conditions are met, x*(t), i(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.1) 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.1) we can select K so that X.i i = 1,2, ., F 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
XiI - F Gr Xi - F + GK G
rank [] = rank L(3-7) G]1G
1 H 0 0 H[Xil - F + GK]1G
This is accomplished by premultiplying and postmultiplying the left-hand side of (3-7) by
Inxn 0 an byixn 0
H[XiI - F + GK]-1 I pxpb y I Imxm
respectively. Here Inxn denotes the identity matrix of dimension nxn. From condition (B.2) and equation (3-7) the matrix H[Xil - F + GK]G must have rank p for all Xi, i = 1,2, ., j. This gives (3) and the proof is complete.
The following example helps to verify the statement given in
Theorem 3.2.