Controllability of a pair of matrices

Question

If the pair $(A,B)$ is controllable, then is the pair $(A^{2},(A+I)B)$ controllable?

The question becomes more interesting if there exists $(A,B)$ is uncontrollable, but the pair $(A^{2},(A+I)B)$ controllable.

Answer 1

No.

What you are asking is, if the following implication holds: $$ (\forall A,B) \: (\text{rk}(P(A,B)) = n) \rightarrow (\text{rk}(P(A^2, (A+I)B)) = n), $$ where $A \in \mathbb{R}^{n \times n}, B\in\mathbb{R}^{n \times m}$ and $P$ the controllability matrix $P(A,B) = (B, AB, ..., A^{n-1}B)$.

Take $n=1$ and $b \neq 0$:
$$ \begin{align} P(A,B) & = P(a,b) = b, \\ P(A^2, (A+I)B) & = P(a^2, (a+1)b) = (a+1)b. \end{align} $$ If, e.g., $a = -1$ we have that $$\text{rk}(P(A^2, (A+I)B) = \text{rk}((-1+1)b) = 0 \neq 1 = \text{rk}(P(A,B)).$$

Hence, $$ (\exists A,B) \: (\text{rk}(P(A,B)) = n) \wedge (\text{rk}(P(A^2, (A+I)B)) \neq n) $$ which is the negation of the above given implication.