Inter. J. MoJ. Math. S.
11978) 13-20
VoZ.
13
EXISTENCE THEOREMS FOR THE
MATRIX RICCATI EQUATION
W’ -I- WP (t)W -I- Q (t)
0
ROBERT A. JONES
Department of Mathematics and Statistics
Louisiana Tech University
Ruston, Louisiana 71272
(Received March 18, 1977 and in revised form September 27, 1977)
ABSTRACT.
Sufficient conditions are established for the matrix Riccati equa-
tion to have a symmetric solution on a given interval.
The criteria involve
The
integral conditions on the coefficient matrices of the Riccati equation.
present results are compared with previously known results.
i.
INTRODUCTION.
Consider the matrix Riccati equation
W’ + WP(t)W + Q(t)
0
(’
.t),
(1.1)
where P(t) and Q(t) are n x n real symmetric matrix functions of the real
variable t.
In what follows sufficient conditions will be established for
(i.I) to have a symmetric solution on a given interval.
integral conditions on the matrices P(t) and Q(t).
The criteria involve
There are very few suffi-
ciency criteria in the literature and even fewer integral criteria.
The
ROBERT A. JONES
14
results will be compared with previously known results.
As for notation, capital letters will denote matrices and P > 0 (P _> 0)
indicates that the matrix P is positive definite (non-negative semi-definlte).
W > 0 and similarly for W < P or
The notation W < P or P > W indicates P
P > W.
The transpose of a matrix P will be denoted by
tity matrix will be denoted by I
shall mean that
< a < b <
shall mean that
< a < b <
2.
pT
and the n x n iden-
When considering the real interval (a,b) we
n
when considering the real interval
and similarly for (a,b] and
[a,b)
we
[a,b].
MAIN RESULTS.
Let P(t), Q(t), and N(t) be n
THEOREM i.
+ Q(s))ds
t(N(s)
Suppose for each t in I that P(t) >_ 0,
(a,b) (or any interval).
matrices on I
n real symmetric and continuous
exists and is finite, and
a
(N(s) + Q(s))d
(N(s) + Q(s))ds P(t)
N(t) >_
a
a
Then (i.i) has a symmetric solution defined on I.
Let V(t)
PROOF.
V’(t)
(N(s) + Q(s))ds for each t in I
(a,b) then
Q(t) and hence on I
-N(t)
V’(t) + V(t)P(t)V(t) + Q(t)
(N(s) + Q(s))ds
(N(s) + Q(s))d P(t)
a
W(c)
W(t)
V(c).
>
a
(a,b) and let W(t) be
I
Let c
a symmetric solution of (i.i) such that
By [l;p. 52] (or [4;p.122]) W(t) is defined and satisfies
V(t) on [c,b).
An extension of the results of [i;p.52] yields W(t)
defined and satisfying W(t)
<_ V(t)
on
(a,c].
The validity of the theorem on intervals
evident.
QED
N(t) <_ 0.
Thus W(t) exists on I
[a,b), (a,b], or [a,b]
is now
(a,b).
MATRIX RICCATI EQUATION
Setting N(t)
15
Q(t) in Theorem i we have the following known sufficiency
criterion, [3;p.344].
Let P(t)
COROLLARY TO THEOREM i.
I--[0,i]
real continuous matrix on
I.
I
QT(t
)_
and let Q(t)
n
with Q(t)>4
[ tQ(s)ds]2
be an n
n
for each t in
Then (1.1) has a symmetric solution defined on I.
The following example illustrates that the matrix Q(t) in Theorem i need
not be non-negative semi-definite.
i, q(t)
Let p(t)
EXAMPLE.
i, and n(t)
t
r is the positive real root of t
0>
t-
+ I
[0,r] where
on I
Note that i < r and on [0,r]
0.
i
t
t
t-l.
Hence
t
+
((s + i) + (s
2sds
i > t
0
l))ds]
2
0
or
(n(s) + q(s))ds
n(t) >
0
w’ +
Thus by Theorem i,
w2
+ q(t)
0 has a solution defined on I
[0,r].
A companion result to Theorem i is the following.
THEOREM 2.
matrices on I
(N(s)
Let P(t), Q(t), and N(t) be n x n real symmetric and continuous
Suppose for each t in I that P(t) > 0,
(a,b) (or any interval).
+ Q(s))ds
exists and is finite, and
(N(s) + Q(s))d
N(t) >
(N(s) + Q(s))d
P(t)
t
t
Then (i.i) has a symmetric solution defined on I.
PROOF.
(N(s) + Q(s))ds for each
Let V(t)
V’ + VP(t)V + Q(t)
VP(t)V
proof of Theorem i.
QED
N(t)
< 0 on
I.
t in I
(a,b).
Then
The proof now proceeds as in the
RDBERT A. JONES
16
The following corollary of Theorem 2 is also a known sufficiency criterion,
[3;p.344].
Let P(t)
COROLLARY TO THEOREM 2:
I
(0,(R)).
real continuous matrix on I
QT(t)
and let Q(t)
n
Q(s)ds
Suppose
be an n
n
Q(s)ds exists
lim
and is finite, and either
Q(t) > 4
()
(s)d
on I or
t
-31
Q(s)ds < I on I.
< 4t
t
-
Then (i.I) has a symmetric solution defined on I.
PROOF.
(u)
()
In Theorem 2 take a
3 I
Since
t
_<
n
I(R)Q
=, P(t)
0, b
(s)ds _<
1
I
and N(t)
n
Q(t).
In then
t
In +
n
(s)ds.
t
Hence
16t 2
(s)ds
n
(s)ds
t
(s)d
t
t
or
[I
Q(s)ds
t
i
+
I
J
n
2
i
<
4t
2
In
Thus
(Q(s)
t
In Theorem 2 take a
THEOREM 3.
matrices on I
0, b
(R),
+- In)d
P(t)
In,
<
4t
and N(t)
Let P(t), Q(t), and M(t) be n
(a,b) (or I
(a,b]).
2
I
-
n"
4t
I
n
QED
n real symmetric and continuous
Suppose for each t in I that P(t) > 0,
MATRIX RICCATI EQUATION
t(M(s)
+ P(s))ds
17
exists and is finite, is invertible, and
a
((s) + P(s))d
Q() <_
((s) + P(s))d
()
a
a
Then (1.1) has a symmetric solution defined on I.
PROOF.
(M(s) + P(s))d
Let V(t)
for each t in I
(a,b) and
ED
proceed as in the proof of Theorem 1.
As an application of the above result we have the followlng corollary.
trIn
The proof follows from Theorem 3 with M(t)
and a well-known comparison
theorem, [3;p.340].
COROLLARY TO THEOREM 3.
tinuous matrices on I
Let P(t) and (t) be n
(O,b) (or I
numbers such that for each t in I 1
(O,b]).
n real symmetric and con-
and r(#-l) are real
Suppose
r
t > O, 0 < P(t) < P(t) < I
+
ut
Q(t) <
( +
and
r-2
2
Ct
-T
n
Then (i.i) has a symmetric solution defined on I.
The following example provides a comparison of the criterion given in
Theorem 3 and the known result found in [3;p.344].
EXAMPLE.
Consider
w’ +
By Corollary 6 with b
I
I, r
w2
+
i/i0
3dt
2t3/2( I
+
2t3/2(1
1/2 and a
(0,i] and hence on [i/i0,i].
I
_
3
ti/2)2
+
O.
tl/2) 2
3/2 (2.2) has a solution defined on
Now
>3
I
1/I0
(2.2)
dt
4t3/2
3
(99) > 8.
RDBEET A. JONES
18
Also
-
1- 1/lO
1110 4
Thus
1/10 4
3dr
I) 2
2t /2 ( + t
z/zo
and hence the sufficient condition as stated in [3;p.344] cannot be satisfied
o.
[/zo",z].
Before stating the final result we note two additional applications of
Theorem 3.
obtain
Kneser’s
dominated by I
criterion (Q(t) <
I/4t21 ).
Then a case when P(t) is not
is the following.
COROLL TO TttN)II 3.
where
in the Corollary to Theorem 3 we
0 and b
i, r
Setting
and q(#-l) are real constants.
continuous matrix on I.
ItqI
Let P(t)
(O,b) (or I
on I
QT(t)
Let Q(t)
be an n
Suppose for each t in I that P(t)
Lr
n real
r
q
O, lt + t > 0
Suppose further that for each t in I
where u and r(#-l) are real constants.
Q(t) _< ut r
O,b])
+
i
+ q
lJ
In.
Then (i.i) has a symmetric solution defined on I.
Finally we have the following companion result to Theorem 3.
THEOREM 4.
Let P(t), M(t), and Q(t) be n
ous matrices on I
P(t)
> 0,
(a,b) (or I
(M(s) + P(s))ds
q(t) <
[
[a,b)).
Suppose for each t in I that
exists, is finite, invertible, and
(M(s) + P(s))d
t
x n real symmetric and continu
(M(s) + P(s))d
(t)
t
19
MATRIX RICCATI EQUATION
Then (i.i) has a symmetric solution defined on I.
PROOF.
Let
(MCs) + P(s))d
V(t)
t
for each t in I and proceed as in the proof of Theorem 3.
QED
It should be noted that in the main results stated above the matrix Q(t)
is not required to be non-negative semi-definite in contrast to that requirement on
Q(t) in (u) of the Coollary to Theorem 2, in [3;p.344], [2;p.89] and
[2;p.83].
REFERENCES
i.
Coppel, W. A. Disconlugacy, Lecture Notes in Mathematics, Springer-Verlag,
New York, 19 71.
2.
Jones, R. A.
3.
Reid, W. T. Ordinary Differential Equations, John Wiley and Sons, Inc.,
New York, 1971.
4.
Reid, W. T.
1972.
5.
Sternberg, R. L. Variational Methods and Non-Oscillatlon Theorems for
Systems of Differential Equations, Duke Math. J. 1__9 (1952) 311-322.
Comparison Theorems for Matrix Riccati Equations, SIAM J.
Appl. Math. 29 (1975).
Riccati Differential Equations., Academic
Press, New York,
KEF WORDS AND PHRASES. Existenae theorems, Symmetnic SolOns, Companison
Thr Dif fetial Equations.
AMS{MOS) SUBJECT CLASSIFICATION (1970).
34AI0.