$($

d

$)$

$$For the matrix

F

$$defined in

$($

c

$)$

$,$

$$let

x$($s$)$

$$and

Y$($s$)$

$$be defined by taking

$($A

$-$

GC

$,$

$[$

GD

$-$

B

$-$

G

$]$

$,$

F

$,$

$[$

I $0$

$]$

$)$

$,$

$$for some matrix

G

$,$

$$as a realization for

x$($s$)$Y$($s$)$

$.$

Motivate that there exists

G

$$such that

x$($s$)$

$$and

Y$($s$)$

$$are stable.

Show that

$(1)$

$$holds for

M$($s$)$

$,$

N$($s$)$

$,$

x$($s$)$

$,$

Y$($s$)$

$$defined through

$(2)$

$$and

$(3)$

$.$