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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,

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Did you mean the algorithm to do it? Single shooting, multiple shooting or something else? Regards –  Amzoti Jan 28 '13 at 21:47
    
Thank you Amzoti Just the single shooting algorithm. Regards –  Samatix Jan 28 '13 at 21:51
    
Try these and see if they provide what you need (I may have a book reference at home and will check). depts.washington.edu/amath/courses/301-autumn-2003/301lec15.pdf tu-ilmenau.de/fileadmin/media/simulation/Lehre/div/… (Slides 55 - end) nm.mathforcollege.com/topics/shooting_method.html (see: mathforcollege.com/nm/mws/gen/08ode/…) en.wikipedia.org/wiki/Shooting_method Regards –  Amzoti Jan 28 '13 at 21:58
    
Thank you so much Amzoti :) –  Samatix Jan 29 '13 at 16:23
    
You are quite welcome. Were you able to converge on your problem using those? Regards –  Amzoti Jan 29 '13 at 16:25
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