Given nonlinear system:
\begin{cases} \dot{x_1}=x_3+u \\ \dot{x_2}=-x_2+\dot{f} \\ \dot{x_3}=-x_3+x_2 \cdot \alpha \sin(\omega t) \\ \dot{x_4}=-x_4+x_2 \cdot (\frac{16}{\alpha^2}(\sin(\omega t)-\frac{1}{2})) \end{cases}
where, $x_1...x_4$ - variables;
$f=-(x_1+\alpha \sin(\omega t))^2$;
$\alpha, \omega >0$ - constants.
Problem: make variable $x_4$ negative in minimal time
How to formulate the criterion for the optimal time?
I would be grateful for advice and help.
