41
conditions are met, x*(t), U(t) and hence u*(t) can be chosen to
satisfy the differential equation (3-2).
The proof to the above statement is quite tedious and will be
omitted; however, an example shall be given later which will make the
statement obvious.
Now it is only necessary to show that when (B.l) and (B.2) hold,
conditions (1), (2), and (3) are satisfied. It immediately follows from
condition (B.2) that (1) must be true. Also, by condition (B.l) we can
select K so that A.., i = 1,2, ..., r is not an eigenvalue of [F GK]
and hence (2) holds. Assuming that such a K has been chosen, it is easy
to show that
rank
X1I F
-H
G
0
rank
A-jI F + GK
H[XiI F + GK]-1G
(3-7)
This is accomplished by premultiplying and postmultiplying the left-hand
side of (3-7) by
HCA-jl F + GK]'1 Ipxp
respectively. Here Inxn denotes the identity matrix of dimension nxn.
From condition (B.2) and equation (3-7) the matrix H[A.jI F + GK]G must
have rank p for all A.ยก, i = 1,2, ..., r. This gives (3) and the proof
is complete.
The following example helps to verify the statement given in
nxn
K
lmxm
Theorem 3.2.