Is a any controlled invariant subspace also a controllability subspace?

Question

If the system $\dot{x}=Ax+Bu \\ y=Cx$, where $x\in R^n, u \in R^M, y\in R^P$ is controllable, can we then say that any controlled invariant subspace is also a controllability subspace?

I would argue the following way:

The system in controllable if $\Gamma=[B\space AB\space ... \space A^{n-1}B]$ has full rank, namely $rank(\Gamma)=n$.
a controllable subspace is defined as $=span\{B,AB,...,A^{n-1}B\}$
by definition we know that subspace $ R$ is a controllability subspace if it is (A,B)-invariant and if there're F and G such that $R=$
A subspace $R$ is a controllability subspace if and and only if there is F such that $R=$.$\\$ now if there is a $G$ s.t. $Im\space B \cap R=Im BG$ and and we insert this in in the controllable subspace $R=$ we get $R=$ which is the reachability subspace. So I'd say the statement is true but I am stuck at this point...

sure, the notation $$ denotes the minimal A-invariant subspace that contains the Image B. — 2017-02-05
I'm not sure what $(A, B)$-invariant is and how you can use $\operatorname{Im}B \cap R$ before $R$ is defined. But controllable subspace is state-feedback invariant as long as $G$ is full rank. With your notation it can be written as $ = $ for any $F$ and full rank $G$. — 2017-02-07
Eigenvector interpretation of (A, B)-invariant and controllability subspaces given in "Computation of Suprema1 (A,B)-Invariant and Controllability Subspaces " by B.C. Moore and A.J.C Laub is a very good reference I guess. And I too think a controlled invariant subspace is a controllability subspace — 2017-03-07

Is a any controlled invariant subspace also a controllability subspace?

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