I got this system state representation:
$\begin{align} \frac{dx}{dt} &= \begin{bmatrix}-3&0&0\\0&\alpha-2&0\\\alpha&0&2+\alpha\end{bmatrix}x+\begin{bmatrix}6\\2\\1\end{bmatrix}u\\ y &= \begin{bmatrix}0&0&1\end{bmatrix}x \end{align} $
I shall find the biggest possible range of values for the system to be BIBO(Bounded Input Bounded Output) stable.
My eigenvalues are $-3, \alpha-2, \alpha +2$.
So my system is asymptotically stable for all values of $\alpha < -2$.
The eigenvalues of the system matrix A are the poles of the transfer function of the system. And if the poles have a negative real part, the system is BIBO stable.
So my system is also BIBO stable for all values of $\alpha < -2$.
Is my guessing here correct?