6. (10pts) Use the PBH test to prove or disapprove that, for any $\alpha \in \mathbb{R}$, the controllability of $(A, B)$ implies the controllability of $(A^2 + \alpha I, B)$.
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.., A^(n-1)B] has full rank, where n is the dimension of the state space. Now, let's apply the PBH test to the pair (A^2 + aI, B): The controllability matrix for (A^2 + aI, B) is [B, (A^2 + aI)B, (A^2 + aI)^2B, ..., (A^2 + aI)^(n-1)B]. To prove or disapprove the Show more…
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