I have the following problem : $$ \begin{aligned} \frac{d x(t)}{dt} &= y(t)\\ \frac{d y(t)}{dt} &= -x(t)+y(t)(1-x(t)^2)+u(t) \end{aligned} $$ with the initial condition $(x,y) = (0,0)$. Those equations represent the dynamic of the mass spring movement. I tried to find the optimal control in minimal time to stabilize the system close to (0,0).
The optimal control $u(t)$ I've found is in this case equal to $sign(p_y(t))$. Where $(p_x(t),p_y(t)$ is the function of the adjoint state.
I tried to apply and simulate the shooting algorithm on this problem but I had no idea what steps should I follow.
I will really appreciate any help of any kind on the use of the shooting algorithm.
Thank you in advance guys,