Q = 0 in the
representation (2.1).
If not all
polynomials
in the
representation
(2.1) are
equal, then they define
a set of
Stokes directions.
Here
we
call
6 a
Stokes direction
if
there exist
two
different polynomials
q
t
,
q
k
in Q
with
the
property that
if
*,(*"')
-
q
k
{z
llP
)
= z
i/p
(a, + O(z-
1/P
)), a,?0,

[4] Matrix-Riccati differential equations 11S
then
|| 0, if p e (0-e, 0),
for some e > 0. The set of all Stokes directions is discrete. If it is not empty,
then we denote the Stokes directions by y^ , n e Z and arrange them so that
y
u
<
y
m+l
.
With this we define (see [10]) the angular domains called normal
sectors
5
|(
= {z|y
/(
_
1
<aigz<y
/l+1
} (2.2)
and we put 5 = C for all fi, if there are no Stokes directions.
If we are given a solution matrix Y of (1.2), then we get from [5] that to
each normal sector S^ as defined in (2.2) there exists a matrix C
p
6
c
("+
m
)*"
such that
Y(z) ~ //(z)C; (2.3)
is an asymptotic representation of Y in S . This asymptotic relation (2.3)
should be understood in the way that to each sector S^ there is a fundamental
solution F (z) of (1.2) such that F (z) ~ H(z). If we truncate the formal
power series T in (2.1) at a power, say —n, and denote the resulting finite
series by T then the last ~ sign means
/
Q(z
1/
"))(z
1/f
')-
L
- T
n
(z) = O(z-
as z-»oo in any closed subsector of S .
In order to define an asymptotic representation for solutions of (1.1) we
divide the n + mxn + m matrix H in (2.1) into two blocks H
{
, H
2
of size
nxn + m,mxn + m respectively, such that
(2.4)
Now we give
DEFINITION
2.1. The mxn matrix *F(z) given by
is said to be an asymptotic representation of a solution W(z) of (1.1) in S ,
if there exists a solution Y = (%) of (1.2) with W = VU~
l
and with the
asymptotic representation
r
i
I
i
/~*
~ \H )
>*
in S
M
. Concisely we write W ~ *F.
This definition is analogous to the scalar case treated in [9].

116 Gerhard Jank
[5]
3.
The distribution
of
poles
Before going into
a
detailed description
of
the asymptotic distribution
of
the poles
of a
given solution
W
of
(1.1),
we
want
to
give
a
crude estimate
on their number
in
a
given circle
of
radius
r.
Suppose that
the
polynomial matrix
Q
in
(1.3) has
the
form
(3.1)
and
Cl
g
is
a
nilpotent matrix
of
degree
k e
N.
Then
it
follows from [8,
p.
214] that
all
solutions
of
(1.2) have
an
order
of
growth less than
or
equal
to
g
+
1 —
\/k. If Q
is not
nilpotent,
one has to put
\/k
—
0.
From this
it
follows
for
the growth order
p
of
the determinant
p(degU)<g+l-l/k.
If
we
denote by
N(r, W)
and
N(r, 0,
dett/) the Nevanlinna counting func-
tions
of
the poles
of
a
solution
W of
(1.1)
and
of
the zeros
of
det
U,
re-
spectively, with
U
from (1.5), then we have
N(r,W)<N(r,0,detU).
(3.2)
Now, with the first fundamental theorem of Nevanlinna theory
(cf.,
e.g., [13],
[8,
p.
50])
and the
definition
of
order (see [8,
p.
96]),
the
right-hand side can
be estimated
for
a
given
e
and for
sufficiently large
r by
N(r,6,detU)<r
g+1
-
i/k+e
.
This yields together with
(3.2)
N(r,W)<r
g+l
-
1/k+e
.
(3.3)
Denoting
by
n(r, W)
the
number
of
poles
of W
in
the
circle with radius
r, we get from [8,
p.
103] that
N(r,
W)
and
n{r, W)
have
the
same order,
and hence
for
e
>
0
and
sufficiently large
r
we have
n(r,W)<r
g+l
-
l/k+E
(3.4)
by (3.3).
After that crude estimation, we want to obtain estimates on the distribution
of the poles.
As we
already pointed out,
the
solutions
Y
of
(1.2)
are
entire
and
the
poles
of
solutions
of
(1.1) represented
by
(1.5)
are
contained
in the

[6]
Matrix-Riccati differential equations
117
set
of
zeros
of
det
U.
Henceforth
we
will give
an
asymptotic estimate
of
detf/.
Suppose that
W
is
a
solution
of
(1.1) and 4* its asymptotic representation
in
S
according
to
Definition 2.1.
In
the following, we work
in
one specific
normal sector
S
and hence
for
simplicity we suppress the index
ft.
Then
we get from Definition 2.1,
U(z)~H
l
(z)C, CeC
{m+n)x
". (3.5)
In
a
first step,
we
write
the
matrix elements
of H
l
in the
form
(tf,)
0
.(z)
=
h
u
(z)expte/z
1
'")),
/ =
1,...,«,
j =
1,...,«
+ m.
(3.6)
The elements
A
(
in
(3.6) are
defined
by (2.1) and
(2.4). Then
the
matrix
elements
of
H
x
(z)C
are
n+m
(H
l
(z)C)
ij
= £
h
ik
(z)exp(q
k
(z
1/p
))c
kj
, i,j=l,...,n,
(3.7)
k=l
where
c. .
denotes the matrix elements
of C.
From this we get formally
n
j=l
\k=\
/
where
a
denotes
a
permutation
of
1,
... , n
.
To
study this determinant
asymptotically, later on we have to truncate all the formal power series
in
the
formal expressions
h,..
k
(z).
Before doing this we define the
set M(Q) as
the
set of
all polynomials
q of
the form
where
q.
denotes the pairwise distinct polynomials from the matrix
Q in
(2.1).
Herewith the determinant
in
(3.8) can
be
written
as
det(/f,C)= Y, h
k
(z)exp(p
k
(z
l/p
)), (3.9)
where the nonvanishing expressions
h
k
are sums
of
products
of
the expres-
sions h
aU)k
(z)
in
(3.8).

118 Gerhard Jank
[7]
To the
set of
Stokes directions
of
the polynomials
in Q,
denoted by
y ,
there is now an additional set of Stokes directions defined by the polynomials
in
M{Q).
This
set
can
be
ordered as before. Assume now that
a is
such
a
Stokes direction with
Vi <
a
<Vi
and
a_ its
predecessor with
Vi <"_<«<
Vi
•
Then we say
a
polynomial
p. in
(3.9)
is
leading
in
the angular domain
S~
=
{z\a_
+
e
<argz
<
a},
e>0
(3.10)
if
hj ^ 0, and if, for
some
&
e
(a_
, a)
with
z =
re
li>
and
for all
polyno-
mials
p
k
in
(3.9) with
h
k
^0,
the quotient
tends
to
zero
for k / j as
r-too.
Notice now that
if k ^
j,
and
if
P
k
(z
1/P
) -Pjiz
1
*)
=
az\\+o{\)), z^oo,
we have
lim
-^ —r^ ^<0.
(3.11)
\z\—»oo,
a_<argz<a
l^l
Assuming now that
in
(3.9)
the
polynomial
/? is
leading
in S~,
then
we
write formally
;
^
j
(3.12)
with
l/p
)
P
(z
1/P
)
E h
k
(z)exp(p
k
(z
l/p
)
-
Pj
(z
1/P
)).
In order to get a useful asymptotic estimate for det
U,
with
U
from (3.5), we
need information about the asymptotic behavior
of
gj(z),
hj(z) and
h
k
{z),
respectively.
If kj is an
eigenvalue
of
multiplicity m,j
of
the matrix
L in
(2.1),
then we get together with (2.1), (2.4)
and
(3.6)
h
ij
{z) = z
X
'
lP
P
ij
{\ogz),
(3.13)
where
P
tj
denotes
a
polynomial with formal power series
as
coefficients.
It
has
the
maximal degree
m.
—
1.

[8] Matrix-Riccati differential equations
119
Analogously
to the
definition
of the set of
polynomials
M(Q), we now
define
the set M(L) as the set of
the complex numbers
where
X
k
denotes
the
eigenvalues
of L in
(2.1).
Herewith, together with (3.9)
and
(3.13), we
can
write
the
formal expres-
sion
h
k
(z) as
**(*)=
E ^
where the P. 's denote polynomials with formal power series as coefficients.
If
we truncate this formal series
at
any power and denote the resulting functions
by Pj
(log
z),
then there exists
K
>
0
such that
for
sufficiently large
\z\ we
have
^(logz)
< \z\
K
;
(3.15)
hence we have together with (3.12)
\g](z)\<
E \z\
K
txp(m[p
k
(z
l/p
)-p
J
(z
l/p
)]).
(3.16)
Here once again
g*(z)
denotes
the
resulting function after truncating
the
formal series
in the
formal expressions
gj(z) in
(3.12).
In the same way as
in
[9] we can now estimate the right-hand side
in
(3.16).
LEMMA
3.1.
If a is a
Stokes direction
of
the polynomials
Pj and p
k
,
with
nonvanishing
h, and h
k
,
respectively,
in
(3.16)
and
leading
p
i
in S~,
then
there exists
d > 0
such that
)
=
o(exp(-\zf)),
as
z -*
oo
in S
e
\A(a), where
A is
defined
by
Me
and ee(0, a-a_).
a) = {z = re
i6
\\e-a\
<
r~
1/2p
logr},
(3.17)
PROOF.
Put
for ;
fixed
p
k
=
±
deg(Pj-p
k
),
p =
max
A
^
o
p
fc
and
Pj(z
l/p
)-
p
k
(z
x
'
p
)
=
d
k
z
Pk
H
(lower order terms). Herewith we define

120
Gerhard
Jank
[9]
Then there exists
A > 0,
such that
for
sufficiently large
\z\ = r
and
for
all
k
in
(3.16) with
h
k
# 0
\p
k
{z
l/p
)-
Pj
(z
1/p
)-d
k
z
p
'\ <Ar
p
-
l/p
.
(3.18)
If
then
2/? > 0,
because
Pj is
leading
and
there
are
no
zeros
of
the functions
y/
k
{6)
in
(a_
, a).
Otherwise
we
would have
a
further Stokes direction
in
this interval. From
y/
k
(6)
=
y/
k
(a)
+
y/'
k
(a)(6-a)+o(6-a), with
\f/
k
{a)
>
0,
we
get
for
sufficiently large
\z\ =
r
outside A(a),
the
estimate
V
k
(0)>{lr-
l/2p
logr.
(3.19)
All these properties
are
easily
to be
seen
if
one keeps
in
mind that
the
func-
tions
y/j and y/
k
are
"cosine-lines". Hence together with (3.18) we
get
with
8
l/4
8 = /p
\z\
K
exp(9Mp
k
(z
i/p
)-p
j
(z
i/p
)])
logr + Ar
p
-
i/p
+ Klogr)
=
o(exp(-r
<5
))
for sufficiently large
\z\
—
r,
outside
A(a).
This proves
the
lemma since
/?
is independent
of k
and
together with (3.16)
we
have
;
|z|')),
z^oo,
(3.21)
in S;\A(a).
DEFINITION
3.1.
Let W be a
solution
of
(1.1)
in
the
form
W =
VU~
X
U
~
H
l
C
in a
normal sector
S^
. The set
of
polynomials
C(W)=\J
{p
k
€
M(Q)
\h
k
*0in (3.9)}
defines
a
subset
of
Stokes directions
of
the
set
of
Stokes directions belonging
to
M(Q).
A
Stokes direction which is defined by the polynomials
p
k
e C{W)
is called
a
permanent critical direction
of W;
the
half ray through
the
origin
in this direction
is
called
a
permanent critical line
of W.
With respect
to the
distribution
of
poles
of a
solution
of
(1.1)
we
have
now
THEOREM 3.1. The poles of a
solution
W
o/(l.l)
are—except
for
a
number
n,(r,
W) <
O(logr)—located
in
a
finite number
of
domains
A,
defined
in

[10] Matrix-Riccati differential equations
121
(3.17),
around
the
permanent critical lines
of W.
Here with
n
x
(r, W) we
denote
the
number of poles
of W in the
circle of radius
r and
outside
all the
domains
A
around the permanent critical directions.
Notice that
the
permanent critical directions
can be
computed from
the
coefficient matrices
A, B, C, D of
(1.1),
by
computing the Stokes directions
of the formal solution
in
(2.1). These directions
do not
depend
on the
initial
value
of
the regarded solution
W.
PROOF.
Suppose that
W is a
solution
(1.1)
given
as W = VU~
X
which,
according
to
Definition
2.1, has an
asymptotic representation
in a
normal
sector
S^
with
the
matrix
C
fl
= C .
In
the
first step
we
derive
the
formal expression (3.12)
and an
asymptotic
estimate of the "remaining term" (3.16).
In
the second step we have
to
find
an
asymptotic estimate
for
the function
hUz)
defined
by the
formal expression
hj(z)
in
(3.12) after truncating
the
formal series
at
certain powers. From
(3.14)
we see
that
A can be
written
as
hj(z)=
£
z
k
"
+K
"{Q
i/
{\ogz)
+
R
v
{\ogz)), (3.22)
where
the
polynomials
Q
v
have constant coefficients
and the
coefficients
of
the polynomials
R
v
^ 0 are
still formal series,
but all of
them
are
starting
with
a
negative power
of z
1/p
. If Q
v
—
0,
then
R
v
= 0.
To
get
asymptotic estimates,
we
have
to
truncate
the
formal series
in R
v
.
We take only
the
first nonvanishing term
of
each series
and
denote
the re-
mainder
by
i?*(logz). Since
by
construction
all
coefficients
of
this function
contain
a
negative power
of z, we get
Rl{logz)
= o(l), z^oo.
(3.23)
This together with (3.22) gives
h]iz)=
£
z
K+K
"{Q
v
{\ogz)
+
o{\)).
(3.24)
If
we
denote by
K the
maximum
of
yK(k
v
+
K
V
)
in
(3.24), then with
x
v
=
3A
V
we
get
from (3.24)
h*{z)
= z
K
£
z'
t
-'((2
I/
(logz)
+
o(l)). (3.25)
Here
we
used
the
boundedness
of the
terms
z'
T
" in an
angular domain.
If
we
now
denote
by s the
maximum degree
of
the polynomials
Q
v
in
(3.25),
then
we can
write (3.25)
as
hj(z) = z
K
(logz)
s
Wc
v
z
ix
- +o(l)| ,
(3.26)

122 Gerhard Jank [11]
where the sum has to be taken over all v with
?K{\
V
+
K
U
)
=
K
, Q
v
^ 0,
deg
Q
v
= s. The function
h
' (3.27)
in (3.26) changes, with the usual transformation z
—
e", into an exponential
sum
O
£/"
T
".
(3.28)
The order of g is one and its type is finite; hence by the first fundamental
theorem of Nevanlinna theory (cf., e.g., [8] p. 47 ff. and p. 103 Theorem
11.4),
we get for the number of zeros of g in a circle of radius R
n(R,0;g)
= O(R), R^oo,
and hence with R
—
logr + o(l)
n(r,0;f)
=
O(\ogr).
(3.29)
From [12, p. 267] we get from a theorem concerning almost periodic func-
tions such as any finite exponential sum g that for any d > 0 there exists
m{6)> 0, such that
\g(u)\
> m(S)
in a strip |3w| < h and outside the circles of radius 8 around the zeros u
k
of g. This gives for / in (3.27) the estimate
1/(2)1
> m{6) (3.30)
outside the sets B
k
{5) = {z\
|
log z
—
u
k
\ <d}.
If we apply Rouche's theorem (cf., e.g., Ahlfors [1], p. 153) to (3.26) and
/ in (3.27), then we get together with (3.27) that h*(z) in (3.26) and / have
the same number of zeros in the sets
B
k
{8).
From (3.5), (3.12) together with
(3.21),
(3.26) and (3.27) we have
= exp(/>.(z
1/p
))[z
K
(logz)
s
(/(z) +
0
(l)) + o{e-^)] (3.31)
- exp(^(z
1/p
))z'
c
(logz)
J
(/(z) + o(l)), z - oo, z e S~\A.
This,
together with (3.29) and (3.30), gives the desired result of Theorem 3.1,
since we have at most a finite number of sectors S~ , and we get analogous
estimates in the corresponding finite number of sectors S* = {z\a < argz <
a
+
— e}
, where a
+
denotes the Stokes direction following a.
It is quite clear from the proof that the existence of an infinite number of
poles,
outside the domains A around the permanent critical lines depends on
the existence of an infinite number of zeros of the sum in (3.26). In general,
this depends on the matrix C in (3.5), and hence on the initial value of the
specific solution W of (1.1). In some special situations, however, the sum in

[12] Matrix-Riccati differential equations
123
(3.26)
has at
most
a
finite number
of
zeros
in any
domain
S~\A and S*\A.
If, for
example,
all the
eigenvalues
X
t
of
the matrix
L in the
formal solution
(2.1)
are
real, then
its
imaginary parts
T
;
are
zero
and the
corresponding
sum
/
in
(3.27) reduces
to a
constant.
4.
Growth estimates
and
critical points
If we
use the
matrix norm
7=1
1=1
for
any
solution
of
(1.1), then
we get a
general estimate
of
||
W|| outside
the
critical domains.
THEOREM
4.1. Suppose that
W is a
solution o/(l.l)
and
max
where
q^
denotes the polynomials
in the
matrix
Q in
(2.1). Then
for
some
constant
K > 0 and
sufficiently large
\z\ = r we
have
log
+
||W(z)||
<K{r"
+ log
r)
(4.2)
for
z
outside
the
domains
A
around
the
permanent critical directions
and
outside
the
domains
B
k
{8)
defined
in
(3.30).
PROOF.
Once again
we
start with
the
representation
W
—
VU~
X
. The ma-
trix elements
v
tj
of V
have, according
to
Definition
2.1, an
asymptotic
representation
n+m
h
n+i,k(
z
)
e
M<l
k
(z
l/P
))c
k
j,
i = 1,
• • •
, m, j = 1, ... , n
t=1
(4.3)
in
a
normal sector
5^ .
This leads
in any
normal sector immediately
to
|t>,.,.(z)|
<
|z|V
|z
",
(4.4)
for some constants
K,
T
> 0 and for
sufficiently large
\z\. Now let the
l
be
represented
as
a
iJ
=
U
ji
/detU,
(4.5)
matrix elements
a
tj
of U~
l
be
represented
as

124
Gerhard
Jank
[13]
where
Uj
t
denotes
the
adjoint elements
to w- of
the matrix
U.
From (3.31)
we
get
(4.6)
From (3.30)
we
have
|/(z)|
> m(S) and
hence
for
sufficiently large
\z\,
\f{z) +
o{\)\>\m{d)
outside
the
domains
B
k
{8).
This gives
|det£/(z)|
>
exp^/z'^JIzrilogzr^,
(4.7)
for sufficiently large
z in S~\A
and
outside
B
k
{6).
Since
the
adjoint elements
U.. are
determinants
of
elements
of U
we get
for each
of
them
an
analogous representation like (4.6).
If
we
notice
now
that
any
function
/ is
bounded
in an
angular domain,
say
by a
constant
c
>
0,
then
we get for
sufficiently large
z in S~\A
and
some
p e
M(Q),
ic
and
s
\U
jt
{z)\
<
2cexpOK/>(z
1/;)
))|z|*|logzr
(4.8)
The estimates
(4.7) and
(4.8), together with (4.5), then give
K,-|<|z|V
|z|
',
(4.9)
for suitable constants
K, x
>
0
and
for
sufficiently large
\z\
with
z out-
side
the
sets
A
around
the
permanent critical directions
and
outside
the
sets
B
k
{8).
Now (4.4) and
(4.9), together with
(1.5) and
(4.1), give (4.2).
Es-
timates
of
this type
for
differential equations with constant coefficients
are
given
in [11].
Notice that
the
exceptional domains
B
k
(S)
can be
omitted
in
Theorem
4.1
if
we are
dealing with
a
differential equation
(1.1)
having
for
example only
real eigenvalues
in
the
matrix
L of
(2.1). Next
we
want
to get
information
on
the
critical point
of
(1.1),
i.e. on the
zeros
of W'.
THEOREM
4.2.
Let a
differential equation (1.1) be given
and
Q
the associated
matrix
in
the formal fundamental system (2.1),
q.-
, i =
1,...,/, thepairwise
Ji
distinct polynomials
in
Q.
Define
the set
of polynomials
M
x
(Q)
as
a
ih
•
^
=
0, ±1,
a] +••
Then the critical points of any solution
W
(i.e.
the points with
W'
= 0)
are—
except for O(logr) many—located
in the
union of a finite number of domains

[14] Matrix-Riccati differential equations 125
A,
defined
in (3.17),
around the Stokes directions defined
by
the polynomials
in M
X
(Q), and the
sets
B
k
(S) as
defined
by (3.30).
PROOF.
Analogously to the proof of Theorem 3.1. we get from (4.3), if #
is thought to be leading, with appropriate constants K
• ,
s.
= const.ex
V
{q
j
{z
Xlp
))z
K
<"'(logz)'"{J
v]
+ o(l)), z - oo, z e S"\A,
(4.10)
where y^. denotes a function as in (3.27). From (3.31) we now get the
determinant of U and, as the adjoint elements in (4.5) are of the same type,
we get a quotient of functions of type (3.31) as an asymptotic representation
of otjj in (4.5). Together with (1.5), if we denote the elements of W by
w
vj
, we see that there exists a polynomial q
vj
e M
{
{Q), leading in S~\X
d h
(logz)
and constants
K
V
., s
vj
such that
l/ K ,, .5
f
v
j\
Z
)
"(logz)
•"
^
^
z)
+
0(1)
(4.11)
where / and f
vj
are functions of type (3.27), respectively. Since we are
dealing with the polynomials q e Af,(Q) we have to partition the normal
sectors (defined by the polynomials Q) by introducing additional Stokes di-
rections defined by the polynomials in M
X
{Q). Now S~ in (4.10), (4.11)
has to be defined with respect to this new set of Stokes directions.
Diiferentiating in (4.11) and collecting the resulting terms yields in the
case of degq^j
<
p
S
v
j\
Z
)
+ °\\)
where g
v
. denotes some new function of the type as defined in (3.27). If in
(4.11) m =
degq
UJ
>
p, then we get analogously
r-K»,
ZG5
£
"\A,
ziB
k
{8).
From (4.12) and (4.13), together (3.30) it follows that—except for a finite
number—the zeros of W
1
which are not in the domain 57
\A,
and not in

126
Gerhard
Jank
[15]
the domains
B
k
(d)
as denned
in
(3.30), have
to
be zeros
of
the functions
S
V
j
+
o(l)
or
f
V
j +
°(1)
•
Since these functions are once again of the type
as denned
in
(3.27), we get analogously
to
the proof
of
Theorem 3.1 that
there are at most 0(log
r)
such zeros. Since we have at most
a
finite number
of sectors
S~
and
at
most
a
finite number
of
sectors
S*,
where we get
analogous estimates, the theorem is proved.
5.
Periodic solutions
In this section, we will study how asymptotic methods can be used to get
results
on
the possible periodicity
of
solutions not only
of
equations with
polynomial coefficients but even
on
Matrix-Riccati equations with certain
periodic coefficients. The existence of periodic positive semidefinite solutions
of
a
Matrix-Riccati equation with periodic coefficients is studied in [7].
We start with an example of an equation
of
type (1.1) with nonconstant
polynomial coefficients, having
a
periodic solution.
The differential equation
(w
l
, w
2
)' = 2{w
x
, w
2
)
-
{w
x
,w
2
)[^
2
°
z
j -
(w
x
,
^
has the periodic solution
(tw,, w
2
) =
(e
z
,
0).
In general the solutions of a Matrix-Riccati equation will not be entire. For
periodic solutions
of
(1.1) with poles we have
THEOREM
5.1.
Let W
be
a
nonentire periodic solution
of
(I. I)
with period
to
e
C\{0}.
Then argeu
= y^,
where
y is
one of the Stokes directions defined in the
preliminary part.
PROOF.
If
co
is a
period and
z
Q
a
pole
of a
given solution
W of
(1.1),
then
W
has poles in z
0
+
lew,
k e
Z. Now define
a
small angular domain
V
E
= {z e
C| arg&>
- e <
argz
<
argto
+
e} and denote by
n
e
(r, W) the
number of poles
of W in
V
e
n {z\ \z\ < r} . Then there exists
k
0
e
N, such
that for all
k > k
0
the poles z
0
+ kw
are
in V
e
and hence there exists
a
constant
K
>
0
such that
n
e
(r,W)>Kr.
(5.1)
Assume that argeo
^ y
for all
fi e
Z. Then
V
e
n
A(y
),
A(y
)
defined in
(3.17),
is
empty
or
bounded for all
fi
and hence contains
at
most
a
finite

[16] Matrix-Riccati differential equations 127
number of poles of W. On the other hand, it follows from Theorem 3.1.
that the number of poles in \z\ < r and ^\A(^) is at most O(logr). This
contradicts (5.1).
If a periodic solution W has no poles, there is one specific situation where
Theorem 5.1. still can be applied.
REMARK 5.1. If W is a nonconstant periodic entire solution of (1.1) with
period to, and if W is a matrix of type n x n, then
where y^ denotes the Stokes directions for the equation Y' = Q, Y with
Q
A + BW
Q
+W
0
C+W
0
DW
0
-D
-C-DW
Q
and
W
Q
= W(z
0
) for some z
0
e C.
PROOF.
The matrix function V
—
{W—W
Q
)~ is w-periodic with poles and
is a solution of
V' = -D-{C + DW
Q
)V - V(B + W
Q
D) - V{A + BW
Q
+ W
Q
C +
W
Q
DW
0
)V,
because V' =
-VW'V,
which is of type (1.1). Using (1.3) we get (5.2).
Finally we want to point out that this result can be applied to a specific class
of Matrix-Riccati differential equation with periodic coefficients. Though the
theory of asymptotic integration would yield results for a larger class of
coef-
ficients
(i.e.
for rational coefficients), we want to stay here within the frame-
work of this paper, that means to ensure that (1.1) has globally meromorphic
solutions.
Suppose the coefficients of (1.1) to be of the form:
CM
.
±C,
(e*p
(^
Z
)y
,
*(.)
-±D
t
(exp
(£
7=1
v
' j=\
where A-, B., C , D- are constant matrices.
With the transformation £ = exp(^z) we get that
are polynomials with a zero in 0.

128 Gerhard Jank
[17]
If we also transform the differential equation (1.1) and (1.2) accordingly,
we get from (1.2)
—CY
1
= Q(QY
with
-C(C) -D(C)\
A(Q
B(Q )•
Since
|^(C) is
still
a
polynomial matrix, our previous theory applies
in
that
case.
If the complex variable
z is
considered
to be
real,
the
treated nonau-
tonomous differential equation (1.1) occurs
for
example
in
linear-quadratic
control problems or in Nash-games.
It
turns out that the asymptotic behavior
of the solutions will be quite different depending on whether the positive real
axis points
in a
Stokes direction
of
(1.1)
or
not. Furthermore
in
the latter
case,
the eigenvalues
of
the matrix
L in
(2.1) can be used
to
decide
if
there
is
a
uniformly asymptotic behavior
of a
solution
or
not. The influence
of
the initial data on the asymptotic behavior
of
solutions
is
not studied
in
this
paper.
6. Examples
To illustrate the results
of
the preceding parts, we shall give two examples.
The first example shows that there are differential equations which may have
solutions with
an
infinite number
of
poles, located outside
the
domains
A
around
the
permanent Stokes directions. Their number must
be of
growth
O(log
r).
The second example shows that there exist equations with solutions
which have
at
most
a
finite number
of
poles outside these regions.
The necessary computational effort is—except
for
some trivial cases—
relatively high; hence
we
use
a
program package (implemented within
the
computer algebra system MAPLE). This package, called ELISE,
is
described
in [4] and implemented
on a
68020-processor based UNIX workstation.
1.
EXAMPLE. The equation
'
\-
9J
A=(
\f \(
w
2
)
\4z
2
+ 2z
+
9J
\4z
2
+
z
+ 12
z
2
+
6j\w
2

[18] Matrix-Riccati differential equations
129
leads together with (1.2) and (1.3) to the linear system
0 1 0
0 0 1
-4z
2
-2z-9 -4z
2
-z-12 -z
2
- 6
and the solutions can be written as
'1/
•*
\
V
2S
Now ELISE gives
the
following expressions
for the
formal fundamental sys-
tem
in
(2.1):
«
-±z
3
-2z
0 0
0
-2z 0
0
0 -2z,
and
for the
eigenvalues
of L
(mod
an
integer),
A,
=
-3, A
2
= -i,
A
3
= i.
The Stokes directions are
n/6, n/2
(mod 27t/3). The CPU-time used from
ELISE was
9
seconds. According
to
Theorem 3.1., the poles
of
solutions are,
except
for a
number
of
O(log
r), in
domains
A
around that
6
directions.
Note that, together with
the
asymptotic existence theorem, there exist
so-
lutions (u,v
l
,v
—
2) with
an
asymptotic representation such that
u(z) = (l+o(l))e-
2z
z
k
(c
l
z
i
+
c
2
z-
i
), c,, c
2
€ C.
This shows that there
are
solutions
of
the Matrix-Riccati equation with
in-
finitely many poles outside
the
domains
A
around
the
critical lines.
The
location
of
this
<9(log r)
many poles depends
on the
constants
c
{
, c
2
,
i.e.
on
the initial value
of
the solution.
EXAMPLE
2.
Here we start with
an
equation
(u;,,
w
2
)' = (iz
3
- 3z -
32/3,
-z
2
+
4/3)
-
(3z
+
8)(w,,
w
2
)
r
+ 8
V
v
i'
w
2>\2z+
1<
This
gives the system
"12
V
/-?^
3
+ ^ + 32/3 z
2
-4/3
z
+
8
\/"
n
«,
2
"21
"22 I = -§z
3
+ 6z + 9/3 2z
2
-2z-8/3 2z+19 (M
21
M
22
v
\
V
2 / \^z
3
-3z-32/3 -z
2
+ 4/3 3z + 8 / \
v
\
v
i

130
Gerhard Jank [19]
with the solutions
M
21
M
22
In the formal fundamental system we have from ELISE
((-&*
+
¥ o o
exp<2(z
I/p
= exp o -z
2
0
U 0 0 -z\
and the 12 Stokes directions from M{Q) are given by y, = 7i/8, y
2
—
n/4,
y
y
= 37T/8 (mod n/2). The eigenvalues of the matrix L in (2.1) are zero
(mod an integer). The CPU-time for ELISE was 17 seconds.
Since we have only real eigenvalues, from the remark at the end of Section
3 and (3.31), for any solution we have at most a finite number of poles
outside the A-domains around the Stokes directions. Notice now that M(Q)
and M
{
(Q) here define the same set of Stokes directions; hence the critical
points of any solution are—except for a finite number—in the same regions
as the poles. Furthermore, neither in Theorem 4.1 nor in Theorem 4.2 the
unpleasant domains B
k
{8) appear in this example.
Acknowledgements
I wish to thank the referees and F. Briiggemann for their helpful remarks
and discussions.
References
[1] L. V. Ahlfors, Complex analysis, [3rd edition], (McGraw Hill, New York-Toronto-
London 1979).
[2] W. Balser, "Einige beitrage zur invariantentheorie meromorpher differential-gleich-
ungen", Habil-Schrift, Ulm 1978.
[3] V. Dietrich, "Newton-Puiseux-Diagramm fur systeme linearer differentialgleichungen",
Complex Variables Theory
Appl.
7 (1987), 265-296.
[4] V. Dietrich, "ELISE, an algorithm to compute asymptotic representations, realized with
the computer algebra system MAPLE", preprint, 1989.
[5] V. Dietrich, "Uber die annahme der mdglichen wachstumsordnungen and typen bei
linearen differentialgleichungen, Habiltationsschrift", RWTH Aachen, 1990.
[6] P. Henrici, Applied and computational complex analysis, Vol. II, (John Wiley, New
York-London-Sydney-Toronto, 1977).
[7] G. A. Hewer, "Periodicity, detectability and the Matrix Riccati equation", SIAM J.
Control 13 (1975) 1235-1251.

[20] Matrix-Riccati differential equations 131
[8] G. Jank and L. Volkmann, Meromorphe funktionen und differentialgleichungen, UTB-
GroBe Reihe, Birkhauser Verlag, Basel-Boston-Stuttgart, 1985.
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