Appendix 2: Proof of Corollary 11.1
429
Appendix 1: Proof of Theorem 11.1
We have ˙
x(t) = W(t), ˙
y(t) = T(t), and ˙
z(t) = R(t). We can rewrite the SEs in the
(ϕ, q)-domain as
M
⎛
⎝
W(t) − W 0
T(t) − T 0
R(t) − R 0
⎞
⎠ = −H
⎛
⎝
F a (X(t)) − F a (X 0 )
Y(t) − Y 0
z(t)
⎞
⎠
−
⎛
⎝
0
F b (Y(t)) − F b (Y 0 )
0
⎞
⎠ − U(t; t 0 ).
We have
z(t) = H
−1
22
− M 22 (R(t) − R 0 )
−H 21
F a (X(t)) − F a (X 0 )
Y(t) − Y 0
− U 2 (t; t 0 )
.
Substituting z(t), we obtain
K( 0 ) = M 11
W(t)
T(t)
+ H 11
F a (X(t))
Y(t)
+
0
F b (Y(t))
+ H 12 H
−1
22
− M 22 R(t)
−H 21
F a (X(t))
Y(t)
− U 2 (t; t 0 )
+ U 1 (t; t 0 )
and the result follows.
Appendix 2: Proof of Corollary 11.1
The property (t; t 0 , 0 ) ∈ M(K( 0 )) for any t ≥ t 0 follows from Theorem 11.1
and (11.29). From (11.23), we obtain
M 11
˙
X(t)
˙
Y(t)
= −H 11
F a (X(t)) − F a (X 0 )
Y(t) − Y 0
−H 12 z(t) −
0
F b (Y(t)) − F b (Y 0 )
+M 11
W 0
T 0
(11.39)