I want to find the time optimal control to the origin of the system:

$$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$

I ran straight into the problem full strength, hit it with all I have got:

$\begin{pmatrix} \dot{x}_1 \\ \dot{x}_2\end{pmatrix} = \begin{pmatrix}3 & 1 \\ 4 & 3\end{pmatrix}\begin{pmatrix}x_1 \\ x_2\end{pmatrix}+ \begin{pmatrix}0\\u \end{pmatrix}$

So we get eigenvalues from $(3-\lambda)^2-4=\lambda^2-6\lambda +5=(\lambda-1)(\lambda-5),\lambda=1,5$

And bam, I got hit by that... Hit straight headfirst with two eigenvalues - same sign - no complex component. But oh no... That means... No it couldn't be?? It is repulsive, the control is going to be hard.

I am running out of time my friends, the oxygen is low and I need to land at base to refill it, but the planets is pulsating with a magenetic field of opposite polarity to the ship at the time being(positive eigenvalues)! How do I get back in the optimal time using my rockets?

Question:How do I setup the optimal control so I can land before my oxygen depletes? I don't know where to go from here. Is the information decoded from the transmission entitled 'Committing to a name' correct?Editing in my current knowledge as an answer as I try it again.