mapping from one invariant subspace to another

Hello to everybody,

I am trying to find a constructive proof for the solution of a problem in control engineering, and I got stuck at some point for the last few days. The problem is as follows:

------------------

For a system defined in state space as:

$\displaystyle

\dot{x}(t)=Ax(t)+Bu(t)

$

$\displaystyle

y(t)=Cx(t)

$

where

$\displaystyle

A \in \mathbb{R}^{nxn}

$

$\displaystyle

B \in \mathbb{R}^{nxm}

$

$\displaystyle

C \in \mathbb{R}^{pxn}

$

There are defined subspaces $\displaystyle \mathcal{S}$ and $\displaystyle \mathcal{V}

$ of $\displaystyle \mathbb{R}^n$ that the rest of the problem deals with, with properties:

$\displaystyle

\mathcal{S} \subset \mathcal{V} \subset \mathbb{R}^n \\

$

$\displaystyle

\exists \text{ a } G \text{ st. } (A+GC)\mathcal{S} \subset \mathcal{S} \\

$

$\displaystyle

\exists \text{ an } F \text{ st. } (A+BF)\mathcal{V} \subset \mathcal{V} \\

$

The problem is to find an $\displaystyle N$ st.

$\displaystyle

(A+BNC)\mathcal{S} \subset \mathcal{V}

$

------------------

Of course, the first thing that comes to mind is to find $\displaystyle N$ st.

$\displaystyle

(A+BNC)\mathcal{S} \subset \mathcal{S}

$

which satisfies the problem requirement as $\displaystyle \mathcal{S} \subset \mathcal{V}$. However, such a method $\displaystyle N$ appears to be too "tight" (meaning that more freedom on $\displaystyle N$ is needed) for many examples from application.

Another idea is to find $\displaystyle N$ st.

$\displaystyle

(A+BNC)\mathcal{V} \subset \mathcal{V}

$

then find a restriction of the resulting $\displaystyle N$ on subspace $\displaystyle \mathcal{S}$. But then my mathematical knowledge is limited on restrictions on linear operators.

In short, I am stuck at finding an elegant approach for the construction of $\displaystyle N$. Any help and ideas on the topic would be of great help. Thanks in advance.