I want to find the time-optimal control to the origin $\underline 0$ for the following:
$\dot{x}_1=-3x_1 + x_2$ and $\dot{x}_2 = x_1 - 3x_2 + u$, $|u|\leq 1$
How do I go about doing this. I simply want an order of required tasks.
I think it is something like:
1) Convert to matrix form
2) Find eigenvalues of system
3) Construct Hamiltonian
4) Solve for variables
But I am unsure. Thank you!
Attempt(editting as I go):
$\begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} = \begin{pmatrix} -3 & 1\\ 1 &-3 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \end{pmatrix} + u\begin{pmatrix} 0 \\ 1 \end{pmatrix}$
$(-3-\lambda)^2 =1$, $\lambda = -2,-4$, since this is real and of the same sign, it is a stable improper node.
Hamiltonian: $H = \psi_0f_0 + \psi_1f_1+\psi_2f_2= \psi_1(-3x_1+x_2) + \psi_2(x_1-3x_2+u)$
Look at $u^* = \pm 1$
$u^*=1$ $$-3x_1+x_2=0,$$ $$x_1-3x_2+1=0$$ $$x_1=\frac18,x_2=\frac38$$
$u^*=-1$ $$-3x_1+x_2=0$$ $$x_1-3x_2-1=0$$ $$x_1=-\frac18,x_2=-\frac38$$