I am working on an optimal control problem,more specifically on minimum-time control under input constraint $|u(t)| \leq 1$, and I need to figure out the commutation curve of a bang-bang control (so $u(t)=\pm 1$). I know that for the double integrator :
$\dot{x}_1=x_2$
$\dot{x}_2=u$
the commutation curve is simply given by $x_1(t)=\pm x_2(t)^2+c$ so it's a family of parabolas. In my problem I have :
$\dot{x}_1=x_2$
$\dot{x}_2=-\omega^2 x_1-2\gamma x_2 + u$
where $\gamma,\omega >0$.
The solution of this system (with suitable initial conditions i.e. $x(0)=(0 \ 0)^T$ ) is :
$x_1(t)= -e^{-\gamma t}-\gamma te^{-\gamma t} \pm 1$
$x_2(t)=\gamma^2 te^{-\gamma t}$
The $\pm 1$ depends on what input I want to start with and w.l.o.g. I can start with $u=1$.
Then, how can I write the commutation curve for these trajectories? I thought about writing :
$x_1(t)=-e^{-\gamma t}-\frac{x_2(t)}{\gamma} \pm 1$
but I don't even know if this is an actual geometric curve. This last formula comes from the fact that $x_2(t)$ is already contained in $x_1(t)$ but the constant multiple $\lambda$.