lnternat. J. Math. & Math. Sci.
(1989) 131-136
VOL. 12 NO.
131
LINEAR AND RICCATI MATRIX EQUATIONS
LLOYD K. WILLIAMS
Department of Mathematics
Texas Southern University
Houston, Texas 77004
(Received July I, 1986 and in revised form April 14, 1987)
In this paper we find exact solutions for linear ordinary differential equations of any order when they are given in matrix form, as well as for classes of Riccati
matrix equations with two or three arbitrary matrix coefficients. Other nonlinear sys-
ABSTRACT.
tems of triangular form are also solved completely.
KEY WORDS AND PHRASES.
Variation of constants technique, associated matrix equations,
linear and Riccati matrix equations.
1980 AMS SUBJECT CLASSIFICATION CODE.
1.
34A05
INTRODUCTION.
A linear equation of finite order can be written as a system having a number of
dependent variables equal to the order. Letting each successive one be the derivative
of its predecessor, the number of equations will be the same as the order. This can
then be put in matrix form.
For example, the linear equation of order n
(n-l)
(n)
+ a
(1.1)
(t)x
a (t)x’ +
x
ao(t)x
n-1
is equivalent to the system
X
X
I
2
X’
n
X
=X
2
(I .2)
3
a0x I
+ a
x
1 2
+
+
an_iXn
which can be written as
X’ =AX
0
0
where
X
(x l,x 2
x
n)
and
A
0
0
ao
a
a
n-1
We will also treat Riccati matrix equations with two or three arbitrary coefficients and some other nonlinear systems.
L.K. WILLIAMS
132
LINEAR ORDINARY DIFFERENTIAL EQUATIONS.
For convenience we will assume that all matrices are nxn with elements which are
continuous functions of the same variable throughout this section.
2.
The general solution of the linear equation
X’
with matric coefficients
A,B
B
A X
(2.1)
can be found from a nonsingular solution
Y
of the cor-
responding homogeneous equation
X’ =AX
by the variation of constants technique as follows:
YC’
Y’C
AYC
YC’
AYC
(2.2)
Subs.titute
B.
X
YC
in {2.1} to get
(2..%)
From (2.3) we get
B
YC’
(2.4)
[ Y-l(s)
Y(I Y-l(s)
B(s)ds
C
X
(2.S)
+ c
B(x) dx
+
c]
{2.6)
Thus it suffices to consider equation of homogeneous forms.
Now when
THEOREM 2.1.
A T is a triangular matrix we have the following.
The equation
X’
TX, T
(tij(x))
is solvable by quadratures if
T
(2.7)
is a triangular matrix with all zeroes above the main
diagonal.
PROOF.
Consider the equation
y’
y
where
Ty
(2.8)
We solve the first equation
II (x)y
for
y
Yl
YI"
+ t
each instance we
etc.
(x)y
In
(x)y
2
22
Y2 12
solve a simple linear first-order equation and in each case we find all solutions. Take
n linearly independent solutions of (2.8) and write them as n columns. This matrix,
called a fundamental matrix of solutions, is a solution of the associated matrix equation (2.7). In the remainder of this section the independent variable x will be understood but not written. Other results for triangular matrices include
is triangular, then
THEOREM 2.2. If S is invertible, Q is nonsingular and
is a vector.
we then solve the next equation
t
Using
t
S-IQs
the equation
SX’
is solvable by quadratures for any matrix
PROOF.
Let
S-1QS
E,
QSX
T.
+
T
a triangular matrix.
X’
(2.9)
Multiplying (2.9) by
S-IT.
EX
S -1
we get
(2.10)
First we can solve
(2.11)
X’ =EX
by Theorem (2.1).
Then we solve (2. i0) by variation of constants.
THEOREM 2.3.
If
A
is invertible,
B
is nonsingular and
A-IB
is triangular, we can
solve
AX’
for any
C.
BX
+ C
(2.12)
LINEAR AND RICCATI MATRIX EQUATIONS
-1
PROOF.
Multiply (2.12) by A
COROLLARY 2.4. If A is invertible
can solve
133
and proceed as in proof of Theorem 2.2.
B
is nonsingular and
A-1B
is trinagular, we
BX.
AX’
(2.13)
We will say a matrix is continuous if each of its elements is a continuous function.
In the next theorem we assume the matrix is continuous.
We now state and prove the main result of this paper.
THEOREM 2.5.
Let
F
be an invertible, continuous matrix.
Then the equation
FY
Y’
(2.14)
is solvable by quadratures.
PROOF.
LEMMA 2.6.
We begin with the following.
T be a nonsingular triangular matrix and let
Let
S
be a nonsingular matrix.
The system
X’ =Y
(2.15)
Y
TX-S
s
solvable by quadratures.
PROOF OF LEM 2.6. First we note that
X’
TX-S.
(2.16)
The solution of (2.16) can be obtained from the solutions of Equation (2.7) by Lagrange’s
variation-of-constants method referred to in the proof of Theorem (2.2).
Each solution of (2.16) yields a unique solution of the system (2.15), since T is
non-singular, by use of the relation
Y
TX-S.
(2.17)
This completes the proof.
For statement of the next lemma let us assume momentarialy that
U and B.
TU-B
is nonsingular
for some matrices
LEMMA 2.7.
A, B, U, V and W the pair of matrices
For any matrices
UV + AW,
X
is a solution of the system
Y
U
BV
W
(2.15) if and only if:
{U(TU-B)-Iu U(TU-B)-Is + [A + U(TU-B)-I(I-TA)]W}’
[I + B(TU-B)-I]u + B(TU-B)-Is + [I + B(TU-B)-I(I-TA)]W.
Here
(2.18)
is the identity matrix.
PROOF OF LEMMA 2.7.
(IF)
Assuming that (2.18) holds we first define
V
by the
equat i on
V
Multiply (2.19) by
TU-B
(TU-B)-Iu
(TU-B)-Is
(TU-B)
-i
AW)
S
{I-TA)W.
{2.19)
and simplify to get
U
BV + W
T(tTV
(2.20)
L.K. WILLIAI
134
Define
X,Y by
X=
UV+AW
(2.21)
Y=U+BV+W
to get
Y=TXoS
(2.22)
X’ =Y.
(2.23)
Then note that (2.18) can be written as
On the other hand, if (2.21) satisfies the sy’stem(2.15) we eliminate
(ONLY IF).
from the equation
U
BV
W
T(UV
+
(2.24)
S
AW)
and substitute its value in (2.21).
We then see that the equation (2.18) is equivalent to (2.23).
This completes the
proof.
PROOF OF THEOREM 2.5. By the foregoing we may assume that F is not triangular.
As usual
stands for the identity.
will denote the determinant of Z and
is non"Since F is nonsingular there is a nonsingular A
such that FA
ZI
TIA
is also nonsingular.
such that
singular and a lower triangular T
1
This is accomplished by imposing nonvanishing restrictions on
(tij)
as in the
ITA-II
aua
P by the equation
Define
(2.25)
(FAI-I) (TIAI-I)
F # T
Now since
coefficient of
tll
is nonzero and in fact the
P is nonsingular, F-PT
can be made nonAccordingly, F-PT
nonzero in
and
is
singular by restricting
Let
C
1
Let
B
F-PTII.
t
II"
TICI-I
are nonsingular. The nonsingularity of
I,
by restricting an element of
can be obtained from that of F-PT
be such that
(F-PTI)C 1
(I-P)
CI-A
be such that the pair
A
U
(FU-B)
The value of
B
must be
1
(TICI-I) [(I-P)-(F-PT1)C1 ]-1
will be nonsingular.
satisfied by
Then
AI+CIBI
U
(CI-AI)B 1
CIB I,
(TIU-B)-I
B
B
is nonsingular an element of
TIA
B
B
+
I.
and
I.
(2.26)
Since
F-PT 1
can be chosen so that
(F-PT1) C1] -I(F-PT1
(I-P)
(TICI-I)BI
is nonsingular and moreover (2.26) is
Finally an element of
is nonsingular.
F-PT
can be chosen so
It follows that
is nonsingular.
At this point we assume that all choices have been made for the elements of
T
C
is a solution of
p.
(F-PT1)C 1] (F-PT1)A 1.
[(I-P)
(CI-A)[(I-P)-(F-PTI)CI]-I(F-PTI)
that
1
+
-1
Tl+ (T1CI-I)
A1 +
FA-I
below.
AI,
C
In summary, the nonsingularity of
is obtained by restricting a
nonsingularity of F-PT
by
restricting tll
nonsingularity
1
and (I-P)-(F-PTI)C
by restricting elements of C
and finally in
the last two cases by use of elements of F-PT
That this is possible can be seen by
single element of
of
l,
AI;
FAI-I TIAI-I
CI-A TICI-I
I.
LINEAR AND RICCATI MATRIX EQUATIONS
requiring that the product of all eight (8)
We define S by the equation
PS
135
determinants be nonzero.
(I-P)U,
U
(2.27)
AI+C1B 1.
REMARK.
Equations (2.25), (2.26) and (2.27) imply that the last expression in (2.18)
of Lemma 2.7 is the result of multiplying the first one on the left by F.
Now let (X ,Y1
be a solution of (2.15) with S
$1, T T 1.
We then seek a pair
such that
(V1,W 1)
X
UV 1
Y1
U +
U
By
(2.28)
AI+CIB1
(C1-A1)B 1
A
Since
AIW
BIV + W
+
[emma 2.7 we
above yields that
is nonsingular, we can solve for V1,W
1.
have that (2.18) is satisfied. This in conjunction with the Remark
U(TIU-B1)-Iu U(TIU-B1)-Is1
[AI+(TIU-B1)(I-TIA1)]W
is a solution
of the equation of the theorem.
EXAMPLE.
Let
n
Both quantities
2
Fal-I!
(flfs(x) f2(x)],
f4(x)
(x)
F
and
and
ITI-
is readily seen by expanding them.
[F-PTI[
the coefficient of t
P
Likewise setting
[Pit 3
is
tl
0
t
t
2
Take
Plf4 + P3f2
Pl P2
P3 P4
In
al a2 [.
A
a
can be made nonzero by restrictions on
-]P[Ip-IF TI[
[F-FFI[
T1
3
a
a
as
and expanding
this connection we note
since P is nonsingular.
We conclude this section with a result needed in the next.
in general that
THEOREM 2.9.
Let U
is a solution of X’
PROOF.
3.
be a nonsingular solution of
X’
AX and let V
U
-1
then
-XA.
I
VU
V’U + V’U 0
V’U + VAU 0
V’
-VA.
RICCATI MATRIX EQUATIONS.
Throughout this section al__l matrices are nxn matrices whose
elements are func-
tions of the same variable in each theorem.
e
theorems of the last section yield
the following.
HEOREM
3.1.
For any nonsingular matrices
X’
XFX
F,G the equation
+
XG
0
(3.1)
is solvable by quadratures.
PROOF.
For given
F,G,
F
let
C-1B,
and consider the set of equations
A,B,C be nonsingular matrices such that
G
C-1AC C-1C
(3.2)
V
4
L.K. ILLIAHS
136
U’ =AU+B
UV=C
From (3.3) we get
VC-IBv V(C-IAc-C-Ic’)
V’
0
so that
V’
+
VFV
+
VG
0.
To solve the given equation choose any C and solve (3.2) for A and B.
Use Theorem 2.S of the last section to solve for U- from which we get
(3.3).
THEOREM 3.2. For nonsingular P,Q,R the equation
X’
PX
+
+
XQX
+
XR
V
from
0
is solvable by quadratures.
For given
PROOF.
P,Q,R
let
F,G,H be such that
H-IH
Note that if
P,
FH
Q,
+
XPX
+ XG
G
R.
(3.4)
is a solution of
X’
0
H-IH’w
+ FHW + WG
W’ +
0. To solve the given equation, use Theorem 2.6 to find
from the first equation of (3.4). Then get F and" G from (3.4).
Finally, use the preceding theorem and its proof to get HW, from which we determine W.
4. NONLINEAR SYSTEMS.
then
H
In the last section some nonlinear equations were solved. Let X be a vector and
F be a vector function. Call the system of differential equations
X’
FCX)
C4.1)
triangular if the k-th component of F is a function of the first k components of X.
let
THEOREM 4.1.
PROOF.
X’
FCX) is triangular we can solve it by quadratures.
F
x{ 1 (Xl) is solved by separation of variables. Then
is solved by the methods used in [i]. x
and all other xi, i
4,5
3
If
x_ F2CXl,X2)
n,
are
determined likewise.
REFERENCES
2.
Willi.ams, L. K. Nonlinear Differential Equations and Algebraic Systems, International Journal of Mathematics and Mathematical Sciences I, (1978), 257-267.
Roxin, E. O. Ordinary Differential Equations, Wadsworth, 1972.
3.
Reid, W. T.
I.
Riccati Differential Equations, Academic
Press, 1972.