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:

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For a system defined in state space as:

where

There are defined subspaces and of that the rest of the problem deals with, with properties:

The problem is to find an st.

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Of course, the first thing that comes to mind is to find st.

which satisfies the problem requirement as . However, such a method appears to be too "tight" (meaning that more freedom on is needed) for many examples from application.

Another idea is to find st.

then find a restriction of the resulting on subspace . 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 . Any help and ideas on the topic would be of great help. Thanks in advance.