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Analysis of the second order matrix Riccati equations
Analysis of the Second Order Matrix Riccati Equations T. Sasagawa
Department of Mechanical Sophia Uniuersfty Tokyo 102, Japan
Engineering
Transmitted by V. Lakshmikantham
ABSTRACT This paper is concerned with periodic solutions of 2 X2 autonomous matrix Riccati differential equations. The author had given a necessary and sufficient condition for periodicity of solutions of matrix Riccati differential equations of general type and some examples. However, it is not so simple to verify whether this condition is satisfied or not. So this paper simplifies the verification by restricting to special cases. In particular, we show that there may exist periodic solutions for any case where the coefficient matrix of the linear part of the equation has complex eigenvalues if we choose an initial value suitably. Many examples having a periodic solution are also shown by systematic analysis; such examples are seldom seen in the literature.
I.
INTRODUCTION
As is well known, matrix Riccati equations (MREJ arise in some optimization problems. Recently, the properties of MRE themselves have been studied intensively by many authors from various points of view. The author has been studying, mainly, periodic solutions of the initial value problem [4,5]. In this paper also, we treat only the existence problem for periodic solutions. We consider the following Riccati differential equation: i’(t)=
AP(t)+P(t)AT-
P(t)RP(t)+Q,
P(O) = p,,
(1)
where A, R, Q, and P,, are n X n real constant matrices. Here we notice that APPLIED
MATHEMATICS
AND COMPUTATION
l&141-152 (1985)
141
0 Elsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017
0096-3003/85/$03.30
142
T. SASAGAWA
Equation (1) can be reduced to the case Q E 0 if one particukrr solution Pi( t ) is known. Namely, by letting P(t) = PI(t)+ U(t), we have l.J(t)=d(t)U(t)+U(t)k?(t)-U(t)RU(t), u(o) = p, - p,(o), where d(t) = A - P,(t)R. For simplicity, we assume in the sequel that Equation (1) has at least one equilibrium point P [ 1,5]. Then, we can take Pi( t ) = li, and we consider only the case Q = 0, i.e., i)(t)=AP(t)+P(t)A-P(t)RP(t), P(0) = PO.
(2)
Following [S], a necessary and sufficient condition for the existence of a unique solution of Equation (2) is given by rank &(t; 0) = n for all t > 0, where Jz(t;o)
=
[
I + jk~T”Re%.sP, 0
(I = identity matrix).
I
Moreover, the unique solution is &periodic that
(3)
[4] iff there exists a 0 > 0 such
Poe-AT8 - e”“Po + Pot?-AT8jee%k”“dsPo
= 0.
0
If Equation (4) is valid for all 8 z 0, we obtain a trivial periodic solution (i.e., constant solution) P(t) = PO, which is not regarded as a periodic solution in the sequel. In addition, this constant solution is isokzted iff the solution P, of the algebraic equation APO+ PoAT - PoRPo = 0
(5)
is isolated. Equations (3) and (4) are not convenient to verify. However, in the following special cases, the integral in Equation (4) can be considerably simplified by simpler calculations: (a) If A = AT (symmetric), AR = - RA, or A = - AT (skew symmetric), AR = RA, then we have I0
eeAT”ReAsds= BR.
Here, we have used the relation ReA* = eeASR if AR = - RA.
Second Orah
143
Matrix Riccati Equutions
(b) IfA=AT,
AR=RA,orA=
Jee
ATqe
,-AT6
-A’, As
AR=
ds = j’cosh A&R.
0
II.
-RA,thenwehave (7)
0
ANALYSIS OF THE SECOND ORDER MRE
Let U be a nonsingular matrix. Letting P = BUT for S yield us
and writing an equation
S=AS+S&-SfiS,
(8)
where d = U- 'AU, a = UTR U. Hence, without loss of generality, we can assume that the matrix A in Equation (2) has Jordan canonical form. Namely, for the second order case, the matrix A is one of the following forms:
[“l J.
[a;I,
[ _; :]
(P#W
In this paper, we are interested in the case where the solution is periodic. Hence, we do not analyze the first and second cases, because in those cases it seems that there does not exist any periodic solution. This is intuitively obvious from the fact that the solution is a function of cut and teSt [5]. Therefore, in the sequel we shall investigate the third case only. We also note that if P(t) in Equation (2) has a periodic solution, then S( t ) in Equation (8) has also a periodic solution. Following [2], the third case is divided into three cases: CaseZ. a=O,AR=RA Since A is skew symmetric, it follows from (6) that the condition (4) is reduced to Poe Ae- eAePo + 9PoeAeRPo = 0.
(9)
If we asme additionally that R is symmetric, we must have R = rZ (r # 0). In this case, the condition (9) is reduced to sinptl [d]+cospt9
[rBPz] =0,
(10)
where d
= P,l - Jp, + dP,JP,,
I=[_;
;I*
144
T. SASAGAWA
In the sequel, we denote the elements of PO as follows:
Pl [ P3
p,=
I
p2 P4
(11)
*
Then we have
1
-p,-p,+d(p,-P,)P,
d=
P, -
P,-P,+r@(p1P4-p22)
P4 + 7$P,2
PO”=
-
PIP41
PT + PZP, (PI
+ %)P,
P,+P,+r@(P,-Pp,)P4
(PI P,P,
+ %)P, +
d
I
(12) l-
Now, we show two examples.
EXAMPLE 1. For simplicity, assume that PO= Pz ( p, = p3) together with above assumptions. (1) The case cOspe = 0. In this case, since sin pe # 0, the periodicity condition (10) can be satisfied only for & = 0. This, together with PO = PO’, implies that pi = 0 (i = 1,2,4), i.e., PO= 0. Hence, we have only a trivial solution P(t) = 0 and no periodic solution exists. (2) The caSe cospe+o. The relation (10) is transformed into
From the (1,l)
element and (2,2) element of this equation and (12), we get - 2p2tan pe + re( pf + ~22)= 0
and 2p2tan pe + re( pij + P,Z) = 0. Summing these equations yields us
d3(pf + 2p,Z+ ~42)= 0.
Second Order Matrix Aiccati Equations
145
This equation at least should be satisfied; however, no cases result in a meaningful periodic solution. Hence, as Sanchez insisted [2], no periodic solution exists if P,, is sy-mmetric, A is skew symmetric, and R = rZ (r # 0). n
EXAMPLE2. Sarmhez did not consider the case PO# P$. We consider such a case here. We shall show first that J# # 0 in this case. If we assume & = 0, from calculations on the elements of the equation & = 0 we have [(l,l)element]+[(2,2)element]
+
(p3-pz)(p,+p4)=0,
[(1,2)
*
2p,p4 = Pz”+ P3”
element] - [(2,1)
element]
provided r8 # 0. These equations cannot be satisfied when p, Z p, as assumed in this example. Hence, we have & # 0. Now, we shall investigate whether the case sin pe = 0 and Pt = 0 in (10) is possible or not. From (12), it follows that Pt = 0 occurs if pf = - p,p,, p, = - pd. This is satisfied by choosing, for instance, PO’
[
;
1;
1
[eigenvalues = 0 (multiple)] .
Therefore, the periodicity condition Pt = 0 and sin pe = 0 is possible. Moreover, as pointed out above, clearly we have for this case
&=
[
24 2(1+ 7e)
20-4 - 2re
I
z.
w, 8.
Now, we investigate the existence condition (3). The final result for existence is obtained from the condition det A( t; 0) = det[ Z + rtP,-J # 0 and is given by detP,>$[trP,12
or
detP,,=O,
t[trP,]aO
(13)
This is a necessary and sufficient condition for existence for case I with P, # POTand R = rZ (r # 0). Hence, for our case, there may exist a periodic solution for some P, with the period 8 = r/p under the condition (13).
T. SASAGAWA
146
Concretely, we shall construct a periodic solution. Let
A=
[
_;
;,
R = I,
1
P,=
[ ;
--;
1.
This Pa satisfies the existence condition (13), and the solution is obtained as cos 2t
P(t)=
2&(
- 2siIq
t + :7r>
- coset
t + SIT)
1
(period = rr)
by using the formula shown in [S].
n
In the sequel, we shall use the relation eAs=${Asinps+(pcosps-asinps)Z}
(A=
[ -;
:])p
which is obtained by a simple calculation (or, of course, the LagrangeSylvester interpolation formula). CaseIZ. a=O,AR=-RA From (7) it follows that the condition (4) is reduced to PoeAe - eAeP o +P o
I0
ecosh AsdsRP,, = 0
(15)
and, moreover, sinp8
[APO + PoAr-
P,RP,]
= 0.
(16)
Here we have used the relations cash As = (e As+ epA”)/2 and (14). If PO is an equilibrium of Equation (2), the condition (16) is always satisfied for all 8 >, 0. This is the case where P(t) = PO for all t > 0. If PO is not an equilibrium of Equation (2), the periodicity condition (16) is satisfied for 8 = r/p, 2r/p,. .. . Now we investigate the existence condition (3). In our case, R should have the form
R=[;:$1 (
eigenvalues = f \/713+).
Second Order Matrix Riccati
For this
147
Equations
R and (ll), the existence condition (3) is equivalent to
# 0.
07)
This condition is satisfied if P2Gx2 + P2)+2p(
“r, + flr2)6 + y&s < p4y-1,
(18)
where
EXAMPLE 3.
The second example in [4] belongs to this case. Let
Then we have clearly AZ’, + POAT - Z’,-,RPoz 0, and the condition (17) [or (IS)] is satisfied. Hence, the ~/%periodic solution exists and the concrete solution is shown in [4].
This
Case ZZZ. A =
1
’ T (up # 0) [ is the ca& khere A has complex conjugate eigenvalues with nonzero real part. The solution of Equation (2) is given by
P(t)=
e2”lU(t)P,[Z+~A~SReA’dsP,] -b(t) 09)
(use the formula shown in [S]), where we assume that rank &(t; 0) = 2 and u(t)=
cos pt - sinpt
sin pt cospt
1 .
T. SASAGAWA
148
Using (14) yields us
JkATSReA”&
= @F(A,
R,sinZpt,cos2pt)
- F(A,
R,O,l),
(20)
0
where F(A,
R,sin2pt,cos2pt)
+ 4p2;fI;2j
+
[(a2-p2)(ATR+RA)-aATRA+(3ap2-a3)R]
pATRA+2up(ATR+RA)+(p3-3pa2)R],
which is essentially the same as obtained by Sanchez (note that AT = - A + 2aZ). Hence, P(t)=U(t)P,[{Z-
F(A, R,0,1)Z’o}e-2”t + F(A, R,sin2pt,cos2pt)Po]
-‘UT(t).
(21)
Can we insist that P(t) is periodic iff F(A, R,O, l)Z’, = I?
(22)
This assertion seems to be correct at a glance, since U(t) and the square bracketed part of P(t) are periodic in this case. However, we must emphasize that this condition (22) is obviously sufficient for periodicity but not necessary. We also notice that the condition (22) implies that P,, must be invertible, which is not satisfied for the first example in [4]. Now we show by detailed analysis of this example that there exists a periodic solution if we choose an initial value suitably, though Sanchez [2,3] insists that in this example there is no periodic solution.
E~AMPLJI4.
Let (O
and
R=
Second Order Math
149
Riccati Eqtuztions
The initial value PO is not fixed at this moment; let us define it as (11) with P, = P,. The matrix A has complex conjugate eigenvalues (a & wi, w = dm), and this example belongs to case III. Since AR # + RA even if a = 0, this example results in neither case I nor II. Here, we directly use the necessary and sufficient condition (4) for periodicity without transforming the matrix A into Jordan canonical form, though we may get a slightly simplified condition from (19) by letting P(B) = PO. After very lengthy calculations of (4), we finally get the following elementwise relations.
(1,l) - element: p,(eae - e-“*)cos~O
-
= v(eae_
UP1 - 3P, w (eae + eKae)sinwO
e-“@)cosa@
_ “‘“%,
pi) (cue + e-“e)sinw@;
(1,2) - element: pz( eae - e-“)
cos id +
3~~ - UP, w
(eae - e-ae)sinoB
+ 5 (e”‘p, - e-aBp,) sin& = (PI + PAP2 4a
+ _
3(P,%
( eoe - emue) cos ue
- P,z>
(eae-
e-ae)sind
4aw
(P1-%)P~(e0e+e-“8)s~~e ; 40
(2,l)
- element:
pz( eoe - eCae) coswe+~(eae-e-ue)sinw~-~(e~ep,-e-~eP~)sin~~
= (P, + PAP, 4a _ (PI-
(eae -
Pd)Ps (ed+
4w
e-ae)cOsd
e-d)shwe
3(P,z - PIPA 4aw
+
;
(eae - e-“e)sind
T. SASAGAWA
150 (2,2) - element:
p4( eae - eKae) c0sw8+~(e”B-e~0e)sinw8-~(eae+e~uB)sinwfI
=
E$l$(eaeepae)cosw*
_
&&&(eae+ eeae)sinw~.
Now we take as 6 = V/W. Then the following relations should be valid: P;+ p=
(PI+
Pz”
4a
“’
PAP, 4a
=
PZ,
Pz” + P4” PC 4a
“*
These are satisfied for one of the following cases:
and
p,2=4aa-a2(>0),
p, = 4a - a,
(23)
(a an arbitrary real number). The first case is the trivial one P(t) = 0. Here we note that the original equation is written in the element form
@dt>=6~2(t)-PZ(t)~
Pm
= Pl
~2(t)=3%(t)-3pl(t)+2uP2(t)-p2(t)%(t)¶
P,(O)
= P2
$4(t)
=
-
O~,(t)+40~4(t)
- p:(t),
P4(O) = P4
Hence, the fourth case also gives us an equilibrium, i.e., trivial periodic solution. This constant solution is isolated, because in (23) there does not exist a such that p, = p4 = 4a. The second and third cases are included as special cases of the fifth one, i.e., (Y= 4a and 0 = 0, respectively.
Second
Or&
151
Matrix Riccati Equations
Here we choose (Y= 2a. Then we get from Equation (23) PO’
2a y. [ f2a
(24
If we take the plus sign in (24), from the formula in [5] we have the following 7r/o_periodic solution:
P(t)=
2a 1+cos~cos2wt
1+ cos(2wt - cp)
[ cosc#J+cos2wt
cos $3+cosewt 1 +cos(2wt
+ gq
1’
(25)
where tan+ = w/a (0 < $I < B, 9 # 7r/2). The remaining cases of Pa give us the same periodic solution as (25) starting at different initial values. Namely, the matrix Pa which is defined by taking the minus sign in (24) or by letting a = 4a or (Y= 0 in (23) is on the trajectory of (25). The trajectories corresponding to these initial values start, respectively, at t = n/20, (a - +)/2w ( > 0), and (a + +)/2w on the trajectory (25). Consequently, it is sure that there exists a periodic solution, though we do not verify the existence condition (3). This example, however, does not satisfy Sanchez’s condition (22). III.
CONCLUSION
We have investigated periodic solutions of autonomous matrix differential Riccati equations, restricting ourselves to the second order case. It seems that Sanchez emphasizes that there do not necessarily exist periodic solutions even if the matrix A has complex eigenvalues. In particular, he insisted that the condition (22) is necessary and sufficient for periodicity of P( t ) and our example in [4] has only two equilibrium points. However, as we noted, this is not correct. In order to deny Sanchez’s assertion in [2], we showed a detailed analysis of our example in [4]. As is easily seen from our examples in this paper, there may exist a periodic solution in most cases where the matrix A has complex eigenvalues, if we choose an initial value Pa suitably. The period of the resulting periodic solution is closely related to the imaginary parts of the eigenvalues of A. REFERENCES 1
A. Bag& and R. C. W. Strijhos, Decoupled decompositionof the Biccati equation, IEEE Trans. Automat. Control AC-27(3):&M-698 (1982).
152 2
3 4 5 6
T. SASAGAWA D. A. SBnchez, Periodic and constant solutions of matrix Riccati differential equations: n = 2, in Zkceedings of the Conference on Ordinary and Partial Diffmential Equations, Dundee, Scotland, 1982, Springer Lecture Notes in Mathematics. D. A. SBnchez, Some preliminary results on periodic solutions of matrix Riccati equations, Prcc. Roy. Sot. Edinburgh Sect. A, to appear. T. Sasagawa, A necessary and sufficient condition for the solution of the Riccati equation to be periodic, ZEEE Tram. Automat. Control AC-25(3):564-566 (1989). T. Sasagawa, On the finite escape phenomena for matrix Riccati equations, ZEEE Trans. Automat. Control AC-27(4):977-979 (1982). T. Sasagawa, Comments on “Decoupled decomposition of the Riccati equation, IEEE Tram. Automat. ControZ AC-28(8):866-867 (1983).
Department of Mechanical Sophia Uniuersfty Tokyo 102, Japan
Engineering
Transmitted by V. Lakshmikantham
ABSTRACT This paper is concerned with periodic solutions of 2 X2 autonomous matrix Riccati differential equations. The author had given a necessary and sufficient condition for periodicity of solutions of matrix Riccati differential equations of general type and some examples. However, it is not so simple to verify whether this condition is satisfied or not. So this paper simplifies the verification by restricting to special cases. In particular, we show that there may exist periodic solutions for any case where the coefficient matrix of the linear part of the equation has complex eigenvalues if we choose an initial value suitably. Many examples having a periodic solution are also shown by systematic analysis; such examples are seldom seen in the literature.
I.
INTRODUCTION
As is well known, matrix Riccati equations (MREJ arise in some optimization problems. Recently, the properties of MRE themselves have been studied intensively by many authors from various points of view. The author has been studying, mainly, periodic solutions of the initial value problem [4,5]. In this paper also, we treat only the existence problem for periodic solutions. We consider the following Riccati differential equation: i’(t)=
AP(t)+P(t)AT-
P(t)RP(t)+Q,
P(O) = p,,
(1)
where A, R, Q, and P,, are n X n real constant matrices. Here we notice that APPLIED
MATHEMATICS
AND COMPUTATION
l&141-152 (1985)
141
0 Elsevier Science Publishing Co., Inc., 1985 52 Vanderbilt Ave., New York, NY 10017
0096-3003/85/$03.30
142
T. SASAGAWA
Equation (1) can be reduced to the case Q E 0 if one particukrr solution Pi( t ) is known. Namely, by letting P(t) = PI(t)+ U(t), we have l.J(t)=d(t)U(t)+U(t)k?(t)-U(t)RU(t), u(o) = p, - p,(o), where d(t) = A - P,(t)R. For simplicity, we assume in the sequel that Equation (1) has at least one equilibrium point P [ 1,5]. Then, we can take Pi( t ) = li, and we consider only the case Q = 0, i.e., i)(t)=AP(t)+P(t)A-P(t)RP(t), P(0) = PO.
(2)
Following [S], a necessary and sufficient condition for the existence of a unique solution of Equation (2) is given by rank &(t; 0) = n for all t > 0, where Jz(t;o)
=
[
I + jk~T”Re%.sP, 0
(I = identity matrix).
I
Moreover, the unique solution is &periodic that
(3)
[4] iff there exists a 0 > 0 such
Poe-AT8 - e”“Po + Pot?-AT8jee%k”“dsPo
= 0.
0
If Equation (4) is valid for all 8 z 0, we obtain a trivial periodic solution (i.e., constant solution) P(t) = PO, which is not regarded as a periodic solution in the sequel. In addition, this constant solution is isokzted iff the solution P, of the algebraic equation APO+ PoAT - PoRPo = 0
(5)
is isolated. Equations (3) and (4) are not convenient to verify. However, in the following special cases, the integral in Equation (4) can be considerably simplified by simpler calculations: (a) If A = AT (symmetric), AR = - RA, or A = - AT (skew symmetric), AR = RA, then we have I0
eeAT”ReAsds= BR.
Here, we have used the relation ReA* = eeASR if AR = - RA.
Second Orah
143
Matrix Riccati Equutions
(b) IfA=AT,
AR=RA,orA=
Jee
ATqe
,-AT6
-A’, As
AR=
ds = j’cosh A&R.
0
II.
-RA,thenwehave (7)
0
ANALYSIS OF THE SECOND ORDER MRE
Let U be a nonsingular matrix. Letting P = BUT for S yield us
and writing an equation
S=AS+S&-SfiS,
(8)
where d = U- 'AU, a = UTR U. Hence, without loss of generality, we can assume that the matrix A in Equation (2) has Jordan canonical form. Namely, for the second order case, the matrix A is one of the following forms:
[“l J.
[a;I,
[ _; :]
(P#W
In this paper, we are interested in the case where the solution is periodic. Hence, we do not analyze the first and second cases, because in those cases it seems that there does not exist any periodic solution. This is intuitively obvious from the fact that the solution is a function of cut and teSt [5]. Therefore, in the sequel we shall investigate the third case only. We also note that if P(t) in Equation (2) has a periodic solution, then S( t ) in Equation (8) has also a periodic solution. Following [2], the third case is divided into three cases: CaseZ. a=O,AR=RA Since A is skew symmetric, it follows from (6) that the condition (4) is reduced to Poe Ae- eAePo + 9PoeAeRPo = 0.
(9)
If we asme additionally that R is symmetric, we must have R = rZ (r # 0). In this case, the condition (9) is reduced to sinptl [d]+cospt9
[rBPz] =0,
(10)
where d
= P,l - Jp, + dP,JP,,
I=[_;
;I*
144
T. SASAGAWA
In the sequel, we denote the elements of PO as follows:
Pl [ P3
p,=
I
p2 P4
(11)
*
Then we have
1
-p,-p,+d(p,-P,)P,
d=
P, -
P,-P,+r@(p1P4-p22)
P4 + 7$P,2
PO”=
-
PIP41
PT + PZP, (PI
+ %)P,
P,+P,+r@(P,-Pp,)P4
(PI P,P,
+ %)P, +
d
I
(12) l-
Now, we show two examples.
EXAMPLE 1. For simplicity, assume that PO= Pz ( p, = p3) together with above assumptions. (1) The case cOspe = 0. In this case, since sin pe # 0, the periodicity condition (10) can be satisfied only for & = 0. This, together with PO = PO’, implies that pi = 0 (i = 1,2,4), i.e., PO= 0. Hence, we have only a trivial solution P(t) = 0 and no periodic solution exists. (2) The caSe cospe+o. The relation (10) is transformed into
From the (1,l)
element and (2,2) element of this equation and (12), we get - 2p2tan pe + re( pf + ~22)= 0
and 2p2tan pe + re( pij + P,Z) = 0. Summing these equations yields us
d3(pf + 2p,Z+ ~42)= 0.
Second Order Matrix Aiccati Equations
145
This equation at least should be satisfied; however, no cases result in a meaningful periodic solution. Hence, as Sanchez insisted [2], no periodic solution exists if P,, is sy-mmetric, A is skew symmetric, and R = rZ (r # 0). n
EXAMPLE2. Sarmhez did not consider the case PO# P$. We consider such a case here. We shall show first that J# # 0 in this case. If we assume & = 0, from calculations on the elements of the equation & = 0 we have [(l,l)element]+[(2,2)element]
+
(p3-pz)(p,+p4)=0,
[(1,2)
*
2p,p4 = Pz”+ P3”
element] - [(2,1)
element]
provided r8 # 0. These equations cannot be satisfied when p, Z p, as assumed in this example. Hence, we have & # 0. Now, we shall investigate whether the case sin pe = 0 and Pt = 0 in (10) is possible or not. From (12), it follows that Pt = 0 occurs if pf = - p,p,, p, = - pd. This is satisfied by choosing, for instance, PO’
[
;
1;
1
[eigenvalues = 0 (multiple)] .
Therefore, the periodicity condition Pt = 0 and sin pe = 0 is possible. Moreover, as pointed out above, clearly we have for this case
&=
[
24 2(1+ 7e)
20-4 - 2re
I
z.
w, 8.
Now, we investigate the existence condition (3). The final result for existence is obtained from the condition det A( t; 0) = det[ Z + rtP,-J # 0 and is given by detP,>$[trP,12
or
detP,,=O,
t[trP,]aO
(13)
This is a necessary and sufficient condition for existence for case I with P, # POTand R = rZ (r # 0). Hence, for our case, there may exist a periodic solution for some P, with the period 8 = r/p under the condition (13).
T. SASAGAWA
146
Concretely, we shall construct a periodic solution. Let
A=
[
_;
;,
R = I,
1
P,=
[ ;
--;
1.
This Pa satisfies the existence condition (13), and the solution is obtained as cos 2t
P(t)=
2&(
- 2siIq
t + :7r>
- coset
t + SIT)
1
(period = rr)
by using the formula shown in [S].
n
In the sequel, we shall use the relation eAs=${Asinps+(pcosps-asinps)Z}
(A=
[ -;
:])p
which is obtained by a simple calculation (or, of course, the LagrangeSylvester interpolation formula). CaseIZ. a=O,AR=-RA From (7) it follows that the condition (4) is reduced to PoeAe - eAeP o +P o
I0
ecosh AsdsRP,, = 0
(15)
and, moreover, sinp8
[APO + PoAr-
P,RP,]
= 0.
(16)
Here we have used the relations cash As = (e As+ epA”)/2 and (14). If PO is an equilibrium of Equation (2), the condition (16) is always satisfied for all 8 >, 0. This is the case where P(t) = PO for all t > 0. If PO is not an equilibrium of Equation (2), the periodicity condition (16) is satisfied for 8 = r/p, 2r/p,. .. . Now we investigate the existence condition (3). In our case, R should have the form
R=[;:$1 (
eigenvalues = f \/713+).
Second Order Matrix Riccati
For this
147
Equations
R and (ll), the existence condition (3) is equivalent to
# 0.
07)
This condition is satisfied if P2Gx2 + P2)+2p(
“r, + flr2)6 + y&s < p4y-1,
(18)
where
EXAMPLE 3.
The second example in [4] belongs to this case. Let
Then we have clearly AZ’, + POAT - Z’,-,RPoz 0, and the condition (17) [or (IS)] is satisfied. Hence, the ~/%periodic solution exists and the concrete solution is shown in [4].
This
Case ZZZ. A =
1
’ T (up # 0) [ is the ca& khere A has complex conjugate eigenvalues with nonzero real part. The solution of Equation (2) is given by
P(t)=
e2”lU(t)P,[Z+~A~SReA’dsP,] -b(t) 09)
(use the formula shown in [S]), where we assume that rank &(t; 0) = 2 and u(t)=
cos pt - sinpt
sin pt cospt
1 .
T. SASAGAWA
148
Using (14) yields us
JkATSReA”&
= @F(A,
R,sinZpt,cos2pt)
- F(A,
R,O,l),
(20)
0
where F(A,
R,sin2pt,cos2pt)
+ 4p2;fI;2j
+
[(a2-p2)(ATR+RA)-aATRA+(3ap2-a3)R]
pATRA+2up(ATR+RA)+(p3-3pa2)R],
which is essentially the same as obtained by Sanchez (note that AT = - A + 2aZ). Hence, P(t)=U(t)P,[{Z-
F(A, R,0,1)Z’o}e-2”t + F(A, R,sin2pt,cos2pt)Po]
-‘UT(t).
(21)
Can we insist that P(t) is periodic iff F(A, R,O, l)Z’, = I?
(22)
This assertion seems to be correct at a glance, since U(t) and the square bracketed part of P(t) are periodic in this case. However, we must emphasize that this condition (22) is obviously sufficient for periodicity but not necessary. We also notice that the condition (22) implies that P,, must be invertible, which is not satisfied for the first example in [4]. Now we show by detailed analysis of this example that there exists a periodic solution if we choose an initial value suitably, though Sanchez [2,3] insists that in this example there is no periodic solution.
E~AMPLJI4.
Let (O
and
R=
Second Order Math
149
Riccati Eqtuztions
The initial value PO is not fixed at this moment; let us define it as (11) with P, = P,. The matrix A has complex conjugate eigenvalues (a & wi, w = dm), and this example belongs to case III. Since AR # + RA even if a = 0, this example results in neither case I nor II. Here, we directly use the necessary and sufficient condition (4) for periodicity without transforming the matrix A into Jordan canonical form, though we may get a slightly simplified condition from (19) by letting P(B) = PO. After very lengthy calculations of (4), we finally get the following elementwise relations.
(1,l) - element: p,(eae - e-“*)cos~O
-
= v(eae_
UP1 - 3P, w (eae + eKae)sinwO
e-“@)cosa@
_ “‘“%,
pi) (cue + e-“e)sinw@;
(1,2) - element: pz( eae - e-“)
cos id +
3~~ - UP, w
(eae - e-ae)sinoB
+ 5 (e”‘p, - e-aBp,) sin& = (PI + PAP2 4a
+ _
3(P,%
( eoe - emue) cos ue
- P,z>
(eae-
e-ae)sind
4aw
(P1-%)P~(e0e+e-“8)s~~e ; 40
(2,l)
- element:
pz( eoe - eCae) coswe+~(eae-e-ue)sinw~-~(e~ep,-e-~eP~)sin~~
= (P, + PAP, 4a _ (PI-
(eae -
Pd)Ps (ed+
4w
e-ae)cOsd
e-d)shwe
3(P,z - PIPA 4aw
+
;
(eae - e-“e)sind
T. SASAGAWA
150 (2,2) - element:
p4( eae - eKae) c0sw8+~(e”B-e~0e)sinw8-~(eae+e~uB)sinwfI
=
E$l$(eaeepae)cosw*
_
&&&(eae+ eeae)sinw~.
Now we take as 6 = V/W. Then the following relations should be valid: P;+ p=
(PI+
Pz”
4a
“’
PAP, 4a
=
PZ,
Pz” + P4” PC 4a
“*
These are satisfied for one of the following cases:
and
p,2=4aa-a2(>0),
p, = 4a - a,
(23)
(a an arbitrary real number). The first case is the trivial one P(t) = 0. Here we note that the original equation is written in the element form
@dt>=6~2(t)-PZ(t)~
Pm
= Pl
~2(t)=3%(t)-3pl(t)+2uP2(t)-p2(t)%(t)¶
P,(O)
= P2
$4(t)
=
-
O~,(t)+40~4(t)
- p:(t),
P4(O) = P4
Hence, the fourth case also gives us an equilibrium, i.e., trivial periodic solution. This constant solution is isolated, because in (23) there does not exist a such that p, = p4 = 4a. The second and third cases are included as special cases of the fifth one, i.e., (Y= 4a and 0 = 0, respectively.
Second
Or&
151
Matrix Riccati Equations
Here we choose (Y= 2a. Then we get from Equation (23) PO’
2a y. [ f2a
(24
If we take the plus sign in (24), from the formula in [5] we have the following 7r/o_periodic solution:
P(t)=
2a 1+cos~cos2wt
1+ cos(2wt - cp)
[ cosc#J+cos2wt
cos $3+cosewt 1 +cos(2wt
+ gq
1’
(25)
where tan+ = w/a (0 < $I < B, 9 # 7r/2). The remaining cases of Pa give us the same periodic solution as (25) starting at different initial values. Namely, the matrix Pa which is defined by taking the minus sign in (24) or by letting a = 4a or (Y= 0 in (23) is on the trajectory of (25). The trajectories corresponding to these initial values start, respectively, at t = n/20, (a - +)/2w ( > 0), and (a + +)/2w on the trajectory (25). Consequently, it is sure that there exists a periodic solution, though we do not verify the existence condition (3). This example, however, does not satisfy Sanchez’s condition (22). III.
CONCLUSION
We have investigated periodic solutions of autonomous matrix differential Riccati equations, restricting ourselves to the second order case. It seems that Sanchez emphasizes that there do not necessarily exist periodic solutions even if the matrix A has complex eigenvalues. In particular, he insisted that the condition (22) is necessary and sufficient for periodicity of P( t ) and our example in [4] has only two equilibrium points. However, as we noted, this is not correct. In order to deny Sanchez’s assertion in [2], we showed a detailed analysis of our example in [4]. As is easily seen from our examples in this paper, there may exist a periodic solution in most cases where the matrix A has complex eigenvalues, if we choose an initial value Pa suitably. The period of the resulting periodic solution is closely related to the imaginary parts of the eigenvalues of A. REFERENCES 1
A. Bag& and R. C. W. Strijhos, Decoupled decompositionof the Biccati equation, IEEE Trans. Automat. Control AC-27(3):&M-698 (1982).
152 2
3 4 5 6
T. SASAGAWA D. A. SBnchez, Periodic and constant solutions of matrix Riccati differential equations: n = 2, in Zkceedings of the Conference on Ordinary and Partial Diffmential Equations, Dundee, Scotland, 1982, Springer Lecture Notes in Mathematics. D. A. SBnchez, Some preliminary results on periodic solutions of matrix Riccati equations, Prcc. Roy. Sot. Edinburgh Sect. A, to appear. T. Sasagawa, A necessary and sufficient condition for the solution of the Riccati equation to be periodic, ZEEE Tram. Automat. Control AC-25(3):564-566 (1989). T. Sasagawa, On the finite escape phenomena for matrix Riccati equations, ZEEE Trans. Automat. Control AC-27(4):977-979 (1982). T. Sasagawa, Comments on “Decoupled decomposition of the Riccati equation, IEEE Tram. Automat. ControZ AC-28(8):866-867 (1983).