Given the dynamical system : $ \dot{x}_1=x_2\\ \dot{x}_2=u\\ \dot{x}_3=x_4\\ \dot{x}_4=\alpha x_3+\beta x_4 + u $
where $\alpha,\beta \in R-\{0\}$ and $|u| \leq 1$. My goal is to find the minimum time control $u$ to bring all the states to the origin from an initial position $x(0)=(x_{i} \ 0 \ 0 \ 0)^T$ where $x_i >0 $. Is this system made of two decoupled subsystems? I found a minimum time controller for the states $(x_1,x_2)$ but doesn't work for $(x_3,x_4)$. Is this problem solvable at all?
EDIT : I know the minimum time controller for the system : $\dot{x}_1=x_2\\ \dot{x}_2=u $
I have verified that such controller does not work for the larger system, and I know that the minimum time controller is also unique, so can I say that the there is no optimal controller for the main system?