Consider a non linear ode in dimension $10$: $\dot x = f(t,x,\lambda)$ where $\lambda$ is a vector of $p$ parameters.
Consider a boundary value problem of the form :
$\dot x(t) = f(t,x(t),\lambda)$ for $t\in [0,T]$ such that $g(x(0), x(T),\lambda)\le 0$ and $h(x(0),x(T),\lambda)=0$ for some vector valued function $g$ and $h$.
Is there exists some numerical scheme to solve this kind of problem.
First I think I can add the parameters $\lambda$ to the state $x$ that is consider the state $\tilde x=(x,\lambda)$ such that $\dot \lambda = 0$ so that I don't consider parametric BVP but BVP.
Second, to handle the inequalities, I am thinking to add also some constants $c$ such that $g(\tilde x(0),\tilde x(T)) + c =0$ and such that $\dot c=0$.
Then I can apply a Newton type algorithm to find a zero of my shooting function $S(\tilde x,c)$.
I can't implement it in python, fortran, matlab, c++ ...
How serious this idea sound ? Do you have other method, maybe a relaxation of my problem leading to an optimization problem ?